χ c1 and χ c2 polarization as a probe of color octet channel
χχ c and χ c polarization as a probe of color octetchannel S.P. Baranov , A.V. Lipatov , April 8, 2020 P.N. Lebedev Institute of Physics, Moscow 119991, Russia Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow119991, Russia Joint Institute for Nuclear Research, Dubna 141980, Moscow Region, Russia
Abstract
We analyze the first LHC data on χ c and χ c polarization obtained very recently bythe CMS Collaboration at √ s = 8 TeV. We describe the perturbative production of c ¯ c pairwith k T -factorization approach and use nonrelativistic QCD formalism for the formation ofbound states. We demonstrate that the polar anisotropy of χ c and χ c mesons is stronglysensitive to the color octet contributions. We extract the long-distance matrix elements for χ c and χ c mesons from the first CMS polarization measurement together with availableLHC data on the χ c and χ c transverse momentum distributions (and their ratios) collectedat √ s = 7 TeV. Our fit points to unequal color singlet wave functions of χ c and χ c states.PACS number(s): 12.38.-t, 13.20.Gd, 14.40.Pq1 a r X i v : . [ h e p - ph ] A p r ery recently, the CMS Collaboration reported on the first measurement [1] of the polar-ization of prompt χ c and χ c mesons produced in pp collisions at the energy √ s = 8 TeV.The polarizations were measured in the decay J/ψ helicity frame through the analysis ofthe χ c to χ c yield ratio as a function of the positive muon polar or azimuthal angle in thecascade χ cJ → J/ψ ( → µ + µ − ) + γ in three bins of J/ψ transverse momentum. No differencehas been seen between the χ c and χ c states in the azimuthal distributions, whereas theywere observed to have significantly different polar anisotropies. Thus, at least one of thesemesons should be strongly polarized along the helicity axis [1]. This result contrasts withthe unpolarized scenario observed for direct S -wave charmonia ( J/ψ , ψ (cid:48) ) and bottomoniaΥ( nS ) at the LHC over a wide transverse momentum range (see, for example, [2, 3] andreferences therein).A commonly accepted framework for the description of heavy quarkonia production anddecay is the non-relativistic Quantum Chromodynamic (NRQCD) [4, 5]. The perturbativelycalculated cross sections for the short distance production of a heavy quark pair Q ¯ Q inan intermediate state S +1 L ( a ) J with spin S , orbital angular momentum L , total angularmomentum J , and color representation a are accompanied with long distance matrix elements(LDMEs) which describe the non-perturbative transition of intermediate Q ¯ Q pair into aphysical meson via soft gluon radiation. The NRQCD calculations at next-to-leading order(NLO) successfully describe charmonia J/ψ , ψ (cid:48) , χ cJ [6–13] and bottomonia Υ( nS ), χ bJ ( mP )[14–18] transverse momenta distributions and agree well with the first CMS data [1] on the χ cJ polarization at the LHC. However, NRQCD has a long-standing challenge in the S -wavecharmonia polarization (see, for example, discussions [19–21] and references therein). Thedescription of η c production data [22] reported recently by the LHCb Collaboration alsoturned out to be rather puzzling [23, 24]. So, at present the overall situation is still far fromthrough understanding, and further theoretical studies are still an urgent task.One possible solution has been proposed in [25]. This solution implies certain modificationof the NRQCD rules. Usually, the final state gluons changing the color and other quantumnumbers of quark pair and bringing it to the observed color singlet (CS) state are regardedas carrying no energy-momentum. This is in obvious contradiction with confinement whichprohibits the emission of infinitely soft colored quanta. In reality, the heavy quark systemmust undergo a kind of final state interaction where the energy-momentum exchange mustbe larger than at least the typical confinement scale. Then, the classical multipole radiationtheory can be applied to describe nonperturbative transformations of the color octet (CO)quark pairs produced in hard subprocesses into observed final state quarkonia. In this way,the polarization puzzle for S -wave charmonia [26] and bottomonia [27,28] and the productionpuzzle for η c mesons [29] have been successfully solved. Further on, a good description of the χ c and χ c production cross sections including their relative rates σ ( χ c ) /σ ( χ c ) has beenachieved and the corresponding LDMEs for χ cJ mesons have been determined [26].The main goal of our present note is to extend the approach [25] to the first and verynew CMS data [1] on χ cJ polarization. We propose a method to implement these data intothe LDMEs fit procedure, thus refining the previously extracted LDMEs for χ cJ mesons.Our study sheds light on the role of CO contributions which were unnecessary or evenunwanted [12] for χ cJ p T spectra or their relative rates σ ( χ c ) /σ ( χ c ), but which revealnow in the measured polar anisotropies. To preserve the consistency with our previousstudies [26–29], we follow mostly the same steps and employ the k T -factorization QCD2pproach [30, 31] to produce the c ¯ c pair in the hard parton scattering. The newly addedcalculations are only for the feeddown contributions from ψ (cid:48) radiative decays.For the reader’s convenience, we briefly recall the calculation details. Our considerationis based on the off-shell gluon-gluon fusion subprocess that represents the true leading order(LO) in QCD: g ∗ ( k ) + g ∗ ( k ) → c ¯ c (cid:104) P [1] J , S [8]1 (cid:105) ( p ) (1)for χ cJ mesons with J = 0, 1, 2. The four-momenta of all particles are indicated in theparentheses and the possible intermediate states of the c ¯ c pair are listed in the brackets.The initial off-shell gluons have non-zero transverse momenta k T = − k T (cid:54) = 0, k T = − k T (cid:54) = 0 and, consequently, an admixture of longitudinal component in the polarizationvectors. According to the k T -factorization prescription [31], the gluon spin density matrix istaken in the form (cid:88) (cid:15) µ (cid:15) ∗ ν = k µT k νT k T , (2)where k T is the component of the gluon momentum perpendicular to the beam axis. In thecollinear limit, where k T →
0, this expression converges to the ordinary − g µν after averagingover the gluon azimuthal angle. In all other respects, we follow the standard QCD Feynmanrules. The hard production amplitudes contain spin and color projection operators [32] thatguarantee the proper quantum numbers of the state under consideration. The respectivecross section σ ( pp → χ cJ + X ) = (cid:90) πx x sF f g ( x , k T , µ ) f g ( x , k T ) , µ ) ×× | ¯ A ( g ∗ + g ∗ → χ cJ ) | d k T d k T dy dφ π dφ π , (3)where φ and φ are the azimuthal angles of incoming off-shell gluons carrying the longitudi-nal momentum fractions x and x , y is the rapidity of produced χ cJ mesons, F is the off-shellflux factor [34] and f g ( x, k T , µ ) is the transverse momentum dependent (TMD, or uninte-grated) gluon density function. More details can be found in our previous papers [26–29].Presently, all of the above formalism is implemented into the newly developed Monte-Carloevent generator pegasus [33].As usual, we have tried several sets of TMD gluon densities in a proton. Three of them,namely, A0 [35], JH’2013 set 1 and JH’2013 set 2 [36] have been obtained from Catani-Ciafaloni-Fiorani-Marchesini (CCFM) evolution equation [37], where the input parametriza-tions (used as boundary conditions) have been fitted to the proton structure function F ( x, Q ).Besides that, we have tested a TMD gluon distribution obtained within the Kimber-Martin-Ryskin (KMR) prescription [38, 39], which provides a method to construct the TMD partondensities from conventional (collinear) ones . Following [41], we set the meson masses to m ( χ c ) = 3 .
51 GeV, m ( χ c ) = 3 .
56 GeV, m ( J/ψ ) = 3 .
096 GeV and branching fractions B ( χ c → J/ψγ ) = 33 . B ( χ c → J/ψγ ) = 19 .
2% and B ( J/ψ → µ + µ − ) = 5 . ψ (cid:48) radiative decays, ψ (cid:48) → χ cJ + γ , we set m ( ψ (cid:48) ) = 3 .
69 GeV, B ( ψ (cid:48) → χ c γ ) = 9 . For the input, we have used LO MMHT’2014 set [40]. B ( ψ (cid:48) → χ c γ ) = 9 . pegasus .As it was mentioned above, to determine the LDMEs of χ cJ mesons a global fit to the χ cJ production data at the LHC was performed [26]. The data on the χ c and χ c trans-verse momentum distributions provided by ATLAS Collaboration [42] at √ s = 7 TeV andthe production rates σ ( χ c ) /σ ( χ c ) reported by CMS [43], ATLAS [42] and LHCb [44, 45]Collaborations were included in the fit. Here we extend our previous consideration andincorporate it with the first data [1] on the χ c and χ c polarization collected by CMS Col-laboration at √ s = 8 TeV. In the original CMS analysis, the χ cJ polarization was extractedfrom the (di)muon angular distributions in the helicity frame of the daughter J/ψ meson.The latter is parametrized as dσd cos θ ∗ dφ ∗ ∼
13 + λ θ (cid:0) λ θ cos θ ∗ + λ φ sin θ ∗ cos 2 φ ∗ + λ θφ sin 2 θ ∗ cos φ ∗ (cid:1) , (4)where θ ∗ and φ ∗ are the positive muon polar and azimuthal angles, so that the χ cJ angularmomentum is encoded in the polarization parameters λ θ , λ φ and λ θφ . The ratio of the yields σ ( χ c ) /σ ( χ c ) has been measured as a function of cos θ ∗ and φ ∗ in three different regions of J/ψ transverse momentum, 8 < p T <
12 GeV, 12 < p T <
18 GeV and 18 < p T <
30 GeV,thus leading to a simple correlation between the λ χ c θ and λ χ c θ parameters: λ χ c θ = ( − .
94 + 0 . λ χ c θ ) ± (0 .
51 + 0 . λ χ c θ ) , < p T <
12 GeV , (5) λ χ c θ = ( − .
76 + 0 . λ χ c θ ) ± (0 .
26 + 0 . λ χ c θ ) , < p T <
18 GeV , (6) λ χ c θ = ( − .
78 + 0 . λ χ c θ ) ± (0 .
26 + 0 . λ χ c θ ) , < p T <
30 GeV . (7)Our main idea is to extract the LDME for S [8]1 contributions, O χ c (cid:2) S [8]1 (cid:3) , from the po-larization data, since it can only be poorly determined from the measured χ cJ transversemomentum distributions. To be precise, a good description of the latter can be achieved for awidely ranging O χ c (cid:2) S [8]1 (cid:3) , always with reasonably good χ /d.o.f. (see, for example, [11–13]).Moreover, its zero value is even preferable for the production rate ratio σ ( χ c ) /σ ( χ c ) [12].However, the reported production rates plotted as functions of cos θ ∗ and φ ∗ have free (in-definite) normalization [46] and thus it is difficult to immediately implement them into theLDMEs fitting procedure. Therefore, we had to use the parametrizations (5) — (7) for ourpurposes.Our fitting procedure is the following. First, we performed a fit of the χ c and χ c transverse momentum distributions and their relative production rates σ ( χ c ) /σ ( χ c ) anddetermined the values of CS wave functions of χ cJ mesons at the origin, |R (cid:48) χ c (0) | and |R (cid:48) χ c (0) | , for a (large) number of fixed guessed O χ c (cid:2) S [8]1 (cid:3) values in the range 10 − < O χ c (cid:2) S [8]1 (cid:3) < − GeV . At this step we employ the fitting algorithm implemented in the gnuplot package [47]. Following [48], we considered the CS wave functions as independent(not necessarily identical) free parameters. The reason for such a suggestion is that treatingthe charmed quarks in the potential models as spinless particles could be an oversimpli-fication, and radiative corrections to the CS wave functions could be large [48] and spindependent. Then, we collected the simulated events in the kinematical region defined by theCMS measurement [1] and generated the decay muon angular distributions according to the4roduction and decay matrix elements. By applying a three-parametric fit based on (4), wedetermined the polarization parameters λ χ c θ and λ χ c θ as functions of O χ c (cid:2) S [8]1 (cid:3) (see Fig. 1).We find that the dependence of these parameters on O χ c (cid:2) S [8]1 (cid:3) is essential and thereforecan be used to extract the latter from the data. One can see that χ c and χ c mesons havesignificantly different polar anisotropies, λ χ c θ > λ χ c θ <
0, which smoothly decreasewhen O χ c (cid:2) S [8]1 (cid:3) grows . It is important to remind that each of the considered O χ c (cid:2) S [8]1 (cid:3) values provides already a good fit to the p T spectra: each value of O χ c (cid:2) S [8]1 (cid:3) is associatedwith a respective set of commonly fitted color-singlet LDMEs. Now, using the relations(5) — (7) between λ χ c θ and λ χ c θ (shown by dashed curves in Fig. 1) one can easily extract O χ c (cid:2) S [8]1 (cid:3) for each of the three p T regions. Finally, the mean-square average is taken asthe fitted value. Thus, this provides us with a complementary way to determine the LDMEsfor χ cJ mesons from the polarization data.It is interesting to note that the determined values of O χ c (cid:2) S [8]1 (cid:3) almost do not dependon the exact polarization of S [8]1 contributions in the CO channel. This can be easilyunderstood because χ c and χ c mesons from the S [8]1 intermediate state produce very close J/ψ polarization, while the measured polar asymmetry is driven by the difference λ χ c θ − λ χ c θ . To illustrate it, we have repeated the calculations treating the S [8]1 contributions asunpolarized (yellow curves in Fig. 1). As one can see, the correlations (5) — (7) obtained inthis toy approximation practically coincide with exact calculations.The mean-square average of the extracted O χ c (cid:2) S [8]1 (cid:3) values and the corresponding CSwave functions at the origin |R (cid:48) χ c (0) | and |R (cid:48) χ c (0) | are shown in Table 1 for all testedTMD gluon densities. The relevant uncertainties are estimated in the conventional way usingStudent’s t-distribution at the confidence level P = 95%. For comparison, we also presentthe LDMEs obtained in the NLO NRQCD by other authors [12, 13]. Our fit shows unequalvalues for the χ c and χ c wave functions with the ratio |R (cid:48) χ c (0) | / |R (cid:48) χ c (0) | ∼ |R (cid:48) χ c (0) | / |R (cid:48) χ c (0) | ∼ χ c and χ c production cross sections. So, the χ c production is dominated by the CScontributions, whereas CO terms are more important for χ c mesons (see Fig. 2).All the LHC data involved in the fits are compared with our predictions in Figs. 2 —4. The green shaded bands represent the theoretical uncertainties of our calculations (re-sponding to JH’2013 set 2 gluon density), which include both the scale uncertainties andthe ones coming from the LDMEs fitting procedure. To estimate the scale uncertainties, thestandard variations in the scale (by a factor of 2) were applied through replacing the JH’2013set 2 gluon density with JH’2013 set 2+, or with JH’2013 set 2 − , respectively. This wasdone to preserve the intrinsic correspondence between the TMD set and scale used in theevolution equation (see [36] for more information). We have achieved quite a nice agreementbetween our calculations and available LHC data. In particular, we obtained a simultane-ous description of the transverse momentum distributions and the relative production rates σ ( χ c ) /σ ( χ c ). There are some deviations from the data at low p T region, where, however, The influence of CO contributions on the χ cJ polarization in the collinear scheme has been investigatedin [49]. χ cJ mesons.The results obtained in the NLO NRQCD fits [12, 13] are shown for comparison.Source |R (cid:48) χ c (0) | /GeV |R (cid:48) χ c (0) | /GeV O χ c (cid:2) S [8]1 (cid:3) /GeV A0 0 . ± .
03 0 . ± . . ± . · − JH’2013 set 1 0 . ± .
03 0 . ± .
004 (7 . ± . · − JH’2013 set 2 0 . ± .
04 0 . ± . . ± . · − KMR (MMHT’2014) 0 . ± .
02 0 . ± .
002 (4 . ± . · − NLO NRQCD fit [12] 0 .
35 0 .
35 4 . · − NLO NRQCD fit [13] 0 .
075 0 .
075 2 . · − an accurate treatment of large logarithms ln m ( χ cJ ) /p T and other nonperturbative effects isneeded.The λ χ c θ values extracted according to (5) — (7) when λ χ c θ is fixed to our predictionsare shown on Fig. 5. As one can see, our fit well agrees with the experimentally determinedcorrelations between λ χ c θ and λ χ c θ . The predicted λ χ cJ θ values are practically independenton the TMD gluon density and are close to the reported NLO NRQCD results [1].To conclude, we have considered first LHC data on χ c and χ c polarizations reported veryrecently by the CMS Collaboration at √ s = 8 TeV. We have demonstrated that the polaranisotropy of χ c and χ c mesons is strongly sensitive to the color octet contributions andproposed a method to extract the corresponding LDMEs from the polarization data. Firsttime with the k T -factorization approach, we have determined the color octet LDMEs and thecolor singlet wave functions at the origin |R (cid:48) χ c (0) | and |R (cid:48) χ c (0) | , thus refining our previousresults based on the measured χ cJ transverse momentum distributions only. Our fit pointsto unequal color singlet wave functions of χ c and χ c states with |R (cid:48) χ c (0) | / |R (cid:48) χ c (0) | ∼ χ cJ production atthe LHC, including their transverse momentum distributions, relative production rates andpolarization observables. Acknowledgements.
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12 < p T (J/ ψ ) < 18 GeV|y(J/ ψ )| < 1.2 λ θχ c1 λ θχ c2 λ θ O χ c0 [ S ] [GeV ]calculated S polarizationunpolarized S -1-0.500.511.510 -4 -3
18 < p T (J/ ψ ) < 30 GeV|y(J/ ψ )| < 1.2 λ θχ c1 λ θχ c2 λ θ O χ c0 [ S ] [GeV ]calculated S polarizationunpolarized S Figure 1: Polarization parameters λ χ c θ and λ χ c θ calculated as a functions of O χ c (cid:2) S [8]1 (cid:3) in the helicity frame at | y ( J/ψ ) | < . √ s = 8 TeV in three different p T regions.Solid green and yellow curves represent the results of exact and approximated (when theintermediate S [8]1 state is taken unpolarized) calculations. Dashed curves correspond to thecorrelations (5) — (7) reported by the CMS Collaboration [1]. Everywhere, the JH’2013 set2 gluon density is used. 10 -1
12 14 16 18 20 22 24 26 28 30ATLAS|y(J/ ψ )| < 0.75 B d σ ( χ c ) / dp T [ pb / G e V ] p T ( χ c ) [GeV]A0KMR (MMHT'2014)JH'2013 set 1JH'2013 set 2 -1
12 14 16 18 20 22 24 26 28 30ATLAS|y(J/ ψ )| < 0.75 B d σ ( χ c ) / dp T [ pb / G e V ] p T ( χ c ) [GeV] P [1] S [8]feeddown from ψ (2S) decaysum -1
12 14 16 18 20 22 24 26 28 30ATLAS|y(J/ ψ )| < 0.75 B d σ ( χ c ) / dp T [ pb / G e V ] p T ( χ c ) [GeV]A0KMR (MMHT'2014)JH'2013 set 1JH'2013 set 2 -1
12 14 16 18 20 22 24 26 28 30ATLAS|y(J/ ψ )| < 0.75 B d σ ( χ c ) / dp T [ pb / G e V ] p T ( χ c ) [GeV] P [1] S [8]feeddown from ψ (2S) decaysum Figure 2: The prompt χ c and χ c production cross sections in pp collisions at √ s = 7 TeVas a function of their transverse momenta. On left panels, the predictions obtained withdifferent TMD gluon densities in a proton are presented. On right panels, the contributionsfrom direct P [1] J , S [8]1 and feeddown production mechanisms are shown separately (theJH’2013 set 2 gluon distribution was used for illustration). The experimental data are fromATLAS [42]. -1
10 15 20 25 30ATLAS|y(J/ ψ )| < 0.75 B d σ ( χ c ) / dp T [ pb / G e V ] p T (J/ ψ ) [GeV]A0KMR (MMHT'2014)JH'2013 set 1JH'2013 set 2 10 -1
10 15 20 25 30ATLAS|y(J/ ψ )| < 0.75 B d σ ( χ c ) / dp T [ pb / G e V ] p T (J/ ψ ) [GeV]A0KMR (MMHT'2014)JH'2013 set 1JH'2013 set 2 Figure 3: The prompt χ c and χ c production cross sections in pp collisions at √ s = 7 TeVas a function of decay J/ψ transverse momenta. Notation of all curves is the same as inFig. 2. The experimental data are from ATLAS [42].11 ψ )| < 0.75 B ( χ c ) σ ( χ c ) / B ( χ c ) σ ( χ c ) p T (J/ ψ ) [GeV]A0KMR (MMHT'2014)JH'2013 set 1JH'2013 set 2 ψ )| < 1 B ( χ c ) σ ( χ c ) / B ( χ c ) σ ( χ c ) p T (J/ ψ ) [GeV]A0KMR (MMHT'2014)JH'2013 set 1JH'2013 set 2CMS ψ ) < 4.5 σ ( χ c ) / σ ( χ c ) p T (J/ ψ ) [GeV]A0KMR (MMHT'2014)JH'2013 set 1JH'2013 set 2LHCb 00.511.52 4 6 8 10 12 14 16 18 202 < y(J/ ψ ) < 4.5 σ ( χ c ) / σ ( χ c ) p T (J/ ψ ) [GeV]A0KMR (MMHT'2014)JH'2013 set 1JH'2013 set 2LHCb Figure 4: The relative production rate σ ( χ c ) /σ ( χ c ) calculated as a function of decay J/ψ transverse momentum at √ s = 7 TeV. Notation of all curves is the same as in Fig. 2. Theexperimental data are from ATLAS [42], CMS [43] and LHCb [44, 45]. -1-0.500.511.5 10 15 20 25 30 λ θχ c1 λ θχ c2 |y(J/ ψ )| < 1.2 λ θ p T (J/ ψ ) [GeV]A0KMR (MMHT'2014)JH'2013 set 1JH'2013 set 2 -1-0.500.511.5 10 15 20 25 30 λ θχ c1 λ θχ c2 |y(J/ ψ )| < 1.2 λ θ p T (J/ ψ ) [GeV]NRQCD Figure 5: The λ χ c θ values determined according to correlations (5) — (7) when the λ χ c θθ