Forward production of $\Upsilon$ mesons in $pp$ collisions at $\sqrt{s}=7$ and 8TeV

The production of $\Upsilon$ mesons in $pp$ collisions at $\sqrt=7$ and $8\,\mathrm{TeV}$ is studied with the LHCb detector using data samples corresponding to an integrated luminosity of $1\,\mathrm{fb}^{-1}$ and $2\,\mathrm{fb}^{-1}$ respectively. The production cross-sections and ratios of cross-sections are measured as functions of the meson transverse momentum $p_T$ and rapidity $y$, for $p_T<30\,\mathrm{GeV}/c$} and $2.0


Introduction
In high energy hadron collisions, the production of heavy quarkonium systems such as the bb states (Υ(1S), Υ(2S) and Υ(3S), represented generically as Υ in the following) probes the dynamics of the colliding partons and provides insight into the non-perturbative regime of quantum chromodynamics (QCD). Despite many models that have been proposed, a complete description of heavy quarkonium production is still not available.
The effective theory of non-relativistic QCD (NRQCD) [1,2] provides the foundation for much of the current theoretical work. According to NRQCD, the production of heavy quarkonium factorises into two steps: a heavy quark-antiquark pair is first created at short distances, and subsequently evolves non-perturbatively into a quarkonium state. The NRQCD calculations include the colour-singlet (CS) and colour-octet (CO) matrix elements for the pertubative stage. The CS model [3,4], which provides a leading-order description of quarkonium production, underestimates the cross-section for single J/ψ production at the Tevatron [5] at high p T , where p T is the component of the meson momentum transverse to the beam. To resolve this discrepancy, the CO mechanism was introduced [6]. The corresponding matrix elements were determined from the high-p T data, as the CO cross-section decreases more slowly with p T than that predicted by the CS model. More recent higher-order calculations [7][8][9][10][11] show better agreement between CS predictions and the experimental data [12], reducing the need for large CO contributions. The production of Υ mesons in proton-proton (pp) collisions can occur either directly in parton scattering or via feed down from the decay of heavier bottomonium states, such as χ b [13][14][15][16][17][18], or higher-mass Υ states, which complicates the theoretical description of bottomonium production [19,20].
The production of the Υ mesons has been studied using pp collision data taken at √ s = 2.76, 7  TeV with a data sample collected in 2010 corresponding to an integrated luminosity of 25 pb −1 , and at √ s = 8 TeV for early 2012 data using 50 pb −1 . Both measurements were differential in p T and y of the Υ mesons in the ranges 2.0 < y < 4.5 and p T < 15 GeV/c. Based on these measurements, an increase of the production cross-section in excess of 30% between √ s = 7 and 8 TeV was observed, which is larger than the increase observed for other quarkonium states such as the J/ψ [23, 28] and larger than the expectations from NRQCD [11].
In this paper we report on the measurement of the inclusive production cross-sections of the Υ states at √ s = 7 and 8 TeV and the ratios of these cross-sections. The Υ cross-section measurement is performed using a data sample corresponding to the complete LHCb Run 1 data set with integrated luminosities of 1 fb −1 and 2 fb −1 , accumulated at √ s = 7 and 8 TeV, respectively. These samples are independent from those used in the previous analyses [22,23]. The increased size of the data sample results in a better statistical precision and allows the measurements to be extended up to p T values of 30 GeV/c.

Detector and simulation
The LHCb detector [29,30] 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 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 [31]. The online event selection is performed by a trigger [32], 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. At the hardware stage, events for this analysis are selected by requiring dimuon candidates with a product of their p T values exceeding 1.7 (2.6) (GeV/c) 2 for data collected at √ s = 7 (8) TeV. In the subsequent software trigger, two well-reconstructed tracks are required to have hits in the muon system, p T > 500 MeV/c, p > 6 GeV/c and to form a common vertex. Only events with a dimuon candidate with a mass m µ + µ − > 4.7 GeV/c 2 are retained for further analysis. In the offline selection, trigger decisions are associated with reconstructed particles. Selection requirements can therefore be made on the trigger selection itself and on whether the decision was due to the signal candidate, the other particles produced in the pp collision, or a combination of both.
In the simulation, pp collisions are generated using Pythia 6 [33] with a specific LHCb configuration [34]. Decays of hadronic particles are described by EvtGen [35], in which final-state radiation is generated using Photos [36]. The interaction of the generated particles with the detector, and its response, are implemented using the Geant4 toolkit [37] as described in Ref. [38].

Selection and cross-section determination
The event selection is based on the criteria described in the previous LHCb Υ analyses [21-23] but slightly modified to improve the signal-to-background ratio. It includes selection criteria that ensure good quality track reconstruction [39], muon identification [40], and the requirement of a good fit quality for the dimuon vertex, where the associated primary vertex position is used as a constraint in the fit [41]. In addition, the muon candidates are required to have 1 < p T < 25 GeV/c, 10 < p < 400 GeV/c and pseudorapidity within the region 2.0 < η < 4.5.
The differential cross-section for the production of an Υ meson decaying into a muon pair is where B Υ is the branching fraction of the Υ → µ + µ − decay, ∆y and ∆p T are the rapidity and p T bin sizes, σ Υ→µ + µ − bin is a production cross-section for Υ → µ + µ − events in the given (p T , y) bin, N Υ→µ + µ − is the efficiency-corrected number of Υ → µ + µ − decays and L is the integrated luminosity. Given the sizeable uncertainty on the dimuon branching fractions of the Υ mesons [42], the measurement of the production cross-section multiplied by the dimuon branching fraction is presented, as in previous LHCb measurements [21][22][23].
A large part of the theoretical and experimental uncertainties cancel in the ratios of production cross-sections of various Υ mesons, defined for a given (p T , y) bin as The evolution of the production cross-sections as a function of pp collision energy is studied using the ratio The signal yields N Υ→µ + µ − in each (p T , y) bin are determined from an unbinned extended maximum likelihood fit to the dimuon mass spectrum of the selected candidates within the range 8.5 < m µ + µ − < 12.5 GeV/c 2 . The correction for efficiency is embedded in the fit procedure. Each dimuon candidate is given a weight calculated as 1/ε tot , where ε tot is the total efficiency, which is determined for each Υ → µ + µ − candidate as where ε rec&sel is the reconstruction and selection efficiency, ε trg is the trigger efficiency and ε µID is the efficiency of the muon identification criteria. The efficiencies ε rec&sel and ε trg are determined using simulation, and corrected using data-driven techniques to account for small differences in the muon reconstruction efficiency between data and simulation [39,40]. The efficiency of the muon identification criteria ε µID is measured directly from data using a large sample of low-background J/ψ → µ + µ − events. All efficiencies are evaluated as functions of the muon and dimuon kinematics. The mean total efficiency ε tot reaches a maximum of about 45% for the region 15 < p T < 20 GeV/c, 3.0 < y < 3.5, and drops down to 10% at high p T and large y, with the average efficiency being about 30%.
In each (p T , y) bin, the dimuon mass distribution is described by the sum of three Crystal Ball functions [43], one for each of the Υ(1S), Υ(2S) and Υ(3S) signals, and the product of an exponential function with a second-order polynomial for the combinatorial background. The mean value and the resolution of the Crystal Ball function describing

LHCb
√ s = 7 TeV 3 < p T < 4 GeV/c 3.0 < y < 3.5 LHCb √ s = 8 TeV 3 < p T < 4 GeV/c 3.0 < y < 3.5 √ s = 8 TeV samples in the region 3 < p T < 4 GeV/c, 3.0 < y < 3.5. The thick dark yellow solid curves show the result of the fits, as described in the text. The three peaks, shown with thin magenta solid lines, correspond to the Υ(1S), Υ(2S) and Υ(3S) signals (left to right). The background component is indicated with a blue dashed line. To show the signal peaks clearly, the range of the dimuon mass shown is narrower than that used in the fit.
the mass distribution of the Υ(1S) meson are free fit parameters. For the Υ(2S) and Υ(3S) mesons the mass differences m(Υ(2S)) − m(Υ(1S)) and m(Υ(3S)) − m(Υ(1S)) are fixed to the known values [42], while the resolutions are fixed to the value of the resolution of the Υ(1S) signal, scaled by the ratio of the masses of the Υ(2S) and Υ(3S) to the Υ(1S) meson. The tail parameters of the Crystal Ball function describing the radiative tail are fixed from studies of simulated samples.
The fits are performed independently on the efficiency-corrected dimuon mass distributions in each (p T ,y) bin. As an example, Fig. 1 shows the results of the fits in the region 3 < p T < 4 GeV/c and 3.0 < y < 3.5. For each bin the position and the resolution of the Υ(1S) signal is found to be consistent between √ s = 7 and 8 TeV data sets. The resolution varies between 33 MeV/c 2 in the region of low p T and small rapidity and 90 MeV/c 2 for the high p T and large y region, with the average value being close to 42 MeV/c 2 . The total signal yields are obtained by summing the signal yields over all (p T , y) bins and are summarised in Table 1.

Systematic uncertainties
The systematic uncertainties are summarised in Table 2, separately for the measurement of the cross-sections and of their ratios.
The uncertainty related to the mass model describing the shape of the dimuon mass distribution is studied by varying the fit range and the signal and background parametrisation used in the fit model. The fit range is varied by moving the upper edge (786.6 ± 2.6) · 10 3 Table 2: Summary of relative systematic uncertainties (in %) for the differential production cross-sections, their ratios, integrated cross-sections and the ratios R 8/7 . The ranges indicate variations depending on the (p T , y) bin and the Υ state.
Fit model and range 0. from 12.5 to 11.5 GeV/c 2 ; the degree of the polynomial function used in the estimation of the background is varied between zeroth and the third order. Also the tail parameters of the Crystal Ball function are allowed to vary in the fit. In addition, the constraints on the difference in the Υ signal peak positions are removed for all bins with high signal yields. The maximum relative difference in the number of signal events is taken as a systematic uncertainty arising from the choice of the fit model. As an alternative to the determination of the signal yields from efficiency-corrected data, the method employed in Ref.
[21] is used. In this method the efficiency-corrected yields for each (p T , y) bin are calculated using the sPlot technique [44]. The difference between this method and the nominal one is taken as a systematic uncertainty on the efficiency correction.
Reconstruction, selection and trigger efficiencies in Eq. (4) are obtained using simulated samples. The uncertainties due to the finite size of these samples are propagated to the measurement using a large number of pseudoexperiments. The same technique is used for the propagation of the uncertainties on the muon identification efficiency determined from large low-background samples of J/ψ → µ + µ − decays.
Several systematic uncertainties are assigned to account for possible imperfections in the simulated samples. The possible mismodelling of the bremsstrahlung simulation for the radiative tail and its effect on the signal shape has been estimated in previous LHCb analyses [23] and leads to an additional uncertainty of 1.0% on the cross-section.
Good agreement between data and simulation is observed for all variables used in the selection. The small differences seen would affect the efficiencies by less than 1.0%, which is conservatively taken as the systematic uncertainty to account for the disagreement between data and simulation.
The efficiency is corrected using data-driven techniques to account for small differences in the tracking efficiency between data and simulation [39,40]. The uncertainty in the correction factor is propagated to the cross-section measurement using pseudoexperiments and results in a global 0.5% systematic uncertainty plus an additional uncertainty of 0.4% per track.
The systematic uncertainty associated with the trigger requirements is assessed by studying the performance of the dimuon trigger for Υ(1S) events selected using the single muon high-p T trigger [32] in data and simulation. The comparison is perfomed in bins of the Υ(1S) meson transverse momentum and rapidity and the largest observed difference of 2.0% is assigned as the systematic uncertainty associated with the imperfection of trigger simulation.
The luminosity measurement was calibrated during dedicated data taking periods, using both van der Meer scans [45] and a beam-gas imaging method [46,47]. The absolute luminosity scale is determined with 1.7 (1.2)% uncertainty for the sample collected at √ s = 7 (8) TeV, of which the beam-gas resolution, the spread of the measurements and the detector alignment are the largest contributions [47][48][49]. The ratio of absolute luminosities for samples accumulated at √ s = 7 and 8 TeV is known with a 1.4% uncertainty. The total systematic uncertainty in each (p T , y) bin is the sum in quadrature of the individual components described above. For the integrated production cross-section the systematic uncertainty is estimated by taking into account bin-to-bin correlations. Several systematic uncertainties cancel or significantly reduce in the measurement of the ratios R i,j and R 8/7 , as shown in Table 2.
The production cross-sections are measured at centre-of-mass energies of 7 and 8 TeV, where the actual beam energy for pp collisions is known with a precision of 0.65% [50]. Assuming a linear dependence of the production cross-section on the pp collision energy, and using the measured production cross-sections at √ s = 7 (8) TeV, the change in the production cross-section due to the imprecise knowledge of the beam energy is estimated to be 1.4 (1.2)%. The effect is strongly correlated between √ s = 7 and 8 TeV data and will therefore mostly cancel in the measurement of the ratio of cross-sections at the two energies.
The efficiency is dependent on the polarisation of the Υ mesons. The polarisation of the Υ mesons produced in pp collisions at √ s = 7 TeV at high p T and central rapidity has been studied by the CMS collaboration [51] in the centre-of-mass helicity, Collins-Soper [52] and the perpendicular helicity frames. No evidence of significant transverse or longitudinal polarisation has been observed for the region 10 < p T < 50 GeV/c, |y| < 1.2. Therefore, results are quoted under the assumption of unpolarised production of Υ mesons and no corresponding systematic uncertainty is assigned on the cross-section. Under the assumption of transversely polarised Υ mesons with λ ϑ = 0.2 in the LHCb kinematic region, 1 the total production cross-section would result in an increase of 3%, with the largest local increase of around 6% occuring in the low p T region (p T < 3 GeV/c), both for small (y < 2.5) and large (y > 4.0) rapidities.

Results
The double-differential production cross-sections multiplied by the dimuon branching fractions for the Υ mesons are shown in Fig. 2. The corresponding production cross-section σ Υ→µ + µ − bin in (p T , y) bins are presented in Tables 3, 4  The transverse momentum spectra are fit using a Tsallis function [53] dσ where E kin T ≡ m 2 Υ + p 2 T − m Υ is the transverse kinetic energy, the power n and the temperature parameter T are free parameters, and m Υ is the known mass of a Υ meson [42]. This function has a power-law asymptotic behaviour ∝ p −n T for high p T as expected for hard scattering processes. It has been successfully applied to fit p T spectra [54][55][56][57] in wide ranges of particle species, processes and kinematics. A fit with the Tsallis distribution for the range 6 < p T < 30 GeV/c is superimposed on the differential cross-sections in Fig. 3. The fit quality is good for all cases. The fitted values of the parameters n and T are listed in Table 9. The parameter n for all cases is close to 8, compatible with the high p T asymptotic behaviour expected by the CS model [3,4,[58][59][60]. The temperature parameters T show little dependence on √ s and increase with the mass of Υ state. The shapes of the rapidity spectra are compared with the CO model prediction in the region 2.5 < y < 4.0 and are fitted using the function given by Eq. (1) of Ref. [63], with free normalisation constants. The fit result, as well as the extrapolation to the full kinematic range 2.0 < y < 4.5, is presented in Fig. 4. The quality of the fit is good for all cases.
The integrated production cross-sections multiplied by the dimuon branching fractions in the full range p T < 30 GeV/c and 2.0 < y < 4.5 at √ s = 7 and 8 TeV are reported in Table 10, where the first uncertainties are statistical and the second systematic. The same measurements are also shown integrated over the reduced range p T < 15 GeV/c in the same rapidity range, to allow the comparison with previous measurements [22,23].
The error bars indicate the sum in quadrature of the statistical and systematic uncertainties. The rapidity ranges 2.0 < y < 2.5, 2.5 < y < 3.0, 3.0 < y < 3.5, 3.5 < y < 4.0 and 4.0 < y < 4.5 are shown with red filled circles, blue open squares, cyan downward triangles, magenta upward triangles and green diamonds, respectively. Some data points are displaced from the bin centres to improve visibility.        d dp   [62,63] in the region 2.5 < y < 4.0, and dashed lines show the extrapolation to the full region 2.0 < y < 4.5. The data points are positioned in the bins according to Eq. (6) in Ref. [61].
The ratios of integrated production cross-section R 8/7 are presented in Table 11 for the full (p T < 30 GeV/c) and reduced (p T < 15 GeV/c) ranges. The results for the reduced range are consistent with the previous measurements, confirming the increase of the bottomonium production cross-section of approximately 30% when the centre-of-mass energy 837 ± 2 ± 25 614 ± 2 ± 20 807 ± 2 ± 24 σ Υ(3S)→µ + µ − 313 ± 2 ± 10 393 ± 1 ± 12 298 ± 1 ± 10 373 ± 1 ± 11 increases from √ s = 7 to 8 TeV [22, 23]. The ratios R 8/7 as a function of p T integrated over the region 2.0 < y < 4.5 are shown in Fig. 5a. The ratios are fitted with a linear function. The fit quality is good, with a p-value exceeding 35% for all cases, and the slopes are found to be 10.8 ± 0.6, 9.5 ± 1.2 and 9.8 ± 1.6 (in units of 10 −3 / (GeV/c)) for Υ(1S), Υ(2S) and Υ(3S), respectively. The measurements are compared with the NRQCD theory predictions [11] in the same kinematic range, where only uncertainties from the CO long distance matrix elements are considered since most other uncertainties are expected to cancel in the ratio. The theory predictions are independent on the Υ state and are consistently lower than the measurements.
The ratio R 8/7 as a function of rapidity, integrated over the region p T < 30 GeV/c is shown in Fig. 5b. The ratios are compared with the expectations from the CO mechanism [62,63] with normalisation factors fixed from the fits of Fig. 4. The trend observed in data does not agree with the pure CO model. It can be noted that also for open beauty hadrons the differential cross-sections exhibit a larger rise as a function of √ s at smaller rapidities [54], while the FONLL calculations [64] predict this behaviour towards larger rapidity.
The ratios R i,j at √ s = 7 and 8 TeV are reported in Fig. 6 and Tables 12, 13, 14 and 15 Table 11: The ratio of production cross-sections for Υ mesons at √ s = 8 to that at √ s = 7 TeV in the full kinematic range p T < 30 GeV/c (left) and reduced range p T < 15 GeV/c (right) for 2.0 < y < 4.5. The first uncertainties are statistical and the second systematic.
LHCb Figure 5: Ratios of the differential cross-sections (left) d dp T σ Υ→µ + µ − and (right) d dy σ Υ→µ + µ − at √ s = 8 and 7 TeV for (red solid circles) Υ(1S), (blue open squares) Υ(2S) and (green solid diamonds) Υ(3S). On the left hand plot, the results of the fit with a linear function are shown with straight thin red solid, blue dotted and green dashed lines. In the same plot, the next-to-leading order NRQCD theory predictions [11] are shown as a thick line. On the right hand plot, the curved red solid, blue dotted and greed dashed lines show the CO model predictions [62,63] with the normalisation fixed from the fits in Fig. 4 for Υ(1S), Υ(2S) and Υ(3S) mesons, respectively. Some data points are displaced from the bin centres to improve visibility.
as a function of p T for different rapidity bins. The same ratios as a function of p T integrated over rapidity, and as a function of y integrated over p T , are shown in Fig. 7. The ratios R i,j show little dependence on rapidity and increase as a function of p T , in agreement with previous observations by LHCb [22,23], ATLAS [25] and CMS [26] at √ s = 7 TeV. The ratios of integrated cross-sections R i,j at √ s = 7 and 8 TeV are reported in Table 16, for the full and the reduced p T kinematic regions. All ratios R i,j agree with previous LHCb measurements. The ratio R 2,1 agrees with the estimates of 0.27 from Refs. [63,65], while R 3,1 significantly exceeds the expected value of 0.04 [63,65] but agrees with the range 0.14 − 0.22, expected for the hypothesis of a large admixture of a hybrid quarkonium state in the Υ(3S) meson state [65].

Summary
The forward production of Υ mesons is studied in pp collisions at centre-of-mass energies of 7 and 8 TeV using data samples corresponding to integrated luminosities of 1 fb −1 and 2 fb −1 respectively, collected with the LHCb detector. The double differential production cross-sections are measured as a function of meson transverse momenta and rapidity for the range p T < 30 GeV/c, 2.0 < y < 4.5. The measured increase in the production cross-sections of Υ mesons between √ s = 8 and 7 TeV significantly exceeds theory expectations and confirms the previous LHCb observations [22,23]. For the region p T < 15 GeV/c the results agree with the previous measurements [22,23], and supersede them.