Constraints on the $\chi_\mathrm{c1}$ versus $\chi_\mathrm{c2}$ polarizations in proton-proton collisions at $\sqrt{s} =$ 8 TeV

The polarizations of promptly produced $\chi_\mathrm{c1}$ and $\chi_\mathrm{c2}$ mesons are studied using data collected by the CMS experiment at the LHC, in proton-proton collisions at $\sqrt{s} = $ 8 TeV. The $\chi_\mathrm{c}$ states are reconstructed via their radiative decays $\chi_\mathrm{c}$ $\to$ $\mathrm{J}/\psi\, \gamma$, with the photons being measured through conversions to e$^+$e$^-$, which allows the two states to be well resolved. The polarizations are measured in the helicity frame, through the analysis of the $\chi_\mathrm{c2}$ to $\chi_\mathrm{c1}$ yield ratio as a function of the polar or azimuthal angle of the positive muon emitted in the $\mathrm{J}/\psi$ $\to$ $\mu^+\mu^-$ decay, in three ranges of $\mathrm{J}/\psi$ transverse momentum. While no differences are seen between the two states in terms of azimuthal decay angle distributions, they are observed to have significantly different polar anisotropies. The measurement favors a scenario where at least one of the two states is strongly polarized along the helicity quantization axis, in agreement with nonrelativistic quantum chromodynamics predictions. This is the first measurement of significantly polarized quarkonia produced at high transverse momentum.


1
Quarkonium production is a benchmark for understanding how quarks combine into hadrons. The heaviness of c and b quarks makes it possible to describe the process in nonrelativistic quantum chromodynamics (NRQCD) [1][2][3][4][5][6][7][8], a framework valid when the quark velocities are small. This theory successfully described quarkonium cross sections measured [9] at high transverse momentum, p T , by complementing the earlier color-singlet model [10,11] with a superposition of several processes where the bound state originates from colored QQ pairs. In contrast to this complex model, the J/ψ, ψ(2S), Υ(1S), Υ(2S), and Υ(3S) differential cross sections measured at central rapidity by ATLAS [12,13] and CMS [14][15][16] have indistinguishable shapes as a function of p T /M, where M is the meson mass [17,18]. This universal momentum scaling pattern is also followed by the χ c1 and χ c2 states [19,20]. The corresponding polarization measurements [21,22] show that the five S-wave states are well compatible with being produced unpolarized, in contrast to the significant polarizations seen for the W and Z [23-30], Drell-Yan dileptons [31][32][33][34][35][36], and low-p T quarkonia [37,38]. The lack of polarization of highp T vector quarkonia was a long-standing challenge for NRQCD [39], until recent global-fit analyses [4,40,41] showed that cross sections and polarizations can be consistently described, unveiling a delicate compensation between terms in the factorization expansion [42]. Among the measurements mentioned above, one piece is clearly missing: the χ c1 and χ c2 polarizations. Contrary to what happens for the vector states, predicting the χ c1 and χ c2 polarizations is rather simple within NRQCD, where they are unequivocally determined by a single coloroctet parameter, which can be extracted from the χ c2 to χ c1 cross section ratio. The analysis of the measured ratios [19, 20] provides a clear result: the polarizations of the two states should be opposite and almost maximal [43] (a result also reached in a parameter-free singlet-only model [44]). Finding that these P-wave states have similar polarizations (following the vector quarkonia in the polarizations, as in the cross sections) would be a challenge to NRQCD, where the two (necessarily different) singlet terms play a leading role.
This Letter reports the first measurement of the polarizations of promptly produced χ c1 and χ c2 mesons, using proton-proton (pp) data collected at the LHC by the CMS experiment at a centerof-mass energy of √ s = 8 TeV, corresponding to an integrated luminosity of 19.1 fb −1 . The central feature of the CMS apparatus is a superconducting solenoid of 6 m internal diameter, providing a magnetic field of 3.8 T. Within the solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter, and a brass and scintillator hadron calorimeter, each composed of a barrel and two endcap sections. Muons are detected in gasionization chambers embedded in the steel flux-return yoke outside the solenoid. A detailed description of the CMS detector, together with a definition of the coordinate system used and relevant kinematic variables, can be found in Ref. [45].
The event sample was collected with a two-level trigger system [46]. At level-1, custom hardware processors select events with two muons. The high-level trigger requires an opposite-sign muon pair of invariant mass 2.8-3.35 GeV, a dimuon vertex fit χ 2 probability larger than 0.5%, and a distance of closest approach between the two muons smaller than 0.5 cm. The trigger also requires that the dimuon has p T > 7.9 GeV and rapidity |y| < 1.25. The offline reconstruction requires two oppositely charged muons matching those that triggered the detector readout. The muon tracks must pass high-purity track quality requirements [47], have p T > 3.5 GeV, |η| < 1.6, and fulfill the soft muon identification requirements [48], which imply, in particular, more than five hits in the silicon tracker, of which at least one is in the pixel layers. The muons are combined to form J/ψ candidates, which are kept for further processing if |y| < 1.2 and 8 < p T < 30 GeV. Promptly produced J/ψ mesons are selected by requiring the distance between the dimuon vertex and the interaction point be smaller than 2.5 times its uncertainty.
The analysis uses χ c → J/ψ γ decays, with the J/ψ decaying to a dimuon. The photons are The measurement of the λ parameters implies knowing the shapes of the χ c1 and χ c2 differential cross sections as functions of |cos ϑ| and ϕ, which crucially depend on the accuracy of the corrections of the muon and photon detection efficiencies. These efficiencies change by an order of magnitude in the low p T bin covered by the present analysis and shape variations within their uncertainties lead to very different λ ϑ values. Increasing the muon p T threshold to avoid the turn-on region of the efficiency function would imply a strong reduction in the number of selected events and a smaller coverage of the |cos ϑ| variable, effectively preventing the evaluation of λ ϑ . Instead, the difference between the χ c1 and χ c2 polarizations, measured from the angular dependence of the χ c2 /χ c1 yield ratio, is essentially insensitive to the detection efficiencies, given that they cancel to a large extent in that ratio.
The |cos ϑ| and ϕ dependences of the yield ratio are independently determined in three J/ψ p T bins: 8-12, 12-18, and 18-30 GeV. For the study of possible azimuthal dependences of the χ c2 /χ c1 yield ratio, the events are split into subsamples corresponding to six equidistant ϕ bins between 0 and 90 • . Folding ϕ into the first quadrant reduces the effect of the statistical fluctuations without any loss of information, given the four-fold ϕ symmetry that the angular distributions obey. For each p T bin, the six J/ψ γ invariant mass distributions are simultaneously fitted with an unbinned maximum likelihood fit. In the mass fit model, identical for all ϕ bins, each of the χ c1 and χ c2 signal peaks is represented by a double-sided Crystal Ball (CB) function [56], which complements a Gaussian core distribution with lower and upper powerlaw tails. The underlying combinatorial background, reflecting uncorrelated J/ψ γ associations, is parametrized by an exponential function multiplied by a term that provides a low-mass turndown, (1 + erf((m − µ bg ) / σ bg )) exp(−m/λ bg ), where m is the J/ψ γ invariant mass and µ bg , σ bg , and λ bg are shape parameters. Although the results of this analysis are insensitive to the presence of a small peak reflecting the χ c0 decays, the fit model includes this background term, represented by a Breit-Wigner convolved with a Gaussian resolution function. To minimize fit instabilities, the χ c0 shape and yield parameters are determined from the corresponding parameters of the χ c1 term. The simultaneous fit has the advantage of reducing by a factor of six the number of free parameters defining the shapes of the signal and background mass models, by requiring that those parameters are independent of ϕ, an assumption validated by studies of simulated and measured event samples.
To study the polar angle dependence of the χ c2 /χ c1 yield ratio, 6, 7, or 5 |cos ϑ| bins are considered, depending on the p T bin. The |cos ϑ| coverage is smaller in the lowest p T bin (up to 0.45 instead of up to 0.625) because those events are the ones most affected by the single-muon p T cut. Analogously to the procedure just described for the ϕ dimension, the χ c2 /χ c1 yield ratios are obtained as a function of |cos ϑ| through a simultaneous fit of the J/ψ γ invariant mass distributions. In this case, however, some of the shape parameters are not required to be independent of |cos ϑ|. More details can be found in Ref. [57].   For each bin in J/ψ p T and |cos ϑ|, or ϕ, the fitted J/ψ γ invariant mass distributions provide functions reflecting the probability that an event of mass m is a χ c1 or a χ c2 . The χ c1 and χ c2 yields, corrected for acceptance and efficiencies, are then computed as the sums, over all events in that bin of J/ψ p T and |cos ϑ|, or ϕ, of the product between the corresponding probabilities and the weights 1/A J (|cos ϑ|, ϕ, p T ), where A J (|cos ϑ|, ϕ, p T ) are the acceptance times efficiency three-dimensional maps, independently evaluated for each χ c J state with large samples of simulated events. By correcting the detector acceptance and efficiency effects on an event-by-event basis, with weights depending on three dimuon observables (|cos ϑ|, ϕ, and p T ), this procedure is immune to integration biases affecting certain one-dimensional analyses [58]. Simulation studies have shown that, if the three-dimensional correction maps are sufficiently fine-grained, the results do not depend on the polarization scenario nor on the p T distributions assumed in the simulation, and that all physically allowed differences between the χ c1 and χ c2 polarizations, in any frame, can be reliably determined from the dependences of the χ c2 /χ c1 yield ratios on |cos ϑ| and ϕ.
The corrected ratios are reported in Tables A.1 and A.2 of Appendix A, and shown in Fig. 2, where it can be seen that the uncorrected and corrected values are almost identical, apart from normalization factors irrelevant for the determination of the polar and azimuthal anisotropies.
Several sources of potential systematic effects have been considered, by redoing the analysis with different inputs and comparing the obtained results with the nominal ones. The results are insensitive to variations of the thresholds used to reject the nonprompt contamination from b hadron decays, estimated to be around 5%, or events with a poor kinematic vertex fit quality in the reconstruction of the χ c candidates. The fits of the mass distributions were redone using alternative options for the low-and high-mass tails of the double-sided CB functions, and by varying the combinatorial background description, both by changing the floating parameters of the nominal function and by using the alternative function (x − x 0 ) λ exp (ν(x − x 0 )), where ν is left free, λ is fitted to a constant, and x 0 = 3.2 GeV, a value determined in fits to the backgroundonly mass distributions obtained by excluding the 3.37-3.6 GeV region. The sensitivity of the results to the acceptance and efficiency corrections was evaluated by redoing the analysis with maps computed with alternative single-muon and photon detection efficiencies, as well as with simulated samples generated with different p T /M shapes for each of the two χ c states. All effects lead to similar variations in the yields of the two states and cancel, to a large extent, in the χ c2 /χ c1 ratio, apart from a normalization shift that has no impact on the angular anisotropies. The total systematic uncertainties are less than 20% of the statistical ones.
The χ c2 to χ c1 yield ratios as a function of ϕ, shown in Fig. 2 (left), are compatible with being flat, excluding large differences in azimuthal anisotropy, as exemplified by the two curves compared to the data points in the second p T bin. These curves represent the simplest conceivable polarization hypotheses leading to large azimuthal effects in the helicity frame: χ c1 and χ c2 have maximally different polar anisotropies in the Collins-Soper frame, corresponding to specific alignments of their angular momentum vectors along the collision direction (J = ±2, for the dotted and dash-dotted curve, respectively). In fact, the change from the Collins-Soper to the helicity quantization axis is almost a 90 • rotation, transforming polarized distributions into azimuthally anisotropic ones. This uniform ϕ behavior confirms the choice of the helicity axis as the one that should reflect most closely the natural alignment of the angular momentum vector, maximizing the polar anisotropy effects.
In Fig. 2 (right) the measured |cos ϑ| dependence of the χ c2 /χ c1 ratio is compared to the analytic    In summary, first experimental constraints on the polarizations of promptly produced χ c1 and χ c2 mesons have been obtained, using pp collisions at √ s = 8 TeV. The analysis uses the J/ψ γ decay channel in three J/ψ p T bins between 8 and 30 GeV. The measurement, made in the helicity frame, shows a significant difference between the polar anisotropy parameters λ χ c1 ϑ and λ χ c2 ϑ , in agreement with the NRQCD prediction. This result is a new step in the experimental studies of quarkonium production and the first significant indication of kinematic differences between the various quarkonia.

Acknowledgments
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.   [11] J.-P. Lansberg, "On the mechanisms of heavy-quarkonium hadroproduction", Eur. Phys. J. C 61 (2009) 693, doi:10.1140/epjc/s10052-008-0826-9, arXiv:0811.4005.