Measurement of the inclusive J/ψ polarization at forward rapidity in pp collisions at √s = 8 TeV

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Measurement of the inclusive J/ψ polarization at forward rapidity in pp collisions at √s = 8 TeV © CERN for the benefit of the ALICE collaboration 2018 Published version ALICE Collaboration ALICE Collaboration. (2018). Measurement of the inclusive J/ψ polarization at forward rapidity in pp collisions at √s = 8 TeV. European Physical Journal C, 78(7), Article 562. https://doi.org/10.1140/epjc/s10052-018-6027-2 2018 Eur. Phys. J. C (2018) 78:562 https://doi.org/10.1140/epjc/s10052-018-6027-2 Regular Article - Experimental Physics Measurement of the inclusive J/ψpolarization at forward rapidity in pp collisions at √s=8TeV ALICE Collaboration CERN, 1211 Geneva 23, Switzerland Received: 23 May 2018 / Accepted: 22 June 2018 © CERN for the benefit of the ALICE collaboration 2018 Abstract We report on the measurement of the inclusive J/ψpolarization parameters in pp collisions at a center of mass energy √s=8 TeV with the ALICE detector at the LHC. The analysis is based on a data sample corresponding to an integrated luminosity of 1.23 pb−1.J/ψresonances are reconstructed in their di-muon decay channel in the rapidity interval 2.5<y<4.0 and over the transverse-momentum interval 2 <pT<15 GeV/c. The three polarization parameters (λθ,λϕ,λθϕ) are measured as a function of pTboth in the helicity and Collins-Soper reference frames. The measured J/ψpolarization parameters are found to be compatible with zero within uncertainties, contrary to expectations from all available predictions. The results are compared with the measurement in pp collisions at √s=7TeV. 1 Introduction More than 40 years after the J/ψdiscovery, its production mechanism in hadronic collisions remains an open issue [1]. Quarkonia states constitute an important test bench for the study of Quantum ChromoDynamics (QCD) both in the vacuum and in high-energy density environments, as those produced in heavy-ion collisions, where the creation of the Quark–Gluon Plasma (QGP) is observed [2]. Consequently, the understanding of the J/ψproduction mechanism is an important scientific question in the sense that it addresses basic concepts of QCD, the theory of the strong interaction, and its application to heavy-ion collisions allows the characterisation of the QGP properties created in the laboratory. Different theoretical models have been developed in an attempt to describe the whole production mechanism from partonic interaction to heavy-quark pair (QQ) hadronisation in quarkonia. All approaches are based on the factorisation hypothesis between hard and soft scales. First phenomenological attempts (e.g. the Color Evaporation Model [3]) have e-mail: [email protected] been replaced by a rigorous effective field theory, the NonRelativistic QCD (NRQCD) [4]. In this framework, two models can be derived according to the sub-processes taken into account: the Color-Singlet Model (CSM) [5,6] and the Color-Octet Mechanism (COM) [4]. The CSM assumes no evolution of the quantum color-singlet state between the QQ production and the quarkonium formation, with a wave function computed at zero QQ separation, i.e. without any free parameter. The COM introduces Long-Distance Matrix Elements (LDMEs) for the hadronisation probability in a quarkonium state. The LDMEs are free parameters of the theory which must be fixed from experimental data. Recent measurements at the LHC confirm that color-octet terms are crucial for a good description of the J/ψand ψ(2S) differential production cross sections [7]. However, the failure in predicting the ηcproduction cross section [8,9] poses serious challenges to the NRQCD approach. In this context, alternative measurements at different energies and in different rapidity regions can help to disentangle tensions between quarkonium measurements and the theoretical predictions. One of the most relevant observables apart from the production cross section is the polarization of quarkonia. The polarization of JPC =1−states like the J/ψis specified by three polarization parameters (λθ,λϕ, λθϕ), which are a function of the three decay amplitudes with respect to the three angular momentum states. The two cases (λθ=1, λϕ=0, λθϕ =0) and (λθ=−1, λϕ=0, λθϕ =0) correspond to the so-called transverse and longitudinal polarizations, respectively. Theoretical models at Next-to-Leading Order (NLO) predict strongly transversemomentum dependent polarization states with a partial longitudinal polarization in the CSM and a partial transverse polarization when color-octet contributions are included in the NRQCD calculation [10]. Experimentally, the polarization parameters can be determined in the quarkonium dilepton decay channel by studying the angular distribution (W) of the leptons in the quarkonium rest-frame [11]: 0123456789().: V,-vol 123 562 Page 2 of 16 Eur. Phys. J. C (2018) 78:562 W(cos θ,ϕ) ∝1 3+λθ1+λθcos2θ+λϕsin2θcos(2ϕ) +λθϕ sin(2θ)cos ϕ(1) where θand ϕare the polar and the azimuthal angles, respectively, defining the orientation of one lepton (for instance the negative one) in the quarkonium rest-frame with respect to a reference axis. In the analysis presented here, the selected reference axes are: (1) the helicity axis corresponding to the quarkonium flight direction in the center-of-mass of the colliding beams, and (2) the Collins-Soper axis defined by the direction of the relative velocity of the colliding beams in the quarkonium rest-frame. In the following, the J/ψrest-frame associated to the helicity axis will be referred to as helicity (HX) frame and the one defined from the Collins-Soper axis will be called Collins-Soper (CS) frame. SincethebeginningoftheLHCoperations,thestudyofthe J/ψpolarization in pp collisions has been carried out at √s= 7 TeV both at midrapidity by the CMS [12] experiment, and atforwardrapidity by the ALICE [13] and LHCb [14] experiments. The midrapidity and forward rapidity results are complementary in terms of the explored transverse-momentum (pT) interval, which is 14 <pT<70 GeV/cfor CMS, 2<pT<15 GeV/cfor LHCb and 2 <pT<8GeV/cfor ALICE. In this paper we present the polarization measurement of inclusively-produced J/ψmesons in pp collisions at √s= 8 TeV in the transverse-momentum interval 2 <pT< 15 GeV/c. This is the first measurement of the J/ψpolarization at this energy, and extends the pTreach of the previous ALICE measurement at √s=7TeV[13]. The paper starts with a brief description of the experimental apparatus and the used data sample in Sect. 2, followed by a description of the analysis in Sect. 3, including a discussion of the systematic uncertainties. The results are presented in Sect. 4and compared with those obtained from √s=7 TeV and with model calculations. Conclusions are finally drawn in Sect. 5. 2 Experimental apparatus and data sample The ALICE apparatus and its performance are described in detail in [15] and [16], respectively. In this paper we focus on the two sub-detectors relevant for the analysis: the forward muon spectrometer [17] and the first two layers of the Inner Tracking System (ITS) [18]. The muon spectrometer detects muons in the pseudorapidity range1−4.0<η<−2.5. It consists of five track1Although the muon spectrometer covers negative pseudorapidities (η) in the ALICE reference frame, we use positive rapidity values when referring to the rapidity (y) of quarkonium states reconstructed via their di-muon decay channel. ing stations with two detection planes of multi-wire proportional chambers with cathode pad readout and two trigger stations, each comprising of two detection planes of resistive plate chambers. A set of absorbers completes the system, to decreasethehadronicbackground:the front-absorber(before the first tracking station) reduces the contamination of light hadron decays, a shield surrounding the beam pipe decreases the background from particles produced in the interaction at large pseudorapidity, and an iron wall shields the trigger stations from residual punch through. The momenta of charged tracks are measured with the help of a 3 T·m dipole magnet surrounding the third tracking station. The ITS consists of six layers of silicon detectors with cylindrical geometry surrounding the beam pipe, with radii ranging from 3.9 to 43 cm from the beam axis. This analysis makes use of the two innermost layers that are equipped with Silicon Pixel Detectors (SPD) and cover the pseudorapidity ranges|η|<2 and |η|<1.4forthe firstandthe secondlayer, respectively. The SPD is used to reconstruct the position of the primary vertex of the collision. The data used for this analysis were collected in 2012. The online event selection is based on the opposite-sign di-muon trigger, with a pTthreshold of about 1 GeV/capplied on each muon candidate. This di-muon trigger runs in coincidence with the crossing of two beam bunches at the interaction point. The data sample recorded with this trigger configuration is the same as in [19] and corresponds to an integrated luminosity of about 1.23 pb−1. 3 Analysis Track selection. The opposite-sign di-muon pair candidates are reconstructed with the following track selection criteria (see [19] for details): – the track pseudorapidity must be in the range corresponding to the muon spectrometer acceptance −4<η< −2.5, – the polar angle θabs measured at the rear-end plane of the front absorber must be in the interval 170 <θ abs <178◦, – the maximum allowed value for the pDCA variable, defined as the product of the total momentum pof the track and its distance of closest approach DCA to the primary vertex in the transverse plane, must be less than 6×σpDCA, where the resolution σpDCA is 54 cm·GeV/c for 170 <θ abs <177◦and 80 cm·GeV/cfor 177 ≤ θabs <178◦, – each track reconstructed in the muon tracking system must match a track in the trigger system and in addition must pass the low-pTtrigger threshold of ∼1GeV/c. 123 Eur. Phys. J. C (2018) 78:562 Page 3 of 16 562 Finally, each unlike-sign di-muon pair is required to be in the rapidity interval 2.5<y<4.0. J/ψpolarization formalism. A polarization analysis performed by fitting for each pTinterval the two-dimensional angular distribution of Eq. (1) requires a large reconstructed J/ψsample. In the present analysis, given the limited statistics, the two-dimensional angular distribution is integrated over one angle at a time, to obtain the three following normalised one-dimensional distributions: W1(cos θ) =3N 2(3+λθ)1+λθcos2θ(2) W2(ϕ) =N 2π1+2λϕ 3+λθ cos(2ϕ)(3) W3(ϕ) =N 2π1+√2λθϕ 3+λθ cosϕ(4) with ϕ=ϕ−3 4πfor cos θ<0 and ϕ=ϕ−1 4πfor cos θ>0, while Ncorresponds to the normalisation factor common to the three distributions. Analysis strategy. In order to extract the polarization parameters as a function of pT, the three angular distributions W1(cos θ),W2(ϕ) and W3(ϕ) are built by classifying the dimuon candidates in cos θ,ϕand ϕintervals, respectively, for each pTinterval. The raw number of J/ψmesons is extracted in each interval of pTand angle via a fit of the corresponding invariant mass distribution. The fit is performed in the invariant mass range 2 <Mμ+μ−<5GeV/c2using a variablewidth Gaussian function to describe the background shape and two extended Crystal Ball functions [20] to describe the J/ψand ψ(2S) resonances. The total number of J/ψin the analyzed data sample is about 50,000 in the transverse momentum range 2 <pT<15 GeV/c. The extracted raw yields are then corrected for the acceptance and efficiency of the detector (A×). Acceptance and efficiency evaluation. This is estimated with Monte Carlo (MC) simulations of unpolarized J/ψmesons with pTand rapidity input distributions parameterized from the measured ones at the same energy [19]. Next, the J/ψ mesons are forced to decay into μ+μ−pairs [21], including a fraction (5.4%) of radiative decays μ+μ−γ[22] in agreement with the prediction from [23]. In the simulation, the particles are propagated through the ALICE apparatus using GEANT 3.21 [24] with a realistic description of the detector response. The (A×) factor is calculated in each interval of pTand angle as the ratio of reconstructed J/ψsatisfying the selection criteria to the number of generated J/ψin the rapidity range 2.5<y<4.0. As an example, Fig. 1(left) shows the (A×) map in the plane (cos θ,pT)fortheCS frame. A similar map is obtained in the HX frame, but with a vanishing (A×) in the interval 0.9<|cos θ|<1for 2<pT<15 GeV/c. The maps as a function of ϕand ˜ϕ in both frames do not exhibit any hole in the (A×), as illustrated in Fig. 1(right) in the plane (ϕ,pT)fortheCS frame. Due to the natural symmetry of the angular distributions the analysis is performed in the intervals 0 ≤cos θ≤1, 0≤ϕ≤π 2and 0 ≤ϕ≤π.ThepTinterval explored in this analysis is constrained by a vanishing (A×)atlow pTand high |cos θ|, and by the limited statistics at high pT. The angular distribution intervals for the analysis are defined in order to have a significance2larger than five. The grid in Fig. 1shows the defined pTranges as well as the cos θ(left plot) and ϕ(right plot) intervals in the CS frame. Extraction of the polarization parameters. After acceptance and efficiency correction of the number of reconstructed J/ψ candidates, a simultaneous fit of the three angular distributions is performed by minimizing the following χ2-function for each pTinterval χ2= ncos θ  i=1NJ/ψ i−W1(cos θ;N,λ θ) σi2 + nϕ  j=1NJ/ψ j−W2(ϕ ;N,λ θ,λ ϕ) σj2 + nϕ  k=1NJ/ψ k−W3(ϕ;N,λ θ,λ θϕ) σk2 (5) with four free parameters: the normalization factor Ncommon to the three distributions and the three polarization parameters (λθ,λϕ,λθϕ). In this expression, NJ/ψ i,j,kand σi,j,k are the corrected numbers of J/ψand their associated statistical uncertainties in the ith, jth and kth bins of the angular distributions W1(cos θ),W2(ϕ) and W3(ϕ), with a total number of bins ncos θ,nϕand nϕ, respectively. Figure 2illustrates the fit results of the angular distributions in the HX frame for thetransverse-momentumrange 4 <pT<5GeV/c(similar fits are obtained in all pTintervals and in both frames). Systematic uncertainty evaluation. TheJ/ψsignalisextracted usingfive differentfitting approaches. The initial approachof the invariant mass fit presented above is varied in the following way. The range of the fit is increased to 1.5<Mμ+μ−< 6GeV/c2or decreased to 2.2<Mμ+μ−<4.5GeV/c2. The product of a Gaussian and an exponential is used as an alternative background shape, and finally the two Crystal Ball functions are replaced by the function used by the NA60 Collaboration [20]. For each approach the analysis is 2The significance is defined as S=S/√S+Bwith Sthe number of signal events and Bthe number of background events in the mass range of ±3σaround the J/ψmass peak, σbeing the J/ψmass resolution. 123 562 Page 4 of 16 Eur. Phys. J. C (2018) 78:562 θ cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 ) c (GeV/ T p 0 2 4 6 8 10 12 14 ε×A 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ϕ 0123456 ) c (GeV/ T p 0 2 4 6 8 10 12 14 ε×A 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 1 (A×) 2-D maps in the planes (cosθ,pT)(left)and(ϕ,pT) (right) in the Collins-Soper frame. The plots illustrate the symmetry with respect to cosθ=0 (left) and with respect to ϕ=πand π/2 (right), while the grid shows the binning used to build the W1(cos θ) and W2(ϕ) distributions in each pTrange θ cos 0 0.2 0.4 0.6 0.8 1 ) θ (cos 1 W 0 1000 2000 3000 4000 5000 6000 7000 8000 = 8 TeVsALICE pp -1 = 1.23 pb int L ϕ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ) ϕ ( 2 W 0 1000 2000 3000 4000 5000 6000 7000 8000 Helicity frame c<5 GeV/ T p<4 , 4<y2.5< ϕ ∼ 0 0.5 1 1.5 2 2.5 3 ) ϕ ∼ ( 3 W 0 1000 2000 3000 4000 5000 6000 7000 8000 ψInclusive J/ Simultaneous fit Fig. 2 AcceptancecorrectedangulardistributionsofJ/ψreconstructed in the di-muon decay channel W1(cos θ),W2(ϕ) and W3(ϕ)in the helicity frame for the transverse momentum interval 4 <pT<5GeV/c, together with the results of the simultaneous fit (see text for details). Vertical bars correspond to statistical uncertainties repeatedandthepolarizationparametersaredetermined.The final values of the polarization parameters λα(with α=θ or ϕor θϕ) correspond to the mean values λαof the five sets of results, and the associated statistical uncertainties are the mean values of the statistical uncertainties returned by each fit. The systematic uncertainties on the λαparameters due to the signal extraction are the sum of the quadratic difference of each configuration result with respect to the mean values. The uncertainties range from 0.012 to 0.108 (see Table 1) with the biggest effect observed on λϕin the HX frame. An exhaustive investigation of potential biases in the (A×) map is carried out. Firstly, the input distributions of the J/ψin the MC simulation are modified by: (1) varying the pTand(2)the yshapesof theJ/ψparameterization withinthe uncertainties of the measured cross sections [19], (3) removing the radiative decay part, (4) varying the λθparameter in the range −0.2<λ θ<0.2, corresponding to 1-sigma deviation of the measured value of λθin the HX frame on average over the whole pTinterval. The four corresponding uncertainties on the λαare summed quadratically to get the total simulation input uncertainties ranging from 0.004 to 0.175. This is the main systematic uncertainty for the λθparameter, dominated by the variation of its input value in simulation, especially at low pT. Secondly, any uncertainty in the simulation of the trigger threshold of ∼1GeV/ccould bias the (A×) estimation. To evaluate this effect, the full simulation of the trigger response function is replaced with a parameterization of the trigger response function as a function of transverse momenta. The parameterization is obtained both in MC and in data by using minimum bias events recorded in parallel with the triggered data sample. The analysis is then repeated using either of the two parameterizations, and the 123 Eur. Phys. J. C (2018) 78:562 Page 5 of 16 562 Table 1 Absolute systematic uncertainties on J/ψpolarization parameters in the HX and CS frames. The four different uncertainty sources are the signal extraction (signal), the input distributions of the J/ψin the MC simulations (inputs), the low-pTtrigger response (trigger) and the detector efficiency (efficiency). The last three sources enter in the computation of the acceptance and efficiency factor (A×) Source λHX θλHX ϕλHX θϕ λCS θλCS ϕλCS θϕ Signal 0.035–0.087 0.021–0.108 0.014–0.032 0.022–0.074 0.022–0.052 0.012–0.063 Inputs 0.076–0.155 0.007–0.024 0.006–0.033 0.013–0.175 0.006–0.040 0.004–0.018 Trigger 0.001–0.064 0.001–0.060 0.005–0.020 0.006–0.036 0.007–0.070 0.006–0.017 Efficiency 0.076–0.133 0.046–0.069 0.064–0.076 0.081–0.121 0.058–0.072 0.073–0.081 resulting difference is taken as the systematic uncertainty. The effect is small (<0.022) for pT>4GeV/c, and a maximum uncertainty of 0.070 is estimated for λϕin the first pTinterval of the CS frame. Thirdly, the uncertainty on the detector efficiency includes the uncertainty on the tracking efficiency, the trigger chamber efficiency and the matching between tracks reconstructed in the tracker and in the trigger system. The resulting uncertainty on the J/ψyields is evaluated with the same procedure as the one described in [25] and is propagated to the corrected yields of the angular distributions by adding it in quadrature with the statistical ones. Finally, the fits are redone and the associated uncertainty on λαparameters is estimated as the square root of the quadratic differencebetweenthenewuncertaintyreturnedbythefitand the statistical one. Its value ranges from 0.046 to 0.133. This is the main uncertainty for the λθϕ parameter. The different sources of systematic uncertainties are summarized in Table 1. The four sources of systematics are independent and can be summed in quadrature to obtain the total systematic uncertainty on each λαparameter. Systematic uncertainties are considered uncorrelated among the three polarization parameters and among the pTintervals. 4 Results The inclusive J/ψpolarization parameters in the interval 2.5<y<4.0 and 2 <pT<15 GeV/cmeasured in pp collisions at √s=8 TeV are shown in Fig. 3for the HX (right) and the CS (left) frames and summarized in Tables 2 and 3, respectively. In the figure, the error bars represent the total uncertainties computed by adding in quadrature the statistical and systematic uncertainties. This is the first measurement of the J/ψpolarization parameters at this energy and extends the pTreach of the previous ALICE measurement at √s=7 TeV from 8 to 15 GeV/c. The results show that the polarization of inclusive J/ψmesons is compatible with zero within uncertainties, with a maximum deviation of 1.8 standard deviations away from zero for the highest pT interval for the λθand λθϕ parameters in the HX frame. As the differences between the J/ψpolarization in pp collisions at √s=7 TeV and 8 TeV are expected to be negligible (see Kniehl et al. predictions in Ref. [14] and in this paper), the measurements at the two energies can be directly compared. This comparison is shown in Fig. 3 with the published results by ALICE [13] (inclusive J/ψ) and LHCb [14] (prompt J/ψ, i.e. without the contribution from b-hadron decays) in the same rapidity interval for pp collisions at √s=7 TeV. The two ALICE measurements agree within one standard deviation. Concerning the comparison between ALICE and LHCb results, a rather good agreement is observed for all polarization parameters over the full pTinterval. The observed agreement between the ALICE and LHCb results seems to indicate that J/ψfrom b-hadron decays do not introduce any observable difference in the polarization parameters. Figure 4shows the comparison of all the measured polarization parameters with the NLO CSM (blue filled band) and NRQCD (red shaded band) predictions from [10] and with another NRQCD (light blue hatched band) prediction from [26]forλθin the helicity frame (labeled as NLO NRQCD2 in Fig. 4). The shown error bands of the models are evaluated by adding in quadrature the uncertainties due to the different scale variations (renormalization, factorization and NRQCD scales) in the calculation and LDME variations. The difference between the two NRQCD calculations originates from the data used to compute the LDMEs. Moreover, in [10] only direct J/ψ(i.e. without feed-down from excited states) are considered, while in [26] feed-down from excited states is included in the J/ψprediction. The CSM and NRQCD calculations from [10] predict an opposite pTtrend for all polarization parameters in the two frames. The pTdependence is relatively small over the considered pTinterval, except for the λθparameter in the HX frame. The NRQCD calculation including both color-singlet and color-octet contributions provides a qualitatively better description of the J/ψpolarization measurement, except for λθin the HX frame where the large transverse J/ψpolarization predicted by the NRQCD [10] is in contradiction with the experimental observations. The NRQCD prediction from [26] favours either zero or small longitudinal polarization, with large theoretical uncertainties, and shows a good agreement with the measurements in the intermediate pT interval (5 <pT<15 GeV/c), but gives no prediction 123 562 Page 6 of 16 Eur. Phys. J. C (2018) 78:562 )c (GeV/ T p 02468101214 ϕθ λ -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 ϕ λ 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 θ λ 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Collins-Soper -1 = 1.23 pb int = 8 TeV, LsALICE pp = 7 TeVsALICE pp = 7 TeVsLHCb pp )c (GeV/ T p 2468101214 0 Helicity ψInclusive J/ < 4y2.5 < Fig. 3 ALICE inclusive J/ψpolarization parameters in pp collisions at √s=8 TeV (black points) compared with ALICE [13] inclusive J/ψ (orange squares, shifted horizontally by −0.3GeV/c)andLHCb[14] prompt J/ψ(blue open diamonds, shifted horizontally by +0.3GeV/c) measurements at √s=7 TeV in the rapidity interval 2.5<y<4.0. The error bars represent the total uncertainties. Left and right plots show results in the Collins-Soper and helicity frames, respectively, for λθ(top plots), λϕ(middle plots) and λθϕ (bottom plots) Table 2 Inclusive J/ψ polarization parameters in the HX frame in the rapidity interval 2.5<y<4.0. The first uncertainty is statistical and the second systematic pT(GeV/c)λHX θλHX ϕλHX θϕ 2–3 0.035 ±0.048 ±0.215 −0.037 ±0.025 ±0.093 −0.024 ±0.032 ±0.082 3–4 −0.085 ±0.053 ±0.189 −0.065 ±0.026 ±0.134 −0.080 ±0.035 ±0.077 4–5 0.083 ±0.066 ±0.188 −0.003 ±0.033 ±0.096 −0.024 ±0.043 ±0.080 5–7 −0.036 ±0.058 ±0.154 0.055 ±0.029 ±0.069 −0.001 ±0.039 ±0.078 7–10 −0.092 ±0.078 ±0.168 0.090 ±0.039 ±0.056 0.089 ±0.055 ±0.082 10–15 −0.329 ±0.121 ±0.130 −0.003 ±0.070 ±0.052 0.222 ±0.099 ±0.079 123 Eur. Phys. J. C (2018) 78:562 Page 7 of 16 562 Table 3 Inclusive J/ψ polarization parameters in the CS frame in the rapidity interval 2.5<y<4.0. The first uncertainty is statistical and the second systematic pT(GeV/c)λCS θλCS ϕλCS θϕ 2–3 0.002 ±0.046 ±0.228 −0.030 ±0.024 ±0.095 0.041 ±0.032 ±0.076 3–4 −0.011 ±0.052 ±0.185 −0.065 ±0.026 ±0.098 −0.075 ±0.035 ±0.084 4–5 0.001 ±0.056 ±0.124 −0.019 ±0.030 ±0.086 0.006 ±0.041 ±0.080 5–7 0.063 ±0.048 ±0.088 −0.020 ±0.031 ±0.087 −0.042 ±0.041 ±0.082 7–10 0.175 ±0.070 ±0.096 0.001 ±0.045 ±0.082 −0.009 ±0.060 ±0.096 10–15 −0.021 ±0.110 ±0.106 −0.052 ±0.084 ±0.077 −0.065 ±0.110 ±0.098 )c (GeV/ T p 02468101214 ϕθ λ -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 ϕ λ 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 θ λ 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Collins-Soper -1 = 1.23 pb int = 8 TeV, LsALICE pp < 4y: 2.5 < ψInclusive J/ )c (GeV/ T p 2 4 6 8 10 12 14 0 Helicity NLO CSM NLO NRQCD NLO NRQCD2 Fig. 4 Inclusive J/ψpolarization parameters in pp collisions at √s= 8 TeV (black points, error bars represent the total uncertainties) compared with model predictions: NLO CSM [10] (blue filled bands), NRQCD [10] (red shaded bands) and NRQCD2 [26] (light blue hatched band). Left and right plots show the results in the Collins-Soper and helicity frames, respectively, for λθ(top plots), λϕ(middle plots) and λθϕ (bottom plots) 123 562 Page 8 of 16 Eur. Phys. J. C (2018) 78:562 Fig. 5 Inclusive J/ψ frame-invariant quantity λin pp collisions at √s=8TeVinthe Collins-Soper (red points, shifted horizontally by −0.1GeV/c) and helicity (green squares, shifted horizontally by +0.1GeV/c) frames compared with the NLO CSM (blue full band) and NRQCD (red shaded band) model predictions from [10] )c (GeV/ T p 02468101214 λ ∼ -1 -0.5 0 0.5 1 1.5 ψ<4, inclusive J/y = 8 TeV, 2.5<sALICE pp Helicity frame Collins-Soper frame NLO CSM NLO NRQCD for pT<5GeV/c. This agreement is not surprising because this model includes the measurements of the J/ψpolarization performed at Tevatron [27,28] to determine the LDMEs. As this model gives no prediction for the other polarization parameters in the HX frame, as well as for the whole set of polarization parameters in the CS frame, it is difficult to draw a clear conclusion about its ability to describe the measurements. As shown by Faccioli et al. [11], frame-invariant observables do exist and the most commonly considered one is  λ=λθ+3λϕ 1−λϕ .(6) Figure 5shows the pTdependence of this invariant quantity for both frames in comparison with the NLO CSM and NRQCD predictions from [10]. To propagate the uncertainties on λθand λϕto the frame-invariant quantity λ, the correlation coefficient ρλθ,λϕreturned by the simultaneous fit of the angular distributions are taken into account to compute the statistical uncertainties, while the systematic uncertainties are assumed to be uncorrelated. For the model predictions, the quoted error bands are computed by adding the uncertainties due to the different scales and LDME variations in quadrature, after propagation of the correlated effects between λθand λϕ. The comparison of the frame-invariant quantity  λshows that the ALICE measurements in both frames are in good agreement within uncertainties, confirming the consistency of the results. Both the CSM and the NRQCDmodelrespecttheframe invariance for λ,but clearly none of them is able to describe the measured pTdependence, even if the NRQCD prediction shows a better agreement with data (χ2 /NDF =1.7 compared to χ2 /NDF =2.0 by CSM), although with large uncertainties especially for pT<6GeV/c. Table 4 Average pT-integrated (over 2 <pT<15 GeV/cin the rapidity range 2.5<y<4.0) inclusive J/ψpolarization parameters λθ,λϕand λθϕin the HX and CS frames Parameter HX frame CS frame λθ−0.006 ±0.115 0.012 ±0.116 λϕ−0.024 ±0.058 −0.036 ±0.053 λθϕ−0.029 ±0.047 −0.006 ±0.047 Using the ALICE inclusive J/ψcross section measurement at √s=8TeV[19], an average value for the polarization parameters over pTcan be computed in the following way λα= 1 σtot 6  j=1 σjλj α,(7) with σtot = 6  j=1 σj.(8) In these equations, jis running over the six pTbins of this analysis, σjis the integrated inclusive J/ψcross section in the pTbin jand λj αis the measured polarization parameter in the corresponding bin. The resulting average values of the polarization parameters over 2 <pT<15 GeV/care summarized in Table 4. The uncertainties are computed by propagating the total uncertainty on the polarization parameters and the uncorrelated uncertainty on the cross section measurements from [19]. All averaged values of the polarization parameters are consistent with zero within uncertainties. 123 Eur. Phys. J. C (2018) 78:562 Page 15 of 16 562 50 Indian Institute of Technology Indore, Indore, India 51 Indonesian Institute of Sciences, Jakarta, Indonesia 52 INFN, Laboratori Nazionali di Frascati, Frascati, Italy 53 INFN, Sezione di Bari, Bari, Italy 54 INFN, Sezione di Bologna, Bologna, Italy 55 INFN, Sezione di Cagliari, Cagliari, Italy 56 INFN, Sezione di Catania, Catania, Italy 57 INFN, Sezione di Padova, Padova, Italy 58 INFN, Sezione di Roma, Rome, Italy 59 INFN, Sezione di Torino, Turin, Italy 60 INFN, Sezione di Trieste, Trieste, Italy 61 Inha University, Incheon, Republic of Korea 62 Institut de Physique Nucléaire d’Orsay (IPNO), Institut National de Physique Nucléaire et de Physique des Particules (IN2P3/CNRS), Université de Paris-Sud, Université Paris-Saclay, Orsay, France 63 Institute for Nuclear Research, Academy of Sciences, Moscow, Russia 64 Institute for Subatomic Physics, Utrecht University/Nikhef, Utrecht, Netherlands 65 Institute for Theoretical and Experimental Physics, Moscow, Russia 66 Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovakia 67 Institute of Physics, Bhubaneswar, India 68 Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic 69 Institute of Space Science (ISS), Bucharest, Romania 70 Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany 71 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico 72 Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil 73 Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico 74 iThemba LABS, National Research Foundation, Somerset West, South Africa 75 Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik, Fachbereich Informatik und Mathematik, Frankfurt, Germany 76 Joint Institute for Nuclear Research (JINR), Dubna, Russia 77 Korea Institute of Science and Technology Information, Daejeon, Republic of Korea 78 KTO Karatay University, Konya, Turkey 79 Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS-IN2P3, Grenoble, France 80 Lawrence Berkeley National Laboratory, Berkeley, CA, USA 81 Lund University Department of Physics, Division of Particle Physics, Lund, Sweden 82 Nagasaki Institute of Applied Science, Nagasaki, Japan 83 Nara Women’s University (NWU), Nara, Japan 84 School of Science, Department of Physics, National and Kapodistrian University of Athens, Athens, Greece 85 National Centre for Nuclear Research, Warsaw, Poland 86 National Institute of Science Education and Research, HBNI, Jatni, India 87 National Nuclear Research Center, Baku, Azerbaijan 88 National Research Centre Kurchatov Institute, Moscow, Russia 89 Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark 90 Nikhef, National institute for subatomic physics, Amsterdam, Netherlands 91 NRC Kurchatov Institute IHEP, Protvino, Russia 92 NRNU Moscow Engineering Physics Institute, Moscow, Russia 93 Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, UK 94 Nuclear Physics Institute of the Czech Academy of Sciences, ˇ Rež u Prahy, Czech Republic 95 Oak Ridge National Laboratory, Oak Ridge, TN, USA 96 Petersburg Nuclear Physics Institute, Gatchina, Russia 97 Physics Department, Faculty of science, University of Zagreb, Zagreb, Croatia 98 Physics Department, Panjab University, Chandigarh, India 99 Physics Department, University of Jammu, Jammu, India 100 Physics Department, University of Rajasthan, Jaipur, India 123 562 Page 16 of 16 Eur. Phys. J. C (2018) 78:562 101 Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen, Germany 102 Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany 103 Physik Department, Technische Universität München, Munich, Germany 104 Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany 105 Rudjer Boškovi´c Institute, Zagreb, Croatia 106 Russian Federal Nuclear Center (VNIIEF), Sarov, Russia 107 Saha Institute of Nuclear Physics, Kolkata, India 108 School of Physics and Astronomy, University of Birmingham, Birmingham, UK 109 Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru 110 Shanghai Institute of Applied Physics, Shanghai, China 111 Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria 112 SUBATECH, IMT Atlantique, Université de Nantes, CNRS-IN2P3, Nantes, France 113 Suranaree University of Technology, Nakhon Ratchasima, Thailand 114 Technical University of Košice, Košice, Slovakia 115 Technische Universität München, Excellence Cluster ’Universe’, Munich, Germany 116 The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland 117 The University of Texas at Austin, Austin, TX, USA 118 Universidad Autónoma de Sinaloa, Culiacán, Mexico 119 Universidade de São Paulo (USP), São Paulo, Brazil 120 Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil 121 Universidade Federal do ABC, Santo Andre, Brazil 122 University College of Southeast Norway, Tonsberg, Norway 123 University of Cape Town, Cape Town, South Africa 124 University of Houston, Houston, TX, USA 125 University of Jyväskylä, Jyväskylä, Finland 126 University of Liverpool, Department of Physics Oliver Lodge Laboratory, Liverpool, UK 127 University of Tennessee, Knoxville, TN, USA 128 University of the Witwatersrand, Johannesburg, South Africa 129 University of Tokyo, Tokyo, Japan 130 University of Tsukuba, Tsukuba, Japan 131 Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France 132 Université de Lyon, Université Lyon 1, CNRS/IN2P3, IPN-Lyon, Villeurbanne, Lyon, France 133 Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France 134 IRFU, Department de Physique Nucléaire (DPhN), Université Paris-Saclay Centre dÉtudes de Saclay (CEA), Saclay, France 135 Università degli Studi di Pavia, Pavia, Italy 136 Università di Brescia, Brescia, Italy 137 V. Fock Institute for Physics, St. Petersburg State University, St. Petersburg, Russia 138 Variable Energy Cyclotron Centre, Kolkata, India 139 Warsaw University of Technology, Warsaw, Poland 140 Wayne State University, Detroit, MI, USA 141 Institut für Kernphysik, Westfälische Wilhelms-Universität Münster, Münster, Germany 142 Wigner Research Centre for Physics, Hungarian Academy of Sciences, Budapest, Hungary 143 Yale University, New Haven, CT, USA 144 Yonsei University, Seoul, Republic of Korea aDeceased bDipartimento DET del Politecnico di Torino, Turin, Italy cM.V. Lomonosov Moscow State University, D.V. Skobeltsyn Institute of Nuclear, Physics, Moscow, Russia dDepartment of Applied Physics, Aligarh Muslim University, Aligarh, India eInstitute of Theoretical Physics, University of Wroclaw, Poland 123