Study of $b\bar{b}$ correlations in high energy proton-proton collisions

Kinematic correlations for pairs of beauty hadrons, produced in high energy proton-proton collisions, are studied. The data sample used was collected with the LHCb experiment at centre-of-mass energies of 7 and 8 TeV and corresponds to an integrated luminosity of 3 fb$^{-1}$. The measurement is performed using inclusive $b\rightarrow J/\psi X$ decays in the rapidity range $2<y^{J/\psi}<4.5$. The observed correlations are in good agreement with theoretical predictions.


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is measured with a resolution of (15 + 29/p T ) µm, where p T is the component of the momentum transverse to the beam, in 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 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 [73].
The online event selection is performed by a trigger [74], which consists of a hardware stage, based on information from the calorimeter and muon systems; followed by a software stage, which applies a full event reconstruction. The hardware trigger selects pairs of opposite-sign muon candidates with a requirement that the product of the muon transverse momenta is larger than 1.7 (2.6) GeV 2 /c 2 for data collected at √ s = 7 (8) TeV. The subsequent software trigger is composed of two stages, the first of which performs a partial event reconstruction. A full event reconstruction is then made at the second stage. In the software trigger, the invariant mass of well-reconstructed pairs of oppositely charged muons that form a vertex with good reconstruction quality is required to exceed 2.7 GeV/c 2 and the vertex is required to be significantly displaced from all PVs.
Simulated samples are used to determine the reconstruction and trigger efficiencies. Proton-proton collisions are generated using Pythia [59,65] with a specific LHCb configuration [75]. Decays of hadronic particles are described by EvtGen [76], in which final-state radiation is generated using Photos [77]. The interaction of the generated particles with the detector, and its response, are implemented using the Geant4 toolkit [78,79] as described in ref. [80].

Signal selection and efficiency determination
Selected events are required to have two reconstructed J/ψ → µ + µ − candidates. In the following these two candidates are marked with subscripts 1 and 2, which are randomly assigned.
The muon candidates must be identified as muons, have good reconstruction quality, p T > 500 MeV/c and 2 < η < 5 [73,81]. Both reconstructed J/ψ candidates are required to have a good-quality vertex, a reconstructed mass in the range 3.00 < m µ + µ − < 3.18 GeV/c 2 , 2 < p J/ψ T < 25 GeV/c and 2 < y J/ψ < 4.5. These criteria ensure a good reconstruction and trigger efficiency. Only events triggered by at least one of the J/ψ candidates are retained. The two J/ψ candidates are required to be associated with the same PV and, in order to suppress background from promptly produced J/ψ mesons, both dimuon vertices are required to be significantly displaced from that PV.
The two-dimensional distribution of the µ + µ − masses, m µ + µ − 1 and m µ + µ − 2 , for the selected pairs of J/ψ → µ + µ − candidates is presented in figure 1 for several requirements on p J/ψ T . A clear signal peak, corresponding to events with two J/ψ mesons detached from the PV, is visible.
The signal yield is determined by performing an extended unbinned maximum likelihood fit to the two-dimensional mass distribution. The distribution is fitted with the func- where the first term corresponds to a signal of two J/ψ mesons, the second term corresponds to a combination of one J/ψ meson and combinatorial background; and the last term describes pure combinatorial background. The coefficients N SS , N SB and N BB are the yields for these three components. The signal component, denoted as S(m), is modelled by a double-sided Crystal Ball function [82,83]. The background component, B (m), is parameterized as the product of an exponential and a first-order polynomial function and the background component B (m 1 , m 2 ) is parameterized as the product of two exponential functions e −τm 1 and e −τm 2 , with the same slope parameter, τ, and a symmetric second-order polynomial. With these parameterizations the overall function is symmetric,    Figure 2 shows the projections of the fit for p J/ψ T > 2 GeV/c. Several background sources potentially contribute to the observed J/ψ -pair signal. The first group of sources involves events where two J/ψ mesons originate from different pp collision vertices: it includes events with two J/ψ mesons from decays of beauty hadrons, events with one J/ψ meson originating from a beauty hadron decay and another J/ψ meson produced promptly and, finally, events with two prompt J/ψ mesons. The second group of sources consists of events where both J/ψ mesons originate from the same pp collision, namely prompt J/ψ -pair production [83,84], and associated production of a prompt J/ψ meson and a bb pair, where one of the b hadrons decays into a J/ψ meson.
The contribution from the first group of background sources is estimated from the measured production cross-sections for b → J/ψ X and prompt J/ψ events [17,18], the multiplicity of pp collision vertices and the size of the beam collision region. Taking from simulation an estimate for the probability of reconstructing two spatially close PVs as a single PV, the total relative contribution from these sources is found to be less than 0.1%.
For the second group of background sources, the contribution from prompt J/ψ -pair production is significantly suppressed by the requirement that both dimuon vertices are -5 -JHEP11(2017)030 displaced from the PV. Using the production cross-section for prompt J/ψ pairs, 1 the relative contribution from this source is estimated to be less than 0.05%. The background from associated production of bb and a prompt J/ψ meson in the same pp collision is calculated assuming double parton scattering is the dominant production mechanism, following ref. [85]. The relative contribution from this source is estimated to be less than 0.05%.
Normalized differential cross-sections [46,85] are presented as a function of kinematic variables, defined below, and here generically denoted as v, where N cor is the total number of efficiency-corrected signal candidates, ∆N cor i is the number of efficiency-corrected signal candidates in bin i, and ∆v i is the corresponding bin width. The efficiency-corrected yields N cor and ∆N cor i are calculated as in refs. [46,86] where the sum runs over all pairs of J/ψ candidates in the case of N cor and all pairs of J/ψ candidates in bin i in the case of ∆N cor i . Here J/ψ J/ψ tot is the total efficiency for the pair of J/ψ candidates and the weights ω j are determined using the sP lot technique [87].
The total efficiency of the J/ψ pair is estimated on an event-by-event basis as in refs. [46, [83][84][85][86] J/ψ J/ψ tot where acc is the geometrical acceptance of the LHCb detector, rec&sel is the reconstruction and selection efficiency for candidates with all final-state muons inside the geometrical acceptance, µID is the muon identification (µID) efficiency for the selected candidates and trg is the trigger efficiency for the selected candidates satisfying the µID requirement. The efficiencies, acc , rec&sel and µID , are factorized as (3.5) The efficiencies J/ψ acc , J/ψ rec&sel and J/ψ trg are estimated as functions of the transverse momentum and rapidity of the J/ψ meson using simulation. The trigger efficiency for single J/ψ mesons, J/ψ trg , has been validated using data. The muon identification efficiency for J/ψ mesons is factorized as The production cross-section of J/ψ pairs is measured at √ s = 7 TeV [83]. The cross-section at √ s = 8 TeV is estimated using a linear interpolation between the measurements at √ s = 7 TeV and √ s = 13 TeV [84].  where the corresponding single-muon identification efficiency, µ ± µID , is determined as a function of muon momentum and pseudorapidity using large samples of prompt J/ψ mesons.

Systematic uncertainties
The systematic uncertainty due to the imprecise determination of the luminosity does not enter in the normalized differential cross-sections. The systematic uncertainties, related to the evaluation of the efficiency-corrected signal yields N cor and ∆N cor i from eq. (3.2) are summarized in table 2 and are discussed in detail below.
Systematic uncertainties associated with the signal determination are studied by varying the signal and background shapes used for the fit function. For the signal parameterization, the power-law tail parameters of the double-sided Crystal Ball function are varied according to the results of fits to large samples of low-background b → J/ψ X and B + → J/ψ K + candidates. The alternative signal shape parameterization from ref. [88] is also used in the fits. For the parameterization of the background functions, B (m) and B (m 1 , m 2 ), the order of the polynomial functions is varied. The difference in the fitted signal yields does not exceed 1% in all of the above cases.
The systematic uncertainty related to the muon identification is estimated to be 0.4%. It is obtained from the uncertainties for the single-particle identification efficiencies, µ ± µID , using pseudoexperiments.
The efficiency J/ψ rec&sel is corrected on a per-track basis for small discrepancies between data and simulation using data-driven techniques [81,89]. The uncertainty in the correction factor is propagated to the determination of the efficiency-corrected signal yields using pseudoexperiments. This results in a systematic uncertainty of 0.6%. Added in quadrature to the (correlated) uncertainty from the track reconstruction of 0.4% per track (1.6% in total) these sources give an overall systematic uncertainty associated with the track reconstruction of 1.7%.
The trigger efficiency has been validated using large low-background samples of B + → J/ψ K + decays and inclusive samples of J/ψ mesons. Taking the largest difference between simulation and data for J/ψ trg , the corresponding systematic uncertainty for the efficiency-corrected yields is 1.2%.
The uncertainties in the efficiencies J/ψ acc , J/ψ rec&sel and J/ψ trg , which are due to the limited size of the simulation samples, are propagated to the efficiency-corrected signal yields using pseudoexperiments and are less than 0.1%.

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Part of the uncertainties, summarized in table 2, cancel in the ratio ∆N cor i N cor and thus do not affect the normalized differential cross-sections. For all bins for which the normalized differential cross-sections are evaluated, the systematic uncertainty is much smaller than the corresponding statistical uncertainty and is therefore neglected hereafter.

Results
The normalized differential production cross-sections defined by eq. (3.2) are presented as a function of the following variables: • |∆φ * |, the difference in the azimuthal angle, φ * , between the two beauty hadrons, where φ * is estimated from the direction of the vector from the PV to the decay vertex of the J/ψ meson; • |∆η * |, the difference in the pseudorapidity, η * , between the two beauty hadrons, where η * is estimated from the direction of the vector from the PV to the decay vertex of the J/ψ meson; , the asymmetry between the transverse momenta of two J/ψ mesons; • m J/ψ J/ψ , the mass of the J/ψ pair; • p J/ψ J/ψ T , the transverse momentum of the J/ψ pair; • y J/ψ J/ψ , the rapidity of the J/ψ pair.
The differential cross-sections with respect to other variables are given in appendix A. The shapes for the differential production cross-sections for |∆φ * | and |∆η * | variables are independent of the decay of the long-lived beauty hadrons and directly probe the production properties of pairs of beauty hadrons. The other variables have a minor dependence both on the branching fractions of different beauty hadrons, as well as on the b → J/ψ X decay kinematics. The normalized differential production cross-sections are shown in figures 3, 4, 5 and 6 for different requirements on the minimum transverse momentum of the J/ψ mesons. Since the distributions obtained for data accumulated at √ s = 7 and 8 TeV are very similar, they are treated together. In general, the width of the resolution function is much smaller than the bin width, i.e. the results are not affected by bin-to-bin migration. The exception to this is a small fraction of events with 2.0 < p J/ψ T < 2.5 GeV/c, where the resolution for |∆φ * | and |∆η * | is close to half of the bin-width.
The normalized differential production cross-sections are compared with expectations from Powheg [66][67][68][69] and Pythia [59,65,75] using the parton distribution functions from CT09MCS [90], CTEQ6L1 [91] and CTEQ6.6 [92] for the samples produced with Powheg, Pythia 6 and Pythia 8, respectively. Since no visible difference between  the default configuration is used except for the b-quark mass, which is set to 4.75 GeV/c 2 . To illustrate the size of the correlations between the two b quarks, predictions from an artificial data-driven model of uncorrelated bb production are also presented. This model is based on the measured transverse momenta and rapidity spectra for b → J/ψ X decays [17,18], assuming uncorrelated production of b and b quarks. The momenta of the two J/ψ mesons are sampled according to the measured (p J/ψ T , y J/ψ ) spectra, assuming a uniform distribution in the azimuthal angle, φ J/ψ . This allows the distributions for all variables except for |∆η * | to be predicted. This model is considered as an extreme case that corresponds to uncorrelated bb production; in contrast, the leading-order collinear approximation, where the transverse momentum of the bb system from the gg → bb process is zero, results in maximum correlation. The smearing of the transverse momenta of the initial gluons could result in significant decorrelations of the initially highly correlated heavy-flavour quarks. It should be noted that the model using uncorrelated bb pairs also mimics a possible small contribution of double parton scattering to bb pair production.
In general, both Powheg and Pythia describe the data well for all distributions, suggesting that NLO effects in bb production in the studied kinematic region are small compared with the experimental precision. Unlike the measurements with open-charm mesons [44][45][46], no significant contribution from gluon splitting is observed at small |∆φ * |. This observation is in agreement with expectations, since the contribution from gluon splitting is suppressed due to the large mass of the beauty quark. For p J/ψ T > 5 and 7 GeV, there is a hint of a small enhancement at small |∆φ * |. This also agrees with the expectation of a larger contribution of gluon splitting at higher p T . Another large enhancement towards the threshold in m J/ψ J/ψ is predicted by Powheg for p J/ψ T > 5 and 7 GeV, due to large leading-logarithm corrections [93]. No evidence for this enhancement is observed in the LHCb data, as can be seen in figures 5d and 6d. The data agree well with the model of uncorrelated bb production for y J/ψ J/ψ and A T , and also for p

Summary and conclusions
Kinematic correlations for pairs of beauty hadrons, produced in high energy proton-proton collisions, are studied. The data sample used was collected with the LHCb experiment at centre-of-mass energies of 7 and 8 TeV and corresponds to an integrated luminosity of 3 fb −1 . The measurement is performed using b → J/ψ X decays in the kinematic range 2 < y J/ψ < 4.5, 2 < p J/ψ T < 25 GeV/c. The observed correlations agree with Pythia (LO) and Powheg (NLO) predictions, suggesting NLO effects in bb production are small. In particular, no large contribution from gluon splitting is observed. The present data do not allow discrimination of theory predictions in the region of large p T of the J/ψ mesons, where the difference between Powheg and Pythia predictions is larger. Such discrimination will be possible with future measurements with larger data samples at higher energy.

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Acknowledgments We would like to thank P. Nason and A.K. Likhoded for interesting and stimulating discussions on the production of heavy-flavours. 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:

A Additional variables
In this appendix the normalized differential production cross-sections are studied for additional variables, namely • ∆φ J/ψ , the difference in the azimuthal angle φ J/ψ between the momentum directions of two J/ψ mesons; • ∆η J/ψ , the difference in the pseudorapidity η J/ψ between the momentum directions of two J/ψ mesons; • ∆y J/ψ , the difference in the rapidity y J/ψ between the two J/ψ mesons.
Unlike |∆φ * | /π and |∆η * |, which are largely independent on the decays of beauty hadrons, all these variables have a minor dependence both on the branching fractions of different beauty hadrons, as well as on the b → J/ψ X decay kinematics. The corresponding differential cross-sections are presented in figures 7 and 8. They are compared with expectations from the Powheg [66][67][68][69] and Pythia [59,65,75] generators and with expectations from the data-driven model of uncorrelated bb production, described in section 4. Also in this case both Powheg and Pythia describe the data well for all distributions, suggesting a small role of next-to-leading order effects in bb production in the studied kinematical range compared to the experimental precision. The data agree Powheg Pythia uncorrelated bb Powheg Pythia uncorrelated bb [7] ALICE collaboration, D + s meson production at central rapidity in proton-proton collisions at √ s = 7