Measurement of the inelastic proton-proton cross section at $\sqrt{s}=$ 13 TeV

A measurement of the inelastic proton-proton cross section with the CMS detector at a center-of-mass energy of $\sqrt{s} =$ 13 TeV is presented. The analysis is based on events with energy deposits in the forward calorimeters, which cover pseudorapidities of -6.6 $<\eta<$ -3.0 and +3.0 $<\eta<$ +5.2. An inelastic cross section of 68.6 $\pm$ 0.5 (syst) $\pm$ 1.6 (lumi) mb is obtained for events with $M_\mathrm{X}>$ 4.1 GeV and/or $M_\mathrm{Y}>$ 13 GeV, where $M_\mathrm{X}$ and $M_\mathrm{Y}$ are the masses of the diffractive dissociation systems at negative and positive pseudorapidities, respectively. The results are compared with those from other experiments as well as to predictions from high-energy hadron-hadron interaction models.


Introduction
At √ s = 13 TeV, the ATLAS Collaboration reported a measurement of the inelastic pp cross section of 68.1 ± 0.6 (syst) ± 1.3 (lumi) mb for ξ > 10 −6 (corresponding to M > 13 GeV) [21]. This value has been extrapolated to the total inelastic phase space, yielding σ inel = 78.1 ± 0.6 (syst) ± 1.3 (lumi) ± 2.6 (extr) mb, with the last number being the extrapolation uncertainty. Finally, the TOTEM Collaboration obtained a value for the inelastic cross section at √ s = 13 TeV of σ inel = 79.5 ± 1.8 (syst) mb [22]. This paper presents a new measurement of the inelastic cross section in pp collisions at √ s = 13 TeV. Data collected with the CMS forward calorimeters HF and CASTOR, described in Section 2 and covering pseudorapidities −6.6 < η < −3.0 and +3.0 < η < +5.2, are analyzed. These detectors provide sensitivity to a large part of the total inelastic cross section, including diffractive events with dissociated protons that produce particles only at forward rapidity, with the exception of low-mass diffraction and events that happen to have rapidity gaps in the regions covered, including central exclusive production, e.g., diffractive dissociation mediated via double pomeron exchange. The fiducial cross section is therefore measured in a phase space region excluding fractional momentum losses of the scattered protons ξ X = M 2 X /s < 10 −7 and ξ Y = M 2 Y /s < 10 −6 , corresponding to M X < 4.1 GeV and M Y < 13 GeV, where M X and M Y are defined as the invariant masses of the dissociated proton systems with negative and positive pseudorapidities, respectively. The use of the CASTOR forward calorimeter allows the extension of this type of measurement to a low mass region so far unexplored.

The CMS detector
CMS uses a right-handed coordinate system, with the origin at the nominal interaction point, the x axis pointing to the center of the LHC, the y axis pointing up (perpendicular to the LHC plane), and the z axis along the anticlockwise-beam direction.
The central feature of the CMS apparatus [23] is a superconducting solenoid of 6 m internal diameter, providing a magnetic field of B = 3.8 T parallel to the beams. Within the solenoid volume are a silicon pixel and strip tracker (covering |η| < 2.5), a lead tungstate crystal electromagnetic calorimeter (|η| < 3) , and a brass and scintillator hadron calorimeter (|η| < 3), each composed of a barrel and two endcap sections.
Extensive forward calorimetry complements the coverage provided by the barrel and endcap detectors. The forward hadron (HF) calorimeters are located on each side of the detector, covering the range 3.0 < |η| < 5.2, and are each composed of 18 steel azimuthal (φ) wedges, with embedded quartz fibers running parallel to the beam direction. Each wedge is subdivided into 13 segments in η, called towers. The very forward angles are covered at one end of CMS (−6.6 < η < −5.2) by the CASTOR calorimeter [24]. This detector, consisting of tungsten absorbers and quartz detection planes, is segmented in 16 φ-sectors and into 2 electromagnetic and 12 hadronic modules along the beam line, corresponding to a total of 224 cells. For operational reasons the CASTOR calorimeter was only partially included in the detector setup during the run periods considered in this analysis, and the data with CASTOR included were taken with the solenoid switched off (B = 0 T).
A more detailed description of the CMS detector can be found in Ref. [23].

Physics models
Various Monte Carlo event generators are used to correct the measured cross section for acceptance and instrumental effects, as well as to compare the final results to different hadron-hadron interaction model predictions.
The PYTHIA MC generator [5,6] uses the Donnachie-Landshoff (DL) parametrization [2] for the total hadron-hadron cross section, and provides different approaches to determine the elastic and diffractive contributions (unless stated otherwise, the default Schuler-Sjöstrand (SS) model [3,4] is used). The difference between the total cross section and the sum of the elastic and diffractive cross sections is used to normalize the nondiffractive part, which is generated through a regularization of perturbative multi-parton interaction cross sections. Event samples are generated with different underlying event tunes: the Z2* [25] tune is used for the PYTHIA 6 (version 6.426) sample, while PYTHIA [8,9].

Event selection
This analysis is based on pp collision data collected with the CMS detector in 2015 at √ s = 13 TeV, during several running periods with low pileup. Runs for which the solenoid was off (B = 0 T), as well as runs with the solenoid at its nominal field strength (B = 3.8 T) are analyzed. The former runs, with CASTOR included in the detector setup, are used because of its larger acceptance for inelastic collisions.
The total integrated luminosity recorded in these runs amounts to 40.8 (28.0) µb −1 for B = 3.8 T (B = 0 T). The measurement of the integrated luminosity for the B = 3.8 T data is based on the pixel tracker, and has been calibrated by means of a dedicated analysis of van der Meer scans [35] with an accuracy of 2.3%. The integrated luminosity for the B = 0 T sample is obtained by normalizing the fiducial cross section found at B = 3.8 T to the one at B = 0 T in the same phase space.
The CMS data acquisition was triggered [36] by the presence of both beams in the interaction point ("zero bias"), signaled by beam pick-up monitors located at ±175 m from the interaction point. Additional triggers requiring the presence of only one beam ("single bunch") or no beams ("empty bunch") were used to study beam-gas, electronic noise, and other backgrounds. These triggers are 100% efficient for the event selection under consideration. A first sample of inelastic events is then selected offline by requiring an energy deposit above 5 GeV in either of the two HF calorimeters. This threshold was optimized by studying detector noise in events without beam. The rate of selected events from single bunches was found to be consistent with the one from empty bunches. This demonstrates that the presence of the beam on one side does not generate more background events compared to no beam, and that the total background contribution can be estimated from the empty bunch events alone.
The presence of the CASTOR calorimeter in the B = 0 T data sample allows a larger coverage of the phase space for inelastic pp collisions. In this case, inelastic events are selected offline by requiring either an energy deposit above 5 GeV in either of the two HF calorimeters, or an energy deposit above 5 GeV in CASTOR.

Correction for noise and pileup
The selected number of inelastic events is first corrected for the contribution of detector noise. The corrected number of interactions is obtained as where N ZB is the number of events triggered by the zero bias trigger and F ZB and F EB are the fractions of events triggered by the zero bias and empty bunch triggers that are selected offline by requiring an energy deposit above threshold, respectively. In Eq. (1), the second term on the right-hand side is a first-order correction for genuine signal events (with occurrence approximated by F ZB − F EB ) overlaid with noise (with occurrence F EB ). Corrections of higher order in F EB are found to be negligible. Table 1 includes an overview of the noise-subtracted fraction of events (N cor /(N ZB F ZB )) found in the various runs. Table 1: Overview of the noise-subtracted fraction of events (N cor /(N ZB F ZB )), average pileup (λ), and fiducial cross section for all runs used in this analysis. The uncertainties in the noisesubtracted fraction of events and pileup have a negligible contribution to the uncertainty in the fiducial cross section, which is quoted with its statistical uncertainty only.  The number of events is further corrected for the effect of pileup. The observed number of pp collisions per bunch crossing, n, follows a Poisson distribution, P(n, λ), with average value λ.
As the probability to find an interaction in a crossing with filled bunches is given by N cor /N ZB , the probability to have no interaction can be obtained as P(0, λ) ≡ exp(−λ) = 1 − N cor /N ZB , which allows λ to be determined from the data. With this information it is possible to correct the inelastic event count using the pileup correction factor: The values of λ in the studied runs range from 0.05 to 0.54 and are given in Table 1. The average pileup value for run 247324 is substantially lower than for the other runs and demonstrates the robustness of the pileup correction, even though the statistical error on the fiducial cross section for this run, as obtained below, is larger due to the smaller number of recorded events. As the beam intensity may vary from one proton bunch to another, the actual pileup correction is applied bunch-by-bunch. The total reconstructed number of interactions, corrected for the contributions of noise and pileup, is then given by with the number of noise-corrected events N b cor (Eq. (1)) and pileup correction factors f b PU (Eq. (2)) calculated for individual bunches.

Extraction of the fiducial inelastic cross section
Before comparison to theoretical predictions, the experimental results need to be corrected for various detector effects, including the event selection efficiency and the resolution in the energy measurement. Corrected results are obtained by means of a simulation of the CMS detector based on GEANT4 [37-39].
The event selection criteria at the detector level are optimized in order to obtain a sample of inelastic events in the largest possible phase space. Low-mass diffractive dissociation events will, however, escape the detector. Central diffractive dissociation is found to contribute at the level of 0.1-0.2 mb, both from predictions obtained with the PYTHIA MBR model and from counting events in the data with activity in the central detectors but without any energy deposited in the forward calorimeters, and is neglected in the further analysis. Adapting the particle-level phase space to the detector acceptance results in a smaller correction of the results, and thus also in a smaller model dependence of the correction factors. A precise definition of the phase space at the level of generated stable particles, for which corrected results are presented, is obtained as follows.
The collection of stable final-state particles (with proper lifetime cτ > 1 cm) is divided into two systems, X and Y, based on the mean rapidity of the two particles separated by the largest rapidity gap in the event. All particles on the negative (positive) side of the largest gap are assigned to the system X (Y). The invariant masses, M X and M Y , of each system are calculated from the four-momenta of the individual particles, and are used to obtain the squared ratios of the mass over the center-of-mass energy, ξ X and ξ Y . For convenience, ξ can be defined by ξ = max(ξ X , ξ Y ). These Lorentz-invariant variables are well defined for any type of events (diffractive and nondiffractive) and are related to the size of the largest rapidity gap [1].
The fiducial phase space can then be quantified at the stable-particle level by appropriate limits on ξ X and ξ Y . These acceptance limits are obtained from a dedicated study based on the hadron-hadron interaction models mentioned in Section 3 using fully simulated events, and are chosen such that the factors required to correct the data to stable-particle level are close to unity, thereby minimizing the model dependence of the correction. An inelastic event is selected at the stable-particle level if ξ > 10 −6 for the selection based on the HF calorimeter alone, and, because the CASTOR calorimeter allows a lower ξ limit on one side, if ξ X > 10 −7 or ξ Y > 10 −6 when the HF and CASTOR calorimeters are combined.
The relationship between the stable-particle level phase space definition and the detector-level offline selection can be quantified through efficiency and contamination factors. The efficiency, ξ , is defined as the fraction of selected stable-particle level events that fulfill the detector-level offline selection criteria, while the contamination, b ξ , is defined as the fraction of detectorlevel offline selected events that are not part of the considered stable-particle level phase space. The efficiency and contamination factors are calculated as the average over MC models and are found to be equal to 97.6% and 0.6% (98.6% and 0.6%) for the HF (HF+CASTOR) fiducial region, respectively.
Finally, the fiducial cross section is calculated as and is given in Table 1 for all runs used in the analysis. The integrated luminosity, L, of the B = 0 T runs has been rescaled in order to yield the same cross section for ξ > 10 −6 as measured in the B = 3.8 T runs, thus exploiting the more accurate luminosity determination in the latter runs.

Systematic uncertainties
The following sources of systematic bias are investigated and the corresponding uncertainties are evaluated.
• Model dependence. The efficiency and contamination factors are obtained from MC simulation. Although the phase space domains are chosen so as to minimize ex-  trapolations, there is a remaining model dependence due to the matching of stableparticle level and detector-level selections. These factors are defined as fractions and are therefore not very sensitive to the absolute magnitude of the inelastic cross section itself; instead they are affected by the way in which a hadronic system of a particular invariant mass fragments into individual particles that deposit energy in the calorimeters. The uncertainty is therefore taken as the standard deviation of the correction factors obtained from all the models discussed in Section 3, even if some models do not describe the measured cross sections well.
• HF and CASTOR energy scale uncertainties. • Run-to-run variation. The cross sections are obtained from various runs and the run-to-run variation is taken as an additional source of systematic uncertainty, estimated as the standard deviation of the cross section distribution.
• Integrated luminosity. The integrated luminosity of the runs at B = 3.8 T is determined with an accuracy of 2.3% [35]. The integrated luminosity of the B = 0 T runs is rescaled using the ratio of the cross sections for ξ > 10 −6 at B = 0 T and B = 3.8 T. Because the energy scale and model uncertainties are fully correlated between the B = 0 T and B = 3.8 T samples, the same accuracy of 2.3% is used to determine the systematic uncertainty due to the integrated luminosity determination at B = 0 T. Table 2 gives an overview of the systematic uncertainties.

Results
The fully corrected cross sections in phase space domains corresponding to the specific detector acceptances (fiducial cross sections) are given in Table 1. The weighted average of the results at B = 3.8 T obtained with the HF calorimeters only is σ(ξ > 10 −6 ) = 67.5 ± 0.8 (syst) ± 1.6 (lumi) mb, with a statistical uncertainty that is smaller than the least significant reported digit. This can be compared to the cross section reported by ATLAS for inelastic pp collisions at 13 TeV with ξ > 10 −6 of 68.1 ± 0.6 (syst) ± 1.3 (lumi) mb [21].
Averaging the cross sections obtained from runs with the HF and CASTOR calorimeters in the Table 3: Relative increase in the cross section from the ξ > 10 −6 to the ξ X > 10 −7 or ξ Y > 10 −6 fiducial region. The uncertainty in the cross section ratio in data is obtained using systematic uncertainties only, varying each uncertainty source simultaneously for both phase space regions and obtaining the total uncertainty from the variation of the ratio, thus assuming full correlation between both measurements.

Relative cross section increase in
also with a statistical uncertainty that is smaller than the least significant reported digit. Figure 1 shows the inelastic cross sections in the two phase space domains compared to the predictions of the various models used in this analysis. Table 3 shows the increase in the cross section from ξ > 10 −6 to ξ X > 10 −7 or ξ Y > 10 −6 for the data and the various models. These models are used with their set of tunable parameters as described in Section 3. Different models and tunes are used to investigate the sensitivity of the fiducial cross section to these parameters, and no uncertainties on individual model predictions are displayed. The variation between model predictions is mainly due to different descriptions of the diffractive contribution to the cross section. This can be concluded from the fact that models with the same approach to diffraction predict very similar cross sections (EPOS LHC and QGSJETII-04, versus PYTHIA 6 Z2* (SS) and PYTHIA 8 CUETP8M1 (SS), respectively). The relative increase in the inelastic cross section observed in the data is three times larger than the experimental uncertainty. Most models describe reasonably well this small, but significant, relative increase in the inelastic cross section, but in general overpredict their value in both ξ ranges. The PYTHIA 8 Monash (DL) model describes data fairly well for ξ > 10 −6 , but predicts a too steep increase for ξ X > 10 −7 or ξ Y > 10 −6 . Moreover, a comparison of the same models to fiducial and total inelastic cross section measurements at √ s = 7 and 13 TeV [12-18, 21, 22] indicates that, while the total inelastic cross section is reasonably well described, the cross section for diffractive dissociation events escaping detection, at masses even lower than those considered here, is substantially underestimated by the models, leading to an overestimation of the fiducial cross sections. One may therefore conclude that a model-based extrapolation of the fiducial cross section to the total inelastic phase space would yield a value that is too low for most models.

Summary
A measurement of the inelastic proton-proton cross section at √ s = 13 TeV with the CMS detector at the LHC has been presented. An inelastic cross section of 67.5 ± 0.8 (syst) ± 1.6 (lumi) mb is obtained for ξ = M 2 /s > 10 −6 (corresponding to M > 13 GeV), with M the larger of M X and M Y , where M X and M Y are the masses of the diffractive dissociation systems with negative and positive pseudorapidities, respectively. This result is consistent with a previous measurement in the same phase space [21]. In addition, an inelastic cross section of 68.6 ± 0.5 (syst) ± 1.6 (lumi) mb is obtained in the enlarged phase space ξ X > 10 −7 and/or ξ Y > 10 −6 (corresponding to M X > 4.1 GeV and/or M Y > 13 GeV). The measured cross sections are smaller than those predicted by the majority of models for hadron-hadron scattering, as previously observed in pp collisions at √ s = 7 TeV [12]. In contrast, the same models generally describe reasonably well the measurements of the total inelastic cross section at √ s = 13 TeV [21,22]. Given that the difference between the two sets of measurements is dominated by the contribution from low-mass diffractive processes, the data-model discrepancies observed here suggest a theoretical underestimation of the cross section for such events.

Acknowledgments
We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centers and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies: BMWFW and FWF (Aus [13] ATLAS Collaboration, "Measurement of the inelastic proton-proton cross-section at √ s = 7 TeV with the ATLAS detector", Nature Commun.  [35] CMS Collaboration, "CMS luminosity measurement for the 2015 data taking period", CMS Physics Analysis Summary CMS-PAS-LUM-15-001, 2015. [36] CMS Collaboration, "The CMS trigger system", JINST 12 (2017), no. 01, P01020, doi:10.1088/1748-0221/12/01/P01020, arXiv:1609.02366.