Measurement of the $B^{\pm}$ production cross-section in pp collisions at $\sqrt{s} =$ 7 and 13 TeV

The production of $B^{\pm}$ mesons is studied in $pp$ collisions at centre-of-mass energies of 7 and 13 TeV, using $B^{\pm}\rightarrow J/\psi K^{\pm}$ decays and data samples corresponding to 1.0 fb$^{-1}$ and 0.3 fb$^{-1}$, respectively. The production cross-sections summed over both charges and integrated over the transverse momentum range $0<p_{\text{T}}< 40$ GeV/$c$ and the rapidity range $2.0<y<4.5$ are measured to be 
$\sigma(pp \rightarrow B^{\pm} X, \sqrt{s} = \text{7 TeV}) = 43.0 \pm 0.2 \pm 2.5 \pm 1.7\mu b,$ 
$\sigma(pp \rightarrow B^{\pm} X, \sqrt{s} = \text{13 TeV}) = 86.6 \pm 0.5 \pm 5.4 \pm 3.4\mu b,$ 
where the first uncertainties are statistical, the second are systematic, and the third are due to the limited knowledge of the $B^{\pm}\rightarrow J/\psi K^{\pm}$ branching fraction. The ratio of the cross-section at 13 TeV to that at 7 TeV is determined to be $2.02\pm0.02\text{(stat)}\pm0.12\text{(syst)}$. Differential cross-sections are also reported as functions of $p_{\text{T}}$ and $y$. All results are in agreement with theoretical calculations based on the state-of-art fixed next-to-leading order quantum chromodynamics.


Introduction
Precise measurements of the production cross-section of B ± mesons in pp collisions provide important tests of perturbative quantum chromodynamics (QCD) calculations, particularly of the state-of-the-art calculations based on the fixed next-to-leading order (NLO) QCD with next-to-leading logarithm (NLL) large transverse momentum resummation (FONLL) approach [1,2]. The FONLL calculations are accurate to the full NLO level at moderate p T values, and to the NLL level at high p T . The FONLL predictions are then achieved by properly merging a 'massless' resummed approach, valid in the high p T region, with a full massive fixed-order calculation, reliable in the small p T region. The ratio of cross-sections between different centre-of-mass energies is of particular interest due to cancellations that occur in the theoretical and experimental uncertainties. Uncertainties arising from assumptions about the values of the FONLL parameters largely cancel in the ratio. Experimentally, uncertainties due to factors such as the branching fractions of decays and the b-quark fragmentation fractions [3] to specific hadrons are highly correlated at different beam energies and their effect is much reduced in the ratio.
Previous measurements of B ± production have been performed in different kinematic regions at the centre-of-mass energy √ s = 7 TeV by several experiments at the Large Hadron Collider. The CMS collaboration reported the integrated and differential B ± production cross-sections in the range p T > 5 GeV/c and |y| < 2.4 [4,5], where p T and -1 -

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y are the component of the momentum transverse to the beam line and the rapidity of the B ± mesons, respectively. The ATLAS collaboration measured the production cross-sections in the range 9 < p T < 120 GeV/c and |y| < 2.25 [6]. The LHCb collaboration measured the integrated and differential cross-sections for B ± with 0 < p T < 40 GeV/c and 2.0 < y < 4.5 using a data sample collected in 2010 that corresponds to an integrated luminosity of 35 pb −1 [7]. This result was later updated using a data sample collected in early 2011, corresponding to an integrated luminosity of 362 pb −1 [8]. This article updates the previous LHCb results using a larger data sample collected in 2011 with the LHCb experiment at √ s = 7 TeV and corresponding to an integrated luminosity of 1.0 fb −1 . The first measurements of the integrated and differential crosssections of B ± mesons at √ s = 13 TeV are also presented, using a data sample collected in 2015 and corresponding to an integrated luminosity of 0.3 fb −1 . Following the previous LHCb measurements [7,8], the B ± mesons are reconstructed in the B ± → J/ψ K ± mode followed by J/ψ → µ + µ − and the production cross-sections are measured in the range 0 < p T < 40 GeV/c and 2.0 < y < 4.5. The ratio of the cross-sections in the 13 TeV and 7 TeV data is also measured as a function of p T and y.

Event selection
The LHCb detector [9,10] 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, 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 placed downstream of the magnet. The tracking system provides a measurement of momentum, p, of charged particles with a relative uncertainty that varies from 0.5% at low momentum to 1.0% at 200 GeV/c. The minimum distance of a track to a primary vertex, the impact parameter, is measured with a resolution of (15 + 29/p T ) µm, where p T is in units of GeV/c. Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov detectors. Photons, electrons and hadrons are identified by a calorimeter system consisting of scintillating-pad (SPD) and preshower detectors, an electromagnetic calorimeter and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers. The online event selection is performed by a trigger, which consists of a hardware stage (L0), based on information from the calorimeter and muon systems, followed by a two-stage software-based high-level trigger (HLT1, HLT2) [11], where the HLT1 stage uses partial event reconstruction to reduce the rate and the HLT2 stage applies a full event reconstruction. At the hardware stage, events are required to have a dimuon candidate with large p T , while at the software stage the dimuon invariant mass is required to be consistent with the known J/ψ mass [12]. Finally, a set of global event cuts (GEC) is applied in order to prevent high-multiplicity events from dominating the processing time of the software trigger, which includes the requirement that the number of hits in the SPD subdetector should be less than 900.

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Simulated events are used to optimise the selection, determine some of the efficiencies and estimate the background contamination. The simulation is based on the Pythia8 generator [13] with a specific LHCb configuration [14]. Decays of hadrons are described by EvtGen [15], in which final-state radiation is generated using Photos [16]. The interaction of the generated particles with the detector, and its response, are implemented using the Geant4 toolkit [17,18], as described in ref. [19].
The B ± candidates are made by combining J/ψ and K ± candidates. The offline event selection forms J/ψ candidates using pairs of muons with opposite charge. The muon candidates must have p T > 700 MeV/c, and satisfy track-reconstruction quality and particle identification (PID) requirements. The muon pair is required to be consistent with originating from a common vertex [20] and have an invariant mass, M (µµ), within the range 3.04 < M (µµ) < 3.14 GeV/c 2 . The K ± candidates are required to have p T greater than 500 MeV/c and satisfy track-reconstruction quality requirements. No PID requirement is applied to select the kaon, as the only topologically similar decay, B ± → J/ψ π ± , is Cabibbo suppressed.
The three tracks in the final state of the B ± decay are required to form a common vertex with a good vertex-fit quality. In order to suppress background due to the random combination of particles produced in the pp interactions, the B ± candidates are required to have a decay time larger than 0.3 ps. Finally, B ± candidates with 0 < p T < 40 GeV/c and 2.0 < y < 4.5 are selected for subsequent analysis.

Cross-section determination
The differential production cross-section of B ± mesons is measured as a function of p T and y using where N B ± is the number of reconstructed B ± → J/ψ K ± candidates in a given (p T , y) bin after background subtraction, L is the integrated luminosity, ε tot is the bin-dependent total efficiency, B (B ± → J/ψ K ± ) is the branching fraction of B ± decays to J/ψ K ± [21,22], B (J/ψ → µ + µ − ) is the branching fraction of J/ψ decays to µ + µ − [12], and ∆y and ∆p T are the bin widths in y and p T . The value of B (B ± → J/ψ K ± ) is calculated to be (1.044 ± 0.040) × 10 −3 by combining the two exclusive measurements from the Belle [21] and BaBar [22] collaborations, under the assumption that only the uncertainty for J/ψ → µ + µ − branching fraction is correlated. The yield of B ± → J/ψ K ± decays in each (p T , y) bin is obtained independently by fitting the invariant mass distribution of the B ± candidates, M (J/ψ K ± ), in the interval 5150 < M (J/ψ K ± ) < 5450 MeV/c 2 , using an extended unbinned maximum likelihood fit. The M (J/ψ K ± ) distribution is described by a probability density function (PDF) consisting of the following three components: a modified Crystal Ball (CB) function [23] to model the signal, an exponential function to model the combinatorial background, and a double CB function to model the contamination from the Cabibbo suppressed decay B ± → J/ψ π ± , -3 - where the charged pion is misidentified as a kaon. The modified CB function used for the signal component has tails on both the low-and the high-mass side of the peak, which are described by separate parameters. In each bin, the tail parameters are determined from simulation, leaving the mean and width as free parameters. The shape of the misidentification background is obtained using simulated B ± → J/ψ π ± decays that satisfy the selection criteria for the decay B ± → J/ψ K ± , and the yield of the misidentification background is determined according to the branching fraction ratio B(B ± → J/ψ π ± )/B(B ± → J/ψ K ± ) from ref. [12]. Figure 1 shows, as an example, the invariant mass distribution of the B ± candidates in the range 3.5 < p T < 4.0 GeV/c and 2.5 < y < 3.0. The total efficiency, ε tot , is the product of several efficiencies and can be written as The acceptance factor, ε acc , is the fraction of signal with all final-state particles within the fiducial region of the detector acceptance, and is calculated from simulation, as shown in appendix A. The efficiency of the particle reconstruction and event selection, ε reco&sel , is also determined from simulation. The efficiency of identifying the two muons in the final state, ε PID , and the track finding efficiency, ε track , are measured using a tag-and-probe method [24,25] on a control data sample of J/ψ → µ + µ − decays. The trigger efficiency, ε trigger , is estimated in two parts, which are ε L0&HLT1 and ε HLT2 . The ε L0&HLT1 efficiency is evaluated by estimating the fraction of events in a trigger-unbiased data sample that satisfy the trigger requirements, and the ε HLT2 efficiency is evaluated using simulated signal events, as the effects of HLT2 trigger are well modelled. The GEC efficiency, ε GEC , is measured to be (99.2 ± 0.1)% for the 7 TeV and (99.3 ± 0.1)% for the 13 TeV data sample, and is independent of p T and y. It is extracted by fitting the SPD multiplicity distribution and extrapolating the function to determine the fraction of events that are accepted. The measured ε tot are tabulated in appendix B.   Table 1. Summary of relative systematic uncertainties on the integrated production cross-sections at √ s = 7 and 13 TeV, and the ratio of the cross-sections R(13 TeV/7 TeV).

Systematic uncertainties
Sources of systematic uncertainty associated with the determination of the luminosity, branching fractions, signal yields, efficiencies, along with their effects on the integrated cross-section measurements, are summarised in table 1. The total systematic uncertainty is obtained from the sum in quadrature of all contributions. Several uncertainties have been reduced in the √ s = 13 TeV measurement, due to a larger simulation sample and a better understanding of the efficiency. Following the procedures used in ref. [26], the relative uncertainty on the luminosity is determined to be 1.7% for the 7 TeV data and 3.9% for the 13 TeV data sample. The relative uncertainty on B (B ± → J/ψ K ± ) is 3.9% [21,22], while the uncertainty on B (J/ψ → µ + µ − ) [12] is negligible.
The variation of the efficiency within a bin induces an uncertainty if the kinematic distributions of the simulated samples do not match those of the data. This uncertainty, which is important close to the edges of the fiducial region, is estimated by increasing or decreasing the bin width in p T and y by a factor of two. The largest variation of the integrated cross-section measurement is taken as a systematic uncertainty on the production cross-sections.
The systematic uncertainty associated with the invariant mass fits is obtained by performing fits using alternative choices for the signal and background functions. The signal PDF is replaced by a Hypatia function [27], while the combinatorial background model is modified to be either a first-order or second-order polynomial function. The largest resulting variation of the cross-section measurement is taken as a systematic uncertainty.
The efficiencies of the tracking, PID and trigger are estimated using control samples, and systematic uncertainties arise due to the limited sample sizes. An additional uncer- tainty arises on the track-reconstruction efficiency due to limited knowledge of the material budget of the detector, which induces a 1.1% uncertainty on the kaon reconstruction efficiency due to modelling of hadronic interactions with the detector material. There is an additional systematic uncertainty on the tracking efficiency from the method [25], which amounts to 0.4% (0.8%) at 7 TeV (13 TeV). For the uncertainty from the PID efficiency, the binning effect is studied by enlarging or decreasing the number of bins by a factor of two in the calculation of the PID efficiency, and taking the largest deviation from the default as the uncertainty. An additional uncertainty on the trigger efficiency is determined by testing the procedure in simulation and taking the deviation as the systematic uncertainty.
The GEC efficiency is obtained from data, and the inefficiency of the global event cuts is taken as a systematic uncertainty. The uncertainty on the offline event selection efficiencies are estimated from data and simulation by varying the selection criteria, comparing the ratios of the selection and reconstruction efficiencies between data and simulation, and taking the largest deviation as the systematic uncertainty.
A weighting procedure is applied to the simulation sample to correct for discrepancies between data and simulation in the track multiplicities. The weighting factors of the simulated events are varied within their statistical uncertainties and the largest deviation of the measured cross-section is taken as a systematic uncertainty.

Results
The measured B ± meson production cross-sections for the 7 TeV and 13 TeV data in the range 0 < p T < 40 GeV/c and 2.0 < y < 4.5 are where the first uncertainties are statistical, the second are systematic, and the third are due to the limited knowledge of the B ± → J/ψ K ± branching fraction. The measured double-differential cross-sections at 7 TeV and 13 TeV as functions of p T and y are shown in figure 2, where the measurements are compared with theoretical predictions based on FONLL calculations [28]. The predictions depend on assumptions on the b-quark mass, the renormalization and factorization scales, and parton distribution functions (PDFs). The b-quark production cross-section calculated with the FONLL approach uses a b-quark mass of 4.75 GeV/c 2 , the renormalization and factorization scales, where m Q and p T are the mass and transverse momenta of the b quark, and the CTEQ6.6 [29] PDFs. Varying the b-quark mass by ±0.25 GeV/c 2 , the µ R /µ F ratio by a factor of two, or using different PDFs introduces an uncertainty in the predicted b-quark production cross-section of up to 50% at low p T [28].
The corresponding single-differential cross-sections are shown in figures 3 and 4. The single-differential cross-sections and the production cross-section in the above range of p T and y are calculated from the measured double-differential cross-sections. All results are in agreement with the FONLL predictions. The results are tabulated in appendix C.   . Measured B ± double-differential production cross-sections at (top) 7 TeV and (bottom) 13 TeV as a function of p T and y. The black points represent the measured values, and the cyan bands are the FONLL predictions [28]. Each set of measurements and predictions in a given rapidity bin is offset by a multiplicative factor 10 −m , where the offset factor is shown after the rapidity range. The error bars include both the statistical and systematic uncertainties.
In the ratio calculation, the systematic uncertainties on the luminosities at 13 and 7 TeV are taken to be 50% correlated, as in ref. [26]; the systematic uncertainties associated with   Figure 4. Measured B ± differential cross-section at 13 TeV as a function of (left) p T or (right) y. The black points represent the measured values, the blue-dashed line and cyan band represent the central values and uncertainties of the FONLL prediction [28]. The error bars include both the statistical and systematic uncertainties. the branching fractions, mass fits, event selection and binning are assumed to be completely correlated; and all other uncertainties are considered to be uncorrelated. The systematic uncertainty on R is summarised in table 1. In figure 5, the ratio of the cross-section at 13 TeV to that at 7 TeV as a function of p T or y is compared with the FONLL predictions. The measured results agree with the FONLL predictions in both the shape and the scale.

Summary
In summary, the double-differential production cross-sections of B ± mesons are measured as functions of the transverse momentum and rapidity, using pp collision data collected with the LHCb detector at the Large Hadron Collider. The integrated luminosities of the data samples are 1.0 fb −1 and 0.3 fb −1 at the centre-of-mass energies of 7 TeV and 13 TeV, respectively. The measurements are performed in the transverse momentum range 0 < p T < 40 GeV/c and the rapidity range 2.0 < y < 4.5. The 7 TeV results are consistent with previously published results [7,8], with improved precision in the low y region. This measurement supersedes previous results. The ratio of the production cross-section at 13 TeV to that at 7 TeV is also measured. All results are in agreement with theoretical calculations based on the FONLL approach.

A Acceptance efficiency
The measured ε acc as a function of p T in different y regions are shown in figure 6 for 7 TeV and 13 TeV, respectively.

B Total efficiencies
The measured ε tot with its statistical uncertainty as a function of p T and y are given in tables 2 and 3, for 7 TeV and 13 TeV, respectively. The ε tot in the low p T region are smaller than that of the high p T region, due to the lifetime cut effects. Also with limited detector coverage, ε tot in the boundary regions (2.0 < y < 2.5 and 4.0 < y < 4.5) are smaller than that in central regions.

C Tabulated results
The measured B ± double-differential cross-section in bins of p T and y is given in tables 4 and 5 for 7 TeV data, and tables 6 and 7 for 13 TeV data. The measured B ± singledifferential cross-section as a function of p T or y is given in tables 8 and 9, respectively. The limited size of the simulation samples gives relative systematic uncertainties in the high p T region that are larger than those in the low p T region.   Table 2. Measured ε tot of B ± events at 7 TeV, in the bins of B ± p T and y. The efficiencies and uncertainties are in percent.   Table 3. Measured ε tot of B ± events at 13 TeV, in the bins of B ± p T and y. The efficiencies and uncertainties are in percent.   Table 4. Measured B ± double-differential cross-section (in units of nb) at 7 TeV, as a function of p T and y, in the rapidity regions of 2.0 < y < 2.5, 2.5 < y < 3.0, and 3.0 < y < 3.5.   Table 5. Measured B ± double-differential cross-section (in units of nb) at 7 TeV, as a function of p T and y, in the rapidity regions of 3.5 < y < 4.0 and 4.0 < y < 4.5.
-  Table 6. Measured B ± double-differential cross-section (in units of nb) at 13 TeV, as a function of p T and y, in the rapidity regions of 2.0 < y < 2.5, 2.5 < y < 3.0, and 3.0 < y < 3.5.
-  Table 7. Measured B ± double-differential cross-section (in units of nb) at 13 TeV, as a function of p T and y, in the rapidity regions of 3.5 < y < 4.0 and 4.0 < y < 4.5.   Table 8. Measured B ± differential cross-sections (in units of nb) at 7 TeV and 13 TeV as functions of p T in the range 2.0 < y < 4.5. The cross-section ratio between 13 TeV and 7 TeV is also presented.  Table 9. Measured B ± differential cross-sections (in units of µb) at 7 TeV and 13 TeV as functions of y in the p T range 0 < p T < 40 GeV/c. The cross-section ratio between 13 TeV and 7 TeV is also presented.

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.