Measurement of ϒ production in pp collisions at s=13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt{s}=13 $$\end{document} TeV

The production cross-sections of ϒ(1S), ϒ(2S) and ϒ(3S) mesons in proton-proton collisions at s=13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt{s}=13 $$\end{document} TeV are measured with a data sample corresponding to an integrated luminosity of 277 ± 11 pb−1 recorded by the LHCb experiment in 2015. The ϒ mesons are reconstructed in the decay mode ϒ → μ+μ−. The differential production cross-sections times the dimuon branching fractions are measured as a function of the ϒ transverse momentum, pT, and rapidity, y, over the range 0 < pT< 30 GeV/c and 2.0 < y < 4.5. The ratios of the cross-sections with respect to the LHCb measurement at s=8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt{s}=8 $$\end{document} TeV are also determined. The measurements are compared with theoretical predictions based on NRQCD.


Introduction
The production of heavy quark-antiquark resonances (quarkonia) at hadron colliders provides important information about the strong interaction, which is described by Quantum Chromodynamics (QCD). Based on the assumption of factorization, the production process can be separated into two steps, the generation of a heavy quark pair (QQ) through the interaction of partons, and the subsequent hadronization of the quark pair into the quarkonium state. The first step can be calculated within perturbative QCD, while the latter one is nonperturbative and needs to be described by phenomenological models. In the Colour Singlet Model [1][2][3][4][5][6][7], the intermediate QQ state is assumed to be colourless, and has the same spin-parity quantum numbers as the final state quarkonium. On the other hand, in the Colour Evaporation Model [4,[8][9][10], the probability of forming a specific quarkonium state is assumed to be independent of the color of the QQ pair. In the nonrelativistic QCD (NRQCD) approach [11][12][13], intermediate QQ states with all possible colour-spin-parity quantum numbers may evolve into the desired quarkonium. The transition probabilities, described by long-distance matrix elements (LDMEs) in NRQCD, are nonperturbative and are determined by fitting to experimental data. The LDMEs are assumed to be independent of the production mechanism and kinematics.

The LHCb detector and event selection
The LHCb detector [22,23] is a single-arm forward spectrometer covering the pseudorapidity range 2 < η < 5, designed for the study of particles containing b or c quarks. The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the pp interaction region [24], a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about 4 Tm, and three stations of silicon-strip detectors and straw drift tubes [25] placed downstream of the magnet.
The tracking system provides a measurement of the momentum, p, of charged particles with a relative uncertainty that varies from 0.5% at low momentum to 1.0% at 200 GeV/c. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers [26].
The online event selection is performed by a trigger [27], which consists of a hardware stage selecting events with at least one muon candidate with p T > 0. 9 GeV/c based on information from the muon systems, followed by a two-stage software stage, which applies a full event reconstruction. In the software trigger, two muon candidates, each having p T > 0.5 GeV/c and p > 3 GeV/c, are required to form a good quality vertex with invariant mass m(µ + µ − ) > 4.7 GeV/c 2 . In between the two software stages, an alignment and calibration of the detector is performed in near real-time and their results are used in the second stage of the trigger [28]. The same alignment and calibration information is propagated to the offline reconstruction, ensuring consistent and high-quality particle identification (PID) information between the trigger and offline software. The identical performance of the online and offline reconstruction offers the opportunity to perform physics analyses directly using candidates reconstructed in the trigger [27,29], which the present analysis exploits. Candidate Υ mesons are further selected offline, where two muon tracks are required to have p T > 0.65 GeV/c, p > 10 GeV/c and 1.9 < η < 4.9, and the invariant mass of the two muons is required to be in the range of 9.0 < m(µ + µ − ) < 10.7 GeV/c 2 .
Simulation is required to model the effects of the detector acceptance and the imposed selection requirements. In the simulation, pp collisions are generated using Pythia [30] with a specific LHCb configuration [31]. Decays of unstable particles are described by EvtGen [32], in which final-state radiation is generated using Photos [33]. The interaction of the generated particles with the detector, and its response, are implemented using the Geant4 toolkit [34] as described in ref. [35].

Cross-section determination
The double-differential cross-section times the Υ → µ + µ − branching fraction (B) of Υ production in a given (p T ,y) bin is defined as is the yield of Υ → µ + µ − signal decays, ε tot is the total efficiency in a (p T ,y) bin, L is the integrated luminosity, and ∆p T and ∆y are the interval widths of the Υ transverse momentum and rapidity, respectively. The Υ yield in each (p T ,y) bin is determined with an extended unbinned maximumlikelihood fit to the invariant-mass distribution of the Υ candidates, denoted as m Υ in the following. Four components are used to describe the distributions, the three Υ signals and a combinatorial background. The mass distribution for each of the Υ (1S), Υ (2S) and Υ (3S) states is described by a Crystal Ball (CB) function [36]. The differences between the means and the ratios of the widths of these CB functions are fixed, respectively, to the differences and the ratios of the world-average Υ (nS) masses [37]. The CB function parameters α and n, which describe the radiative tail of the Υ mass distribution, are fixed to the values of α = 2 and n = 1, following the same approach of the previous analyses [19,20]. The mass distribution of the combinatorial background contribution is described by an exponential function.
The total efficiency ε tot is expressed as the product of four components: the geometrical acceptance, Acc , the reconstruction and selection efficiency, Rec&Sel , the muon identification efficiency, PID , and the trigger efficiency, Trg , as These efficiencies in general depend on the Υ polarisation. The measurement of the Υ polarisation by LHCb at √ s = 7 and 8 TeV [38] in pp collisions indicates that the three polarisation parameters λ θ , λ θφ and λ φ are consistent with zero. Therefore, the efficiencies are determined in each (p T , y) bin for the Υ (1S), Υ (2S) and Υ (3S) mesons, assuming the Υ states are unpolarised. Variations of cross-sections due to polarisation are studied for different scenarios of the polarisation parameter λ θ , as the dependence of the efficiency on the λ θφ and λ φ parameters is smaller [39].
The quantity Acc is calculated using the simulation sample without a prior geometrical acceptance requirement on the muons. The efficiency Rec&Sel is determined using full simulation samples for which the distributions of charged track multiplicity of pp collisions and Υ rapidity are corrected to match data. The efficiency PID is calculated using the single muon PID efficiency in bins of (p, η) weighted by the muon (p, η) distribution in simulation samples. The single-muon PID efficiency is obtained using J/ψ → µ + µ − decays in calibration data [40,41]. The efficiency Trg is determined on simulated events.

Systematic uncertainties
Several sources of systematic uncertainty are expected to affect the cross-section measurements: determination of the signal yields, efficiency calculations and computation of the integrated luminosity. The efficiency-related uncertainties are determined with independent simulation samples for each Υ (nS) meson, and are expected to be similar for the three states. They are reported in table 1 and discussed in detail in the following. The total uncertainty is calculated as the quadrature sum of these uncertainties.
The systematic uncertainty from the determination of the signal yields is affected by the invariant-mass fit model and is estimated using pseudoexperiments. The pseudoexperiments employ a mixture of Υ (1S) signal in the full simulation sample and background  generated according to the background model fitted in data. In the pseudoexperiments, the relative fraction of the two components is chosen such that the level of background matches the data in the Υ signal region. This signal-plus-background sample is fitted with the same model as for the data. The systematic uncertainty is estimated as the relative difference between the signal yield obtained from the fit and that of the input. This study is repeated for the Υ (1S), Υ (2S) and Υ (3S) states, with different fractions of background, yielding 2.5%, 2.8% and 3.1% as the systematic uncertainty for the three states, respectively. The small difference between the uncertainties for the three states is attributed to different background levels under each signal peak.
Systematic uncertainties on both hardware and software trigger efficiency are studied in this analysis. To validate systematic effects on the hardware trigger, the trigger efficiency is determined for µ + and µ − individually and is then used to calculate the efficiency for Υ signals following the method described in ref. [42]. The difference between the trigger efficiencies obtained on data and simulation is quoted as the systematic uncertainty on the hardware-trigger efficiency. The systematic uncertainty, estimated to be in the range 0.0-1.9%, depends on the (p T , y) bin of the Υ mesons and is assumed to be correlated among the bins. For the software trigger, alternative efficiency values are determined using a subset of events triggered independently of the Υ signals [43]. Within this sample, the fraction of Υ signal events that passes the software requirement is quoted as the trigger efficiency. The efficiency is calculated for both data and simulation samples in (p T , y) bins of the Υ mesons, and their difference is taken as the systematic uncertainty.
The track reconstruction efficiency is evaluated for each muon using calibrated simulation and data samples [44], and their difference is used to correct the Rec&Sel efficiency of simulated Υ decays. A systematic uncertainty of 0.8% per track is assigned to account for the dependency of the correction on the event multiplicity variable that is used to weight simulation to match data. Furthermore, an uncertainty is associated with the correction due to the limited size of the calibration samples, which is propagated to the Υ cross-section measurement using a large number of pseudoexperiments.

JHEP07(2023)069
The muon identification efficiency is affected by the statistical fluctuation of the PID calibration constants originating from the limited size of calibration samples and from the choice of the (p, η) binning scheme for muons. The statistical uncertainties in the PID efficiencies are propagated to the Υ cross-sections using pseudoexperiments. The variation is found to be below 1% and is therefore neglected. The systematic uncertainty due to the kinematic binning scheme of the calibration sample is studied by measuring the PID efficiency using alternative binning schemes. The difference between the alternative efficiency and the default one is quoted as the systematic uncertainty.
The kinematic distributions of the Υ mesons in data and simulation samples could be different, which may cause a mis-modelling in the underlying detection efficiency. Efficiencies measured in (p T , y) intervals help to reduce the discrepancy, however the (p T , y) bin size can cause systematic effects in the efficiency determination. To check for possible residual effects, the y distributions of the Υ states in simulation are reweighted to match those in data, for which the background has been subtracted using the sPlot technique [45]. In this analysis, the y distribution is studied as the efficiencies are found to have only a weak dependence on the p T of the Υ candidates. All the efficiencies are re-calculated with the reweighted y spectrum in the simulation sample, and relative differences between the new results and the default are taken as systematic uncertainties.
Due to the missing energy caused by final-state radiation that is not reconstructed, some signal events are removed by the requirement on the µ + µ − invariant mass. The efficiency of this requirement is studied using simulation samples. A systematic uncertainty of 1.0% is assigned to take into account possible differences between data and simulated Υ decays, following the method of ref. [20].
The limited size of the simulation samples used to determine the efficiencies is a source of systematic uncertainty. The relative statistical uncertainties in the efficiencies are propagated to the cross-section measurements. The integrated luminosity has a relative uncertainty of 2.0% [21], which is propagated to the final results as a systematic uncertainty.

Cross-sections
The double-differential cross-sections times the branching fraction of Υ → µ + µ − are shown in figure 2 and the numeric results are listed in tables 2, 3 and 4 in appendix A for Υ (1S), Υ (2S) and Υ (3S), respectively. The cross-section times branching fraction as a function of p T and integrated over y or vice-versa are shown in figure 3, and listed in tables 5 and 6 in appendix A for the three Υ states. The change of Υ cross-sections for alternative polarisation relative to the zero polarisation is studied for λ θ = −1, 0.1 and +1 in the helicity frame, which is reported in appendix B. The result for λ θ = ±1 corresponds to the extreme transverse and longitudinal polarisation and λ θ = 0.1 is the approximation of LHCb measurement [38]. Results for other λ θ values can be interpolated.
The differential cross-section times branching fraction as a function of p T for Υ (1S) mesons for y integrated over 2.0-4.5 are compared with the NRQCD predictions [46] in figure 4. The data points are plotted at the mean p T of each bin. Good agreement is observed between NRQCD predictions and the measurement of this analysis for p T >     [pb] 5 GeV/c. Another NRQCD calculation that tends to describe high-p T data at rapidity y ∼ 0 is found to overestimate the LHCb result [47]. The difference between the two NRQCD calculations is due to different values of LDMEs used in their analyses.    The double-differential cross-sections integrated over the kinematic range of p T < 20 GeV/c and 2.0 < y < 4.5 are The total cross-section in the LHCb acceptance as a function of pp centre-of-mass energy is shown in figure 5 for Υ (1S), Υ (2S) and Υ (3S) mesons. All three distributions show approximately a linear rise as a function of √ s in the range 2.76-13 TeV.  Figure 6. Ratios of the Υ (2S) or Υ (3S) production cross-section over that of the Υ (1S) state, (left) as a function of p T for 2.0 < y < 4.5 and (right) as a function of y for p T < 20 GeV/c. Statistical and systematic uncertainties are added in quadrature.

Cross-section ratios
The cross-section ratios between the different Υ states are calculated as: and the results are shown in figure 6. The ratios increase slightly with p T but show no obvious dependence on y.
In the evaluation of the cross-section ratios, the statistical and systematic uncertainties due to the limited simulation sample sizes are assumed to be uncorrelated between the two Υ states and, hence, cancel in the ratios. The systematic uncertainties due to the invariant mass fit model, and those originating from the trigger as well as tracking efficiencies are assumed to be fully correlated between cross-sections of two Υ states.  figure 7 and are summarized in tables 7 and 8 in appendix A, from which it is seen that the ratio increases from about 2 at low p T to about 4 at high p T , but is almost independent of y.
For the cross-section ratios between measurements at different √ s, the systematic uncertainties due to the invariant mass fit model and final-state radiation are considered to be fully correlated. The systematic uncertainties originating from uncertainties of trigger, muon identification, tracking efficiencies and luminosity are assumed to be partially correlated.

Update of the Υ (1S) nuclear modification factor
To quantify nuclear effects for particle production in heavy-ion collisions, the nuclear modification factor The measurement of the Υ (1S) cross-section times the Υ (1S) → µ + µ − branching fraction in the same kinematic range from this analysis is given by where the first uncertainty is statistical and the second systematic. While the two results are consistent, the direct measurement has an uncertainty that is a factor of two smaller. The nuclear modification factor R pPb is updated based on the direct measurement and the results are 0.76 ± 0.08 ± 0.05 (2.5 < y < 4.0), (5.4) where the positive (negative) rapidity is for particles produced in the p (Pb) beam direction. Figure 8 compares the updated result with theoretical calculations considering the EPS09 nuclear parton distribution function [50, 51] or parton energy loss [52], quoted in ref. [17], and good agreements are found between data and these calculations.

Conclusion
The Υ production cross-sections in proton-proton collisions at a centre-of-mass energy of √ s = 5 TeV are studied using a data sample with an integrated luminosity of 9.13 ± 0.18 pb −1 , collected by the LHCb detector. The double-differential cross-sections, as function of the transverse momentum p T and the rapidity y of the Υ mesons, are determined in the range p T < 20 GeV/c and 2 < y < 4.5 for the Υ (1S), Υ (2S) and Υ (3S) states. The Υ (1S) cross-section as a function of p T is well described by the NRQCD calculation for p T > 5 GeV/c with appropriate long distance matrix elements. The ratios of cross-sections of Υ (2S) and Υ (3S) mesons with respect to Υ (1S) mesons are given in bins of p T or y. The ratios of cross-sections measured at different pp collision energies are also determined. The nuclear modification factor R pPb for Υ (1S) in proton-lead collisions at √ s NN = 5 TeV is updated using the Υ (1S) production in pp data measured in this analysis. This result is consistent with the previous result, while improving the precision by a factor of two.

Acknowledgments
We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies:

B Dependence of cross-sections on the polarisation
The angular distribution of Υ → µ + µ − decays is described by where θ and φ are the polar and azimuthal angles of the µ + momentum in the rest frame of the Υ meson for a given polarisation coordinate system, and λ θ , λ θφ and λ φ are the three polarisation parameters [53]. Zero polarisation implies λ θ = λ θφ = λ φ = 0. The detection efficiency of the Υ mesons is a function of the polarisation, especially of λ θ . The Υ (nS) mesons are assumed to be unpolarised in this measurement, which is also the case for the simulated signal samples used to determine the efficiencies. Such an assumption is supported by the measurements of the Υ polarization by LHCb in pp collisions at √ s = 7 TeV and 8 TeV [38] in the same kinematic range as in this analysis, and by the CMS experiment [54] for √ s = 8 TeV. To evaluate the change of results assuming non-zero polarisation, the angular distributions of muons in the Υ rest frame in the simulated signal samples are reweighted, and the total efficiencies are recomputed. The relative changes of the double-differential cross-sections for a polarisation of λ θ = 0.1 [38] in the helicity frame compared to zero polarisation in each (p T ,y) interval are given in tables 9, 10 and 11 for Υ (1S), Υ (2S) and Υ (3S) mesons, respectively. In addition, the relative changes of the double-differential cross-sections for a polarisation of λ θ = +1 (−1) are also evaluated, and are shown in tables 12 (15), 13(16) and 14 (17) for Υ (1S), Υ (2S) and Υ (3S) mesons, respectively. The helicity frame uses the Υ momentum in the laboratory frame as the spin quantization axis.
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