I compare current approaches to quarkonium production with regard to what
they tell us about quarkonium polarization. Predictions for $J/\psi$
polarization in hadron-hadron and photon-hadron collisions are summarized.Comment: 6 pages, LaTeX, 3 figures included via epsf.sty, uses sprocl.sty
  (provided

Beneke, M.

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I compare current approaches to quarkonium production with regard to what they tell us about quarkonium polarization. Predictions for $J/\psi$ polarization in hadron-hadron and photon-hadron collisions are summarized.I compare current approaches to quarkonium production with regard to what they tell us about quarkonium polarization. Predictions for $J/\psi$ polarization in hadron-hadron and photon-hadron collisions are summarized

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CERN-TH/97-355hep-ph/9712298December 1997QUARKONIUM POLARIZATION AS A TEST OFNON-RELATIVISTIC EFFECTIVE THEORYM. BENEKETheory Division, CERN, CH-1211 Geneva 23I compare current approaches to quarkonium production with regard to what theytell us about quarkonium polarization. Predictions for J= polarization in hadron-hadron and photon-hadron collisions are summarized.To appear in the Proceedings of theInternational Europhysics Conferenceon High Energy Physics,Jerusalem, Israel, 19-26 August 1997The production of charmonium involves physics at short and long dis-tances. The short-distance part is given, uncontroversially, by the productioncross section for a charm-anticharm-quark pair at small relative velocity v ofthe two quarks. The process through which the cc pair binds into a particu-lar charmonium state is sensitive to long times   1=(mcv2)  1=(500 MeV)and therefore it is non-perturbative. Various descriptions, with more or lesscontact to QCD, and relying on quite dierent physical pictures of this pro-cess, have been proposed, and used, over the years. The colour-singlet model(CSM) 1 assumes that only those cc pairs form J= which are produced ina colour-singlet 3S1 state already at short distances. The long-distance partis Coulomb-binding, accountable for by the wave-function at the origin. Nogluons with energy less than O(mc) in the J= rest frame are emitted. Thecolour-evaporation model (CEM) 2 assumes that soft gluon emission from thecc pair is unsuppressed. The colour and spin quantum numbers of the cc pairat short distances are irrelevant. The long-distance physics is supposed to bedescribed by a phenomenological parameter fJ= , the fraction of ‘open’ cc pairsbelow threshold that bind into J= . The non-relativistic QCD (NRQCD) ap-proach 3 synthesizes elements of both approaches. J= can be produced fromcc pairs in any colour or angular momentum state at short distances but withprobabilities that follow denite scaling rules 4 in v2. Soft gluon emissiondoes take place, but the interaction of soft gluons with the heavy quarks isdetermined by the NRQCD eective Lagrangian. Spin symmetry holds toleading order in v2. There is a price to pay for the more detailed description ofthe long-distance part in NRQCD: It depends on (at least) four (rather thanone) non-perturbative parameters, which have to be extracted from experi-ment. They are hOJ= 1 (3S1)i, hOJ= 8 (3S8)i, hOJ= 8 (1S0)i, hOJ= 8 (3P0)i, wherethe colour and angular momentum state indicated refers to the cc pair at shortdistances. The precise denition of these matrix elements is given in 3.The following deals exclusively with polarization phenomena in J= pro-duction. We discuss predictions for J= production in hadron-hadron andphoton-proton collisions, based on the CSM and the NRQCD approach. Theprediction by the CEM is straightforward and universal: Because the modelassumes that transitions 3S1 $ 1S0 are unsuppressed, we expect that J= isalways produced unpolarized. A polarization measurement has various dis-criminative powers. One can learn to what degree spin-flip transitions are sup-pressed and thereby check the basic assumption that distinguishes the CEMfrom the NRQCD approach. Since the octet production matrix elements ofNRQCD (see above) lead to a polarization pattern dierent from the CSM,one can learn about the importance of colour-octet production mechanisms.In particular, production through a 1S(8)0 state yields unpolarized quarkonium.1See 5;6;7 for the other production channels.J= production in xed target hadron-hadron collisions. Polarizationmeasurements exist for  and  0 production in pion scattering xed target ex-periments 8. Both experiments observe an essentially flat angular distributionin the decay  ! +− ( = J= ;  0),dd cos / 1 +  cos2 ;where the angle  is dened as the angle between the three-momentum vectorof the positively charged muon and the beam axis in the rest frame of thequarkonium. The observed values for  are 0:02  0:14 for  0, measured atps = 21:8 GeV in the region xF > 0:25 and 0:028 0:04 for J= measured atps = 15:3 GeV in the region xF > 0.The colour-singlet contribution alone yields   0:25 for the direct S-waveproduction cross section 9. However, the total cross section is largely due tocolour-octet production. The polarization in the colour-octet channels has beenconsidered in 6 (see also 10). If hO 08 (1S0)i and hO 08 (3P0)i are constrained to bepositive, 0:15 <  < 0:44 is obtained for  0 production atps = 21:8 GeV. Thelower bound is obtained if production through a cc[1S(8)0 ] intermediate statedominates. The analysis of J= polarization is complicated by indirect J= production through c decays, which are not separated in the measurementabove. In Fig. 1 the polar angle parameter  is plotted as a function of rL, thelongitudinal polarization fraction of indirectly produced J= (i.e. rL = 1=3,if c feed-down gives unpolarized J= ). rL is dicult to obtain theoreticallyas c1 and perhaps even c2 is dominantly produced through colour-octetstates, whose polarization yield is described by too many phenomenologicalparameters to be predictable. The wide band in Fig. 1 is obtained by saturatingthe direct J= production cross section by either hOJ= 8 (1S0)i (lower curve) orhOJ= 8 (3P0)i (upper curve). If the indirectly produced J= are unpolarized,one would again have to assume that hOJ= 8 (1S0)i  hOJ= 8 (3P0)i in order toreproduce the data (horizontal band in Fig. 1). A measurement of rL couldclarify the situation.Since the total cross section is dominated by J= production at smalltransverse momentum, non-factorizable nal state interactions may be signi-cant (though formally suppressed) and invalidate the predictions based on theCSM or NRQCD.J= polarization at the Tevatron. At transverse momentum pt  2mcJ= production in hadron-hadron collisions is now regarded as a gluon frag-2polar angle asymmetry λ in ψ→µ+µ-rL-0.5-0.2500.250.50 0.2 0.4 0.6 0.8 1Figure 1: Polar angle asymmetry for J= production in pion-nucleus collisions atps =15 GeV as a function of the longitudinal polarization fraction of indirect J= from radiativefeed-down. The horizontal band shows the measurement of .polar angle asymmetry λ in ψ→µ+µ-CTEQ4LpT (GeV)3S (8)1   only00.20.40.60.814 6 8 10 12 14 16 18 20 22 24Figure 2:  as a function of pt in p+ p ! J= + X at the Tevatron cms energyps = 1:8TeV. From 13.3mentation process g ! cc[3S(8)1 ] ! J= + X11. Since the fragmenting gluonis nearly on-shell, this implies transversely polarized J= as pt ! 1 12 upto spin-symmetry breaking corrections of order v4 5. At nite pt longitudi-nally polarized J= can be produced, if a hard gluon is radiated in the frag-mentation process 5 or the fragmentation approximation is relaxed 13;14. Thenon-fragmentation terms turn out to be particularly important. The predictedpolar angle asymmetry  is shown in Fig. 2. At pt  5 GeV no trace of trans-verse polarization remains. As pt increases, the angular distribution becomesrapidly more anisotropic. The observation of this pattern, even qualitatively,would already constitute strong support for the gluon fragmentation mech-anism and the relevance of spin symmetry in quarkonium production. Thepolarization measurement will therefore rule out either the CEM or the appli-cability of NRQCD velocity power counting at the charmonium scale. Becauseof this, this measurement is probably the single most important one that canbe done in the near future.J= photo-production at HERA. This production process deserves specialattention in the context of NRQCD (and the CEM), since the colour-octetcontributions to the energy distribution of inelastically produced J= seem tobe too large close to the point z = 1 of maximal energy transfer 15. Since theNRQCD velocity expansion is not valid in this endpoint region 16, one wouldlike to infer the relevance of the colour-octet contribution from the data them-selves. Polar and azimuthal J= decay angular distributions may provide aclue to the answer to this problem, since the distributions are predicted to bequite dierent in the CSM and in the NRQCD approach 17. [Recall that allangular distributions are isotropic in the J= rest frame in the CEM.] Theazimuthal dependence, characterized by two additional angular parameters and , is particularly instructive, as a function of both energy fraction z ortransverse momentum as shown in Fig. 3. The shaded band reflects the varia-tion that follows if either one of hOJ= 8 (1S0)i, hOJ= 8 (3P0)i is set to zero, whilethe other saturates the sum which is constrained by the total production rate(in other production processes, primarily hadron-hadron collisions as discussedabove). A measurement of angular distributions in inelastic J= productioncomparable to the measurement of polarization in diractive J= productionat HERA 18 could resolve the controversy whether the measured energy dis-tribution is described by the CSM alone and/or is in conflict with the size ofcolour-octet contributions suggested by the NRQCD approach.In the framework of NRQCD, predictions of J= polarization have alsobeen obtained for B ! J= +X 19, direct J= production in Z0 decay 20 and4ν (Collins-Soper)√sγP = 100 GeV, pT > 1 GeVz-1-0.500.510 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1ν (Collins-Soper)√sγP = 100 GeV, z > 0.1pt-1-0.500.511 2 3 4 5 6 7 8 9 10Figure 3: Azimuthal angle parameter  in the Collins-Soper frame for J= energy andtransverse momentum distributions in J= photo-production at a typical HERA energy.From 17 where more details can be found.5lepto-production of J= 21. Because of lack of space, the reader is referred tothe original papers.Acknowledgements. I would like to thank M. Kra¨mer, I.Z. Rothstein andM. Va¨nttinen for their collaboration on this topic.1. For a review, see G.A. Schuler, CERN-TH-7170-94 [hep-ph/9403387],and references therein.2. H. Fritzsch, Phys. Lett. B67, 217 (1977); F. Halzen, Phys. Lett. B69, 105(1977).3. G.T. Bodwin, E. Braaten and G.P. Lepage, Phys. Rev. D51, 1125 (1995)[Erratum: ibid. D55, 5853 (1997)].4. G.P. Lepage et al., Phys. Rev. D46, 4052 (1992).5. M. Beneke and I.Z. Rothstein, Phys. Lett. B372, 157 (1996) [Erratum:ibid. B389, 789 (1996)].6. M. Beneke and I.Z. Rothstein, Phys. Rev. D54, 2005 (1996) [Erratum:ibid. D54, 7082 (1996)].7. E. Braaten and Y.Q. Chen, Phys. Rev. D54, 3216 (1996).8. J.G. Heinrich et al., Phys. Rev. D44, 1909 (1991); C. Akerlof et al., Phys.Rev. D48, 5067 (1993).9. M. Va¨nttinen et al., Phys. Rev. D51, 3332 (1995).10. W.-K. Tang and M. Va¨nttinen, Phys. Rev. D54, 4349 (1996).11. E. Braaten and S. Fleming, Phys. Rev. Lett. 74, 3327 (1995).12. P. Cho and M.B. Wise, Phys. Lett. B346, 129 (1995).13. M. Beneke and M. Kra¨mer, Phys. Rev. D55, 5269 (1997).14. A.K. Leibovich, Phys. Rev. D56, 4412 (1997).15. M. Cacciari and M. Kra¨mer, Phys. Rev. Lett. 76, 4128 (1996); P. Ko,J. Lee and H.S. Song, Phys. Rev. D54, 4312 (1996).16. M. Beneke, I.Z. Rothstein and M.B. Wise, Phys. Lett. B408, 373 (1997).17. M. Beneke, M. Kra¨mer and M. Va¨nttinen, [hep-ph/9709376].18. H1 Collaboration, S. Aid et al., Nucl. Phys. B472, 3 (1996) and theupdate Paper pa02-085, submitted to the 28th International Conferenceon High Energy Physics, ICHEP’96, Warsaw, Poland, July 1996; ZEUSCollaboration, J. Breitweg et al., Z. Phys. C75, 215 (1997).19. S. Fleming et al., Phys. Rev. D55, 4098 (1997).20. S. Baek et al., Phys. Rev. D55, 6839 (1997).21. S. Fleming and T. Mehen, JHU-TIPAC-96022 [hep-ph/9707365].6

Quarkonium polarization as a test of non-relativistic effective theory

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