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Calculated Shard: 129 (from laksa045)

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curl -X POST \
  'http://laksa129.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8\')), getAhrefsUnparsedNoserviceFromURL(\'https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8\'))) FORMAT JSONEachRow'

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{"found_url":"https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8","crawl_time":1773798542,"first_indexed_time":0,"http_code":200,"src_unparsed":"com,springer!link,\/article\/10.1140\/epjc\/s10052-026-15476-8 s443","src_root_hash":"17645177711233004329","history_drop_reason":null,"meta_title":"First measurements of beam-beam effects in beam-separation, luminosity-calibration scans at the LHC | The European Physical Journal C | Springer Nature Link","meta_descriptions":["At the CERN Large Hadron Collider (LHC), absolute luminosity calibrations obtained by the van der Meer (vdM) method are affected by the mutual electromagne"],"attrs_boilerpipe_text":"1\nIntroduction\nA precise determination of the absolute luminosity scale is essential for a wide range of LHC measurements, as it directly affects the normalization of many key physics cross sections. At the LHC, this calibration relies primarily on the van der Meer (vdM) method [\n1\n,\n2\n], which uses dedicated beam-separation scans performed under specially tailored beam conditions to relate measured interaction rates to the absolute luminosity inferred from beam parameters. While sub-percent uncertainties on the integrated luminosity have been achieved for Run-2 data by ATLAS [\n3\n] and CMS [\n4\n], further improvements and robust experimental validation of beam-dynamical effects remain essential.\nFor relativistic proton beams colliding with zero crossing angle, which is the typical configuration for the\nvdM\ncalibration, the luminosity at a given interaction point (IP) of the LHC is proportional to the overlap integral of the particle-density distributions\n\$\\rho _{\\text {1,i}}\$\n,\n\$\\rho _{\\text {2,i}}\$\nin bunch-pair\ni\n [\n5\n]:\n$$\\begin{aligned} {\\mathcal {L}}_{inst}&= 2c f_{\\text {rev}} \\sum ^{N_b}_i n_{\\text {1,i}}, n_{\\text {2,i}} \\nonumber \\\\&\\quad \\times \\iiiint ^{+\\infty }_{-\\infty }\\rho _{\\text {1,i}}(x,y,z-ct) \\nonumber \\\\&\\quad \\times \\rho _{\\text {2,i}}(x,y,z+ct)\\,dxdydzdt, \\end{aligned}$$\n(1)\nwhere\nc\nis the speed of light,\n\$f_{\\text {rev}}\$\nis the LHC revolution frequency,\n\$N_b\$\nis the number of colliding bunches, and\n\$n_{\\text {1,i}}, n_{\\text {2,i}}\$\nare the corresponding total charges per bunch for each colliding pair. Although only a very small fraction of the beam particles actually collide, the two opposing beams interact with each other electromagnetically, a dynamical process known as the beam-beam interaction [\n5\n,\n6\n]. In the LHC, as in any other synchrotron, the beams are not continuous but are divided into discrete “bunches”, each with a high proton density, containing approximately\n\$10^{11}\$\nprotons over a bunch length of 7.5–9 cm (in terms of RMS). This electromagnetic interaction occurs when the two beams share a common beam pipe, and is governed by a non-linear force that depends on the radial distance\nr\nof a test particle to the center of the opposing “source” bunch with\nn\nprotons Gaussian-distributed in the transverse planes (valid approximation for the LHC beams):\n$$\\begin{aligned} F = - \\frac{ne^2}{2\\pi \\epsilon _0r}\\biggl (1-\\exp \\biggl [ -\\frac{r^2}{2\\sigma ^2}\\biggr ] \\biggr ), \\end{aligned}$$\n(2)\nwhere\ne\nis the elementary proton charge,\n\$\\epsilon _0\$\nis the vacuum permittivity, and\n\$\\sigma \$\nis the RMS radius of the transverse charge distribution within the bunch, in a simplified case of a round shape\n\$(\\sigma = \\sigma _x = \\sigma _y)\$\n. At very small amplitudes\n\$(r\\rightarrow 0)\$\nthe force can be linearized giving the expression for the so called beam-beam parameter, which is often used to assess the strength of the force:\n$$\\begin{aligned} \\xi = \\frac{nr_p\\beta ^*}{4\\pi \\gamma \\sigma ^2} = \\frac{nr_p}{4\\pi \\epsilon _n}, \\end{aligned}$$\n(3)\nwhere\n\$r_p\$\nis the classical proton radius,\n\$\\beta ^*\$\nis the optical beta-function at the interaction point (IP) – one of the Courant–Snyder parameters [\n7\n],\n\$\\gamma \$\nis the Lorentz factor, and\n\$\\epsilon _n\$\nis the normalized emittance.\nBeam-beam effects were first observed almost 40 years ago [\n8\n] at the lepton colliders, and have been used extensively for accelerator diagnostics and optimization at both SLC and LEP [\n9\n,\n10\n,\n11\n]. At the LHC, they were first observed in late Run 1 [\n12\n], but only in terms of the beam-beam induced coherent deflections. Single particle effects, in contrast, were presumed too small in hadron colliders that until recently, there was no motivation to measure them with a precision that would be meaningful in the context evoked here.\nNonetheless, the full complexity of this interaction must be accounted for when performing the luminosity calibration via the vdM method, using the separation scans. These are designed to measure the detector-specific constant\n\$\\sigma _{vis}\$\nthat relates the observed rate\n\$\\mu ^{vis}\$\nto the absolute instantaneous luminosity\n\${\\mathcal {L}}_{inst}\$\n:\n$$\\begin{aligned} \\mu ^{vis} = \\frac{{\\mathcal {L}}_{inst}\\sigma _{vis}}{f_{rev}}. \\end{aligned}$$\n(4)\nUnder the assumption of uncorrelated particle densities in\nx\nand\ny\nplanes, the transverse convolved bunch widths\n\$\\Sigma _x, \\Sigma _y\$\ncan be extracted from the measured beam-separation dependence of the collision rate in the corresponding direction. The combined information from these scans, and bunch intensities can be used to calculate the instantaneous luminosity at the head-on position. The proportionality between the measured rate at the centered head-on position\n\$\\mu _{pk}\$\nand the reconstructed luminosity from measured bunch parameters during the vdM calibration defines the luminometer-specific visible cross-section:\n$$\\begin{aligned} \\sigma _{vis} = \\frac{2\\pi \\Sigma _x\\Sigma _y}{n_1 n_2} \\mu _{pk}. \\end{aligned}$$\n(5)\nIf left uncorrected, the\n\$\\sigma _{vis}\$\nmeasurement by the\nvdM\nmethod is biased by the beam-separation dependence of the mutual electromagnetic interaction of the two beams: the colliding bunches experience deflection-induced orbit shifts, as well as optical distortions akin to the dynamic-\n\$\\beta \$\neffect, that both depend on the transverse beam separation and must therefore be accounted for when deriving the absolute luminosity scale. The correction strategy discussed extensively in Ref. [\n13\n] relies on simulation studies only, and thus requires experimental validation to establish its reliability.\nThe present report summarizes a campaign of dedicated measurements carried out in the spring of 2022 at the LHC, aimed at confirming the accuracy of the simulation studies detailed in Ref. [\n13\n], and at validating the beam-beam correction strategy used in the luminosity-calibration analyses of the ALICE, ATLAS, CMS and LHCb experiments. More specifically, the goal is to quantify the impact of the beam-beam interaction on the tune spectra, orbit, the transverse-density distribution of the colliding bunches, as well as, for the first time, on the luminosity, by systematically varying the strength or number of the beam-beam interactions. The measurements are repeated for different values of the beam-beam parameter to verify the scaling proposed in [\n13\n], for various choices of the scanning IP and for several multi-IP configurations. The beam conditions are chosen to be representative of\nvdM\ncalibration sessions at the LHC, but optimized so as to maximize both the sensitivity of the measurements, and the operational efficiency.\nThis paper is organized as follows. The measurement strategy is detailed in Sect. \n2\n, followed by the main ingredients of this accelerator experiment: beam conditions (Sect. \n3\n), luminometers and accelerator instrumentation (Sect. \n4\n), and optimization of the ring lattice to maximize the sensitivity of the measurements to beam–beam-induced optical distortions of the colliding bunches (Sect. \n5\n). Section \n6\nis devoted to the experimental characterization of the impact of beam-beam effects on the tune spectra, on the luminosity at different IPs, as well as on other observables such as transverse single-bunch widths and IP orbits. At each step, the results are confronted with the predictions of the COherent Multibunch Beam-beam Interactions (COMBI) tracking code [\n14\n], that produced most of the results detailed in Ref. [\n13\n]. A collection of macroparticles is simulated turn by turn, based on a simplified model of the accelerator lattice using linear transfer and the self-consistent computation of the beam-beam forces at the IP. The conclusions are presented in Sect. \n7\n.\n2\nMeasurement strategy\nThe beam-beam interaction at an interaction point, referred to here as the\nscanning\nIP, induces measurable changes in key beam parameters such as the tune spectrum, closed orbit, and transverse single-beam profiles. These effects can be observed, albeit with limited precision, using standard accelerator diagnostics by comparing measurements obtained under varying transverse beam separation at the scanning IP. However, the resulting modifications to the transverse density distribution of the colliding bunches remain below the sensitivity threshold of conventional instrumentation, including synchrotron-radiation-based beam-profile monitors. Beam-beam-induced variations typically lie in the 0.5–1% range, small enough that only a highly precise measurement of collision-rate changes can provide adequate sensitivity.\nFig. 1\nScheme of the LHC and its interaction points (IP) with indicated IP1 as witness IP while performing a scan at IP5\nFull size image\nImportantly, the beam-beam interaction induces changes in beam properties that are intrinsically entangled with the\npp\ncollisions when observing luminosity at the scanning IP. Consequently, under realistic collider conditions, it is not possible to directly access a reference luminosity signal that is free of beam-beam effects – unlike in simulation studies, where such effects can be explicitly disabled. In a multi-collision configuration, however, luminosity shifts at one or more other IPs (Fig. \n1\n), where the beams are kept in continuous head-on collision and therefore act as non-scanning, or\nwitness\n, IPs, can be used to monitor the separation-dependent beam–beam effects. These effects are induced at the scanning IP, where the beams are deliberately brought in and out of collision, and propagate around the rings to the witness IPs, where they manifest as changes in the measured luminosity. In principle, any of the four LHC IPs can serve as a scanning IP, and any non-scanning IP can be designated as the witness IP for a given measurement. In practice, instrumental and operational constraints at the time the experiment was carried out restrict the choice of witness IP to the ATLAS (IP1) and CMS (IP5) collision points; IPs 1, 2 or 5, or combinations thereof, are used as scanning IP(s). Collisions at IP8 were deliberately avoided because the large longitudinal offset of the LHCb collision point, that breaks the eight-fold symmetry of the LHC rings, precludes the possibility of both members of a colliding-bunch pair to collide at all four IPs. In addition, the large crossing angle at IP8 would complicate the interpretation of the beam-beam effects that would occur at that IP if some of the bunches collided there.\nThe propagation of the beam–beam-induced, amplitude-dependent\n\$\\beta \$\n-beating from the scanning to a witness IP is controlled by the betatron phase advance between these two IPs. Since the targeted signatures are delicate to measure at best, it is natural to try and enhance them by adjusting this phase advance so as to maximize the sensitivity of luminosity shifts at the witness IP to beam-beam effects at the scanning IP; in doing so, however, the overall tunes must be preserved. Given the central and symmetric roles played by IP1 and IP5 in this experiment, the adjustment of their relative phase advance drove the optimization procedure that is detailed in Sect. \n5\n, and that achieved a threefold improvement in measurement sensitivity.\nCentral to the quality of the measurements is the controlled variation of the beam-beam parameter at the scanning IP. This can be achieved either:\nby using\nstep scans\n,\ni.e.\nby fully separating the beams at the scanning IP in either the horizontal or the vertical plane, and then bringing them back into head-on collision; or\nby using\nseparation scans\n,\ni.e.\nby scanning the beams transversely with respect to each other in either the horizontal or the vertical plane; or\nby taking advantage of the natural beam-intensity decay and emittance growth to progressively reduce the beam-beam parameter.\nEqually important is to ensure that the beams remain in head-on collision at the witness IP(s), such that reproducible luminosity shifts measured at these locations can be unambiguously correlated with controlled changes in the strength of the beam-beam interaction at the scanning IP. It was verified that the beam-beam-induced orbit shift and the potential non-closure of the orbit bumps used to control the beam separation at the scanning IP, did not significantly affect the actual separation, and thereby the measured collision rate, at the witness IP. In a few cases however, orbit drifts of uncontrolled origin ended up degrading the quality of some of the measurements.\nSimulations demonstrate [\n13\n] that multi-IP effects strongly influence the magnitude of beam-beam biases to vdM calibrations. Their characterization, therefore, constitutes an essential component of this experiment. The scan protocol is a generalization of that in the two-IP case. One IP (say, IP1) is designated as the witness IP, with beams colliding head-on there and at IP2, and IP5; the beams are then taken out of collision at IP2 only, then at both IP2 and IP5, and finally returned to a three-collision configuration in the reverse order. Since the phase-advance combinations are different in each configuration, the corresponding luminosity shifts predicted at the witness IP are also different and can be meaningfully confronted with the data.\n3\nBeam conditions\nIn the round-beam approximation, the beam-beam parameter\n\$\\xi \$\ndoes not depend on the beam energy (Eq. (\n3\n)). Therefore, and in order to minimize the overhead associated with commissioning and operating the LHC in a non-standard optical configuration, as well as to allow for frequent refilling with different bunch patterns, the measurements were carried out at injection energy (\n\$450\\ \\text {GeV}\$\nper beam), and under non-standard machine-protection conditions. The improvement in operational efficiency came at a cost:\nthe counting rate of the luminometers (or equivalently their visible cross-section) is about an order of magnitude smaller at\n\$\\sqrt{s} = 900\\ \\text {GeV}\$\nthan at\n\$13\\ \\text {TeV}\$\n, reflecting the combination of a 40% drop in the inelastic\npp\ncross-section, a factor of two to three drop in the multiplicity of the final-state particles, and a significant softening of their momentum spectrum;\nbecause of the total-intensity constraints dictated by machine-protection requirements, the injected beam could not exceed 4 bunches per beam (compared to 150 during a routine\nvdM\n-calibration session), with a maximum allowed population of\n\$1.25\\times 10^{11}~p\$\n\/bunch;\nthe combination of the low beam energy, that suppresses synchrotron-radiation damping, and of the large bunch intensity, that enhances intra-beam scattering, resulted in relatively rapid emittance growth and rather short single-beam lifetimes. To partially mitigate these effects, all collision-rate measurements are expressed in terms of specific luminosity, thereby automatically accounting for the beam-intensity decay.\nThe bunch intensity was deliberately increased by about 25% beyond the\nvdM\n-scan values typical of LHC Run 2, to maximize both the luminosity and the beam-beam parameter\n\$\\xi \$\n. The latter reached 0.010 per IP (Table \n1\n), compared to 0.003–0.006 during normal\npp\nvdM\nsessions. With a target injected emittance of\n\$1.5 \\ \\upmu \\text {m} \\cdot \\text {rad}\$\n, the\n\$\\beta \$\nfunction at the IP set to\n\$\\beta ^* = 11\\ \\text {m}\$\n, and zero crossing angle at IP1 and IP5, the statistical uncertainty affecting a typically 60-second long per-bunch luminosity measurement in the presence of head-on collisions, lay around 0.5%.\nTable 1 Range of beam-beam parameter values during the various stages of the experiment\nFull size table\nThe bunch patterns were chosen such that all bunches in each beam collided either at IP1 and IP5 only, or at all three of IP1, 2 and 5. The orbit-stabilization feedback system was turned off during the data-taking periods to prevent it from interfering with the beam–beam-induced orbit shift. The chromaticity was set to its standard value in physics fills, of +10 units, to guarantee coherent stability against the machine impedance, and in view of the very small number of bunches and of their large longitudinal spacing, the beam-stabilizing Landau octupoles were set to their minimum current (1A). The damping time of the bunch-by-bunch transverse feedback was set to 1000 turns, a rather loose setting intended to preserve longer the natural beam oscillation and therefore improve the precision of the tune measurements.\n4\nBeam instrumentation\n4.1\nLuminometers\nThe luminometer systems in use by the ATLAS and CMS collaborations at IP1 and IP5 are described in Refs. [\n15\n,\n16\n] respectively; no luminosity measurements were available at IP2 during the collider experiment described in this paper. Maximizing the statistical sensitivity leads to choosing the luminosity algorithm with the highest possible acceptance, namely:\nthe hit rate per bunch crossing in the ATLAS Minimum Bias Trigger Scintillators (MBTS) [\n15\n], and\nthe occupancy in the CMS Hadron Forward Calorimeter (HFOC) [\n16\n].\nIn both cases, raw luminometer counts are converted to collision rates using the Poisson formalism [\n15\n,\n16\n], and the instantaneous luminosity is averaged over typically 60-second time bins, that during the scans are synchronized with the scan steps. The statistical uncertainties are estimated from either the total number of raw luminometer counts per time bin (ATLAS), or from the RMS of the approximately 40 luminosity samples recorded in a given time bin (CMS). Since the absolute luminosity scale is irrelevant, and in order to simplify the interpretation of the results, all measurements in this paper are presented in terms of fractional shifts in the bunch-averaged specific luminosity, relative to a reference time that depends on the type of measurement considered.\n4.2\nBunch-charge measurement\nThe total intensity of each beam is measured by a direct-current current transformer (DCCT), the absolute scale of which is calibrated against a very high precision pulse generator [\n17\n]. The bunch-by-bunch charge fractions, in turn, normalized to the total stored intensity, are measured, separately for the two beams, by Fast Beam Current Transformers (FBCT) [\n18\n]. The latter devices also provide the fill pattern,\ni.e.\nthe relative location, at a given instant and around the two rings, of the nominally filled bunches. Since the absolute scale of the DCCTs is known to much better accuracy than that of the FBCTs, the bunch charges are typically computed as the product of the FBCT bunch-charge fractions and the total circulating beam intensity reported by the corresponding DCCT.\n4.3\nEmittance measurement\nThe synchrotron-light beam-profile monitors [\n19\n], dubbed BSRTs (“Beam Synchrotron Radiation Telescope”), are mainly used to track emittance evolution over time, but they can also measure the relative changes in transverse single-beam size that result from beam–beam-induced\n\$\\beta \$\n-beating. BSRT data are recorded every second; in what follows, they are presented averaged over one-minute intervals for easier comparison with other measurements.\nThe absolute length scale of the BSRT profiles suffers from significant uncertainties, if only because the online optical corrections to the images can be updated only a few times per year, and are unable to track the evolution of the efficiency of the light sensors, that depends both on time and on the position of the light spot on the sensor array. For each beam therefore, the BSRT is complemented by a wire-scanner (WS) profile monitor that is located in the same straight section. The advantage of the WS is that its accuracy is significantly better than that of the BSRT; its down side, in the present context, is that the WS cannot acquire data continuously and must be triggered manually. For the results presented in this paper, the wire scanners were flown through the beams at the start of each group of measurements to provide single-beam emittance measurements that are as accurate as possible, and then at regular intervals thereafter. The BSRT is used to interpolate the time evolution of the emittance between two sets of WS measurements.\nTo interpret measured RMS bunch widths in terms of emittance requires the knowledge of the\n\$\\beta \$\nfunctions at the locations of the BSRT and the WS. These are determined by the phase-advance method, and compared in Table \n2\nto their model value computed using MAD-X; the agreement is typically better than 5%, and the worst disagreement amounts to 9%. These discrepancies are attributed to imperfections in the magnetic lattice. Uncertainties in the measurement mainly arise from the interpolation of the lattice functions between the two closest beam-position monitors (BPMs) where the\n\$\\beta \$\n-functions are measured [\n20\n,\n21\n], and either the BSRT or the WS location. Based on the MAD-X LHC lattice model, these uncertainties are estimated not to exceed 3%, and this value is assigned as the systematic uncertainty on the\n\$\\beta \$\nfunctions used in BSRT- and WS-based emittance measurements.\nTable 2\n\$\\beta \$\nfunctions at the BSRT and WS locations, for the optical configuration detailed in Sect. \n5.3\nFull size table\nThe beam-averaged emittance,\ni.e.\nthe average of the emittances of the beam-1 bunch and of the corresponding beam-2 bunch, can be obtained directly from emittance scans [\n22\n] at IP1 and\/or IP5. In this approach, the convolved transverse bunch sizes measured using beam-separation scans are translated into emittances using the\n\$\\beta ^*\$\nvalues determined by\nk\n-modulation [\n23\n] at the relevant IP (see Sect. \n4.3\n). The associated uncertainty (Table \n3\n) is combined with the statistical uncertainty in convolved transverse width to estimate the error affecting the measured emittance. Comparison with beam-averaged emittances extracted from the single-beam profile monitors reveals excellent agreement between WS and emittance-scan results (Fig. \n2\n). The relative time-evolution of the BSRT emittances is consistent with that observed using the WS or emittance scans, but the absolute magnitudes differ significantly, especially in the horizontal plane.\nTable 3\n\$\\beta ^*\$\nfunctions at the ATLAS and CMS IPs measured by\nk\n-modulation. The target value in all cases is\n\$\\beta ^*=\$\n11 m\nFull size table\nFig. 2\nTime evolution of the horizontal (top) and vertical (bottom) beam-averaged normalized emittances during LHC fill 8037, as reported by the BSRT (blue), the WS (orange), and using beam-separation scans at IP1 (red) and IP5 (purple). Different shades of the same color correspond to two different bunches present in this fill. For ATLAS emittance scans, only the bunch-averaged value is shown\nFull size image\n4.4\nBeam-beam parameter determination\nThe systematic and quantitative comparison of the measured and of the predicted beam-beam impact on the observables detailed in Sect. \n2\nrequires continuous monitoring of the actual beam-beam parameter throughout the duration of the experiment. Only the BSRT provides uninterrupted emittance determination throughout the fill. In view of the scale biases apparent in Fig. \n2\n, however, WS-based emittances are used, whenever possible, as input to the determination of the beam-beam parameter; BSRT emittances re-scaled to close-in-time, absolute WS measurements provide time-interpolated emittance values whenever WS data are unavailable.\nThe beam-beam parameter evolution during the first fill of the experiment is illustrated in Fig. \n3\n. The error bands are dominated by the 3% systematic uncertainty in the measured optical functions. The strength of the beam-beam interaction drops by a factor of two over a couple of hours, from the combined effect of beam-intensity decay and of emittance growth.\nFig. 3\nTime evolution of the beam- and plane-averaged beam-beam parameter\n\$\\xi \$\ninferred from the measured bunch charges and emittances, separately for the two colliding-bunch pairs present in the fill pattern. The color bands indicate the systematic uncertainty (see text)\nFull size image\n4.5\nTune measurements\nCoherent spectra can be monitored using either the LHC Transverse Damper (ADT) [\n24\n], or the Base-Band Tune (BBQ) [\n25\n] system; their comparison typically yields consistent results. Spectrograms with a tune resolution of 0.0001 are computed using bunch positions recorded at every turn, and averaged over the one-minute time bins mentioned in Sect. \n4.1\n. To mitigate the influence of noise on the tune measurement, a median filter is applied at the pre-processing stage with a self-defined local window size. Additionally, the 50 Hz noise lines present in the spectra are masked. The filtered data are fitted by the sum of a Gaussian function and a constant baseline, and the frequency at the peak of the Gaussian is interpreted as the measured mean tune. The systematic uncertainty on measured tune shifts\n\$\\Delta Q\$\nis estimated empirically to be\n\$\\sigma _{\\Delta Q}=0.001\$\n.\n4.6\nIP-orbit monitoring\nOrbit displacements at each IP are measured using the Diode ORbit and OScillation (DOROS) BPM system, that provides sub-micrometer beam-position resolution [\n26\n]. These strip-line BPMs are located in the two quadrupoles on either side of and closest to the IP, allowing the position of both beams to be measured simultaneously. The position and the angle of each beam at the IP are inferred from, respectively, the average and the difference of the positions measured in the two final-triplet quadrupoles.\n5\nOptimization of the phase advance between interaction points\nIn the two-IP configuration, the basic theory of linearized beam-beam\n\$\\beta \$\n-beating provides an analytical determination of the optimum phase advance between the two IPs,\ni.e.\nof the setting that maximizes the sensitivity, at the witness IP, to beam-beam effects at the scanning IP (Sect. \n5.1\n). A generalization to the three- and four-IP configurations is discussed in Sect. \n5.2\n. The required modifications to the baseline LHC optics, as well as their implementation and their experimental validation, are detailed in Sect. \n5.3\n.\n5.1\nLinear beam-beam\n\$\\beta \$\n-beating\nIn circular colliders, the\n\$\\beta \$\n-beating induced by beam-beam effects at the IP(s) leads to changes in transverse beam size [\n5\n,\n27\n], thereby impacting the luminosity. For a single collision point, the change in\n\$\\beta ^*\$\nof a single small-amplitude particle in (for instance) the horizontal plane\nx\n, is given by:\n$$\\begin{aligned} \\frac{\\beta _x^*}{\\beta _{0,x}^*} = \\frac{1}{\\sqrt{1-4\\pi \\xi \\cot {(2\\pi Q_x)} - 4\\pi ^2\\xi ^2}}. \\end{aligned}$$\n(6)\nThe magnitude of the effect depends on the absolute value\n\$\\xi \$\nof the beam-beam parameter and on the nominal betatron tune\n\$Q_x\$\n; its periodicity is \n\$ [\\pi ]\$\n. It can be shown that approaching the half-integer betatron tune from below minimizes the\n\$\\beta \$\nfunction at the IP, thereby maximizing the luminosity; this phenomenon has been exploited with great success in (among others) the KEKB and PEP-II\nB\nfactories. In the LHC, where during collisions the nominal fractional betatron tunes are fixed at\n\$q_x=0.31,\\,q_y=0.32\$\n, this dynamic-\n\$\\beta \$\neffect depends only on the beam-beam parameter.\nThis conclusion no longer strictly holds in the presence of more than one collision point. In the case of two IPs with identical values of\n\$\\xi \$\nand\n\$\\beta ^*\$\n, the dynamic-\n\$\\beta \$\neffect can be described analytically [\n28\n]:\n$$\\begin{aligned} \\frac{\\beta _x^*}{\\beta ^*_{0,x}} = \\frac{\\sin {2\\pi Q_x} + 4\\pi \\xi (\\cos (2\\pi Q_x-2\\mu _{1,x})-\\cos 2\\pi Q_x )}{\\pm \\sqrt{1-\\cos ^2{2\\pi (Q_x+\\Delta Q_x)}}}, \\end{aligned}$$\n(7)\nwhere\n\$\\mu _{1,x}\$\nis the phase advance from the scanning IP to the witness IP. The sign in the denominator is linked to that of the numerator so as to ensure that the ratio remains positive; the denominator must be positive (resp. negative) below (resp. above) the half-integer tune,\ni.e.\nwhen\n\$m< Q_x <m+\\frac{1}{2}\$\n(resp.\n\$m+\\frac{1}{2}< Q_x < m+1\$\n), where\nm\nis a positive integer. The tune shift\n\$\\Delta Q_x\$\nis related to\n\$Q_x\$\n,\n\$\\xi \$\nand\n\$\\mu _{1,x}\$\nby:\n$$\\begin{aligned} \\cos {2\\pi (Q_x+\\Delta Q_x)}&=(1-16\\pi ^2\\xi ^2)\\cos {2\\pi Q_x} + 8\\pi \\xi \\sin {2\\pi Q_x} \\nonumber \\\\&\\quad + 16\\pi ^2\\xi ^2\\cos {(2\\pi Q_x-2\\mu _{1,x})}. \\end{aligned}$$\n(8)\nThe phase-advance dependence of the resulting\n\$\\beta \$\n-beating is illustrated in Fig. \n4\n. The minimum of each curve corresponds to the phase advance that yields the largest mutual beam–beam-induced luminosity enhancement between the two IPs. The optimal settings are\n\$\\mu _{1,x}^{min}\/2\\pi =0.405,\\,\\mu _{1,y}^{min}\/2\\pi =0.410\$\n; their difference reflects that between the nominal horizontal and vertical fractional tunes\n\$q_x\$\nand\n\$q_y\$\n.\nFig. 4\nAnalytically computed beam-beam induced\n\$\\beta \$\n-beating in a two-IP configuration, as a function of the phase advance\n\$ \\mu _1\$\nbetween the 2 IPs (\n\$q_x=0.31\$\n,\n\$q_y=0.32\$\n, head-on collisions at both IPs,\n\$\\xi =7\\times 10^{-3}\$\nper IP)\nFull size image\nThe extrema of the curves in Fig. \n4\ncan be determined analytically by differentiating Eq. (\n7\n) with respect to the phase advance\n\$\\mu _{1,x}\$\n. The general expression for this derivative\n\$\\frac{d}{d\\mu _{1,x}} \\biggl ( \\frac{\\beta _x^*}{\\beta ^*_{0,x}} \\biggr )\$\n, detailed in Appendix A, is the product of two factors, each of which can be zero:\nimposing that\n\$\\sin {(2\\pi Q_x - 2\\mu _{1,x})} = 0\$\nyields the solution\n$$\\begin{aligned} \\frac{\\mu _{1,x}}{2\\pi } =\\frac{Q_x}{2}-\\frac{m}{4} \\, \\end{aligned}$$\n(9)\nwhere\nm\nis an integer of either sign, or zero;\nthe other factor is zero for either\n$$\\begin{aligned} \\frac{\\mu _{1,x}}{2\\pi } = \\frac{Q_x}{2} - A(\\xi , Q_x)\\, \\end{aligned}$$\n(10)\nor\n$$\\begin{aligned} \\frac{\\mu _{1,x}}{2\\pi } = \\frac{Q_x - 1}{2} + A(\\xi , Q_x) \\, \\end{aligned}$$\n(11)\nwith\n\$A(\\xi , Q_x)\$\ndefined in Eq. (\n12\n)\n$$\\begin{aligned} A(\\xi , Q_x) = \\arccos \\left( \\frac{ 32\\pi ^3\\xi ^3 \\sin (4\\pi Q_x) + 48\\pi ^2\\xi ^2 \\cos ^2(2\\pi Q_x) - 32\\pi ^2\\xi ^2 - 6\\pi \\xi \\sin (4\\pi Q_x) + \\cos ^2(2\\pi Q_x) - 1}{16\\pi ^2\\xi ^2 \\left( 4\\pi \\xi \\sin (2\\pi Q_x) + \\cos (2\\pi Q_x) \\right) } \\right) \/ (4\\pi ) \\end{aligned}$$\n(12)\nThe two sets of solutions do not overlap. The subset of solutions that minimize (rather than maximize)\n\$\\beta ^*\$\nis identified by requiring that the second derivative be positive. This restricts the main solution to:\n$$\\begin{aligned} \\frac{\\mu _{1,x}}{2\\pi } =\\frac{Q_x}{2}+\\frac{m+1}{4} . \\end{aligned}$$\n(13)\nThe optimal phase\n\$\\mu _{x,1}\$\nthat minimizes the\n\$\\beta ^*\$\nfor range of\n\$(\\xi , Q_x)\$\nvalues is shown in Fig. \n5\n.\nFig. 5\nOptimal phase advance value\n\$\\mu _{x,1}\$\nfor various values of\n\$(\\xi , Q_x)\$\nin the two-IPs configuration\nFull size image\nThe phase-advance dependence of the dynamic-\n\$\\beta \$\neffect can also be quantified in terms of the luminosity shift, at the witness IP, that is associated with the electromagnetic interaction of the two bunches at the scanning IP (Fig. \n6\n). The optimal setting,\ni.e.\nthat which maximizes the sensitivity, at the witness IP, to beam-beam effects at the scanning IP, is indicated by the green dotted vertical line at\n\$\\Delta \\mu _{\\text {IP1-IP5}}=0.41\\ [2\\pi ]\$\n; the full suppression of the phase-related luminosity enhancement is indicated by the red dotted line. These results are consistent with the predictions of the analytical model (Fig. \n4\n). In the two-IP configuration considered here, in which IP1 and IP5 can both play the role of either the scanning or the witness IP, and due to the periodicity of this curve (\n\$ \\pi \$\n) and to the fractional-tune values used during collisions (\n\$q_x \/ q_y = 0.31 \/ 0.32)\$\n, the phase–advance-related luminosity enhancement is the same at the two IPs; at the optimum setting, it is three times larger than it would be using the nominal LHC lattice. The maximum effect depends mainly on the transverse tunes, and thus could have been further enhanced if it had been possible to move these away from their nominal values.\nA more comprehensive simulation study, that evaluates the impact of the beam-beam interaction on the luminosity for more realistic Gaussian proton-density distributions, multiple collision points and as a function of the beam separation, is reported in Ref. [\n29\n].\nFig. 6\nBeam–beam-induced luminosity shift at IP5 predicted by COMBI simulations, as a function of the phase advance\n\$ \\mu _1=\\Delta \\mu _{\\text {IP1-IP5}}\$\nbetween the two IPs (\n\$q_x=0.31\$\n,\n\$q_y=0.32\$\n, head-on collisions at both IPs,\n\$\\xi =7\\times 10^{-3}\$\nper IP)\nFull size image\n5.2\nOptimized phases for multi-IP configurations\nThe general expression to obtain the first-order change of the\n\$\\beta \$\nfunction at an arbitrarily chosen reference IP (at\n\$s=\\mu _{0,x}=0\$\n) can be approximated as multiple quadrupole errors [\n30\n]:\n$$\\begin{aligned} \\frac{\\beta ^*_x}{\\beta ^*_{0,x}} = \\frac{2\\pi \\xi }{\\sin {2\\pi Q_x}} \\sum _{i\\in IPs}^{N-1} \\cos (2\\pi Q_x-2\\mu _{i,x})\\, \\end{aligned}$$\n(14)\nwhere\n\$\\mu _{i,x}\$\nis the horizontal phase advance between the reference and the\n\$i^{th}\$\nIP for a total of\nN\nIPs. Therefore, the optimal configuration, which minimizes the\n\$\\beta ^*\$\nchange at all IPs, must satisfy the condition:\n$$\\begin{aligned} \\sum _{i=1}^{N-1} \\cos (2\\pi Q_x-2\\mu _{i,x}) = -(N-1)\\, \\end{aligned}$$\n(15)\nwhich yields the same solution for all IPs as in the 2 IPs configuration (Eq. \n13\n):\n$$\\begin{aligned} \\frac{\\mu _{i,x}}{2\\pi } =\\frac{Q_x}{2}+\\frac{n+1}{4} . \\end{aligned}$$\n(16)\nwhere\nn\nis an integer of either sign, or zero. To maximize the effect on luminosity, the phase advance between the reference interaction point at\n\$ i = 0 \$\nand the nearest IP, at\n\$ i = 1 \$\nmust satisfy\n\$ \\mu _{1,x}\/2\\pi = (Q_x + 0.5)\/2 \$\n, as given by the first solution (for\n\$n=0\$\n) of Eq. (\n13\n). This condition implies that the phase advance to any subsequent adjacent IPs (\n\$ i \\ge 2 \$\n) must be an integer multiple of \n\$ \\pi \$\n. Thus, this requirement exposes a fundamental constraint: it is not possible to minimize\n\$ \\beta ^* \$\nsimultaneously at all IPs. This would only be possible for (\n\$ Q_x = 0.5 \$\n) which is practically not achievable, as it would require operating the storage ring precisely at the half-integer resonance.\n5.3\nOptics validation\nThe optimum phase advance from IP1 to IP5 determined in Sect. \n5.1\nwas implemented using tuning trim quadrupoles in the LHC arcs. The targeted fractional phase advance was\n\$0.9 \\, [2 \\pi ]\$\nfor both beams and both planes (Table \n4\n). This setting, which differs from that in Fig. \n6\nby one period (\n\$ \\pi \$\n), is equivalent in terms of sensitivity optimization but has the advantage that it minimizes the absolute magnitude of the perturbations inflicted upon the nominal lattice.\nSeveral measurements were carried out to validate this adjustment: comparison of the targeted and achieved shifts in phase advance (Sect. \n5.3.1\n),\n\$\\beta \$\n-function measurements across the full LHC circumference (Sect. \n5.3.2\n) as well as at the IPs (Sect. \n5.3.3\n), and dispersion measurements to estimate the dispersive contribution to the transverse beam size at the IP (Sect. \n5.3.4\n).\n5.3.1\nPhase advance\nThe phase advance around each LHC ring is determined using turn-by-turn orbit measurements [\n20\n,\n21\n]. The targeted and achieved phase-advance shifts agree within\n\$0.01\\, [2\\pi ]\$\nor better (Table \n4\n). The phase advance change from IP1 and IP5 was compensated on the opposite side of the ring, ensuring that the transverse tunes remain unchanged. The target fractional local phase advance is set to\n\$0.9,[2\\pi ]\$\n. Table \n4\nlists the corresponding phase shifts applied to the nominal lattice to reach this target on one side of the ring; this is consistent with the analytically derived value of\n\$0.41\\,[2\\pi ]\$\nfor the compensating shift on the opposite side. The change in phase advance was also measured at the BSRT location, where the discrepancy between the model and measurements is more pronounced, reaching up to\n\$0.05\\,[2\\pi ]\$\n.\nTable 4 Nominal and optimized IP1 to IP5 phase-advance settings. Column 4 (resp. 5) displays the targeted (resp. measured) shift in phase advance after the optimized settings have been applied\nFull size table\n5.3.2\nResidual\n\$\\beta \$\n-beating\nThe\n\$\\beta \$\n-beating was quantified and found to be well within acceptable tolerances for various configurations of the trim strength. The measurement results, relative to the reference measurements without any phase advance adjustments, are illustrated in Fig. \n7\n. The maximal observed deviation is approximately 25% around IP5 for Beam 1 in the vertical plane. Such significant deviations in proximity to the IPs are anticipated and are frequently attributed to erroneous BPM readings. Overall, the measurements indicated that\n\$\\beta \$\n-beating remained within acceptable bounds, predominantly within a 10% margin.\nFig. 7\nMeasured\n\$\\beta \$\n-function change from the full strength of the phase advance knob with reference to nominal lattice along the LHC ring, for Beam 1 (top two figures for\nx\nand\ny\n) and Beam 2 (bottom two figures for\nx\nand\ny\n)\nFull size image\nThe measured\n\$\\beta \$\nfunctions are also compared to the MADX LHC lattice model predictions, which include the phase advance adjustment. The results, illustrated along the LHC lattice in Fig. \n8\n, indicate that the measurements align with the model within an acceptable margin of 10%.\nFig. 8\nMeasured deviations in the\n\$\\beta \$\n-function along the LHC ring relative to the MADX model incorporating the phase advance knob, for Beam 1 (top two figures for\nx\nand\ny\n) and Beam 2 (bottom two figures for\nx\nand\ny\n). The error bars represent solely the statistical uncertainties as determined in the measurements\nFull size image\n5.3.3\n\$\\beta \$\nfunctions at the interaction points\nThe DOROS BPMs are strategically located at the end of the inner triplets on both sides of each of the experiments. Due to the phase advance of\n\$\\pi \$\nbetween the DOROS on the left and right sides, these are inadequate for the precise determination of the measurement of\n\$\\beta ^*\$\nat the IP. This can only be achieved with a special\nk\n-modulation technique [\n31\n], which was performed at the interaction points used in the experiment. The results are summarized in Table \n3\n. While the achieved precision of the\n\$\\beta ^*\$\ndetermination varies between measurements, uncertainties at the level of\n\${\\mathcal {O}}(1\\%)\$\nare not uncommon and are not systematic to a specific beam or interaction point. Such uncertainties would propagate to only a few-percent effect on derived quantities such as the emittance or the beam–beam parameter.\n5.3.4\nDispersion\nIt was essential to perform dispersion\n\$D^*\$\nmeasurements at the interaction points, under the new optics configurations, as it can contribute to the measured beam size. Assuming the LHC design momentum spread value of\n\$\\frac{\\Delta p}{p}=10^{-4}\$\n, the average dispersion measurement around IP1 and IP5 is estimated to contribute up to 2% to the observed bunch sizes (worst case measured for B2Y). This effect is considered static throughout the experiment.\n6\nExperimental results\nThis section presents experimental measurements of beam–beam effects in configurations involving multiple collision points. The impact of the optics modifications introduced in Sect. \n5\nis first assessed through tune spectra measurements, as detailed in Sect. \n6.1\n. To quantify the cumulative effect of one or more additional collisions and assess the reproducibility of the results, a stepwise collision scan was employed. In this procedure, beams were sequentially brought into head-on collision at selected interaction points (IPs), followed by full transverse separation (\n\$\\Delta > 6\\sigma \$\n) in either the horizontal or vertical plane at all but one witness IP. Two distinct filling schemes were used to isolate the influence of a single additional collision (2-IP configuration: ATLAS and CMS, see Sect. \n6.2\n) and to investigate more complex multi-collision scenarios (3-IP configuration: including ALICE, see Sect. \n6.3\n). The first measurement of separation-dependent beam–beam effects, analogous to those encountered during luminosity calibration scans, is presented in Sect. \n6.4\n, alongside comparisons with numerical predictions. An additional measurement conducted during the 2022 van der Meer calibration is detailed in Sect. \n6.5\n. A consolidated overview of all findings is provided in Sect.\n7\n.\n6.1\nMeasured effect on the tune spectra of the phase advance changes\nThe coherent transverse tune spectra serve as a diagnostic for evaluating the effectiveness of the phase advance modifications. COMBI simulations were carried out for two extreme lattice configurations, one designed to maximize and the other to suppress beam–beam effects. In the head-on scenario (Fig. \n9\n), the coherent modes are suppressed for the maximizing configuration, as the phase advance between the two collision points is close to\n\$\\pi \$\n, as was previously shown in simulation in Ref. [\n32\n]. In this configuration, the effect of the first collision is counteracted by the second. This setting is particularly suitable for studying incoherent spectra, as the diminished coherent signal allows for a clearer extraction of the mean tune shift from fitted spectra. During separation scans, the symmetry between horizontal (\nx\n) and vertical (\ny\n) planes is broken. In the scanning plane (horizontal), coherent modes re-emerge at\n\$1.5\\,\\sigma \$\nseparation (Fig. \n10\n), with their frequency shifting in response to the separation distance (additional example for\n\$3\\,\\sigma \$\nis shown in Fig. \n11\n).\nFig. 9\nComparison of COMBI simulated tune spectra in\nx\n-plane (left) and\ny\n-plane (right) for the suppressing (in blue) and maximizing (in green) phase configurations, for head-on collision\nFull size image\nFig. 10\nComparison of COMBI simulated tune spectra in\nx\n-plane (left) and\ny\n-plane (right) for the suppressing (in blue) and maximizing (in green) phase configurations, at\n\$1.5\\,\\sigma \$\nseparation in the horizontal plane\nFull size image\nFig. 11\nComparison of COMBI simulated tune spectra in\nx\n-plane (left) and\ny\n-plane (right) for the suppressing (in blue) and maximizing (in green) phase configurations, at\n\$3\\,\\sigma \$\nseparation in the horizontal plane\nFull size image\nThese simulated trends of the coherent spectra are also observed in experimental data. Transverse spectra recorded for head-on conditions confirm the suppression of coherent modes in the maximizing phase configuration (Fig. \n12\n). Additional comparisons for separated beams at the IP (see Figs. \n13\nand \n14\n), shown here for the case of vertical separation, reveal similar features. In this configuration, the non-scanning (horizontal) plane exhibits spectra that qualitatively agree with simulation predictions (see Figs. \n10\nand \n11\n), while in the scanning (vertical) plane the coherent modes do not reappear as expected. An equivalent behavior is observed when the separation is applied in the horizontal plane, with the roles of the two transverse planes interchanged; this case is not shown for brevity. This discrepancy is attributed to the effect of the transverse damper, which acts independently on the two beam centroids and suppresses coherent oscillations arising from beam–beam coupling, consistent with earlier observations [\n33\n].\nFig. 12\nMeasured tune spectra before and after phase change, corresponding to the suppressing (in blue) and maximizing (in green) phase configurations, at the head-on position, for each of the transverse planes (Beam 1 shown as an example). Dash-dotted lines show Gaussian fits to the data\nFull size image\nFig. 13\nComparison of the measured tune spectra at suppressing (blue) and maximizing (green) phase configurations with\n\$1.3\\,\\sigma \$\nseparation in the vertical plane\nFull size image\nFig. 14\nComparison of the measured tune spectra at suppressing (blue) and maximizing (green) phase configurations with\n\$2.6\\,\\sigma \$\nseparation in the vertical plane\nFull size image\n6.2\nHead-on collisions in a two-IP configuration at\n\$\\sqrt{s} = 900\\ \\text {GeV}\$\nEach test consisted of four measurements designed to quantify the beam–beam effect induced by a single additional head-on collision. The sequence was executed with CMS as the witness IP, followed by ATLAS, and concluded with a repeat at CMS. Over the course of the fill, the beam–beam parameter decreased from approximately 0.01 to 0.005. The outcome of each scan is analyzed using three independent observables: luminosity, beam size, and tune spectra, discussed separately below.\n6.2.1\nImpact of beam-beam effects on head-on luminosity\nAn overview of the luminosity measurements during the first test with CMS as the witness IP is shown in Fig. \n15\n, where the specific luminosity at CMS is normalized to the initial head-on step to highlight relative changes. The uncertainty at each step is given by the standard error of the mean; shaded bands represent the standard deviation. A comparable dataset for ATLAS is presented in Fig. \n16\n, confirming the expected luminosity change due to beam–beam effects originating from another IP. Relative luminosity changes for each step in all tests are summarized in Table \n5\n. These values are computed relative to the average of adjacent head-on points, which removes the linear luminosity decay over time. Assuming slow change of beam parameters, the results within each test are expected to be consistent.\nFig. 15\nCMS specific luminosity, normalized to the first measured both IPs head-on point (blue), during the first step-function scan at IP1. The measurements with ATLAS fully separated are shown in red. The error bars indicate error on the mean\nFull size image\nFig. 16\nATLAS specific luminosity, normalized to the first measured both IPs head-on point (blue), during the step-function scan at IP5. The points for which IP5 was separated are shown in red. The error bars indicate the statistical error\nFull size image\nTable 5 Measured luminosity shifts in each of full separation steps – summary of all three step-function scans performed for IP1 and IP5 as the witness IP. The beam-beam parameter at the start of each scan\n\$\\xi _\\text {start}\$\nis included for reference\nFull size table\nAll measurements are compiled in Fig. \n17\nand compared to COMBI simulation predictions. The results are plotted as a function of the beam–beam parameter (estimated as described in Sect. \n4.4\n), following a reverse chronological order. Agreement with simulation is excellent across the entire range.\nFig. 17\nCOMBI simulated luminosity enhancement induced by a single head-on collision as a function of the beam-beam parameter (dashed line), compared to the test results at both ATLAS (red points) and CMS (green points)\nFull size image\nFig. 18\nMeasurements of beam-beam effects on beam widths shown with red points for each beam and plane, during the experiment with IP5 as the witness IP at\n\$\\xi =0.0056\$\n. COMBI predictions are shown in blue with calculated systematic error from the phase advance error between the IP and BSRT locations\nFull size image\n6.2.2\nImpact of beam-beam effects on single-beam sizes\nComplementary measurements of the bunch size from the BSRT are shown in Fig. \n18\n. Data from all steps and bunches are combined, and a linear trend corresponding to continuous emittance growth is subtracted using the difference between the first and last head-on steps. For comparison, COMBI simulation results are also included in the figure. The dominant source of uncertainty in COMBI predictions arises from discrepancies between the measured phase advance and the MADX model, with the largest observed deviation being\n\$0.05\\,[2\\pi ]\$\n(as discussed in Sect. \n5.3.1\n). This uncertainty is propagated using a modified form of Eq. (\n6\n) adapted to the BSRT location. The magnitude of this uncertainty depends on the specific phase advance configuration, as some are more sensitive to small deviations. This is reflected in the varying sizes of the error bands associated with each prediction. The most accurate predictions are obtained in the vertical plane of Beam 2, where the beam–beam–induced\n\$\\beta \$\n-beating propagates to the BSRT with a particularly favorable phase advance. In this configuration, the measured effect is also the most pronounced, and excellent agreement with COMBI simulations is observed. In other cases, the expected absolute effect is smaller, typically below 1%, and the influence of phase advance uncertainty becomes more significant, limiting the predictive reliability of the simulations. Experimentally, it is also possible that in these configurations the beam–beam–induced effect is too subtle to resolve, potentially masked by instrumental limitations.\n6.2.3\nImpact of beam-beam effects on tune spectra\nFigure \n19\nshows representative ADT spectra recorded during the step-function scans. These spectra are affected by coupling, noise, and various nonlinearities, so they do not follow an idealized analytical distribution. Nonetheless, the peak position can be extracted using a Gaussian fit (indicated by dashed lines), yielding an estimate of the mean tune shift. In the single-collision configuration (blue lines), unsuppressed coherent modes dominate the spectrum and partially obscure the underlying incoherent component. The Gaussian fit is therefore applied to the incoherent contribution between coherent lines, rather than to the global spectral maximum, which results in a larger uncertainty on the extracted peak position. Despite this, in each of the performed step-scans at both collision points, the single-collision tune shift, defined as twice the fitted mean shift, is found to be consistent within\n\$\\pm 5\\%\$\nwith the expected coherent beam–beam tune shift corresponding to the conditions of the respective scan. For head-on collisions with Gaussian transverse beam distributions, the coherent tune shift of the dipole mode is expected to be\n\$Y\\xi \$\n, where\nY\nis the Yokoya factor (\n\$Y = 1.21\$\nfor round beams) [\n34\n]. As expected, the two-collision configuration yields a tune shift approximately double that of the single-collision case [\n13\n].\nFig. 19\nMeasured vertical tune spectra for different collision configurations: reference nominal tune (red), without the presence of any collisions, compared to single collision (blue) and two collisions (green) at\n\$\\xi \\simeq 0.0086\$\n. The dash-dotted lines in corresponding colors indicate Gaussian fits to the data used to extract mean tunes\nFull size image\nFig. 20\nSpecific luminosity, normalized to the first head-on point, measured at ATLAS (top) and CMS (bottom) during beam separation at the other IPs, with horizontal separation applied in steps 2–4 and vertical separation in steps 6–8. The tested configurations are indicated in the legend. The blue line shows an interpolation of the head-on measurements, serving as a reference for relative luminosity changes\nFull size image\n6.3\nHead-on collisions in a three-IP configuration at\n\$\\sqrt{s} = 900\\ \\text {GeV}\$\nAn additional head-on collision was included in the configuration using IP2 (ALICE), as it was the only IP where this was feasible. A specially designed filling scheme ensured uniform collision exposure across all bunches, allowing measurements to be combined and thereby reducing statistical uncertainties. However, no reliable luminosity data were available from IP2 during this experiment.\nTable 6 Luminosity shift per full separation step, calculated with respect to an interpolated reference obtained from the two closest head-on points. Results for horizontal and vertical separations are shown as Shift part 1 and Shift part 2, respectively\nFull size table\n6.3.1\nImpact of beam-beam effects on head-on luminosity\nA step-function scan was conducted by sequentially collapsing or separating one IP at a time, with the separation applied first in the horizontal plane and, in a second stage of the experiment, in the vertical plane. A comprehensive summary of the measured specific luminosity at ATLAS and CMS is presented in Fig. \n20\n. Exceptionally stable beam conditions during this test enabled measurements at both experiments under nearly identical beam–beam parameters. The beam–beam-induced luminosity changes observed during each step are quantified in Table \n6\n, referenced to the time-interpolated specific luminosity from three head-on points.\nOrbit-drift corrections are applied only to these head-on reference points and result in changes of the extracted luminosity shifts of at most 0.2%. Good agreement with COMBI simulations is observed across the separation steps, with measured shifts closely matching the simulated values for configurations involving IP1 and IP5. For single-IP separation, the level of agreement depends on the witness IP and the separation step. When IP1 is used as the witness, the luminosity shift in the first separation step is reproduced almost exactly by COMBI, while a smaller shift is observed in the second step. This behaviour is expected, as the beam–beam parameter decreases during the scan and the measurement becomes increasingly sensitive to residual orbit perturbations. For configurations involving IP2, the comparison is less conclusive. As no reliable luminosity data were available from IP2 during this experiment, it was not possible to precisely optimize the collision conditions at this IP. As a result, the actual beam–beam conditions at IP2 may differ from those assumed in the simulations, affecting the comparison. Residual orbit drifts affecting the separation steps cannot be fully corrected. As illustrated by the DOROS measurements in Fig. \n21\n, the orbit data contain both slow drifts and beam–beam–induced orbit offsets arising from changes in the collision configuration when other IPs are separated. While the beam–beam–induced offsets are included in the COMBI simulations, the residual orbit drifts are not. Since these contributions cannot be disentangled in the measurements, a full correction is not possible. Consequently, the effective beam–beam impact measured during the experiment may be weaker than that assumed in the simulations.\nFig. 21\nResidual separation between the two beams at all three IPs in horizontal (top) and vertical (bottom) planes, for test with observations at ATLAS, where the separation was performed first in the horizontal and then in the vertical plane at the other IPs\nFull size image\n6.3.2\nTune shift scaling with the number of beam–beam interactions\nAnalysis of the transverse tune spectra in this configuration is further complicated by the reappearance of coherent modes when beams are separated at one or more IPs, as predicted by simulation. In such cases, the maximum tune shift is determined from the frequency difference between the\n\$\\pi \$\nand\n\$\\sigma \$\nmodes, rather than from fitting the incoherent peak. For configurations where all three IPs were in collision, coherent modes remained suppressed due to the retained symmetry between IP1 and IP5. Figure \n22\npresents the measured tune shifts for all tested collision configurations. The unperturbed tune measurement, shown at ‘0’ collisions, serves as the reference from which all tune shifts are calculated. An empirical uncertainty of 0.001 is assumed for each independent measurement. Both transverse planes and separation directions are included to demonstrate consistency. As expected, the measured tune shift scales linearly with the number of head-on collisions.\nFig. 22\nMeasured mean tune shift as a function of the number of head-on collisions, shown for both transverse planes and for both separation directions (horizontal and vertical) used in the experiment. The separation refers to how collisions were removed: it was applied in only one transverse direction at a time (H sep. or V sep.) at the non-colliding interaction points, which are not explicitly labeled on the x-axis\nFull size image\n6.4\nBeam-beam signatures during transverse-separation scans at\n\$\\sqrt{s} = 900\\ \\text {GeV}\$\nIn the van der Meer (vdM) calibration method, beam–beam corrections are applied separately at each separation step. Therefore, it is essential to validate the simulated dependence of luminosity shifts on nominal beam separation. Full vdM-like scans were performed in both transverse planes at ATLAS and CMS, using a configuration where only these two IPs were in collision.\n6.4.1\nImpact of beam–beam effects on the luminosity scan-curves\nFigure \n23\nshows the luminosity curves measured at ATLAS, acting as the witness IP, during a separation scan performed at CMS. The curves are fitted with Gaussian profiles for visual guidance. The beam separation at CMS is expressed in units of the measured transverse beam size,\n\$\\sigma _{\\textrm{meas}}\$\n, which was found to be smaller than the nominal beam size,\n\$\\sigma _{\\textrm{nom}}\$\n, used to define the scan steps. As a result, scan points corresponding to nominal separations beyond\n\$6\\,\\sigma _{\\textrm{nom}}\$\nextend to larger values when expressed in units of\n\$\\sigma _{\\textrm{meas}}\$\n.\nTo account for emittance evolution during the scan, both linear and exponential fits are applied to the head-on luminosity points before, after, and in the middle of the scan. These trends are subtracted from the data in Fig. \n23\n, and all measurements are presented relative to this fitted head-on reference (normalized to unity).\nFig. 23\nATLAS specific luminosity relative changes (blue points) during separation scans in\nx\n-plane (top) and\ny\n-plane (bottom) at CMS, corrected for exponential (exp. corr.) luminosity decay. Gaussian function with a constant fitted to these points is shown with blue dashed line and compared to fit performed on the data corrected for linear luminosity change (lin. corr.) in time (dotted line). These are compared to COMBI simulation predictions (purple solid line)\nFull size image\nAt CMS, similar scans were performed; however, degraded beam stability was observed during this period of the fill. The scan results exhibit systematic deviations, especially at large separation, which could not be symmetrically corrected using simple linear or exponential models. Despite these issues, all four scan cases (two transverse planes at two IPs) yield a total specific luminosity reduction of approximately 2.4%, consistent with expectations based on the measured beam–beam parameter\n\$\\xi \$\n. Comparison with COMBI simulations reveal up to 30% discrepancy for IP5 as the witness IP, while agreement improves to within 10% for measurements at IP1. These differences are comparable to those observed in Table \n6\n, suggesting a common underlying origin. While a finer scan granularity can improve the description of the steep-response region below\n\$3\\,\\sigma _{\\textrm{meas}}\$\n, it would not address the dominant discrepancy between head-on and fully separated luminosity, which is likely driven by simulation assumptions and scan-related systematics.\n6.4.2\nBeam–beam-induced orbit deflections\nBeam–beam deflections are clearly observed during separation scans. Figure \n24\nshows the fully corrected residual horizontal beam displacement at IP1, measured with the DOROS beam position monitors, as a function of the beam separation. This representation allows a direct comparison with the analytical beam–beam deflection prediction based on the Bassetti–Erskine formalism [\n35\n].\nThe raw orbit data were corrected for slow orbit drifts by aligning the beam positions at the luminosity optimization points before and after each scan. Deviations of the applied beam separation from the nominal settings were accounted for by fitting a common length scale simultaneously with the beam–beam deflection. This approach avoids an overestimation of the separation calibration. The measured orbit displacement at the DOROS location was corrected for the contribution arising from the beam–beam deflection angle by applying a geometric amplification factor, proportional to the distance between the IP and the BPM in the small-angle approximation. After all corrections, the measured beam–beam deflection as a function of beam separation closely follows the analytical prediction. The maximum horizontal deflection observed at IP1 is 10.2 \n\$\\upmu \$\nm, compared to an expected value of 8.4 \n\$\\upmu \$\nm, with agreement within the total measurement uncertainty. The dominant uncertainty contribution (3.5 \n\$\\upmu \$\nm, corresponding to about 90% of the total) arises from a possible residual half crossing angle at the IP of up to 0.5 \n\$\\upmu \$\nrad. Additional systematic uncertainties include the fitted separation length scale, tune uncertainty of 0.005, and an extended 2 % uncertainty on\n\$\\beta ^*\$\n. Statistical uncertainties are negligible for most separation steps.\nFig. 24\nOrbit offset caused by the beam–beam interaction measured by DOROS BPMs in the horizontal plane at IP1 during the horizontal separation scan. Analytical prediction is shown for comparison [\n35\n]\nFull size image\n6.4.3\nImpact of the beam-beam effects on beam sizes and tune shift\nSeparation-dependent changes in beam size were measured using the synchrotron light monitor. Figure \n25\nshows good consistency across individual bunch measurements, as well as strong agreement with COMBI simulations. This reinforces the utility of the BSRT as a reliable diagnostic for beam–beam effects. The uncertainties shown are statistical only and are treated as uncorrelated between scan points. The normalization to the zero-separation value is applied for visualization purposes and does not impose a constraint on that point.\nFig. 25\nBSRT beam-size measurement in the vertical plane for each bunch (points) during horizontal (top) and vertical (bottom) separation scans at IP1. The exponential evolution correction is applied based on adjacent head-on points (before, in the middle and after the scan) to highlight the relative changes. COMBI predictions are shown (gray dashed line) for comparison\nFull size image\nTransverse tune shifts were extracted from ADT-measured spectra. As illustrated in Fig. \n26\n, the mean tune shift evolution is reconstructed for both the scanning and non-scanning planes. The measured tune shifts closely follow the expected COMBI trends, further validating the simulation models used for beam–beam corrections.\nFig. 26\nMeasured tune shift in units of the beam-beam parameter\n\$\\xi _{BB}\$\nfor the horizontal (blue) and vertical (orange) planes, during a horizontal separation scan at IP1. The error bars indicate typically assumed conservative empirical systematic error on the tune measurement of 0.001. The solid curves represent the COMBI prediction\nFull size image\n6.5\nBeam-beam signatures during vdM scans at\n\$\\sqrt{s} = 13.6\\ \\text {TeV}\$\nDuring the first vdM calibration of LHC Run 3, an unusually high beam–beam parameter was recorded, offering a unique opportunity to study beam–beam effects under enhanced interaction conditions. These effects are examined using online ATLAS luminosity data, with representative results shown in Fig. \n27\n.\nThe example focuses on a standard vdM scan pair conducted at CMS, while monitoring luminosity at the non-scanning IP (ATLAS). The plotted luminosity changes are normalized to the central scan point, where both IPs are in head-on collision. Variations in this normalized luminosity across the scan directly reflect the impact of beam–beam interaction at the scanning IP on the luminosity at the non-scanning IP. These patterns are reproducible across multiple scan pairs acquired within the same fill. At the largest nominal separations (approximately\n\$\\pm 0.6\\,\$\nmm), the beams are sufficiently separated to eliminate beam–beam effects entirely. Notably, a qualitative difference is observed between the horizontal and vertical scan planes, an asymmetry that is not present in the above presented dedicated experiment. This behavior is attributed to differences in the phase advance between IPs in the standard vdM optics configuration. For vertical (\ny\n) scans, experimental data agree well with simulation, confirming the robustness of the modeling under these conditions. In contrast, the horizontal (\nx\n) scans exhibit a systematic discrepancy: the observed beam–beam impact is broader than that predicted by simulation. The model assumes that the separation steps are accurately defined using the beam size measured during the vdM scans, but this mismatch may suggest an unmodeled phase advance deviation or another optics imperfection. The total difference between data and simulation is within 0.2%. This dataset originates from a standard vdM calibration, for which only online luminosity measurement was available, and many contributions to the overall vdM uncertainty, such as bunch-to-bunch variations, averaging over different bunch families, and emittance evolution during the scan, are not disentangled. Therefore, the 0.2% difference should be interpreted as a consistency check under typical operational conditions rather than a precision validation of the beam–beam model. Nevertheless, this indicates that the model remains sufficiently accurate for beam–beam corrections in vdM analyses.\nFig. 27\nChanges from the separation-dependent beam-beam effects at the non-scanning IP – ATLAS online luminosity shown with points. Data comes from a standard vdM scan pair during the 2022\npp\nluminosity calibration. Separation steps and direction are indicated with gray markers. The last CMS scan is shown, with the highest beam-beam parameter of\n\$ \\xi =5.3 \\times 10^{-3} \$\n. The corresponding COMBI simulation results are also shown (triangles)\nFull size image\n7\nConclusions\nBeam–beam effects were directly observed and quantitatively validated at the LHC. The impact of beam–beam interactions on head-on luminosity, bunch sizes, tune spectra, and beam orbit positions was measured across various collision configurations. The observed effects align closely with COMBI simulations. In particular, the luminosity enhancement due to amplitude-dependent\n\$\\beta \$\n-beating was measured for the first time and is found to agree with expectations over a wide range of the beam–beam parameter\n\$\\xi \$\n. Together with the observed linear scaling with respect to the beam–beam parameter, this provides experimental confirmation of the numerical studies presented in [\n13\n]. Consistent signatures were found across multiple scan configurations, including two- and three-IP setups, and the characteristic tune shifts were well reproduced by both measurement and simulation.\nAccurate instrumentation and experimental design were essential. Key to the results was the stepwise collision scheme, isolating individual beam–beam contributions, with the implementation of the witness IP. Synchrotron light measurements from the BSRT, especially in the vertical plane of Beam 2, showed excellent agreement with COMBI predictions and bunch-by-bunch consistency. Tune spectra from the ADT system provided direct access to mean tune shifts, while orbit data from the DOROS system enabled measurement of beam–beam deflection down to 1 \n\$\\upmu \$\nm, matching analytical estimates. These tools, combined with precise control of IP configurations, enabled a comprehensive cross-check of beam–beam models.\nSome systematic limitations were identified. During one of the tests, beam stability during the scans was insufficient to allow for symmetric modeling of luminosity decay in both transverse planes. In several scans, the nominal separation steps were based on overestimated beam sizes, resulting in larger-than-expected scan intervals and less sensitivity at low separation. Beam–beam deflection measurements showed sign reversals at high separation, attributed to nonlinear orbit distortions from hysteresis effects in superconducting magnets. Phase advance errors between IPs and diagnostic systems further reduced the accuracy of simulations in some configurations.\nTargeted improvements would allow for higher precision and new insights. Finer separation steps, especially below\n\$3\\,\\sigma \$\n, would improve sensitivity to nonlinear beam–beam effects and allow more accurate comparison with models. Suppressing coherent beam–beam modes via phase advance tuning of not only two but all three IPs would improve tune shift measurements. Moreover, such optimal phase advance configuration between the IPs can be exploited to enhance luminosity. This effect could be directly propagated into at least a few percent increase in the total collected integrated luminosity, as discussed in Ref. [\n36\n]. Aligning the phase advance between IPs and diagnostic devices, particularly the BSRT, would reduce sensitivity to optics errors, enabling better beam size measurements for both beams and transverse planes. Moreover, the suppression of the coherent beam–beam modes facilitates the observation and analysis of the central, incoherent part of the tune spectrum, which is otherwise obscured by the dominant coherent peaks.\nFurther experimental strategies should also focus on directly probing the beam–beam interaction at its point of occurrence. This would require access to the full transverse beam distributions with high spatial resolution and sensitivity, going beyond projected sizes or centroids. 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Article\n# First measurements of beam-beam effects in beam-separation, luminosity-calibration scans at the LHC\n- Regular Article - Experimental Physics\n- Experimental Physics\n- [Open access](https:\/\/www.springernature.com\/gp\/open-science\/about\/the-fundamentals-of-open-access-and-open-research)\n- Published:\n  17 March 2026\n\n- Volume 86, article number 282, (2026)\n- [Cite this article](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#citeas)\n\nYou have full access to this [open access](https:\/\/www.springernature.com\/gp\/open-science\/about\/the-fundamentals-of-open-access-and-open-research) article\n\n[Download PDF](https:\/\/link.springer.com\/content\/pdf\/10.1140\/epjc\/s10052-026-15476-8.pdf)\n\n[Save article](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/save-research?_csrf=skr420D_Tsy443125TKQwosqfCsJrZRn)\n\n[View saved research](https:\/\/link.springer.com\/saved-research)\n\n[![](https:\/\/media.springernature.com\/w72\/springer-static\/cover-hires\/journal\/10052?as=webp) The European Physical Journal C](https:\/\/link.springer.com\/journal\/10052)\n\n[Aims and scope](https:\/\/link.springer.com\/journal\/10052\/aims-and-scope)\n\n[Submit manuscript](https:\/\/mc.manuscriptcentral.com\/epjc)\n\nFirst measurements of beam-beam effects in beam-separation, luminosity-calibration scans at the LHC\n\n[Download PDF](https:\/\/link.springer.com\/content\/pdf\/10.1140\/epjc\/s10052-026-15476-8.pdf)\n\n- [Joanna Wańczyk](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#auth-Joanna-Wa_czyk-Aff1-Aff2)\n  [ORCID: orcid.org\/0000-0002-8562-1863](https:\/\/orcid.org\/0000-0002-8562-1863)[1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Aff1),[2](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Aff2),\n- [Tatiana Pieloni](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#auth-Tatiana-Pieloni-Aff2) [ORCID: orcid.org\/0000-0002-3218-0048](https:\/\/orcid.org\/0000-0002-3218-0048)[2](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Aff2),\n- [Xavier Buffat](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#auth-Xavier-Buffat-Aff1) [ORCID: orcid.org\/0000-0002-2338-8808](https:\/\/orcid.org\/0000-0002-2338-8808)[1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Aff1),\n- [Anne Dabrowski](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#auth-Anne-Dabrowski-Aff1) [ORCID: orcid.org\/0000-0003-2570-9676](https:\/\/orcid.org\/0000-0003-2570-9676)[1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Aff1),\n- [David Stickland](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#auth-David-Stickland-Aff3) [ORCID: orcid.org\/0000-0003-4702-8820](https:\/\/orcid.org\/0000-0003-4702-8820)[3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Aff3) &\n- …\n- [Rogelio Tomas](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#auth-Rogelio-Tomas-Aff1) [ORCID: orcid.org\/0000-0002-9857-1703](https:\/\/orcid.org\/0000-0002-9857-1703)[1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Aff1)\n\nShow authors\n\n## Abstract\nAt the CERN Large Hadron Collider (LHC), absolute luminosity calibrations obtained by the van der Meer (*vdM*) method are affected by the mutual electromagnetic interaction of the two beams. In the last few years, increasingly stringent, physics-driven requirements on the accuracy of the absolute luminosity scale, combined with the unparalleled precision of the luminosity measurements achieved by the LHC experiments, motivated comprehensive simulation studies to accurately model beam–beam effects and provide simulation-based corrections to *vdM*\\-based luminosity calibrations. This paper reports the first attempt at an experimental validation of that model, in dedicated accelerator measurements under controlled conditions, of the predicted impact of beam-beam effects on the luminosity measured at the LHC under luminosity-calibration conditions. This is the first measurement of beam-beam interaction effects on luminosity at the LHC. The results show good agreement with the predictions of multiparticle simulations, and justify the recently proposed strategy for correcting beam-beam biases on absolute luminosity calibrations at hadron colliders. This marks a critical step forward in precise luminosity calibration for past, current, and future datasets, including the High-Luminosity Large Hadron Collider era.\n\n### Explore related subjects\nDiscover the latest articles, books and news in related subjects, suggested using machine learning.\n\n- [Accelerator Physics](https:\/\/link.springer.com\/subjects\/accelerator-physics)\n- [Experimental Particle Physics](https:\/\/link.springer.com\/subjects\/experimental-particle-physics)\n- [Measurement Science and Instrumentation](https:\/\/link.springer.com\/subjects\/measurement-science-and-instrumentation)\n- [Metrology](https:\/\/link.springer.com\/subjects\/metrology)\n- [Optical Metrology](https:\/\/link.springer.com\/subjects\/optical-metrology)\n- [Particle Physics](https:\/\/link.springer.com\/subjects\/particle-physics)\n\n## 1 Introduction\nA precise determination of the absolute luminosity scale is essential for a wide range of LHC measurements, as it directly affects the normalization of many key physics cross sections. At the LHC, this calibration relies primarily on the van der Meer (vdM) method \\[[1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR1 \"S. Meer, Calibration of the effective beam height in the ISR. Technical report, CERN, Geneva (1968). \n                  https:\/\/cds.cern.ch\/record\/296752\n                  \n                \"), [2](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR2 \"C. Rubbia, Measurement of the luminosity of \n                  $$p{{\\overline{p}}}$$\n                  \n                    \n                      p\n                      \n                        p\n                        ¯\n                      \n                    \n                  \n                 collider with a (generalized) Van der Meer Method. Technical report, CERN, Geneva (1977). \n                  https:\/\/cds.cern.ch\/record\/1025746\n                  \n                \")\\], which uses dedicated beam-separation scans performed under specially tailored beam conditions to relate measured interaction rates to the absolute luminosity inferred from beam parameters. While sub-percent uncertainties on the integrated luminosity have been achieved for Run-2 data by ATLAS \\[[3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR3 \"ATLAS collaboration, Luminosity determination in \n                  $$pp$$\n                  \n                    \n                      pp\n                    \n                  \n                 collisions at \n                  $$\\sqrt{s}=13$$\n                  \n                    \n                      \n                        s\n                      \n                      =\n                      13\n                    \n                  \n                 TeV using the ATLAS detector at the LHC. Eur. Phys. J. C 83(10), 982 (2023). \n                  https:\/\/doi.org\/10.1140\/epjc\/s10052-023-11747-w\n                  \n                . \n                  arXiv:2212.09379\n                  \n                \")\\] and CMS \\[[4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR4 \"CMS Collaboration, Precision luminosity measurement in proton-proton collisions at sqrts = 13 TeV with the CMS detector. Technical report, CERN, Geneva (2025). \n                  https:\/\/cds.cern.ch\/record\/2940794\n                  \n                \")\\], further improvements and robust experimental validation of beam-dynamical effects remain essential.\n\nFor relativistic proton beams colliding with zero crossing angle, which is the typical configuration for the *vdM* calibration, the luminosity at a given interaction point (IP) of the LHC is proportional to the overlap integral of the particle-density distributions \\\$\\\\rho \\_{\\\\text {1,i}}\\\$, \\\$\\\\rho \\_{\\\\text {2,i}}\\\$ in bunch-pair *i* \\[[5](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR5 \"W. Herr, T. Pieloni, Beam-Beam Effects. Technical report, CERN (2014). Contribution to the CAS–CERN Accelerator School: Advanced Accelerator Physics Course, Trondheim, Norway, 18–29 Aug 2013, p. 29. \n                  https:\/\/doi.org\/10.5170\/CERN-2014-009.431\n                  \n                . \n                  https:\/\/cds.cern.ch\/record\/1982430\n                  \n                \")\\]:\n\n\\$\\$\\\\begin{aligned} {\\\\mathcal {L}}\\_{inst}&= 2c f\\_{\\\\text {rev}} \\\\sum ^{N\\_b}\\_i n\\_{\\\\text {1,i}}, n\\_{\\\\text {2,i}} \\\\nonumber \\\\\\\\&\\\\quad \\\\times \\\\iiiint ^{+\\\\infty }\\_{-\\\\infty }\\\\rho \\_{\\\\text {1,i}}(x,y,z-ct) \\\\nonumber \\\\\\\\&\\\\quad \\\\times \\\\rho \\_{\\\\text {2,i}}(x,y,z+ct)\\\\,dxdydzdt, \\\\end{aligned}\\$\\$\n\n(1)\n\nwhere *c* is the speed of light, \\\$f\\_{\\\\text {rev}}\\\$ is the LHC revolution frequency, \\\$N\\_b\\\$ is the number of colliding bunches, and \\\$n\\_{\\\\text {1,i}}, n\\_{\\\\text {2,i}}\\\$ are the corresponding total charges per bunch for each colliding pair. Although only a very small fraction of the beam particles actually collide, the two opposing beams interact with each other electromagnetically, a dynamical process known as the beam-beam interaction \\[[5](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR5 \"W. Herr, T. Pieloni, Beam-Beam Effects. Technical report, CERN (2014). Contribution to the CAS–CERN Accelerator School: Advanced Accelerator Physics Course, Trondheim, Norway, 18–29 Aug 2013, p. 29. \n                  https:\/\/doi.org\/10.5170\/CERN-2014-009.431\n                  \n                . \n                  https:\/\/cds.cern.ch\/record\/1982430\n                  \n                \"), [6](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR6 \"A.W. Chao, Beam-beam instability. AIP Conf. Proc. 127, 201–242 (1985). \n                  https:\/\/doi.org\/10.1063\/1.35187\n                  \n                \")\\]. In the LHC, as in any other synchrotron, the beams are not continuous but are divided into discrete “bunches”, each with a high proton density, containing approximately \\\$10^{11}\\\$ protons over a bunch length of 7.5–9 cm (in terms of RMS). This electromagnetic interaction occurs when the two beams share a common beam pipe, and is governed by a non-linear force that depends on the radial distance *r* of a test particle to the center of the opposing “source” bunch with *n* protons Gaussian-distributed in the transverse planes (valid approximation for the LHC beams):\n\n\\$\\$\\\\begin{aligned} F = - \\\\frac{ne^2}{2\\\\pi \\\\epsilon \\_0r}\\\\biggl (1-\\\\exp \\\\biggl \\[ -\\\\frac{r^2}{2\\\\sigma ^2}\\\\biggr \\] \\\\biggr ), \\\\end{aligned}\\$\\$\n\n(2)\n\nwhere *e* is the elementary proton charge, \\\$\\\\epsilon \\_0\\\$ is the vacuum permittivity, and \\\$\\\\sigma \\\$ is the RMS radius of the transverse charge distribution within the bunch, in a simplified case of a round shape \\\$(\\\\sigma = \\\\sigma \\_x = \\\\sigma \\_y)\\\$. At very small amplitudes \\\$(r\\\\rightarrow 0)\\\$ the force can be linearized giving the expression for the so called beam-beam parameter, which is often used to assess the strength of the force:\n\n\\$\\$\\\\begin{aligned} \\\\xi = \\\\frac{nr\\_p\\\\beta ^\\*}{4\\\\pi \\\\gamma \\\\sigma ^2} = \\\\frac{nr\\_p}{4\\\\pi \\\\epsilon \\_n}, \\\\end{aligned}\\$\\$\n\n(3)\n\nwhere \\\$r\\_p\\\$ is the classical proton radius, \\\$\\\\beta ^\\*\\\$ is the optical beta-function at the interaction point (IP) – one of the Courant–Snyder parameters \\[[7](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR7 \"E.D. Courant, H.S. Snyder, Theory of the alternating-gradient synchrotron. Ann. Phys. 3, 1–48 (1958). \n                  https:\/\/doi.org\/10.1016\/0003-4916(58)90012-5\n                  \n                \")\\], \\\$\\\\gamma \\\$ is the Lorentz factor, and \\\$\\\\epsilon \\_n\\\$ is the normalized emittance.\n\nBeam-beam effects were first observed almost 40 years ago \\[[8](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR8 \"P. Bambade, R. Erickson, W.A. Koska, W. Kozanecki, N. Phinney, S.R. Wagner, Observation of beam-beam deflections at the interaction point of the SLAC Linear Collider. Phys. Rev. Lett. 62(25), 2949–2952 (1989). \n                  https:\/\/doi.org\/10.1103\/PhysRevLett.62.2949\n                  \n                \")\\] at the lepton colliders, and have been used extensively for accelerator diagnostics and optimization at both SLC and LEP \\[[9](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR9 \"W.A. Koska, P. Bambade, W. Kozanecki, N. Phinney, S.R. Wagner, Beam-beam deflection as a beam tuning tool at the slac linear collider. Nucl. Instrum. Methods A 286, 32 (1990). \n                  https:\/\/doi.org\/10.1016\/0168-9002(90)90203-I\n                  \n                \"),[10](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR10 \"C. Bovet, M.D. Hildreth, M. Lamont, H. Schmickler, J. Wenninger, Luminosity optimisation using beam-beam deflections at LEP, in Conf. Proc. C (1996). \n                  https:\/\/cds.cern.ch\/record\/306910\n                  \n                \"),[11](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR11 \"D. Brandt, W. Herr, M. Meddahi, A. Verdier, Is lep beam-beam limited at its highest energy? in Proceedings of the 1999 Particle Accelerator Conference (PAC ’99), New York, pp. 3005–3007. IEEE\/JACoW. THP25; CERN report CERN-SL-99-030-AP (1999). \n                  https:\/\/accelconf.web.cern.ch\/p99\/PAPERS\/THP25.pdf\n                  \n                \")\\]. At the LHC, they were first observed in late Run 1 \\[[12](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR12 \"W. Kozanecki, T. Pieloni, J. Wenninger, Observation of Beam-beam Deflections with LHC Orbit Data. Technical report, CERN (2013). \n                  https:\/\/cds.cern.ch\/record\/1581723?ln=en\n                  \n                \")\\], but only in terms of the beam-beam induced coherent deflections. Single particle effects, in contrast, were presumed too small in hadron colliders that until recently, there was no motivation to measure them with a precision that would be meaningful in the context evoked here.\n\nNonetheless, the full complexity of this interaction must be accounted for when performing the luminosity calibration via the vdM method, using the separation scans. These are designed to measure the detector-specific constant \\\$\\\\sigma \\_{vis}\\\$ that relates the observed rate \\\$\\\\mu ^{vis}\\\$ to the absolute instantaneous luminosity \\\${\\\\mathcal {L}}\\_{inst}\\\$:\n\n\\$\\$\\\\begin{aligned} \\\\mu ^{vis} = \\\\frac{{\\\\mathcal {L}}\\_{inst}\\\\sigma \\_{vis}}{f\\_{rev}}. \\\\end{aligned}\\$\\$\n\n(4)\n\nUnder the assumption of uncorrelated particle densities in *x* and *y* planes, the transverse convolved bunch widths \\\$\\\\Sigma \\_x, \\\\Sigma \\_y\\\$ can be extracted from the measured beam-separation dependence of the collision rate in the corresponding direction. The combined information from these scans, and bunch intensities can be used to calculate the instantaneous luminosity at the head-on position. The proportionality between the measured rate at the centered head-on position \\\$\\\\mu \\_{pk}\\\$ and the reconstructed luminosity from measured bunch parameters during the vdM calibration defines the luminometer-specific visible cross-section:\n\n\\$\\$\\\\begin{aligned} \\\\sigma \\_{vis} = \\\\frac{2\\\\pi \\\\Sigma \\_x\\\\Sigma \\_y}{n\\_1 n\\_2} \\\\mu \\_{pk}. \\\\end{aligned}\\$\\$\n\n(5)\n\nIf left uncorrected, the \\\$\\\\sigma \\_{vis}\\\$ measurement by the *vdM* method is biased by the beam-separation dependence of the mutual electromagnetic interaction of the two beams: the colliding bunches experience deflection-induced orbit shifts, as well as optical distortions akin to the dynamic-\\\$\\\\beta \\\$ effect, that both depend on the transverse beam separation and must therefore be accounted for when deriving the absolute luminosity scale. The correction strategy discussed extensively in Ref. \\[[13](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR13 \"A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). \n                  https:\/\/doi.org\/10.48550\/arXiv.2306.10394\n                  \n                \")\\] relies on simulation studies only, and thus requires experimental validation to establish its reliability.\n\nThe present report summarizes a campaign of dedicated measurements carried out in the spring of 2022 at the LHC, aimed at confirming the accuracy of the simulation studies detailed in Ref. \\[[13](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR13 \"A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). \n                  https:\/\/doi.org\/10.48550\/arXiv.2306.10394\n                  \n                \")\\], and at validating the beam-beam correction strategy used in the luminosity-calibration analyses of the ALICE, ATLAS, CMS and LHCb experiments. More specifically, the goal is to quantify the impact of the beam-beam interaction on the tune spectra, orbit, the transverse-density distribution of the colliding bunches, as well as, for the first time, on the luminosity, by systematically varying the strength or number of the beam-beam interactions. The measurements are repeated for different values of the beam-beam parameter to verify the scaling proposed in \\[[13](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR13 \"A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). \n                  https:\/\/doi.org\/10.48550\/arXiv.2306.10394\n                  \n                \")\\], for various choices of the scanning IP and for several multi-IP configurations. The beam conditions are chosen to be representative of *vdM* calibration sessions at the LHC, but optimized so as to maximize both the sensitivity of the measurements, and the operational efficiency.\n\nThis paper is organized as follows. The measurement strategy is detailed in Sect. [2](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec2), followed by the main ingredients of this accelerator experiment: beam conditions (Sect. [3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec3)), luminometers and accelerator instrumentation (Sect. [4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec4)), and optimization of the ring lattice to maximize the sensitivity of the measurements to beam–beam-induced optical distortions of the colliding bunches (Sect. [5](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec11)). Section [6](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec19) is devoted to the experimental characterization of the impact of beam-beam effects on the tune spectra, on the luminosity at different IPs, as well as on other observables such as transverse single-bunch widths and IP orbits. At each step, the results are confronted with the predictions of the COherent Multibunch Beam-beam Interactions (COMBI) tracking code \\[[14](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR14 \"T. Pieloni, A study of beam-beam effects in hadron colliders with a large number of bunches. PhD Thesis, École Polytechnique Fédérale de Lausanne (2008). \n                  https:\/\/doi.org\/10.5075\/epfl-thesis-4211\n                  \n                \")\\], that produced most of the results detailed in Ref. \\[[13](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR13 \"A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). \n                  https:\/\/doi.org\/10.48550\/arXiv.2306.10394\n                  \n                \")\\]. A collection of macroparticles is simulated turn by turn, based on a simplified model of the accelerator lattice using linear transfer and the self-consistent computation of the beam-beam forces at the IP. The conclusions are presented in Sect. [7](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec33).\n\n## 2 Measurement strategy\nThe beam-beam interaction at an interaction point, referred to here as the *scanning* IP, induces measurable changes in key beam parameters such as the tune spectrum, closed orbit, and transverse single-beam profiles. These effects can be observed, albeit with limited precision, using standard accelerator diagnostics by comparing measurements obtained under varying transverse beam separation at the scanning IP. However, the resulting modifications to the transverse density distribution of the colliding bunches remain below the sensitivity threshold of conventional instrumentation, including synchrotron-radiation-based beam-profile monitors. Beam-beam-induced variations typically lie in the 0.5–1% range, small enough that only a highly precise measurement of collision-rate changes can provide adequate sensitivity.\n\n**Fig. 1**\n\n[![Fig. 1](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig1_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/1)\n\nScheme of the LHC and its interaction points (IP) with indicated IP1 as witness IP while performing a scan at IP5\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/1)\n\nImportantly, the beam-beam interaction induces changes in beam properties that are intrinsically entangled with the *pp* collisions when observing luminosity at the scanning IP. Consequently, under realistic collider conditions, it is not possible to directly access a reference luminosity signal that is free of beam-beam effects – unlike in simulation studies, where such effects can be explicitly disabled. In a multi-collision configuration, however, luminosity shifts at one or more other IPs (Fig. [1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig1)), where the beams are kept in continuous head-on collision and therefore act as non-scanning, or *witness*, IPs, can be used to monitor the separation-dependent beam–beam effects. These effects are induced at the scanning IP, where the beams are deliberately brought in and out of collision, and propagate around the rings to the witness IPs, where they manifest as changes in the measured luminosity. In principle, any of the four LHC IPs can serve as a scanning IP, and any non-scanning IP can be designated as the witness IP for a given measurement. In practice, instrumental and operational constraints at the time the experiment was carried out restrict the choice of witness IP to the ATLAS (IP1) and CMS (IP5) collision points; IPs 1, 2 or 5, or combinations thereof, are used as scanning IP(s). Collisions at IP8 were deliberately avoided because the large longitudinal offset of the LHCb collision point, that breaks the eight-fold symmetry of the LHC rings, precludes the possibility of both members of a colliding-bunch pair to collide at all four IPs. In addition, the large crossing angle at IP8 would complicate the interpretation of the beam-beam effects that would occur at that IP if some of the bunches collided there.\n\nThe propagation of the beam–beam-induced, amplitude-dependent \\\$\\\\beta \\\$\\-beating from the scanning to a witness IP is controlled by the betatron phase advance between these two IPs. Since the targeted signatures are delicate to measure at best, it is natural to try and enhance them by adjusting this phase advance so as to maximize the sensitivity of luminosity shifts at the witness IP to beam-beam effects at the scanning IP; in doing so, however, the overall tunes must be preserved. Given the central and symmetric roles played by IP1 and IP5 in this experiment, the adjustment of their relative phase advance drove the optimization procedure that is detailed in Sect. [5](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec11), and that achieved a threefold improvement in measurement sensitivity.\n\nCentral to the quality of the measurements is the controlled variation of the beam-beam parameter at the scanning IP. This can be achieved either:\n\n- by using *step scans*, *i.e.* by fully separating the beams at the scanning IP in either the horizontal or the vertical plane, and then bringing them back into head-on collision; or\n- by using *separation scans*, *i.e.* by scanning the beams transversely with respect to each other in either the horizontal or the vertical plane; or\n- by taking advantage of the natural beam-intensity decay and emittance growth to progressively reduce the beam-beam parameter.\n\nEqually important is to ensure that the beams remain in head-on collision at the witness IP(s), such that reproducible luminosity shifts measured at these locations can be unambiguously correlated with controlled changes in the strength of the beam-beam interaction at the scanning IP. It was verified that the beam-beam-induced orbit shift and the potential non-closure of the orbit bumps used to control the beam separation at the scanning IP, did not significantly affect the actual separation, and thereby the measured collision rate, at the witness IP. In a few cases however, orbit drifts of uncontrolled origin ended up degrading the quality of some of the measurements.\n\nSimulations demonstrate \\[[13](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR13 \"A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). \n                  https:\/\/doi.org\/10.48550\/arXiv.2306.10394\n                  \n                \")\\] that multi-IP effects strongly influence the magnitude of beam-beam biases to vdM calibrations. Their characterization, therefore, constitutes an essential component of this experiment. The scan protocol is a generalization of that in the two-IP case. One IP (say, IP1) is designated as the witness IP, with beams colliding head-on there and at IP2, and IP5; the beams are then taken out of collision at IP2 only, then at both IP2 and IP5, and finally returned to a three-collision configuration in the reverse order. Since the phase-advance combinations are different in each configuration, the corresponding luminosity shifts predicted at the witness IP are also different and can be meaningfully confronted with the data.\n\n## 3 Beam conditions\nIn the round-beam approximation, the beam-beam parameter \\\$\\\\xi \\\$ does not depend on the beam energy (Eq. ([3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Equ3))). Therefore, and in order to minimize the overhead associated with commissioning and operating the LHC in a non-standard optical configuration, as well as to allow for frequent refilling with different bunch patterns, the measurements were carried out at injection energy (\\\$450\\\\ \\\\text {GeV}\\\$ per beam), and under non-standard machine-protection conditions. The improvement in operational efficiency came at a cost:\n\n- the counting rate of the luminometers (or equivalently their visible cross-section) is about an order of magnitude smaller at \\\$\\\\sqrt{s} = 900\\\\ \\\\text {GeV}\\\$ than at \\\$13\\\\ \\\\text {TeV}\\\$, reflecting the combination of a 40% drop in the inelastic *pp* cross-section, a factor of two to three drop in the multiplicity of the final-state particles, and a significant softening of their momentum spectrum;\n- because of the total-intensity constraints dictated by machine-protection requirements, the injected beam could not exceed 4 bunches per beam (compared to 150 during a routine *vdM*\\-calibration session), with a maximum allowed population of \\\$1.25\\\\times 10^{11}~p\\\$\/bunch;\n- the combination of the low beam energy, that suppresses synchrotron-radiation damping, and of the large bunch intensity, that enhances intra-beam scattering, resulted in relatively rapid emittance growth and rather short single-beam lifetimes. To partially mitigate these effects, all collision-rate measurements are expressed in terms of specific luminosity, thereby automatically accounting for the beam-intensity decay.\n\nThe bunch intensity was deliberately increased by about 25% beyond the *vdM*\\-scan values typical of LHC Run 2, to maximize both the luminosity and the beam-beam parameter \\\$\\\\xi \\\$. The latter reached 0.010 per IP (Table [1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Tab1)), compared to 0.003–0.006 during normal *pp* *vdM* sessions. With a target injected emittance of \\\$1.5 \\\\ \\\\upmu \\\\text {m} \\\\cdot \\\\text {rad}\\\$, the \\\$\\\\beta \\\$ function at the IP set to \\\$\\\\beta ^\\* = 11\\\\ \\\\text {m}\\\$, and zero crossing angle at IP1 and IP5, the statistical uncertainty affecting a typically 60-second long per-bunch luminosity measurement in the presence of head-on collisions, lay around 0.5%.\n\n**Table 1 Range of beam-beam parameter values during the various stages of the experiment**\n\n[Full size table](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/tables\/1)\n\nThe bunch patterns were chosen such that all bunches in each beam collided either at IP1 and IP5 only, or at all three of IP1, 2 and 5. The orbit-stabilization feedback system was turned off during the data-taking periods to prevent it from interfering with the beam–beam-induced orbit shift. The chromaticity was set to its standard value in physics fills, of +10 units, to guarantee coherent stability against the machine impedance, and in view of the very small number of bunches and of their large longitudinal spacing, the beam-stabilizing Landau octupoles were set to their minimum current (1A). The damping time of the bunch-by-bunch transverse feedback was set to 1000 turns, a rather loose setting intended to preserve longer the natural beam oscillation and therefore improve the precision of the tune measurements.\n\n## 4 Beam instrumentation\n### 4\\.1 Luminometers\nThe luminometer systems in use by the ATLAS and CMS collaborations at IP1 and IP5 are described in Refs. \\[[15](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR15 \"ATLAS Collaboration, Improved luminosity determination in pp collisions at \n                  $$\\sqrt{s} = 7$$\n                  \n                    \n                      \n                        s\n                      \n                      =\n                      7\n                    \n                  \n                 TeV using the ATLAS detector at the LHC. Eur. Phys. J. C 73, 2518 (2013). \n                  https:\/\/doi.org\/10.1140\/epjc\/s10052-013-2518-3\n                  \n                \"), [16](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR16 \"C.M.S. Collaboration, Development of the cms detector for the cern lhc run 3. J. Instrum. 19(05), 05064 (2024). \n                  https:\/\/doi.org\/10.1088\/1748-0221\/19\/05\/p05064\n                  \n                \")\\] respectively; no luminosity measurements were available at IP2 during the collider experiment described in this paper. Maximizing the statistical sensitivity leads to choosing the luminosity algorithm with the highest possible acceptance, namely:\n\n- the hit rate per bunch crossing in the ATLAS Minimum Bias Trigger Scintillators (MBTS) \\[[15](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR15 \"ATLAS Collaboration, Improved luminosity determination in pp collisions at \n                    $$\\sqrt{s} = 7$$\n                    \n                      \n                        \n                          s\n                        \n                        =\n                        7\n                      \n                    \n                   TeV using the ATLAS detector at the LHC. Eur. Phys. J. C 73, 2518 (2013). \n                    https:\/\/doi.org\/10.1140\/epjc\/s10052-013-2518-3\n                    \n                  \")\\], and\n- the occupancy in the CMS Hadron Forward Calorimeter (HFOC) \\[[16](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR16 \"C.M.S. Collaboration, Development of the cms detector for the cern lhc run 3. J. Instrum. 19(05), 05064 (2024). \n                    https:\/\/doi.org\/10.1088\/1748-0221\/19\/05\/p05064\n                    \n                  \")\\].\n\nIn both cases, raw luminometer counts are converted to collision rates using the Poisson formalism \\[[15](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR15 \"ATLAS Collaboration, Improved luminosity determination in pp collisions at \n                  $$\\sqrt{s} = 7$$\n                  \n                    \n                      \n                        s\n                      \n                      =\n                      7\n                    \n                  \n                 TeV using the ATLAS detector at the LHC. Eur. Phys. J. C 73, 2518 (2013). \n                  https:\/\/doi.org\/10.1140\/epjc\/s10052-013-2518-3\n                  \n                \"), [16](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR16 \"C.M.S. Collaboration, Development of the cms detector for the cern lhc run 3. J. Instrum. 19(05), 05064 (2024). \n                  https:\/\/doi.org\/10.1088\/1748-0221\/19\/05\/p05064\n                  \n                \")\\], and the instantaneous luminosity is averaged over typically 60-second time bins, that during the scans are synchronized with the scan steps. The statistical uncertainties are estimated from either the total number of raw luminometer counts per time bin (ATLAS), or from the RMS of the approximately 40 luminosity samples recorded in a given time bin (CMS). Since the absolute luminosity scale is irrelevant, and in order to simplify the interpretation of the results, all measurements in this paper are presented in terms of fractional shifts in the bunch-averaged specific luminosity, relative to a reference time that depends on the type of measurement considered.\n\n### 4\\.2 Bunch-charge measurement\nThe total intensity of each beam is measured by a direct-current current transformer (DCCT), the absolute scale of which is calibrated against a very high precision pulse generator \\[[17](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR17 \"C. Barschel, M. Ferro-Luzzi, J.-J. Gras, M. Ludwig, P. Odier, S. Thoulet, Results of the LHC DCCT calibration studies. Technical Report CERN-ATS-Note-2012-026 PERF (2012). \n                  https:\/\/cds.cern.ch\/record\/1425904?ln=en\n                  \n                \")\\]. The bunch-by-bunch charge fractions, in turn, normalized to the total stored intensity, are measured, separately for the two beams, by Fast Beam Current Transformers (FBCT) \\[[18](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR18 \"G. Anders et al., Study of the relative LHC bunch populations for luminosity calibration. Technical Report. CERN-ATS-Note-2012-028 PERF (2012). \n                  https:\/\/cds.cern.ch\/record\/1427726\n                  \n                \")\\]. The latter devices also provide the fill pattern, *i.e.* the relative location, at a given instant and around the two rings, of the nominally filled bunches. Since the absolute scale of the DCCTs is known to much better accuracy than that of the FBCTs, the bunch charges are typically computed as the product of the FBCT bunch-charge fractions and the total circulating beam intensity reported by the corresponding DCCT.\n\n### 4\\.3 Emittance measurement\nThe synchrotron-light beam-profile monitors \\[[19](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR19 \"G. Trad, Development and Optimisation of the SPS and LHC Beam Diagnostics Based on Synchrotron Radiation Monitors Presented 22 Jan 2015. Presented 22 Jan 2015. \n                  https:\/\/cds.cern.ch\/record\/2266055\n                  \n                \")\\], dubbed BSRTs (“Beam Synchrotron Radiation Telescope”), are mainly used to track emittance evolution over time, but they can also measure the relative changes in transverse single-beam size that result from beam–beam-induced \\\$\\\\beta \\\$\\-beating. BSRT data are recorded every second; in what follows, they are presented averaged over one-minute intervals for easier comparison with other measurements.\n\nThe absolute length scale of the BSRT profiles suffers from significant uncertainties, if only because the online optical corrections to the images can be updated only a few times per year, and are unable to track the evolution of the efficiency of the light sensors, that depends both on time and on the position of the light spot on the sensor array. For each beam therefore, the BSRT is complemented by a wire-scanner (WS) profile monitor that is located in the same straight section. The advantage of the WS is that its accuracy is significantly better than that of the BSRT; its down side, in the present context, is that the WS cannot acquire data continuously and must be triggered manually. For the results presented in this paper, the wire scanners were flown through the beams at the start of each group of measurements to provide single-beam emittance measurements that are as accurate as possible, and then at regular intervals thereafter. The BSRT is used to interpolate the time evolution of the emittance between two sets of WS measurements.\n\nTo interpret measured RMS bunch widths in terms of emittance requires the knowledge of the \\\$\\\\beta \\\$ functions at the locations of the BSRT and the WS. These are determined by the phase-advance method, and compared in Table [2](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Tab2) to their model value computed using MAD-X; the agreement is typically better than 5%, and the worst disagreement amounts to 9%. These discrepancies are attributed to imperfections in the magnetic lattice. Uncertainties in the measurement mainly arise from the interpolation of the lattice functions between the two closest beam-position monitors (BPMs) where the \\\$\\\\beta \\\$\\-functions are measured \\[[20](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR20 \"R. Tomás, O. Brüning, M. Giovannozzi, P. Hagen, M. Lamont, F. Schmidt, G. Vanbavinckhove, M. Aiba, R. Calaga, R. Miyamoto, Cern large hadron collider optics model, measurements, and corrections. Phys. Rev. ST Accel. Beams 13, 121004 (2010). \n                  https:\/\/doi.org\/10.1103\/PhysRevSTAB.13.121004\n                  \n                \"), [21](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR21 \"R. Tomás, M. Aiba, A. Franchi, U. Iriso, Review of linear optics measurement and correction for charged particle accelerators. Phys. Rev. Accel. Beams 20, 054801 (2017). \n                  https:\/\/doi.org\/10.1103\/PhysRevAccelBeams.20.054801\n                  \n                \")\\], and either the BSRT or the WS location. Based on the MAD-X LHC lattice model, these uncertainties are estimated not to exceed 3%, and this value is assigned as the systematic uncertainty on the \\\$\\\\beta \\\$ functions used in BSRT- and WS-based emittance measurements.\n\n**Table 2 \\\$\\\\beta \\\$ functions at the BSRT and WS locations, for the optical configuration detailed in Sect. [5\\.3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec14)**\n\n[Full size table](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/tables\/2)\n\nThe beam-averaged emittance, *i.e.* the average of the emittances of the beam-1 bunch and of the corresponding beam-2 bunch, can be obtained directly from emittance scans \\[[22](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR22 \"O. Karacheban, P. Tsrunchev, on behalf of CMS, Emittance scans for cms luminosity calibration. EPJ Web Conf. 201, 04001 (2019). \n                  https:\/\/doi.org\/10.1051\/epjconf\/201920104001\n                  \n                \")\\] at IP1 and\/or IP5. In this approach, the convolved transverse bunch sizes measured using beam-separation scans are translated into emittances using the \\\$\\\\beta ^\\*\\\$ values determined by *k*\\-modulation \\[[23](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR23 \"F. Carlier, R. Tomás, Accuracy and feasibility of the \n                  $${\\beta }^{*}$$\n                  \n                    \n                      \n                        β\n                      \n                      \n                      \n                        \n                        ∗\n                      \n                    \n                  \n                 measurement for lhc and high luminosity lhc using \n                  $$k$$\n                  \n                    k\n                  \n                 modulation. Phys. Rev. Accel. Beams 20, 011005 (2017). \n                  https:\/\/doi.org\/10.1103\/PhysRevAccelBeams.20.011005\n                  \n                \")\\] at the relevant IP (see Sect. [4\\.3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec7)). The associated uncertainty (Table [3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Tab3)) is combined with the statistical uncertainty in convolved transverse width to estimate the error affecting the measured emittance. Comparison with beam-averaged emittances extracted from the single-beam profile monitors reveals excellent agreement between WS and emittance-scan results (Fig. [2](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig2)). The relative time-evolution of the BSRT emittances is consistent with that observed using the WS or emittance scans, but the absolute magnitudes differ significantly, especially in the horizontal plane.\n\n**Table 3 \\\$\\\\beta ^\\*\\\$ functions at the ATLAS and CMS IPs measured by *k*\\-modulation. The target value in all cases is \\\$\\\\beta ^\\*=\\\$ 11 m**\n\n[Full size table](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/tables\/3)\n\n**Fig. 2**\n\n[![Fig. 2](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig2_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/2)\n\nTime evolution of the horizontal (top) and vertical (bottom) beam-averaged normalized emittances during LHC fill 8037, as reported by the BSRT (blue), the WS (orange), and using beam-separation scans at IP1 (red) and IP5 (purple). Different shades of the same color correspond to two different bunches present in this fill. For ATLAS emittance scans, only the bunch-averaged value is shown\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/2)\n### 4\\.4 Beam-beam parameter determination\nThe systematic and quantitative comparison of the measured and of the predicted beam-beam impact on the observables detailed in Sect. [2](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec2) requires continuous monitoring of the actual beam-beam parameter throughout the duration of the experiment. Only the BSRT provides uninterrupted emittance determination throughout the fill. In view of the scale biases apparent in Fig. [2](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig2), however, WS-based emittances are used, whenever possible, as input to the determination of the beam-beam parameter; BSRT emittances re-scaled to close-in-time, absolute WS measurements provide time-interpolated emittance values whenever WS data are unavailable.\n\nThe beam-beam parameter evolution during the first fill of the experiment is illustrated in Fig. [3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig3). The error bands are dominated by the 3% systematic uncertainty in the measured optical functions. The strength of the beam-beam interaction drops by a factor of two over a couple of hours, from the combined effect of beam-intensity decay and of emittance growth.\n\n**Fig. 3**\n\n[![Fig. 3](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig3_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/3)\n\nTime evolution of the beam- and plane-averaged beam-beam parameter \\\$\\\\xi \\\$ inferred from the measured bunch charges and emittances, separately for the two colliding-bunch pairs present in the fill pattern. The color bands indicate the systematic uncertainty (see text)\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/3)\n### 4\\.5 Tune measurements\nCoherent spectra can be monitored using either the LHC Transverse Damper (ADT) \\[[24](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR24 \"M. Söderén, J. Komppula, G. Kotzian, S. Rains, D. Valuch, ADT and Obsbox in LHC Run 2, plans for LS2, pp. 165–171 (2019)\")\\], or the Base-Band Tune (BBQ) \\[[25](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR25 \"A. Boccardi, M. Gasior, R. Jones, P. Karlsson, R. Steinhagen, First results from the lhc bbq tune and chromaticity systems. Technical report, CERN, Geneva (2009). \n                  https:\/\/cds.cern.ch\/record\/1156349\n                  \n                \")\\] system; their comparison typically yields consistent results. Spectrograms with a tune resolution of 0.0001 are computed using bunch positions recorded at every turn, and averaged over the one-minute time bins mentioned in Sect. [4\\.1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec5). To mitigate the influence of noise on the tune measurement, a median filter is applied at the pre-processing stage with a self-defined local window size. Additionally, the 50 Hz noise lines present in the spectra are masked. The filtered data are fitted by the sum of a Gaussian function and a constant baseline, and the frequency at the peak of the Gaussian is interpreted as the measured mean tune. The systematic uncertainty on measured tune shifts \\\$\\\\Delta Q\\\$ is estimated empirically to be \\\$\\\\sigma \\_{\\\\Delta Q}=0.001\\\$.\n\n### 4\\.6 IP-orbit monitoring\nOrbit displacements at each IP are measured using the Diode ORbit and OScillation (DOROS) BPM system, that provides sub-micrometer beam-position resolution \\[[26](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR26 \"M. Gasior, G. Baud, J. Olexa, G. Valentino, First Operational Experience with the LHC Diode ORbit and OScillation (DOROS) System, p. 07 (2017) \n                  https:\/\/doi.org\/10.18429\/JACoW-IBIC2016-MOPG07\n                  \n                \")\\]. These strip-line BPMs are located in the two quadrupoles on either side of and closest to the IP, allowing the position of both beams to be measured simultaneously. The position and the angle of each beam at the IP are inferred from, respectively, the average and the difference of the positions measured in the two final-triplet quadrupoles.\n\n## 5 Optimization of the phase advance between interaction points\nIn the two-IP configuration, the basic theory of linearized beam-beam \\\$\\\\beta \\\$\\-beating provides an analytical determination of the optimum phase advance between the two IPs, *i.e.* of the setting that maximizes the sensitivity, at the witness IP, to beam-beam effects at the scanning IP (Sect. [5\\.1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec12)). A generalization to the three- and four-IP configurations is discussed in Sect. [5\\.2](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec13). The required modifications to the baseline LHC optics, as well as their implementation and their experimental validation, are detailed in Sect. [5\\.3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec14).\n\n### 5\\.1 Linear beam-beam \\\$\\\\beta \\\$\\-beating\nIn circular colliders, the \\\$\\\\beta \\\$\\-beating induced by beam-beam effects at the IP(s) leads to changes in transverse beam size \\[[5](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR5 \"W. Herr, T. Pieloni, Beam-Beam Effects. Technical report, CERN (2014). Contribution to the CAS–CERN Accelerator School: Advanced Accelerator Physics Course, Trondheim, Norway, 18–29 Aug 2013, p. 29. \n                  https:\/\/doi.org\/10.5170\/CERN-2014-009.431\n                  \n                . \n                  https:\/\/cds.cern.ch\/record\/1982430\n                  \n                \"), [27](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR27 \"A.W. Chao, Coherent beam-beam effects, in Frontiers of Particle Beams: Intensity Limitations (Springer, Berlin, Heidelberg, 1992), pp. 363–414. \n                  https:\/\/doi.org\/10.1007\/3-540-55250-2_36\n                  \n                \")\\], thereby impacting the luminosity. For a single collision point, the change in \\\$\\\\beta ^\\*\\\$ of a single small-amplitude particle in (for instance) the horizontal plane *x*, is given by:\n\n\\$\\$\\\\begin{aligned} \\\\frac{\\\\beta \\_x^\\*}{\\\\beta \\_{0,x}^\\*} = \\\\frac{1}{\\\\sqrt{1-4\\\\pi \\\\xi \\\\cot {(2\\\\pi Q\\_x)} - 4\\\\pi ^2\\\\xi ^2}}. \\\\end{aligned}\\$\\$\n\n(6)\n\nThe magnitude of the effect depends on the absolute value \\\$\\\\xi \\\$ of the beam-beam parameter and on the nominal betatron tune \\\$Q\\_x\\\$; its periodicity is \\\$ \\[\\\\pi \\]\\\$. It can be shown that approaching the half-integer betatron tune from below minimizes the \\\$\\\\beta \\\$ function at the IP, thereby maximizing the luminosity; this phenomenon has been exploited with great success in (among others) the KEKB and PEP-II *B* factories. In the LHC, where during collisions the nominal fractional betatron tunes are fixed at \\\$q\\_x=0.31,\\\\,q\\_y=0.32\\\$, this dynamic-\\\$\\\\beta \\\$ effect depends only on the beam-beam parameter.\n\nThis conclusion no longer strictly holds in the presence of more than one collision point. In the case of two IPs with identical values of \\\$\\\\xi \\\$ and \\\$\\\\beta ^\\*\\\$, the dynamic-\\\$\\\\beta \\\$ effect can be described analytically \\[[28](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR28 \"J. Wańczyk, A. Dabrowski, D. Stickland, T. Pieloni, W. Kozanecki, X. Buffat, Impact of multiple beam-beam encounters on LHC absolute-luminosity calibrations by the van der Meer method. JACoW IPAC 2023, 053 (2023). \n                  https:\/\/doi.org\/10.18429\/JACoW-IPAC2023-WEPA053\n                  \n                \")\\]:\n\n\\$\\$\\\\begin{aligned} \\\\frac{\\\\beta \\_x^\\*}{\\\\beta ^\\*\\_{0,x}} = \\\\frac{\\\\sin {2\\\\pi Q\\_x} + 4\\\\pi \\\\xi (\\\\cos (2\\\\pi Q\\_x-2\\\\mu \\_{1,x})-\\\\cos 2\\\\pi Q\\_x )}{\\\\pm \\\\sqrt{1-\\\\cos ^2{2\\\\pi (Q\\_x+\\\\Delta Q\\_x)}}}, \\\\end{aligned}\\$\\$\n\n(7)\n\nwhere \\\$\\\\mu \\_{1,x}\\\$ is the phase advance from the scanning IP to the witness IP. The sign in the denominator is linked to that of the numerator so as to ensure that the ratio remains positive; the denominator must be positive (resp. negative) below (resp. above) the half-integer tune, *i.e.* when \\\$m\\< Q\\_x \\<m+\\\\frac{1}{2}\\\$ (resp. \\\$m+\\\\frac{1}{2}\\< Q\\_x \\< m+1\\\$), where *m* is a positive integer. The tune shift \\\$\\\\Delta Q\\_x\\\$ is related to \\\$Q\\_x\\\$, \\\$\\\\xi \\\$ and \\\$\\\\mu \\_{1,x}\\\$ by:\n\n\\$\\$\\\\begin{aligned} \\\\cos {2\\\\pi (Q\\_x+\\\\Delta Q\\_x)}&=(1-16\\\\pi ^2\\\\xi ^2)\\\\cos {2\\\\pi Q\\_x} + 8\\\\pi \\\\xi \\\\sin {2\\\\pi Q\\_x} \\\\nonumber \\\\\\\\&\\\\quad + 16\\\\pi ^2\\\\xi ^2\\\\cos {(2\\\\pi Q\\_x-2\\\\mu \\_{1,x})}. \\\\end{aligned}\\$\\$\n\n(8)\n\nThe phase-advance dependence of the resulting \\\$\\\\beta \\\$\\-beating is illustrated in Fig. [4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig4). The minimum of each curve corresponds to the phase advance that yields the largest mutual beam–beam-induced luminosity enhancement between the two IPs. The optimal settings are \\\$\\\\mu \\_{1,x}^{min}\/2\\\\pi =0.405,\\\\,\\\\mu \\_{1,y}^{min}\/2\\\\pi =0.410\\\$; their difference reflects that between the nominal horizontal and vertical fractional tunes \\\$q\\_x\\\$ and \\\$q\\_y\\\$.\n\n**Fig. 4**\n\n[![Fig. 4](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig4_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/4)\n\nAnalytically computed beam-beam induced \\\$\\\\beta \\\$\\-beating in a two-IP configuration, as a function of the phase advance \\\$ \\\\mu \\_1\\\$ between the 2 IPs (\\\$q\\_x=0.31\\\$, \\\$q\\_y=0.32\\\$, head-on collisions at both IPs, \\\$\\\\xi =7\\\\times 10^{-3}\\\$ per IP)\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/4)\n\nThe extrema of the curves in Fig. [4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig4) can be determined analytically by differentiating Eq. ([7](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Equ7)) with respect to the phase advance \\\$\\\\mu \\_{1,x}\\\$. The general expression for this derivative \\\$\\\\frac{d}{d\\\\mu \\_{1,x}} \\\\biggl ( \\\\frac{\\\\beta \\_x^\\*}{\\\\beta ^\\*\\_{0,x}} \\\\biggr )\\\$, detailed in Appendix A, is the product of two factors, each of which can be zero:\n\n- imposing that \\\$\\\\sin {(2\\\\pi Q\\_x - 2\\\\mu \\_{1,x})} = 0\\\$ yields the solution\n  \\$\\$\\\\begin{aligned} \\\\frac{\\\\mu \\_{1,x}}{2\\\\pi } =\\\\frac{Q\\_x}{2}-\\\\frac{m}{4} \\\\, \\\\end{aligned}\\$\\$\n  \n  (9)\n  where *m* is an integer of either sign, or zero;\n- the other factor is zero for either\n  \\$\\$\\\\begin{aligned} \\\\frac{\\\\mu \\_{1,x}}{2\\\\pi } = \\\\frac{Q\\_x}{2} - A(\\\\xi , Q\\_x)\\\\, \\\\end{aligned}\\$\\$\n  \n  (10)\n  or\n  \\$\\$\\\\begin{aligned} \\\\frac{\\\\mu \\_{1,x}}{2\\\\pi } = \\\\frac{Q\\_x - 1}{2} + A(\\\\xi , Q\\_x) \\\\, \\\\end{aligned}\\$\\$\n  \n  (11)\n  with \\\$A(\\\\xi , Q\\_x)\\\$ defined in Eq. ([12](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Equ12))\n\n\\$\\$\\\\begin{aligned} A(\\\\xi , Q\\_x) = \\\\arccos \\\\left( \\\\frac{ 32\\\\pi ^3\\\\xi ^3 \\\\sin (4\\\\pi Q\\_x) + 48\\\\pi ^2\\\\xi ^2 \\\\cos ^2(2\\\\pi Q\\_x) - 32\\\\pi ^2\\\\xi ^2 - 6\\\\pi \\\\xi \\\\sin (4\\\\pi Q\\_x) + \\\\cos ^2(2\\\\pi Q\\_x) - 1}{16\\\\pi ^2\\\\xi ^2 \\\\left( 4\\\\pi \\\\xi \\\\sin (2\\\\pi Q\\_x) + \\\\cos (2\\\\pi Q\\_x) \\\\right) } \\\\right) \/ (4\\\\pi ) \\\\end{aligned}\\$\\$\n\n(12)\n\nThe two sets of solutions do not overlap. The subset of solutions that minimize (rather than maximize) \\\$\\\\beta ^\\*\\\$ is identified by requiring that the second derivative be positive. This restricts the main solution to:\n\n\\$\\$\\\\begin{aligned} \\\\frac{\\\\mu \\_{1,x}}{2\\\\pi } =\\\\frac{Q\\_x}{2}+\\\\frac{m+1}{4} . \\\\end{aligned}\\$\\$\n\n(13)\n\nThe optimal phase \\\$\\\\mu \\_{x,1}\\\$ that minimizes the \\\$\\\\beta ^\\*\\\$ for range of \\\$(\\\\xi , Q\\_x)\\\$ values is shown in Fig. [5](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig5).\n\n**Fig. 5**\n\n[![Fig. 5](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig5_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/5)\n\nOptimal phase advance value \\\$\\\\mu \\_{x,1}\\\$ for various values of \\\$(\\\\xi , Q\\_x)\\\$ in the two-IPs configuration\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/5)\n\nThe phase-advance dependence of the dynamic-\\\$\\\\beta \\\$ effect can also be quantified in terms of the luminosity shift, at the witness IP, that is associated with the electromagnetic interaction of the two bunches at the scanning IP (Fig. [6](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig6)). The optimal setting, *i.e.* that which maximizes the sensitivity, at the witness IP, to beam-beam effects at the scanning IP, is indicated by the green dotted vertical line at \\\$\\\\Delta \\\\mu \\_{\\\\text {IP1-IP5}}=0.41\\\\ \\[2\\\\pi \\]\\\$; the full suppression of the phase-related luminosity enhancement is indicated by the red dotted line. These results are consistent with the predictions of the analytical model (Fig. [4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig4)). In the two-IP configuration considered here, in which IP1 and IP5 can both play the role of either the scanning or the witness IP, and due to the periodicity of this curve (\\\$ \\\\pi \\\$) and to the fractional-tune values used during collisions (\\\$q\\_x \/ q\\_y = 0.31 \/ 0.32)\\\$, the phase–advance-related luminosity enhancement is the same at the two IPs; at the optimum setting, it is three times larger than it would be using the nominal LHC lattice. The maximum effect depends mainly on the transverse tunes, and thus could have been further enhanced if it had been possible to move these away from their nominal values.\n\nA more comprehensive simulation study, that evaluates the impact of the beam-beam interaction on the luminosity for more realistic Gaussian proton-density distributions, multiple collision points and as a function of the beam separation, is reported in Ref. \\[[29](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR29 \"J. Wańczyk, Precision luminosity measurement at hadron colliders. PhD Thesis, École Polytechnique Fédérale de Lausanne (2024). \n                  https:\/\/doi.org\/10.5075\/epfl-thesis-10500\n                  \n                \")\\].\n\n**Fig. 6**\n\n[![Fig. 6](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig6_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/6)\n\nBeam–beam-induced luminosity shift at IP5 predicted by COMBI simulations, as a function of the phase advance \\\$ \\\\mu \\_1=\\\\Delta \\\\mu \\_{\\\\text {IP1-IP5}}\\\$ between the two IPs (\\\$q\\_x=0.31\\\$, \\\$q\\_y=0.32\\\$, head-on collisions at both IPs, \\\$\\\\xi =7\\\\times 10^{-3}\\\$ per IP)\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/6)\n### 5\\.2 Optimized phases for multi-IP configurations\nThe general expression to obtain the first-order change of the \\\$\\\\beta \\\$ function at an arbitrarily chosen reference IP (at \\\$s=\\\\mu \\_{0,x}=0\\\$) can be approximated as multiple quadrupole errors \\[[30](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR30 \"T. Pieloni, J. Barranco, X. Buffat, P.C. Jorge, L.M. Medrano, C. Tambasco, R. Tomás, Dynamic beta and beta-beating effects in the presence of the beam-beam interactions in 57th ICFA Advanced Beam Dynamics Workshop on High-Intensity and High-Brightness Hadron Beams, p. 027 (2016). \n                  https:\/\/doi.org\/10.18429\/JACoW-HB2016-MOPR027\n                  \n                \")\\]:\n\n\\$\\$\\\\begin{aligned} \\\\frac{\\\\beta ^\\*\\_x}{\\\\beta ^\\*\\_{0,x}} = \\\\frac{2\\\\pi \\\\xi }{\\\\sin {2\\\\pi Q\\_x}} \\\\sum \\_{i\\\\in IPs}^{N-1} \\\\cos (2\\\\pi Q\\_x-2\\\\mu \\_{i,x})\\\\, \\\\end{aligned}\\$\\$\n\n(14)\n\nwhere \\\$\\\\mu \\_{i,x}\\\$ is the horizontal phase advance between the reference and the \\\$i^{th}\\\$ IP for a total of *N* IPs. Therefore, the optimal configuration, which minimizes the \\\$\\\\beta ^\\*\\\$ change at all IPs, must satisfy the condition:\n\n\\$\\$\\\\begin{aligned} \\\\sum \\_{i=1}^{N-1} \\\\cos (2\\\\pi Q\\_x-2\\\\mu \\_{i,x}) = -(N-1)\\\\, \\\\end{aligned}\\$\\$\n\n(15)\n\nwhich yields the same solution for all IPs as in the 2 IPs configuration (Eq. [13](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Equ13)):\n\n\\$\\$\\\\begin{aligned} \\\\frac{\\\\mu \\_{i,x}}{2\\\\pi } =\\\\frac{Q\\_x}{2}+\\\\frac{n+1}{4} . \\\\end{aligned}\\$\\$\n\n(16)\n\nwhere *n* is an integer of either sign, or zero. To maximize the effect on luminosity, the phase advance between the reference interaction point at \\\$ i = 0 \\\$ and the nearest IP, at \\\$ i = 1 \\\$ must satisfy \\\$ \\\\mu \\_{1,x}\/2\\\\pi = (Q\\_x + 0.5)\/2 \\\$, as given by the first solution (for \\\$n=0\\\$) of Eq. ([13](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Equ13)). This condition implies that the phase advance to any subsequent adjacent IPs (\\\$ i \\\\ge 2 \\\$) must be an integer multiple of \\\$ \\\\pi \\\$. Thus, this requirement exposes a fundamental constraint: it is not possible to minimize \\\$ \\\\beta ^\\* \\\$ simultaneously at all IPs. This would only be possible for (\\\$ Q\\_x = 0.5 \\\$) which is practically not achievable, as it would require operating the storage ring precisely at the half-integer resonance.\n\n### 5\\.3 Optics validation\nThe optimum phase advance from IP1 to IP5 determined in Sect. [5\\.1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec12) was implemented using tuning trim quadrupoles in the LHC arcs. The targeted fractional phase advance was \\\$0.9 \\\\, \\[2 \\\\pi \\]\\\$ for both beams and both planes (Table [4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Tab4)). This setting, which differs from that in Fig. [6](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig6) by one period (\\\$ \\\\pi \\\$), is equivalent in terms of sensitivity optimization but has the advantage that it minimizes the absolute magnitude of the perturbations inflicted upon the nominal lattice.\n\nSeveral measurements were carried out to validate this adjustment: comparison of the targeted and achieved shifts in phase advance (Sect. [5\\.3.1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec15)), \\\$\\\\beta \\\$\\-function measurements across the full LHC circumference (Sect. [5\\.3.2](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec16)) as well as at the IPs (Sect. [5\\.3.3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec17)), and dispersion measurements to estimate the dispersive contribution to the transverse beam size at the IP (Sect. [5\\.3.4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec18)).\n\n#### 5\\.3.1 Phase advance\nThe phase advance around each LHC ring is determined using turn-by-turn orbit measurements \\[[20](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR20 \"R. Tomás, O. Brüning, M. Giovannozzi, P. Hagen, M. Lamont, F. Schmidt, G. Vanbavinckhove, M. Aiba, R. Calaga, R. Miyamoto, Cern large hadron collider optics model, measurements, and corrections. Phys. Rev. ST Accel. Beams 13, 121004 (2010). \n                  https:\/\/doi.org\/10.1103\/PhysRevSTAB.13.121004\n                  \n                \"), [21](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR21 \"R. Tomás, M. Aiba, A. Franchi, U. Iriso, Review of linear optics measurement and correction for charged particle accelerators. Phys. Rev. Accel. Beams 20, 054801 (2017). \n                  https:\/\/doi.org\/10.1103\/PhysRevAccelBeams.20.054801\n                  \n                \")\\]. The targeted and achieved phase-advance shifts agree within \\\$0.01\\\\, \\[2\\\\pi \\]\\\$ or better (Table [4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Tab4)). The phase advance change from IP1 and IP5 was compensated on the opposite side of the ring, ensuring that the transverse tunes remain unchanged. The target fractional local phase advance is set to \\\$0.9,\\[2\\\\pi \\]\\\$. Table [4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Tab4) lists the corresponding phase shifts applied to the nominal lattice to reach this target on one side of the ring; this is consistent with the analytically derived value of \\\$0.41\\\\,\\[2\\\\pi \\]\\\$ for the compensating shift on the opposite side. The change in phase advance was also measured at the BSRT location, where the discrepancy between the model and measurements is more pronounced, reaching up to \\\$0.05\\\\,\\[2\\\\pi \\]\\\$.\n\n**Table 4 Nominal and optimized IP1 to IP5 phase-advance settings. Column 4 (resp. 5) displays the targeted (resp. measured) shift in phase advance after the optimized settings have been applied**\n\n[Full size table](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/tables\/4)\n#### 5\\.3.2 Residual \\\$\\\\beta \\\$\\-beating\nThe \\\$\\\\beta \\\$\\-beating was quantified and found to be well within acceptable tolerances for various configurations of the trim strength. The measurement results, relative to the reference measurements without any phase advance adjustments, are illustrated in Fig. [7](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig7). The maximal observed deviation is approximately 25% around IP5 for Beam 1 in the vertical plane. Such significant deviations in proximity to the IPs are anticipated and are frequently attributed to erroneous BPM readings. Overall, the measurements indicated that \\\$\\\\beta \\\$\\-beating remained within acceptable bounds, predominantly within a 10% margin.\n\n**Fig. 7**\n\n[![Fig. 7](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig7_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/7)\n\nMeasured \\\$\\\\beta \\\$\\-function change from the full strength of the phase advance knob with reference to nominal lattice along the LHC ring, for Beam 1 (top two figures for *x* and *y*) and Beam 2 (bottom two figures for *x* and *y*)\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/7)\n\nThe measured \\\$\\\\beta \\\$ functions are also compared to the MADX LHC lattice model predictions, which include the phase advance adjustment. The results, illustrated along the LHC lattice in Fig. [8](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig8), indicate that the measurements align with the model within an acceptable margin of 10%.\n\n**Fig. 8**\n\n[![Fig. 8](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig8_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/8)\n\nMeasured deviations in the \\\$\\\\beta \\\$\\-function along the LHC ring relative to the MADX model incorporating the phase advance knob, for Beam 1 (top two figures for *x* and *y*) and Beam 2 (bottom two figures for *x* and *y*). The error bars represent solely the statistical uncertainties as determined in the measurements\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/8)\n#### 5\\.3.3  \\\$\\\\beta \\\$ functions at the interaction points\nThe DOROS BPMs are strategically located at the end of the inner triplets on both sides of each of the experiments. Due to the phase advance of \\\$\\\\pi \\\$ between the DOROS on the left and right sides, these are inadequate for the precise determination of the measurement of \\\$\\\\beta ^\\*\\\$ at the IP. This can only be achieved with a special *k*\\-modulation technique \\[[31](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR31 \"F. Carlier, et al. Challenges of K-modulation measurements in the LHC Run 3, in Proceedings of IPAC’23. IPAC’23—14th International Particle Accelerator Conference (JACoW Publishing, Geneva, Venice, 2023), pp. 531–534. \n                  https:\/\/doi.org\/10.18429\/JACoW-IPAC2023-MOPL014\n                  \n                \")\\], which was performed at the interaction points used in the experiment. The results are summarized in Table [3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Tab3). While the achieved precision of the \\\$\\\\beta ^\\*\\\$ determination varies between measurements, uncertainties at the level of \\\${\\\\mathcal {O}}(1\\\\%)\\\$ are not uncommon and are not systematic to a specific beam or interaction point. Such uncertainties would propagate to only a few-percent effect on derived quantities such as the emittance or the beam–beam parameter.\n\n#### 5\\.3.4 Dispersion\nIt was essential to perform dispersion \\\$D^\\*\\\$ measurements at the interaction points, under the new optics configurations, as it can contribute to the measured beam size. Assuming the LHC design momentum spread value of \\\$\\\\frac{\\\\Delta p}{p}=10^{-4}\\\$, the average dispersion measurement around IP1 and IP5 is estimated to contribute up to 2% to the observed bunch sizes (worst case measured for B2Y). This effect is considered static throughout the experiment.\n\n## 6 Experimental results\nThis section presents experimental measurements of beam–beam effects in configurations involving multiple collision points. The impact of the optics modifications introduced in Sect. [5](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec11) is first assessed through tune spectra measurements, as detailed in Sect. [6\\.1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec20). To quantify the cumulative effect of one or more additional collisions and assess the reproducibility of the results, a stepwise collision scan was employed. In this procedure, beams were sequentially brought into head-on collision at selected interaction points (IPs), followed by full transverse separation (\\\$\\\\Delta \\> 6\\\\sigma \\\$) in either the horizontal or vertical plane at all but one witness IP. Two distinct filling schemes were used to isolate the influence of a single additional collision (2-IP configuration: ATLAS and CMS, see Sect. [6\\.2](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec21)) and to investigate more complex multi-collision scenarios (3-IP configuration: including ALICE, see Sect. [6\\.3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec25)). The first measurement of separation-dependent beam–beam effects, analogous to those encountered during luminosity calibration scans, is presented in Sect. [6\\.4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec28), alongside comparisons with numerical predictions. An additional measurement conducted during the 2022 van der Meer calibration is detailed in Sect. [6\\.5](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec32). A consolidated overview of all findings is provided in Sect. [7](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec33).\n\n### 6\\.1 Measured effect on the tune spectra of the phase advance changes\nThe coherent transverse tune spectra serve as a diagnostic for evaluating the effectiveness of the phase advance modifications. COMBI simulations were carried out for two extreme lattice configurations, one designed to maximize and the other to suppress beam–beam effects. In the head-on scenario (Fig. [9](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig9)), the coherent modes are suppressed for the maximizing configuration, as the phase advance between the two collision points is close to \\\$\\\\pi \\\$, as was previously shown in simulation in Ref. \\[[32](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR32 \"Y. Alexahin, A study of the coherent beam-beam effect in the framework of the Vlasov perturbation theory. Nucl. Instrum. Methods Phys. Res., Sect. A 480(1–3), 253–288 (2002). \n                  https:\/\/doi.org\/10.1016\/S0168-9002(01)01219-0\n                  \n                \")\\]. In this configuration, the effect of the first collision is counteracted by the second. This setting is particularly suitable for studying incoherent spectra, as the diminished coherent signal allows for a clearer extraction of the mean tune shift from fitted spectra. During separation scans, the symmetry between horizontal (*x*) and vertical (*y*) planes is broken. In the scanning plane (horizontal), coherent modes re-emerge at \\\$1.5\\\\,\\\\sigma \\\$ separation (Fig. [10](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig10)), with their frequency shifting in response to the separation distance (additional example for \\\$3\\\\,\\\\sigma \\\$ is shown in Fig. [11](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig11)).\n\n**Fig. 9**\n\n[![Fig. 9](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig9_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/9)\n\nComparison of COMBI simulated tune spectra in *x*\\-plane (left) and *y*\\-plane (right) for the suppressing (in blue) and maximizing (in green) phase configurations, for head-on collision\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/9)\n\n**Fig. 10**\n\n[![Fig. 10](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig10_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/10)\n\nComparison of COMBI simulated tune spectra in *x*\\-plane (left) and *y*\\-plane (right) for the suppressing (in blue) and maximizing (in green) phase configurations, at \\\$1.5\\\\,\\\\sigma \\\$ separation in the horizontal plane\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/10)\n\n**Fig. 11**\n\n[![Fig. 11](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig11_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/11)\n\nComparison of COMBI simulated tune spectra in *x*\\-plane (left) and *y*\\-plane (right) for the suppressing (in blue) and maximizing (in green) phase configurations, at \\\$3\\\\,\\\\sigma \\\$ separation in the horizontal plane\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/11)\n\nThese simulated trends of the coherent spectra are also observed in experimental data. Transverse spectra recorded for head-on conditions confirm the suppression of coherent modes in the maximizing phase configuration (Fig. [12](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig12)). Additional comparisons for separated beams at the IP (see Figs. [13](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig13) and [14](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig14)), shown here for the case of vertical separation, reveal similar features. In this configuration, the non-scanning (horizontal) plane exhibits spectra that qualitatively agree with simulation predictions (see Figs. [10](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig10) and [11](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig11)), while in the scanning (vertical) plane the coherent modes do not reappear as expected. An equivalent behavior is observed when the separation is applied in the horizontal plane, with the roles of the two transverse planes interchanged; this case is not shown for brevity. This discrepancy is attributed to the effect of the transverse damper, which acts independently on the two beam centroids and suppresses coherent oscillations arising from beam–beam coupling, consistent with earlier observations \\[[33](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR33 \"R. Calaga, W. Herr, G. Papotti, T. Pieloni, X. Buffat, S. White, R. Giachino, Coherent beam-beam mode in the LHC. CERN (2014). \n                  https:\/\/doi.org\/10.5170\/CERN-2014-004.227\n                  \n                . \n                  http:\/\/cds.cern.ch\/record\/1957039\n                  \n                \")\\].\n\n**Fig. 12**\n\n[![Fig. 12](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig12_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/12)\n\nMeasured tune spectra before and after phase change, corresponding to the suppressing (in blue) and maximizing (in green) phase configurations, at the head-on position, for each of the transverse planes (Beam 1 shown as an example). Dash-dotted lines show Gaussian fits to the data\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/12)\n\n**Fig. 13**\n\n[![Fig. 13](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig13_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/13)\n\nComparison of the measured tune spectra at suppressing (blue) and maximizing (green) phase configurations with \\\$1.3\\\\,\\\\sigma \\\$ separation in the vertical plane\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/13)\n\n**Fig. 14**\n\n[![Fig. 14](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig14_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/14)\n\nComparison of the measured tune spectra at suppressing (blue) and maximizing (green) phase configurations with \\\$2.6\\\\,\\\\sigma \\\$ separation in the vertical plane\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/14)\n### 6\\.2 Head-on collisions in a two-IP configuration at \\\$\\\\sqrt{s} = 900\\\\ \\\\text {GeV}\\\$\nEach test consisted of four measurements designed to quantify the beam–beam effect induced by a single additional head-on collision. The sequence was executed with CMS as the witness IP, followed by ATLAS, and concluded with a repeat at CMS. Over the course of the fill, the beam–beam parameter decreased from approximately 0.01 to 0.005. The outcome of each scan is analyzed using three independent observables: luminosity, beam size, and tune spectra, discussed separately below.\n\n#### 6\\.2.1 Impact of beam-beam effects on head-on luminosity\nAn overview of the luminosity measurements during the first test with CMS as the witness IP is shown in Fig. [15](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig15), where the specific luminosity at CMS is normalized to the initial head-on step to highlight relative changes. The uncertainty at each step is given by the standard error of the mean; shaded bands represent the standard deviation. A comparable dataset for ATLAS is presented in Fig. [16](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig16), confirming the expected luminosity change due to beam–beam effects originating from another IP. Relative luminosity changes for each step in all tests are summarized in Table [5](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Tab5). These values are computed relative to the average of adjacent head-on points, which removes the linear luminosity decay over time. Assuming slow change of beam parameters, the results within each test are expected to be consistent.\n\n**Fig. 15**\n\n[![Fig. 15](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig15_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/15)\n\nCMS specific luminosity, normalized to the first measured both IPs head-on point (blue), during the first step-function scan at IP1. The measurements with ATLAS fully separated are shown in red. The error bars indicate error on the mean\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/15)\n\n**Fig. 16**\n\n[![Fig. 16](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig16_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/16)\n\nATLAS specific luminosity, normalized to the first measured both IPs head-on point (blue), during the step-function scan at IP5. The points for which IP5 was separated are shown in red. The error bars indicate the statistical error\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/16)\n\n**Table 5 Measured luminosity shifts in each of full separation steps – summary of all three step-function scans performed for IP1 and IP5 as the witness IP. The beam-beam parameter at the start of each scan \\\$\\\\xi \\_\\\\text {start}\\\$ is included for reference**\n\n[Full size table](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/tables\/5)\n\nAll measurements are compiled in Fig. [17](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig17) and compared to COMBI simulation predictions. The results are plotted as a function of the beam–beam parameter (estimated as described in Sect. [4\\.4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec8)), following a reverse chronological order. Agreement with simulation is excellent across the entire range.\n\n**Fig. 17**\n\n[![Fig. 17](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig17_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/17)\n\nCOMBI simulated luminosity enhancement induced by a single head-on collision as a function of the beam-beam parameter (dashed line), compared to the test results at both ATLAS (red points) and CMS (green points)\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/17)\n\n**Fig. 18**\n\n[![Fig. 18](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig18_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/18)\n\nMeasurements of beam-beam effects on beam widths shown with red points for each beam and plane, during the experiment with IP5 as the witness IP at \\\$\\\\xi =0.0056\\\$. COMBI predictions are shown in blue with calculated systematic error from the phase advance error between the IP and BSRT locations\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/18)\n#### 6\\.2.2 Impact of beam-beam effects on single-beam sizes\nComplementary measurements of the bunch size from the BSRT are shown in Fig. [18](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig18). Data from all steps and bunches are combined, and a linear trend corresponding to continuous emittance growth is subtracted using the difference between the first and last head-on steps. For comparison, COMBI simulation results are also included in the figure. The dominant source of uncertainty in COMBI predictions arises from discrepancies between the measured phase advance and the MADX model, with the largest observed deviation being \\\$0.05\\\\,\\[2\\\\pi \\]\\\$ (as discussed in Sect. [5\\.3.1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec15)). This uncertainty is propagated using a modified form of Eq. ([6](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Equ6)) adapted to the BSRT location. The magnitude of this uncertainty depends on the specific phase advance configuration, as some are more sensitive to small deviations. This is reflected in the varying sizes of the error bands associated with each prediction. The most accurate predictions are obtained in the vertical plane of Beam 2, where the beam–beam–induced \\\$\\\\beta \\\$\\-beating propagates to the BSRT with a particularly favorable phase advance. In this configuration, the measured effect is also the most pronounced, and excellent agreement with COMBI simulations is observed. In other cases, the expected absolute effect is smaller, typically below 1%, and the influence of phase advance uncertainty becomes more significant, limiting the predictive reliability of the simulations. Experimentally, it is also possible that in these configurations the beam–beam–induced effect is too subtle to resolve, potentially masked by instrumental limitations.\n\n#### 6\\.2.3 Impact of beam-beam effects on tune spectra\nFigure [19](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig19) shows representative ADT spectra recorded during the step-function scans. These spectra are affected by coupling, noise, and various nonlinearities, so they do not follow an idealized analytical distribution. Nonetheless, the peak position can be extracted using a Gaussian fit (indicated by dashed lines), yielding an estimate of the mean tune shift. In the single-collision configuration (blue lines), unsuppressed coherent modes dominate the spectrum and partially obscure the underlying incoherent component. The Gaussian fit is therefore applied to the incoherent contribution between coherent lines, rather than to the global spectral maximum, which results in a larger uncertainty on the extracted peak position. Despite this, in each of the performed step-scans at both collision points, the single-collision tune shift, defined as twice the fitted mean shift, is found to be consistent within \\\$\\\\pm 5\\\\%\\\$ with the expected coherent beam–beam tune shift corresponding to the conditions of the respective scan. For head-on collisions with Gaussian transverse beam distributions, the coherent tune shift of the dipole mode is expected to be \\\$Y\\\\xi \\\$, where *Y* is the Yokoya factor (\\\$Y = 1.21\\\$ for round beams) \\[[34](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR34 \"K. Yokoya, Y. Funakoshi, E. Kikutani, H. Koiso, J. Urakawa, Tune shift of coherent beam-beam oscillations. Part. Accel. 27, 181–186 (1990)\")\\]. As expected, the two-collision configuration yields a tune shift approximately double that of the single-collision case \\[[13](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR13 \"A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). \n                  https:\/\/doi.org\/10.48550\/arXiv.2306.10394\n                  \n                \")\\].\n\n**Fig. 19**\n\n[![Fig. 19](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig19_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/19)\n\nMeasured vertical tune spectra for different collision configurations: reference nominal tune (red), without the presence of any collisions, compared to single collision (blue) and two collisions (green) at \\\$\\\\xi \\\\simeq 0.0086\\\$. The dash-dotted lines in corresponding colors indicate Gaussian fits to the data used to extract mean tunes\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/19)\n\n**Fig. 20**\n\n[![Fig. 20](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig20_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/20)\n\nSpecific luminosity, normalized to the first head-on point, measured at ATLAS (top) and CMS (bottom) during beam separation at the other IPs, with horizontal separation applied in steps 2–4 and vertical separation in steps 6–8. The tested configurations are indicated in the legend. The blue line shows an interpolation of the head-on measurements, serving as a reference for relative luminosity changes\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/20)\n### 6\\.3 Head-on collisions in a three-IP configuration at \\\$\\\\sqrt{s} = 900\\\\ \\\\text {GeV}\\\$\nAn additional head-on collision was included in the configuration using IP2 (ALICE), as it was the only IP where this was feasible. A specially designed filling scheme ensured uniform collision exposure across all bunches, allowing measurements to be combined and thereby reducing statistical uncertainties. However, no reliable luminosity data were available from IP2 during this experiment.\n\n**Table 6 Luminosity shift per full separation step, calculated with respect to an interpolated reference obtained from the two closest head-on points. Results for horizontal and vertical separations are shown as Shift part 1 and Shift part 2, respectively**\n\n[Full size table](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/tables\/6)\n#### 6\\.3.1 Impact of beam-beam effects on head-on luminosity\nA step-function scan was conducted by sequentially collapsing or separating one IP at a time, with the separation applied first in the horizontal plane and, in a second stage of the experiment, in the vertical plane. A comprehensive summary of the measured specific luminosity at ATLAS and CMS is presented in Fig. [20](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig20). Exceptionally stable beam conditions during this test enabled measurements at both experiments under nearly identical beam–beam parameters. The beam–beam-induced luminosity changes observed during each step are quantified in Table [6](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Tab6), referenced to the time-interpolated specific luminosity from three head-on points.\n\nOrbit-drift corrections are applied only to these head-on reference points and result in changes of the extracted luminosity shifts of at most 0.2%. Good agreement with COMBI simulations is observed across the separation steps, with measured shifts closely matching the simulated values for configurations involving IP1 and IP5. For single-IP separation, the level of agreement depends on the witness IP and the separation step. When IP1 is used as the witness, the luminosity shift in the first separation step is reproduced almost exactly by COMBI, while a smaller shift is observed in the second step. This behaviour is expected, as the beam–beam parameter decreases during the scan and the measurement becomes increasingly sensitive to residual orbit perturbations. For configurations involving IP2, the comparison is less conclusive. As no reliable luminosity data were available from IP2 during this experiment, it was not possible to precisely optimize the collision conditions at this IP. As a result, the actual beam–beam conditions at IP2 may differ from those assumed in the simulations, affecting the comparison. Residual orbit drifts affecting the separation steps cannot be fully corrected. As illustrated by the DOROS measurements in Fig. [21](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig21), the orbit data contain both slow drifts and beam–beam–induced orbit offsets arising from changes in the collision configuration when other IPs are separated. While the beam–beam–induced offsets are included in the COMBI simulations, the residual orbit drifts are not. Since these contributions cannot be disentangled in the measurements, a full correction is not possible. Consequently, the effective beam–beam impact measured during the experiment may be weaker than that assumed in the simulations.\n\n**Fig. 21**\n\n[![Fig. 21](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig21_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/21)\n\nResidual separation between the two beams at all three IPs in horizontal (top) and vertical (bottom) planes, for test with observations at ATLAS, where the separation was performed first in the horizontal and then in the vertical plane at the other IPs\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/21)\n#### 6\\.3.2 Tune shift scaling with the number of beam–beam interactions\nAnalysis of the transverse tune spectra in this configuration is further complicated by the reappearance of coherent modes when beams are separated at one or more IPs, as predicted by simulation. In such cases, the maximum tune shift is determined from the frequency difference between the \\\$\\\\pi \\\$ and \\\$\\\\sigma \\\$ modes, rather than from fitting the incoherent peak. For configurations where all three IPs were in collision, coherent modes remained suppressed due to the retained symmetry between IP1 and IP5. Figure [22](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig22) presents the measured tune shifts for all tested collision configurations. The unperturbed tune measurement, shown at ‘0’ collisions, serves as the reference from which all tune shifts are calculated. An empirical uncertainty of 0.001 is assumed for each independent measurement. Both transverse planes and separation directions are included to demonstrate consistency. As expected, the measured tune shift scales linearly with the number of head-on collisions.\n\n**Fig. 22**\n\n[![Fig. 22](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig22_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/22)\n\nMeasured mean tune shift as a function of the number of head-on collisions, shown for both transverse planes and for both separation directions (horizontal and vertical) used in the experiment. The separation refers to how collisions were removed: it was applied in only one transverse direction at a time (H sep. or V sep.) at the non-colliding interaction points, which are not explicitly labeled on the x-axis\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/22)\n### 6\\.4 Beam-beam signatures during transverse-separation scans at \\\$\\\\sqrt{s} = 900\\\\ \\\\text {GeV}\\\$\nIn the van der Meer (vdM) calibration method, beam–beam corrections are applied separately at each separation step. Therefore, it is essential to validate the simulated dependence of luminosity shifts on nominal beam separation. Full vdM-like scans were performed in both transverse planes at ATLAS and CMS, using a configuration where only these two IPs were in collision.\n\n#### 6\\.4.1 Impact of beam–beam effects on the luminosity scan-curves\nFigure [23](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig23) shows the luminosity curves measured at ATLAS, acting as the witness IP, during a separation scan performed at CMS. The curves are fitted with Gaussian profiles for visual guidance. The beam separation at CMS is expressed in units of the measured transverse beam size, \\\$\\\\sigma \\_{\\\\textrm{meas}}\\\$, which was found to be smaller than the nominal beam size, \\\$\\\\sigma \\_{\\\\textrm{nom}}\\\$, used to define the scan steps. As a result, scan points corresponding to nominal separations beyond \\\$6\\\\,\\\\sigma \\_{\\\\textrm{nom}}\\\$ extend to larger values when expressed in units of \\\$\\\\sigma \\_{\\\\textrm{meas}}\\\$.\n\nTo account for emittance evolution during the scan, both linear and exponential fits are applied to the head-on luminosity points before, after, and in the middle of the scan. These trends are subtracted from the data in Fig. [23](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig23), and all measurements are presented relative to this fitted head-on reference (normalized to unity).\n\n**Fig. 23**\n\n[![Fig. 23](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig23_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/23)\n\nATLAS specific luminosity relative changes (blue points) during separation scans in *x*\\-plane (top) and *y*\\-plane (bottom) at CMS, corrected for exponential (exp. corr.) luminosity decay. Gaussian function with a constant fitted to these points is shown with blue dashed line and compared to fit performed on the data corrected for linear luminosity change (lin. corr.) in time (dotted line). These are compared to COMBI simulation predictions (purple solid line)\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/23)\n\nAt CMS, similar scans were performed; however, degraded beam stability was observed during this period of the fill. The scan results exhibit systematic deviations, especially at large separation, which could not be symmetrically corrected using simple linear or exponential models. Despite these issues, all four scan cases (two transverse planes at two IPs) yield a total specific luminosity reduction of approximately 2.4%, consistent with expectations based on the measured beam–beam parameter \\\$\\\\xi \\\$. Comparison with COMBI simulations reveal up to 30% discrepancy for IP5 as the witness IP, while agreement improves to within 10% for measurements at IP1. These differences are comparable to those observed in Table [6](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Tab6), suggesting a common underlying origin. While a finer scan granularity can improve the description of the steep-response region below \\\$3\\\\,\\\\sigma \\_{\\\\textrm{meas}}\\\$, it would not address the dominant discrepancy between head-on and fully separated luminosity, which is likely driven by simulation assumptions and scan-related systematics.\n\n#### 6\\.4.2 Beam–beam-induced orbit deflections\nBeam–beam deflections are clearly observed during separation scans. Figure [24](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig24) shows the fully corrected residual horizontal beam displacement at IP1, measured with the DOROS beam position monitors, as a function of the beam separation. This representation allows a direct comparison with the analytical beam–beam deflection prediction based on the Bassetti–Erskine formalism \\[[35](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR35 \"M. Bassetti, G.A. Erskine, Closed expression for the electrical field of a two-dimensional gaussian charge. Technical report, CERN, Geneva (1980). \n                  https:\/\/cds.cern.ch\/record\/122227\n                  \n                \")\\].\n\nThe raw orbit data were corrected for slow orbit drifts by aligning the beam positions at the luminosity optimization points before and after each scan. Deviations of the applied beam separation from the nominal settings were accounted for by fitting a common length scale simultaneously with the beam–beam deflection. This approach avoids an overestimation of the separation calibration. The measured orbit displacement at the DOROS location was corrected for the contribution arising from the beam–beam deflection angle by applying a geometric amplification factor, proportional to the distance between the IP and the BPM in the small-angle approximation. After all corrections, the measured beam–beam deflection as a function of beam separation closely follows the analytical prediction. The maximum horizontal deflection observed at IP1 is 10.2 \\\$\\\\upmu \\\$m, compared to an expected value of 8.4 \\\$\\\\upmu \\\$m, with agreement within the total measurement uncertainty. The dominant uncertainty contribution (3.5 \\\$\\\\upmu \\\$m, corresponding to about 90% of the total) arises from a possible residual half crossing angle at the IP of up to 0.5 \\\$\\\\upmu \\\$rad. Additional systematic uncertainties include the fitted separation length scale, tune uncertainty of 0.005, and an extended 2 % uncertainty on \\\$\\\\beta ^\\*\\\$. Statistical uncertainties are negligible for most separation steps.\n\n**Fig. 24**\n\n[![Fig. 24](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig24_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/24)\n\nOrbit offset caused by the beam–beam interaction measured by DOROS BPMs in the horizontal plane at IP1 during the horizontal separation scan. Analytical prediction is shown for comparison \\[[35](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR35 \"M. Bassetti, G.A. Erskine, Closed expression for the electrical field of a two-dimensional gaussian charge. Technical report, CERN, Geneva (1980). \n                  https:\/\/cds.cern.ch\/record\/122227\n                  \n                \")\\]\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/24)\n#### 6\\.4.3 Impact of the beam-beam effects on beam sizes and tune shift\nSeparation-dependent changes in beam size were measured using the synchrotron light monitor. Figure [25](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig25) shows good consistency across individual bunch measurements, as well as strong agreement with COMBI simulations. This reinforces the utility of the BSRT as a reliable diagnostic for beam–beam effects. The uncertainties shown are statistical only and are treated as uncorrelated between scan points. The normalization to the zero-separation value is applied for visualization purposes and does not impose a constraint on that point.\n\n**Fig. 25**\n\n[![Fig. 25](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig25_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/25)\n\nBSRT beam-size measurement in the vertical plane for each bunch (points) during horizontal (top) and vertical (bottom) separation scans at IP1. The exponential evolution correction is applied based on adjacent head-on points (before, in the middle and after the scan) to highlight the relative changes. COMBI predictions are shown (gray dashed line) for comparison\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/25)\n\nTransverse tune shifts were extracted from ADT-measured spectra. As illustrated in Fig. [26](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig26), the mean tune shift evolution is reconstructed for both the scanning and non-scanning planes. The measured tune shifts closely follow the expected COMBI trends, further validating the simulation models used for beam–beam corrections.\n\n**Fig. 26**\n\n[![Fig. 26](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig26_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/26)\n\nMeasured tune shift in units of the beam-beam parameter \\\$\\\\xi \\_{BB}\\\$ for the horizontal (blue) and vertical (orange) planes, during a horizontal separation scan at IP1. The error bars indicate typically assumed conservative empirical systematic error on the tune measurement of 0.001. The solid curves represent the COMBI prediction\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/26)\n### 6\\.5 Beam-beam signatures during vdM scans at \\\$\\\\sqrt{s} = 13.6\\\\ \\\\text {TeV}\\\$\nDuring the first vdM calibration of LHC Run 3, an unusually high beam–beam parameter was recorded, offering a unique opportunity to study beam–beam effects under enhanced interaction conditions. These effects are examined using online ATLAS luminosity data, with representative results shown in Fig. [27](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig27).\n\nThe example focuses on a standard vdM scan pair conducted at CMS, while monitoring luminosity at the non-scanning IP (ATLAS). The plotted luminosity changes are normalized to the central scan point, where both IPs are in head-on collision. Variations in this normalized luminosity across the scan directly reflect the impact of beam–beam interaction at the scanning IP on the luminosity at the non-scanning IP. These patterns are reproducible across multiple scan pairs acquired within the same fill. At the largest nominal separations (approximately \\\$\\\\pm 0.6\\\\,\\\$mm), the beams are sufficiently separated to eliminate beam–beam effects entirely. Notably, a qualitative difference is observed between the horizontal and vertical scan planes, an asymmetry that is not present in the above presented dedicated experiment. This behavior is attributed to differences in the phase advance between IPs in the standard vdM optics configuration. For vertical (*y*) scans, experimental data agree well with simulation, confirming the robustness of the modeling under these conditions. In contrast, the horizontal (*x*) scans exhibit a systematic discrepancy: the observed beam–beam impact is broader than that predicted by simulation. The model assumes that the separation steps are accurately defined using the beam size measured during the vdM scans, but this mismatch may suggest an unmodeled phase advance deviation or another optics imperfection. The total difference between data and simulation is within 0.2%. This dataset originates from a standard vdM calibration, for which only online luminosity measurement was available, and many contributions to the overall vdM uncertainty, such as bunch-to-bunch variations, averaging over different bunch families, and emittance evolution during the scan, are not disentangled. Therefore, the 0.2% difference should be interpreted as a consistency check under typical operational conditions rather than a precision validation of the beam–beam model. Nevertheless, this indicates that the model remains sufficiently accurate for beam–beam corrections in vdM analyses.\n\n**Fig. 27**\n\n[![Fig. 27](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig27_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/27)\n\nChanges from the separation-dependent beam-beam effects at the non-scanning IP – ATLAS online luminosity shown with points. Data comes from a standard vdM scan pair during the 2022 *pp* luminosity calibration. Separation steps and direction are indicated with gray markers. The last CMS scan is shown, with the highest beam-beam parameter of \\\$ \\\\xi =5.3 \\\\times 10^{-3} \\\$. The corresponding COMBI simulation results are also shown (triangles)\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/27)\n\n## 7 Conclusions\nBeam–beam effects were directly observed and quantitatively validated at the LHC. The impact of beam–beam interactions on head-on luminosity, bunch sizes, tune spectra, and beam orbit positions was measured across various collision configurations. The observed effects align closely with COMBI simulations. In particular, the luminosity enhancement due to amplitude-dependent \\\$\\\\beta \\\$\\-beating was measured for the first time and is found to agree with expectations over a wide range of the beam–beam parameter \\\$\\\\xi \\\$. Together with the observed linear scaling with respect to the beam–beam parameter, this provides experimental confirmation of the numerical studies presented in \\[[13](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR13 \"A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). \n                  https:\/\/doi.org\/10.48550\/arXiv.2306.10394\n                  \n                \")\\]. Consistent signatures were found across multiple scan configurations, including two- and three-IP setups, and the characteristic tune shifts were well reproduced by both measurement and simulation.\n\nAccurate instrumentation and experimental design were essential. Key to the results was the stepwise collision scheme, isolating individual beam–beam contributions, with the implementation of the witness IP. Synchrotron light measurements from the BSRT, especially in the vertical plane of Beam 2, showed excellent agreement with COMBI predictions and bunch-by-bunch consistency. Tune spectra from the ADT system provided direct access to mean tune shifts, while orbit data from the DOROS system enabled measurement of beam–beam deflection down to 1 \\\$\\\\upmu \\\$m, matching analytical estimates. These tools, combined with precise control of IP configurations, enabled a comprehensive cross-check of beam–beam models.\n\nSome systematic limitations were identified. During one of the tests, beam stability during the scans was insufficient to allow for symmetric modeling of luminosity decay in both transverse planes. In several scans, the nominal separation steps were based on overestimated beam sizes, resulting in larger-than-expected scan intervals and less sensitivity at low separation. Beam–beam deflection measurements showed sign reversals at high separation, attributed to nonlinear orbit distortions from hysteresis effects in superconducting magnets. Phase advance errors between IPs and diagnostic systems further reduced the accuracy of simulations in some configurations.\n\nTargeted improvements would allow for higher precision and new insights. Finer separation steps, especially below \\\$3\\\\,\\\\sigma \\\$, would improve sensitivity to nonlinear beam–beam effects and allow more accurate comparison with models. Suppressing coherent beam–beam modes via phase advance tuning of not only two but all three IPs would improve tune shift measurements. Moreover, such optimal phase advance configuration between the IPs can be exploited to enhance luminosity. This effect could be directly propagated into at least a few percent increase in the total collected integrated luminosity, as discussed in Ref. \\[[36](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR36 \"J. Wańczyk, T. Pieloni, Optimizing beam-beam beta-beating for luminosity enhancement at the LHC. JACoW IPAC 2025, 018 (2025)\")\\]. Aligning the phase advance between IPs and diagnostic devices, particularly the BSRT, would reduce sensitivity to optics errors, enabling better beam size measurements for both beams and transverse planes. Moreover, the suppression of the coherent beam–beam modes facilitates the observation and analysis of the central, incoherent part of the tune spectrum, which is otherwise obscured by the dominant coherent peaks.\n\nFurther experimental strategies should also focus on directly probing the beam–beam interaction at its point of occurrence. This would require access to the full transverse beam distributions with high spatial resolution and sensitivity, going beyond projected sizes or centroids. Such measurements could directly reveal the nonlinear distortions and tails induced by the beam–beam force. In parallel, validating correction schemes that parametrize multi-IP beam–beam effects on luminosity via equivalent tune shifts remains essential. These models underpin practical luminosity corrections in complex fill configurations and would benefit from systematic benchmarking.\n\n## Data Availability Statement\nData will be made available on reasonable request. \\[Authors’ comment: The datasets generated during and\/or analysed during the current study are available from the corresponding author on reasonable request.\\]\n\n## Code Availability Statement\nCode\/software will be made available on reasonable request. \\[Authors’ comment: The code\/software generated during and\/or analysed during the current study is available from the corresponding author on reasonable request.\\]\n\n## References\n1. S. Meer, Calibration of the effective beam height in the ISR. Technical report, CERN, Geneva (1968). <https:\/\/cds.cern.ch\/record\/296752>\n2. C. Rubbia, Measurement of the luminosity of \\\$p{{\\\\overline{p}}}\\\$ collider with a (generalized) Van der Meer Method. 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JACoW IPAC **2025**, 018 (2025)\n    [Google Scholar](http:\/\/scholar.google.com\/scholar_lookup?&title=Optimizing%20beam-beam%20beta-beating%20for%20luminosity%20enhancement%20at%20the%20LHC&journal=JACoW%20IPAC&volume=2025&publication_year=2025&author=Wa%C5%84czyk%2CJ&author=Pieloni%2CT)\n\n[Download references](https:\/\/citation-needed.springer.com\/v2\/references\/10.1140\/epjc\/s10052-026-15476-8?format=refman&flavour=references)\n\n## Acknowledgements\nWe would especially like to acknowledge W. Kozanecki for his substantial contributions to the development of this work and for carefully reviewing the manuscript, and Y. Wu for participation in the experiment and online data checks. We are also indebted to the ATLAS and CMS luminosity experts, in particular R. Hawkings, E. Torrence and A. Shevelev. We thank LHC experts for help with setting up and carrying out the experiment, in particular M. Hostettler, T. Persson, M. Solfaroli Camillocci, G. Trad, J. Wenninger. As well as optics team, in particular F. Carlier and M. Le Garrec. Some of the authors have received support from the Swiss Accelerator Research and Technology Institute (CHART).\n\n## Author information\n### Authors and Affiliations\n1. CERN, Geneva, Switzerland\n   Joanna Wańczyk, Xavier Buffat, Anne Dabrowski & Rogelio Tomas\n2. Laboratory of Particle Accelerator Physics, EPFL, Lausanne, Switzerland\n   Joanna Wańczyk & Tatiana Pieloni\n3. Princeton University, Princeton, NJ, USA\n   David Stickland\n\nAuthors\n\n1. 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Rogelio Tomas\n   [View author publications](https:\/\/link.springer.com\/search?sortBy=newestFirst&contributor=Rogelio%20Tomas)\n   \n   Search author on:[PubMed](https:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=search&term=Rogelio%20Tomas) [Google Scholar](https:\/\/scholar.google.co.uk\/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Rogelio%20Tomas%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en)\n### Corresponding author\nCorrespondence to [Joanna Wańczyk](mailto:jwanczyk@cern.ch).\n\n## Appendix A: \\\$\\\\beta \\\$-beating derivative\n### Appendix A: \\\$\\\\beta \\\$\\-beating derivative\nDerivative of Eq. ([7](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Equ7)) with respect to the phase advance \\\$\\\\mu \\_{1,x}\\\$, which characterizes the change in the linear beam-beam \\\$\\\\beta \\\$\\-beating due to variations in phase advance:\n\n![](https:\/\/media.springernature.com\/lw540\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Equ17_HTML.png)\n\n## Rights and permissions\n**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. 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article\nWańczyk, J., Pieloni, T., Buffat, X. *et al.* First measurements of beam-beam effects in beam-separation, luminosity-calibration scans at the LHC. *Eur. Phys. J. C* **86**, 282 (2026). https:\/\/doi.org\/10.1140\/epjc\/s10052-026-15476-8\n\n[Download citation](https:\/\/citation-needed.springer.com\/v2\/references\/10.1140\/epjc\/s10052-026-15476-8?format=refman&flavour=citation)\n\n- Received: 13 October 2025\n- Accepted: 21 February 2026\n- Published: 17 March 2026\n- Version of record: 17 March 2026\n- DOI: https:\/\/doi.org\/10.1140\/epjc\/s10052-026-15476-8\n\n### Share this article\nAnyone you share the following link with will be able to read this content:\n\nGet shareable link\n\nSorry, a shareable link is not currently available for this article.\n\nCopy shareable link to clipboard\n\nProvided by the Springer Nature SharedIt content-sharing initiative\n\nAdvertisement\n\n## Search\n## Navigation\n- [Find a journal](https:\/\/link.springer.com\/journals\/)\n- [Publish with us](https:\/\/www.springernature.com\/gp\/authors)\n- [Track your research](https:\/\/link.springernature.com\/home\/)\n\n### Discover content\n- [Journals A-Z](https:\/\/link.springer.com\/journals\/a\/1)\n- [Books A-Z](https:\/\/link.springer.com\/books\/a\/1)\n\n### Publish with us\n- [Journal finder](https:\/\/link.springer.com\/journals)\n- [Publish your research](https:\/\/www.springernature.com\/gp\/authors)\n- [Language editing](https:\/\/authorservices.springernature.com\/go\/sn\/?utm_source=SNLinkfooter&utm_medium=Web&utm_campaign=SNReferral)\n- [Open access publishing](https:\/\/www.springernature.com\/gp\/open-science\/about\/the-fundamentals-of-open-access-and-open-research)\n\n### Products and services\n- [Our products](https:\/\/www.springernature.com\/gp\/products)\n- [Librarians](https:\/\/www.springernature.com\/gp\/librarians)\n- [Societies](https:\/\/www.springernature.com\/gp\/societies)\n- [Partners and advertisers](https:\/\/www.springernature.com\/gp\/partners)\n\n### Our brands\n- [Springer](https:\/\/link.springer.com\/brands\/springer)\n- [Nature Portfolio](https:\/\/www.nature.com\/)\n- [BMC](https:\/\/link.springer.com\/brands\/bmc)\n- [Palgrave Macmillan](https:\/\/link.springer.com\/brands\/palgrave)\n- [Apress](https:\/\/link.springer.com\/brands\/apress)\n- [Discover](https:\/\/link.springer.com\/brands\/discover)\n\n- Your privacy choices\/Manage cookies\n- [Your US state privacy rights](https:\/\/www.springernature.com\/gp\/legal\/ccpa)\n- [Accessibility statement](https:\/\/link.springer.com\/accessibility)\n- [Terms and conditions](https:\/\/link.springer.com\/termsandconditions)\n- [Privacy policy](https:\/\/link.springer.com\/privacystatement)\n- [Help and support](https:\/\/support.springernature.com\/en\/support\/home)\n- [Legal notice](https:\/\/link.springer.com\/legal-notice)\n- [Cancel contracts here](https:\/\/support.springernature.com\/en\/support\/solutions\/articles\/6000255911-subscription-cancellations)\n\nNot affiliated\n\n[![Springer Nature](https:\/\/link.springer.com\/oscar-static\/images\/logo-springernature-white-dbadd2cbd6.svg)](https:\/\/www.springernature.com\/)\n\n© 2026 Springer Nature","attrs_readable_markdown":"## 1 Introduction\nA precise determination of the absolute luminosity scale is essential for a wide range of LHC measurements, as it directly affects the normalization of many key physics cross sections. At the LHC, this calibration relies primarily on the van der Meer (vdM) method \\[[1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR1 \"S. Meer, Calibration of the effective beam height in the ISR. Technical report, CERN, Geneva (1968). \n                  https:\/\/cds.cern.ch\/record\/296752\n                  \n                \"), [2](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR2 \"C. Rubbia, Measurement of the luminosity of \n                  $$p{{\\overline{p}}}$$\n                  \n                    \n                      p\n                      \n                        p\n                        ¯\n                      \n                    \n                  \n                 collider with a (generalized) Van der Meer Method. Technical report, CERN, Geneva (1977). \n                  https:\/\/cds.cern.ch\/record\/1025746\n                  \n                \")\\], which uses dedicated beam-separation scans performed under specially tailored beam conditions to relate measured interaction rates to the absolute luminosity inferred from beam parameters. While sub-percent uncertainties on the integrated luminosity have been achieved for Run-2 data by ATLAS \\[[3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR3 \"ATLAS collaboration, Luminosity determination in \n                  $$pp$$\n                  \n                    \n                      pp\n                    \n                  \n                 collisions at \n                  $$\\sqrt{s}=13$$\n                  \n                    \n                      \n                        s\n                      \n                      =\n                      13\n                    \n                  \n                 TeV using the ATLAS detector at the LHC. Eur. Phys. J. C 83(10), 982 (2023). \n                  https:\/\/doi.org\/10.1140\/epjc\/s10052-023-11747-w\n                  \n                . \n                  arXiv:2212.09379\n                  \n                \")\\] and CMS \\[[4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR4 \"CMS Collaboration, Precision luminosity measurement in proton-proton collisions at sqrts = 13 TeV with the CMS detector. Technical report, CERN, Geneva (2025). \n                  https:\/\/cds.cern.ch\/record\/2940794\n                  \n                \")\\], further improvements and robust experimental validation of beam-dynamical effects remain essential.\n\nFor relativistic proton beams colliding with zero crossing angle, which is the typical configuration for the *vdM* calibration, the luminosity at a given interaction point (IP) of the LHC is proportional to the overlap integral of the particle-density distributions \\\$\\\\rho \\_{\\\\text {1,i}}\\\$, \\\$\\\\rho \\_{\\\\text {2,i}}\\\$ in bunch-pair *i* \\[[5](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR5 \"W. Herr, T. Pieloni, Beam-Beam Effects. Technical report, CERN (2014). Contribution to the CAS–CERN Accelerator School: Advanced Accelerator Physics Course, Trondheim, Norway, 18–29 Aug 2013, p. 29. \n                  https:\/\/doi.org\/10.5170\/CERN-2014-009.431\n                  \n                . \n                  https:\/\/cds.cern.ch\/record\/1982430\n                  \n                \")\\]:\n\n\\$\\$\\\\begin{aligned} {\\\\mathcal {L}}\\_{inst}&= 2c f\\_{\\\\text {rev}} \\\\sum ^{N\\_b}\\_i n\\_{\\\\text {1,i}}, n\\_{\\\\text {2,i}} \\\\nonumber \\\\\\\\&\\\\quad \\\\times \\\\iiiint ^{+\\\\infty }\\_{-\\\\infty }\\\\rho \\_{\\\\text {1,i}}(x,y,z-ct) \\\\nonumber \\\\\\\\&\\\\quad \\\\times \\\\rho \\_{\\\\text {2,i}}(x,y,z+ct)\\\\,dxdydzdt, \\\\end{aligned}\\$\\$\n\n(1)\n\nwhere *c* is the speed of light, \\\$f\\_{\\\\text {rev}}\\\$ is the LHC revolution frequency, \\\$N\\_b\\\$ is the number of colliding bunches, and \\\$n\\_{\\\\text {1,i}}, n\\_{\\\\text {2,i}}\\\$ are the corresponding total charges per bunch for each colliding pair. Although only a very small fraction of the beam particles actually collide, the two opposing beams interact with each other electromagnetically, a dynamical process known as the beam-beam interaction \\[[5](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR5 \"W. Herr, T. Pieloni, Beam-Beam Effects. Technical report, CERN (2014). Contribution to the CAS–CERN Accelerator School: Advanced Accelerator Physics Course, Trondheim, Norway, 18–29 Aug 2013, p. 29. \n                  https:\/\/doi.org\/10.5170\/CERN-2014-009.431\n                  \n                . \n                  https:\/\/cds.cern.ch\/record\/1982430\n                  \n                \"), [6](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR6 \"A.W. Chao, Beam-beam instability. AIP Conf. Proc. 127, 201–242 (1985). \n                  https:\/\/doi.org\/10.1063\/1.35187\n                  \n                \")\\]. In the LHC, as in any other synchrotron, the beams are not continuous but are divided into discrete “bunches”, each with a high proton density, containing approximately \\\$10^{11}\\\$ protons over a bunch length of 7.5–9 cm (in terms of RMS). This electromagnetic interaction occurs when the two beams share a common beam pipe, and is governed by a non-linear force that depends on the radial distance *r* of a test particle to the center of the opposing “source” bunch with *n* protons Gaussian-distributed in the transverse planes (valid approximation for the LHC beams):\n\n\\$\\$\\\\begin{aligned} F = - \\\\frac{ne^2}{2\\\\pi \\\\epsilon \\_0r}\\\\biggl (1-\\\\exp \\\\biggl \\[ -\\\\frac{r^2}{2\\\\sigma ^2}\\\\biggr \\] \\\\biggr ), \\\\end{aligned}\\$\\$\n\n(2)\n\nwhere *e* is the elementary proton charge, \\\$\\\\epsilon \\_0\\\$ is the vacuum permittivity, and \\\$\\\\sigma \\\$ is the RMS radius of the transverse charge distribution within the bunch, in a simplified case of a round shape \\\$(\\\\sigma = \\\\sigma \\_x = \\\\sigma \\_y)\\\$. At very small amplitudes \\\$(r\\\\rightarrow 0)\\\$ the force can be linearized giving the expression for the so called beam-beam parameter, which is often used to assess the strength of the force:\n\n\\$\\$\\\\begin{aligned} \\\\xi = \\\\frac{nr\\_p\\\\beta ^\\*}{4\\\\pi \\\\gamma \\\\sigma ^2} = \\\\frac{nr\\_p}{4\\\\pi \\\\epsilon \\_n}, \\\\end{aligned}\\$\\$\n\n(3)\n\nwhere \\\$r\\_p\\\$ is the classical proton radius, \\\$\\\\beta ^\\*\\\$ is the optical beta-function at the interaction point (IP) – one of the Courant–Snyder parameters \\[[7](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR7 \"E.D. Courant, H.S. Snyder, Theory of the alternating-gradient synchrotron. Ann. Phys. 3, 1–48 (1958). \n                  https:\/\/doi.org\/10.1016\/0003-4916(58)90012-5\n                  \n                \")\\], \\\$\\\\gamma \\\$ is the Lorentz factor, and \\\$\\\\epsilon \\_n\\\$ is the normalized emittance.\n\nBeam-beam effects were first observed almost 40 years ago \\[[8](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR8 \"P. Bambade, R. Erickson, W.A. Koska, W. Kozanecki, N. Phinney, S.R. Wagner, Observation of beam-beam deflections at the interaction point of the SLAC Linear Collider. Phys. Rev. Lett. 62(25), 2949–2952 (1989). \n                  https:\/\/doi.org\/10.1103\/PhysRevLett.62.2949\n                  \n                \")\\] at the lepton colliders, and have been used extensively for accelerator diagnostics and optimization at both SLC and LEP \\[[9](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR9 \"W.A. Koska, P. Bambade, W. Kozanecki, N. Phinney, S.R. Wagner, Beam-beam deflection as a beam tuning tool at the slac linear collider. Nucl. Instrum. Methods A 286, 32 (1990). \n                  https:\/\/doi.org\/10.1016\/0168-9002(90)90203-I\n                  \n                \"),[10](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR10 \"C. Bovet, M.D. Hildreth, M. Lamont, H. Schmickler, J. Wenninger, Luminosity optimisation using beam-beam deflections at LEP, in Conf. Proc. C (1996). \n                  https:\/\/cds.cern.ch\/record\/306910\n                  \n                \"),[11](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR11 \"D. Brandt, W. Herr, M. Meddahi, A. Verdier, Is lep beam-beam limited at its highest energy? in Proceedings of the 1999 Particle Accelerator Conference (PAC ’99), New York, pp. 3005–3007. IEEE\/JACoW. THP25; CERN report CERN-SL-99-030-AP (1999). \n                  https:\/\/accelconf.web.cern.ch\/p99\/PAPERS\/THP25.pdf\n                  \n                \")\\]. At the LHC, they were first observed in late Run 1 \\[[12](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR12 \"W. Kozanecki, T. Pieloni, J. Wenninger, Observation of Beam-beam Deflections with LHC Orbit Data. Technical report, CERN (2013). \n                  https:\/\/cds.cern.ch\/record\/1581723?ln=en\n                  \n                \")\\], but only in terms of the beam-beam induced coherent deflections. Single particle effects, in contrast, were presumed too small in hadron colliders that until recently, there was no motivation to measure them with a precision that would be meaningful in the context evoked here.\n\nNonetheless, the full complexity of this interaction must be accounted for when performing the luminosity calibration via the vdM method, using the separation scans. These are designed to measure the detector-specific constant \\\$\\\\sigma \\_{vis}\\\$ that relates the observed rate \\\$\\\\mu ^{vis}\\\$ to the absolute instantaneous luminosity \\\${\\\\mathcal {L}}\\_{inst}\\\$:\n\n\\$\\$\\\\begin{aligned} \\\\mu ^{vis} = \\\\frac{{\\\\mathcal {L}}\\_{inst}\\\\sigma \\_{vis}}{f\\_{rev}}. \\\\end{aligned}\\$\\$\n\n(4)\n\nUnder the assumption of uncorrelated particle densities in *x* and *y* planes, the transverse convolved bunch widths \\\$\\\\Sigma \\_x, \\\\Sigma \\_y\\\$ can be extracted from the measured beam-separation dependence of the collision rate in the corresponding direction. The combined information from these scans, and bunch intensities can be used to calculate the instantaneous luminosity at the head-on position. The proportionality between the measured rate at the centered head-on position \\\$\\\\mu \\_{pk}\\\$ and the reconstructed luminosity from measured bunch parameters during the vdM calibration defines the luminometer-specific visible cross-section:\n\n\\$\\$\\\\begin{aligned} \\\\sigma \\_{vis} = \\\\frac{2\\\\pi \\\\Sigma \\_x\\\\Sigma \\_y}{n\\_1 n\\_2} \\\\mu \\_{pk}. \\\\end{aligned}\\$\\$\n\n(5)\n\nIf left uncorrected, the \\\$\\\\sigma \\_{vis}\\\$ measurement by the *vdM* method is biased by the beam-separation dependence of the mutual electromagnetic interaction of the two beams: the colliding bunches experience deflection-induced orbit shifts, as well as optical distortions akin to the dynamic-\\\$\\\\beta \\\$ effect, that both depend on the transverse beam separation and must therefore be accounted for when deriving the absolute luminosity scale. The correction strategy discussed extensively in Ref. \\[[13](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR13 \"A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). \n                  https:\/\/doi.org\/10.48550\/arXiv.2306.10394\n                  \n                \")\\] relies on simulation studies only, and thus requires experimental validation to establish its reliability.\n\nThe present report summarizes a campaign of dedicated measurements carried out in the spring of 2022 at the LHC, aimed at confirming the accuracy of the simulation studies detailed in Ref. \\[[13](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR13 \"A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). \n                  https:\/\/doi.org\/10.48550\/arXiv.2306.10394\n                  \n                \")\\], and at validating the beam-beam correction strategy used in the luminosity-calibration analyses of the ALICE, ATLAS, CMS and LHCb experiments. More specifically, the goal is to quantify the impact of the beam-beam interaction on the tune spectra, orbit, the transverse-density distribution of the colliding bunches, as well as, for the first time, on the luminosity, by systematically varying the strength or number of the beam-beam interactions. The measurements are repeated for different values of the beam-beam parameter to verify the scaling proposed in \\[[13](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR13 \"A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). \n                  https:\/\/doi.org\/10.48550\/arXiv.2306.10394\n                  \n                \")\\], for various choices of the scanning IP and for several multi-IP configurations. The beam conditions are chosen to be representative of *vdM* calibration sessions at the LHC, but optimized so as to maximize both the sensitivity of the measurements, and the operational efficiency.\n\nThis paper is organized as follows. The measurement strategy is detailed in Sect. [2](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec2), followed by the main ingredients of this accelerator experiment: beam conditions (Sect. [3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec3)), luminometers and accelerator instrumentation (Sect. [4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec4)), and optimization of the ring lattice to maximize the sensitivity of the measurements to beam–beam-induced optical distortions of the colliding bunches (Sect. [5](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec11)). Section [6](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec19) is devoted to the experimental characterization of the impact of beam-beam effects on the tune spectra, on the luminosity at different IPs, as well as on other observables such as transverse single-bunch widths and IP orbits. At each step, the results are confronted with the predictions of the COherent Multibunch Beam-beam Interactions (COMBI) tracking code \\[[14](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR14 \"T. Pieloni, A study of beam-beam effects in hadron colliders with a large number of bunches. PhD Thesis, École Polytechnique Fédérale de Lausanne (2008). \n                  https:\/\/doi.org\/10.5075\/epfl-thesis-4211\n                  \n                \")\\], that produced most of the results detailed in Ref. \\[[13](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR13 \"A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). \n                  https:\/\/doi.org\/10.48550\/arXiv.2306.10394\n                  \n                \")\\]. A collection of macroparticles is simulated turn by turn, based on a simplified model of the accelerator lattice using linear transfer and the self-consistent computation of the beam-beam forces at the IP. The conclusions are presented in Sect. [7](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec33).\n\n## 2 Measurement strategy\nThe beam-beam interaction at an interaction point, referred to here as the *scanning* IP, induces measurable changes in key beam parameters such as the tune spectrum, closed orbit, and transverse single-beam profiles. These effects can be observed, albeit with limited precision, using standard accelerator diagnostics by comparing measurements obtained under varying transverse beam separation at the scanning IP. However, the resulting modifications to the transverse density distribution of the colliding bunches remain below the sensitivity threshold of conventional instrumentation, including synchrotron-radiation-based beam-profile monitors. Beam-beam-induced variations typically lie in the 0.5–1% range, small enough that only a highly precise measurement of collision-rate changes can provide adequate sensitivity.\n\n**Fig. 1**\n\n[![Fig. 1](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig1_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/1)\n\nScheme of the LHC and its interaction points (IP) with indicated IP1 as witness IP while performing a scan at IP5\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/1)\n\nImportantly, the beam-beam interaction induces changes in beam properties that are intrinsically entangled with the *pp* collisions when observing luminosity at the scanning IP. Consequently, under realistic collider conditions, it is not possible to directly access a reference luminosity signal that is free of beam-beam effects – unlike in simulation studies, where such effects can be explicitly disabled. In a multi-collision configuration, however, luminosity shifts at one or more other IPs (Fig. [1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig1)), where the beams are kept in continuous head-on collision and therefore act as non-scanning, or *witness*, IPs, can be used to monitor the separation-dependent beam–beam effects. These effects are induced at the scanning IP, where the beams are deliberately brought in and out of collision, and propagate around the rings to the witness IPs, where they manifest as changes in the measured luminosity. In principle, any of the four LHC IPs can serve as a scanning IP, and any non-scanning IP can be designated as the witness IP for a given measurement. In practice, instrumental and operational constraints at the time the experiment was carried out restrict the choice of witness IP to the ATLAS (IP1) and CMS (IP5) collision points; IPs 1, 2 or 5, or combinations thereof, are used as scanning IP(s). Collisions at IP8 were deliberately avoided because the large longitudinal offset of the LHCb collision point, that breaks the eight-fold symmetry of the LHC rings, precludes the possibility of both members of a colliding-bunch pair to collide at all four IPs. In addition, the large crossing angle at IP8 would complicate the interpretation of the beam-beam effects that would occur at that IP if some of the bunches collided there.\n\nThe propagation of the beam–beam-induced, amplitude-dependent \\\$\\\\beta \\\$\\-beating from the scanning to a witness IP is controlled by the betatron phase advance between these two IPs. Since the targeted signatures are delicate to measure at best, it is natural to try and enhance them by adjusting this phase advance so as to maximize the sensitivity of luminosity shifts at the witness IP to beam-beam effects at the scanning IP; in doing so, however, the overall tunes must be preserved. Given the central and symmetric roles played by IP1 and IP5 in this experiment, the adjustment of their relative phase advance drove the optimization procedure that is detailed in Sect. [5](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec11), and that achieved a threefold improvement in measurement sensitivity.\n\nCentral to the quality of the measurements is the controlled variation of the beam-beam parameter at the scanning IP. This can be achieved either:\n\n- by using *step scans*, *i.e.* by fully separating the beams at the scanning IP in either the horizontal or the vertical plane, and then bringing them back into head-on collision; or\n- by using *separation scans*, *i.e.* by scanning the beams transversely with respect to each other in either the horizontal or the vertical plane; or\n- by taking advantage of the natural beam-intensity decay and emittance growth to progressively reduce the beam-beam parameter.\n\nEqually important is to ensure that the beams remain in head-on collision at the witness IP(s), such that reproducible luminosity shifts measured at these locations can be unambiguously correlated with controlled changes in the strength of the beam-beam interaction at the scanning IP. It was verified that the beam-beam-induced orbit shift and the potential non-closure of the orbit bumps used to control the beam separation at the scanning IP, did not significantly affect the actual separation, and thereby the measured collision rate, at the witness IP. In a few cases however, orbit drifts of uncontrolled origin ended up degrading the quality of some of the measurements.\n\nSimulations demonstrate \\[[13](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR13 \"A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). \n                  https:\/\/doi.org\/10.48550\/arXiv.2306.10394\n                  \n                \")\\] that multi-IP effects strongly influence the magnitude of beam-beam biases to vdM calibrations. Their characterization, therefore, constitutes an essential component of this experiment. The scan protocol is a generalization of that in the two-IP case. One IP (say, IP1) is designated as the witness IP, with beams colliding head-on there and at IP2, and IP5; the beams are then taken out of collision at IP2 only, then at both IP2 and IP5, and finally returned to a three-collision configuration in the reverse order. Since the phase-advance combinations are different in each configuration, the corresponding luminosity shifts predicted at the witness IP are also different and can be meaningfully confronted with the data.\n\n## 3 Beam conditions\nIn the round-beam approximation, the beam-beam parameter \\\$\\\\xi \\\$ does not depend on the beam energy (Eq. ([3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Equ3))). Therefore, and in order to minimize the overhead associated with commissioning and operating the LHC in a non-standard optical configuration, as well as to allow for frequent refilling with different bunch patterns, the measurements were carried out at injection energy (\\\$450\\\\ \\\\text {GeV}\\\$ per beam), and under non-standard machine-protection conditions. The improvement in operational efficiency came at a cost:\n\n- the counting rate of the luminometers (or equivalently their visible cross-section) is about an order of magnitude smaller at \\\$\\\\sqrt{s} = 900\\\\ \\\\text {GeV}\\\$ than at \\\$13\\\\ \\\\text {TeV}\\\$, reflecting the combination of a 40% drop in the inelastic *pp* cross-section, a factor of two to three drop in the multiplicity of the final-state particles, and a significant softening of their momentum spectrum;\n- because of the total-intensity constraints dictated by machine-protection requirements, the injected beam could not exceed 4 bunches per beam (compared to 150 during a routine *vdM*\\-calibration session), with a maximum allowed population of \\\$1.25\\\\times 10^{11}~p\\\$\/bunch;\n- the combination of the low beam energy, that suppresses synchrotron-radiation damping, and of the large bunch intensity, that enhances intra-beam scattering, resulted in relatively rapid emittance growth and rather short single-beam lifetimes. To partially mitigate these effects, all collision-rate measurements are expressed in terms of specific luminosity, thereby automatically accounting for the beam-intensity decay.\n\nThe bunch intensity was deliberately increased by about 25% beyond the *vdM*\\-scan values typical of LHC Run 2, to maximize both the luminosity and the beam-beam parameter \\\$\\\\xi \\\$. The latter reached 0.010 per IP (Table [1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Tab1)), compared to 0.003–0.006 during normal *pp* *vdM* sessions. With a target injected emittance of \\\$1.5 \\\\ \\\\upmu \\\\text {m} \\\\cdot \\\\text {rad}\\\$, the \\\$\\\\beta \\\$ function at the IP set to \\\$\\\\beta ^\\* = 11\\\\ \\\\text {m}\\\$, and zero crossing angle at IP1 and IP5, the statistical uncertainty affecting a typically 60-second long per-bunch luminosity measurement in the presence of head-on collisions, lay around 0.5%.\n\n**Table 1 Range of beam-beam parameter values during the various stages of the experiment**\n\n[Full size table](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/tables\/1)\n\nThe bunch patterns were chosen such that all bunches in each beam collided either at IP1 and IP5 only, or at all three of IP1, 2 and 5. The orbit-stabilization feedback system was turned off during the data-taking periods to prevent it from interfering with the beam–beam-induced orbit shift. The chromaticity was set to its standard value in physics fills, of +10 units, to guarantee coherent stability against the machine impedance, and in view of the very small number of bunches and of their large longitudinal spacing, the beam-stabilizing Landau octupoles were set to their minimum current (1A). The damping time of the bunch-by-bunch transverse feedback was set to 1000 turns, a rather loose setting intended to preserve longer the natural beam oscillation and therefore improve the precision of the tune measurements.\n\n## 4 Beam instrumentation\n### 4\\.1 Luminometers\nThe luminometer systems in use by the ATLAS and CMS collaborations at IP1 and IP5 are described in Refs. \\[[15](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR15 \"ATLAS Collaboration, Improved luminosity determination in pp collisions at \n                  $$\\sqrt{s} = 7$$\n                  \n                    \n                      \n                        s\n                      \n                      =\n                      7\n                    \n                  \n                 TeV using the ATLAS detector at the LHC. Eur. Phys. J. C 73, 2518 (2013). \n                  https:\/\/doi.org\/10.1140\/epjc\/s10052-013-2518-3\n                  \n                \"), [16](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR16 \"C.M.S. Collaboration, Development of the cms detector for the cern lhc run 3. J. Instrum. 19(05), 05064 (2024). \n                  https:\/\/doi.org\/10.1088\/1748-0221\/19\/05\/p05064\n                  \n                \")\\] respectively; no luminosity measurements were available at IP2 during the collider experiment described in this paper. Maximizing the statistical sensitivity leads to choosing the luminosity algorithm with the highest possible acceptance, namely:\n\n- the hit rate per bunch crossing in the ATLAS Minimum Bias Trigger Scintillators (MBTS) \\[[15](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR15 \"ATLAS Collaboration, Improved luminosity determination in pp collisions at \n                    $$\\sqrt{s} = 7$$\n                    \n                      \n                        \n                          s\n                        \n                        =\n                        7\n                      \n                    \n                   TeV using the ATLAS detector at the LHC. Eur. Phys. J. C 73, 2518 (2013). \n                    https:\/\/doi.org\/10.1140\/epjc\/s10052-013-2518-3\n                    \n                  \")\\], and\n- the occupancy in the CMS Hadron Forward Calorimeter (HFOC) \\[[16](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR16 \"C.M.S. Collaboration, Development of the cms detector for the cern lhc run 3. J. Instrum. 19(05), 05064 (2024). \n                    https:\/\/doi.org\/10.1088\/1748-0221\/19\/05\/p05064\n                    \n                  \")\\].\n\nIn both cases, raw luminometer counts are converted to collision rates using the Poisson formalism \\[[15](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR15 \"ATLAS Collaboration, Improved luminosity determination in pp collisions at \n                  $$\\sqrt{s} = 7$$\n                  \n                    \n                      \n                        s\n                      \n                      =\n                      7\n                    \n                  \n                 TeV using the ATLAS detector at the LHC. Eur. Phys. J. C 73, 2518 (2013). \n                  https:\/\/doi.org\/10.1140\/epjc\/s10052-013-2518-3\n                  \n                \"), [16](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR16 \"C.M.S. Collaboration, Development of the cms detector for the cern lhc run 3. J. Instrum. 19(05), 05064 (2024). \n                  https:\/\/doi.org\/10.1088\/1748-0221\/19\/05\/p05064\n                  \n                \")\\], and the instantaneous luminosity is averaged over typically 60-second time bins, that during the scans are synchronized with the scan steps. The statistical uncertainties are estimated from either the total number of raw luminometer counts per time bin (ATLAS), or from the RMS of the approximately 40 luminosity samples recorded in a given time bin (CMS). Since the absolute luminosity scale is irrelevant, and in order to simplify the interpretation of the results, all measurements in this paper are presented in terms of fractional shifts in the bunch-averaged specific luminosity, relative to a reference time that depends on the type of measurement considered.\n\n### 4\\.2 Bunch-charge measurement\nThe total intensity of each beam is measured by a direct-current current transformer (DCCT), the absolute scale of which is calibrated against a very high precision pulse generator \\[[17](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR17 \"C. Barschel, M. Ferro-Luzzi, J.-J. Gras, M. Ludwig, P. Odier, S. Thoulet, Results of the LHC DCCT calibration studies. Technical Report CERN-ATS-Note-2012-026 PERF (2012). \n                  https:\/\/cds.cern.ch\/record\/1425904?ln=en\n                  \n                \")\\]. The bunch-by-bunch charge fractions, in turn, normalized to the total stored intensity, are measured, separately for the two beams, by Fast Beam Current Transformers (FBCT) \\[[18](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR18 \"G. Anders et al., Study of the relative LHC bunch populations for luminosity calibration. Technical Report. CERN-ATS-Note-2012-028 PERF (2012). \n                  https:\/\/cds.cern.ch\/record\/1427726\n                  \n                \")\\]. The latter devices also provide the fill pattern, *i.e.* the relative location, at a given instant and around the two rings, of the nominally filled bunches. Since the absolute scale of the DCCTs is known to much better accuracy than that of the FBCTs, the bunch charges are typically computed as the product of the FBCT bunch-charge fractions and the total circulating beam intensity reported by the corresponding DCCT.\n\n### 4\\.3 Emittance measurement\nThe synchrotron-light beam-profile monitors \\[[19](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR19 \"G. Trad, Development and Optimisation of the SPS and LHC Beam Diagnostics Based on Synchrotron Radiation Monitors Presented 22 Jan 2015. Presented 22 Jan 2015. \n                  https:\/\/cds.cern.ch\/record\/2266055\n                  \n                \")\\], dubbed BSRTs (“Beam Synchrotron Radiation Telescope”), are mainly used to track emittance evolution over time, but they can also measure the relative changes in transverse single-beam size that result from beam–beam-induced \\\$\\\\beta \\\$\\-beating. BSRT data are recorded every second; in what follows, they are presented averaged over one-minute intervals for easier comparison with other measurements.\n\nThe absolute length scale of the BSRT profiles suffers from significant uncertainties, if only because the online optical corrections to the images can be updated only a few times per year, and are unable to track the evolution of the efficiency of the light sensors, that depends both on time and on the position of the light spot on the sensor array. For each beam therefore, the BSRT is complemented by a wire-scanner (WS) profile monitor that is located in the same straight section. The advantage of the WS is that its accuracy is significantly better than that of the BSRT; its down side, in the present context, is that the WS cannot acquire data continuously and must be triggered manually. For the results presented in this paper, the wire scanners were flown through the beams at the start of each group of measurements to provide single-beam emittance measurements that are as accurate as possible, and then at regular intervals thereafter. The BSRT is used to interpolate the time evolution of the emittance between two sets of WS measurements.\n\nTo interpret measured RMS bunch widths in terms of emittance requires the knowledge of the \\\$\\\\beta \\\$ functions at the locations of the BSRT and the WS. These are determined by the phase-advance method, and compared in Table [2](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Tab2) to their model value computed using MAD-X; the agreement is typically better than 5%, and the worst disagreement amounts to 9%. These discrepancies are attributed to imperfections in the magnetic lattice. Uncertainties in the measurement mainly arise from the interpolation of the lattice functions between the two closest beam-position monitors (BPMs) where the \\\$\\\\beta \\\$\\-functions are measured \\[[20](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR20 \"R. Tomás, O. Brüning, M. Giovannozzi, P. Hagen, M. Lamont, F. Schmidt, G. Vanbavinckhove, M. Aiba, R. Calaga, R. Miyamoto, Cern large hadron collider optics model, measurements, and corrections. Phys. Rev. ST Accel. Beams 13, 121004 (2010). \n                  https:\/\/doi.org\/10.1103\/PhysRevSTAB.13.121004\n                  \n                \"), [21](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR21 \"R. Tomás, M. Aiba, A. Franchi, U. Iriso, Review of linear optics measurement and correction for charged particle accelerators. Phys. Rev. Accel. Beams 20, 054801 (2017). \n                  https:\/\/doi.org\/10.1103\/PhysRevAccelBeams.20.054801\n                  \n                \")\\], and either the BSRT or the WS location. Based on the MAD-X LHC lattice model, these uncertainties are estimated not to exceed 3%, and this value is assigned as the systematic uncertainty on the \\\$\\\\beta \\\$ functions used in BSRT- and WS-based emittance measurements.\n\n**Table 2 \\\$\\\\beta \\\$ functions at the BSRT and WS locations, for the optical configuration detailed in Sect. [5\\.3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec14)**\n\n[Full size table](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/tables\/2)\n\nThe beam-averaged emittance, *i.e.* the average of the emittances of the beam-1 bunch and of the corresponding beam-2 bunch, can be obtained directly from emittance scans \\[[22](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR22 \"O. Karacheban, P. Tsrunchev, on behalf of CMS, Emittance scans for cms luminosity calibration. EPJ Web Conf. 201, 04001 (2019). \n                  https:\/\/doi.org\/10.1051\/epjconf\/201920104001\n                  \n                \")\\] at IP1 and\/or IP5. In this approach, the convolved transverse bunch sizes measured using beam-separation scans are translated into emittances using the \\\$\\\\beta ^\\*\\\$ values determined by *k*\\-modulation \\[[23](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR23 \"F. Carlier, R. Tomás, Accuracy and feasibility of the \n                  $${\\beta }^{*}$$\n                  \n                    \n                      \n                        β\n                      \n                      \n                      \n                        \n                        ∗\n                      \n                    \n                  \n                 measurement for lhc and high luminosity lhc using \n                  $$k$$\n                  \n                    k\n                  \n                 modulation. Phys. Rev. Accel. Beams 20, 011005 (2017). \n                  https:\/\/doi.org\/10.1103\/PhysRevAccelBeams.20.011005\n                  \n                \")\\] at the relevant IP (see Sect. [4\\.3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec7)). The associated uncertainty (Table [3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Tab3)) is combined with the statistical uncertainty in convolved transverse width to estimate the error affecting the measured emittance. Comparison with beam-averaged emittances extracted from the single-beam profile monitors reveals excellent agreement between WS and emittance-scan results (Fig. [2](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig2)). The relative time-evolution of the BSRT emittances is consistent with that observed using the WS or emittance scans, but the absolute magnitudes differ significantly, especially in the horizontal plane.\n\n**Table 3 \\\$\\\\beta ^\\*\\\$ functions at the ATLAS and CMS IPs measured by *k*\\-modulation. The target value in all cases is \\\$\\\\beta ^\\*=\\\$ 11 m**\n\n[Full size table](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/tables\/3)\n\n**Fig. 2**\n\n[![Fig. 2](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig2_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/2)\n\nTime evolution of the horizontal (top) and vertical (bottom) beam-averaged normalized emittances during LHC fill 8037, as reported by the BSRT (blue), the WS (orange), and using beam-separation scans at IP1 (red) and IP5 (purple). Different shades of the same color correspond to two different bunches present in this fill. For ATLAS emittance scans, only the bunch-averaged value is shown\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/2)\n### 4\\.4 Beam-beam parameter determination\nThe systematic and quantitative comparison of the measured and of the predicted beam-beam impact on the observables detailed in Sect. [2](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec2) requires continuous monitoring of the actual beam-beam parameter throughout the duration of the experiment. Only the BSRT provides uninterrupted emittance determination throughout the fill. In view of the scale biases apparent in Fig. [2](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig2), however, WS-based emittances are used, whenever possible, as input to the determination of the beam-beam parameter; BSRT emittances re-scaled to close-in-time, absolute WS measurements provide time-interpolated emittance values whenever WS data are unavailable.\n\nThe beam-beam parameter evolution during the first fill of the experiment is illustrated in Fig. [3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig3). The error bands are dominated by the 3% systematic uncertainty in the measured optical functions. The strength of the beam-beam interaction drops by a factor of two over a couple of hours, from the combined effect of beam-intensity decay and of emittance growth.\n\n**Fig. 3**\n\n[![Fig. 3](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig3_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/3)\n\nTime evolution of the beam- and plane-averaged beam-beam parameter \\\$\\\\xi \\\$ inferred from the measured bunch charges and emittances, separately for the two colliding-bunch pairs present in the fill pattern. The color bands indicate the systematic uncertainty (see text)\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/3)\n### 4\\.5 Tune measurements\nCoherent spectra can be monitored using either the LHC Transverse Damper (ADT) \\[[24](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR24 \"M. Söderén, J. Komppula, G. Kotzian, S. Rains, D. Valuch, ADT and Obsbox in LHC Run 2, plans for LS2, pp. 165–171 (2019)\")\\], or the Base-Band Tune (BBQ) \\[[25](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR25 \"A. Boccardi, M. Gasior, R. Jones, P. Karlsson, R. Steinhagen, First results from the lhc bbq tune and chromaticity systems. Technical report, CERN, Geneva (2009). \n                  https:\/\/cds.cern.ch\/record\/1156349\n                  \n                \")\\] system; their comparison typically yields consistent results. Spectrograms with a tune resolution of 0.0001 are computed using bunch positions recorded at every turn, and averaged over the one-minute time bins mentioned in Sect. [4\\.1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec5). To mitigate the influence of noise on the tune measurement, a median filter is applied at the pre-processing stage with a self-defined local window size. Additionally, the 50 Hz noise lines present in the spectra are masked. The filtered data are fitted by the sum of a Gaussian function and a constant baseline, and the frequency at the peak of the Gaussian is interpreted as the measured mean tune. The systematic uncertainty on measured tune shifts \\\$\\\\Delta Q\\\$ is estimated empirically to be \\\$\\\\sigma \\_{\\\\Delta Q}=0.001\\\$.\n\n### 4\\.6 IP-orbit monitoring\nOrbit displacements at each IP are measured using the Diode ORbit and OScillation (DOROS) BPM system, that provides sub-micrometer beam-position resolution \\[[26](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR26 \"M. Gasior, G. Baud, J. Olexa, G. Valentino, First Operational Experience with the LHC Diode ORbit and OScillation (DOROS) System, p. 07 (2017) \n                  https:\/\/doi.org\/10.18429\/JACoW-IBIC2016-MOPG07\n                  \n                \")\\]. These strip-line BPMs are located in the two quadrupoles on either side of and closest to the IP, allowing the position of both beams to be measured simultaneously. The position and the angle of each beam at the IP are inferred from, respectively, the average and the difference of the positions measured in the two final-triplet quadrupoles.\n\n## 5 Optimization of the phase advance between interaction points\nIn the two-IP configuration, the basic theory of linearized beam-beam \\\$\\\\beta \\\$\\-beating provides an analytical determination of the optimum phase advance between the two IPs, *i.e.* of the setting that maximizes the sensitivity, at the witness IP, to beam-beam effects at the scanning IP (Sect. [5\\.1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec12)). A generalization to the three- and four-IP configurations is discussed in Sect. [5\\.2](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec13). The required modifications to the baseline LHC optics, as well as their implementation and their experimental validation, are detailed in Sect. [5\\.3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec14).\n\n### 5\\.1 Linear beam-beam \\\$\\\\beta \\\$\\-beating\nIn circular colliders, the \\\$\\\\beta \\\$\\-beating induced by beam-beam effects at the IP(s) leads to changes in transverse beam size \\[[5](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR5 \"W. Herr, T. Pieloni, Beam-Beam Effects. Technical report, CERN (2014). Contribution to the CAS–CERN Accelerator School: Advanced Accelerator Physics Course, Trondheim, Norway, 18–29 Aug 2013, p. 29. \n                  https:\/\/doi.org\/10.5170\/CERN-2014-009.431\n                  \n                . \n                  https:\/\/cds.cern.ch\/record\/1982430\n                  \n                \"), [27](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR27 \"A.W. Chao, Coherent beam-beam effects, in Frontiers of Particle Beams: Intensity Limitations (Springer, Berlin, Heidelberg, 1992), pp. 363–414. \n                  https:\/\/doi.org\/10.1007\/3-540-55250-2_36\n                  \n                \")\\], thereby impacting the luminosity. For a single collision point, the change in \\\$\\\\beta ^\\*\\\$ of a single small-amplitude particle in (for instance) the horizontal plane *x*, is given by:\n\n\\$\\$\\\\begin{aligned} \\\\frac{\\\\beta \\_x^\\*}{\\\\beta \\_{0,x}^\\*} = \\\\frac{1}{\\\\sqrt{1-4\\\\pi \\\\xi \\\\cot {(2\\\\pi Q\\_x)} - 4\\\\pi ^2\\\\xi ^2}}. \\\\end{aligned}\\$\\$\n\n(6)\n\nThe magnitude of the effect depends on the absolute value \\\$\\\\xi \\\$ of the beam-beam parameter and on the nominal betatron tune \\\$Q\\_x\\\$; its periodicity is \\\$ \\[\\\\pi \\]\\\$. It can be shown that approaching the half-integer betatron tune from below minimizes the \\\$\\\\beta \\\$ function at the IP, thereby maximizing the luminosity; this phenomenon has been exploited with great success in (among others) the KEKB and PEP-II *B* factories. In the LHC, where during collisions the nominal fractional betatron tunes are fixed at \\\$q\\_x=0.31,\\\\,q\\_y=0.32\\\$, this dynamic-\\\$\\\\beta \\\$ effect depends only on the beam-beam parameter.\n\nThis conclusion no longer strictly holds in the presence of more than one collision point. In the case of two IPs with identical values of \\\$\\\\xi \\\$ and \\\$\\\\beta ^\\*\\\$, the dynamic-\\\$\\\\beta \\\$ effect can be described analytically \\[[28](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR28 \"J. Wańczyk, A. Dabrowski, D. Stickland, T. Pieloni, W. Kozanecki, X. Buffat, Impact of multiple beam-beam encounters on LHC absolute-luminosity calibrations by the van der Meer method. JACoW IPAC 2023, 053 (2023). \n                  https:\/\/doi.org\/10.18429\/JACoW-IPAC2023-WEPA053\n                  \n                \")\\]:\n\n\\$\\$\\\\begin{aligned} \\\\frac{\\\\beta \\_x^\\*}{\\\\beta ^\\*\\_{0,x}} = \\\\frac{\\\\sin {2\\\\pi Q\\_x} + 4\\\\pi \\\\xi (\\\\cos (2\\\\pi Q\\_x-2\\\\mu \\_{1,x})-\\\\cos 2\\\\pi Q\\_x )}{\\\\pm \\\\sqrt{1-\\\\cos ^2{2\\\\pi (Q\\_x+\\\\Delta Q\\_x)}}}, \\\\end{aligned}\\$\\$\n\n(7)\n\nwhere \\\$\\\\mu \\_{1,x}\\\$ is the phase advance from the scanning IP to the witness IP. The sign in the denominator is linked to that of the numerator so as to ensure that the ratio remains positive; the denominator must be positive (resp. negative) below (resp. above) the half-integer tune, *i.e.* when \\\$m\\< Q\\_x \\<m+\\\\frac{1}{2}\\\$ (resp. \\\$m+\\\\frac{1}{2}\\< Q\\_x \\< m+1\\\$), where *m* is a positive integer. The tune shift \\\$\\\\Delta Q\\_x\\\$ is related to \\\$Q\\_x\\\$, \\\$\\\\xi \\\$ and \\\$\\\\mu \\_{1,x}\\\$ by:\n\n\\$\\$\\\\begin{aligned} \\\\cos {2\\\\pi (Q\\_x+\\\\Delta Q\\_x)}&=(1-16\\\\pi ^2\\\\xi ^2)\\\\cos {2\\\\pi Q\\_x} + 8\\\\pi \\\\xi \\\\sin {2\\\\pi Q\\_x} \\\\nonumber \\\\\\\\&\\\\quad + 16\\\\pi ^2\\\\xi ^2\\\\cos {(2\\\\pi Q\\_x-2\\\\mu \\_{1,x})}. \\\\end{aligned}\\$\\$\n\n(8)\n\nThe phase-advance dependence of the resulting \\\$\\\\beta \\\$\\-beating is illustrated in Fig. [4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig4). The minimum of each curve corresponds to the phase advance that yields the largest mutual beam–beam-induced luminosity enhancement between the two IPs. The optimal settings are \\\$\\\\mu \\_{1,x}^{min}\/2\\\\pi =0.405,\\\\,\\\\mu \\_{1,y}^{min}\/2\\\\pi =0.410\\\$; their difference reflects that between the nominal horizontal and vertical fractional tunes \\\$q\\_x\\\$ and \\\$q\\_y\\\$.\n\n**Fig. 4**\n\n[![Fig. 4](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig4_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/4)\n\nAnalytically computed beam-beam induced \\\$\\\\beta \\\$\\-beating in a two-IP configuration, as a function of the phase advance \\\$ \\\\mu \\_1\\\$ between the 2 IPs (\\\$q\\_x=0.31\\\$, \\\$q\\_y=0.32\\\$, head-on collisions at both IPs, \\\$\\\\xi =7\\\\times 10^{-3}\\\$ per IP)\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/4)\n\nThe extrema of the curves in Fig. [4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig4) can be determined analytically by differentiating Eq. ([7](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Equ7)) with respect to the phase advance \\\$\\\\mu \\_{1,x}\\\$. The general expression for this derivative \\\$\\\\frac{d}{d\\\\mu \\_{1,x}} \\\\biggl ( \\\\frac{\\\\beta \\_x^\\*}{\\\\beta ^\\*\\_{0,x}} \\\\biggr )\\\$, detailed in Appendix A, is the product of two factors, each of which can be zero:\n\n- imposing that \\\$\\\\sin {(2\\\\pi Q\\_x - 2\\\\mu \\_{1,x})} = 0\\\$ yields the solution\n  \\$\\$\\\\begin{aligned} \\\\frac{\\\\mu \\_{1,x}}{2\\\\pi } =\\\\frac{Q\\_x}{2}-\\\\frac{m}{4} \\\\, \\\\end{aligned}\\$\\$\n  \n  (9)\n  where *m* is an integer of either sign, or zero;\n- the other factor is zero for either\n  \\$\\$\\\\begin{aligned} \\\\frac{\\\\mu \\_{1,x}}{2\\\\pi } = \\\\frac{Q\\_x}{2} - A(\\\\xi , Q\\_x)\\\\, \\\\end{aligned}\\$\\$\n  \n  (10)\n  or\n  \\$\\$\\\\begin{aligned} \\\\frac{\\\\mu \\_{1,x}}{2\\\\pi } = \\\\frac{Q\\_x - 1}{2} + A(\\\\xi , Q\\_x) \\\\, \\\\end{aligned}\\$\\$\n  \n  (11)\n  with \\\$A(\\\\xi , Q\\_x)\\\$ defined in Eq. ([12](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Equ12))\n\n\\$\\$\\\\begin{aligned} A(\\\\xi , Q\\_x) = \\\\arccos \\\\left( \\\\frac{ 32\\\\pi ^3\\\\xi ^3 \\\\sin (4\\\\pi Q\\_x) + 48\\\\pi ^2\\\\xi ^2 \\\\cos ^2(2\\\\pi Q\\_x) - 32\\\\pi ^2\\\\xi ^2 - 6\\\\pi \\\\xi \\\\sin (4\\\\pi Q\\_x) + \\\\cos ^2(2\\\\pi Q\\_x) - 1}{16\\\\pi ^2\\\\xi ^2 \\\\left( 4\\\\pi \\\\xi \\\\sin (2\\\\pi Q\\_x) + \\\\cos (2\\\\pi Q\\_x) \\\\right) } \\\\right) \/ (4\\\\pi ) \\\\end{aligned}\\$\\$\n\n(12)\n\nThe two sets of solutions do not overlap. The subset of solutions that minimize (rather than maximize) \\\$\\\\beta ^\\*\\\$ is identified by requiring that the second derivative be positive. This restricts the main solution to:\n\n\\$\\$\\\\begin{aligned} \\\\frac{\\\\mu \\_{1,x}}{2\\\\pi } =\\\\frac{Q\\_x}{2}+\\\\frac{m+1}{4} . \\\\end{aligned}\\$\\$\n\n(13)\n\nThe optimal phase \\\$\\\\mu \\_{x,1}\\\$ that minimizes the \\\$\\\\beta ^\\*\\\$ for range of \\\$(\\\\xi , Q\\_x)\\\$ values is shown in Fig. [5](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig5).\n\n**Fig. 5**\n\n[![Fig. 5](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig5_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/5)\n\nOptimal phase advance value \\\$\\\\mu \\_{x,1}\\\$ for various values of \\\$(\\\\xi , Q\\_x)\\\$ in the two-IPs configuration\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/5)\n\nThe phase-advance dependence of the dynamic-\\\$\\\\beta \\\$ effect can also be quantified in terms of the luminosity shift, at the witness IP, that is associated with the electromagnetic interaction of the two bunches at the scanning IP (Fig. [6](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig6)). The optimal setting, *i.e.* that which maximizes the sensitivity, at the witness IP, to beam-beam effects at the scanning IP, is indicated by the green dotted vertical line at \\\$\\\\Delta \\\\mu \\_{\\\\text {IP1-IP5}}=0.41\\\\ \\[2\\\\pi \\]\\\$; the full suppression of the phase-related luminosity enhancement is indicated by the red dotted line. These results are consistent with the predictions of the analytical model (Fig. [4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig4)). In the two-IP configuration considered here, in which IP1 and IP5 can both play the role of either the scanning or the witness IP, and due to the periodicity of this curve (\\\$ \\\\pi \\\$) and to the fractional-tune values used during collisions (\\\$q\\_x \/ q\\_y = 0.31 \/ 0.32)\\\$, the phase–advance-related luminosity enhancement is the same at the two IPs; at the optimum setting, it is three times larger than it would be using the nominal LHC lattice. The maximum effect depends mainly on the transverse tunes, and thus could have been further enhanced if it had been possible to move these away from their nominal values.\n\nA more comprehensive simulation study, that evaluates the impact of the beam-beam interaction on the luminosity for more realistic Gaussian proton-density distributions, multiple collision points and as a function of the beam separation, is reported in Ref. \\[[29](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR29 \"J. Wańczyk, Precision luminosity measurement at hadron colliders. PhD Thesis, École Polytechnique Fédérale de Lausanne (2024). \n                  https:\/\/doi.org\/10.5075\/epfl-thesis-10500\n                  \n                \")\\].\n\n**Fig. 6**\n\n[![Fig. 6](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig6_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/6)\n\nBeam–beam-induced luminosity shift at IP5 predicted by COMBI simulations, as a function of the phase advance \\\$ \\\\mu \\_1=\\\\Delta \\\\mu \\_{\\\\text {IP1-IP5}}\\\$ between the two IPs (\\\$q\\_x=0.31\\\$, \\\$q\\_y=0.32\\\$, head-on collisions at both IPs, \\\$\\\\xi =7\\\\times 10^{-3}\\\$ per IP)\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/6)\n### 5\\.2 Optimized phases for multi-IP configurations\nThe general expression to obtain the first-order change of the \\\$\\\\beta \\\$ function at an arbitrarily chosen reference IP (at \\\$s=\\\\mu \\_{0,x}=0\\\$) can be approximated as multiple quadrupole errors \\[[30](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR30 \"T. Pieloni, J. Barranco, X. Buffat, P.C. Jorge, L.M. Medrano, C. Tambasco, R. Tomás, Dynamic beta and beta-beating effects in the presence of the beam-beam interactions in 57th ICFA Advanced Beam Dynamics Workshop on High-Intensity and High-Brightness Hadron Beams, p. 027 (2016). \n                  https:\/\/doi.org\/10.18429\/JACoW-HB2016-MOPR027\n                  \n                \")\\]:\n\n\\$\\$\\\\begin{aligned} \\\\frac{\\\\beta ^\\*\\_x}{\\\\beta ^\\*\\_{0,x}} = \\\\frac{2\\\\pi \\\\xi }{\\\\sin {2\\\\pi Q\\_x}} \\\\sum \\_{i\\\\in IPs}^{N-1} \\\\cos (2\\\\pi Q\\_x-2\\\\mu \\_{i,x})\\\\, \\\\end{aligned}\\$\\$\n\n(14)\n\nwhere \\\$\\\\mu \\_{i,x}\\\$ is the horizontal phase advance between the reference and the \\\$i^{th}\\\$ IP for a total of *N* IPs. Therefore, the optimal configuration, which minimizes the \\\$\\\\beta ^\\*\\\$ change at all IPs, must satisfy the condition:\n\n\\$\\$\\\\begin{aligned} \\\\sum \\_{i=1}^{N-1} \\\\cos (2\\\\pi Q\\_x-2\\\\mu \\_{i,x}) = -(N-1)\\\\, \\\\end{aligned}\\$\\$\n\n(15)\n\nwhich yields the same solution for all IPs as in the 2 IPs configuration (Eq. [13](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Equ13)):\n\n\\$\\$\\\\begin{aligned} \\\\frac{\\\\mu \\_{i,x}}{2\\\\pi } =\\\\frac{Q\\_x}{2}+\\\\frac{n+1}{4} . \\\\end{aligned}\\$\\$\n\n(16)\n\nwhere *n* is an integer of either sign, or zero. To maximize the effect on luminosity, the phase advance between the reference interaction point at \\\$ i = 0 \\\$ and the nearest IP, at \\\$ i = 1 \\\$ must satisfy \\\$ \\\\mu \\_{1,x}\/2\\\\pi = (Q\\_x + 0.5)\/2 \\\$, as given by the first solution (for \\\$n=0\\\$) of Eq. ([13](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Equ13)). This condition implies that the phase advance to any subsequent adjacent IPs (\\\$ i \\\\ge 2 \\\$) must be an integer multiple of \\\$ \\\\pi \\\$. Thus, this requirement exposes a fundamental constraint: it is not possible to minimize \\\$ \\\\beta ^\\* \\\$ simultaneously at all IPs. This would only be possible for (\\\$ Q\\_x = 0.5 \\\$) which is practically not achievable, as it would require operating the storage ring precisely at the half-integer resonance.\n\n### 5\\.3 Optics validation\nThe optimum phase advance from IP1 to IP5 determined in Sect. [5\\.1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec12) was implemented using tuning trim quadrupoles in the LHC arcs. The targeted fractional phase advance was \\\$0.9 \\\\, \\[2 \\\\pi \\]\\\$ for both beams and both planes (Table [4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Tab4)). This setting, which differs from that in Fig. [6](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig6) by one period (\\\$ \\\\pi \\\$), is equivalent in terms of sensitivity optimization but has the advantage that it minimizes the absolute magnitude of the perturbations inflicted upon the nominal lattice.\n\nSeveral measurements were carried out to validate this adjustment: comparison of the targeted and achieved shifts in phase advance (Sect. [5\\.3.1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec15)), \\\$\\\\beta \\\$\\-function measurements across the full LHC circumference (Sect. [5\\.3.2](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec16)) as well as at the IPs (Sect. [5\\.3.3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec17)), and dispersion measurements to estimate the dispersive contribution to the transverse beam size at the IP (Sect. [5\\.3.4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec18)).\n\n#### 5\\.3.1 Phase advance\nThe phase advance around each LHC ring is determined using turn-by-turn orbit measurements \\[[20](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR20 \"R. Tomás, O. Brüning, M. Giovannozzi, P. Hagen, M. Lamont, F. Schmidt, G. Vanbavinckhove, M. Aiba, R. Calaga, R. Miyamoto, Cern large hadron collider optics model, measurements, and corrections. Phys. Rev. ST Accel. Beams 13, 121004 (2010). \n                  https:\/\/doi.org\/10.1103\/PhysRevSTAB.13.121004\n                  \n                \"), [21](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR21 \"R. Tomás, M. Aiba, A. Franchi, U. Iriso, Review of linear optics measurement and correction for charged particle accelerators. Phys. Rev. Accel. Beams 20, 054801 (2017). \n                  https:\/\/doi.org\/10.1103\/PhysRevAccelBeams.20.054801\n                  \n                \")\\]. The targeted and achieved phase-advance shifts agree within \\\$0.01\\\\, \\[2\\\\pi \\]\\\$ or better (Table [4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Tab4)). The phase advance change from IP1 and IP5 was compensated on the opposite side of the ring, ensuring that the transverse tunes remain unchanged. The target fractional local phase advance is set to \\\$0.9,\\[2\\\\pi \\]\\\$. Table [4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Tab4) lists the corresponding phase shifts applied to the nominal lattice to reach this target on one side of the ring; this is consistent with the analytically derived value of \\\$0.41\\\\,\\[2\\\\pi \\]\\\$ for the compensating shift on the opposite side. The change in phase advance was also measured at the BSRT location, where the discrepancy between the model and measurements is more pronounced, reaching up to \\\$0.05\\\\,\\[2\\\\pi \\]\\\$.\n\n**Table 4 Nominal and optimized IP1 to IP5 phase-advance settings. Column 4 (resp. 5) displays the targeted (resp. measured) shift in phase advance after the optimized settings have been applied**\n\n[Full size table](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/tables\/4)\n#### 5\\.3.2 Residual \\\$\\\\beta \\\$\\-beating\nThe \\\$\\\\beta \\\$\\-beating was quantified and found to be well within acceptable tolerances for various configurations of the trim strength. The measurement results, relative to the reference measurements without any phase advance adjustments, are illustrated in Fig. [7](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig7). The maximal observed deviation is approximately 25% around IP5 for Beam 1 in the vertical plane. Such significant deviations in proximity to the IPs are anticipated and are frequently attributed to erroneous BPM readings. Overall, the measurements indicated that \\\$\\\\beta \\\$\\-beating remained within acceptable bounds, predominantly within a 10% margin.\n\n**Fig. 7**\n\n[![Fig. 7](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig7_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/7)\n\nMeasured \\\$\\\\beta \\\$\\-function change from the full strength of the phase advance knob with reference to nominal lattice along the LHC ring, for Beam 1 (top two figures for *x* and *y*) and Beam 2 (bottom two figures for *x* and *y*)\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/7)\n\nThe measured \\\$\\\\beta \\\$ functions are also compared to the MADX LHC lattice model predictions, which include the phase advance adjustment. The results, illustrated along the LHC lattice in Fig. [8](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig8), indicate that the measurements align with the model within an acceptable margin of 10%.\n\n**Fig. 8**\n\n[![Fig. 8](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig8_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/8)\n\nMeasured deviations in the \\\$\\\\beta \\\$\\-function along the LHC ring relative to the MADX model incorporating the phase advance knob, for Beam 1 (top two figures for *x* and *y*) and Beam 2 (bottom two figures for *x* and *y*). The error bars represent solely the statistical uncertainties as determined in the measurements\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/8)\n#### 5\\.3.3  \\\$\\\\beta \\\$ functions at the interaction points\nThe DOROS BPMs are strategically located at the end of the inner triplets on both sides of each of the experiments. Due to the phase advance of \\\$\\\\pi \\\$ between the DOROS on the left and right sides, these are inadequate for the precise determination of the measurement of \\\$\\\\beta ^\\*\\\$ at the IP. This can only be achieved with a special *k*\\-modulation technique \\[[31](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR31 \"F. Carlier, et al. Challenges of K-modulation measurements in the LHC Run 3, in Proceedings of IPAC’23. IPAC’23—14th International Particle Accelerator Conference (JACoW Publishing, Geneva, Venice, 2023), pp. 531–534. \n                  https:\/\/doi.org\/10.18429\/JACoW-IPAC2023-MOPL014\n                  \n                \")\\], which was performed at the interaction points used in the experiment. The results are summarized in Table [3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Tab3). While the achieved precision of the \\\$\\\\beta ^\\*\\\$ determination varies between measurements, uncertainties at the level of \\\${\\\\mathcal {O}}(1\\\\%)\\\$ are not uncommon and are not systematic to a specific beam or interaction point. Such uncertainties would propagate to only a few-percent effect on derived quantities such as the emittance or the beam–beam parameter.\n\n#### 5\\.3.4 Dispersion\nIt was essential to perform dispersion \\\$D^\\*\\\$ measurements at the interaction points, under the new optics configurations, as it can contribute to the measured beam size. Assuming the LHC design momentum spread value of \\\$\\\\frac{\\\\Delta p}{p}=10^{-4}\\\$, the average dispersion measurement around IP1 and IP5 is estimated to contribute up to 2% to the observed bunch sizes (worst case measured for B2Y). This effect is considered static throughout the experiment.\n\n## 6 Experimental results\nThis section presents experimental measurements of beam–beam effects in configurations involving multiple collision points. The impact of the optics modifications introduced in Sect. [5](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec11) is first assessed through tune spectra measurements, as detailed in Sect. [6\\.1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec20). To quantify the cumulative effect of one or more additional collisions and assess the reproducibility of the results, a stepwise collision scan was employed. In this procedure, beams were sequentially brought into head-on collision at selected interaction points (IPs), followed by full transverse separation (\\\$\\\\Delta \\> 6\\\\sigma \\\$) in either the horizontal or vertical plane at all but one witness IP. Two distinct filling schemes were used to isolate the influence of a single additional collision (2-IP configuration: ATLAS and CMS, see Sect. [6\\.2](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec21)) and to investigate more complex multi-collision scenarios (3-IP configuration: including ALICE, see Sect. [6\\.3](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec25)). The first measurement of separation-dependent beam–beam effects, analogous to those encountered during luminosity calibration scans, is presented in Sect. [6\\.4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec28), alongside comparisons with numerical predictions. An additional measurement conducted during the 2022 van der Meer calibration is detailed in Sect. [6\\.5](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec32). A consolidated overview of all findings is provided in Sect. [7](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec33).\n\n### 6\\.1 Measured effect on the tune spectra of the phase advance changes\nThe coherent transverse tune spectra serve as a diagnostic for evaluating the effectiveness of the phase advance modifications. COMBI simulations were carried out for two extreme lattice configurations, one designed to maximize and the other to suppress beam–beam effects. In the head-on scenario (Fig. [9](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig9)), the coherent modes are suppressed for the maximizing configuration, as the phase advance between the two collision points is close to \\\$\\\\pi \\\$, as was previously shown in simulation in Ref. \\[[32](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR32 \"Y. Alexahin, A study of the coherent beam-beam effect in the framework of the Vlasov perturbation theory. Nucl. Instrum. Methods Phys. Res., Sect. A 480(1–3), 253–288 (2002). \n                  https:\/\/doi.org\/10.1016\/S0168-9002(01)01219-0\n                  \n                \")\\]. In this configuration, the effect of the first collision is counteracted by the second. This setting is particularly suitable for studying incoherent spectra, as the diminished coherent signal allows for a clearer extraction of the mean tune shift from fitted spectra. During separation scans, the symmetry between horizontal (*x*) and vertical (*y*) planes is broken. In the scanning plane (horizontal), coherent modes re-emerge at \\\$1.5\\\\,\\\\sigma \\\$ separation (Fig. [10](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig10)), with their frequency shifting in response to the separation distance (additional example for \\\$3\\\\,\\\\sigma \\\$ is shown in Fig. [11](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig11)).\n\n**Fig. 9**\n\n[![Fig. 9](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig9_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/9)\n\nComparison of COMBI simulated tune spectra in *x*\\-plane (left) and *y*\\-plane (right) for the suppressing (in blue) and maximizing (in green) phase configurations, for head-on collision\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/9)\n\n**Fig. 10**\n\n[![Fig. 10](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig10_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/10)\n\nComparison of COMBI simulated tune spectra in *x*\\-plane (left) and *y*\\-plane (right) for the suppressing (in blue) and maximizing (in green) phase configurations, at \\\$1.5\\\\,\\\\sigma \\\$ separation in the horizontal plane\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/10)\n\n**Fig. 11**\n\n[![Fig. 11](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig11_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/11)\n\nComparison of COMBI simulated tune spectra in *x*\\-plane (left) and *y*\\-plane (right) for the suppressing (in blue) and maximizing (in green) phase configurations, at \\\$3\\\\,\\\\sigma \\\$ separation in the horizontal plane\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/11)\n\nThese simulated trends of the coherent spectra are also observed in experimental data. Transverse spectra recorded for head-on conditions confirm the suppression of coherent modes in the maximizing phase configuration (Fig. [12](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig12)). Additional comparisons for separated beams at the IP (see Figs. [13](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig13) and [14](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig14)), shown here for the case of vertical separation, reveal similar features. In this configuration, the non-scanning (horizontal) plane exhibits spectra that qualitatively agree with simulation predictions (see Figs. [10](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig10) and [11](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig11)), while in the scanning (vertical) plane the coherent modes do not reappear as expected. An equivalent behavior is observed when the separation is applied in the horizontal plane, with the roles of the two transverse planes interchanged; this case is not shown for brevity. This discrepancy is attributed to the effect of the transverse damper, which acts independently on the two beam centroids and suppresses coherent oscillations arising from beam–beam coupling, consistent with earlier observations \\[[33](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR33 \"R. Calaga, W. Herr, G. Papotti, T. Pieloni, X. Buffat, S. White, R. Giachino, Coherent beam-beam mode in the LHC. CERN (2014). \n                  https:\/\/doi.org\/10.5170\/CERN-2014-004.227\n                  \n                . \n                  http:\/\/cds.cern.ch\/record\/1957039\n                  \n                \")\\].\n\n**Fig. 12**\n\n[![Fig. 12](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig12_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/12)\n\nMeasured tune spectra before and after phase change, corresponding to the suppressing (in blue) and maximizing (in green) phase configurations, at the head-on position, for each of the transverse planes (Beam 1 shown as an example). Dash-dotted lines show Gaussian fits to the data\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/12)\n\n**Fig. 13**\n\n[![Fig. 13](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig13_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/13)\n\nComparison of the measured tune spectra at suppressing (blue) and maximizing (green) phase configurations with \\\$1.3\\\\,\\\\sigma \\\$ separation in the vertical plane\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/13)\n\n**Fig. 14**\n\n[![Fig. 14](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig14_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/14)\n\nComparison of the measured tune spectra at suppressing (blue) and maximizing (green) phase configurations with \\\$2.6\\\\,\\\\sigma \\\$ separation in the vertical plane\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/14)\n### 6\\.2 Head-on collisions in a two-IP configuration at \\\$\\\\sqrt{s} = 900\\\\ \\\\text {GeV}\\\$\nEach test consisted of four measurements designed to quantify the beam–beam effect induced by a single additional head-on collision. The sequence was executed with CMS as the witness IP, followed by ATLAS, and concluded with a repeat at CMS. Over the course of the fill, the beam–beam parameter decreased from approximately 0.01 to 0.005. The outcome of each scan is analyzed using three independent observables: luminosity, beam size, and tune spectra, discussed separately below.\n\n#### 6\\.2.1 Impact of beam-beam effects on head-on luminosity\nAn overview of the luminosity measurements during the first test with CMS as the witness IP is shown in Fig. [15](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig15), where the specific luminosity at CMS is normalized to the initial head-on step to highlight relative changes. The uncertainty at each step is given by the standard error of the mean; shaded bands represent the standard deviation. A comparable dataset for ATLAS is presented in Fig. [16](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig16), confirming the expected luminosity change due to beam–beam effects originating from another IP. Relative luminosity changes for each step in all tests are summarized in Table [5](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Tab5). These values are computed relative to the average of adjacent head-on points, which removes the linear luminosity decay over time. Assuming slow change of beam parameters, the results within each test are expected to be consistent.\n\n**Fig. 15**\n\n[![Fig. 15](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig15_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/15)\n\nCMS specific luminosity, normalized to the first measured both IPs head-on point (blue), during the first step-function scan at IP1. The measurements with ATLAS fully separated are shown in red. The error bars indicate error on the mean\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/15)\n\n**Fig. 16**\n\n[![Fig. 16](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig16_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/16)\n\nATLAS specific luminosity, normalized to the first measured both IPs head-on point (blue), during the step-function scan at IP5. The points for which IP5 was separated are shown in red. The error bars indicate the statistical error\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/16)\n\n**Table 5 Measured luminosity shifts in each of full separation steps – summary of all three step-function scans performed for IP1 and IP5 as the witness IP. The beam-beam parameter at the start of each scan \\\$\\\\xi \\_\\\\text {start}\\\$ is included for reference**\n\n[Full size table](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/tables\/5)\n\nAll measurements are compiled in Fig. [17](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig17) and compared to COMBI simulation predictions. The results are plotted as a function of the beam–beam parameter (estimated as described in Sect. [4\\.4](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec8)), following a reverse chronological order. Agreement with simulation is excellent across the entire range.\n\n**Fig. 17**\n\n[![Fig. 17](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig17_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/17)\n\nCOMBI simulated luminosity enhancement induced by a single head-on collision as a function of the beam-beam parameter (dashed line), compared to the test results at both ATLAS (red points) and CMS (green points)\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/17)\n\n**Fig. 18**\n\n[![Fig. 18](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig18_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/18)\n\nMeasurements of beam-beam effects on beam widths shown with red points for each beam and plane, during the experiment with IP5 as the witness IP at \\\$\\\\xi =0.0056\\\$. COMBI predictions are shown in blue with calculated systematic error from the phase advance error between the IP and BSRT locations\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/18)\n#### 6\\.2.2 Impact of beam-beam effects on single-beam sizes\nComplementary measurements of the bunch size from the BSRT are shown in Fig. [18](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig18). Data from all steps and bunches are combined, and a linear trend corresponding to continuous emittance growth is subtracted using the difference between the first and last head-on steps. For comparison, COMBI simulation results are also included in the figure. The dominant source of uncertainty in COMBI predictions arises from discrepancies between the measured phase advance and the MADX model, with the largest observed deviation being \\\$0.05\\\\,\\[2\\\\pi \\]\\\$ (as discussed in Sect. [5\\.3.1](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Sec15)). This uncertainty is propagated using a modified form of Eq. ([6](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Equ6)) adapted to the BSRT location. The magnitude of this uncertainty depends on the specific phase advance configuration, as some are more sensitive to small deviations. This is reflected in the varying sizes of the error bands associated with each prediction. The most accurate predictions are obtained in the vertical plane of Beam 2, where the beam–beam–induced \\\$\\\\beta \\\$\\-beating propagates to the BSRT with a particularly favorable phase advance. In this configuration, the measured effect is also the most pronounced, and excellent agreement with COMBI simulations is observed. In other cases, the expected absolute effect is smaller, typically below 1%, and the influence of phase advance uncertainty becomes more significant, limiting the predictive reliability of the simulations. Experimentally, it is also possible that in these configurations the beam–beam–induced effect is too subtle to resolve, potentially masked by instrumental limitations.\n\n#### 6\\.2.3 Impact of beam-beam effects on tune spectra\nFigure [19](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig19) shows representative ADT spectra recorded during the step-function scans. These spectra are affected by coupling, noise, and various nonlinearities, so they do not follow an idealized analytical distribution. Nonetheless, the peak position can be extracted using a Gaussian fit (indicated by dashed lines), yielding an estimate of the mean tune shift. In the single-collision configuration (blue lines), unsuppressed coherent modes dominate the spectrum and partially obscure the underlying incoherent component. The Gaussian fit is therefore applied to the incoherent contribution between coherent lines, rather than to the global spectral maximum, which results in a larger uncertainty on the extracted peak position. Despite this, in each of the performed step-scans at both collision points, the single-collision tune shift, defined as twice the fitted mean shift, is found to be consistent within \\\$\\\\pm 5\\\\%\\\$ with the expected coherent beam–beam tune shift corresponding to the conditions of the respective scan. For head-on collisions with Gaussian transverse beam distributions, the coherent tune shift of the dipole mode is expected to be \\\$Y\\\\xi \\\$, where *Y* is the Yokoya factor (\\\$Y = 1.21\\\$ for round beams) \\[[34](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR34 \"K. Yokoya, Y. Funakoshi, E. Kikutani, H. Koiso, J. Urakawa, Tune shift of coherent beam-beam oscillations. Part. Accel. 27, 181–186 (1990)\")\\]. As expected, the two-collision configuration yields a tune shift approximately double that of the single-collision case \\[[13](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR13 \"A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). \n                  https:\/\/doi.org\/10.48550\/arXiv.2306.10394\n                  \n                \")\\].\n\n**Fig. 19**\n\n[![Fig. 19](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig19_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/19)\n\nMeasured vertical tune spectra for different collision configurations: reference nominal tune (red), without the presence of any collisions, compared to single collision (blue) and two collisions (green) at \\\$\\\\xi \\\\simeq 0.0086\\\$. The dash-dotted lines in corresponding colors indicate Gaussian fits to the data used to extract mean tunes\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/19)\n\n**Fig. 20**\n\n[![Fig. 20](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig20_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/20)\n\nSpecific luminosity, normalized to the first head-on point, measured at ATLAS (top) and CMS (bottom) during beam separation at the other IPs, with horizontal separation applied in steps 2–4 and vertical separation in steps 6–8. The tested configurations are indicated in the legend. The blue line shows an interpolation of the head-on measurements, serving as a reference for relative luminosity changes\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/20)\n### 6\\.3 Head-on collisions in a three-IP configuration at \\\$\\\\sqrt{s} = 900\\\\ \\\\text {GeV}\\\$\nAn additional head-on collision was included in the configuration using IP2 (ALICE), as it was the only IP where this was feasible. A specially designed filling scheme ensured uniform collision exposure across all bunches, allowing measurements to be combined and thereby reducing statistical uncertainties. However, no reliable luminosity data were available from IP2 during this experiment.\n\n**Table 6 Luminosity shift per full separation step, calculated with respect to an interpolated reference obtained from the two closest head-on points. Results for horizontal and vertical separations are shown as Shift part 1 and Shift part 2, respectively**\n\n[Full size table](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/tables\/6)\n#### 6\\.3.1 Impact of beam-beam effects on head-on luminosity\nA step-function scan was conducted by sequentially collapsing or separating one IP at a time, with the separation applied first in the horizontal plane and, in a second stage of the experiment, in the vertical plane. A comprehensive summary of the measured specific luminosity at ATLAS and CMS is presented in Fig. [20](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig20). Exceptionally stable beam conditions during this test enabled measurements at both experiments under nearly identical beam–beam parameters. The beam–beam-induced luminosity changes observed during each step are quantified in Table [6](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Tab6), referenced to the time-interpolated specific luminosity from three head-on points.\n\nOrbit-drift corrections are applied only to these head-on reference points and result in changes of the extracted luminosity shifts of at most 0.2%. Good agreement with COMBI simulations is observed across the separation steps, with measured shifts closely matching the simulated values for configurations involving IP1 and IP5. For single-IP separation, the level of agreement depends on the witness IP and the separation step. When IP1 is used as the witness, the luminosity shift in the first separation step is reproduced almost exactly by COMBI, while a smaller shift is observed in the second step. This behaviour is expected, as the beam–beam parameter decreases during the scan and the measurement becomes increasingly sensitive to residual orbit perturbations. For configurations involving IP2, the comparison is less conclusive. As no reliable luminosity data were available from IP2 during this experiment, it was not possible to precisely optimize the collision conditions at this IP. As a result, the actual beam–beam conditions at IP2 may differ from those assumed in the simulations, affecting the comparison. Residual orbit drifts affecting the separation steps cannot be fully corrected. As illustrated by the DOROS measurements in Fig. [21](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig21), the orbit data contain both slow drifts and beam–beam–induced orbit offsets arising from changes in the collision configuration when other IPs are separated. While the beam–beam–induced offsets are included in the COMBI simulations, the residual orbit drifts are not. Since these contributions cannot be disentangled in the measurements, a full correction is not possible. Consequently, the effective beam–beam impact measured during the experiment may be weaker than that assumed in the simulations.\n\n**Fig. 21**\n\n[![Fig. 21](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig21_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/21)\n\nResidual separation between the two beams at all three IPs in horizontal (top) and vertical (bottom) planes, for test with observations at ATLAS, where the separation was performed first in the horizontal and then in the vertical plane at the other IPs\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/21)\n#### 6\\.3.2 Tune shift scaling with the number of beam–beam interactions\nAnalysis of the transverse tune spectra in this configuration is further complicated by the reappearance of coherent modes when beams are separated at one or more IPs, as predicted by simulation. In such cases, the maximum tune shift is determined from the frequency difference between the \\\$\\\\pi \\\$ and \\\$\\\\sigma \\\$ modes, rather than from fitting the incoherent peak. For configurations where all three IPs were in collision, coherent modes remained suppressed due to the retained symmetry between IP1 and IP5. Figure [22](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig22) presents the measured tune shifts for all tested collision configurations. The unperturbed tune measurement, shown at ‘0’ collisions, serves as the reference from which all tune shifts are calculated. An empirical uncertainty of 0.001 is assumed for each independent measurement. Both transverse planes and separation directions are included to demonstrate consistency. As expected, the measured tune shift scales linearly with the number of head-on collisions.\n\n**Fig. 22**\n\n[![Fig. 22](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig22_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/22)\n\nMeasured mean tune shift as a function of the number of head-on collisions, shown for both transverse planes and for both separation directions (horizontal and vertical) used in the experiment. The separation refers to how collisions were removed: it was applied in only one transverse direction at a time (H sep. or V sep.) at the non-colliding interaction points, which are not explicitly labeled on the x-axis\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/22)\n### 6\\.4 Beam-beam signatures during transverse-separation scans at \\\$\\\\sqrt{s} = 900\\\\ \\\\text {GeV}\\\$\nIn the van der Meer (vdM) calibration method, beam–beam corrections are applied separately at each separation step. Therefore, it is essential to validate the simulated dependence of luminosity shifts on nominal beam separation. Full vdM-like scans were performed in both transverse planes at ATLAS and CMS, using a configuration where only these two IPs were in collision.\n\n#### 6\\.4.1 Impact of beam–beam effects on the luminosity scan-curves\nFigure [23](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig23) shows the luminosity curves measured at ATLAS, acting as the witness IP, during a separation scan performed at CMS. The curves are fitted with Gaussian profiles for visual guidance. The beam separation at CMS is expressed in units of the measured transverse beam size, \\\$\\\\sigma \\_{\\\\textrm{meas}}\\\$, which was found to be smaller than the nominal beam size, \\\$\\\\sigma \\_{\\\\textrm{nom}}\\\$, used to define the scan steps. As a result, scan points corresponding to nominal separations beyond \\\$6\\\\,\\\\sigma \\_{\\\\textrm{nom}}\\\$ extend to larger values when expressed in units of \\\$\\\\sigma \\_{\\\\textrm{meas}}\\\$.\n\nTo account for emittance evolution during the scan, both linear and exponential fits are applied to the head-on luminosity points before, after, and in the middle of the scan. These trends are subtracted from the data in Fig. [23](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig23), and all measurements are presented relative to this fitted head-on reference (normalized to unity).\n\n**Fig. 23**\n\n[![Fig. 23](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig23_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/23)\n\nATLAS specific luminosity relative changes (blue points) during separation scans in *x*\\-plane (top) and *y*\\-plane (bottom) at CMS, corrected for exponential (exp. corr.) luminosity decay. Gaussian function with a constant fitted to these points is shown with blue dashed line and compared to fit performed on the data corrected for linear luminosity change (lin. corr.) in time (dotted line). These are compared to COMBI simulation predictions (purple solid line)\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/23)\n\nAt CMS, similar scans were performed; however, degraded beam stability was observed during this period of the fill. The scan results exhibit systematic deviations, especially at large separation, which could not be symmetrically corrected using simple linear or exponential models. Despite these issues, all four scan cases (two transverse planes at two IPs) yield a total specific luminosity reduction of approximately 2.4%, consistent with expectations based on the measured beam–beam parameter \\\$\\\\xi \\\$. Comparison with COMBI simulations reveal up to 30% discrepancy for IP5 as the witness IP, while agreement improves to within 10% for measurements at IP1. These differences are comparable to those observed in Table [6](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Tab6), suggesting a common underlying origin. While a finer scan granularity can improve the description of the steep-response region below \\\$3\\\\,\\\\sigma \\_{\\\\textrm{meas}}\\\$, it would not address the dominant discrepancy between head-on and fully separated luminosity, which is likely driven by simulation assumptions and scan-related systematics.\n\n#### 6\\.4.2 Beam–beam-induced orbit deflections\nBeam–beam deflections are clearly observed during separation scans. Figure [24](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig24) shows the fully corrected residual horizontal beam displacement at IP1, measured with the DOROS beam position monitors, as a function of the beam separation. This representation allows a direct comparison with the analytical beam–beam deflection prediction based on the Bassetti–Erskine formalism \\[[35](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR35 \"M. Bassetti, G.A. Erskine, Closed expression for the electrical field of a two-dimensional gaussian charge. Technical report, CERN, Geneva (1980). \n                  https:\/\/cds.cern.ch\/record\/122227\n                  \n                \")\\].\n\nThe raw orbit data were corrected for slow orbit drifts by aligning the beam positions at the luminosity optimization points before and after each scan. Deviations of the applied beam separation from the nominal settings were accounted for by fitting a common length scale simultaneously with the beam–beam deflection. This approach avoids an overestimation of the separation calibration. The measured orbit displacement at the DOROS location was corrected for the contribution arising from the beam–beam deflection angle by applying a geometric amplification factor, proportional to the distance between the IP and the BPM in the small-angle approximation. After all corrections, the measured beam–beam deflection as a function of beam separation closely follows the analytical prediction. The maximum horizontal deflection observed at IP1 is 10.2 \\\$\\\\upmu \\\$m, compared to an expected value of 8.4 \\\$\\\\upmu \\\$m, with agreement within the total measurement uncertainty. The dominant uncertainty contribution (3.5 \\\$\\\\upmu \\\$m, corresponding to about 90% of the total) arises from a possible residual half crossing angle at the IP of up to 0.5 \\\$\\\\upmu \\\$rad. Additional systematic uncertainties include the fitted separation length scale, tune uncertainty of 0.005, and an extended 2 % uncertainty on \\\$\\\\beta ^\\*\\\$. Statistical uncertainties are negligible for most separation steps.\n\n**Fig. 24**\n\n[![Fig. 24](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig24_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/24)\n\nOrbit offset caused by the beam–beam interaction measured by DOROS BPMs in the horizontal plane at IP1 during the horizontal separation scan. Analytical prediction is shown for comparison \\[[35](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR35 \"M. Bassetti, G.A. Erskine, Closed expression for the electrical field of a two-dimensional gaussian charge. Technical report, CERN, Geneva (1980). \n                  https:\/\/cds.cern.ch\/record\/122227\n                  \n                \")\\]\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/24)\n#### 6\\.4.3 Impact of the beam-beam effects on beam sizes and tune shift\nSeparation-dependent changes in beam size were measured using the synchrotron light monitor. Figure [25](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig25) shows good consistency across individual bunch measurements, as well as strong agreement with COMBI simulations. This reinforces the utility of the BSRT as a reliable diagnostic for beam–beam effects. The uncertainties shown are statistical only and are treated as uncorrelated between scan points. The normalization to the zero-separation value is applied for visualization purposes and does not impose a constraint on that point.\n\n**Fig. 25**\n\n[![Fig. 25](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig25_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/25)\n\nBSRT beam-size measurement in the vertical plane for each bunch (points) during horizontal (top) and vertical (bottom) separation scans at IP1. The exponential evolution correction is applied based on adjacent head-on points (before, in the middle and after the scan) to highlight the relative changes. COMBI predictions are shown (gray dashed line) for comparison\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/25)\n\nTransverse tune shifts were extracted from ADT-measured spectra. As illustrated in Fig. [26](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig26), the mean tune shift evolution is reconstructed for both the scanning and non-scanning planes. The measured tune shifts closely follow the expected COMBI trends, further validating the simulation models used for beam–beam corrections.\n\n**Fig. 26**\n\n[![Fig. 26](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig26_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/26)\n\nMeasured tune shift in units of the beam-beam parameter \\\$\\\\xi \\_{BB}\\\$ for the horizontal (blue) and vertical (orange) planes, during a horizontal separation scan at IP1. The error bars indicate typically assumed conservative empirical systematic error on the tune measurement of 0.001. The solid curves represent the COMBI prediction\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/26)\n### 6\\.5 Beam-beam signatures during vdM scans at \\\$\\\\sqrt{s} = 13.6\\\\ \\\\text {TeV}\\\$\nDuring the first vdM calibration of LHC Run 3, an unusually high beam–beam parameter was recorded, offering a unique opportunity to study beam–beam effects under enhanced interaction conditions. These effects are examined using online ATLAS luminosity data, with representative results shown in Fig. [27](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#Fig27).\n\nThe example focuses on a standard vdM scan pair conducted at CMS, while monitoring luminosity at the non-scanning IP (ATLAS). The plotted luminosity changes are normalized to the central scan point, where both IPs are in head-on collision. Variations in this normalized luminosity across the scan directly reflect the impact of beam–beam interaction at the scanning IP on the luminosity at the non-scanning IP. These patterns are reproducible across multiple scan pairs acquired within the same fill. At the largest nominal separations (approximately \\\$\\\\pm 0.6\\\\,\\\$mm), the beams are sufficiently separated to eliminate beam–beam effects entirely. Notably, a qualitative difference is observed between the horizontal and vertical scan planes, an asymmetry that is not present in the above presented dedicated experiment. This behavior is attributed to differences in the phase advance between IPs in the standard vdM optics configuration. For vertical (*y*) scans, experimental data agree well with simulation, confirming the robustness of the modeling under these conditions. In contrast, the horizontal (*x*) scans exhibit a systematic discrepancy: the observed beam–beam impact is broader than that predicted by simulation. The model assumes that the separation steps are accurately defined using the beam size measured during the vdM scans, but this mismatch may suggest an unmodeled phase advance deviation or another optics imperfection. The total difference between data and simulation is within 0.2%. This dataset originates from a standard vdM calibration, for which only online luminosity measurement was available, and many contributions to the overall vdM uncertainty, such as bunch-to-bunch variations, averaging over different bunch families, and emittance evolution during the scan, are not disentangled. Therefore, the 0.2% difference should be interpreted as a consistency check under typical operational conditions rather than a precision validation of the beam–beam model. Nevertheless, this indicates that the model remains sufficiently accurate for beam–beam corrections in vdM analyses.\n\n**Fig. 27**\n\n[![Fig. 27](https:\/\/media.springernature.com\/lw685\/springer-static\/image\/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8\/MediaObjects\/10052_2026_15476_Fig27_HTML.png)](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/27)\n\nChanges from the separation-dependent beam-beam effects at the non-scanning IP – ATLAS online luminosity shown with points. Data comes from a standard vdM scan pair during the 2022 *pp* luminosity calibration. Separation steps and direction are indicated with gray markers. The last CMS scan is shown, with the highest beam-beam parameter of \\\$ \\\\xi =5.3 \\\\times 10^{-3} \\\$. The corresponding COMBI simulation results are also shown (triangles)\n\n[Full size image](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8\/figures\/27)\n\n## 7 Conclusions\nBeam–beam effects were directly observed and quantitatively validated at the LHC. The impact of beam–beam interactions on head-on luminosity, bunch sizes, tune spectra, and beam orbit positions was measured across various collision configurations. The observed effects align closely with COMBI simulations. In particular, the luminosity enhancement due to amplitude-dependent \\\$\\\\beta \\\$\\-beating was measured for the first time and is found to agree with expectations over a wide range of the beam–beam parameter \\\$\\\\xi \\\$. Together with the observed linear scaling with respect to the beam–beam parameter, this provides experimental confirmation of the numerical studies presented in \\[[13](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR13 \"A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). \n                  https:\/\/doi.org\/10.48550\/arXiv.2306.10394\n                  \n                \")\\]. Consistent signatures were found across multiple scan configurations, including two- and three-IP setups, and the characteristic tune shifts were well reproduced by both measurement and simulation.\n\nAccurate instrumentation and experimental design were essential. Key to the results was the stepwise collision scheme, isolating individual beam–beam contributions, with the implementation of the witness IP. Synchrotron light measurements from the BSRT, especially in the vertical plane of Beam 2, showed excellent agreement with COMBI predictions and bunch-by-bunch consistency. Tune spectra from the ADT system provided direct access to mean tune shifts, while orbit data from the DOROS system enabled measurement of beam–beam deflection down to 1 \\\$\\\\upmu \\\$m, matching analytical estimates. These tools, combined with precise control of IP configurations, enabled a comprehensive cross-check of beam–beam models.\n\nSome systematic limitations were identified. During one of the tests, beam stability during the scans was insufficient to allow for symmetric modeling of luminosity decay in both transverse planes. In several scans, the nominal separation steps were based on overestimated beam sizes, resulting in larger-than-expected scan intervals and less sensitivity at low separation. Beam–beam deflection measurements showed sign reversals at high separation, attributed to nonlinear orbit distortions from hysteresis effects in superconducting magnets. Phase advance errors between IPs and diagnostic systems further reduced the accuracy of simulations in some configurations.\n\nTargeted improvements would allow for higher precision and new insights. Finer separation steps, especially below \\\$3\\\\,\\\\sigma \\\$, would improve sensitivity to nonlinear beam–beam effects and allow more accurate comparison with models. Suppressing coherent beam–beam modes via phase advance tuning of not only two but all three IPs would improve tune shift measurements. Moreover, such optimal phase advance configuration between the IPs can be exploited to enhance luminosity. This effect could be directly propagated into at least a few percent increase in the total collected integrated luminosity, as discussed in Ref. \\[[36](https:\/\/link.springer.com\/article\/10.1140\/epjc\/s10052-026-15476-8#ref-CR36 \"J. Wańczyk, T. Pieloni, Optimizing beam-beam beta-beating for luminosity enhancement at the LHC. JACoW IPAC 2025, 018 (2025)\")\\]. Aligning the phase advance between IPs and diagnostic devices, particularly the BSRT, would reduce sensitivity to optics errors, enabling better beam size measurements for both beams and transverse planes. Moreover, the suppression of the coherent beam–beam modes facilitates the observation and analysis of the central, incoherent part of the tune spectrum, which is otherwise obscured by the dominant coherent peaks.\n\nFurther experimental strategies should also focus on directly probing the beam–beam interaction at its point of occurrence. This would require access to the full transverse beam distributions with high spatial resolution and sensitivity, going beyond projected sizes or centroids. Such measurements could directly reveal the nonlinear distortions and tails induced by the beam–beam force. In parallel, validating correction schemes that parametrize multi-IP beam–beam effects on luminosity via equivalent tune shifts remains essential. 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Meta Description	At the CERN Large Hadron Collider (LHC), absolute luminosity calibrations obtained by the van der Meer (vdM) method are affected by the mutual electromagne
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Boilerpipe Text	1 Introduction A precise determination of the absolute luminosity scale is essential for a wide range of LHC measurements, as it directly affects the normalization of many key physics cross sections. At the LHC, this calibration relies primarily on the van der Meer (vdM) method [ 1 , 2 ], which uses dedicated beam-separation scans performed under specially tailored beam conditions to relate measured interaction rates to the absolute luminosity inferred from beam parameters. While sub-percent uncertainties on the integrated luminosity have been achieved for Run-2 data by ATLAS [ 3 ] and CMS [ 4 ], further improvements and robust experimental validation of beam-dynamical effects remain essential. For relativistic proton beams colliding with zero crossing angle, which is the typical configuration for the vdM calibration, the luminosity at a given interaction point (IP) of the LHC is proportional to the overlap integral of the particle-density distributions $\rho _{\text {1,i}}$ , $\rho _{\text {2,i}}$ in bunch-pair i [ 5 ]: $$\begin{aligned} {\mathcal {L}}_{inst}&= 2c f_{\text {rev}} \sum ^{N_b}_i n_{\text {1,i}}, n_{\text {2,i}} \nonumber \\&\quad \times \iiiint ^{+\infty }_{-\infty }\rho _{\text {1,i}}(x,y,z-ct) \nonumber \\&\quad \times \rho _{\text {2,i}}(x,y,z+ct)\,dxdydzdt, \end{aligned}$$ (1) where c is the speed of light, $f_{\text {rev}}$ is the LHC revolution frequency, $N_b$ is the number of colliding bunches, and $n_{\text {1,i}}, n_{\text {2,i}}$ are the corresponding total charges per bunch for each colliding pair. Although only a very small fraction of the beam particles actually collide, the two opposing beams interact with each other electromagnetically, a dynamical process known as the beam-beam interaction [ 5 , 6 ]. In the LHC, as in any other synchrotron, the beams are not continuous but are divided into discrete “bunches”, each with a high proton density, containing approximately $10^{11}$ protons over a bunch length of 7.5–9 cm (in terms of RMS). This electromagnetic interaction occurs when the two beams share a common beam pipe, and is governed by a non-linear force that depends on the radial distance r of a test particle to the center of the opposing “source” bunch with n protons Gaussian-distributed in the transverse planes (valid approximation for the LHC beams): $$\begin{aligned} F = - \frac{ne^2}{2\pi \epsilon _0r}\biggl (1-\exp \biggl [ -\frac{r^2}{2\sigma ^2}\biggr ] \biggr ), \end{aligned}$$ (2) where e is the elementary proton charge, $\epsilon _0$ is the vacuum permittivity, and $\sigma $ is the RMS radius of the transverse charge distribution within the bunch, in a simplified case of a round shape $(\sigma = \sigma _x = \sigma _y)$ . At very small amplitudes $(r\rightarrow 0)$ the force can be linearized giving the expression for the so called beam-beam parameter, which is often used to assess the strength of the force: $$\begin{aligned} \xi = \frac{nr_p\beta ^}{4\pi \gamma \sigma ^2} = \frac{nr_p}{4\pi \epsilon _n}, \end{aligned}$$ (3) where $r_p$ is the classical proton radius, $\beta ^$ is the optical beta-function at the interaction point (IP) – one of the Courant–Snyder parameters [ 7 ], $\gamma $ is the Lorentz factor, and $\epsilon _n$ is the normalized emittance. Beam-beam effects were first observed almost 40 years ago [ 8 ] at the lepton colliders, and have been used extensively for accelerator diagnostics and optimization at both SLC and LEP [ 9 , 10 , 11 ]. At the LHC, they were first observed in late Run 1 [ 12 ], but only in terms of the beam-beam induced coherent deflections. Single particle effects, in contrast, were presumed too small in hadron colliders that until recently, there was no motivation to measure them with a precision that would be meaningful in the context evoked here. Nonetheless, the full complexity of this interaction must be accounted for when performing the luminosity calibration via the vdM method, using the separation scans. These are designed to measure the detector-specific constant $\sigma _{vis}$ that relates the observed rate $\mu ^{vis}$ to the absolute instantaneous luminosity ${\mathcal {L}}_{inst}$ : $$\begin{aligned} \mu ^{vis} = \frac{{\mathcal {L}}_{inst}\sigma _{vis}}{f_{rev}}. \end{aligned}$$ (4) Under the assumption of uncorrelated particle densities in x and y planes, the transverse convolved bunch widths $\Sigma _x, \Sigma _y$ can be extracted from the measured beam-separation dependence of the collision rate in the corresponding direction. The combined information from these scans, and bunch intensities can be used to calculate the instantaneous luminosity at the head-on position. The proportionality between the measured rate at the centered head-on position $\mu _{pk}$ and the reconstructed luminosity from measured bunch parameters during the vdM calibration defines the luminometer-specific visible cross-section: $$\begin{aligned} \sigma _{vis} = \frac{2\pi \Sigma _x\Sigma _y}{n_1 n_2} \mu _{pk}. \end{aligned}$$ (5) If left uncorrected, the $\sigma _{vis}$ measurement by the vdM method is biased by the beam-separation dependence of the mutual electromagnetic interaction of the two beams: the colliding bunches experience deflection-induced orbit shifts, as well as optical distortions akin to the dynamic- $\beta $ effect, that both depend on the transverse beam separation and must therefore be accounted for when deriving the absolute luminosity scale. The correction strategy discussed extensively in Ref. [ 13 ] relies on simulation studies only, and thus requires experimental validation to establish its reliability. The present report summarizes a campaign of dedicated measurements carried out in the spring of 2022 at the LHC, aimed at confirming the accuracy of the simulation studies detailed in Ref. [ 13 ], and at validating the beam-beam correction strategy used in the luminosity-calibration analyses of the ALICE, ATLAS, CMS and LHCb experiments. More specifically, the goal is to quantify the impact of the beam-beam interaction on the tune spectra, orbit, the transverse-density distribution of the colliding bunches, as well as, for the first time, on the luminosity, by systematically varying the strength or number of the beam-beam interactions. The measurements are repeated for different values of the beam-beam parameter to verify the scaling proposed in [ 13 ], for various choices of the scanning IP and for several multi-IP configurations. The beam conditions are chosen to be representative of vdM calibration sessions at the LHC, but optimized so as to maximize both the sensitivity of the measurements, and the operational efficiency. This paper is organized as follows. The measurement strategy is detailed in Sect. 2 , followed by the main ingredients of this accelerator experiment: beam conditions (Sect. 3 ), luminometers and accelerator instrumentation (Sect. 4 ), and optimization of the ring lattice to maximize the sensitivity of the measurements to beam–beam-induced optical distortions of the colliding bunches (Sect. 5 ). Section 6 is devoted to the experimental characterization of the impact of beam-beam effects on the tune spectra, on the luminosity at different IPs, as well as on other observables such as transverse single-bunch widths and IP orbits. At each step, the results are confronted with the predictions of the COherent Multibunch Beam-beam Interactions (COMBI) tracking code [ 14 ], that produced most of the results detailed in Ref. [ 13 ]. A collection of macroparticles is simulated turn by turn, based on a simplified model of the accelerator lattice using linear transfer and the self-consistent computation of the beam-beam forces at the IP. The conclusions are presented in Sect. 7 . 2 Measurement strategy The beam-beam interaction at an interaction point, referred to here as the scanning IP, induces measurable changes in key beam parameters such as the tune spectrum, closed orbit, and transverse single-beam profiles. These effects can be observed, albeit with limited precision, using standard accelerator diagnostics by comparing measurements obtained under varying transverse beam separation at the scanning IP. However, the resulting modifications to the transverse density distribution of the colliding bunches remain below the sensitivity threshold of conventional instrumentation, including synchrotron-radiation-based beam-profile monitors. Beam-beam-induced variations typically lie in the 0.5–1% range, small enough that only a highly precise measurement of collision-rate changes can provide adequate sensitivity. Fig. 1 Scheme of the LHC and its interaction points (IP) with indicated IP1 as witness IP while performing a scan at IP5 Full size image Importantly, the beam-beam interaction induces changes in beam properties that are intrinsically entangled with the pp collisions when observing luminosity at the scanning IP. Consequently, under realistic collider conditions, it is not possible to directly access a reference luminosity signal that is free of beam-beam effects – unlike in simulation studies, where such effects can be explicitly disabled. In a multi-collision configuration, however, luminosity shifts at one or more other IPs (Fig. 1 ), where the beams are kept in continuous head-on collision and therefore act as non-scanning, or witness , IPs, can be used to monitor the separation-dependent beam–beam effects. These effects are induced at the scanning IP, where the beams are deliberately brought in and out of collision, and propagate around the rings to the witness IPs, where they manifest as changes in the measured luminosity. In principle, any of the four LHC IPs can serve as a scanning IP, and any non-scanning IP can be designated as the witness IP for a given measurement. In practice, instrumental and operational constraints at the time the experiment was carried out restrict the choice of witness IP to the ATLAS (IP1) and CMS (IP5) collision points; IPs 1, 2 or 5, or combinations thereof, are used as scanning IP(s). Collisions at IP8 were deliberately avoided because the large longitudinal offset of the LHCb collision point, that breaks the eight-fold symmetry of the LHC rings, precludes the possibility of both members of a colliding-bunch pair to collide at all four IPs. In addition, the large crossing angle at IP8 would complicate the interpretation of the beam-beam effects that would occur at that IP if some of the bunches collided there. The propagation of the beam–beam-induced, amplitude-dependent $\beta $ -beating from the scanning to a witness IP is controlled by the betatron phase advance between these two IPs. Since the targeted signatures are delicate to measure at best, it is natural to try and enhance them by adjusting this phase advance so as to maximize the sensitivity of luminosity shifts at the witness IP to beam-beam effects at the scanning IP; in doing so, however, the overall tunes must be preserved. Given the central and symmetric roles played by IP1 and IP5 in this experiment, the adjustment of their relative phase advance drove the optimization procedure that is detailed in Sect. 5 , and that achieved a threefold improvement in measurement sensitivity. Central to the quality of the measurements is the controlled variation of the beam-beam parameter at the scanning IP. This can be achieved either: by using step scans , i.e. by fully separating the beams at the scanning IP in either the horizontal or the vertical plane, and then bringing them back into head-on collision; or by using separation scans , i.e. by scanning the beams transversely with respect to each other in either the horizontal or the vertical plane; or by taking advantage of the natural beam-intensity decay and emittance growth to progressively reduce the beam-beam parameter. Equally important is to ensure that the beams remain in head-on collision at the witness IP(s), such that reproducible luminosity shifts measured at these locations can be unambiguously correlated with controlled changes in the strength of the beam-beam interaction at the scanning IP. It was verified that the beam-beam-induced orbit shift and the potential non-closure of the orbit bumps used to control the beam separation at the scanning IP, did not significantly affect the actual separation, and thereby the measured collision rate, at the witness IP. In a few cases however, orbit drifts of uncontrolled origin ended up degrading the quality of some of the measurements. Simulations demonstrate [ 13 ] that multi-IP effects strongly influence the magnitude of beam-beam biases to vdM calibrations. Their characterization, therefore, constitutes an essential component of this experiment. The scan protocol is a generalization of that in the two-IP case. One IP (say, IP1) is designated as the witness IP, with beams colliding head-on there and at IP2, and IP5; the beams are then taken out of collision at IP2 only, then at both IP2 and IP5, and finally returned to a three-collision configuration in the reverse order. Since the phase-advance combinations are different in each configuration, the corresponding luminosity shifts predicted at the witness IP are also different and can be meaningfully confronted with the data. 3 Beam conditions In the round-beam approximation, the beam-beam parameter $\xi $ does not depend on the beam energy (Eq. ( 3 )). Therefore, and in order to minimize the overhead associated with commissioning and operating the LHC in a non-standard optical configuration, as well as to allow for frequent refilling with different bunch patterns, the measurements were carried out at injection energy ( $450\ \text {GeV}$ per beam), and under non-standard machine-protection conditions. The improvement in operational efficiency came at a cost: the counting rate of the luminometers (or equivalently their visible cross-section) is about an order of magnitude smaller at $\sqrt{s} = 900\ \text {GeV}$ than at $13\ \text {TeV}$ , reflecting the combination of a 40% drop in the inelastic pp cross-section, a factor of two to three drop in the multiplicity of the final-state particles, and a significant softening of their momentum spectrum; because of the total-intensity constraints dictated by machine-protection requirements, the injected beam could not exceed 4 bunches per beam (compared to 150 during a routine vdM -calibration session), with a maximum allowed population of $1.25\times 10^{11}~p$ /bunch; the combination of the low beam energy, that suppresses synchrotron-radiation damping, and of the large bunch intensity, that enhances intra-beam scattering, resulted in relatively rapid emittance growth and rather short single-beam lifetimes. To partially mitigate these effects, all collision-rate measurements are expressed in terms of specific luminosity, thereby automatically accounting for the beam-intensity decay. The bunch intensity was deliberately increased by about 25% beyond the vdM -scan values typical of LHC Run 2, to maximize both the luminosity and the beam-beam parameter $\xi $ . The latter reached 0.010 per IP (Table 1 ), compared to 0.003–0.006 during normal pp vdM sessions. With a target injected emittance of $1.5 \ \upmu \text {m} \cdot \text {rad}$ , the $\beta $ function at the IP set to $\beta ^* = 11\ \text {m}$ , and zero crossing angle at IP1 and IP5, the statistical uncertainty affecting a typically 60-second long per-bunch luminosity measurement in the presence of head-on collisions, lay around 0.5%. Table 1 Range of beam-beam parameter values during the various stages of the experiment Full size table The bunch patterns were chosen such that all bunches in each beam collided either at IP1 and IP5 only, or at all three of IP1, 2 and 5. The orbit-stabilization feedback system was turned off during the data-taking periods to prevent it from interfering with the beam–beam-induced orbit shift. The chromaticity was set to its standard value in physics fills, of +10 units, to guarantee coherent stability against the machine impedance, and in view of the very small number of bunches and of their large longitudinal spacing, the beam-stabilizing Landau octupoles were set to their minimum current (1A). The damping time of the bunch-by-bunch transverse feedback was set to 1000 turns, a rather loose setting intended to preserve longer the natural beam oscillation and therefore improve the precision of the tune measurements. 4 Beam instrumentation 4.1 Luminometers The luminometer systems in use by the ATLAS and CMS collaborations at IP1 and IP5 are described in Refs. [ 15 , 16 ] respectively; no luminosity measurements were available at IP2 during the collider experiment described in this paper. Maximizing the statistical sensitivity leads to choosing the luminosity algorithm with the highest possible acceptance, namely: the hit rate per bunch crossing in the ATLAS Minimum Bias Trigger Scintillators (MBTS) [ 15 ], and the occupancy in the CMS Hadron Forward Calorimeter (HFOC) [ 16 ]. In both cases, raw luminometer counts are converted to collision rates using the Poisson formalism [ 15 , 16 ], and the instantaneous luminosity is averaged over typically 60-second time bins, that during the scans are synchronized with the scan steps. The statistical uncertainties are estimated from either the total number of raw luminometer counts per time bin (ATLAS), or from the RMS of the approximately 40 luminosity samples recorded in a given time bin (CMS). Since the absolute luminosity scale is irrelevant, and in order to simplify the interpretation of the results, all measurements in this paper are presented in terms of fractional shifts in the bunch-averaged specific luminosity, relative to a reference time that depends on the type of measurement considered. 4.2 Bunch-charge measurement The total intensity of each beam is measured by a direct-current current transformer (DCCT), the absolute scale of which is calibrated against a very high precision pulse generator [ 17 ]. The bunch-by-bunch charge fractions, in turn, normalized to the total stored intensity, are measured, separately for the two beams, by Fast Beam Current Transformers (FBCT) [ 18 ]. The latter devices also provide the fill pattern, i.e. the relative location, at a given instant and around the two rings, of the nominally filled bunches. Since the absolute scale of the DCCTs is known to much better accuracy than that of the FBCTs, the bunch charges are typically computed as the product of the FBCT bunch-charge fractions and the total circulating beam intensity reported by the corresponding DCCT. 4.3 Emittance measurement The synchrotron-light beam-profile monitors [ 19 ], dubbed BSRTs (“Beam Synchrotron Radiation Telescope”), are mainly used to track emittance evolution over time, but they can also measure the relative changes in transverse single-beam size that result from beam–beam-induced $\beta $ -beating. BSRT data are recorded every second; in what follows, they are presented averaged over one-minute intervals for easier comparison with other measurements. The absolute length scale of the BSRT profiles suffers from significant uncertainties, if only because the online optical corrections to the images can be updated only a few times per year, and are unable to track the evolution of the efficiency of the light sensors, that depends both on time and on the position of the light spot on the sensor array. For each beam therefore, the BSRT is complemented by a wire-scanner (WS) profile monitor that is located in the same straight section. The advantage of the WS is that its accuracy is significantly better than that of the BSRT; its down side, in the present context, is that the WS cannot acquire data continuously and must be triggered manually. For the results presented in this paper, the wire scanners were flown through the beams at the start of each group of measurements to provide single-beam emittance measurements that are as accurate as possible, and then at regular intervals thereafter. The BSRT is used to interpolate the time evolution of the emittance between two sets of WS measurements. To interpret measured RMS bunch widths in terms of emittance requires the knowledge of the $\beta $ functions at the locations of the BSRT and the WS. These are determined by the phase-advance method, and compared in Table 2 to their model value computed using MAD-X; the agreement is typically better than 5%, and the worst disagreement amounts to 9%. These discrepancies are attributed to imperfections in the magnetic lattice. Uncertainties in the measurement mainly arise from the interpolation of the lattice functions between the two closest beam-position monitors (BPMs) where the $\beta $ -functions are measured [ 20 , 21 ], and either the BSRT or the WS location. Based on the MAD-X LHC lattice model, these uncertainties are estimated not to exceed 3%, and this value is assigned as the systematic uncertainty on the $\beta $ functions used in BSRT- and WS-based emittance measurements. Table 2 $\beta $ functions at the BSRT and WS locations, for the optical configuration detailed in Sect. 5.3 Full size table The beam-averaged emittance, i.e. the average of the emittances of the beam-1 bunch and of the corresponding beam-2 bunch, can be obtained directly from emittance scans [ 22 ] at IP1 and/or IP5. In this approach, the convolved transverse bunch sizes measured using beam-separation scans are translated into emittances using the $\beta ^$ values determined by k -modulation [ 23 ] at the relevant IP (see Sect. 4.3 ). The associated uncertainty (Table 3 ) is combined with the statistical uncertainty in convolved transverse width to estimate the error affecting the measured emittance. Comparison with beam-averaged emittances extracted from the single-beam profile monitors reveals excellent agreement between WS and emittance-scan results (Fig. 2 ). The relative time-evolution of the BSRT emittances is consistent with that observed using the WS or emittance scans, but the absolute magnitudes differ significantly, especially in the horizontal plane. Table 3 $\beta ^$ functions at the ATLAS and CMS IPs measured by k -modulation. The target value in all cases is $\beta ^=$ 11 m Full size table Fig. 2 Time evolution of the horizontal (top) and vertical (bottom) beam-averaged normalized emittances during LHC fill 8037, as reported by the BSRT (blue), the WS (orange), and using beam-separation scans at IP1 (red) and IP5 (purple). Different shades of the same color correspond to two different bunches present in this fill. For ATLAS emittance scans, only the bunch-averaged value is shown Full size image 4.4 Beam-beam parameter determination The systematic and quantitative comparison of the measured and of the predicted beam-beam impact on the observables detailed in Sect. 2 requires continuous monitoring of the actual beam-beam parameter throughout the duration of the experiment. Only the BSRT provides uninterrupted emittance determination throughout the fill. In view of the scale biases apparent in Fig. 2 , however, WS-based emittances are used, whenever possible, as input to the determination of the beam-beam parameter; BSRT emittances re-scaled to close-in-time, absolute WS measurements provide time-interpolated emittance values whenever WS data are unavailable. The beam-beam parameter evolution during the first fill of the experiment is illustrated in Fig. 3 . The error bands are dominated by the 3% systematic uncertainty in the measured optical functions. The strength of the beam-beam interaction drops by a factor of two over a couple of hours, from the combined effect of beam-intensity decay and of emittance growth. Fig. 3 Time evolution of the beam- and plane-averaged beam-beam parameter $\xi $ inferred from the measured bunch charges and emittances, separately for the two colliding-bunch pairs present in the fill pattern. The color bands indicate the systematic uncertainty (see text) Full size image 4.5 Tune measurements Coherent spectra can be monitored using either the LHC Transverse Damper (ADT) [ 24 ], or the Base-Band Tune (BBQ) [ 25 ] system; their comparison typically yields consistent results. Spectrograms with a tune resolution of 0.0001 are computed using bunch positions recorded at every turn, and averaged over the one-minute time bins mentioned in Sect. 4.1 . To mitigate the influence of noise on the tune measurement, a median filter is applied at the pre-processing stage with a self-defined local window size. Additionally, the 50 Hz noise lines present in the spectra are masked. The filtered data are fitted by the sum of a Gaussian function and a constant baseline, and the frequency at the peak of the Gaussian is interpreted as the measured mean tune. The systematic uncertainty on measured tune shifts $\Delta Q$ is estimated empirically to be $\sigma _{\Delta Q}=0.001$ . 4.6 IP-orbit monitoring Orbit displacements at each IP are measured using the Diode ORbit and OScillation (DOROS) BPM system, that provides sub-micrometer beam-position resolution [ 26 ]. These strip-line BPMs are located in the two quadrupoles on either side of and closest to the IP, allowing the position of both beams to be measured simultaneously. The position and the angle of each beam at the IP are inferred from, respectively, the average and the difference of the positions measured in the two final-triplet quadrupoles. 5 Optimization of the phase advance between interaction points In the two-IP configuration, the basic theory of linearized beam-beam $\beta $ -beating provides an analytical determination of the optimum phase advance between the two IPs, i.e. of the setting that maximizes the sensitivity, at the witness IP, to beam-beam effects at the scanning IP (Sect. 5.1 ). A generalization to the three- and four-IP configurations is discussed in Sect. 5.2 . The required modifications to the baseline LHC optics, as well as their implementation and their experimental validation, are detailed in Sect. 5.3 . 5.1 Linear beam-beam $\beta $ -beating In circular colliders, the $\beta $ -beating induced by beam-beam effects at the IP(s) leads to changes in transverse beam size [ 5 , 27 ], thereby impacting the luminosity. For a single collision point, the change in $\beta ^$ of a single small-amplitude particle in (for instance) the horizontal plane x , is given by: $$\begin{aligned} \frac{\beta _x^}{\beta _{0,x}^} = \frac{1}{\sqrt{1-4\pi \xi \cot {(2\pi Q_x)} - 4\pi ^2\xi ^2}}. \end{aligned}$$ (6) The magnitude of the effect depends on the absolute value $\xi $ of the beam-beam parameter and on the nominal betatron tune $Q_x$ ; its periodicity is $ [\pi ]$ . It can be shown that approaching the half-integer betatron tune from below minimizes the $\beta $ function at the IP, thereby maximizing the luminosity; this phenomenon has been exploited with great success in (among others) the KEKB and PEP-II B factories. In the LHC, where during collisions the nominal fractional betatron tunes are fixed at $q_x=0.31,\,q_y=0.32$ , this dynamic- $\beta $ effect depends only on the beam-beam parameter. This conclusion no longer strictly holds in the presence of more than one collision point. In the case of two IPs with identical values of $\xi $ and $\beta ^$ , the dynamic- $\beta $ effect can be described analytically [ 28 ]: $$\begin{aligned} \frac{\beta _x^}{\beta ^_{0,x}} = \frac{\sin {2\pi Q_x} + 4\pi \xi (\cos (2\pi Q_x-2\mu _{1,x})-\cos 2\pi Q_x )}{\pm \sqrt{1-\cos ^2{2\pi (Q_x+\Delta Q_x)}}}, \end{aligned}$$ (7) where $\mu _{1,x}$ is the phase advance from the scanning IP to the witness IP. The sign in the denominator is linked to that of the numerator so as to ensure that the ratio remains positive; the denominator must be positive (resp. negative) below (resp. above) the half-integer tune, i.e. when $m< Q_x <m+\frac{1}{2}$ (resp. $m+\frac{1}{2}< Q_x < m+1$ ), where m is a positive integer. The tune shift $\Delta Q_x$ is related to $Q_x$ , $\xi $ and $\mu _{1,x}$ by: $$\begin{aligned} \cos {2\pi (Q_x+\Delta Q_x)}&=(1-16\pi ^2\xi ^2)\cos {2\pi Q_x} + 8\pi \xi \sin {2\pi Q_x} \nonumber \\&\quad + 16\pi ^2\xi ^2\cos {(2\pi Q_x-2\mu _{1,x})}. \end{aligned}$$ (8) The phase-advance dependence of the resulting $\beta $ -beating is illustrated in Fig. 4 . The minimum of each curve corresponds to the phase advance that yields the largest mutual beam–beam-induced luminosity enhancement between the two IPs. The optimal settings are $\mu _{1,x}^{min}/2\pi =0.405,\,\mu _{1,y}^{min}/2\pi =0.410$ ; their difference reflects that between the nominal horizontal and vertical fractional tunes $q_x$ and $q_y$ . Fig. 4 Analytically computed beam-beam induced $\beta $ -beating in a two-IP configuration, as a function of the phase advance $ \mu _1$ between the 2 IPs ( $q_x=0.31$ , $q_y=0.32$ , head-on collisions at both IPs, $\xi =7\times 10^{-3}$ per IP) Full size image The extrema of the curves in Fig. 4 can be determined analytically by differentiating Eq. ( 7 ) with respect to the phase advance $\mu _{1,x}$ . The general expression for this derivative $\frac{d}{d\mu _{1,x}} \biggl ( \frac{\beta _x^}{\beta ^_{0,x}} \biggr )$ , detailed in Appendix A, is the product of two factors, each of which can be zero: imposing that $\sin {(2\pi Q_x - 2\mu _{1,x})} = 0$ yields the solution $$\begin{aligned} \frac{\mu _{1,x}}{2\pi } =\frac{Q_x}{2}-\frac{m}{4} \, \end{aligned}$$ (9) where m is an integer of either sign, or zero; the other factor is zero for either $$\begin{aligned} \frac{\mu _{1,x}}{2\pi } = \frac{Q_x}{2} - A(\xi , Q_x)\, \end{aligned}$$ (10) or $$\begin{aligned} \frac{\mu _{1,x}}{2\pi } = \frac{Q_x - 1}{2} + A(\xi , Q_x) \, \end{aligned}$$ (11) with $A(\xi , Q_x)$ defined in Eq. ( 12 ) $$\begin{aligned} A(\xi , Q_x) = \arccos \left( \frac{ 32\pi ^3\xi ^3 \sin (4\pi Q_x) + 48\pi ^2\xi ^2 \cos ^2(2\pi Q_x) - 32\pi ^2\xi ^2 - 6\pi \xi \sin (4\pi Q_x) + \cos ^2(2\pi Q_x) - 1}{16\pi ^2\xi ^2 \left( 4\pi \xi \sin (2\pi Q_x) + \cos (2\pi Q_x) \right) } \right) / (4\pi ) \end{aligned}$$ (12) The two sets of solutions do not overlap. The subset of solutions that minimize (rather than maximize) $\beta ^$ is identified by requiring that the second derivative be positive. This restricts the main solution to: $$\begin{aligned} \frac{\mu _{1,x}}{2\pi } =\frac{Q_x}{2}+\frac{m+1}{4} . \end{aligned}$$ (13) The optimal phase $\mu _{x,1}$ that minimizes the $\beta ^$ for range of $(\xi , Q_x)$ values is shown in Fig. 5 . Fig. 5 Optimal phase advance value $\mu _{x,1}$ for various values of $(\xi , Q_x)$ in the two-IPs configuration Full size image The phase-advance dependence of the dynamic- $\beta $ effect can also be quantified in terms of the luminosity shift, at the witness IP, that is associated with the electromagnetic interaction of the two bunches at the scanning IP (Fig. 6 ). The optimal setting, i.e. that which maximizes the sensitivity, at the witness IP, to beam-beam effects at the scanning IP, is indicated by the green dotted vertical line at $\Delta \mu _{\text {IP1-IP5}}=0.41\ [2\pi ]$ ; the full suppression of the phase-related luminosity enhancement is indicated by the red dotted line. These results are consistent with the predictions of the analytical model (Fig. 4 ). In the two-IP configuration considered here, in which IP1 and IP5 can both play the role of either the scanning or the witness IP, and due to the periodicity of this curve ( $ \pi $ ) and to the fractional-tune values used during collisions ( $q_x / q_y = 0.31 / 0.32)$ , the phase–advance-related luminosity enhancement is the same at the two IPs; at the optimum setting, it is three times larger than it would be using the nominal LHC lattice. The maximum effect depends mainly on the transverse tunes, and thus could have been further enhanced if it had been possible to move these away from their nominal values. A more comprehensive simulation study, that evaluates the impact of the beam-beam interaction on the luminosity for more realistic Gaussian proton-density distributions, multiple collision points and as a function of the beam separation, is reported in Ref. [ 29 ]. Fig. 6 Beam–beam-induced luminosity shift at IP5 predicted by COMBI simulations, as a function of the phase advance $ \mu _1=\Delta \mu _{\text {IP1-IP5}}$ between the two IPs ( $q_x=0.31$ , $q_y=0.32$ , head-on collisions at both IPs, $\xi =7\times 10^{-3}$ per IP) Full size image 5.2 Optimized phases for multi-IP configurations The general expression to obtain the first-order change of the $\beta $ function at an arbitrarily chosen reference IP (at $s=\mu _{0,x}=0$ ) can be approximated as multiple quadrupole errors [ 30 ]: $$\begin{aligned} \frac{\beta ^_x}{\beta ^_{0,x}} = \frac{2\pi \xi }{\sin {2\pi Q_x}} \sum _{i\in IPs}^{N-1} \cos (2\pi Q_x-2\mu _{i,x})\, \end{aligned}$$ (14) where $\mu _{i,x}$ is the horizontal phase advance between the reference and the $i^{th}$ IP for a total of N IPs. Therefore, the optimal configuration, which minimizes the $\beta ^$ change at all IPs, must satisfy the condition: $$\begin{aligned} \sum _{i=1}^{N-1} \cos (2\pi Q_x-2\mu _{i,x}) = -(N-1)\, \end{aligned}$$ (15) which yields the same solution for all IPs as in the 2 IPs configuration (Eq. 13 ): $$\begin{aligned} \frac{\mu _{i,x}}{2\pi } =\frac{Q_x}{2}+\frac{n+1}{4} . \end{aligned}$$ (16) where n is an integer of either sign, or zero. To maximize the effect on luminosity, the phase advance between the reference interaction point at $ i = 0 $ and the nearest IP, at $ i = 1 $ must satisfy $ \mu _{1,x}/2\pi = (Q_x + 0.5)/2 $ , as given by the first solution (for $n=0$ ) of Eq. ( 13 ). This condition implies that the phase advance to any subsequent adjacent IPs ( $ i \ge 2 $ ) must be an integer multiple of $ \pi $ . Thus, this requirement exposes a fundamental constraint: it is not possible to minimize $ \beta ^* $ simultaneously at all IPs. This would only be possible for ( $ Q_x = 0.5 $ ) which is practically not achievable, as it would require operating the storage ring precisely at the half-integer resonance. 5.3 Optics validation The optimum phase advance from IP1 to IP5 determined in Sect. 5.1 was implemented using tuning trim quadrupoles in the LHC arcs. The targeted fractional phase advance was $0.9 \, [2 \pi ]$ for both beams and both planes (Table 4 ). This setting, which differs from that in Fig. 6 by one period ( $ \pi $ ), is equivalent in terms of sensitivity optimization but has the advantage that it minimizes the absolute magnitude of the perturbations inflicted upon the nominal lattice. Several measurements were carried out to validate this adjustment: comparison of the targeted and achieved shifts in phase advance (Sect. 5.3.1 ), $\beta $ -function measurements across the full LHC circumference (Sect. 5.3.2 ) as well as at the IPs (Sect. 5.3.3 ), and dispersion measurements to estimate the dispersive contribution to the transverse beam size at the IP (Sect. 5.3.4 ). 5.3.1 Phase advance The phase advance around each LHC ring is determined using turn-by-turn orbit measurements [ 20 , 21 ]. The targeted and achieved phase-advance shifts agree within $0.01\, [2\pi ]$ or better (Table 4 ). The phase advance change from IP1 and IP5 was compensated on the opposite side of the ring, ensuring that the transverse tunes remain unchanged. The target fractional local phase advance is set to $0.9,[2\pi ]$ . Table 4 lists the corresponding phase shifts applied to the nominal lattice to reach this target on one side of the ring; this is consistent with the analytically derived value of $0.41\,[2\pi ]$ for the compensating shift on the opposite side. The change in phase advance was also measured at the BSRT location, where the discrepancy between the model and measurements is more pronounced, reaching up to $0.05\,[2\pi ]$ . Table 4 Nominal and optimized IP1 to IP5 phase-advance settings. Column 4 (resp. 5) displays the targeted (resp. measured) shift in phase advance after the optimized settings have been applied Full size table 5.3.2 Residual $\beta $ -beating The $\beta $ -beating was quantified and found to be well within acceptable tolerances for various configurations of the trim strength. The measurement results, relative to the reference measurements without any phase advance adjustments, are illustrated in Fig. 7 . The maximal observed deviation is approximately 25% around IP5 for Beam 1 in the vertical plane. Such significant deviations in proximity to the IPs are anticipated and are frequently attributed to erroneous BPM readings. Overall, the measurements indicated that $\beta $ -beating remained within acceptable bounds, predominantly within a 10% margin. Fig. 7 Measured $\beta $ -function change from the full strength of the phase advance knob with reference to nominal lattice along the LHC ring, for Beam 1 (top two figures for x and y ) and Beam 2 (bottom two figures for x and y ) Full size image The measured $\beta $ functions are also compared to the MADX LHC lattice model predictions, which include the phase advance adjustment. The results, illustrated along the LHC lattice in Fig. 8 , indicate that the measurements align with the model within an acceptable margin of 10%. Fig. 8 Measured deviations in the $\beta $ -function along the LHC ring relative to the MADX model incorporating the phase advance knob, for Beam 1 (top two figures for x and y ) and Beam 2 (bottom two figures for x and y ). The error bars represent solely the statistical uncertainties as determined in the measurements Full size image 5.3.3 $\beta $ functions at the interaction points The DOROS BPMs are strategically located at the end of the inner triplets on both sides of each of the experiments. Due to the phase advance of $\pi $ between the DOROS on the left and right sides, these are inadequate for the precise determination of the measurement of $\beta ^$ at the IP. This can only be achieved with a special k -modulation technique [ 31 ], which was performed at the interaction points used in the experiment. The results are summarized in Table 3 . While the achieved precision of the $\beta ^$ determination varies between measurements, uncertainties at the level of ${\mathcal {O}}(1\%)$ are not uncommon and are not systematic to a specific beam or interaction point. Such uncertainties would propagate to only a few-percent effect on derived quantities such as the emittance or the beam–beam parameter. 5.3.4 Dispersion It was essential to perform dispersion $D^$ measurements at the interaction points, under the new optics configurations, as it can contribute to the measured beam size. Assuming the LHC design momentum spread value of $\frac{\Delta p}{p}=10^{-4}$ , the average dispersion measurement around IP1 and IP5 is estimated to contribute up to 2% to the observed bunch sizes (worst case measured for B2Y). This effect is considered static throughout the experiment. 6 Experimental results This section presents experimental measurements of beam–beam effects in configurations involving multiple collision points. The impact of the optics modifications introduced in Sect. 5 is first assessed through tune spectra measurements, as detailed in Sect. 6.1 . To quantify the cumulative effect of one or more additional collisions and assess the reproducibility of the results, a stepwise collision scan was employed. In this procedure, beams were sequentially brought into head-on collision at selected interaction points (IPs), followed by full transverse separation ( $\Delta > 6\sigma $ ) in either the horizontal or vertical plane at all but one witness IP. Two distinct filling schemes were used to isolate the influence of a single additional collision (2-IP configuration: ATLAS and CMS, see Sect. 6.2 ) and to investigate more complex multi-collision scenarios (3-IP configuration: including ALICE, see Sect. 6.3 ). The first measurement of separation-dependent beam–beam effects, analogous to those encountered during luminosity calibration scans, is presented in Sect. 6.4 , alongside comparisons with numerical predictions. An additional measurement conducted during the 2022 van der Meer calibration is detailed in Sect. 6.5 . A consolidated overview of all findings is provided in Sect. 7 . 6.1 Measured effect on the tune spectra of the phase advance changes The coherent transverse tune spectra serve as a diagnostic for evaluating the effectiveness of the phase advance modifications. COMBI simulations were carried out for two extreme lattice configurations, one designed to maximize and the other to suppress beam–beam effects. In the head-on scenario (Fig. 9 ), the coherent modes are suppressed for the maximizing configuration, as the phase advance between the two collision points is close to $\pi $ , as was previously shown in simulation in Ref. [ 32 ]. In this configuration, the effect of the first collision is counteracted by the second. This setting is particularly suitable for studying incoherent spectra, as the diminished coherent signal allows for a clearer extraction of the mean tune shift from fitted spectra. During separation scans, the symmetry between horizontal ( x ) and vertical ( y ) planes is broken. In the scanning plane (horizontal), coherent modes re-emerge at $1.5\,\sigma $ separation (Fig. 10 ), with their frequency shifting in response to the separation distance (additional example for $3\,\sigma $ is shown in Fig. 11 ). Fig. 9 Comparison of COMBI simulated tune spectra in x -plane (left) and y -plane (right) for the suppressing (in blue) and maximizing (in green) phase configurations, for head-on collision Full size image Fig. 10 Comparison of COMBI simulated tune spectra in x -plane (left) and y -plane (right) for the suppressing (in blue) and maximizing (in green) phase configurations, at $1.5\,\sigma $ separation in the horizontal plane Full size image Fig. 11 Comparison of COMBI simulated tune spectra in x -plane (left) and y -plane (right) for the suppressing (in blue) and maximizing (in green) phase configurations, at $3\,\sigma $ separation in the horizontal plane Full size image These simulated trends of the coherent spectra are also observed in experimental data. Transverse spectra recorded for head-on conditions confirm the suppression of coherent modes in the maximizing phase configuration (Fig. 12 ). Additional comparisons for separated beams at the IP (see Figs. 13 and 14 ), shown here for the case of vertical separation, reveal similar features. In this configuration, the non-scanning (horizontal) plane exhibits spectra that qualitatively agree with simulation predictions (see Figs. 10 and 11 ), while in the scanning (vertical) plane the coherent modes do not reappear as expected. An equivalent behavior is observed when the separation is applied in the horizontal plane, with the roles of the two transverse planes interchanged; this case is not shown for brevity. This discrepancy is attributed to the effect of the transverse damper, which acts independently on the two beam centroids and suppresses coherent oscillations arising from beam–beam coupling, consistent with earlier observations [ 33 ]. Fig. 12 Measured tune spectra before and after phase change, corresponding to the suppressing (in blue) and maximizing (in green) phase configurations, at the head-on position, for each of the transverse planes (Beam 1 shown as an example). Dash-dotted lines show Gaussian fits to the data Full size image Fig. 13 Comparison of the measured tune spectra at suppressing (blue) and maximizing (green) phase configurations with $1.3\,\sigma $ separation in the vertical plane Full size image Fig. 14 Comparison of the measured tune spectra at suppressing (blue) and maximizing (green) phase configurations with $2.6\,\sigma $ separation in the vertical plane Full size image 6.2 Head-on collisions in a two-IP configuration at $\sqrt{s} = 900\ \text {GeV}$ Each test consisted of four measurements designed to quantify the beam–beam effect induced by a single additional head-on collision. The sequence was executed with CMS as the witness IP, followed by ATLAS, and concluded with a repeat at CMS. Over the course of the fill, the beam–beam parameter decreased from approximately 0.01 to 0.005. The outcome of each scan is analyzed using three independent observables: luminosity, beam size, and tune spectra, discussed separately below. 6.2.1 Impact of beam-beam effects on head-on luminosity An overview of the luminosity measurements during the first test with CMS as the witness IP is shown in Fig. 15 , where the specific luminosity at CMS is normalized to the initial head-on step to highlight relative changes. The uncertainty at each step is given by the standard error of the mean; shaded bands represent the standard deviation. A comparable dataset for ATLAS is presented in Fig. 16 , confirming the expected luminosity change due to beam–beam effects originating from another IP. Relative luminosity changes for each step in all tests are summarized in Table 5 . These values are computed relative to the average of adjacent head-on points, which removes the linear luminosity decay over time. Assuming slow change of beam parameters, the results within each test are expected to be consistent. Fig. 15 CMS specific luminosity, normalized to the first measured both IPs head-on point (blue), during the first step-function scan at IP1. The measurements with ATLAS fully separated are shown in red. The error bars indicate error on the mean Full size image Fig. 16 ATLAS specific luminosity, normalized to the first measured both IPs head-on point (blue), during the step-function scan at IP5. The points for which IP5 was separated are shown in red. The error bars indicate the statistical error Full size image Table 5 Measured luminosity shifts in each of full separation steps – summary of all three step-function scans performed for IP1 and IP5 as the witness IP. The beam-beam parameter at the start of each scan $\xi _\text {start}$ is included for reference Full size table All measurements are compiled in Fig. 17 and compared to COMBI simulation predictions. The results are plotted as a function of the beam–beam parameter (estimated as described in Sect. 4.4 ), following a reverse chronological order. Agreement with simulation is excellent across the entire range. Fig. 17 COMBI simulated luminosity enhancement induced by a single head-on collision as a function of the beam-beam parameter (dashed line), compared to the test results at both ATLAS (red points) and CMS (green points) Full size image Fig. 18 Measurements of beam-beam effects on beam widths shown with red points for each beam and plane, during the experiment with IP5 as the witness IP at $\xi =0.0056$ . COMBI predictions are shown in blue with calculated systematic error from the phase advance error between the IP and BSRT locations Full size image 6.2.2 Impact of beam-beam effects on single-beam sizes Complementary measurements of the bunch size from the BSRT are shown in Fig. 18 . Data from all steps and bunches are combined, and a linear trend corresponding to continuous emittance growth is subtracted using the difference between the first and last head-on steps. For comparison, COMBI simulation results are also included in the figure. The dominant source of uncertainty in COMBI predictions arises from discrepancies between the measured phase advance and the MADX model, with the largest observed deviation being $0.05\,[2\pi ]$ (as discussed in Sect. 5.3.1 ). This uncertainty is propagated using a modified form of Eq. ( 6 ) adapted to the BSRT location. The magnitude of this uncertainty depends on the specific phase advance configuration, as some are more sensitive to small deviations. This is reflected in the varying sizes of the error bands associated with each prediction. The most accurate predictions are obtained in the vertical plane of Beam 2, where the beam–beam–induced $\beta $ -beating propagates to the BSRT with a particularly favorable phase advance. In this configuration, the measured effect is also the most pronounced, and excellent agreement with COMBI simulations is observed. In other cases, the expected absolute effect is smaller, typically below 1%, and the influence of phase advance uncertainty becomes more significant, limiting the predictive reliability of the simulations. Experimentally, it is also possible that in these configurations the beam–beam–induced effect is too subtle to resolve, potentially masked by instrumental limitations. 6.2.3 Impact of beam-beam effects on tune spectra Figure 19 shows representative ADT spectra recorded during the step-function scans. These spectra are affected by coupling, noise, and various nonlinearities, so they do not follow an idealized analytical distribution. Nonetheless, the peak position can be extracted using a Gaussian fit (indicated by dashed lines), yielding an estimate of the mean tune shift. In the single-collision configuration (blue lines), unsuppressed coherent modes dominate the spectrum and partially obscure the underlying incoherent component. The Gaussian fit is therefore applied to the incoherent contribution between coherent lines, rather than to the global spectral maximum, which results in a larger uncertainty on the extracted peak position. Despite this, in each of the performed step-scans at both collision points, the single-collision tune shift, defined as twice the fitted mean shift, is found to be consistent within $\pm 5\%$ with the expected coherent beam–beam tune shift corresponding to the conditions of the respective scan. For head-on collisions with Gaussian transverse beam distributions, the coherent tune shift of the dipole mode is expected to be $Y\xi $ , where Y is the Yokoya factor ( $Y = 1.21$ for round beams) [ 34 ]. As expected, the two-collision configuration yields a tune shift approximately double that of the single-collision case [ 13 ]. Fig. 19 Measured vertical tune spectra for different collision configurations: reference nominal tune (red), without the presence of any collisions, compared to single collision (blue) and two collisions (green) at $\xi \simeq 0.0086$ . The dash-dotted lines in corresponding colors indicate Gaussian fits to the data used to extract mean tunes Full size image Fig. 20 Specific luminosity, normalized to the first head-on point, measured at ATLAS (top) and CMS (bottom) during beam separation at the other IPs, with horizontal separation applied in steps 2–4 and vertical separation in steps 6–8. The tested configurations are indicated in the legend. The blue line shows an interpolation of the head-on measurements, serving as a reference for relative luminosity changes Full size image 6.3 Head-on collisions in a three-IP configuration at $\sqrt{s} = 900\ \text {GeV}$ An additional head-on collision was included in the configuration using IP2 (ALICE), as it was the only IP where this was feasible. A specially designed filling scheme ensured uniform collision exposure across all bunches, allowing measurements to be combined and thereby reducing statistical uncertainties. However, no reliable luminosity data were available from IP2 during this experiment. Table 6 Luminosity shift per full separation step, calculated with respect to an interpolated reference obtained from the two closest head-on points. Results for horizontal and vertical separations are shown as Shift part 1 and Shift part 2, respectively Full size table 6.3.1 Impact of beam-beam effects on head-on luminosity A step-function scan was conducted by sequentially collapsing or separating one IP at a time, with the separation applied first in the horizontal plane and, in a second stage of the experiment, in the vertical plane. A comprehensive summary of the measured specific luminosity at ATLAS and CMS is presented in Fig. 20 . Exceptionally stable beam conditions during this test enabled measurements at both experiments under nearly identical beam–beam parameters. The beam–beam-induced luminosity changes observed during each step are quantified in Table 6 , referenced to the time-interpolated specific luminosity from three head-on points. Orbit-drift corrections are applied only to these head-on reference points and result in changes of the extracted luminosity shifts of at most 0.2%. Good agreement with COMBI simulations is observed across the separation steps, with measured shifts closely matching the simulated values for configurations involving IP1 and IP5. For single-IP separation, the level of agreement depends on the witness IP and the separation step. When IP1 is used as the witness, the luminosity shift in the first separation step is reproduced almost exactly by COMBI, while a smaller shift is observed in the second step. This behaviour is expected, as the beam–beam parameter decreases during the scan and the measurement becomes increasingly sensitive to residual orbit perturbations. For configurations involving IP2, the comparison is less conclusive. As no reliable luminosity data were available from IP2 during this experiment, it was not possible to precisely optimize the collision conditions at this IP. As a result, the actual beam–beam conditions at IP2 may differ from those assumed in the simulations, affecting the comparison. Residual orbit drifts affecting the separation steps cannot be fully corrected. As illustrated by the DOROS measurements in Fig. 21 , the orbit data contain both slow drifts and beam–beam–induced orbit offsets arising from changes in the collision configuration when other IPs are separated. While the beam–beam–induced offsets are included in the COMBI simulations, the residual orbit drifts are not. Since these contributions cannot be disentangled in the measurements, a full correction is not possible. Consequently, the effective beam–beam impact measured during the experiment may be weaker than that assumed in the simulations. Fig. 21 Residual separation between the two beams at all three IPs in horizontal (top) and vertical (bottom) planes, for test with observations at ATLAS, where the separation was performed first in the horizontal and then in the vertical plane at the other IPs Full size image 6.3.2 Tune shift scaling with the number of beam–beam interactions Analysis of the transverse tune spectra in this configuration is further complicated by the reappearance of coherent modes when beams are separated at one or more IPs, as predicted by simulation. In such cases, the maximum tune shift is determined from the frequency difference between the $\pi $ and $\sigma $ modes, rather than from fitting the incoherent peak. For configurations where all three IPs were in collision, coherent modes remained suppressed due to the retained symmetry between IP1 and IP5. Figure 22 presents the measured tune shifts for all tested collision configurations. The unperturbed tune measurement, shown at ‘0’ collisions, serves as the reference from which all tune shifts are calculated. An empirical uncertainty of 0.001 is assumed for each independent measurement. Both transverse planes and separation directions are included to demonstrate consistency. As expected, the measured tune shift scales linearly with the number of head-on collisions. Fig. 22 Measured mean tune shift as a function of the number of head-on collisions, shown for both transverse planes and for both separation directions (horizontal and vertical) used in the experiment. The separation refers to how collisions were removed: it was applied in only one transverse direction at a time (H sep. or V sep.) at the non-colliding interaction points, which are not explicitly labeled on the x-axis Full size image 6.4 Beam-beam signatures during transverse-separation scans at $\sqrt{s} = 900\ \text {GeV}$ In the van der Meer (vdM) calibration method, beam–beam corrections are applied separately at each separation step. Therefore, it is essential to validate the simulated dependence of luminosity shifts on nominal beam separation. Full vdM-like scans were performed in both transverse planes at ATLAS and CMS, using a configuration where only these two IPs were in collision. 6.4.1 Impact of beam–beam effects on the luminosity scan-curves Figure 23 shows the luminosity curves measured at ATLAS, acting as the witness IP, during a separation scan performed at CMS. The curves are fitted with Gaussian profiles for visual guidance. The beam separation at CMS is expressed in units of the measured transverse beam size, $\sigma _{\textrm{meas}}$ , which was found to be smaller than the nominal beam size, $\sigma _{\textrm{nom}}$ , used to define the scan steps. As a result, scan points corresponding to nominal separations beyond $6\,\sigma _{\textrm{nom}}$ extend to larger values when expressed in units of $\sigma _{\textrm{meas}}$ . To account for emittance evolution during the scan, both linear and exponential fits are applied to the head-on luminosity points before, after, and in the middle of the scan. These trends are subtracted from the data in Fig. 23 , and all measurements are presented relative to this fitted head-on reference (normalized to unity). Fig. 23 ATLAS specific luminosity relative changes (blue points) during separation scans in x -plane (top) and y -plane (bottom) at CMS, corrected for exponential (exp. corr.) luminosity decay. Gaussian function with a constant fitted to these points is shown with blue dashed line and compared to fit performed on the data corrected for linear luminosity change (lin. corr.) in time (dotted line). These are compared to COMBI simulation predictions (purple solid line) Full size image At CMS, similar scans were performed; however, degraded beam stability was observed during this period of the fill. The scan results exhibit systematic deviations, especially at large separation, which could not be symmetrically corrected using simple linear or exponential models. Despite these issues, all four scan cases (two transverse planes at two IPs) yield a total specific luminosity reduction of approximately 2.4%, consistent with expectations based on the measured beam–beam parameter $\xi $ . Comparison with COMBI simulations reveal up to 30% discrepancy for IP5 as the witness IP, while agreement improves to within 10% for measurements at IP1. These differences are comparable to those observed in Table 6 , suggesting a common underlying origin. While a finer scan granularity can improve the description of the steep-response region below $3\,\sigma _{\textrm{meas}}$ , it would not address the dominant discrepancy between head-on and fully separated luminosity, which is likely driven by simulation assumptions and scan-related systematics. 6.4.2 Beam–beam-induced orbit deflections Beam–beam deflections are clearly observed during separation scans. Figure 24 shows the fully corrected residual horizontal beam displacement at IP1, measured with the DOROS beam position monitors, as a function of the beam separation. This representation allows a direct comparison with the analytical beam–beam deflection prediction based on the Bassetti–Erskine formalism [ 35 ]. The raw orbit data were corrected for slow orbit drifts by aligning the beam positions at the luminosity optimization points before and after each scan. Deviations of the applied beam separation from the nominal settings were accounted for by fitting a common length scale simultaneously with the beam–beam deflection. This approach avoids an overestimation of the separation calibration. The measured orbit displacement at the DOROS location was corrected for the contribution arising from the beam–beam deflection angle by applying a geometric amplification factor, proportional to the distance between the IP and the BPM in the small-angle approximation. After all corrections, the measured beam–beam deflection as a function of beam separation closely follows the analytical prediction. The maximum horizontal deflection observed at IP1 is 10.2 $\upmu $ m, compared to an expected value of 8.4 $\upmu $ m, with agreement within the total measurement uncertainty. The dominant uncertainty contribution (3.5 $\upmu $ m, corresponding to about 90% of the total) arises from a possible residual half crossing angle at the IP of up to 0.5 $\upmu $ rad. Additional systematic uncertainties include the fitted separation length scale, tune uncertainty of 0.005, and an extended 2 % uncertainty on $\beta ^$ . Statistical uncertainties are negligible for most separation steps. Fig. 24 Orbit offset caused by the beam–beam interaction measured by DOROS BPMs in the horizontal plane at IP1 during the horizontal separation scan. Analytical prediction is shown for comparison [ 35 ] Full size image 6.4.3 Impact of the beam-beam effects on beam sizes and tune shift Separation-dependent changes in beam size were measured using the synchrotron light monitor. Figure 25 shows good consistency across individual bunch measurements, as well as strong agreement with COMBI simulations. This reinforces the utility of the BSRT as a reliable diagnostic for beam–beam effects. The uncertainties shown are statistical only and are treated as uncorrelated between scan points. The normalization to the zero-separation value is applied for visualization purposes and does not impose a constraint on that point. Fig. 25 BSRT beam-size measurement in the vertical plane for each bunch (points) during horizontal (top) and vertical (bottom) separation scans at IP1. The exponential evolution correction is applied based on adjacent head-on points (before, in the middle and after the scan) to highlight the relative changes. COMBI predictions are shown (gray dashed line) for comparison Full size image Transverse tune shifts were extracted from ADT-measured spectra. As illustrated in Fig. 26 , the mean tune shift evolution is reconstructed for both the scanning and non-scanning planes. The measured tune shifts closely follow the expected COMBI trends, further validating the simulation models used for beam–beam corrections. Fig. 26 Measured tune shift in units of the beam-beam parameter $\xi _{BB}$ for the horizontal (blue) and vertical (orange) planes, during a horizontal separation scan at IP1. The error bars indicate typically assumed conservative empirical systematic error on the tune measurement of 0.001. The solid curves represent the COMBI prediction Full size image 6.5 Beam-beam signatures during vdM scans at $\sqrt{s} = 13.6\ \text {TeV}$ During the first vdM calibration of LHC Run 3, an unusually high beam–beam parameter was recorded, offering a unique opportunity to study beam–beam effects under enhanced interaction conditions. These effects are examined using online ATLAS luminosity data, with representative results shown in Fig. 27 . The example focuses on a standard vdM scan pair conducted at CMS, while monitoring luminosity at the non-scanning IP (ATLAS). The plotted luminosity changes are normalized to the central scan point, where both IPs are in head-on collision. Variations in this normalized luminosity across the scan directly reflect the impact of beam–beam interaction at the scanning IP on the luminosity at the non-scanning IP. These patterns are reproducible across multiple scan pairs acquired within the same fill. At the largest nominal separations (approximately $\pm 0.6\,$ mm), the beams are sufficiently separated to eliminate beam–beam effects entirely. Notably, a qualitative difference is observed between the horizontal and vertical scan planes, an asymmetry that is not present in the above presented dedicated experiment. This behavior is attributed to differences in the phase advance between IPs in the standard vdM optics configuration. For vertical ( y ) scans, experimental data agree well with simulation, confirming the robustness of the modeling under these conditions. In contrast, the horizontal ( x ) scans exhibit a systematic discrepancy: the observed beam–beam impact is broader than that predicted by simulation. The model assumes that the separation steps are accurately defined using the beam size measured during the vdM scans, but this mismatch may suggest an unmodeled phase advance deviation or another optics imperfection. The total difference between data and simulation is within 0.2%. This dataset originates from a standard vdM calibration, for which only online luminosity measurement was available, and many contributions to the overall vdM uncertainty, such as bunch-to-bunch variations, averaging over different bunch families, and emittance evolution during the scan, are not disentangled. Therefore, the 0.2% difference should be interpreted as a consistency check under typical operational conditions rather than a precision validation of the beam–beam model. Nevertheless, this indicates that the model remains sufficiently accurate for beam–beam corrections in vdM analyses. Fig. 27 Changes from the separation-dependent beam-beam effects at the non-scanning IP – ATLAS online luminosity shown with points. Data comes from a standard vdM scan pair during the 2022 pp luminosity calibration. Separation steps and direction are indicated with gray markers. The last CMS scan is shown, with the highest beam-beam parameter of $ \xi =5.3 \times 10^{-3} $ . The corresponding COMBI simulation results are also shown (triangles) Full size image 7 Conclusions Beam–beam effects were directly observed and quantitatively validated at the LHC. The impact of beam–beam interactions on head-on luminosity, bunch sizes, tune spectra, and beam orbit positions was measured across various collision configurations. The observed effects align closely with COMBI simulations. In particular, the luminosity enhancement due to amplitude-dependent $\beta $ -beating was measured for the first time and is found to agree with expectations over a wide range of the beam–beam parameter $\xi $ . Together with the observed linear scaling with respect to the beam–beam parameter, this provides experimental confirmation of the numerical studies presented in [ 13 ]. Consistent signatures were found across multiple scan configurations, including two- and three-IP setups, and the characteristic tune shifts were well reproduced by both measurement and simulation. Accurate instrumentation and experimental design were essential. Key to the results was the stepwise collision scheme, isolating individual beam–beam contributions, with the implementation of the witness IP. Synchrotron light measurements from the BSRT, especially in the vertical plane of Beam 2, showed excellent agreement with COMBI predictions and bunch-by-bunch consistency. Tune spectra from the ADT system provided direct access to mean tune shifts, while orbit data from the DOROS system enabled measurement of beam–beam deflection down to 1 $\upmu $ m, matching analytical estimates. These tools, combined with precise control of IP configurations, enabled a comprehensive cross-check of beam–beam models. Some systematic limitations were identified. During one of the tests, beam stability during the scans was insufficient to allow for symmetric modeling of luminosity decay in both transverse planes. In several scans, the nominal separation steps were based on overestimated beam sizes, resulting in larger-than-expected scan intervals and less sensitivity at low separation. Beam–beam deflection measurements showed sign reversals at high separation, attributed to nonlinear orbit distortions from hysteresis effects in superconducting magnets. Phase advance errors between IPs and diagnostic systems further reduced the accuracy of simulations in some configurations. Targeted improvements would allow for higher precision and new insights. Finer separation steps, especially below $3\,\sigma $ , would improve sensitivity to nonlinear beam–beam effects and allow more accurate comparison with models. Suppressing coherent beam–beam modes via phase advance tuning of not only two but all three IPs would improve tune shift measurements. Moreover, such optimal phase advance configuration between the IPs can be exploited to enhance luminosity. This effect could be directly propagated into at least a few percent increase in the total collected integrated luminosity, as discussed in Ref. [ 36 ]. Aligning the phase advance between IPs and diagnostic devices, particularly the BSRT, would reduce sensitivity to optics errors, enabling better beam size measurements for both beams and transverse planes. Moreover, the suppression of the coherent beam–beam modes facilitates the observation and analysis of the central, incoherent part of the tune spectrum, which is otherwise obscured by the dominant coherent peaks. Further experimental strategies should also focus on directly probing the beam–beam interaction at its point of occurrence. This would require access to the full transverse beam distributions with high spatial resolution and sensitivity, going beyond projected sizes or centroids. Such measurements could directly reveal the nonlinear distortions and tails induced by the beam–beam force. In parallel, validating correction schemes that parametrize multi-IP beam–beam effects on luminosity via equivalent tune shifts remains essential. These models underpin practical luminosity corrections in complex fill configurations and would benefit from systematic benchmarking. References S. Meer, Calibration of the effective beam height in the ISR. Technical report, CERN, Geneva (1968). https://cds.cern.ch/record/296752 C. Rubbia, Measurement of the luminosity of $p{{\overline{p}}}$ collider with a (generalized) Van der Meer Method. Technical report, CERN, Geneva (1977). https://cds.cern.ch/record/1025746 ATLAS collaboration, Luminosity determination in $pp$ collisions at $\sqrt{s}=13$ TeV using the ATLAS detector at the LHC. Eur. Phys. J. C 83 (10), 982 (2023). https://doi.org/10.1140/epjc/s10052-023-11747-w . arXiv:2212.09379 CMS Collaboration, Precision luminosity measurement in proton-proton collisions at sqrts = 13 TeV with the CMS detector. Technical report, CERN, Geneva (2025). https://cds.cern.ch/record/2940794 W. Herr, T. Pieloni, Beam-Beam Effects. Technical report, CERN (2014). Contribution to the CAS–CERN Accelerator School: Advanced Accelerator Physics Course, Trondheim, Norway, 18–29 Aug 2013, p. 29. https://doi.org/10.5170/CERN-2014-009.431 . https://cds.cern.ch/record/1982430 A.W. Chao, Beam-beam instability. AIP Conf. Proc. 127 , 201–242 (1985). https://doi.org/10.1063/1.35187 Article ADS Google Scholar E.D. Courant, H.S. Snyder, Theory of the alternating-gradient synchrotron. Ann. Phys. 3 , 1–48 (1958). https://doi.org/10.1016/0003-4916(58)90012-5 Article ADS Google Scholar P. Bambade, R. Erickson, W.A. Koska, W. Kozanecki, N. Phinney, S.R. Wagner, Observation of beam-beam deflections at the interaction point of the SLAC Linear Collider. Phys. Rev. Lett. 62 (25), 2949–2952 (1989). https://doi.org/10.1103/PhysRevLett.62.2949 Article ADS Google Scholar W.A. Koska, P. Bambade, W. Kozanecki, N. Phinney, S.R. Wagner, Beam-beam deflection as a beam tuning tool at the slac linear collider. Nucl. Instrum. Methods A 286 , 32 (1990). https://doi.org/10.1016/0168-9002(90)90203-I Article ADS Google Scholar C. Bovet, M.D. Hildreth, M. Lamont, H. Schmickler, J. Wenninger, Luminosity optimisation using beam-beam deflections at LEP, in Conf. Proc. C (1996). https://cds.cern.ch/record/306910 D. Brandt, W. Herr, M. Meddahi, A. Verdier, Is lep beam-beam limited at its highest energy? in Proceedings of the 1999 Particle Accelerator Conference (PAC ’99), New York , pp. 3005–3007. IEEE/JACoW. THP25; CERN report CERN-SL-99-030-AP (1999). https://accelconf.web.cern.ch/p99/PAPERS/THP25.pdf W. Kozanecki, T. Pieloni, J. Wenninger, Observation of Beam-beam Deflections with LHC Orbit Data. Technical report, CERN (2013). https://cds.cern.ch/record/1581723?ln=en A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84 , 17 (2024). https://doi.org/10.48550/arXiv.2306.10394 Article ADS Google Scholar T. Pieloni, A study of beam-beam effects in hadron colliders with a large number of bunches. PhD Thesis, École Polytechnique Fédérale de Lausanne (2008). https://doi.org/10.5075/epfl-thesis-4211 ATLAS Collaboration, Improved luminosity determination in pp collisions at $\sqrt{s} = 7$ TeV using the ATLAS detector at the LHC. Eur. Phys. J. C 73 , 2518 (2013). https://doi.org/10.1140/epjc/s10052-013-2518-3 C.M.S. Collaboration, Development of the cms detector for the cern lhc run 3. J. Instrum. 19 (05), 05064 (2024). https://doi.org/10.1088/1748-0221/19/05/p05064 Article Google Scholar C. Barschel, M. Ferro-Luzzi, J.-J. Gras, M. Ludwig, P. Odier, S. Thoulet, Results of the LHC DCCT calibration studies. Technical Report CERN-ATS-Note-2012-026 PERF (2012). https://cds.cern.ch/record/1425904?ln=en G. Anders et al., Study of the relative LHC bunch populations for luminosity calibration. Technical Report. CERN-ATS-Note-2012-028 PERF (2012). https://cds.cern.ch/record/1427726 G. Trad, Development and Optimisation of the SPS and LHC Beam Diagnostics Based on Synchrotron Radiation Monitors Presented 22 Jan 2015. Presented 22 Jan 2015. https://cds.cern.ch/record/2266055 R. Tomás, O. Brüning, M. Giovannozzi, P. Hagen, M. Lamont, F. Schmidt, G. Vanbavinckhove, M. Aiba, R. Calaga, R. Miyamoto, Cern large hadron collider optics model, measurements, and corrections. Phys. Rev. ST Accel. Beams 13 , 121004 (2010). https://doi.org/10.1103/PhysRevSTAB.13.121004 R. Tomás, M. Aiba, A. Franchi, U. Iriso, Review of linear optics measurement and correction for charged particle accelerators. Phys. Rev. Accel. Beams 20 , 054801 (2017). https://doi.org/10.1103/PhysRevAccelBeams.20.054801 Article ADS Google Scholar O. Karacheban, P. Tsrunchev, on behalf of CMS, Emittance scans for cms luminosity calibration. EPJ Web Conf. 201 , 04001 (2019). https://doi.org/10.1051/epjconf/201920104001 F. Carlier, R. Tomás, Accuracy and feasibility of the ${\beta }^{*}$ measurement for lhc and high luminosity lhc using $k$ modulation. Phys. Rev. Accel. Beams 20 , 011005 (2017). https://doi.org/10.1103/PhysRevAccelBeams.20.011005 Article ADS Google Scholar M. Söderén, J. Komppula, G. Kotzian, S. Rains, D. Valuch, ADT and Obsbox in LHC Run 2, plans for LS2, pp. 165–171 (2019) A. Boccardi, M. Gasior, R. Jones, P. Karlsson, R. Steinhagen, First results from the lhc bbq tune and chromaticity systems. Technical report, CERN, Geneva (2009). https://cds.cern.ch/record/1156349 M. Gasior, G. Baud, J. Olexa, G. Valentino, First Operational Experience with the LHC Diode ORbit and OScillation (DOROS) System, p. 07 (2017) https://doi.org/10.18429/JACoW-IBIC2016-MOPG07 A.W. Chao, Coherent beam-beam effects, in Frontiers of Particle Beams: Intensity Limitations (Springer, Berlin, Heidelberg, 1992), pp. 363–414. https://doi.org/10.1007/3-540-55250-2_36 J. Wańczyk, A. Dabrowski, D. Stickland, T. Pieloni, W. Kozanecki, X. Buffat, Impact of multiple beam-beam encounters on LHC absolute-luminosity calibrations by the van der Meer method. JACoW IPAC 2023 , 053 (2023). https://doi.org/10.18429/JACoW-IPAC2023-WEPA053 Article Google Scholar J. Wańczyk, Precision luminosity measurement at hadron colliders. PhD Thesis, École Polytechnique Fédérale de Lausanne (2024). https://doi.org/10.5075/epfl-thesis-10500 T. Pieloni, J. Barranco, X. Buffat, P.C. Jorge, L.M. Medrano, C. Tambasco, R. Tomás, Dynamic beta and beta-beating effects in the presence of the beam-beam interactions in 57th ICFA Advanced Beam Dynamics Workshop on High-Intensity and High-Brightness Hadron Beams , p. 027 (2016). https://doi.org/10.18429/JACoW-HB2016-MOPR027 F. Carlier, et al. Challenges of K-modulation measurements in the LHC Run 3, in Proceedings of IPAC’23. IPAC’23—14th International Particle Accelerator Conference (JACoW Publishing, Geneva, Venice, 2023), pp. 531–534. https://doi.org/10.18429/JACoW-IPAC2023-MOPL014 Y. Alexahin, A study of the coherent beam-beam effect in the framework of the Vlasov perturbation theory. Nucl. Instrum. Methods Phys. Res., Sect. A 480 (1–3), 253–288 (2002). https://doi.org/10.1016/S0168-9002(01)01219-0 Article ADS Google Scholar R. Calaga, W. Herr, G. Papotti, T. Pieloni, X. Buffat, S. White, R. Giachino, Coherent beam-beam mode in the LHC. CERN (2014). https://doi.org/10.5170/CERN-2014-004.227 . http://cds.cern.ch/record/1957039 K. Yokoya, Y. Funakoshi, E. Kikutani, H. Koiso, J. Urakawa, Tune shift of coherent beam-beam oscillations. Part. Accel. 27 , 181–186 (1990) Google Scholar M. Bassetti, G.A. Erskine, Closed expression for the electrical field of a two-dimensional gaussian charge. Technical report, CERN, Geneva (1980). https://cds.cern.ch/record/122227 J. Wańczyk, T. Pieloni, Optimizing beam-beam beta-beating for luminosity enhancement at the LHC. JACoW IPAC 2025 , 018 (2025) Google Scholar Download references
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Article # First measurements of beam-beam effects in beam-separation, luminosity-calibration scans at the LHC - Regular Article - Experimental Physics - Experimental Physics - [Open access](https://www.springernature.com/gp/open-science/about/the-fundamentals-of-open-access-and-open-research) - Published: 17 March 2026 - Volume 86, article number 282, (2026) - [Cite this article](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#citeas) You have full access to this [open access](https://www.springernature.com/gp/open-science/about/the-fundamentals-of-open-access-and-open-research) article [Download PDF](https://link.springer.com/content/pdf/10.1140/epjc/s10052-026-15476-8.pdf) [Save article](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/save-research?_csrf=skr420D_Tsy443125TKQwosqfCsJrZRn) [View saved research](https://link.springer.com/saved-research) [![](https://media.springernature.com/w72/springer-static/cover-hires/journal/10052?as=webp) The European Physical Journal C](https://link.springer.com/journal/10052) [Aims and scope](https://link.springer.com/journal/10052/aims-and-scope) [Submit manuscript](https://mc.manuscriptcentral.com/epjc) First measurements of beam-beam effects in beam-separation, luminosity-calibration scans at the LHC [Download PDF](https://link.springer.com/content/pdf/10.1140/epjc/s10052-026-15476-8.pdf) - [Joanna Wańczyk](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#auth-Joanna-Wa_czyk-Aff1-Aff2) [ORCID: orcid.org/0000-0002-8562-1863](https://orcid.org/0000-0002-8562-1863)[1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Aff1),[2](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Aff2), - [Tatiana Pieloni](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#auth-Tatiana-Pieloni-Aff2) [ORCID: orcid.org/0000-0002-3218-0048](https://orcid.org/0000-0002-3218-0048)[2](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Aff2), - [Xavier Buffat](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#auth-Xavier-Buffat-Aff1) [ORCID: orcid.org/0000-0002-2338-8808](https://orcid.org/0000-0002-2338-8808)[1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Aff1), - [Anne Dabrowski](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#auth-Anne-Dabrowski-Aff1) [ORCID: orcid.org/0000-0003-2570-9676](https://orcid.org/0000-0003-2570-9676)[1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Aff1), - [David Stickland](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#auth-David-Stickland-Aff3) [ORCID: orcid.org/0000-0003-4702-8820](https://orcid.org/0000-0003-4702-8820)[3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Aff3) & - … - [Rogelio Tomas](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#auth-Rogelio-Tomas-Aff1) [ORCID: orcid.org/0000-0002-9857-1703](https://orcid.org/0000-0002-9857-1703)[1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Aff1) Show authors ## Abstract At the CERN Large Hadron Collider (LHC), absolute luminosity calibrations obtained by the van der Meer (vdM) method are affected by the mutual electromagnetic interaction of the two beams. In the last few years, increasingly stringent, physics-driven requirements on the accuracy of the absolute luminosity scale, combined with the unparalleled precision of the luminosity measurements achieved by the LHC experiments, motivated comprehensive simulation studies to accurately model beam–beam effects and provide simulation-based corrections to vdM\-based luminosity calibrations. This paper reports the first attempt at an experimental validation of that model, in dedicated accelerator measurements under controlled conditions, of the predicted impact of beam-beam effects on the luminosity measured at the LHC under luminosity-calibration conditions. This is the first measurement of beam-beam interaction effects on luminosity at the LHC. The results show good agreement with the predictions of multiparticle simulations, and justify the recently proposed strategy for correcting beam-beam biases on absolute luminosity calibrations at hadron colliders. This marks a critical step forward in precise luminosity calibration for past, current, and future datasets, including the High-Luminosity Large Hadron Collider era. ### Explore related subjects Discover the latest articles, books and news in related subjects, suggested using machine learning. - [Accelerator Physics](https://link.springer.com/subjects/accelerator-physics) - [Experimental Particle Physics](https://link.springer.com/subjects/experimental-particle-physics) - [Measurement Science and Instrumentation](https://link.springer.com/subjects/measurement-science-and-instrumentation) - [Metrology](https://link.springer.com/subjects/metrology) - [Optical Metrology](https://link.springer.com/subjects/optical-metrology) - [Particle Physics](https://link.springer.com/subjects/particle-physics) ## 1 Introduction A precise determination of the absolute luminosity scale is essential for a wide range of LHC measurements, as it directly affects the normalization of many key physics cross sections. At the LHC, this calibration relies primarily on the van der Meer (vdM) method \[[1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR1 "S. Meer, Calibration of the effective beam height in the ISR. Technical report, CERN, Geneva (1968). https://cds.cern.ch/record/296752 "), [2](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR2 "C. Rubbia, Measurement of the luminosity of $$p{{\overline{p}}}$$ p p ¯ collider with a (generalized) Van der Meer Method. Technical report, CERN, Geneva (1977). https://cds.cern.ch/record/1025746 ")\], which uses dedicated beam-separation scans performed under specially tailored beam conditions to relate measured interaction rates to the absolute luminosity inferred from beam parameters. While sub-percent uncertainties on the integrated luminosity have been achieved for Run-2 data by ATLAS \[[3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR3 "ATLAS collaboration, Luminosity determination in $$pp$$ pp collisions at $$\sqrt{s}=13$$ s = 13 TeV using the ATLAS detector at the LHC. Eur. Phys. J. C 83(10), 982 (2023). https://doi.org/10.1140/epjc/s10052-023-11747-w . arXiv:2212.09379 ")\] and CMS \[[4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR4 "CMS Collaboration, Precision luminosity measurement in proton-proton collisions at sqrts = 13 TeV with the CMS detector. Technical report, CERN, Geneva (2025). https://cds.cern.ch/record/2940794 ")\], further improvements and robust experimental validation of beam-dynamical effects remain essential. For relativistic proton beams colliding with zero crossing angle, which is the typical configuration for the vdM calibration, the luminosity at a given interaction point (IP) of the LHC is proportional to the overlap integral of the particle-density distributions \$\\rho \_{\\text {1,i}}\$, \$\\rho \_{\\text {2,i}}\$ in bunch-pair i \[[5](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR5 "W. Herr, T. Pieloni, Beam-Beam Effects. Technical report, CERN (2014). Contribution to the CAS–CERN Accelerator School: Advanced Accelerator Physics Course, Trondheim, Norway, 18–29 Aug 2013, p. 29. https://doi.org/10.5170/CERN-2014-009.431 . https://cds.cern.ch/record/1982430 ")\]: \$\$\\begin{aligned} {\\mathcal {L}}\_{inst}&= 2c f\_{\\text {rev}} \\sum ^{N\_b}\_i n\_{\\text {1,i}}, n\_{\\text {2,i}} \\nonumber \\\\&\\quad \\times \\iiiint ^{+\\infty }\_{-\\infty }\\rho \_{\\text {1,i}}(x,y,z-ct) \\nonumber \\\\&\\quad \\times \\rho \_{\\text {2,i}}(x,y,z+ct)\\,dxdydzdt, \\end{aligned}\$\$ (1) where c is the speed of light, \$f\_{\\text {rev}}\$ is the LHC revolution frequency, \$N\_b\$ is the number of colliding bunches, and \$n\_{\\text {1,i}}, n\_{\\text {2,i}}\$ are the corresponding total charges per bunch for each colliding pair. Although only a very small fraction of the beam particles actually collide, the two opposing beams interact with each other electromagnetically, a dynamical process known as the beam-beam interaction \[[5](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR5 "W. Herr, T. Pieloni, Beam-Beam Effects. Technical report, CERN (2014). Contribution to the CAS–CERN Accelerator School: Advanced Accelerator Physics Course, Trondheim, Norway, 18–29 Aug 2013, p. 29. https://doi.org/10.5170/CERN-2014-009.431 . https://cds.cern.ch/record/1982430 "), [6](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR6 "A.W. Chao, Beam-beam instability. AIP Conf. Proc. 127, 201–242 (1985). https://doi.org/10.1063/1.35187 ")\]. In the LHC, as in any other synchrotron, the beams are not continuous but are divided into discrete “bunches”, each with a high proton density, containing approximately \$10^{11}\$ protons over a bunch length of 7.5–9 cm (in terms of RMS). This electromagnetic interaction occurs when the two beams share a common beam pipe, and is governed by a non-linear force that depends on the radial distance r of a test particle to the center of the opposing “source” bunch with n protons Gaussian-distributed in the transverse planes (valid approximation for the LHC beams): \$\$\\begin{aligned} F = - \\frac{ne^2}{2\\pi \\epsilon \_0r}\\biggl (1-\\exp \\biggl \[ -\\frac{r^2}{2\\sigma ^2}\\biggr \] \\biggr ), \\end{aligned}\$\$ (2) where e is the elementary proton charge, \$\\epsilon \_0\$ is the vacuum permittivity, and \$\\sigma \$ is the RMS radius of the transverse charge distribution within the bunch, in a simplified case of a round shape \$(\\sigma = \\sigma \_x = \\sigma \_y)\$. At very small amplitudes \$(r\\rightarrow 0)\$ the force can be linearized giving the expression for the so called beam-beam parameter, which is often used to assess the strength of the force: \$\$\\begin{aligned} \\xi = \\frac{nr\_p\\beta ^\}{4\\pi \\gamma \\sigma ^2} = \\frac{nr\_p}{4\\pi \\epsilon \_n}, \\end{aligned}\$\$ (3) where \$r\_p\$ is the classical proton radius, \$\\beta ^\\$ is the optical beta-function at the interaction point (IP) – one of the Courant–Snyder parameters \[[7](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR7 "E.D. Courant, H.S. Snyder, Theory of the alternating-gradient synchrotron. Ann. Phys. 3, 1–48 (1958). https://doi.org/10.1016/0003-4916(58)90012-5 ")\], \$\\gamma \$ is the Lorentz factor, and \$\\epsilon \_n\$ is the normalized emittance. Beam-beam effects were first observed almost 40 years ago \[[8](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR8 "P. Bambade, R. Erickson, W.A. Koska, W. Kozanecki, N. Phinney, S.R. Wagner, Observation of beam-beam deflections at the interaction point of the SLAC Linear Collider. Phys. Rev. Lett. 62(25), 2949–2952 (1989). https://doi.org/10.1103/PhysRevLett.62.2949 ")\] at the lepton colliders, and have been used extensively for accelerator diagnostics and optimization at both SLC and LEP \[[9](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR9 "W.A. Koska, P. Bambade, W. Kozanecki, N. Phinney, S.R. Wagner, Beam-beam deflection as a beam tuning tool at the slac linear collider. Nucl. Instrum. Methods A 286, 32 (1990). https://doi.org/10.1016/0168-9002(90)90203-I "),[10](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR10 "C. Bovet, M.D. Hildreth, M. Lamont, H. Schmickler, J. Wenninger, Luminosity optimisation using beam-beam deflections at LEP, in Conf. Proc. C (1996). https://cds.cern.ch/record/306910 "),[11](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR11 "D. Brandt, W. Herr, M. Meddahi, A. Verdier, Is lep beam-beam limited at its highest energy? in Proceedings of the 1999 Particle Accelerator Conference (PAC ’99), New York, pp. 3005–3007. IEEE/JACoW. THP25; CERN report CERN-SL-99-030-AP (1999). https://accelconf.web.cern.ch/p99/PAPERS/THP25.pdf ")\]. At the LHC, they were first observed in late Run 1 \[[12](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR12 "W. Kozanecki, T. Pieloni, J. Wenninger, Observation of Beam-beam Deflections with LHC Orbit Data. Technical report, CERN (2013). https://cds.cern.ch/record/1581723?ln=en ")\], but only in terms of the beam-beam induced coherent deflections. Single particle effects, in contrast, were presumed too small in hadron colliders that until recently, there was no motivation to measure them with a precision that would be meaningful in the context evoked here. Nonetheless, the full complexity of this interaction must be accounted for when performing the luminosity calibration via the vdM method, using the separation scans. These are designed to measure the detector-specific constant \$\\sigma \_{vis}\$ that relates the observed rate \$\\mu ^{vis}\$ to the absolute instantaneous luminosity \${\\mathcal {L}}\_{inst}\$: \$\$\\begin{aligned} \\mu ^{vis} = \\frac{{\\mathcal {L}}\_{inst}\\sigma \_{vis}}{f\_{rev}}. \\end{aligned}\$\$ (4) Under the assumption of uncorrelated particle densities in x and y planes, the transverse convolved bunch widths \$\\Sigma \_x, \\Sigma \_y\$ can be extracted from the measured beam-separation dependence of the collision rate in the corresponding direction. The combined information from these scans, and bunch intensities can be used to calculate the instantaneous luminosity at the head-on position. The proportionality between the measured rate at the centered head-on position \$\\mu \_{pk}\$ and the reconstructed luminosity from measured bunch parameters during the vdM calibration defines the luminometer-specific visible cross-section: \$\$\\begin{aligned} \\sigma \_{vis} = \\frac{2\\pi \\Sigma \_x\\Sigma \_y}{n\_1 n\_2} \\mu \_{pk}. \\end{aligned}\$\$ (5) If left uncorrected, the \$\\sigma \_{vis}\$ measurement by the vdM method is biased by the beam-separation dependence of the mutual electromagnetic interaction of the two beams: the colliding bunches experience deflection-induced orbit shifts, as well as optical distortions akin to the dynamic-\$\\beta \$ effect, that both depend on the transverse beam separation and must therefore be accounted for when deriving the absolute luminosity scale. The correction strategy discussed extensively in Ref. \[[13](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR13 "A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). https://doi.org/10.48550/arXiv.2306.10394 ")\] relies on simulation studies only, and thus requires experimental validation to establish its reliability. The present report summarizes a campaign of dedicated measurements carried out in the spring of 2022 at the LHC, aimed at confirming the accuracy of the simulation studies detailed in Ref. \[[13](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR13 "A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). https://doi.org/10.48550/arXiv.2306.10394 ")\], and at validating the beam-beam correction strategy used in the luminosity-calibration analyses of the ALICE, ATLAS, CMS and LHCb experiments. More specifically, the goal is to quantify the impact of the beam-beam interaction on the tune spectra, orbit, the transverse-density distribution of the colliding bunches, as well as, for the first time, on the luminosity, by systematically varying the strength or number of the beam-beam interactions. The measurements are repeated for different values of the beam-beam parameter to verify the scaling proposed in \[[13](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR13 "A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). https://doi.org/10.48550/arXiv.2306.10394 ")\], for various choices of the scanning IP and for several multi-IP configurations. The beam conditions are chosen to be representative of vdM calibration sessions at the LHC, but optimized so as to maximize both the sensitivity of the measurements, and the operational efficiency. This paper is organized as follows. The measurement strategy is detailed in Sect. [2](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec2), followed by the main ingredients of this accelerator experiment: beam conditions (Sect. [3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec3)), luminometers and accelerator instrumentation (Sect. [4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec4)), and optimization of the ring lattice to maximize the sensitivity of the measurements to beam–beam-induced optical distortions of the colliding bunches (Sect. [5](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec11)). Section [6](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec19) is devoted to the experimental characterization of the impact of beam-beam effects on the tune spectra, on the luminosity at different IPs, as well as on other observables such as transverse single-bunch widths and IP orbits. At each step, the results are confronted with the predictions of the COherent Multibunch Beam-beam Interactions (COMBI) tracking code \[[14](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR14 "T. Pieloni, A study of beam-beam effects in hadron colliders with a large number of bunches. PhD Thesis, École Polytechnique Fédérale de Lausanne (2008). https://doi.org/10.5075/epfl-thesis-4211 ")\], that produced most of the results detailed in Ref. \[[13](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR13 "A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). https://doi.org/10.48550/arXiv.2306.10394 ")\]. A collection of macroparticles is simulated turn by turn, based on a simplified model of the accelerator lattice using linear transfer and the self-consistent computation of the beam-beam forces at the IP. The conclusions are presented in Sect. [7](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec33). ## 2 Measurement strategy The beam-beam interaction at an interaction point, referred to here as the scanning IP, induces measurable changes in key beam parameters such as the tune spectrum, closed orbit, and transverse single-beam profiles. These effects can be observed, albeit with limited precision, using standard accelerator diagnostics by comparing measurements obtained under varying transverse beam separation at the scanning IP. However, the resulting modifications to the transverse density distribution of the colliding bunches remain below the sensitivity threshold of conventional instrumentation, including synchrotron-radiation-based beam-profile monitors. Beam-beam-induced variations typically lie in the 0.5–1% range, small enough that only a highly precise measurement of collision-rate changes can provide adequate sensitivity. Fig. 1 [![Fig. 1](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig1_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/1) Scheme of the LHC and its interaction points (IP) with indicated IP1 as witness IP while performing a scan at IP5 [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/1) Importantly, the beam-beam interaction induces changes in beam properties that are intrinsically entangled with the pp collisions when observing luminosity at the scanning IP. Consequently, under realistic collider conditions, it is not possible to directly access a reference luminosity signal that is free of beam-beam effects – unlike in simulation studies, where such effects can be explicitly disabled. In a multi-collision configuration, however, luminosity shifts at one or more other IPs (Fig. [1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig1)), where the beams are kept in continuous head-on collision and therefore act as non-scanning, or witness, IPs, can be used to monitor the separation-dependent beam–beam effects. These effects are induced at the scanning IP, where the beams are deliberately brought in and out of collision, and propagate around the rings to the witness IPs, where they manifest as changes in the measured luminosity. In principle, any of the four LHC IPs can serve as a scanning IP, and any non-scanning IP can be designated as the witness IP for a given measurement. In practice, instrumental and operational constraints at the time the experiment was carried out restrict the choice of witness IP to the ATLAS (IP1) and CMS (IP5) collision points; IPs 1, 2 or 5, or combinations thereof, are used as scanning IP(s). Collisions at IP8 were deliberately avoided because the large longitudinal offset of the LHCb collision point, that breaks the eight-fold symmetry of the LHC rings, precludes the possibility of both members of a colliding-bunch pair to collide at all four IPs. In addition, the large crossing angle at IP8 would complicate the interpretation of the beam-beam effects that would occur at that IP if some of the bunches collided there. The propagation of the beam–beam-induced, amplitude-dependent \$\\beta \$\-beating from the scanning to a witness IP is controlled by the betatron phase advance between these two IPs. Since the targeted signatures are delicate to measure at best, it is natural to try and enhance them by adjusting this phase advance so as to maximize the sensitivity of luminosity shifts at the witness IP to beam-beam effects at the scanning IP; in doing so, however, the overall tunes must be preserved. Given the central and symmetric roles played by IP1 and IP5 in this experiment, the adjustment of their relative phase advance drove the optimization procedure that is detailed in Sect. [5](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec11), and that achieved a threefold improvement in measurement sensitivity. Central to the quality of the measurements is the controlled variation of the beam-beam parameter at the scanning IP. This can be achieved either: - by using step scans, i.e. by fully separating the beams at the scanning IP in either the horizontal or the vertical plane, and then bringing them back into head-on collision; or - by using separation scans, i.e. by scanning the beams transversely with respect to each other in either the horizontal or the vertical plane; or - by taking advantage of the natural beam-intensity decay and emittance growth to progressively reduce the beam-beam parameter. Equally important is to ensure that the beams remain in head-on collision at the witness IP(s), such that reproducible luminosity shifts measured at these locations can be unambiguously correlated with controlled changes in the strength of the beam-beam interaction at the scanning IP. It was verified that the beam-beam-induced orbit shift and the potential non-closure of the orbit bumps used to control the beam separation at the scanning IP, did not significantly affect the actual separation, and thereby the measured collision rate, at the witness IP. In a few cases however, orbit drifts of uncontrolled origin ended up degrading the quality of some of the measurements. Simulations demonstrate \[[13](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR13 "A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). https://doi.org/10.48550/arXiv.2306.10394 ")\] that multi-IP effects strongly influence the magnitude of beam-beam biases to vdM calibrations. Their characterization, therefore, constitutes an essential component of this experiment. The scan protocol is a generalization of that in the two-IP case. One IP (say, IP1) is designated as the witness IP, with beams colliding head-on there and at IP2, and IP5; the beams are then taken out of collision at IP2 only, then at both IP2 and IP5, and finally returned to a three-collision configuration in the reverse order. Since the phase-advance combinations are different in each configuration, the corresponding luminosity shifts predicted at the witness IP are also different and can be meaningfully confronted with the data. ## 3 Beam conditions In the round-beam approximation, the beam-beam parameter \$\\xi \$ does not depend on the beam energy (Eq. ([3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Equ3))). Therefore, and in order to minimize the overhead associated with commissioning and operating the LHC in a non-standard optical configuration, as well as to allow for frequent refilling with different bunch patterns, the measurements were carried out at injection energy (\$450\\ \\text {GeV}\$ per beam), and under non-standard machine-protection conditions. The improvement in operational efficiency came at a cost: - the counting rate of the luminometers (or equivalently their visible cross-section) is about an order of magnitude smaller at \$\\sqrt{s} = 900\\ \\text {GeV}\$ than at \$13\\ \\text {TeV}\$, reflecting the combination of a 40% drop in the inelastic pp cross-section, a factor of two to three drop in the multiplicity of the final-state particles, and a significant softening of their momentum spectrum; - because of the total-intensity constraints dictated by machine-protection requirements, the injected beam could not exceed 4 bunches per beam (compared to 150 during a routine vdM\-calibration session), with a maximum allowed population of \$1.25\\times 10^{11}~p\$/bunch; - the combination of the low beam energy, that suppresses synchrotron-radiation damping, and of the large bunch intensity, that enhances intra-beam scattering, resulted in relatively rapid emittance growth and rather short single-beam lifetimes. To partially mitigate these effects, all collision-rate measurements are expressed in terms of specific luminosity, thereby automatically accounting for the beam-intensity decay. The bunch intensity was deliberately increased by about 25% beyond the vdM\-scan values typical of LHC Run 2, to maximize both the luminosity and the beam-beam parameter \$\\xi \$. The latter reached 0.010 per IP (Table [1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Tab1)), compared to 0.003–0.006 during normal pp vdM sessions. With a target injected emittance of \$1.5 \\ \\upmu \\text {m} \\cdot \\text {rad}\$, the \$\\beta \$ function at the IP set to \$\\beta ^\* = 11\\ \\text {m}\$, and zero crossing angle at IP1 and IP5, the statistical uncertainty affecting a typically 60-second long per-bunch luminosity measurement in the presence of head-on collisions, lay around 0.5%. Table 1 Range of beam-beam parameter values during the various stages of the experiment [Full size table](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/tables/1) The bunch patterns were chosen such that all bunches in each beam collided either at IP1 and IP5 only, or at all three of IP1, 2 and 5. The orbit-stabilization feedback system was turned off during the data-taking periods to prevent it from interfering with the beam–beam-induced orbit shift. The chromaticity was set to its standard value in physics fills, of +10 units, to guarantee coherent stability against the machine impedance, and in view of the very small number of bunches and of their large longitudinal spacing, the beam-stabilizing Landau octupoles were set to their minimum current (1A). The damping time of the bunch-by-bunch transverse feedback was set to 1000 turns, a rather loose setting intended to preserve longer the natural beam oscillation and therefore improve the precision of the tune measurements. ## 4 Beam instrumentation ### 4\.1 Luminometers The luminometer systems in use by the ATLAS and CMS collaborations at IP1 and IP5 are described in Refs. \[[15](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR15 "ATLAS Collaboration, Improved luminosity determination in pp collisions at $$\sqrt{s} = 7$$ s = 7 TeV using the ATLAS detector at the LHC. Eur. Phys. J. C 73, 2518 (2013). https://doi.org/10.1140/epjc/s10052-013-2518-3 "), [16](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR16 "C.M.S. Collaboration, Development of the cms detector for the cern lhc run 3. J. Instrum. 19(05), 05064 (2024). https://doi.org/10.1088/1748-0221/19/05/p05064 ")\] respectively; no luminosity measurements were available at IP2 during the collider experiment described in this paper. Maximizing the statistical sensitivity leads to choosing the luminosity algorithm with the highest possible acceptance, namely: - the hit rate per bunch crossing in the ATLAS Minimum Bias Trigger Scintillators (MBTS) \[[15](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR15 "ATLAS Collaboration, Improved luminosity determination in pp collisions at $$\sqrt{s} = 7$$ s = 7 TeV using the ATLAS detector at the LHC. Eur. Phys. J. C 73, 2518 (2013). https://doi.org/10.1140/epjc/s10052-013-2518-3 ")\], and - the occupancy in the CMS Hadron Forward Calorimeter (HFOC) \[[16](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR16 "C.M.S. Collaboration, Development of the cms detector for the cern lhc run 3. J. Instrum. 19(05), 05064 (2024). https://doi.org/10.1088/1748-0221/19/05/p05064 ")\]. In both cases, raw luminometer counts are converted to collision rates using the Poisson formalism \[[15](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR15 "ATLAS Collaboration, Improved luminosity determination in pp collisions at $$\sqrt{s} = 7$$ s = 7 TeV using the ATLAS detector at the LHC. Eur. Phys. J. C 73, 2518 (2013). https://doi.org/10.1140/epjc/s10052-013-2518-3 "), [16](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR16 "C.M.S. Collaboration, Development of the cms detector for the cern lhc run 3. J. Instrum. 19(05), 05064 (2024). https://doi.org/10.1088/1748-0221/19/05/p05064 ")\], and the instantaneous luminosity is averaged over typically 60-second time bins, that during the scans are synchronized with the scan steps. The statistical uncertainties are estimated from either the total number of raw luminometer counts per time bin (ATLAS), or from the RMS of the approximately 40 luminosity samples recorded in a given time bin (CMS). Since the absolute luminosity scale is irrelevant, and in order to simplify the interpretation of the results, all measurements in this paper are presented in terms of fractional shifts in the bunch-averaged specific luminosity, relative to a reference time that depends on the type of measurement considered. ### 4\.2 Bunch-charge measurement The total intensity of each beam is measured by a direct-current current transformer (DCCT), the absolute scale of which is calibrated against a very high precision pulse generator \[[17](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR17 "C. Barschel, M. Ferro-Luzzi, J.-J. Gras, M. Ludwig, P. Odier, S. Thoulet, Results of the LHC DCCT calibration studies. Technical Report CERN-ATS-Note-2012-026 PERF (2012). https://cds.cern.ch/record/1425904?ln=en ")\]. The bunch-by-bunch charge fractions, in turn, normalized to the total stored intensity, are measured, separately for the two beams, by Fast Beam Current Transformers (FBCT) \[[18](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR18 "G. Anders et al., Study of the relative LHC bunch populations for luminosity calibration. Technical Report. CERN-ATS-Note-2012-028 PERF (2012). https://cds.cern.ch/record/1427726 ")\]. The latter devices also provide the fill pattern, i.e. the relative location, at a given instant and around the two rings, of the nominally filled bunches. Since the absolute scale of the DCCTs is known to much better accuracy than that of the FBCTs, the bunch charges are typically computed as the product of the FBCT bunch-charge fractions and the total circulating beam intensity reported by the corresponding DCCT. ### 4\.3 Emittance measurement The synchrotron-light beam-profile monitors \[[19](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR19 "G. Trad, Development and Optimisation of the SPS and LHC Beam Diagnostics Based on Synchrotron Radiation Monitors Presented 22 Jan 2015. Presented 22 Jan 2015. https://cds.cern.ch/record/2266055 ")\], dubbed BSRTs (“Beam Synchrotron Radiation Telescope”), are mainly used to track emittance evolution over time, but they can also measure the relative changes in transverse single-beam size that result from beam–beam-induced \$\\beta \$\-beating. BSRT data are recorded every second; in what follows, they are presented averaged over one-minute intervals for easier comparison with other measurements. The absolute length scale of the BSRT profiles suffers from significant uncertainties, if only because the online optical corrections to the images can be updated only a few times per year, and are unable to track the evolution of the efficiency of the light sensors, that depends both on time and on the position of the light spot on the sensor array. For each beam therefore, the BSRT is complemented by a wire-scanner (WS) profile monitor that is located in the same straight section. The advantage of the WS is that its accuracy is significantly better than that of the BSRT; its down side, in the present context, is that the WS cannot acquire data continuously and must be triggered manually. For the results presented in this paper, the wire scanners were flown through the beams at the start of each group of measurements to provide single-beam emittance measurements that are as accurate as possible, and then at regular intervals thereafter. The BSRT is used to interpolate the time evolution of the emittance between two sets of WS measurements. To interpret measured RMS bunch widths in terms of emittance requires the knowledge of the \$\\beta \$ functions at the locations of the BSRT and the WS. These are determined by the phase-advance method, and compared in Table [2](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Tab2) to their model value computed using MAD-X; the agreement is typically better than 5%, and the worst disagreement amounts to 9%. These discrepancies are attributed to imperfections in the magnetic lattice. Uncertainties in the measurement mainly arise from the interpolation of the lattice functions between the two closest beam-position monitors (BPMs) where the \$\\beta \$\-functions are measured \[[20](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR20 "R. Tomás, O. Brüning, M. Giovannozzi, P. Hagen, M. Lamont, F. Schmidt, G. Vanbavinckhove, M. Aiba, R. Calaga, R. Miyamoto, Cern large hadron collider optics model, measurements, and corrections. Phys. Rev. ST Accel. Beams 13, 121004 (2010). https://doi.org/10.1103/PhysRevSTAB.13.121004 "), [21](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR21 "R. Tomás, M. Aiba, A. Franchi, U. Iriso, Review of linear optics measurement and correction for charged particle accelerators. Phys. Rev. Accel. Beams 20, 054801 (2017). https://doi.org/10.1103/PhysRevAccelBeams.20.054801 ")\], and either the BSRT or the WS location. Based on the MAD-X LHC lattice model, these uncertainties are estimated not to exceed 3%, and this value is assigned as the systematic uncertainty on the \$\\beta \$ functions used in BSRT- and WS-based emittance measurements. Table 2 \$\\beta \$ functions at the BSRT and WS locations, for the optical configuration detailed in Sect. [5\.3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec14) [Full size table](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/tables/2) The beam-averaged emittance, i.e. the average of the emittances of the beam-1 bunch and of the corresponding beam-2 bunch, can be obtained directly from emittance scans \[[22](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR22 "O. Karacheban, P. Tsrunchev, on behalf of CMS, Emittance scans for cms luminosity calibration. EPJ Web Conf. 201, 04001 (2019). https://doi.org/10.1051/epjconf/201920104001 ")\] at IP1 and/or IP5. In this approach, the convolved transverse bunch sizes measured using beam-separation scans are translated into emittances using the \$\\beta ^\\$ values determined by k\-modulation \[[23](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR23 "F. Carlier, R. Tomás, Accuracy and feasibility of the $${\beta }^{}$$ β ∗ measurement for lhc and high luminosity lhc using $$k$$ k modulation. Phys. Rev. Accel. Beams 20, 011005 (2017). https://doi.org/10.1103/PhysRevAccelBeams.20.011005 ")\] at the relevant IP (see Sect. [4\.3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec7)). The associated uncertainty (Table [3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Tab3)) is combined with the statistical uncertainty in convolved transverse width to estimate the error affecting the measured emittance. Comparison with beam-averaged emittances extracted from the single-beam profile monitors reveals excellent agreement between WS and emittance-scan results (Fig. [2](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig2)). The relative time-evolution of the BSRT emittances is consistent with that observed using the WS or emittance scans, but the absolute magnitudes differ significantly, especially in the horizontal plane. *Table 3 \$\\beta ^\\$ functions at the ATLAS and CMS IPs measured by k\-modulation. The target value in all cases is \$\\beta ^\=\$ 11 m* [Full size table](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/tables/3) Fig. 2 [![Fig. 2](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig2_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/2) Time evolution of the horizontal (top) and vertical (bottom) beam-averaged normalized emittances during LHC fill 8037, as reported by the BSRT (blue), the WS (orange), and using beam-separation scans at IP1 (red) and IP5 (purple). Different shades of the same color correspond to two different bunches present in this fill. For ATLAS emittance scans, only the bunch-averaged value is shown [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/2) ### 4\.4 Beam-beam parameter determination The systematic and quantitative comparison of the measured and of the predicted beam-beam impact on the observables detailed in Sect. [2](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec2) requires continuous monitoring of the actual beam-beam parameter throughout the duration of the experiment. Only the BSRT provides uninterrupted emittance determination throughout the fill. In view of the scale biases apparent in Fig. [2](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig2), however, WS-based emittances are used, whenever possible, as input to the determination of the beam-beam parameter; BSRT emittances re-scaled to close-in-time, absolute WS measurements provide time-interpolated emittance values whenever WS data are unavailable. The beam-beam parameter evolution during the first fill of the experiment is illustrated in Fig. [3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig3). The error bands are dominated by the 3% systematic uncertainty in the measured optical functions. The strength of the beam-beam interaction drops by a factor of two over a couple of hours, from the combined effect of beam-intensity decay and of emittance growth. Fig. 3 [![Fig. 3](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig3_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/3) Time evolution of the beam- and plane-averaged beam-beam parameter \$\\xi \$ inferred from the measured bunch charges and emittances, separately for the two colliding-bunch pairs present in the fill pattern. The color bands indicate the systematic uncertainty (see text) [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/3) ### 4\.5 Tune measurements Coherent spectra can be monitored using either the LHC Transverse Damper (ADT) \[[24](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR24 "M. Söderén, J. Komppula, G. Kotzian, S. Rains, D. Valuch, ADT and Obsbox in LHC Run 2, plans for LS2, pp. 165–171 (2019)")\], or the Base-Band Tune (BBQ) \[[25](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR25 "A. Boccardi, M. Gasior, R. Jones, P. Karlsson, R. Steinhagen, First results from the lhc bbq tune and chromaticity systems. Technical report, CERN, Geneva (2009). https://cds.cern.ch/record/1156349 ")\] system; their comparison typically yields consistent results. Spectrograms with a tune resolution of 0.0001 are computed using bunch positions recorded at every turn, and averaged over the one-minute time bins mentioned in Sect. [4\.1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec5). To mitigate the influence of noise on the tune measurement, a median filter is applied at the pre-processing stage with a self-defined local window size. Additionally, the 50 Hz noise lines present in the spectra are masked. The filtered data are fitted by the sum of a Gaussian function and a constant baseline, and the frequency at the peak of the Gaussian is interpreted as the measured mean tune. The systematic uncertainty on measured tune shifts \$\\Delta Q\$ is estimated empirically to be \$\\sigma \_{\\Delta Q}=0.001\$. ### 4\.6 IP-orbit monitoring Orbit displacements at each IP are measured using the Diode ORbit and OScillation (DOROS) BPM system, that provides sub-micrometer beam-position resolution \[[26](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR26 "M. Gasior, G. Baud, J. Olexa, G. Valentino, First Operational Experience with the LHC Diode ORbit and OScillation (DOROS) System, p. 07 (2017) https://doi.org/10.18429/JACoW-IBIC2016-MOPG07 ")\]. These strip-line BPMs are located in the two quadrupoles on either side of and closest to the IP, allowing the position of both beams to be measured simultaneously. The position and the angle of each beam at the IP are inferred from, respectively, the average and the difference of the positions measured in the two final-triplet quadrupoles. ## 5 Optimization of the phase advance between interaction points In the two-IP configuration, the basic theory of linearized beam-beam \$\\beta \$\-beating provides an analytical determination of the optimum phase advance between the two IPs, i.e. of the setting that maximizes the sensitivity, at the witness IP, to beam-beam effects at the scanning IP (Sect. [5\.1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec12)). A generalization to the three- and four-IP configurations is discussed in Sect. [5\.2](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec13). The required modifications to the baseline LHC optics, as well as their implementation and their experimental validation, are detailed in Sect. [5\.3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec14). ### 5\.1 Linear beam-beam \$\\beta \$\-beating In circular colliders, the \$\\beta \$\-beating induced by beam-beam effects at the IP(s) leads to changes in transverse beam size \[[5](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR5 "W. Herr, T. Pieloni, Beam-Beam Effects. Technical report, CERN (2014). Contribution to the CAS–CERN Accelerator School: Advanced Accelerator Physics Course, Trondheim, Norway, 18–29 Aug 2013, p. 29. https://doi.org/10.5170/CERN-2014-009.431 . https://cds.cern.ch/record/1982430 "), [27](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR27 "A.W. Chao, Coherent beam-beam effects, in Frontiers of Particle Beams: Intensity Limitations (Springer, Berlin, Heidelberg, 1992), pp. 363–414. https://doi.org/10.1007/3-540-55250-2_36 ")\], thereby impacting the luminosity. For a single collision point, the change in \$\\beta ^\\$ of a single small-amplitude particle in (for instance) the horizontal plane x, is given by: \$\$\\begin{aligned} \\frac{\\beta \_x^\}{\\beta \_{0,x}^\} = \\frac{1}{\\sqrt{1-4\\pi \\xi \\cot {(2\\pi Q\_x)} - 4\\pi ^2\\xi ^2}}. \\end{aligned}\$\$ (6) The magnitude of the effect depends on the absolute value \$\\xi \$ of the beam-beam parameter and on the nominal betatron tune \$Q\_x\$; its periodicity is \$ \[\\pi \]\$. It can be shown that approaching the half-integer betatron tune from below minimizes the \$\\beta \$ function at the IP, thereby maximizing the luminosity; this phenomenon has been exploited with great success in (among others) the KEKB and PEP-II B* factories. In the LHC, where during collisions the nominal fractional betatron tunes are fixed at \$q\_x=0.31,\\,q\_y=0.32\$, this dynamic-\$\\beta \$ effect depends only on the beam-beam parameter. This conclusion no longer strictly holds in the presence of more than one collision point. In the case of two IPs with identical values of \$\\xi \$ and \$\\beta ^\\$, the dynamic-\$\\beta \$ effect can be described analytically \[[28](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR28 "J. Wańczyk, A. Dabrowski, D. Stickland, T. Pieloni, W. Kozanecki, X. Buffat, Impact of multiple beam-beam encounters on LHC absolute-luminosity calibrations by the van der Meer method. JACoW IPAC 2023, 053 (2023). https://doi.org/10.18429/JACoW-IPAC2023-WEPA053 ")\]: \$\$\\begin{aligned} \\frac{\\beta \_x^\}{\\beta ^\\_{0,x}} = \\frac{\\sin {2\\pi Q\_x} + 4\\pi \\xi (\\cos (2\\pi Q\_x-2\\mu \_{1,x})-\\cos 2\\pi Q\_x )}{\\pm \\sqrt{1-\\cos ^2{2\\pi (Q\_x+\\Delta Q\_x)}}}, \\end{aligned}\$\$ (7) where \$\\mu \_{1,x}\$ is the phase advance from the scanning IP to the witness IP. The sign in the denominator is linked to that of the numerator so as to ensure that the ratio remains positive; the denominator must be positive (resp. negative) below (resp. above) the half-integer tune, i.e.* when \$m\< Q\_x \<m+\\frac{1}{2}\$ (resp. \$m+\\frac{1}{2}\< Q\_x \< m+1\$), where m is a positive integer. The tune shift \$\\Delta Q\_x\$ is related to \$Q\_x\$, \$\\xi \$ and \$\\mu \_{1,x}\$ by: \$\$\\begin{aligned} \\cos {2\\pi (Q\_x+\\Delta Q\_x)}&=(1-16\\pi ^2\\xi ^2)\\cos {2\\pi Q\_x} + 8\\pi \\xi \\sin {2\\pi Q\_x} \\nonumber \\\\&\\quad + 16\\pi ^2\\xi ^2\\cos {(2\\pi Q\_x-2\\mu \_{1,x})}. \\end{aligned}\$\$ (8) The phase-advance dependence of the resulting \$\\beta \$\-beating is illustrated in Fig. [4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig4). The minimum of each curve corresponds to the phase advance that yields the largest mutual beam–beam-induced luminosity enhancement between the two IPs. The optimal settings are \$\\mu \_{1,x}^{min}/2\\pi =0.405,\\,\\mu \_{1,y}^{min}/2\\pi =0.410\$; their difference reflects that between the nominal horizontal and vertical fractional tunes \$q\_x\$ and \$q\_y\$. Fig. 4 [![Fig. 4](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig4_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/4) Analytically computed beam-beam induced \$\\beta \$\-beating in a two-IP configuration, as a function of the phase advance \$ \\mu \_1\$ between the 2 IPs (\$q\_x=0.31\$, \$q\_y=0.32\$, head-on collisions at both IPs, \$\\xi =7\\times 10^{-3}\$ per IP) [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/4) The extrema of the curves in Fig. [4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig4) can be determined analytically by differentiating Eq. ([7](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Equ7)) with respect to the phase advance \$\\mu \_{1,x}\$. The general expression for this derivative \$\\frac{d}{d\\mu \_{1,x}} \\biggl ( \\frac{\\beta \_x^\}{\\beta ^\\_{0,x}} \\biggr )\$, detailed in Appendix A, is the product of two factors, each of which can be zero: - imposing that \$\\sin {(2\\pi Q\_x - 2\\mu \_{1,x})} = 0\$ yields the solution \$\$\\begin{aligned} \\frac{\\mu \_{1,x}}{2\\pi } =\\frac{Q\_x}{2}-\\frac{m}{4} \\, \\end{aligned}\$\$ (9) where m is an integer of either sign, or zero; - the other factor is zero for either \$\$\\begin{aligned} \\frac{\\mu \_{1,x}}{2\\pi } = \\frac{Q\_x}{2} - A(\\xi , Q\_x)\\, \\end{aligned}\$\$ (10) or \$\$\\begin{aligned} \\frac{\\mu \_{1,x}}{2\\pi } = \\frac{Q\_x - 1}{2} + A(\\xi , Q\_x) \\, \\end{aligned}\$\$ (11) with \$A(\\xi , Q\_x)\$ defined in Eq. ([12](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Equ12)) \$\$\\begin{aligned} A(\\xi , Q\_x) = \\arccos \\left( \\frac{ 32\\pi ^3\\xi ^3 \\sin (4\\pi Q\_x) + 48\\pi ^2\\xi ^2 \\cos ^2(2\\pi Q\_x) - 32\\pi ^2\\xi ^2 - 6\\pi \\xi \\sin (4\\pi Q\_x) + \\cos ^2(2\\pi Q\_x) - 1}{16\\pi ^2\\xi ^2 \\left( 4\\pi \\xi \\sin (2\\pi Q\_x) + \\cos (2\\pi Q\_x) \\right) } \\right) / (4\\pi ) \\end{aligned}\$\$ (12) The two sets of solutions do not overlap. The subset of solutions that minimize (rather than maximize) \$\\beta ^\\$ is identified by requiring that the second derivative be positive. This restricts the main solution to: \$\$\\begin{aligned} \\frac{\\mu \_{1,x}}{2\\pi } =\\frac{Q\_x}{2}+\\frac{m+1}{4} . \\end{aligned}\$\$ (13) The optimal phase \$\\mu \_{x,1}\$ that minimizes the \$\\beta ^\\$ for range of \$(\\xi , Q\_x)\$ values is shown in Fig. [5](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig5). Fig. 5 [![Fig. 5](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig5_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/5) Optimal phase advance value \$\\mu \_{x,1}\$ for various values of \$(\\xi , Q\_x)\$ in the two-IPs configuration [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/5) The phase-advance dependence of the dynamic-\$\\beta \$ effect can also be quantified in terms of the luminosity shift, at the witness IP, that is associated with the electromagnetic interaction of the two bunches at the scanning IP (Fig. [6](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig6)). The optimal setting, i.e. that which maximizes the sensitivity, at the witness IP, to beam-beam effects at the scanning IP, is indicated by the green dotted vertical line at \$\\Delta \\mu \_{\\text {IP1-IP5}}=0.41\\ \[2\\pi \]\$; the full suppression of the phase-related luminosity enhancement is indicated by the red dotted line. These results are consistent with the predictions of the analytical model (Fig. [4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig4)). In the two-IP configuration considered here, in which IP1 and IP5 can both play the role of either the scanning or the witness IP, and due to the periodicity of this curve (\$ \\pi \$) and to the fractional-tune values used during collisions (\$q\_x / q\_y = 0.31 / 0.32)\$, the phase–advance-related luminosity enhancement is the same at the two IPs; at the optimum setting, it is three times larger than it would be using the nominal LHC lattice. The maximum effect depends mainly on the transverse tunes, and thus could have been further enhanced if it had been possible to move these away from their nominal values. A more comprehensive simulation study, that evaluates the impact of the beam-beam interaction on the luminosity for more realistic Gaussian proton-density distributions, multiple collision points and as a function of the beam separation, is reported in Ref. \[[29](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR29 "J. Wańczyk, Precision luminosity measurement at hadron colliders. PhD Thesis, École Polytechnique Fédérale de Lausanne (2024). https://doi.org/10.5075/epfl-thesis-10500 ")\]. Fig. 6 [![Fig. 6](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig6_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/6) Beam–beam-induced luminosity shift at IP5 predicted by COMBI simulations, as a function of the phase advance \$ \\mu \_1=\\Delta \\mu \_{\\text {IP1-IP5}}\$ between the two IPs (\$q\_x=0.31\$, \$q\_y=0.32\$, head-on collisions at both IPs, \$\\xi =7\\times 10^{-3}\$ per IP) [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/6) ### 5\.2 Optimized phases for multi-IP configurations The general expression to obtain the first-order change of the \$\\beta \$ function at an arbitrarily chosen reference IP (at \$s=\\mu \_{0,x}=0\$) can be approximated as multiple quadrupole errors \[[30](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR30 "T. Pieloni, J. Barranco, X. Buffat, P.C. Jorge, L.M. Medrano, C. Tambasco, R. Tomás, Dynamic beta and beta-beating effects in the presence of the beam-beam interactions in 57th ICFA Advanced Beam Dynamics Workshop on High-Intensity and High-Brightness Hadron Beams, p. 027 (2016). https://doi.org/10.18429/JACoW-HB2016-MOPR027 ")\]: \$\$\\begin{aligned} \\frac{\\beta ^\\_x}{\\beta ^\\_{0,x}} = \\frac{2\\pi \\xi }{\\sin {2\\pi Q\_x}} \\sum \_{i\\in IPs}^{N-1} \\cos (2\\pi Q\_x-2\\mu \_{i,x})\\, \\end{aligned}\$\$ (14) where \$\\mu \_{i,x}\$ is the horizontal phase advance between the reference and the \$i^{th}\$ IP for a total of N IPs. Therefore, the optimal configuration, which minimizes the \$\\beta ^\\$ change at all IPs, must satisfy the condition: \$\$\\begin{aligned} \\sum \_{i=1}^{N-1} \\cos (2\\pi Q\_x-2\\mu \_{i,x}) = -(N-1)\\, \\end{aligned}\$\$ (15) which yields the same solution for all IPs as in the 2 IPs configuration (Eq. [13](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Equ13)): \$\$\\begin{aligned} \\frac{\\mu \_{i,x}}{2\\pi } =\\frac{Q\_x}{2}+\\frac{n+1}{4} . \\end{aligned}\$\$ (16) where n* is an integer of either sign, or zero. To maximize the effect on luminosity, the phase advance between the reference interaction point at \$ i = 0 \$ and the nearest IP, at \$ i = 1 \$ must satisfy \$ \\mu \_{1,x}/2\\pi = (Q\_x + 0.5)/2 \$, as given by the first solution (for \$n=0\$) of Eq. ([13](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Equ13)). This condition implies that the phase advance to any subsequent adjacent IPs (\$ i \\ge 2 \$) must be an integer multiple of \$ \\pi \$. Thus, this requirement exposes a fundamental constraint: it is not possible to minimize \$ \\beta ^\* \$ simultaneously at all IPs. This would only be possible for (\$ Q\_x = 0.5 \$) which is practically not achievable, as it would require operating the storage ring precisely at the half-integer resonance. ### 5\.3 Optics validation The optimum phase advance from IP1 to IP5 determined in Sect. [5\.1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec12) was implemented using tuning trim quadrupoles in the LHC arcs. The targeted fractional phase advance was \$0.9 \\, \[2 \\pi \]\$ for both beams and both planes (Table [4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Tab4)). This setting, which differs from that in Fig. [6](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig6) by one period (\$ \\pi \$), is equivalent in terms of sensitivity optimization but has the advantage that it minimizes the absolute magnitude of the perturbations inflicted upon the nominal lattice. Several measurements were carried out to validate this adjustment: comparison of the targeted and achieved shifts in phase advance (Sect. [5\.3.1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec15)), \$\\beta \$\-function measurements across the full LHC circumference (Sect. [5\.3.2](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec16)) as well as at the IPs (Sect. [5\.3.3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec17)), and dispersion measurements to estimate the dispersive contribution to the transverse beam size at the IP (Sect. [5\.3.4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec18)). #### 5\.3.1 Phase advance The phase advance around each LHC ring is determined using turn-by-turn orbit measurements \[[20](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR20 "R. Tomás, O. Brüning, M. Giovannozzi, P. Hagen, M. Lamont, F. Schmidt, G. Vanbavinckhove, M. Aiba, R. Calaga, R. Miyamoto, Cern large hadron collider optics model, measurements, and corrections. Phys. Rev. ST Accel. Beams 13, 121004 (2010). https://doi.org/10.1103/PhysRevSTAB.13.121004 "), [21](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR21 "R. Tomás, M. Aiba, A. Franchi, U. Iriso, Review of linear optics measurement and correction for charged particle accelerators. Phys. Rev. Accel. Beams 20, 054801 (2017). https://doi.org/10.1103/PhysRevAccelBeams.20.054801 ")\]. The targeted and achieved phase-advance shifts agree within \$0.01\\, \[2\\pi \]\$ or better (Table [4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Tab4)). The phase advance change from IP1 and IP5 was compensated on the opposite side of the ring, ensuring that the transverse tunes remain unchanged. The target fractional local phase advance is set to \$0.9,\[2\\pi \]\$. Table [4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Tab4) lists the corresponding phase shifts applied to the nominal lattice to reach this target on one side of the ring; this is consistent with the analytically derived value of \$0.41\\,\[2\\pi \]\$ for the compensating shift on the opposite side. The change in phase advance was also measured at the BSRT location, where the discrepancy between the model and measurements is more pronounced, reaching up to \$0.05\\,\[2\\pi \]\$. Table 4 Nominal and optimized IP1 to IP5 phase-advance settings. Column 4 (resp. 5) displays the targeted (resp. measured) shift in phase advance after the optimized settings have been applied [Full size table](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/tables/4) #### 5\.3.2 Residual \$\\beta \$\-beating The \$\\beta \$\-beating was quantified and found to be well within acceptable tolerances for various configurations of the trim strength. The measurement results, relative to the reference measurements without any phase advance adjustments, are illustrated in Fig. [7](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig7). The maximal observed deviation is approximately 25% around IP5 for Beam 1 in the vertical plane. Such significant deviations in proximity to the IPs are anticipated and are frequently attributed to erroneous BPM readings. Overall, the measurements indicated that \$\\beta \$\-beating remained within acceptable bounds, predominantly within a 10% margin. Fig. 7 [![Fig. 7](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig7_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/7) Measured \$\\beta \$\-function change from the full strength of the phase advance knob with reference to nominal lattice along the LHC ring, for Beam 1 (top two figures for x and y) and Beam 2 (bottom two figures for x and y) [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/7) The measured \$\\beta \$ functions are also compared to the MADX LHC lattice model predictions, which include the phase advance adjustment. The results, illustrated along the LHC lattice in Fig. [8](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig8), indicate that the measurements align with the model within an acceptable margin of 10%. Fig. 8 [![Fig. 8](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig8_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/8) Measured deviations in the \$\\beta \$\-function along the LHC ring relative to the MADX model incorporating the phase advance knob, for Beam 1 (top two figures for x and y) and Beam 2 (bottom two figures for x and y). The error bars represent solely the statistical uncertainties as determined in the measurements [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/8) #### 5\.3.3 \$\\beta \$ functions at the interaction points The DOROS BPMs are strategically located at the end of the inner triplets on both sides of each of the experiments. Due to the phase advance of \$\\pi \$ between the DOROS on the left and right sides, these are inadequate for the precise determination of the measurement of \$\\beta ^\\$ at the IP. This can only be achieved with a special k\-modulation technique \[[31](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR31 "F. Carlier, et al. Challenges of K-modulation measurements in the LHC Run 3, in Proceedings of IPAC’23. IPAC’23—14th International Particle Accelerator Conference (JACoW Publishing, Geneva, Venice, 2023), pp. 531–534. https://doi.org/10.18429/JACoW-IPAC2023-MOPL014 ")\], which was performed at the interaction points used in the experiment. The results are summarized in Table [3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Tab3). While the achieved precision of the \$\\beta ^\\$ determination varies between measurements, uncertainties at the level of \${\\mathcal {O}}(1\\%)\$ are not uncommon and are not systematic to a specific beam or interaction point. Such uncertainties would propagate to only a few-percent effect on derived quantities such as the emittance or the beam–beam parameter. #### 5\.3.4 Dispersion It was essential to perform dispersion \$D^\\$ measurements at the interaction points, under the new optics configurations, as it can contribute to the measured beam size. Assuming the LHC design momentum spread value of \$\\frac{\\Delta p}{p}=10^{-4}\$, the average dispersion measurement around IP1 and IP5 is estimated to contribute up to 2% to the observed bunch sizes (worst case measured for B2Y). This effect is considered static throughout the experiment. ## 6 Experimental results This section presents experimental measurements of beam–beam effects in configurations involving multiple collision points. The impact of the optics modifications introduced in Sect. [5](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec11) is first assessed through tune spectra measurements, as detailed in Sect. [6\.1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec20). To quantify the cumulative effect of one or more additional collisions and assess the reproducibility of the results, a stepwise collision scan was employed. In this procedure, beams were sequentially brought into head-on collision at selected interaction points (IPs), followed by full transverse separation (\$\\Delta \> 6\\sigma \$) in either the horizontal or vertical plane at all but one witness IP. Two distinct filling schemes were used to isolate the influence of a single additional collision (2-IP configuration: ATLAS and CMS, see Sect. [6\.2](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec21)) and to investigate more complex multi-collision scenarios (3-IP configuration: including ALICE, see Sect. [6\.3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec25)). The first measurement of separation-dependent beam–beam effects, analogous to those encountered during luminosity calibration scans, is presented in Sect. [6\.4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec28), alongside comparisons with numerical predictions. An additional measurement conducted during the 2022 van der Meer calibration is detailed in Sect. [6\.5](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec32). A consolidated overview of all findings is provided in Sect. [7](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec33). ### 6\.1 Measured effect on the tune spectra of the phase advance changes The coherent transverse tune spectra serve as a diagnostic for evaluating the effectiveness of the phase advance modifications. COMBI simulations were carried out for two extreme lattice configurations, one designed to maximize and the other to suppress beam–beam effects. In the head-on scenario (Fig. [9](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig9)), the coherent modes are suppressed for the maximizing configuration, as the phase advance between the two collision points is close to \$\\pi \$, as was previously shown in simulation in Ref. \[[32](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR32 "Y. Alexahin, A study of the coherent beam-beam effect in the framework of the Vlasov perturbation theory. Nucl. Instrum. Methods Phys. Res., Sect. A 480(1–3), 253–288 (2002). https://doi.org/10.1016/S0168-9002(01)01219-0 ")\]. In this configuration, the effect of the first collision is counteracted by the second. This setting is particularly suitable for studying incoherent spectra, as the diminished coherent signal allows for a clearer extraction of the mean tune shift from fitted spectra. During separation scans, the symmetry between horizontal (x) and vertical (y) planes is broken. In the scanning plane (horizontal), coherent modes re-emerge at \$1.5\\,\\sigma \$ separation (Fig. [10](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig10)), with their frequency shifting in response to the separation distance (additional example for \$3\\,\\sigma \$ is shown in Fig. [11](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig11)). Fig. 9* [![Fig. 9](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig9_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/9) Comparison of COMBI simulated tune spectra in x\-plane (left) and y\-plane (right) for the suppressing (in blue) and maximizing (in green) phase configurations, for head-on collision [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/9) Fig. 10 [![Fig. 10](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig10_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/10) Comparison of COMBI simulated tune spectra in x\-plane (left) and y\-plane (right) for the suppressing (in blue) and maximizing (in green) phase configurations, at \$1.5\\,\\sigma \$ separation in the horizontal plane [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/10) Fig. 11 [![Fig. 11](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig11_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/11) Comparison of COMBI simulated tune spectra in x\-plane (left) and y\-plane (right) for the suppressing (in blue) and maximizing (in green) phase configurations, at \$3\\,\\sigma \$ separation in the horizontal plane [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/11) These simulated trends of the coherent spectra are also observed in experimental data. Transverse spectra recorded for head-on conditions confirm the suppression of coherent modes in the maximizing phase configuration (Fig. [12](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig12)). Additional comparisons for separated beams at the IP (see Figs. [13](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig13) and [14](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig14)), shown here for the case of vertical separation, reveal similar features. In this configuration, the non-scanning (horizontal) plane exhibits spectra that qualitatively agree with simulation predictions (see Figs. [10](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig10) and [11](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig11)), while in the scanning (vertical) plane the coherent modes do not reappear as expected. An equivalent behavior is observed when the separation is applied in the horizontal plane, with the roles of the two transverse planes interchanged; this case is not shown for brevity. This discrepancy is attributed to the effect of the transverse damper, which acts independently on the two beam centroids and suppresses coherent oscillations arising from beam–beam coupling, consistent with earlier observations \[[33](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR33 "R. Calaga, W. Herr, G. Papotti, T. Pieloni, X. Buffat, S. White, R. Giachino, Coherent beam-beam mode in the LHC. CERN (2014). https://doi.org/10.5170/CERN-2014-004.227 . http://cds.cern.ch/record/1957039 ")\]. Fig. 12 [![Fig. 12](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig12_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/12) Measured tune spectra before and after phase change, corresponding to the suppressing (in blue) and maximizing (in green) phase configurations, at the head-on position, for each of the transverse planes (Beam 1 shown as an example). Dash-dotted lines show Gaussian fits to the data [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/12) Fig. 13 [![Fig. 13](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig13_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/13) Comparison of the measured tune spectra at suppressing (blue) and maximizing (green) phase configurations with \$1.3\\,\\sigma \$ separation in the vertical plane [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/13) Fig. 14 [![Fig. 14](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig14_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/14) Comparison of the measured tune spectra at suppressing (blue) and maximizing (green) phase configurations with \$2.6\\,\\sigma \$ separation in the vertical plane [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/14) ### 6\.2 Head-on collisions in a two-IP configuration at \$\\sqrt{s} = 900\\ \\text {GeV}\$ Each test consisted of four measurements designed to quantify the beam–beam effect induced by a single additional head-on collision. The sequence was executed with CMS as the witness IP, followed by ATLAS, and concluded with a repeat at CMS. Over the course of the fill, the beam–beam parameter decreased from approximately 0.01 to 0.005. The outcome of each scan is analyzed using three independent observables: luminosity, beam size, and tune spectra, discussed separately below. #### 6\.2.1 Impact of beam-beam effects on head-on luminosity An overview of the luminosity measurements during the first test with CMS as the witness IP is shown in Fig. [15](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig15), where the specific luminosity at CMS is normalized to the initial head-on step to highlight relative changes. The uncertainty at each step is given by the standard error of the mean; shaded bands represent the standard deviation. A comparable dataset for ATLAS is presented in Fig. [16](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig16), confirming the expected luminosity change due to beam–beam effects originating from another IP. Relative luminosity changes for each step in all tests are summarized in Table [5](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Tab5). These values are computed relative to the average of adjacent head-on points, which removes the linear luminosity decay over time. Assuming slow change of beam parameters, the results within each test are expected to be consistent. Fig. 15 [![Fig. 15](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig15_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/15) CMS specific luminosity, normalized to the first measured both IPs head-on point (blue), during the first step-function scan at IP1. The measurements with ATLAS fully separated are shown in red. The error bars indicate error on the mean [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/15) Fig. 16 [![Fig. 16](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig16_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/16) ATLAS specific luminosity, normalized to the first measured both IPs head-on point (blue), during the step-function scan at IP5. The points for which IP5 was separated are shown in red. The error bars indicate the statistical error [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/16) Table 5 Measured luminosity shifts in each of full separation steps – summary of all three step-function scans performed for IP1 and IP5 as the witness IP. The beam-beam parameter at the start of each scan \$\\xi \_\\text {start}\$ is included for reference [Full size table](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/tables/5) All measurements are compiled in Fig. [17](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig17) and compared to COMBI simulation predictions. The results are plotted as a function of the beam–beam parameter (estimated as described in Sect. [4\.4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec8)), following a reverse chronological order. Agreement with simulation is excellent across the entire range. Fig. 17 [![Fig. 17](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig17_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/17) COMBI simulated luminosity enhancement induced by a single head-on collision as a function of the beam-beam parameter (dashed line), compared to the test results at both ATLAS (red points) and CMS (green points) [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/17) Fig. 18 [![Fig. 18](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig18_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/18) Measurements of beam-beam effects on beam widths shown with red points for each beam and plane, during the experiment with IP5 as the witness IP at \$\\xi =0.0056\$. COMBI predictions are shown in blue with calculated systematic error from the phase advance error between the IP and BSRT locations [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/18) #### 6\.2.2 Impact of beam-beam effects on single-beam sizes Complementary measurements of the bunch size from the BSRT are shown in Fig. [18](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig18). Data from all steps and bunches are combined, and a linear trend corresponding to continuous emittance growth is subtracted using the difference between the first and last head-on steps. For comparison, COMBI simulation results are also included in the figure. The dominant source of uncertainty in COMBI predictions arises from discrepancies between the measured phase advance and the MADX model, with the largest observed deviation being \$0.05\\,\[2\\pi \]\$ (as discussed in Sect. [5\.3.1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec15)). This uncertainty is propagated using a modified form of Eq. ([6](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Equ6)) adapted to the BSRT location. The magnitude of this uncertainty depends on the specific phase advance configuration, as some are more sensitive to small deviations. This is reflected in the varying sizes of the error bands associated with each prediction. The most accurate predictions are obtained in the vertical plane of Beam 2, where the beam–beam–induced \$\\beta \$\-beating propagates to the BSRT with a particularly favorable phase advance. In this configuration, the measured effect is also the most pronounced, and excellent agreement with COMBI simulations is observed. In other cases, the expected absolute effect is smaller, typically below 1%, and the influence of phase advance uncertainty becomes more significant, limiting the predictive reliability of the simulations. Experimentally, it is also possible that in these configurations the beam–beam–induced effect is too subtle to resolve, potentially masked by instrumental limitations. #### 6\.2.3 Impact of beam-beam effects on tune spectra Figure [19](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig19) shows representative ADT spectra recorded during the step-function scans. These spectra are affected by coupling, noise, and various nonlinearities, so they do not follow an idealized analytical distribution. Nonetheless, the peak position can be extracted using a Gaussian fit (indicated by dashed lines), yielding an estimate of the mean tune shift. In the single-collision configuration (blue lines), unsuppressed coherent modes dominate the spectrum and partially obscure the underlying incoherent component. The Gaussian fit is therefore applied to the incoherent contribution between coherent lines, rather than to the global spectral maximum, which results in a larger uncertainty on the extracted peak position. Despite this, in each of the performed step-scans at both collision points, the single-collision tune shift, defined as twice the fitted mean shift, is found to be consistent within \$\\pm 5\\%\$ with the expected coherent beam–beam tune shift corresponding to the conditions of the respective scan. For head-on collisions with Gaussian transverse beam distributions, the coherent tune shift of the dipole mode is expected to be \$Y\\xi \$, where Y is the Yokoya factor (\$Y = 1.21\$ for round beams) \[[34](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR34 "K. Yokoya, Y. Funakoshi, E. Kikutani, H. Koiso, J. Urakawa, Tune shift of coherent beam-beam oscillations. Part. Accel. 27, 181–186 (1990)")\]. As expected, the two-collision configuration yields a tune shift approximately double that of the single-collision case \[[13](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR13 "A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). https://doi.org/10.48550/arXiv.2306.10394 ")\]. Fig. 19 [![Fig. 19](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig19_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/19) Measured vertical tune spectra for different collision configurations: reference nominal tune (red), without the presence of any collisions, compared to single collision (blue) and two collisions (green) at \$\\xi \\simeq 0.0086\$. The dash-dotted lines in corresponding colors indicate Gaussian fits to the data used to extract mean tunes [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/19) Fig. 20 [![Fig. 20](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig20_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/20) Specific luminosity, normalized to the first head-on point, measured at ATLAS (top) and CMS (bottom) during beam separation at the other IPs, with horizontal separation applied in steps 2–4 and vertical separation in steps 6–8. The tested configurations are indicated in the legend. The blue line shows an interpolation of the head-on measurements, serving as a reference for relative luminosity changes [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/20) ### 6\.3 Head-on collisions in a three-IP configuration at \$\\sqrt{s} = 900\\ \\text {GeV}\$ An additional head-on collision was included in the configuration using IP2 (ALICE), as it was the only IP where this was feasible. A specially designed filling scheme ensured uniform collision exposure across all bunches, allowing measurements to be combined and thereby reducing statistical uncertainties. However, no reliable luminosity data were available from IP2 during this experiment. Table 6 Luminosity shift per full separation step, calculated with respect to an interpolated reference obtained from the two closest head-on points. Results for horizontal and vertical separations are shown as Shift part 1 and Shift part 2, respectively [Full size table](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/tables/6) #### 6\.3.1 Impact of beam-beam effects on head-on luminosity A step-function scan was conducted by sequentially collapsing or separating one IP at a time, with the separation applied first in the horizontal plane and, in a second stage of the experiment, in the vertical plane. A comprehensive summary of the measured specific luminosity at ATLAS and CMS is presented in Fig. [20](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig20). Exceptionally stable beam conditions during this test enabled measurements at both experiments under nearly identical beam–beam parameters. The beam–beam-induced luminosity changes observed during each step are quantified in Table [6](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Tab6), referenced to the time-interpolated specific luminosity from three head-on points. Orbit-drift corrections are applied only to these head-on reference points and result in changes of the extracted luminosity shifts of at most 0.2%. Good agreement with COMBI simulations is observed across the separation steps, with measured shifts closely matching the simulated values for configurations involving IP1 and IP5. For single-IP separation, the level of agreement depends on the witness IP and the separation step. When IP1 is used as the witness, the luminosity shift in the first separation step is reproduced almost exactly by COMBI, while a smaller shift is observed in the second step. This behaviour is expected, as the beam–beam parameter decreases during the scan and the measurement becomes increasingly sensitive to residual orbit perturbations. For configurations involving IP2, the comparison is less conclusive. As no reliable luminosity data were available from IP2 during this experiment, it was not possible to precisely optimize the collision conditions at this IP. As a result, the actual beam–beam conditions at IP2 may differ from those assumed in the simulations, affecting the comparison. Residual orbit drifts affecting the separation steps cannot be fully corrected. As illustrated by the DOROS measurements in Fig. [21](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig21), the orbit data contain both slow drifts and beam–beam–induced orbit offsets arising from changes in the collision configuration when other IPs are separated. While the beam–beam–induced offsets are included in the COMBI simulations, the residual orbit drifts are not. Since these contributions cannot be disentangled in the measurements, a full correction is not possible. Consequently, the effective beam–beam impact measured during the experiment may be weaker than that assumed in the simulations. Fig. 21 [![Fig. 21](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig21_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/21) Residual separation between the two beams at all three IPs in horizontal (top) and vertical (bottom) planes, for test with observations at ATLAS, where the separation was performed first in the horizontal and then in the vertical plane at the other IPs [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/21) #### 6\.3.2 Tune shift scaling with the number of beam–beam interactions Analysis of the transverse tune spectra in this configuration is further complicated by the reappearance of coherent modes when beams are separated at one or more IPs, as predicted by simulation. In such cases, the maximum tune shift is determined from the frequency difference between the \$\\pi \$ and \$\\sigma \$ modes, rather than from fitting the incoherent peak. For configurations where all three IPs were in collision, coherent modes remained suppressed due to the retained symmetry between IP1 and IP5. Figure [22](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig22) presents the measured tune shifts for all tested collision configurations. The unperturbed tune measurement, shown at ‘0’ collisions, serves as the reference from which all tune shifts are calculated. An empirical uncertainty of 0.001 is assumed for each independent measurement. Both transverse planes and separation directions are included to demonstrate consistency. As expected, the measured tune shift scales linearly with the number of head-on collisions. Fig. 22 [![Fig. 22](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig22_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/22) Measured mean tune shift as a function of the number of head-on collisions, shown for both transverse planes and for both separation directions (horizontal and vertical) used in the experiment. The separation refers to how collisions were removed: it was applied in only one transverse direction at a time (H sep. or V sep.) at the non-colliding interaction points, which are not explicitly labeled on the x-axis [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/22) ### 6\.4 Beam-beam signatures during transverse-separation scans at \$\\sqrt{s} = 900\\ \\text {GeV}\$ In the van der Meer (vdM) calibration method, beam–beam corrections are applied separately at each separation step. Therefore, it is essential to validate the simulated dependence of luminosity shifts on nominal beam separation. Full vdM-like scans were performed in both transverse planes at ATLAS and CMS, using a configuration where only these two IPs were in collision. #### 6\.4.1 Impact of beam–beam effects on the luminosity scan-curves Figure [23](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig23) shows the luminosity curves measured at ATLAS, acting as the witness IP, during a separation scan performed at CMS. The curves are fitted with Gaussian profiles for visual guidance. The beam separation at CMS is expressed in units of the measured transverse beam size, \$\\sigma \_{\\textrm{meas}}\$, which was found to be smaller than the nominal beam size, \$\\sigma \_{\\textrm{nom}}\$, used to define the scan steps. As a result, scan points corresponding to nominal separations beyond \$6\\,\\sigma \_{\\textrm{nom}}\$ extend to larger values when expressed in units of \$\\sigma \_{\\textrm{meas}}\$. To account for emittance evolution during the scan, both linear and exponential fits are applied to the head-on luminosity points before, after, and in the middle of the scan. These trends are subtracted from the data in Fig. [23](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig23), and all measurements are presented relative to this fitted head-on reference (normalized to unity). Fig. 23 [![Fig. 23](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig23_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/23) ATLAS specific luminosity relative changes (blue points) during separation scans in x\-plane (top) and y\-plane (bottom) at CMS, corrected for exponential (exp. corr.) luminosity decay. Gaussian function with a constant fitted to these points is shown with blue dashed line and compared to fit performed on the data corrected for linear luminosity change (lin. corr.) in time (dotted line). These are compared to COMBI simulation predictions (purple solid line) [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/23) At CMS, similar scans were performed; however, degraded beam stability was observed during this period of the fill. The scan results exhibit systematic deviations, especially at large separation, which could not be symmetrically corrected using simple linear or exponential models. Despite these issues, all four scan cases (two transverse planes at two IPs) yield a total specific luminosity reduction of approximately 2.4%, consistent with expectations based on the measured beam–beam parameter \$\\xi \$. Comparison with COMBI simulations reveal up to 30% discrepancy for IP5 as the witness IP, while agreement improves to within 10% for measurements at IP1. These differences are comparable to those observed in Table [6](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Tab6), suggesting a common underlying origin. While a finer scan granularity can improve the description of the steep-response region below \$3\\,\\sigma \_{\\textrm{meas}}\$, it would not address the dominant discrepancy between head-on and fully separated luminosity, which is likely driven by simulation assumptions and scan-related systematics. #### 6\.4.2 Beam–beam-induced orbit deflections Beam–beam deflections are clearly observed during separation scans. Figure [24](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig24) shows the fully corrected residual horizontal beam displacement at IP1, measured with the DOROS beam position monitors, as a function of the beam separation. This representation allows a direct comparison with the analytical beam–beam deflection prediction based on the Bassetti–Erskine formalism \[[35](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR35 "M. Bassetti, G.A. Erskine, Closed expression for the electrical field of a two-dimensional gaussian charge. Technical report, CERN, Geneva (1980). https://cds.cern.ch/record/122227 ")\]. The raw orbit data were corrected for slow orbit drifts by aligning the beam positions at the luminosity optimization points before and after each scan. Deviations of the applied beam separation from the nominal settings were accounted for by fitting a common length scale simultaneously with the beam–beam deflection. This approach avoids an overestimation of the separation calibration. The measured orbit displacement at the DOROS location was corrected for the contribution arising from the beam–beam deflection angle by applying a geometric amplification factor, proportional to the distance between the IP and the BPM in the small-angle approximation. After all corrections, the measured beam–beam deflection as a function of beam separation closely follows the analytical prediction. The maximum horizontal deflection observed at IP1 is 10.2 \$\\upmu \$m, compared to an expected value of 8.4 \$\\upmu \$m, with agreement within the total measurement uncertainty. The dominant uncertainty contribution (3.5 \$\\upmu \$m, corresponding to about 90% of the total) arises from a possible residual half crossing angle at the IP of up to 0.5 \$\\upmu \$rad. Additional systematic uncertainties include the fitted separation length scale, tune uncertainty of 0.005, and an extended 2 % uncertainty on \$\\beta ^\\$. Statistical uncertainties are negligible for most separation steps. Fig. 24* [![Fig. 24](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig24_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/24) Orbit offset caused by the beam–beam interaction measured by DOROS BPMs in the horizontal plane at IP1 during the horizontal separation scan. Analytical prediction is shown for comparison \[[35](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR35 "M. Bassetti, G.A. Erskine, Closed expression for the electrical field of a two-dimensional gaussian charge. Technical report, CERN, Geneva (1980). https://cds.cern.ch/record/122227 ")\] [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/24) #### 6\.4.3 Impact of the beam-beam effects on beam sizes and tune shift Separation-dependent changes in beam size were measured using the synchrotron light monitor. Figure [25](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig25) shows good consistency across individual bunch measurements, as well as strong agreement with COMBI simulations. This reinforces the utility of the BSRT as a reliable diagnostic for beam–beam effects. The uncertainties shown are statistical only and are treated as uncorrelated between scan points. The normalization to the zero-separation value is applied for visualization purposes and does not impose a constraint on that point. Fig. 25 [![Fig. 25](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig25_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/25) BSRT beam-size measurement in the vertical plane for each bunch (points) during horizontal (top) and vertical (bottom) separation scans at IP1. The exponential evolution correction is applied based on adjacent head-on points (before, in the middle and after the scan) to highlight the relative changes. COMBI predictions are shown (gray dashed line) for comparison [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/25) Transverse tune shifts were extracted from ADT-measured spectra. As illustrated in Fig. [26](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig26), the mean tune shift evolution is reconstructed for both the scanning and non-scanning planes. The measured tune shifts closely follow the expected COMBI trends, further validating the simulation models used for beam–beam corrections. Fig. 26 [![Fig. 26](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig26_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/26) Measured tune shift in units of the beam-beam parameter \$\\xi \_{BB}\$ for the horizontal (blue) and vertical (orange) planes, during a horizontal separation scan at IP1. The error bars indicate typically assumed conservative empirical systematic error on the tune measurement of 0.001. The solid curves represent the COMBI prediction [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/26) ### 6\.5 Beam-beam signatures during vdM scans at \$\\sqrt{s} = 13.6\\ \\text {TeV}\$ During the first vdM calibration of LHC Run 3, an unusually high beam–beam parameter was recorded, offering a unique opportunity to study beam–beam effects under enhanced interaction conditions. These effects are examined using online ATLAS luminosity data, with representative results shown in Fig. [27](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig27). The example focuses on a standard vdM scan pair conducted at CMS, while monitoring luminosity at the non-scanning IP (ATLAS). The plotted luminosity changes are normalized to the central scan point, where both IPs are in head-on collision. Variations in this normalized luminosity across the scan directly reflect the impact of beam–beam interaction at the scanning IP on the luminosity at the non-scanning IP. These patterns are reproducible across multiple scan pairs acquired within the same fill. At the largest nominal separations (approximately \$\\pm 0.6\\,\$mm), the beams are sufficiently separated to eliminate beam–beam effects entirely. Notably, a qualitative difference is observed between the horizontal and vertical scan planes, an asymmetry that is not present in the above presented dedicated experiment. This behavior is attributed to differences in the phase advance between IPs in the standard vdM optics configuration. For vertical (y) scans, experimental data agree well with simulation, confirming the robustness of the modeling under these conditions. In contrast, the horizontal (x) scans exhibit a systematic discrepancy: the observed beam–beam impact is broader than that predicted by simulation. The model assumes that the separation steps are accurately defined using the beam size measured during the vdM scans, but this mismatch may suggest an unmodeled phase advance deviation or another optics imperfection. The total difference between data and simulation is within 0.2%. This dataset originates from a standard vdM calibration, for which only online luminosity measurement was available, and many contributions to the overall vdM uncertainty, such as bunch-to-bunch variations, averaging over different bunch families, and emittance evolution during the scan, are not disentangled. Therefore, the 0.2% difference should be interpreted as a consistency check under typical operational conditions rather than a precision validation of the beam–beam model. Nevertheless, this indicates that the model remains sufficiently accurate for beam–beam corrections in vdM analyses. Fig. 27 [![Fig. 27](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig27_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/27) Changes from the separation-dependent beam-beam effects at the non-scanning IP – ATLAS online luminosity shown with points. Data comes from a standard vdM scan pair during the 2022 pp luminosity calibration. Separation steps and direction are indicated with gray markers. The last CMS scan is shown, with the highest beam-beam parameter of \$ \\xi =5.3 \\times 10^{-3} \$. The corresponding COMBI simulation results are also shown (triangles) [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/27) ## 7 Conclusions Beam–beam effects were directly observed and quantitatively validated at the LHC. The impact of beam–beam interactions on head-on luminosity, bunch sizes, tune spectra, and beam orbit positions was measured across various collision configurations. The observed effects align closely with COMBI simulations. In particular, the luminosity enhancement due to amplitude-dependent \$\\beta \$\-beating was measured for the first time and is found to agree with expectations over a wide range of the beam–beam parameter \$\\xi \$. Together with the observed linear scaling with respect to the beam–beam parameter, this provides experimental confirmation of the numerical studies presented in \[[13](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR13 "A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). https://doi.org/10.48550/arXiv.2306.10394 ")\]. Consistent signatures were found across multiple scan configurations, including two- and three-IP setups, and the characteristic tune shifts were well reproduced by both measurement and simulation. Accurate instrumentation and experimental design were essential. Key to the results was the stepwise collision scheme, isolating individual beam–beam contributions, with the implementation of the witness IP. Synchrotron light measurements from the BSRT, especially in the vertical plane of Beam 2, showed excellent agreement with COMBI predictions and bunch-by-bunch consistency. Tune spectra from the ADT system provided direct access to mean tune shifts, while orbit data from the DOROS system enabled measurement of beam–beam deflection down to 1 \$\\upmu \$m, matching analytical estimates. These tools, combined with precise control of IP configurations, enabled a comprehensive cross-check of beam–beam models. Some systematic limitations were identified. During one of the tests, beam stability during the scans was insufficient to allow for symmetric modeling of luminosity decay in both transverse planes. In several scans, the nominal separation steps were based on overestimated beam sizes, resulting in larger-than-expected scan intervals and less sensitivity at low separation. Beam–beam deflection measurements showed sign reversals at high separation, attributed to nonlinear orbit distortions from hysteresis effects in superconducting magnets. Phase advance errors between IPs and diagnostic systems further reduced the accuracy of simulations in some configurations. Targeted improvements would allow for higher precision and new insights. Finer separation steps, especially below \$3\\,\\sigma \$, would improve sensitivity to nonlinear beam–beam effects and allow more accurate comparison with models. Suppressing coherent beam–beam modes via phase advance tuning of not only two but all three IPs would improve tune shift measurements. Moreover, such optimal phase advance configuration between the IPs can be exploited to enhance luminosity. This effect could be directly propagated into at least a few percent increase in the total collected integrated luminosity, as discussed in Ref. \[[36](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR36 "J. Wańczyk, T. Pieloni, Optimizing beam-beam beta-beating for luminosity enhancement at the LHC. JACoW IPAC 2025, 018 (2025)")\]. Aligning the phase advance between IPs and diagnostic devices, particularly the BSRT, would reduce sensitivity to optics errors, enabling better beam size measurements for both beams and transverse planes. Moreover, the suppression of the coherent beam–beam modes facilitates the observation and analysis of the central, incoherent part of the tune spectrum, which is otherwise obscured by the dominant coherent peaks. Further experimental strategies should also focus on directly probing the beam–beam interaction at its point of occurrence. This would require access to the full transverse beam distributions with high spatial resolution and sensitivity, going beyond projected sizes or centroids. Such measurements could directly reveal the nonlinear distortions and tails induced by the beam–beam force. In parallel, validating correction schemes that parametrize multi-IP beam–beam effects on luminosity via equivalent tune shifts remains essential. These models underpin practical luminosity corrections in complex fill configurations and would benefit from systematic benchmarking. ## Data Availability Statement Data will be made available on reasonable request. \[Authors’ comment: The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.\] ## Code Availability Statement Code/software will be made available on reasonable request. \[Authors’ comment: The code/software generated during and/or analysed during the current study is available from the corresponding author on reasonable request.\] ## References 1. S. Meer, Calibration of the effective beam height in the ISR. Technical report, CERN, Geneva (1968). <https://cds.cern.ch/record/296752> 2. C. Rubbia, Measurement of the luminosity of \$p{{\\overline{p}}}\$ collider with a (generalized) Van der Meer Method. Technical report, CERN, Geneva (1977). <https://cds.cern.ch/record/1025746> 3. 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JACoW IPAC 2025, 018 (2025) [Google Scholar](http://scholar.google.com/scholar_lookup?&title=Optimizing%20beam-beam%20beta-beating%20for%20luminosity%20enhancement%20at%20the%20LHC&journal=JACoW%20IPAC&volume=2025&publication_year=2025&author=Wa%C5%84czyk%2CJ&author=Pieloni%2CT) [Download references](https://citation-needed.springer.com/v2/references/10.1140/epjc/s10052-026-15476-8?format=refman&flavour=references) ## Acknowledgements We would especially like to acknowledge W. Kozanecki for his substantial contributions to the development of this work and for carefully reviewing the manuscript, and Y. Wu for participation in the experiment and online data checks. We are also indebted to the ATLAS and CMS luminosity experts, in particular R. Hawkings, E. Torrence and A. Shevelev. We thank LHC experts for help with setting up and carrying out the experiment, in particular M. Hostettler, T. Persson, M. Solfaroli Camillocci, G. Trad, J. Wenninger. As well as optics team, in particular F. Carlier and M. Le Garrec. Some of the authors have received support from the Swiss Accelerator Research and Technology Institute (CHART). ## Author information ### Authors and Affiliations 1. CERN, Geneva, Switzerland Joanna Wańczyk, Xavier Buffat, Anne Dabrowski & Rogelio Tomas 2. Laboratory of Particle Accelerator Physics, EPFL, Lausanne, Switzerland Joanna Wańczyk & Tatiana Pieloni 3. Princeton University, Princeton, NJ, USA David Stickland Authors 1. Joanna Wańczyk [View author publications](https://link.springer.com/search?sortBy=newestFirst&contributor=Joanna%20Wa%C5%84czyk) Search author on:[PubMed](https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=Joanna%20Wa%C5%84czyk) [Google Scholar](https://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Joanna%20Wa%C5%84czyk%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en) 2. 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Rogelio Tomas [View author publications](https://link.springer.com/search?sortBy=newestFirst&contributor=Rogelio%20Tomas) Search author on:[PubMed](https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=Rogelio%20Tomas) [Google Scholar](https://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Rogelio%20Tomas%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en) ### Corresponding author Correspondence to [Joanna Wańczyk](mailto:jwanczyk@cern.ch). ## Appendix A: \$\\beta \$-beating derivative ### Appendix A: \$\\beta \$\-beating derivative Derivative of Eq. ([7](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Equ7)) with respect to the phase advance \$\\mu \_{1,x}\$, which characterizes the change in the linear beam-beam \$\\beta \$\-beating due to variations in phase advance: ![](https://media.springernature.com/lw540/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Equ17_HTML.png) ## Rights and permissions Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. 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Wańczyk, J., Pieloni, T., Buffat, X. et al. First measurements of beam-beam effects in beam-separation, luminosity-calibration scans at the LHC. Eur. Phys. J. C 86, 282 (2026). https://doi.org/10.1140/epjc/s10052-026-15476-8 [Download citation](https://citation-needed.springer.com/v2/references/10.1140/epjc/s10052-026-15476-8?format=refman&flavour=citation) - Received: 13 October 2025 - Accepted: 21 February 2026 - Published: 17 March 2026 - Version of record: 17 March 2026 - DOI: https://doi.org/10.1140/epjc/s10052-026-15476-8 ### Share this article Anyone you share the following link with will be able to read this content: Get shareable link Sorry, a shareable link is not currently available for this article. 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Readable Markdown	## 1 Introduction A precise determination of the absolute luminosity scale is essential for a wide range of LHC measurements, as it directly affects the normalization of many key physics cross sections. At the LHC, this calibration relies primarily on the van der Meer (vdM) method \[[1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR1 "S. Meer, Calibration of the effective beam height in the ISR. Technical report, CERN, Geneva (1968). https://cds.cern.ch/record/296752 "), [2](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR2 "C. Rubbia, Measurement of the luminosity of $$p{{\overline{p}}}$$ p p ¯ collider with a (generalized) Van der Meer Method. Technical report, CERN, Geneva (1977). https://cds.cern.ch/record/1025746 ")\], which uses dedicated beam-separation scans performed under specially tailored beam conditions to relate measured interaction rates to the absolute luminosity inferred from beam parameters. While sub-percent uncertainties on the integrated luminosity have been achieved for Run-2 data by ATLAS \[[3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR3 "ATLAS collaboration, Luminosity determination in $$pp$$ pp collisions at $$\sqrt{s}=13$$ s = 13 TeV using the ATLAS detector at the LHC. Eur. Phys. J. C 83(10), 982 (2023). https://doi.org/10.1140/epjc/s10052-023-11747-w . arXiv:2212.09379 ")\] and CMS \[[4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR4 "CMS Collaboration, Precision luminosity measurement in proton-proton collisions at sqrts = 13 TeV with the CMS detector. Technical report, CERN, Geneva (2025). https://cds.cern.ch/record/2940794 ")\], further improvements and robust experimental validation of beam-dynamical effects remain essential. For relativistic proton beams colliding with zero crossing angle, which is the typical configuration for the vdM calibration, the luminosity at a given interaction point (IP) of the LHC is proportional to the overlap integral of the particle-density distributions \$\\rho \_{\\text {1,i}}\$, \$\\rho \_{\\text {2,i}}\$ in bunch-pair i \[[5](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR5 "W. Herr, T. Pieloni, Beam-Beam Effects. Technical report, CERN (2014). Contribution to the CAS–CERN Accelerator School: Advanced Accelerator Physics Course, Trondheim, Norway, 18–29 Aug 2013, p. 29. https://doi.org/10.5170/CERN-2014-009.431 . https://cds.cern.ch/record/1982430 ")\]: \$\$\\begin{aligned} {\\mathcal {L}}\_{inst}&= 2c f\_{\\text {rev}} \\sum ^{N\_b}\_i n\_{\\text {1,i}}, n\_{\\text {2,i}} \\nonumber \\\\&\\quad \\times \\iiiint ^{+\\infty }\_{-\\infty }\\rho \_{\\text {1,i}}(x,y,z-ct) \\nonumber \\\\&\\quad \\times \\rho \_{\\text {2,i}}(x,y,z+ct)\\,dxdydzdt, \\end{aligned}\$\$ (1) where c is the speed of light, \$f\_{\\text {rev}}\$ is the LHC revolution frequency, \$N\_b\$ is the number of colliding bunches, and \$n\_{\\text {1,i}}, n\_{\\text {2,i}}\$ are the corresponding total charges per bunch for each colliding pair. Although only a very small fraction of the beam particles actually collide, the two opposing beams interact with each other electromagnetically, a dynamical process known as the beam-beam interaction \[[5](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR5 "W. Herr, T. Pieloni, Beam-Beam Effects. Technical report, CERN (2014). Contribution to the CAS–CERN Accelerator School: Advanced Accelerator Physics Course, Trondheim, Norway, 18–29 Aug 2013, p. 29. https://doi.org/10.5170/CERN-2014-009.431 . https://cds.cern.ch/record/1982430 "), [6](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR6 "A.W. Chao, Beam-beam instability. AIP Conf. Proc. 127, 201–242 (1985). https://doi.org/10.1063/1.35187 ")\]. In the LHC, as in any other synchrotron, the beams are not continuous but are divided into discrete “bunches”, each with a high proton density, containing approximately \$10^{11}\$ protons over a bunch length of 7.5–9 cm (in terms of RMS). This electromagnetic interaction occurs when the two beams share a common beam pipe, and is governed by a non-linear force that depends on the radial distance r of a test particle to the center of the opposing “source” bunch with n protons Gaussian-distributed in the transverse planes (valid approximation for the LHC beams): \$\$\\begin{aligned} F = - \\frac{ne^2}{2\\pi \\epsilon \_0r}\\biggl (1-\\exp \\biggl \[ -\\frac{r^2}{2\\sigma ^2}\\biggr \] \\biggr ), \\end{aligned}\$\$ (2) where e is the elementary proton charge, \$\\epsilon \_0\$ is the vacuum permittivity, and \$\\sigma \$ is the RMS radius of the transverse charge distribution within the bunch, in a simplified case of a round shape \$(\\sigma = \\sigma \_x = \\sigma \_y)\$. At very small amplitudes \$(r\\rightarrow 0)\$ the force can be linearized giving the expression for the so called beam-beam parameter, which is often used to assess the strength of the force: \$\$\\begin{aligned} \\xi = \\frac{nr\_p\\beta ^\}{4\\pi \\gamma \\sigma ^2} = \\frac{nr\_p}{4\\pi \\epsilon \_n}, \\end{aligned}\$\$ (3) where \$r\_p\$ is the classical proton radius, \$\\beta ^\\$ is the optical beta-function at the interaction point (IP) – one of the Courant–Snyder parameters \[[7](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR7 "E.D. Courant, H.S. Snyder, Theory of the alternating-gradient synchrotron. Ann. Phys. 3, 1–48 (1958). https://doi.org/10.1016/0003-4916(58)90012-5 ")\], \$\\gamma \$ is the Lorentz factor, and \$\\epsilon \_n\$ is the normalized emittance. Beam-beam effects were first observed almost 40 years ago \[[8](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR8 "P. Bambade, R. Erickson, W.A. Koska, W. Kozanecki, N. Phinney, S.R. Wagner, Observation of beam-beam deflections at the interaction point of the SLAC Linear Collider. Phys. Rev. Lett. 62(25), 2949–2952 (1989). https://doi.org/10.1103/PhysRevLett.62.2949 ")\] at the lepton colliders, and have been used extensively for accelerator diagnostics and optimization at both SLC and LEP \[[9](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR9 "W.A. Koska, P. Bambade, W. Kozanecki, N. Phinney, S.R. Wagner, Beam-beam deflection as a beam tuning tool at the slac linear collider. Nucl. Instrum. Methods A 286, 32 (1990). https://doi.org/10.1016/0168-9002(90)90203-I "),[10](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR10 "C. Bovet, M.D. Hildreth, M. Lamont, H. Schmickler, J. Wenninger, Luminosity optimisation using beam-beam deflections at LEP, in Conf. Proc. C (1996). https://cds.cern.ch/record/306910 "),[11](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR11 "D. Brandt, W. Herr, M. Meddahi, A. Verdier, Is lep beam-beam limited at its highest energy? in Proceedings of the 1999 Particle Accelerator Conference (PAC ’99), New York, pp. 3005–3007. IEEE/JACoW. THP25; CERN report CERN-SL-99-030-AP (1999). https://accelconf.web.cern.ch/p99/PAPERS/THP25.pdf ")\]. At the LHC, they were first observed in late Run 1 \[[12](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR12 "W. Kozanecki, T. Pieloni, J. Wenninger, Observation of Beam-beam Deflections with LHC Orbit Data. Technical report, CERN (2013). https://cds.cern.ch/record/1581723?ln=en ")\], but only in terms of the beam-beam induced coherent deflections. Single particle effects, in contrast, were presumed too small in hadron colliders that until recently, there was no motivation to measure them with a precision that would be meaningful in the context evoked here. Nonetheless, the full complexity of this interaction must be accounted for when performing the luminosity calibration via the vdM method, using the separation scans. These are designed to measure the detector-specific constant \$\\sigma \_{vis}\$ that relates the observed rate \$\\mu ^{vis}\$ to the absolute instantaneous luminosity \${\\mathcal {L}}\_{inst}\$: \$\$\\begin{aligned} \\mu ^{vis} = \\frac{{\\mathcal {L}}\_{inst}\\sigma \_{vis}}{f\_{rev}}. \\end{aligned}\$\$ (4) Under the assumption of uncorrelated particle densities in x and y planes, the transverse convolved bunch widths \$\\Sigma \_x, \\Sigma \_y\$ can be extracted from the measured beam-separation dependence of the collision rate in the corresponding direction. The combined information from these scans, and bunch intensities can be used to calculate the instantaneous luminosity at the head-on position. The proportionality between the measured rate at the centered head-on position \$\\mu \_{pk}\$ and the reconstructed luminosity from measured bunch parameters during the vdM calibration defines the luminometer-specific visible cross-section: \$\$\\begin{aligned} \\sigma \_{vis} = \\frac{2\\pi \\Sigma \_x\\Sigma \_y}{n\_1 n\_2} \\mu \_{pk}. \\end{aligned}\$\$ (5) If left uncorrected, the \$\\sigma \_{vis}\$ measurement by the vdM method is biased by the beam-separation dependence of the mutual electromagnetic interaction of the two beams: the colliding bunches experience deflection-induced orbit shifts, as well as optical distortions akin to the dynamic-\$\\beta \$ effect, that both depend on the transverse beam separation and must therefore be accounted for when deriving the absolute luminosity scale. The correction strategy discussed extensively in Ref. \[[13](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR13 "A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). https://doi.org/10.48550/arXiv.2306.10394 ")\] relies on simulation studies only, and thus requires experimental validation to establish its reliability. The present report summarizes a campaign of dedicated measurements carried out in the spring of 2022 at the LHC, aimed at confirming the accuracy of the simulation studies detailed in Ref. \[[13](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR13 "A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). https://doi.org/10.48550/arXiv.2306.10394 ")\], and at validating the beam-beam correction strategy used in the luminosity-calibration analyses of the ALICE, ATLAS, CMS and LHCb experiments. More specifically, the goal is to quantify the impact of the beam-beam interaction on the tune spectra, orbit, the transverse-density distribution of the colliding bunches, as well as, for the first time, on the luminosity, by systematically varying the strength or number of the beam-beam interactions. The measurements are repeated for different values of the beam-beam parameter to verify the scaling proposed in \[[13](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR13 "A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). https://doi.org/10.48550/arXiv.2306.10394 ")\], for various choices of the scanning IP and for several multi-IP configurations. The beam conditions are chosen to be representative of vdM calibration sessions at the LHC, but optimized so as to maximize both the sensitivity of the measurements, and the operational efficiency. This paper is organized as follows. The measurement strategy is detailed in Sect. [2](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec2), followed by the main ingredients of this accelerator experiment: beam conditions (Sect. [3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec3)), luminometers and accelerator instrumentation (Sect. [4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec4)), and optimization of the ring lattice to maximize the sensitivity of the measurements to beam–beam-induced optical distortions of the colliding bunches (Sect. [5](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec11)). Section [6](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec19) is devoted to the experimental characterization of the impact of beam-beam effects on the tune spectra, on the luminosity at different IPs, as well as on other observables such as transverse single-bunch widths and IP orbits. At each step, the results are confronted with the predictions of the COherent Multibunch Beam-beam Interactions (COMBI) tracking code \[[14](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR14 "T. Pieloni, A study of beam-beam effects in hadron colliders with a large number of bunches. PhD Thesis, École Polytechnique Fédérale de Lausanne (2008). https://doi.org/10.5075/epfl-thesis-4211 ")\], that produced most of the results detailed in Ref. \[[13](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR13 "A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). https://doi.org/10.48550/arXiv.2306.10394 ")\]. A collection of macroparticles is simulated turn by turn, based on a simplified model of the accelerator lattice using linear transfer and the self-consistent computation of the beam-beam forces at the IP. The conclusions are presented in Sect. [7](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec33). ## 2 Measurement strategy The beam-beam interaction at an interaction point, referred to here as the scanning IP, induces measurable changes in key beam parameters such as the tune spectrum, closed orbit, and transverse single-beam profiles. These effects can be observed, albeit with limited precision, using standard accelerator diagnostics by comparing measurements obtained under varying transverse beam separation at the scanning IP. However, the resulting modifications to the transverse density distribution of the colliding bunches remain below the sensitivity threshold of conventional instrumentation, including synchrotron-radiation-based beam-profile monitors. Beam-beam-induced variations typically lie in the 0.5–1% range, small enough that only a highly precise measurement of collision-rate changes can provide adequate sensitivity. Fig. 1 [![Fig. 1](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig1_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/1) Scheme of the LHC and its interaction points (IP) with indicated IP1 as witness IP while performing a scan at IP5 [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/1) Importantly, the beam-beam interaction induces changes in beam properties that are intrinsically entangled with the pp collisions when observing luminosity at the scanning IP. Consequently, under realistic collider conditions, it is not possible to directly access a reference luminosity signal that is free of beam-beam effects – unlike in simulation studies, where such effects can be explicitly disabled. In a multi-collision configuration, however, luminosity shifts at one or more other IPs (Fig. [1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig1)), where the beams are kept in continuous head-on collision and therefore act as non-scanning, or witness, IPs, can be used to monitor the separation-dependent beam–beam effects. These effects are induced at the scanning IP, where the beams are deliberately brought in and out of collision, and propagate around the rings to the witness IPs, where they manifest as changes in the measured luminosity. In principle, any of the four LHC IPs can serve as a scanning IP, and any non-scanning IP can be designated as the witness IP for a given measurement. In practice, instrumental and operational constraints at the time the experiment was carried out restrict the choice of witness IP to the ATLAS (IP1) and CMS (IP5) collision points; IPs 1, 2 or 5, or combinations thereof, are used as scanning IP(s). Collisions at IP8 were deliberately avoided because the large longitudinal offset of the LHCb collision point, that breaks the eight-fold symmetry of the LHC rings, precludes the possibility of both members of a colliding-bunch pair to collide at all four IPs. In addition, the large crossing angle at IP8 would complicate the interpretation of the beam-beam effects that would occur at that IP if some of the bunches collided there. The propagation of the beam–beam-induced, amplitude-dependent \$\\beta \$\-beating from the scanning to a witness IP is controlled by the betatron phase advance between these two IPs. Since the targeted signatures are delicate to measure at best, it is natural to try and enhance them by adjusting this phase advance so as to maximize the sensitivity of luminosity shifts at the witness IP to beam-beam effects at the scanning IP; in doing so, however, the overall tunes must be preserved. Given the central and symmetric roles played by IP1 and IP5 in this experiment, the adjustment of their relative phase advance drove the optimization procedure that is detailed in Sect. [5](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec11), and that achieved a threefold improvement in measurement sensitivity. Central to the quality of the measurements is the controlled variation of the beam-beam parameter at the scanning IP. This can be achieved either: - by using step scans, i.e. by fully separating the beams at the scanning IP in either the horizontal or the vertical plane, and then bringing them back into head-on collision; or - by using separation scans, i.e. by scanning the beams transversely with respect to each other in either the horizontal or the vertical plane; or - by taking advantage of the natural beam-intensity decay and emittance growth to progressively reduce the beam-beam parameter. Equally important is to ensure that the beams remain in head-on collision at the witness IP(s), such that reproducible luminosity shifts measured at these locations can be unambiguously correlated with controlled changes in the strength of the beam-beam interaction at the scanning IP. It was verified that the beam-beam-induced orbit shift and the potential non-closure of the orbit bumps used to control the beam separation at the scanning IP, did not significantly affect the actual separation, and thereby the measured collision rate, at the witness IP. In a few cases however, orbit drifts of uncontrolled origin ended up degrading the quality of some of the measurements. Simulations demonstrate \[[13](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR13 "A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). https://doi.org/10.48550/arXiv.2306.10394 ")\] that multi-IP effects strongly influence the magnitude of beam-beam biases to vdM calibrations. Their characterization, therefore, constitutes an essential component of this experiment. The scan protocol is a generalization of that in the two-IP case. One IP (say, IP1) is designated as the witness IP, with beams colliding head-on there and at IP2, and IP5; the beams are then taken out of collision at IP2 only, then at both IP2 and IP5, and finally returned to a three-collision configuration in the reverse order. Since the phase-advance combinations are different in each configuration, the corresponding luminosity shifts predicted at the witness IP are also different and can be meaningfully confronted with the data. ## 3 Beam conditions In the round-beam approximation, the beam-beam parameter \$\\xi \$ does not depend on the beam energy (Eq. ([3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Equ3))). Therefore, and in order to minimize the overhead associated with commissioning and operating the LHC in a non-standard optical configuration, as well as to allow for frequent refilling with different bunch patterns, the measurements were carried out at injection energy (\$450\\ \\text {GeV}\$ per beam), and under non-standard machine-protection conditions. The improvement in operational efficiency came at a cost: - the counting rate of the luminometers (or equivalently their visible cross-section) is about an order of magnitude smaller at \$\\sqrt{s} = 900\\ \\text {GeV}\$ than at \$13\\ \\text {TeV}\$, reflecting the combination of a 40% drop in the inelastic pp cross-section, a factor of two to three drop in the multiplicity of the final-state particles, and a significant softening of their momentum spectrum; - because of the total-intensity constraints dictated by machine-protection requirements, the injected beam could not exceed 4 bunches per beam (compared to 150 during a routine vdM\-calibration session), with a maximum allowed population of \$1.25\\times 10^{11}~p\$/bunch; - the combination of the low beam energy, that suppresses synchrotron-radiation damping, and of the large bunch intensity, that enhances intra-beam scattering, resulted in relatively rapid emittance growth and rather short single-beam lifetimes. To partially mitigate these effects, all collision-rate measurements are expressed in terms of specific luminosity, thereby automatically accounting for the beam-intensity decay. The bunch intensity was deliberately increased by about 25% beyond the vdM\-scan values typical of LHC Run 2, to maximize both the luminosity and the beam-beam parameter \$\\xi \$. The latter reached 0.010 per IP (Table [1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Tab1)), compared to 0.003–0.006 during normal pp vdM sessions. With a target injected emittance of \$1.5 \\ \\upmu \\text {m} \\cdot \\text {rad}\$, the \$\\beta \$ function at the IP set to \$\\beta ^\* = 11\\ \\text {m}\$, and zero crossing angle at IP1 and IP5, the statistical uncertainty affecting a typically 60-second long per-bunch luminosity measurement in the presence of head-on collisions, lay around 0.5%. Table 1 Range of beam-beam parameter values during the various stages of the experiment [Full size table](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/tables/1) The bunch patterns were chosen such that all bunches in each beam collided either at IP1 and IP5 only, or at all three of IP1, 2 and 5. The orbit-stabilization feedback system was turned off during the data-taking periods to prevent it from interfering with the beam–beam-induced orbit shift. The chromaticity was set to its standard value in physics fills, of +10 units, to guarantee coherent stability against the machine impedance, and in view of the very small number of bunches and of their large longitudinal spacing, the beam-stabilizing Landau octupoles were set to their minimum current (1A). The damping time of the bunch-by-bunch transverse feedback was set to 1000 turns, a rather loose setting intended to preserve longer the natural beam oscillation and therefore improve the precision of the tune measurements. ## 4 Beam instrumentation ### 4\.1 Luminometers The luminometer systems in use by the ATLAS and CMS collaborations at IP1 and IP5 are described in Refs. \[[15](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR15 "ATLAS Collaboration, Improved luminosity determination in pp collisions at $$\sqrt{s} = 7$$ s = 7 TeV using the ATLAS detector at the LHC. Eur. Phys. J. C 73, 2518 (2013). https://doi.org/10.1140/epjc/s10052-013-2518-3 "), [16](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR16 "C.M.S. Collaboration, Development of the cms detector for the cern lhc run 3. J. Instrum. 19(05), 05064 (2024). https://doi.org/10.1088/1748-0221/19/05/p05064 ")\] respectively; no luminosity measurements were available at IP2 during the collider experiment described in this paper. Maximizing the statistical sensitivity leads to choosing the luminosity algorithm with the highest possible acceptance, namely: - the hit rate per bunch crossing in the ATLAS Minimum Bias Trigger Scintillators (MBTS) \[[15](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR15 "ATLAS Collaboration, Improved luminosity determination in pp collisions at $$\sqrt{s} = 7$$ s = 7 TeV using the ATLAS detector at the LHC. Eur. Phys. J. C 73, 2518 (2013). https://doi.org/10.1140/epjc/s10052-013-2518-3 ")\], and - the occupancy in the CMS Hadron Forward Calorimeter (HFOC) \[[16](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR16 "C.M.S. Collaboration, Development of the cms detector for the cern lhc run 3. J. Instrum. 19(05), 05064 (2024). https://doi.org/10.1088/1748-0221/19/05/p05064 ")\]. In both cases, raw luminometer counts are converted to collision rates using the Poisson formalism \[[15](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR15 "ATLAS Collaboration, Improved luminosity determination in pp collisions at $$\sqrt{s} = 7$$ s = 7 TeV using the ATLAS detector at the LHC. Eur. Phys. J. C 73, 2518 (2013). https://doi.org/10.1140/epjc/s10052-013-2518-3 "), [16](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR16 "C.M.S. Collaboration, Development of the cms detector for the cern lhc run 3. J. Instrum. 19(05), 05064 (2024). https://doi.org/10.1088/1748-0221/19/05/p05064 ")\], and the instantaneous luminosity is averaged over typically 60-second time bins, that during the scans are synchronized with the scan steps. The statistical uncertainties are estimated from either the total number of raw luminometer counts per time bin (ATLAS), or from the RMS of the approximately 40 luminosity samples recorded in a given time bin (CMS). Since the absolute luminosity scale is irrelevant, and in order to simplify the interpretation of the results, all measurements in this paper are presented in terms of fractional shifts in the bunch-averaged specific luminosity, relative to a reference time that depends on the type of measurement considered. ### 4\.2 Bunch-charge measurement The total intensity of each beam is measured by a direct-current current transformer (DCCT), the absolute scale of which is calibrated against a very high precision pulse generator \[[17](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR17 "C. Barschel, M. Ferro-Luzzi, J.-J. Gras, M. Ludwig, P. Odier, S. Thoulet, Results of the LHC DCCT calibration studies. Technical Report CERN-ATS-Note-2012-026 PERF (2012). https://cds.cern.ch/record/1425904?ln=en ")\]. The bunch-by-bunch charge fractions, in turn, normalized to the total stored intensity, are measured, separately for the two beams, by Fast Beam Current Transformers (FBCT) \[[18](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR18 "G. Anders et al., Study of the relative LHC bunch populations for luminosity calibration. Technical Report. CERN-ATS-Note-2012-028 PERF (2012). https://cds.cern.ch/record/1427726 ")\]. The latter devices also provide the fill pattern, i.e. the relative location, at a given instant and around the two rings, of the nominally filled bunches. Since the absolute scale of the DCCTs is known to much better accuracy than that of the FBCTs, the bunch charges are typically computed as the product of the FBCT bunch-charge fractions and the total circulating beam intensity reported by the corresponding DCCT. ### 4\.3 Emittance measurement The synchrotron-light beam-profile monitors \[[19](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR19 "G. Trad, Development and Optimisation of the SPS and LHC Beam Diagnostics Based on Synchrotron Radiation Monitors Presented 22 Jan 2015. Presented 22 Jan 2015. https://cds.cern.ch/record/2266055 ")\], dubbed BSRTs (“Beam Synchrotron Radiation Telescope”), are mainly used to track emittance evolution over time, but they can also measure the relative changes in transverse single-beam size that result from beam–beam-induced \$\\beta \$\-beating. BSRT data are recorded every second; in what follows, they are presented averaged over one-minute intervals for easier comparison with other measurements. The absolute length scale of the BSRT profiles suffers from significant uncertainties, if only because the online optical corrections to the images can be updated only a few times per year, and are unable to track the evolution of the efficiency of the light sensors, that depends both on time and on the position of the light spot on the sensor array. For each beam therefore, the BSRT is complemented by a wire-scanner (WS) profile monitor that is located in the same straight section. The advantage of the WS is that its accuracy is significantly better than that of the BSRT; its down side, in the present context, is that the WS cannot acquire data continuously and must be triggered manually. For the results presented in this paper, the wire scanners were flown through the beams at the start of each group of measurements to provide single-beam emittance measurements that are as accurate as possible, and then at regular intervals thereafter. The BSRT is used to interpolate the time evolution of the emittance between two sets of WS measurements. To interpret measured RMS bunch widths in terms of emittance requires the knowledge of the \$\\beta \$ functions at the locations of the BSRT and the WS. These are determined by the phase-advance method, and compared in Table [2](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Tab2) to their model value computed using MAD-X; the agreement is typically better than 5%, and the worst disagreement amounts to 9%. These discrepancies are attributed to imperfections in the magnetic lattice. Uncertainties in the measurement mainly arise from the interpolation of the lattice functions between the two closest beam-position monitors (BPMs) where the \$\\beta \$\-functions are measured \[[20](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR20 "R. Tomás, O. Brüning, M. Giovannozzi, P. Hagen, M. Lamont, F. Schmidt, G. Vanbavinckhove, M. Aiba, R. Calaga, R. Miyamoto, Cern large hadron collider optics model, measurements, and corrections. Phys. Rev. ST Accel. Beams 13, 121004 (2010). https://doi.org/10.1103/PhysRevSTAB.13.121004 "), [21](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR21 "R. Tomás, M. Aiba, A. Franchi, U. Iriso, Review of linear optics measurement and correction for charged particle accelerators. Phys. Rev. Accel. Beams 20, 054801 (2017). https://doi.org/10.1103/PhysRevAccelBeams.20.054801 ")\], and either the BSRT or the WS location. Based on the MAD-X LHC lattice model, these uncertainties are estimated not to exceed 3%, and this value is assigned as the systematic uncertainty on the \$\\beta \$ functions used in BSRT- and WS-based emittance measurements. Table 2 \$\\beta \$ functions at the BSRT and WS locations, for the optical configuration detailed in Sect. [5\.3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec14) [Full size table](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/tables/2) The beam-averaged emittance, i.e. the average of the emittances of the beam-1 bunch and of the corresponding beam-2 bunch, can be obtained directly from emittance scans \[[22](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR22 "O. Karacheban, P. Tsrunchev, on behalf of CMS, Emittance scans for cms luminosity calibration. EPJ Web Conf. 201, 04001 (2019). https://doi.org/10.1051/epjconf/201920104001 ")\] at IP1 and/or IP5. In this approach, the convolved transverse bunch sizes measured using beam-separation scans are translated into emittances using the \$\\beta ^\\$ values determined by k\-modulation \[[23](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR23 "F. Carlier, R. Tomás, Accuracy and feasibility of the $${\beta }^{}$$ β ∗ measurement for lhc and high luminosity lhc using $$k$$ k modulation. Phys. Rev. Accel. Beams 20, 011005 (2017). https://doi.org/10.1103/PhysRevAccelBeams.20.011005 ")\] at the relevant IP (see Sect. [4\.3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec7)). The associated uncertainty (Table [3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Tab3)) is combined with the statistical uncertainty in convolved transverse width to estimate the error affecting the measured emittance. Comparison with beam-averaged emittances extracted from the single-beam profile monitors reveals excellent agreement between WS and emittance-scan results (Fig. [2](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig2)). The relative time-evolution of the BSRT emittances is consistent with that observed using the WS or emittance scans, but the absolute magnitudes differ significantly, especially in the horizontal plane. *Table 3 \$\\beta ^\\$ functions at the ATLAS and CMS IPs measured by k\-modulation. The target value in all cases is \$\\beta ^\=\$ 11 m* [Full size table](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/tables/3) Fig. 2 [![Fig. 2](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig2_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/2) Time evolution of the horizontal (top) and vertical (bottom) beam-averaged normalized emittances during LHC fill 8037, as reported by the BSRT (blue), the WS (orange), and using beam-separation scans at IP1 (red) and IP5 (purple). Different shades of the same color correspond to two different bunches present in this fill. For ATLAS emittance scans, only the bunch-averaged value is shown [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/2) ### 4\.4 Beam-beam parameter determination The systematic and quantitative comparison of the measured and of the predicted beam-beam impact on the observables detailed in Sect. [2](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec2) requires continuous monitoring of the actual beam-beam parameter throughout the duration of the experiment. Only the BSRT provides uninterrupted emittance determination throughout the fill. In view of the scale biases apparent in Fig. [2](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig2), however, WS-based emittances are used, whenever possible, as input to the determination of the beam-beam parameter; BSRT emittances re-scaled to close-in-time, absolute WS measurements provide time-interpolated emittance values whenever WS data are unavailable. The beam-beam parameter evolution during the first fill of the experiment is illustrated in Fig. [3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig3). The error bands are dominated by the 3% systematic uncertainty in the measured optical functions. The strength of the beam-beam interaction drops by a factor of two over a couple of hours, from the combined effect of beam-intensity decay and of emittance growth. Fig. 3 [![Fig. 3](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig3_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/3) Time evolution of the beam- and plane-averaged beam-beam parameter \$\\xi \$ inferred from the measured bunch charges and emittances, separately for the two colliding-bunch pairs present in the fill pattern. The color bands indicate the systematic uncertainty (see text) [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/3) ### 4\.5 Tune measurements Coherent spectra can be monitored using either the LHC Transverse Damper (ADT) \[[24](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR24 "M. Söderén, J. Komppula, G. Kotzian, S. Rains, D. Valuch, ADT and Obsbox in LHC Run 2, plans for LS2, pp. 165–171 (2019)")\], or the Base-Band Tune (BBQ) \[[25](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR25 "A. Boccardi, M. Gasior, R. Jones, P. Karlsson, R. Steinhagen, First results from the lhc bbq tune and chromaticity systems. Technical report, CERN, Geneva (2009). https://cds.cern.ch/record/1156349 ")\] system; their comparison typically yields consistent results. Spectrograms with a tune resolution of 0.0001 are computed using bunch positions recorded at every turn, and averaged over the one-minute time bins mentioned in Sect. [4\.1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec5). To mitigate the influence of noise on the tune measurement, a median filter is applied at the pre-processing stage with a self-defined local window size. Additionally, the 50 Hz noise lines present in the spectra are masked. The filtered data are fitted by the sum of a Gaussian function and a constant baseline, and the frequency at the peak of the Gaussian is interpreted as the measured mean tune. The systematic uncertainty on measured tune shifts \$\\Delta Q\$ is estimated empirically to be \$\\sigma \_{\\Delta Q}=0.001\$. ### 4\.6 IP-orbit monitoring Orbit displacements at each IP are measured using the Diode ORbit and OScillation (DOROS) BPM system, that provides sub-micrometer beam-position resolution \[[26](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR26 "M. Gasior, G. Baud, J. Olexa, G. Valentino, First Operational Experience with the LHC Diode ORbit and OScillation (DOROS) System, p. 07 (2017) https://doi.org/10.18429/JACoW-IBIC2016-MOPG07 ")\]. These strip-line BPMs are located in the two quadrupoles on either side of and closest to the IP, allowing the position of both beams to be measured simultaneously. The position and the angle of each beam at the IP are inferred from, respectively, the average and the difference of the positions measured in the two final-triplet quadrupoles. ## 5 Optimization of the phase advance between interaction points In the two-IP configuration, the basic theory of linearized beam-beam \$\\beta \$\-beating provides an analytical determination of the optimum phase advance between the two IPs, i.e. of the setting that maximizes the sensitivity, at the witness IP, to beam-beam effects at the scanning IP (Sect. [5\.1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec12)). A generalization to the three- and four-IP configurations is discussed in Sect. [5\.2](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec13). The required modifications to the baseline LHC optics, as well as their implementation and their experimental validation, are detailed in Sect. [5\.3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec14). ### 5\.1 Linear beam-beam \$\\beta \$\-beating In circular colliders, the \$\\beta \$\-beating induced by beam-beam effects at the IP(s) leads to changes in transverse beam size \[[5](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR5 "W. Herr, T. Pieloni, Beam-Beam Effects. Technical report, CERN (2014). Contribution to the CAS–CERN Accelerator School: Advanced Accelerator Physics Course, Trondheim, Norway, 18–29 Aug 2013, p. 29. https://doi.org/10.5170/CERN-2014-009.431 . https://cds.cern.ch/record/1982430 "), [27](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR27 "A.W. Chao, Coherent beam-beam effects, in Frontiers of Particle Beams: Intensity Limitations (Springer, Berlin, Heidelberg, 1992), pp. 363–414. https://doi.org/10.1007/3-540-55250-2_36 ")\], thereby impacting the luminosity. For a single collision point, the change in \$\\beta ^\\$ of a single small-amplitude particle in (for instance) the horizontal plane x, is given by: \$\$\\begin{aligned} \\frac{\\beta \_x^\}{\\beta \_{0,x}^\} = \\frac{1}{\\sqrt{1-4\\pi \\xi \\cot {(2\\pi Q\_x)} - 4\\pi ^2\\xi ^2}}. \\end{aligned}\$\$ (6) The magnitude of the effect depends on the absolute value \$\\xi \$ of the beam-beam parameter and on the nominal betatron tune \$Q\_x\$; its periodicity is \$ \[\\pi \]\$. It can be shown that approaching the half-integer betatron tune from below minimizes the \$\\beta \$ function at the IP, thereby maximizing the luminosity; this phenomenon has been exploited with great success in (among others) the KEKB and PEP-II B* factories. In the LHC, where during collisions the nominal fractional betatron tunes are fixed at \$q\_x=0.31,\\,q\_y=0.32\$, this dynamic-\$\\beta \$ effect depends only on the beam-beam parameter. This conclusion no longer strictly holds in the presence of more than one collision point. In the case of two IPs with identical values of \$\\xi \$ and \$\\beta ^\\$, the dynamic-\$\\beta \$ effect can be described analytically \[[28](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR28 "J. Wańczyk, A. Dabrowski, D. Stickland, T. Pieloni, W. Kozanecki, X. Buffat, Impact of multiple beam-beam encounters on LHC absolute-luminosity calibrations by the van der Meer method. JACoW IPAC 2023, 053 (2023). https://doi.org/10.18429/JACoW-IPAC2023-WEPA053 ")\]: \$\$\\begin{aligned} \\frac{\\beta \_x^\}{\\beta ^\\_{0,x}} = \\frac{\\sin {2\\pi Q\_x} + 4\\pi \\xi (\\cos (2\\pi Q\_x-2\\mu \_{1,x})-\\cos 2\\pi Q\_x )}{\\pm \\sqrt{1-\\cos ^2{2\\pi (Q\_x+\\Delta Q\_x)}}}, \\end{aligned}\$\$ (7) where \$\\mu \_{1,x}\$ is the phase advance from the scanning IP to the witness IP. The sign in the denominator is linked to that of the numerator so as to ensure that the ratio remains positive; the denominator must be positive (resp. negative) below (resp. above) the half-integer tune, i.e.* when \$m\< Q\_x \<m+\\frac{1}{2}\$ (resp. \$m+\\frac{1}{2}\< Q\_x \< m+1\$), where m is a positive integer. The tune shift \$\\Delta Q\_x\$ is related to \$Q\_x\$, \$\\xi \$ and \$\\mu \_{1,x}\$ by: \$\$\\begin{aligned} \\cos {2\\pi (Q\_x+\\Delta Q\_x)}&=(1-16\\pi ^2\\xi ^2)\\cos {2\\pi Q\_x} + 8\\pi \\xi \\sin {2\\pi Q\_x} \\nonumber \\\\&\\quad + 16\\pi ^2\\xi ^2\\cos {(2\\pi Q\_x-2\\mu \_{1,x})}. \\end{aligned}\$\$ (8) The phase-advance dependence of the resulting \$\\beta \$\-beating is illustrated in Fig. [4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig4). The minimum of each curve corresponds to the phase advance that yields the largest mutual beam–beam-induced luminosity enhancement between the two IPs. The optimal settings are \$\\mu \_{1,x}^{min}/2\\pi =0.405,\\,\\mu \_{1,y}^{min}/2\\pi =0.410\$; their difference reflects that between the nominal horizontal and vertical fractional tunes \$q\_x\$ and \$q\_y\$. Fig. 4 [![Fig. 4](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig4_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/4) Analytically computed beam-beam induced \$\\beta \$\-beating in a two-IP configuration, as a function of the phase advance \$ \\mu \_1\$ between the 2 IPs (\$q\_x=0.31\$, \$q\_y=0.32\$, head-on collisions at both IPs, \$\\xi =7\\times 10^{-3}\$ per IP) [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/4) The extrema of the curves in Fig. [4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig4) can be determined analytically by differentiating Eq. ([7](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Equ7)) with respect to the phase advance \$\\mu \_{1,x}\$. The general expression for this derivative \$\\frac{d}{d\\mu \_{1,x}} \\biggl ( \\frac{\\beta \_x^\}{\\beta ^\\_{0,x}} \\biggr )\$, detailed in Appendix A, is the product of two factors, each of which can be zero: - imposing that \$\\sin {(2\\pi Q\_x - 2\\mu \_{1,x})} = 0\$ yields the solution \$\$\\begin{aligned} \\frac{\\mu \_{1,x}}{2\\pi } =\\frac{Q\_x}{2}-\\frac{m}{4} \\, \\end{aligned}\$\$ (9) where m is an integer of either sign, or zero; - the other factor is zero for either \$\$\\begin{aligned} \\frac{\\mu \_{1,x}}{2\\pi } = \\frac{Q\_x}{2} - A(\\xi , Q\_x)\\, \\end{aligned}\$\$ (10) or \$\$\\begin{aligned} \\frac{\\mu \_{1,x}}{2\\pi } = \\frac{Q\_x - 1}{2} + A(\\xi , Q\_x) \\, \\end{aligned}\$\$ (11) with \$A(\\xi , Q\_x)\$ defined in Eq. ([12](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Equ12)) \$\$\\begin{aligned} A(\\xi , Q\_x) = \\arccos \\left( \\frac{ 32\\pi ^3\\xi ^3 \\sin (4\\pi Q\_x) + 48\\pi ^2\\xi ^2 \\cos ^2(2\\pi Q\_x) - 32\\pi ^2\\xi ^2 - 6\\pi \\xi \\sin (4\\pi Q\_x) + \\cos ^2(2\\pi Q\_x) - 1}{16\\pi ^2\\xi ^2 \\left( 4\\pi \\xi \\sin (2\\pi Q\_x) + \\cos (2\\pi Q\_x) \\right) } \\right) / (4\\pi ) \\end{aligned}\$\$ (12) The two sets of solutions do not overlap. The subset of solutions that minimize (rather than maximize) \$\\beta ^\\$ is identified by requiring that the second derivative be positive. This restricts the main solution to: \$\$\\begin{aligned} \\frac{\\mu \_{1,x}}{2\\pi } =\\frac{Q\_x}{2}+\\frac{m+1}{4} . \\end{aligned}\$\$ (13) The optimal phase \$\\mu \_{x,1}\$ that minimizes the \$\\beta ^\\$ for range of \$(\\xi , Q\_x)\$ values is shown in Fig. [5](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig5). Fig. 5 [![Fig. 5](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig5_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/5) Optimal phase advance value \$\\mu \_{x,1}\$ for various values of \$(\\xi , Q\_x)\$ in the two-IPs configuration [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/5) The phase-advance dependence of the dynamic-\$\\beta \$ effect can also be quantified in terms of the luminosity shift, at the witness IP, that is associated with the electromagnetic interaction of the two bunches at the scanning IP (Fig. [6](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig6)). The optimal setting, i.e. that which maximizes the sensitivity, at the witness IP, to beam-beam effects at the scanning IP, is indicated by the green dotted vertical line at \$\\Delta \\mu \_{\\text {IP1-IP5}}=0.41\\ \[2\\pi \]\$; the full suppression of the phase-related luminosity enhancement is indicated by the red dotted line. These results are consistent with the predictions of the analytical model (Fig. [4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig4)). In the two-IP configuration considered here, in which IP1 and IP5 can both play the role of either the scanning or the witness IP, and due to the periodicity of this curve (\$ \\pi \$) and to the fractional-tune values used during collisions (\$q\_x / q\_y = 0.31 / 0.32)\$, the phase–advance-related luminosity enhancement is the same at the two IPs; at the optimum setting, it is three times larger than it would be using the nominal LHC lattice. The maximum effect depends mainly on the transverse tunes, and thus could have been further enhanced if it had been possible to move these away from their nominal values. A more comprehensive simulation study, that evaluates the impact of the beam-beam interaction on the luminosity for more realistic Gaussian proton-density distributions, multiple collision points and as a function of the beam separation, is reported in Ref. \[[29](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR29 "J. Wańczyk, Precision luminosity measurement at hadron colliders. PhD Thesis, École Polytechnique Fédérale de Lausanne (2024). https://doi.org/10.5075/epfl-thesis-10500 ")\]. Fig. 6 [![Fig. 6](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig6_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/6) Beam–beam-induced luminosity shift at IP5 predicted by COMBI simulations, as a function of the phase advance \$ \\mu \_1=\\Delta \\mu \_{\\text {IP1-IP5}}\$ between the two IPs (\$q\_x=0.31\$, \$q\_y=0.32\$, head-on collisions at both IPs, \$\\xi =7\\times 10^{-3}\$ per IP) [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/6) ### 5\.2 Optimized phases for multi-IP configurations The general expression to obtain the first-order change of the \$\\beta \$ function at an arbitrarily chosen reference IP (at \$s=\\mu \_{0,x}=0\$) can be approximated as multiple quadrupole errors \[[30](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR30 "T. Pieloni, J. Barranco, X. Buffat, P.C. Jorge, L.M. Medrano, C. Tambasco, R. Tomás, Dynamic beta and beta-beating effects in the presence of the beam-beam interactions in 57th ICFA Advanced Beam Dynamics Workshop on High-Intensity and High-Brightness Hadron Beams, p. 027 (2016). https://doi.org/10.18429/JACoW-HB2016-MOPR027 ")\]: \$\$\\begin{aligned} \\frac{\\beta ^\\_x}{\\beta ^\\_{0,x}} = \\frac{2\\pi \\xi }{\\sin {2\\pi Q\_x}} \\sum \_{i\\in IPs}^{N-1} \\cos (2\\pi Q\_x-2\\mu \_{i,x})\\, \\end{aligned}\$\$ (14) where \$\\mu \_{i,x}\$ is the horizontal phase advance between the reference and the \$i^{th}\$ IP for a total of N IPs. Therefore, the optimal configuration, which minimizes the \$\\beta ^\\$ change at all IPs, must satisfy the condition: \$\$\\begin{aligned} \\sum \_{i=1}^{N-1} \\cos (2\\pi Q\_x-2\\mu \_{i,x}) = -(N-1)\\, \\end{aligned}\$\$ (15) which yields the same solution for all IPs as in the 2 IPs configuration (Eq. [13](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Equ13)): \$\$\\begin{aligned} \\frac{\\mu \_{i,x}}{2\\pi } =\\frac{Q\_x}{2}+\\frac{n+1}{4} . \\end{aligned}\$\$ (16) where n* is an integer of either sign, or zero. To maximize the effect on luminosity, the phase advance between the reference interaction point at \$ i = 0 \$ and the nearest IP, at \$ i = 1 \$ must satisfy \$ \\mu \_{1,x}/2\\pi = (Q\_x + 0.5)/2 \$, as given by the first solution (for \$n=0\$) of Eq. ([13](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Equ13)). This condition implies that the phase advance to any subsequent adjacent IPs (\$ i \\ge 2 \$) must be an integer multiple of \$ \\pi \$. Thus, this requirement exposes a fundamental constraint: it is not possible to minimize \$ \\beta ^\* \$ simultaneously at all IPs. This would only be possible for (\$ Q\_x = 0.5 \$) which is practically not achievable, as it would require operating the storage ring precisely at the half-integer resonance. ### 5\.3 Optics validation The optimum phase advance from IP1 to IP5 determined in Sect. [5\.1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec12) was implemented using tuning trim quadrupoles in the LHC arcs. The targeted fractional phase advance was \$0.9 \\, \[2 \\pi \]\$ for both beams and both planes (Table [4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Tab4)). This setting, which differs from that in Fig. [6](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig6) by one period (\$ \\pi \$), is equivalent in terms of sensitivity optimization but has the advantage that it minimizes the absolute magnitude of the perturbations inflicted upon the nominal lattice. Several measurements were carried out to validate this adjustment: comparison of the targeted and achieved shifts in phase advance (Sect. [5\.3.1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec15)), \$\\beta \$\-function measurements across the full LHC circumference (Sect. [5\.3.2](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec16)) as well as at the IPs (Sect. [5\.3.3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec17)), and dispersion measurements to estimate the dispersive contribution to the transverse beam size at the IP (Sect. [5\.3.4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec18)). #### 5\.3.1 Phase advance The phase advance around each LHC ring is determined using turn-by-turn orbit measurements \[[20](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR20 "R. Tomás, O. Brüning, M. Giovannozzi, P. Hagen, M. Lamont, F. Schmidt, G. Vanbavinckhove, M. Aiba, R. Calaga, R. Miyamoto, Cern large hadron collider optics model, measurements, and corrections. Phys. Rev. ST Accel. Beams 13, 121004 (2010). https://doi.org/10.1103/PhysRevSTAB.13.121004 "), [21](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR21 "R. Tomás, M. Aiba, A. Franchi, U. Iriso, Review of linear optics measurement and correction for charged particle accelerators. Phys. Rev. Accel. Beams 20, 054801 (2017). https://doi.org/10.1103/PhysRevAccelBeams.20.054801 ")\]. The targeted and achieved phase-advance shifts agree within \$0.01\\, \[2\\pi \]\$ or better (Table [4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Tab4)). The phase advance change from IP1 and IP5 was compensated on the opposite side of the ring, ensuring that the transverse tunes remain unchanged. The target fractional local phase advance is set to \$0.9,\[2\\pi \]\$. Table [4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Tab4) lists the corresponding phase shifts applied to the nominal lattice to reach this target on one side of the ring; this is consistent with the analytically derived value of \$0.41\\,\[2\\pi \]\$ for the compensating shift on the opposite side. The change in phase advance was also measured at the BSRT location, where the discrepancy between the model and measurements is more pronounced, reaching up to \$0.05\\,\[2\\pi \]\$. Table 4 Nominal and optimized IP1 to IP5 phase-advance settings. Column 4 (resp. 5) displays the targeted (resp. measured) shift in phase advance after the optimized settings have been applied [Full size table](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/tables/4) #### 5\.3.2 Residual \$\\beta \$\-beating The \$\\beta \$\-beating was quantified and found to be well within acceptable tolerances for various configurations of the trim strength. The measurement results, relative to the reference measurements without any phase advance adjustments, are illustrated in Fig. [7](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig7). The maximal observed deviation is approximately 25% around IP5 for Beam 1 in the vertical plane. Such significant deviations in proximity to the IPs are anticipated and are frequently attributed to erroneous BPM readings. Overall, the measurements indicated that \$\\beta \$\-beating remained within acceptable bounds, predominantly within a 10% margin. Fig. 7 [![Fig. 7](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig7_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/7) Measured \$\\beta \$\-function change from the full strength of the phase advance knob with reference to nominal lattice along the LHC ring, for Beam 1 (top two figures for x and y) and Beam 2 (bottom two figures for x and y) [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/7) The measured \$\\beta \$ functions are also compared to the MADX LHC lattice model predictions, which include the phase advance adjustment. The results, illustrated along the LHC lattice in Fig. [8](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig8), indicate that the measurements align with the model within an acceptable margin of 10%. Fig. 8 [![Fig. 8](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig8_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/8) Measured deviations in the \$\\beta \$\-function along the LHC ring relative to the MADX model incorporating the phase advance knob, for Beam 1 (top two figures for x and y) and Beam 2 (bottom two figures for x and y). The error bars represent solely the statistical uncertainties as determined in the measurements [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/8) #### 5\.3.3 \$\\beta \$ functions at the interaction points The DOROS BPMs are strategically located at the end of the inner triplets on both sides of each of the experiments. Due to the phase advance of \$\\pi \$ between the DOROS on the left and right sides, these are inadequate for the precise determination of the measurement of \$\\beta ^\\$ at the IP. This can only be achieved with a special k\-modulation technique \[[31](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR31 "F. Carlier, et al. Challenges of K-modulation measurements in the LHC Run 3, in Proceedings of IPAC’23. IPAC’23—14th International Particle Accelerator Conference (JACoW Publishing, Geneva, Venice, 2023), pp. 531–534. https://doi.org/10.18429/JACoW-IPAC2023-MOPL014 ")\], which was performed at the interaction points used in the experiment. The results are summarized in Table [3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Tab3). While the achieved precision of the \$\\beta ^\\$ determination varies between measurements, uncertainties at the level of \${\\mathcal {O}}(1\\%)\$ are not uncommon and are not systematic to a specific beam or interaction point. Such uncertainties would propagate to only a few-percent effect on derived quantities such as the emittance or the beam–beam parameter. #### 5\.3.4 Dispersion It was essential to perform dispersion \$D^\\$ measurements at the interaction points, under the new optics configurations, as it can contribute to the measured beam size. Assuming the LHC design momentum spread value of \$\\frac{\\Delta p}{p}=10^{-4}\$, the average dispersion measurement around IP1 and IP5 is estimated to contribute up to 2% to the observed bunch sizes (worst case measured for B2Y). This effect is considered static throughout the experiment. ## 6 Experimental results This section presents experimental measurements of beam–beam effects in configurations involving multiple collision points. The impact of the optics modifications introduced in Sect. [5](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec11) is first assessed through tune spectra measurements, as detailed in Sect. [6\.1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec20). To quantify the cumulative effect of one or more additional collisions and assess the reproducibility of the results, a stepwise collision scan was employed. In this procedure, beams were sequentially brought into head-on collision at selected interaction points (IPs), followed by full transverse separation (\$\\Delta \> 6\\sigma \$) in either the horizontal or vertical plane at all but one witness IP. Two distinct filling schemes were used to isolate the influence of a single additional collision (2-IP configuration: ATLAS and CMS, see Sect. [6\.2](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec21)) and to investigate more complex multi-collision scenarios (3-IP configuration: including ALICE, see Sect. [6\.3](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec25)). The first measurement of separation-dependent beam–beam effects, analogous to those encountered during luminosity calibration scans, is presented in Sect. [6\.4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec28), alongside comparisons with numerical predictions. An additional measurement conducted during the 2022 van der Meer calibration is detailed in Sect. [6\.5](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec32). A consolidated overview of all findings is provided in Sect. [7](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec33). ### 6\.1 Measured effect on the tune spectra of the phase advance changes The coherent transverse tune spectra serve as a diagnostic for evaluating the effectiveness of the phase advance modifications. COMBI simulations were carried out for two extreme lattice configurations, one designed to maximize and the other to suppress beam–beam effects. In the head-on scenario (Fig. [9](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig9)), the coherent modes are suppressed for the maximizing configuration, as the phase advance between the two collision points is close to \$\\pi \$, as was previously shown in simulation in Ref. \[[32](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR32 "Y. Alexahin, A study of the coherent beam-beam effect in the framework of the Vlasov perturbation theory. Nucl. Instrum. Methods Phys. Res., Sect. A 480(1–3), 253–288 (2002). https://doi.org/10.1016/S0168-9002(01)01219-0 ")\]. In this configuration, the effect of the first collision is counteracted by the second. This setting is particularly suitable for studying incoherent spectra, as the diminished coherent signal allows for a clearer extraction of the mean tune shift from fitted spectra. During separation scans, the symmetry between horizontal (x) and vertical (y) planes is broken. In the scanning plane (horizontal), coherent modes re-emerge at \$1.5\\,\\sigma \$ separation (Fig. [10](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig10)), with their frequency shifting in response to the separation distance (additional example for \$3\\,\\sigma \$ is shown in Fig. [11](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig11)). Fig. 9* [![Fig. 9](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig9_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/9) Comparison of COMBI simulated tune spectra in x\-plane (left) and y\-plane (right) for the suppressing (in blue) and maximizing (in green) phase configurations, for head-on collision [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/9) Fig. 10 [![Fig. 10](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig10_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/10) Comparison of COMBI simulated tune spectra in x\-plane (left) and y\-plane (right) for the suppressing (in blue) and maximizing (in green) phase configurations, at \$1.5\\,\\sigma \$ separation in the horizontal plane [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/10) Fig. 11 [![Fig. 11](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig11_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/11) Comparison of COMBI simulated tune spectra in x\-plane (left) and y\-plane (right) for the suppressing (in blue) and maximizing (in green) phase configurations, at \$3\\,\\sigma \$ separation in the horizontal plane [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/11) These simulated trends of the coherent spectra are also observed in experimental data. Transverse spectra recorded for head-on conditions confirm the suppression of coherent modes in the maximizing phase configuration (Fig. [12](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig12)). Additional comparisons for separated beams at the IP (see Figs. [13](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig13) and [14](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig14)), shown here for the case of vertical separation, reveal similar features. In this configuration, the non-scanning (horizontal) plane exhibits spectra that qualitatively agree with simulation predictions (see Figs. [10](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig10) and [11](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig11)), while in the scanning (vertical) plane the coherent modes do not reappear as expected. An equivalent behavior is observed when the separation is applied in the horizontal plane, with the roles of the two transverse planes interchanged; this case is not shown for brevity. This discrepancy is attributed to the effect of the transverse damper, which acts independently on the two beam centroids and suppresses coherent oscillations arising from beam–beam coupling, consistent with earlier observations \[[33](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR33 "R. Calaga, W. Herr, G. Papotti, T. Pieloni, X. Buffat, S. White, R. Giachino, Coherent beam-beam mode in the LHC. CERN (2014). https://doi.org/10.5170/CERN-2014-004.227 . http://cds.cern.ch/record/1957039 ")\]. Fig. 12 [![Fig. 12](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig12_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/12) Measured tune spectra before and after phase change, corresponding to the suppressing (in blue) and maximizing (in green) phase configurations, at the head-on position, for each of the transverse planes (Beam 1 shown as an example). Dash-dotted lines show Gaussian fits to the data [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/12) Fig. 13 [![Fig. 13](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig13_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/13) Comparison of the measured tune spectra at suppressing (blue) and maximizing (green) phase configurations with \$1.3\\,\\sigma \$ separation in the vertical plane [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/13) Fig. 14 [![Fig. 14](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig14_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/14) Comparison of the measured tune spectra at suppressing (blue) and maximizing (green) phase configurations with \$2.6\\,\\sigma \$ separation in the vertical plane [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/14) ### 6\.2 Head-on collisions in a two-IP configuration at \$\\sqrt{s} = 900\\ \\text {GeV}\$ Each test consisted of four measurements designed to quantify the beam–beam effect induced by a single additional head-on collision. The sequence was executed with CMS as the witness IP, followed by ATLAS, and concluded with a repeat at CMS. Over the course of the fill, the beam–beam parameter decreased from approximately 0.01 to 0.005. The outcome of each scan is analyzed using three independent observables: luminosity, beam size, and tune spectra, discussed separately below. #### 6\.2.1 Impact of beam-beam effects on head-on luminosity An overview of the luminosity measurements during the first test with CMS as the witness IP is shown in Fig. [15](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig15), where the specific luminosity at CMS is normalized to the initial head-on step to highlight relative changes. The uncertainty at each step is given by the standard error of the mean; shaded bands represent the standard deviation. A comparable dataset for ATLAS is presented in Fig. [16](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig16), confirming the expected luminosity change due to beam–beam effects originating from another IP. Relative luminosity changes for each step in all tests are summarized in Table [5](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Tab5). These values are computed relative to the average of adjacent head-on points, which removes the linear luminosity decay over time. Assuming slow change of beam parameters, the results within each test are expected to be consistent. Fig. 15 [![Fig. 15](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig15_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/15) CMS specific luminosity, normalized to the first measured both IPs head-on point (blue), during the first step-function scan at IP1. The measurements with ATLAS fully separated are shown in red. The error bars indicate error on the mean [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/15) Fig. 16 [![Fig. 16](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig16_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/16) ATLAS specific luminosity, normalized to the first measured both IPs head-on point (blue), during the step-function scan at IP5. The points for which IP5 was separated are shown in red. The error bars indicate the statistical error [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/16) Table 5 Measured luminosity shifts in each of full separation steps – summary of all three step-function scans performed for IP1 and IP5 as the witness IP. The beam-beam parameter at the start of each scan \$\\xi \_\\text {start}\$ is included for reference [Full size table](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/tables/5) All measurements are compiled in Fig. [17](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig17) and compared to COMBI simulation predictions. The results are plotted as a function of the beam–beam parameter (estimated as described in Sect. [4\.4](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec8)), following a reverse chronological order. Agreement with simulation is excellent across the entire range. Fig. 17 [![Fig. 17](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig17_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/17) COMBI simulated luminosity enhancement induced by a single head-on collision as a function of the beam-beam parameter (dashed line), compared to the test results at both ATLAS (red points) and CMS (green points) [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/17) Fig. 18 [![Fig. 18](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig18_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/18) Measurements of beam-beam effects on beam widths shown with red points for each beam and plane, during the experiment with IP5 as the witness IP at \$\\xi =0.0056\$. COMBI predictions are shown in blue with calculated systematic error from the phase advance error between the IP and BSRT locations [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/18) #### 6\.2.2 Impact of beam-beam effects on single-beam sizes Complementary measurements of the bunch size from the BSRT are shown in Fig. [18](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig18). Data from all steps and bunches are combined, and a linear trend corresponding to continuous emittance growth is subtracted using the difference between the first and last head-on steps. For comparison, COMBI simulation results are also included in the figure. The dominant source of uncertainty in COMBI predictions arises from discrepancies between the measured phase advance and the MADX model, with the largest observed deviation being \$0.05\\,\[2\\pi \]\$ (as discussed in Sect. [5\.3.1](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Sec15)). This uncertainty is propagated using a modified form of Eq. ([6](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Equ6)) adapted to the BSRT location. The magnitude of this uncertainty depends on the specific phase advance configuration, as some are more sensitive to small deviations. This is reflected in the varying sizes of the error bands associated with each prediction. The most accurate predictions are obtained in the vertical plane of Beam 2, where the beam–beam–induced \$\\beta \$\-beating propagates to the BSRT with a particularly favorable phase advance. In this configuration, the measured effect is also the most pronounced, and excellent agreement with COMBI simulations is observed. In other cases, the expected absolute effect is smaller, typically below 1%, and the influence of phase advance uncertainty becomes more significant, limiting the predictive reliability of the simulations. Experimentally, it is also possible that in these configurations the beam–beam–induced effect is too subtle to resolve, potentially masked by instrumental limitations. #### 6\.2.3 Impact of beam-beam effects on tune spectra Figure [19](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig19) shows representative ADT spectra recorded during the step-function scans. These spectra are affected by coupling, noise, and various nonlinearities, so they do not follow an idealized analytical distribution. Nonetheless, the peak position can be extracted using a Gaussian fit (indicated by dashed lines), yielding an estimate of the mean tune shift. In the single-collision configuration (blue lines), unsuppressed coherent modes dominate the spectrum and partially obscure the underlying incoherent component. The Gaussian fit is therefore applied to the incoherent contribution between coherent lines, rather than to the global spectral maximum, which results in a larger uncertainty on the extracted peak position. Despite this, in each of the performed step-scans at both collision points, the single-collision tune shift, defined as twice the fitted mean shift, is found to be consistent within \$\\pm 5\\%\$ with the expected coherent beam–beam tune shift corresponding to the conditions of the respective scan. For head-on collisions with Gaussian transverse beam distributions, the coherent tune shift of the dipole mode is expected to be \$Y\\xi \$, where Y is the Yokoya factor (\$Y = 1.21\$ for round beams) \[[34](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR34 "K. Yokoya, Y. Funakoshi, E. Kikutani, H. Koiso, J. Urakawa, Tune shift of coherent beam-beam oscillations. Part. Accel. 27, 181–186 (1990)")\]. As expected, the two-collision configuration yields a tune shift approximately double that of the single-collision case \[[13](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR13 "A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). https://doi.org/10.48550/arXiv.2306.10394 ")\]. Fig. 19 [![Fig. 19](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig19_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/19) Measured vertical tune spectra for different collision configurations: reference nominal tune (red), without the presence of any collisions, compared to single collision (blue) and two collisions (green) at \$\\xi \\simeq 0.0086\$. The dash-dotted lines in corresponding colors indicate Gaussian fits to the data used to extract mean tunes [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/19) Fig. 20 [![Fig. 20](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig20_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/20) Specific luminosity, normalized to the first head-on point, measured at ATLAS (top) and CMS (bottom) during beam separation at the other IPs, with horizontal separation applied in steps 2–4 and vertical separation in steps 6–8. The tested configurations are indicated in the legend. The blue line shows an interpolation of the head-on measurements, serving as a reference for relative luminosity changes [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/20) ### 6\.3 Head-on collisions in a three-IP configuration at \$\\sqrt{s} = 900\\ \\text {GeV}\$ An additional head-on collision was included in the configuration using IP2 (ALICE), as it was the only IP where this was feasible. A specially designed filling scheme ensured uniform collision exposure across all bunches, allowing measurements to be combined and thereby reducing statistical uncertainties. However, no reliable luminosity data were available from IP2 during this experiment. Table 6 Luminosity shift per full separation step, calculated with respect to an interpolated reference obtained from the two closest head-on points. Results for horizontal and vertical separations are shown as Shift part 1 and Shift part 2, respectively [Full size table](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/tables/6) #### 6\.3.1 Impact of beam-beam effects on head-on luminosity A step-function scan was conducted by sequentially collapsing or separating one IP at a time, with the separation applied first in the horizontal plane and, in a second stage of the experiment, in the vertical plane. A comprehensive summary of the measured specific luminosity at ATLAS and CMS is presented in Fig. [20](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig20). Exceptionally stable beam conditions during this test enabled measurements at both experiments under nearly identical beam–beam parameters. The beam–beam-induced luminosity changes observed during each step are quantified in Table [6](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Tab6), referenced to the time-interpolated specific luminosity from three head-on points. Orbit-drift corrections are applied only to these head-on reference points and result in changes of the extracted luminosity shifts of at most 0.2%. Good agreement with COMBI simulations is observed across the separation steps, with measured shifts closely matching the simulated values for configurations involving IP1 and IP5. For single-IP separation, the level of agreement depends on the witness IP and the separation step. When IP1 is used as the witness, the luminosity shift in the first separation step is reproduced almost exactly by COMBI, while a smaller shift is observed in the second step. This behaviour is expected, as the beam–beam parameter decreases during the scan and the measurement becomes increasingly sensitive to residual orbit perturbations. For configurations involving IP2, the comparison is less conclusive. As no reliable luminosity data were available from IP2 during this experiment, it was not possible to precisely optimize the collision conditions at this IP. As a result, the actual beam–beam conditions at IP2 may differ from those assumed in the simulations, affecting the comparison. Residual orbit drifts affecting the separation steps cannot be fully corrected. As illustrated by the DOROS measurements in Fig. [21](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig21), the orbit data contain both slow drifts and beam–beam–induced orbit offsets arising from changes in the collision configuration when other IPs are separated. While the beam–beam–induced offsets are included in the COMBI simulations, the residual orbit drifts are not. Since these contributions cannot be disentangled in the measurements, a full correction is not possible. Consequently, the effective beam–beam impact measured during the experiment may be weaker than that assumed in the simulations. Fig. 21 [![Fig. 21](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig21_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/21) Residual separation between the two beams at all three IPs in horizontal (top) and vertical (bottom) planes, for test with observations at ATLAS, where the separation was performed first in the horizontal and then in the vertical plane at the other IPs [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/21) #### 6\.3.2 Tune shift scaling with the number of beam–beam interactions Analysis of the transverse tune spectra in this configuration is further complicated by the reappearance of coherent modes when beams are separated at one or more IPs, as predicted by simulation. In such cases, the maximum tune shift is determined from the frequency difference between the \$\\pi \$ and \$\\sigma \$ modes, rather than from fitting the incoherent peak. For configurations where all three IPs were in collision, coherent modes remained suppressed due to the retained symmetry between IP1 and IP5. Figure [22](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig22) presents the measured tune shifts for all tested collision configurations. The unperturbed tune measurement, shown at ‘0’ collisions, serves as the reference from which all tune shifts are calculated. An empirical uncertainty of 0.001 is assumed for each independent measurement. Both transverse planes and separation directions are included to demonstrate consistency. As expected, the measured tune shift scales linearly with the number of head-on collisions. Fig. 22 [![Fig. 22](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig22_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/22) Measured mean tune shift as a function of the number of head-on collisions, shown for both transverse planes and for both separation directions (horizontal and vertical) used in the experiment. The separation refers to how collisions were removed: it was applied in only one transverse direction at a time (H sep. or V sep.) at the non-colliding interaction points, which are not explicitly labeled on the x-axis [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/22) ### 6\.4 Beam-beam signatures during transverse-separation scans at \$\\sqrt{s} = 900\\ \\text {GeV}\$ In the van der Meer (vdM) calibration method, beam–beam corrections are applied separately at each separation step. Therefore, it is essential to validate the simulated dependence of luminosity shifts on nominal beam separation. Full vdM-like scans were performed in both transverse planes at ATLAS and CMS, using a configuration where only these two IPs were in collision. #### 6\.4.1 Impact of beam–beam effects on the luminosity scan-curves Figure [23](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig23) shows the luminosity curves measured at ATLAS, acting as the witness IP, during a separation scan performed at CMS. The curves are fitted with Gaussian profiles for visual guidance. The beam separation at CMS is expressed in units of the measured transverse beam size, \$\\sigma \_{\\textrm{meas}}\$, which was found to be smaller than the nominal beam size, \$\\sigma \_{\\textrm{nom}}\$, used to define the scan steps. As a result, scan points corresponding to nominal separations beyond \$6\\,\\sigma \_{\\textrm{nom}}\$ extend to larger values when expressed in units of \$\\sigma \_{\\textrm{meas}}\$. To account for emittance evolution during the scan, both linear and exponential fits are applied to the head-on luminosity points before, after, and in the middle of the scan. These trends are subtracted from the data in Fig. [23](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig23), and all measurements are presented relative to this fitted head-on reference (normalized to unity). Fig. 23 [![Fig. 23](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig23_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/23) ATLAS specific luminosity relative changes (blue points) during separation scans in x\-plane (top) and y\-plane (bottom) at CMS, corrected for exponential (exp. corr.) luminosity decay. Gaussian function with a constant fitted to these points is shown with blue dashed line and compared to fit performed on the data corrected for linear luminosity change (lin. corr.) in time (dotted line). These are compared to COMBI simulation predictions (purple solid line) [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/23) At CMS, similar scans were performed; however, degraded beam stability was observed during this period of the fill. The scan results exhibit systematic deviations, especially at large separation, which could not be symmetrically corrected using simple linear or exponential models. Despite these issues, all four scan cases (two transverse planes at two IPs) yield a total specific luminosity reduction of approximately 2.4%, consistent with expectations based on the measured beam–beam parameter \$\\xi \$. Comparison with COMBI simulations reveal up to 30% discrepancy for IP5 as the witness IP, while agreement improves to within 10% for measurements at IP1. These differences are comparable to those observed in Table [6](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Tab6), suggesting a common underlying origin. While a finer scan granularity can improve the description of the steep-response region below \$3\\,\\sigma \_{\\textrm{meas}}\$, it would not address the dominant discrepancy between head-on and fully separated luminosity, which is likely driven by simulation assumptions and scan-related systematics. #### 6\.4.2 Beam–beam-induced orbit deflections Beam–beam deflections are clearly observed during separation scans. Figure [24](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig24) shows the fully corrected residual horizontal beam displacement at IP1, measured with the DOROS beam position monitors, as a function of the beam separation. This representation allows a direct comparison with the analytical beam–beam deflection prediction based on the Bassetti–Erskine formalism \[[35](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR35 "M. Bassetti, G.A. Erskine, Closed expression for the electrical field of a two-dimensional gaussian charge. Technical report, CERN, Geneva (1980). https://cds.cern.ch/record/122227 ")\]. The raw orbit data were corrected for slow orbit drifts by aligning the beam positions at the luminosity optimization points before and after each scan. Deviations of the applied beam separation from the nominal settings were accounted for by fitting a common length scale simultaneously with the beam–beam deflection. This approach avoids an overestimation of the separation calibration. The measured orbit displacement at the DOROS location was corrected for the contribution arising from the beam–beam deflection angle by applying a geometric amplification factor, proportional to the distance between the IP and the BPM in the small-angle approximation. After all corrections, the measured beam–beam deflection as a function of beam separation closely follows the analytical prediction. The maximum horizontal deflection observed at IP1 is 10.2 \$\\upmu \$m, compared to an expected value of 8.4 \$\\upmu \$m, with agreement within the total measurement uncertainty. The dominant uncertainty contribution (3.5 \$\\upmu \$m, corresponding to about 90% of the total) arises from a possible residual half crossing angle at the IP of up to 0.5 \$\\upmu \$rad. Additional systematic uncertainties include the fitted separation length scale, tune uncertainty of 0.005, and an extended 2 % uncertainty on \$\\beta ^\\$. Statistical uncertainties are negligible for most separation steps. Fig. 24* [![Fig. 24](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig24_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/24) Orbit offset caused by the beam–beam interaction measured by DOROS BPMs in the horizontal plane at IP1 during the horizontal separation scan. Analytical prediction is shown for comparison \[[35](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR35 "M. Bassetti, G.A. Erskine, Closed expression for the electrical field of a two-dimensional gaussian charge. Technical report, CERN, Geneva (1980). https://cds.cern.ch/record/122227 ")\] [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/24) #### 6\.4.3 Impact of the beam-beam effects on beam sizes and tune shift Separation-dependent changes in beam size were measured using the synchrotron light monitor. Figure [25](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig25) shows good consistency across individual bunch measurements, as well as strong agreement with COMBI simulations. This reinforces the utility of the BSRT as a reliable diagnostic for beam–beam effects. The uncertainties shown are statistical only and are treated as uncorrelated between scan points. The normalization to the zero-separation value is applied for visualization purposes and does not impose a constraint on that point. Fig. 25 [![Fig. 25](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig25_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/25) BSRT beam-size measurement in the vertical plane for each bunch (points) during horizontal (top) and vertical (bottom) separation scans at IP1. The exponential evolution correction is applied based on adjacent head-on points (before, in the middle and after the scan) to highlight the relative changes. COMBI predictions are shown (gray dashed line) for comparison [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/25) Transverse tune shifts were extracted from ADT-measured spectra. As illustrated in Fig. [26](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig26), the mean tune shift evolution is reconstructed for both the scanning and non-scanning planes. The measured tune shifts closely follow the expected COMBI trends, further validating the simulation models used for beam–beam corrections. Fig. 26 [![Fig. 26](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig26_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/26) Measured tune shift in units of the beam-beam parameter \$\\xi \_{BB}\$ for the horizontal (blue) and vertical (orange) planes, during a horizontal separation scan at IP1. The error bars indicate typically assumed conservative empirical systematic error on the tune measurement of 0.001. The solid curves represent the COMBI prediction [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/26) ### 6\.5 Beam-beam signatures during vdM scans at \$\\sqrt{s} = 13.6\\ \\text {TeV}\$ During the first vdM calibration of LHC Run 3, an unusually high beam–beam parameter was recorded, offering a unique opportunity to study beam–beam effects under enhanced interaction conditions. These effects are examined using online ATLAS luminosity data, with representative results shown in Fig. [27](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#Fig27). The example focuses on a standard vdM scan pair conducted at CMS, while monitoring luminosity at the non-scanning IP (ATLAS). The plotted luminosity changes are normalized to the central scan point, where both IPs are in head-on collision. Variations in this normalized luminosity across the scan directly reflect the impact of beam–beam interaction at the scanning IP on the luminosity at the non-scanning IP. These patterns are reproducible across multiple scan pairs acquired within the same fill. At the largest nominal separations (approximately \$\\pm 0.6\\,\$mm), the beams are sufficiently separated to eliminate beam–beam effects entirely. Notably, a qualitative difference is observed between the horizontal and vertical scan planes, an asymmetry that is not present in the above presented dedicated experiment. This behavior is attributed to differences in the phase advance between IPs in the standard vdM optics configuration. For vertical (y) scans, experimental data agree well with simulation, confirming the robustness of the modeling under these conditions. In contrast, the horizontal (x) scans exhibit a systematic discrepancy: the observed beam–beam impact is broader than that predicted by simulation. The model assumes that the separation steps are accurately defined using the beam size measured during the vdM scans, but this mismatch may suggest an unmodeled phase advance deviation or another optics imperfection. The total difference between data and simulation is within 0.2%. This dataset originates from a standard vdM calibration, for which only online luminosity measurement was available, and many contributions to the overall vdM uncertainty, such as bunch-to-bunch variations, averaging over different bunch families, and emittance evolution during the scan, are not disentangled. Therefore, the 0.2% difference should be interpreted as a consistency check under typical operational conditions rather than a precision validation of the beam–beam model. Nevertheless, this indicates that the model remains sufficiently accurate for beam–beam corrections in vdM analyses. Fig. 27 [![Fig. 27](https://media.springernature.com/lw685/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-026-15476-8/MediaObjects/10052_2026_15476_Fig27_HTML.png)](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/27) Changes from the separation-dependent beam-beam effects at the non-scanning IP – ATLAS online luminosity shown with points. Data comes from a standard vdM scan pair during the 2022 pp luminosity calibration. Separation steps and direction are indicated with gray markers. The last CMS scan is shown, with the highest beam-beam parameter of \$ \\xi =5.3 \\times 10^{-3} \$. The corresponding COMBI simulation results are also shown (triangles) [Full size image](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8/figures/27) ## 7 Conclusions Beam–beam effects were directly observed and quantitatively validated at the LHC. The impact of beam–beam interactions on head-on luminosity, bunch sizes, tune spectra, and beam orbit positions was measured across various collision configurations. The observed effects align closely with COMBI simulations. In particular, the luminosity enhancement due to amplitude-dependent \$\\beta \$\-beating was measured for the first time and is found to agree with expectations over a wide range of the beam–beam parameter \$\\xi \$. Together with the observed linear scaling with respect to the beam–beam parameter, this provides experimental confirmation of the numerical studies presented in \[[13](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR13 "A. Babaev, T. Barklow, O. Karacheban, W. Kozanecki, I. Kralik, A. Mehta, G. Pasztor, T. Pieloni, D. Stickland, C. Tambasco, R. Tomas, J. Wańczyk, Impact of beam-beam effects on absolute luminosity calibrations at the CERN large hadron collider. Eur. Phys. J. C 84, 17 (2024). https://doi.org/10.48550/arXiv.2306.10394 ")\]. Consistent signatures were found across multiple scan configurations, including two- and three-IP setups, and the characteristic tune shifts were well reproduced by both measurement and simulation. Accurate instrumentation and experimental design were essential. Key to the results was the stepwise collision scheme, isolating individual beam–beam contributions, with the implementation of the witness IP. Synchrotron light measurements from the BSRT, especially in the vertical plane of Beam 2, showed excellent agreement with COMBI predictions and bunch-by-bunch consistency. Tune spectra from the ADT system provided direct access to mean tune shifts, while orbit data from the DOROS system enabled measurement of beam–beam deflection down to 1 \$\\upmu \$m, matching analytical estimates. These tools, combined with precise control of IP configurations, enabled a comprehensive cross-check of beam–beam models. Some systematic limitations were identified. During one of the tests, beam stability during the scans was insufficient to allow for symmetric modeling of luminosity decay in both transverse planes. In several scans, the nominal separation steps were based on overestimated beam sizes, resulting in larger-than-expected scan intervals and less sensitivity at low separation. Beam–beam deflection measurements showed sign reversals at high separation, attributed to nonlinear orbit distortions from hysteresis effects in superconducting magnets. Phase advance errors between IPs and diagnostic systems further reduced the accuracy of simulations in some configurations. Targeted improvements would allow for higher precision and new insights. Finer separation steps, especially below \$3\\,\\sigma \$, would improve sensitivity to nonlinear beam–beam effects and allow more accurate comparison with models. Suppressing coherent beam–beam modes via phase advance tuning of not only two but all three IPs would improve tune shift measurements. Moreover, such optimal phase advance configuration between the IPs can be exploited to enhance luminosity. This effect could be directly propagated into at least a few percent increase in the total collected integrated luminosity, as discussed in Ref. \[[36](https://link.springer.com/article/10.1140/epjc/s10052-026-15476-8#ref-CR36 "J. Wańczyk, T. Pieloni, Optimizing beam-beam beta-beating for luminosity enhancement at the LHC. JACoW IPAC 2025, 018 (2025)")\]. Aligning the phase advance between IPs and diagnostic devices, particularly the BSRT, would reduce sensitivity to optics errors, enabling better beam size measurements for both beams and transverse planes. Moreover, the suppression of the coherent beam–beam modes facilitates the observation and analysis of the central, incoherent part of the tune spectrum, which is otherwise obscured by the dominant coherent peaks. Further experimental strategies should also focus on directly probing the beam–beam interaction at its point of occurrence. This would require access to the full transverse beam distributions with high spatial resolution and sensitivity, going beyond projected sizes or centroids. Such measurements could directly reveal the nonlinear distortions and tails induced by the beam–beam force. In parallel, validating correction schemes that parametrize multi-IP beam–beam effects on luminosity via equivalent tune shifts remains essential. These models underpin practical luminosity corrections in complex fill configurations and would benefit from systematic benchmarking. ## References 1. S. Meer, Calibration of the effective beam height in the ISR. 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