J/$\psi$ polarization in pp collisions at $\sqrt{s}=7$ TeV

We have studied J/psi production in pp collisions at $\sqrt{s}=7$ TeV at the LHC through its muon pair decay. The polar and azimuthal angle distributions of the decay muons were measured, and results on the J/$\psi$ polarization parameters $\lambda_{\theta}$ and $\lambda_\phi$ were obtained. The study was performed in the kinematic region 2.5<y<4, 2<$p_{\rm T}$<8 GeV/$c$, in the helicity and Collins-Soper reference frames. In both frames, the polarization parameters are compatible with zero, within uncertainties.

The distribution of the J/ψ decay products can be expressed in its general form [21] as W (θ , φ ) ∝ 1 3 + λ θ 1 + λ θ cos 2 θ + λ φ sin 2 θ cos 2φ +λ θ φ sin 2θ cos φ , where θ (φ ) are the polar (azimuthal) angles in a given reference frame. In this analysis, the Collins-Soper (CS) and helicity (HE) frames were considered. In the CS frame the z-axis is defined as the bisector of the angle between the direction of one beam and the opposite of the direction of the other one, in the rest frame of the decaying particle. In the HE reference frame the z-axis is given by the direction of the decaying particle in the center of mass frame of the collision. The φ = 0 plane is the one containing the two beams, in the J/ψ rest frame. Equation 1 contains the three parameters λ θ , λ φ and λ θ φ , which quantify the degree of polarization. In particular, λ θ > 0 values indicate transverse polarization, while a longitudinal polarization gives λ θ < 0. In principle the values of the parameters could be extracted by means of a fit to the acceptance-corrected 2-D distributions for cos θ vs φ . However, the limited J/ψ statistics (about 6.8 · 10 3 signal events in the p t range under study) makes a 2-D binning impossible. Therefore the study of the angular distributions was separately performed on the polar and azimuthal variables. In particular, λ θ and λ φ were obtained by studying the distributions obtained by integrating Eq. 1 in the φ and cos θ variables, respectively.
The distributions of the angular variables for the J/ψ decay products were obtained starting from the study of the dimuon invariant mass spectra. The study was performed in five bins for the | cos θ | variable (the angular distribution is symmetric with respect to cos θ = 0), in the range 0 < | cos θ | < 0.8. For the azimuthal variable four bins in |φ | were defined, in the range 0 < |φ | < π/2 (values between π/2 and π were mirrored around |φ | = π/2, due to the period of the cos 2φ function). The analysis was carried out in three transverse momentum intervals (2 < p t < 3 GeV/c, 3 < p t < 4 GeV/c and 4 < p t < 8 GeV/c). The limits of the explored p t range are related to the strong decrease of the acceptance for large | cos θ | values at low p t and to the limited statistics at high p t .
The number of J/ψ signal events for the various bins in | cos θ | and |φ | were obtained by means of fits to the corresponding dimuon invariant mass spectra performed in the range 1.5 < m µ µ < 5 GeV/c 2 , and in Fig. 1 we show one of them as an example. The J/ψ signal was described by a Crystal Ball function (CB) [22] while for the background an empirical function, corresponding to a Gaussian with a width linearly depending on mass, was adopted. The position of the CB peak was left as a free parameter in the fits, and was found to correspond to the nominal J/ψ pole mass within at most 1%. The width of the CB function obtained from the data (between 72 and 120 MeV/c 2 , depending on the kinematics) was found to be in agreement with the Monte Carlo (MC) within ∼ 8 − 10 MeV/c 2 . In the fits, the width of the CB function for each bin i (where i represents a certain | cos θ | or |φ | interval for the J/ψ p t bin under study) was fixed to σ i J/ψ = σ J/ψ · (σ i,MC J/ψ /σ MC J/ψ ), i.e. by scaling the measured width for the angle-integrated spectrum with the MC ratio between the widths for the bin i and for the integrated spectrum. The quality of all the fits is satisfactory, with χ 2 /ndf in a range between 0.63 and 1.34. Signal over background ratios in a ±3σ mass window around the CB peak vary between 0.5 and 3.5. The number of signal events per bin ranges from ∼100 (for 2 < p t < 3 GeV/c, 0.6 < | cos θ CS | < 0.8) to ∼1000 (for 2 < p t < 3 GeV/c, 0 < | cos θ CS | < 0.15).
The polarization parameters for the J/ψ were obtained by correcting the number of signal events N i J/ψ for each bin for the product A i × ε i of acceptance times detection efficiency, calculated via MC simulation, J/ψ polarization in pp collisions at √ s=7 TeV 5 and then fitting the corrected angular distributions with the functions shown in Eq. 2. The simulation includes, for the tracking chambers, a map of dead channels and the residual misalignment of the detection elements and, for the trigger chambers, an evaluation of their efficiency based on data. It also includes a random misalignment of the tracking detector elements, of the same size of the resolution obtained by the offline alignment procedure [17]. For both tracking and triggering detectors, the time variation of the efficiencies during the data taking period was accounted for (see [17] for details). Since the cos θ -and φ -acceptances are strongly correlated, the acceptance values as a function of one variable strongly depend on the input distribution used for the other variable. Given the fact that the correct input distributions are not known a priori, but rather represent the outcome of the data analysis, an iterative procedure was followed in order to determine them. In the first iteration a flat distribution of the angular variables (equivalent to a totally unpolarized J/ψ distribution) was adopted to calculate the acceptances. After correcting the signal with those acceptances, a first determination of the polarization parameters is performed, and the results are then used in a second determination of the acceptance values. The procedure is then repeated until convergence is reached, i.e. the extracted polarization parameters do not vary by more than 0.005 between two successive iterations. This occurs, for this analysis, after at most three steps. It was also checked that using polarized MC input distributions in the first iteration the procedure converges towards the same results as in the default, unpolarized, case. Typical A i × ε i values vary between ∼0.22 (0.05) at low p t and large | cos θ | and ∼0.41 (0.63) at large p t and small | cos θ | for the HE (CS) frame.
A simultaneous study of the J/ψ polarization variables in several reference frames, as first carried out in hadroproduction studies by the HERA-B experiment [23], is particularly interesting since consistency checks on the results can be performed, using combinations of the polarization parameters which are frame-invariant. In particular we made use of the invariant F = (λ θ + 3λ φ )/(1 − λ φ ) [21], performing a simultaneous fit of the | cos θ | and |φ | distributions in the two reference systems and further constraining the fit by imposing F to be the same in the CS and HE frames. In Fig. 2 we present, as an example, the result of such a fit relative to the last iteration of the A i ×ε i calculation, for 2 < p t < 3 GeV/c. The χ 2 /nd f values (nd f = 10) are 1.08, 1.00, 1.32 for 2 < p t < 3, 3 < p t < 4 and 4 < p t < 8 GeV/c, respectively, showing that the quality of the fits is good. Compatible results are obtained when the constraint on F is released.
In the analysis described so far, the λ θ φ parameter was implicitly assumed to be zero in the iterative acceptance calculation. In the one-dimensional approach followed in this analysis λ θ φ could be estimated from the data, defining an ad-hoc variableφ which is a function of cos θ and φ and contains λ θ φ as a parameter (see [21] for details). In principle, the iterative procedure applied to λ θ and λ φ determination could be extended to include λ θ φ ; however, in some cases, relatively small statistical fluctuations in the distributions of the measured variables tend to induce large variations of the fitted values in the following iterations, leading to convergence problems. A check of the λ θ φ = 0 assumption was done a posteriori for each p t bin, by fitting theφ distributions, corrected with an acceptance which makes use of the measured λ θ and λ φ values as inputs. In this way we get for all the p t bins λ θ φ values compatible with zero for both CS and HE reference frames. We also note that all the previous experiments assumed λ θ φ = 0 in their analysis, with the exception of HERA-B [23] who measured it in pA collisions at √ s = 41.6 GeV and found values ranging from 0 to 0.05.
Various sources of systematic uncertainty on the measurement of the polarization parameters have been investigated. The uncertainty on the signal extraction was studied by leaving in the fits the width of the CB function as a free parameter. This choice leads to an absolute variation of the polarization parameters between 0.02 and 0.10. Another sizeable source of systematic uncertainty is the choice of the input distributions for p t and y in the simulation. It was evaluated by comparing the results obtained with a parameterization of our 7 TeV results on differential J/ψ cross sections [17] with those obtained using an extrapolation of lower energy results [24]. The absolute effect on the polarization parameters varies between 0.01 and 0.07. For the lowest p t bin, the acceptance in the HE frame drops by about 40% in the highest | cos θ | bin used in the analysis (0.6 < | cos θ | < 0.8), and has also a strong variation inside the bin itself. We therefore followed an alternative approach, fitting the angular spectrum in the restricted interval 0 < | cos θ | < 0.6 (instead of the default choice 0 < | cos θ | < 0.8) and we conservatively considered the variation in the result of the fit (0.15) as an additional systematic uncertainty on λ θ . For consistency, the same evaluation was performed in the CS frame. The role of the systematic uncertainties on the trigger and tracking efficiency [17] was also studied. The first was evaluated by varying the efficiency values for each detector element by 2% with respect to the default values in the simulation. This choice is related to the estimated uncertainty on the detector efficiency calculation. For the second we have used the rather conservative choice of comparing the reference results, obtained with realistic dead channel maps, with those relative to an ideal detector set-up. The result is typically 0.03-0.04. Finally, by quadratically combining the results for the various sources, values between 0.04 and 0.21 are obtained for the global systematic uncertainties. J/ψ polarization in pp collisions at √ s=7 TeV 7 In Fig. 3 we show the results on λ θ and λ φ for inclusive J/ψ production. In both frames all the parameters are compatible with zero, with a possible hint for a longitudinal polarization at low p t (at a 1.6σ level) in the HE frame. The numerical values are given in Table 1.  p t ( p t ) λ θ λ φ (GeV/c) 2-3 (2.5) −0.36 ± 0.09 ± 0.21 0.05 ± 0.04 ± 0.04 HE 3-4 (3.4) −0.20 ± 0.11 ± 0.13 0.01 ± 0.05 ± 0.05 4-8 (5.1) 0.00 ± 0.10 ± 0.10 0.00 ± 0.04 ± 0.04 2-3 (2.5) −0.10 ± 0.14 ± 0.13 −0.04 ± 0.08 ± 0.07 CS 3-4 (3.4) −0.06 ± 0.14 ± 0.07 −0.03 ± 0.08 ± 0.05 4-8 (5.1) −0.09 ± 0.10 ± 0.08 0.03 ± 0.06 ± 0.07 The inclusive J/ψ yield is composed of a "prompt" component (direct J/ψ + decay of the ψ(2S) and χ c resonances) and of a component from B-meson decays. In the p t range accessed in this analysis the B-meson decay component accounts for 10% (2 < p T < 3 GeV/c), 12% (3 < p T < 4 GeV/c) and 15% (4 < p T < 8 GeV/c) of the inclusive yield, according to the LHCb measurements carried out in our same kinematical domain [15]. The polarization of the non-prompt component is expected to be quite small. In fact, even if a sizeable polarization were observed when the polarization axis refers to the B-meson direction [25], it would be strongly smeared when it is calculated with respect to the direction of the decay J/ψ [15], as observed by CDF, who measured in this way λ θ (J/ψ ← B) ∼ −0.1 in the HE frame [5]. Assuming conservatively |λ θ (J/ψ ← B)| < 0.2 for both frames, and taking into account the fraction of the inclusive yield coming from B-meson decays [15], the difference between prompt and inclusive J/ψ polarization was estimated and found to be at most 0.05, a value smaller than the systematic uncertainties of our measurements. Concerning higher-mass charmonia, the χ c → J/ψ + γ decay cannot be reconstructed in the muon spectrometer, and the ψ(2S) → µ µ statistics is currently too low. Values of the feed-down ratios measured mainly by lower energy experiments range from ∼10% for the ψ(2S) [26] to 25-30% for the χ c [27], implying that there could be a sizeable difference between direct and prompt J/ψ polarization.
The results presented in Fig. 3 extend the study of the J/ψ polarization to LHC energies and therefore open up a new testing ground for theoretical models. At present, NLO calculations for direct J/ψ polarization at the LHC via the color-singlet channel [12,10] predict a large longitudinal polarization in the HE frame (λ θ ∼ −0.6) at p t ∼ 5 GeV/c, which is in contrast with the vanishing polarization that we observe in such a transverse momentum region. The contribution of the S-wave color-octet channels was also worked out [9] and indicates a significantly different trend (large transverse polarization) with respect to the color-singlet contribution, but again in contrast with our result. In this situation, a rigorous treatment on the theory side of all the color-octet terms (including P-wave contributions) is mandatory, as well as a study of the contribution of χ c and ψ(2S) feed-down which, as outlined before, is important for a quantitative comparison with our result [28]. Such studies are presently in progress and the comparison of their outcome with the results presented in this Letter will allow a very significant test of the understanding of the heavy-quarkonium production mechanisms in QCD-based models.
In summary, we have measured the polarization parameters λ θ and λ φ for inclusive J/ψ production in √ s = 7 TeV pp collisions at the LHC. The measurement was carried out in the kinematical region 2.5 < y < 4, 2 < p t < 8 GeV/c. The polarization parameters λ θ and λ φ are consistent with zero, in both the helicity and Collins-Soper reference frames. These results can be used as a stringent constraint on the commonly adopted QCD framework for heavy quarkonium production.

Acknowledgements
The ALICE Collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The ALICE Collaboration acknowledges the following funding agencies for their support in building and running the ALICE detector: