Measurement of J = ψ and ψ ð 2 S Þ Prompt Double-Differential Cross Sections in pp Collisions at ﬃﬃ s p ¼ 7 TeV

The double-differential cross sections of promptly produced J= ψ and ψ ð 2 S Þ mesons are measured in pp collisions at ﬃﬃﬃ s p ¼ 7 TeV, as a function of transverse momentum p T and absolute rapidity j y j . The analysis uses J= ψ and ψ ð 2 S Þ dimuon samples collected by the CMS experiment, corresponding to integrated luminosities of 4.55 and 4 . 90 fb − 1 , respectively. The results are based on a two-dimensional analysis of the dimuon invariant mass and decay length, and extend to p T ¼ 120 and 100 GeV for the J= ψ and ψ ð 2 S Þ , respectively, when integrated over the interval j y j < 1 . 2 . The ratio of the ψ ð 2 S Þ to J= ψ cross sections is also reported for j y j < 1 . 2 , over the range 10 < p T < 100 GeV. These are the highest p T values for which the cross sections and ratio have been measured.

J , with spin S, orbital angular momentum L, and total angular momentum J that are either identical to (color singlet, a ¼ 1) or different from (color octet, a ¼ 8) those of the corresponding quarkonium state. The QQ cross sections are determined by short-distance coefficients (SDCs), kinematic-dependent functions calculable perturbatively as expansions in the strong-coupling constant α s . Then this "preresonant" QQ pair binds into the physically observable quarkonium through a nonperturbative evolution that may change L and S, with bound-state formation probabilities proportional to long-distance matrix elements (LDMEs). The LDMEs are conjectured to be constant (i.e., independent of the QQ momentum) and universal (i.e., process independent). The color-octet terms are expected to scale with powers of the heavy-quark velocity in the QQ rest frame. In the nonrelativistic limit, an S-wave vector quarkonium state should be formed from a QQ pair produced as a color singlet ( 3 S ½1 1 ) or as one of three color octets ( 1 S ½8 0 , 3 S ½8 1 , and 3 P ½8 J ).
Three "global fits" to measured quarkonium data [3][4][5] obtained incompatible octet LDMEs, despite the use of essentially identical theory inputs: next-to-leading-order (NLO) QCD calculations of the singlet and octet SDCs. The disagreement stems from the fact that different sets of measurements were considered. In particular, the results crucially depend on the minimum p T of the fitted measurements [6], because the octet SDCs have different p T dependences. Fits including low-p T cross sections lead to the conclusion that, at high p T , quarkonium production should be dominated by transversely polarized octet terms. This prediction is in stark contradiction with the unpolarized production seen by the CDF [7,8] and CMS [9,10] experiments, an observation known as the "quarkonium polarization puzzle." As shown in Ref. [6], the puzzle is seemingly solved by restricting the NRQCD global fits to high-p T quarkonia, indicating that the presently available fixed-order calculations provide SDCs that are unable to reproduce reality at lower p T values or that NRQCD factorization only holds for p T values much larger than the quarkonium mass. The polarization measurements add a crucial dimension to the global fits because the various channels have remarkably distinct polarization properties: in the helicity frame, 3 S ½1 1 is longitudinally polarized, 1 S ½8 0 is unpolarized, 3 S ½8 1 is transversely polarized, and 3 P ½8 J has a polarization that changes significantly with p T . Bottomonium and prompt charmonium polarizations reaching or exceeding p T ¼ 50 GeV were measured by CMS [9,10], using a very robust analysis framework [11,12], on the basis of event samples collected in 2011. Instead, the differential charmonium cross sections published by CMS [13] are based on data collected in 2010 and have a much lower p T reach. Measurements of prompt charmonium cross sections extending well beyond p T ¼ 50 GeV will trigger improved NRQCD global fits, restricted to a kinematic domain where the factorization formalism is unquestioned, and will provide more accurate and reliable LDMEs.
This Letter presents measurements of the doubledifferential cross sections of J=ψ and ψð2SÞ mesons promptly produced in pp collisions at a center-of-mass energy of 7 TeV, based on dimuon event samples collected by CMS in 2011. They complement other prompt charmonium cross sections measured at the LHC, by ATLAS [14,15], LHCb [16,17], and ALICE [18]. The analysis is made in four bins of absolute rapidity (jyj < 0.3, 0.3 < jyj < 0.6, 0.6 < jyj < 0.9, and 0.9 < jyj < 1.2) and in the p T ranges 10-95 GeV for the J=ψ and 10-75 GeV for the ψð2SÞ. A rapidity-integrated result in the range jyj < 1.2 is also provided, extending the p T reach to 120 GeV for the J=ψ and 100 GeV for the ψð2SÞ. The corresponding ψð2SÞ over J=ψ cross section ratios are also reported. The dimuon invariant mass distribution is used to separate the J=ψ and ψð2SÞ signals from other processes, mostly pairs of uncorrelated muons, while the dimuon decay length is used to separate the nonprompt charmonia, coming from decays of b hadrons,from the prompt component. Feed-down from decays of heavier charmonium states, approximately 33% of the prompt J=ψ cross section [19], is not distinguished from the directly produced charmonia.
The CMS apparatus is based on a superconducting solenoid of 6 m internal diameter, providing a 3.8 T field. Within the solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter, and a brass and scintillator hadron calorimeter. Muons are measured with three kinds of gas-ionization detectors: drift tubes, cathode strip chambers, and resistive-plate chambers. The main subdetectors used in this analysis are the silicon tracker and the muon system, which enable the measurement of muon momenta over the pseudorapidity range jηj < 2.4. A more detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematic variables, can be found in Ref. [20].
The events were collected using a two-level trigger system. The first level, made of custom hardware processors, uses data from the muon system to select events with two muon candidates. The high-level trigger, adding information from the silicon tracker, reduces the rate of stored events by requiring an opposite-sign muon pair of invariant mass 2.8 < M < 3.35 GeV, p T > 9.9 GeV, and jyj < 1.25 for the J=ψ trigger, and 3.35 < M < 4.05 GeV and p T > 6.9 GeV for the ψð2SÞ trigger. No p T requirement is imposed on the single muons at trigger level. Both triggers require a dimuon vertex fit χ 2 probability greater than 0.5% and a distance of closest approach between the two muons less than 5 mm. Events where the muons bend towards each other in the magnetic field are rejected to lower the trigger rate while retaining the highest-quality dimuons. The J=ψ and ψð2SÞ analyses are conducted independently, using event samples separated at the trigger level. The ψð2SÞ sample corresponds to an integrated luminosity of 4.90 fb −1 , while the J=ψ sample has a reduced value, 4.55 fb −1 , because the p T threshold of the J=ψ trigger was raised to 12.9 GeV in a fraction of the data-taking period; the integrated luminosities have an uncertainty of 2.2% [21].
The muon tracks are required to have hits in at least eleven tracker layers, with at least two in the silicon pixel detector, and to be matched with at least one segment in the muon system. They must have a good track fit quality (χ 2 per degree of freedom smaller than 1.8) and point to the interaction region. The selected muons must also match in pseudorapidity and azimuthal angle with the muon objects responsible for triggering the event. The analysis is restricted to muons produced within a fiducial phase-space window where the muon detection efficiencies are accurately measured: p T > 4.5, 3.5, and 3.0 GeV for the regions jηj < 1.2, 1.2 < jηj < 1.4, and 1.4 < jηj < 1.6, respectively. The combinatorial dimuon background is reduced by requiring a dimuon vertex fit χ 2 probability larger than 1%. After applying the event selection criteria, the combined yields of prompt and nonprompt charmonia in the range jyj < 1.2 are 5.45 M for the J=ψ and 266 k for the ψð2SÞ. The prompt charmonia are separated from those resulting from decays of b hadrons through the use of the dimuon pseudo-proper-decay-length [22], l ¼ L xy M=p T , where L xy is the transverse decay length in the laboratory frame, measured after removing the two muon tracks from the calculation of the primary vertex position. For events with multiple collision vertices, L xy is calculated with respect to the vertex closest to the direction of the dimuon momentum, extrapolated towards the beam line.
For each ðjyj; p T Þ bin, the prompt charmonium yields are evaluated through an extended unbinned maximumlikelihood fit to the two-dimensional ðM; lÞ event distribution. In the mass dimension, the shape of each signal peak is represented by a Crystal Ball (CB) function [23], with free mean (μ CB ) and width (σ CB ) parameters. Given the strong correlation between the two CB tail parameters, α CB and n CB , they are fixed to values evaluated from fits to event samples integrated in broader p T ranges. A single CB function provides a good description of the signal mass peaks, given that the dimuon mass distributions are studied in narrow ðjyj; p T Þ bins, within which the dimuon invariant mass resolution has a negligible variation. The mass distribution of the underlying continuum background is described by an exponential function. Concerning the pseudo-proper-decay-length variable, the prompt signal component is modeled by a resolution function, which exploits the per-event uncertainty information provided by the vertex reconstruction algorithm, while the nonprompt charmonium term is modeled by an exponential function convolved with the resolution function. The continuum background component is represented by a sum of prompt and nonprompt empirical forms. The distributions are well described with a relatively small number of free parameters. Figure 1 shows the J=ψ and ψð2SÞ dimuon invariant mass and pseudo-proper-decay-length projections for two representative ðjyj; p T Þ bins. The decay length projections are shown for events with dimuon invariant mass within AE3σ CB of the pole mass. In the highest p T bins, where the number of dimuons is relatively small, stable results are obtained by fixing μ CB and the slope of the exponential-like function describing the nonprompt combinatorial background to values extrapolated from the trend found from the lower-p T bins. The systematic uncertainties in the signal yields are evaluated by repeating the fit with different functional forms, varying the values of the fixed parameters, and allowing for more free parameters in the fit. The fit results are robust with respect to changes in the procedure; the corresponding systematic uncertainties are negligible at low p T and increase to ≈2% for the J=ψ and ≈6% for the ψð2SÞ in the highest p T bins.
The single-muon detection efficiencies ϵ μ are measured with a "tag-and-probe" (T&P) technique [24], using event samples collected with triggers specifically designed for this purpose, including a sample enriched in dimuons from J=ψ decays where a muon is combined with another track and the pair is required to have an invariant mass within the range 2.8-3.4 GeV. The procedure was validated in the phase-space window of the analysis with detailed Monte Carlo (MC) simulation studies. The measured efficiencies are parametrized as a function of muon p T , in eight bins of muon jηj. Their uncertainties, reflecting the statistical precision of the T&P samples and possible imperfections of the parametrization, are ≈2%-3%. The efficiency of the dimuon vertex fit χ 2 probability requirement is also measured with the T&P approach, using a sample of events collected with a dedicated (prescaled) trigger. It is around 95%-97%, improving with increasing p T , with a 2% systematic uncertainty. At high p T , when the two muons might be emitted relatively close to each other, the efficiency of the dimuon trigger ϵ μμ is smaller than the product of the two single-muon efficiencies [13], ϵ μμ ¼ ϵ μ 1 ϵ μ 2 ρ. The correction factor ρ is evaluated with MC simulations, validated from data collected with singlemuon triggers. For p T < 35 GeV, ρ is consistent with being unity, within a systematic uncertainty estimated as  2%, except in the 0.9 < jyj < 1.2 bin, where the uncertainty increases to 4.3% for the J=ψ if p T < 12 GeV, and to 2.7% for the ψð2SÞ if p T < 11 GeV. For p T > 35 GeV, ρ decreases approximately linearly with p T , reaching 60%-70% for p T ∼ 85 GeV, with systematic uncertainties evaluated by comparing the MC simulation results with estimations made using data collected with single-muon triggers: 5% up to p T ¼ 50 (55) GeV for the J=ψ [ψð2SÞ] and 10% for higher p T . The total dimuon detection efficiency increases from ϵ μμ ≈ 78% at p T ¼ 15 GeV to ≈85% at 30 GeV, and then decreases to ≈65% at 80 GeV.
To obtain the charmonium cross sections in each ðjyj; p T Þ bin without any restrictions on the kinematic variables of the two muons, we correct for the corresponding dimuon acceptance, defined as the fraction of dimuon decays having both muons emitted within the single-muon fiducial phase space. These acceptances are calculated using a detailed MC simulation of the CMS experiment. Charmonia are generated using a flat rapidity distribution and p T distributions based on previous measurements [13]; using flat p T distributions leads to negligible changes. The particles are decayed by EVTGEN [25] interfaced to PYTHIA 6.4 [26], while PHOTOS [27] is used to simulate final-state radiation. The fractions of J=ψ and ψð2SÞ dimuon events in a given ðjyj; p T Þ bin with both muons surviving the fiducial selections depend on the decay kinematics and, in particular, on the polarization of the mother particle. Acceptances are calculated using polarization scenarios corresponding to different values of the polar anisotropy parameter in the helicity frame, λ HX ϑ : 0 (unpolarized), þ1 (transverse), and −1 (longitudinal). A fourth scenario, corresponding to λ HX ϑ ¼ þ0.10 for the J=ψ and þ0.03 for the ψð2SÞ, reflects the results published by CMS [10]. The two other parameters characterizing the dimuon angular distributions [28], λ φ and λ ϑφ , have been measured to be essentially zero [10] and have a negligible influence on the acceptance. The acceptances are essentially identical for the two charmonia and are almost rapidity independent for jyj < 1.2. The two-dimensional acceptance maps are calculated with large MC simulation samples, so that statistical fluctuations are small, and in narrow jyj bins, so that variations within the bins can be neglected. Since the efficiencies and acceptances are evaluated for events where the two muons bend away from each other, a factor of 2 is applied to obtain the final cross sections.
The double-differential cross sections of promptly produced J=ψ and ψð2SÞ in the dimuon channel, Bd 2 σ=dp T dy, where B is the J=ψ or ψð2SÞ dimuon branching fraction, are obtained by dividing the fitted prompt-signal yields, already corrected on an event-byevent basis for efficiencies and acceptance, by the integrated luminosity and the widths of the p T and jyj bins. The numerical values, including the relative statistical and systematic uncertainties, are reported for both charmonia, five rapidity intervals, and four polarization scenarios in Tables 1-4 of the Supplemental Material [29]. Figure 2 shows the results obtained in the unpolarized scenario. With respect to the jyj < 0.3 bin, the cross sections drop by ≈5% for 0.6 < jyj < 0.9 and ≈15% for 0.9 < jyj < 1.2. Measuring the charmonium production cross sections in the broader rapidity range jyj < 1.2 has the advantage that the increased statistical accuracy allows the measurement to be extended to higher-p T values, where comparisons with theoretical calculations are particularly informative. Figure 3 compares the rapidity-integrated (unpolarized) cross sections, after rescaling with the branching fraction B of the dimuon decay channels [30], with results reported by ATLAS [14,15]. The curve represents a fit of the J=ψ cross section measured in this analysis to a power-law function [31]. The band labeled FKLSW represents the result of a global fit [6] comparing SDCs calculated at NLO [3] with ψð2SÞ cross sections and polarizations previously reported by CMS [10,13] and LHCb [17]. According to that fit, ψð2SÞ mesons are produced predominantly unpolarized. At high p T , the values reported in this Letter tend to be higher than the band, which is essentially determined from results for p T < 30 GeV.
The ratio of the ψð2SÞ to J=ψ differential cross sections is also measured in the jyj < 1.2 range, recomputing the J=ψ values in the p T bins of the ψð2SÞ analysis.
The measured values are reported in Table 5 of the Supplemental Material [29]. The corrections owing to the integrated luminosity, acceptances, and efficiencies cancel to a large extent in the measurement of the ratio. The total systematic uncertainty, dominated by the ρ correction for p T > 30 GeV and by the acceptance and  2 (color online). The J=ψ and ψð2SÞ differential p T cross sections times the dimuon branching fractions for four rapidity bins and integrated over the range jyj < 1.2 (scaled up by a factor of 2 for presentation purposes), assuming the unpolarized scenario. The vertical bars show the statistical and systematic uncertainties added in quadrature. efficiency corrections for p T < 20 GeV, does not exceed 3%, except for p T > 75 GeV, where it reaches 5%. Larger event samples are needed to clarify the trend of the ratio for p T above ≈35 GeV.
In summary, the double-differential cross sections of the J=ψ and ψð2SÞ mesons promptly produced in pp collisions at ffiffi ffi s p ¼ 7 TeV have been measured as a function of p T in four jyj bins, as well as integrated over the jyj < 1.2 range, extending up to or beyond p T ¼ 100 GeV. New global fits of cross sections and polarizations, including these high-p T measurements, will probe the theoretical calculations in a kinematical region where NRQCD factorization is believed to be most reliable. The new data should also provide input to stringent tests of recent theory developments, such as those described in Refs. [32][33][34].
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   The J=ψ (open symbols) and ψð2SÞ (closed symbols) differential (unpolarized) cross sections from this analysis (circles) and from ATLAS (squares) [14,15]. The vertical bars show the statistical and systematic uncertainties added in quadrature, not including the uncertainties from integrated luminosities and branching fractions, which are indicated by the percentages given in the legend. The curve shows a fit of the J=ψ cross section measured in this analysis to a power-law function, while the band labeled FKLSW represents a calculation of the ψð2SÞ cross section using LDMEs determined with lowerp T LHC data [6].