Heavy Quarkonium Production in Single Transverse Polarized High Energy Scattering
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Heavy Quarkonium Production in Single Transverse PolarizedHigh Energy Scattering
Feng Yuan ∗ Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 andRIKEN BNL Research Center, Building 510A,Brookhaven National Laboratory, Upton, NY 11973
Abstract
We formulate the single transverse spin asymmetry in heavy quarkonium production in lepton-nucleon and nucleon-nucleon collisions in the non-relativistic limit. We find that the asymmetry isvery sensitive to the production mechanism. The final state interactions with the heavy quark andantiquark cancel out among themselves when the pair are produced in a color-single configuration,or cancel out with the initial state interaction in pp scattering when they are in color-octet. As aconsequence, the asymmetry is nonzero in ep collisions only in the color-octet model, whereas in pp collisions only in the color-singlet model. ∗ Electronic address: [email protected] . Introduction. Single-transverse spin asymmetry (SSA) is a novel phenomena inhadronic reactions [1, 2], and has long been observed in various processes. In these pro-cesses, a transversely polarized nucleon scatters off a unpolarized nucleon (or virtual photon)target, and the final observed hadron have asymmetric distribution in the transverse planeperpendicular to the beam direction depending on the polarization vector of the scatteringnucleon. Recent experimental studies have motivated much theoretical developments forunderstanding the underlying physics associated with the SSA phenomena.An important theoretical progress made in the last few years is the uncover of the crucialrole of the initial/final state interactions [3], which leads to a non-vanishing SSA in theBjorken limit in the semi-inclusive hadron production in deep inelastic scattering (SIDIS)and Drell-Yan lepton pair production processes. These initial/final state interactions wereunderstood as a result of the gauge link in the gauge invariant transverse momentum de-pendent (TMD) quark distributions [4, 5, 6], and the SSA can be traced back to a naivetime-reversal-odd distribution, the so-called Sivers function [7]. The difference between theinitial and final state interaction in the above two processes results into a sign change forthe associated quark Sivers function and the SSAs. The Sivers function has been extendedto the gluon sector, and the associated prospects have been investigated in recent years[8, 9, 10, 11, 12, 13].Heavy quark and quarkonium productions are natural probes for the gluon Sivers func-tion [10, 14]. Especially at low transverse momentum, their production will be sensitiveto the intrinsic transverse momentum [15], and can be used to study the gluon Sivers ef-fect. Meanwhile, the heavy quarkonium production has attracted many experimental andtheoretical investigations in the last decade, starting from the anomalous production dis-covered at the Tevatron p ¯ p experiment [16] and a theoretical framework for studying theheavy quarkonium system, the so-called non-relativistic QCD (NRQCD) [17]. The basicargument for the NRQCD factorization is that the heavy quark pair are produced at shortdistance in a color-singlet or color-octet configurations. The hadronization of the pair (insinglet or octet) is described by the associated matrix elements, which can be character-ized according to the velocity expansion [17]. This framework had success in explainingsome experimental observations. Yet, we do not have a definitive answer for the productionmechanism [18, 19, 20, 21].In this paper, we formulate the SSA in heavy quarkonium production by carefully exam-2ning the initial and final state interaction effects. We follow the general arguments of theNRQCD factorization. The SSA depends on the color configuration of the pair producedat short distance, although the hadronization details are not relevant to obtaining a non-zero SSA. Therefore, the experimental study of the SSAs shall shed light on the productionmechanism for the heavy quarkonium. We will focus on the gluon-gluon (photon-gluon)fusion contributions in two processes: one is the lepton-nucleon ep scattering and one isnucleon-nucleon pp scattering, schematically as A ( P A , S ⊥ ) + B ( P B ) → [ Q ¯ Q ] (1 , + X → H + X , (1)where A is the polarized nucleon with momentum P A , B the un-polarized hadron or photonwith momentum P B . The heavy quark pair Q ¯ Q produced at short distance ( ∼ /M Q , M Q the heavy quark mass) can be in a color-singlet (labeled with (1)) or color-octet (labeledwith (8)) configurations, and H represents the final physical quarkonium state. The SSAscoming from the initial/final state interactions are calculated based on the interactionsbetween the quark pair or the incident gluon with the remanet of the polarized target. Weare interested in a kinematic region of low transverse momentum for the heavy quarkonium, P ⊥ ≪ M Q , where the TMD parton distribution is relevant. In the following, we will presenta general analysis for the SSAs coming from these interactions in this kinematic region. Adetailed argument for low P ⊥ factorization in terms of the TMD parton distributions is alsoimportant. Especially, when P ⊥ is the same order of the heavy quark kinetic energy in thequarkonium’s rest frame, the hadronization of the heavy quark pair might interfere with theinitial gluon radiation and break the TMD as well as NRQCD factorization [15, 22]. Wewill come to this important issue in the future.
2. Final state interactions with the quark pair in the non-relativistic limit.
Wefirst discuss how to formulate the final state interactions effects with the quark pair, takingthe lepton-nucleon scattering as an example. In this process, the dominant subprocess isthe photon-gluon fusion channel, where the photon comes from the lepton radiation. Aswe discussed in the introduction, in order to obtain a nonzero Sivers-type SSA, we haveto generate a phase from either the initial or final state interactions. Because the lepton(photon) is colorless, there is no initial state interaction in this process.In Figs. 1 and 2 we plot the generic final state interaction diagrams, for color-singlet andcolor-octet cases, respectively. If the pair are produced in a color-single configuration, a3 ( a ) ( b )( c ) ( d ) c aij jia [ QQ ] (1) [ QQ ] (1) a [ QQ ] (1) [ QQ ] (1) k k k ′ k ′ k , b k , b FIG. 1:
Vanishing SSA in photo(lepto)-production of heavy quarkonium when the heavy quark pairare produced in a color-singlet configuration. The two final state interactions with the quark (a)and antiquark (b) cancel out each other; the final state interactions with the unobserved particlescancel out among different cuts. gluon has to be radiated. In general, we will have final state interactions with the quarkpair (Fig. 1(a,b)) and the radiated gluon (Fig. 1(c,d)). The interaction with the unobservedparticle (here is the radiated gluon) vanishes after we summing different cut diagrams [2, 23].For example, the contribution from Fig. 1(c) is proportional to pole contribution from thepropagator labeled by a short bar with momentum k ′ + k ,1( k ′ + k ) + iǫ δ (( k ′ ) ) | pole = − iπδ (( k ′ + k ) ) δ (( k ′ ) ) , whereas the contribution from Fig. 1(d) will be1( k ′ ) − iǫ δ (( k ′ + k ) ) | pole = + iπδ (( k ′ ) ) δ (( k ′ + k ) ) . Clearly, these two contributions cancel out each other, because the other parts of the scat-tering amplitudes for these two diagrams are the same except the above pole contributionswhich are opposite to each other. The above result is quite general, and shows that all thefinal state interactions with the unobserved particles do not contribute to the SSA for theassociated process. 4herefore, we only need to consider the final state interactions with the quark pair. Twoexample diagrams are shown in Figs. 1(a) and (b). In the non-relativistic limit, the finalstate interaction with the quark in Fig. 1(a) can be derived as follows,Φ( k − P ij )( − ig ) γ ρ T b i k − 6 k + M Q ( k − k ) − M Q + iǫ Γ ≈ g − k +1 + iǫ Φ( k − P ij ) T b Γ , (2)where k and P are momenta for the quark and the quark pair, respectively, Φ representsthe wave function for the pair, ij are color indices for the quark and antiquark, b is thecolor index for the gluon attaching to the quark line, ρ is the index contracted with thegauge potential A ρ , and Γ represents other hard part for the scattering amplitude. Thelight-cone momentum components are defined as k ± = ( k ± k z ) / √
2, and we assume thatthe polarized nucleon is moving along the +ˆ z direction: P A = ( P + A , − , ⊥ ). In the non-relativistic limit, each of the quark and antiquark carries half of the pair’s momentum, k ≈ P/
2, and they are on mass shell: k ≈ M Q . More over, the dominant contribution fromthe gluon interaction with the nucleon remanet comes from the gauge potential A + in thecovariant gauge, and where the gluon momentum is collinear to the polarized nucleon. Inabove derivation, we have also only taken the leading power contributions, and neglected allhigher order correction in terms of 1 /M Q . This derivation shows that we can simplify thefinal state interaction as an eikonal propagator. Similarly, when the gluon attaches to theantiquark, Fig. 1(b), the contribution will be proportional to − g − k +1 + iǫ T b Φ( k − P ij )Γ , (3)where the Γ is the same as above. The minus sign of this result comes from the interactionwith the antiquark. We can combine the contributions from these two diagrams together,and it is proportional to g − k +1 + iǫ (cid:18) Φ( k − P ij ) T b − T b Φ( k − P ij ) (cid:19) Γ . (4)From this, we immediate find that the SSA contribution vanishes if the pair are producedin a color-singlet configuration, for which the color matrix for Φ will be Φ (1) ij ∝ δ ij , and theabove two terms cancel out each other completely. This observation is generic, and onlydepends on the non-relativistic approximation we made and it is valid for higher orders5 j ( b ) ji ( a ) [ QQ ] (8) c [ QQ ] (8) c a ak , b k , b FIG. 2:
Nonzero SSA in ep collisions arise when the heavy quark pair are produced in a color-octetconfiguration. The two final state interactions with the quark (a) and antiquark (b) have net effects. too. Therefore, we conclude that the SSA for heavy quarkonium production in ep scatteringvanishes in the non-relativistic limit in the color-singlet model.However, the above arguments do not hold when the pair are produced in a color-octetconfiguration, for which we will have net effects from the two final state interactions. Weshow these interactions in Fig. 2. Because the pair are in color-octet, the color matrix ofthe wave function can be parameterized as Φ (8 ,c ) ij ∝ T cij where c = 1 , · · · , g − k +1 + iǫ ( − if bcd )Φ (8 ,d ) Γ , (5)where the latter factor is the same hard part in the above scattering amplitude. This resultcan be summarized into a diagram shown in Fig. 3(a), and can be easily extended to atwo-gluon exchange contributions (b), which is proportional to g − k +2 + iǫ g − k +1 − k +2 + iǫ ( − if dce )( − if bef )Φ (8 ,f ) Γ . (6)When it is generalized to all orders, we will find the final state interactions can be summedinto a gauge link associated with the gluon distribution, and the SSA depends on the gluonSivers function, which is the spin-dependent part of the following distribution [25], xG ⊥ T ( x, k ⊥ ) = Z dξ − d ξ ⊥ P + (2 π ) e − ixP + ξ − + i~k ⊥ · ~ξ ⊥ (7) × D P S ⊥ | F + µ ( ξ − , ξ ⊥ ) L † ξ − ,ξ ⊥ L , ⊥ F µ + (0) | P S ⊥ E , where sum over color indices is implicit. F µν is the gluon field strength tensor, F µνa = ∂ µ A νa − ∂ ν A µa − gf abc A µb A νc , x the momentum fraction carried by the gluon, k ⊥ the transversemomentum. L ξ the process-dependent gauge link [4, 5]. For the diagrams in Fig. 3, the6 QQ ] (8) c ( b ) k , b [ QQ ] (8) c b k , d ( a ) k aa FIG. 3:
Summarize the two final state interactions in Fig. 2 into a gauge link associated with thegluon interaction. gauge link sums all the final state interactions with the quark pair (in color-octet state), forwhich we have future pointing gauge link going to + ∞ , L ξ − ,ξ ⊥ → L ∞ ξ = P exp (cid:18) − ig Z ∞ dζ − A + ( ζ − + ξ ) (cid:19) , (8)in the covariant gauge, where A µ = A µc t c is the gluon potential in the adjoint representation,with t cab = − if abc . A transverse gauge link at the spatial infinity is needed to retain thegauge invariance in a singular gauge [5].A potential contribution to the above gluon Sivers function comes from the quark Siversfunction by splitting. Following the calculations in [24], we find that the large k ⊥ gluon Siversfunction can be generated from the twist-three quark-gluon correlation function T F ( x ) [2], G ⊥ T = α s π ǫ αβ S α ⊥ k β ⊥ ( ~k ⊥ ) N c Z dx ′ x ′ (cid:26)(cid:18) x ′ ∂∂x ′ T F ( x ′ , x ′ ) (cid:19) × (1 − ξ ) (cid:2) − ξ ) (cid:3) + T F ( x ′ , x ′ )2(1 − ξ ) (cid:9) , (9)where ξ = x/x ′ and a sum over all quark flavor is implicit. The overall sign of this distributiondepends on the relative contribution from the derivative and non-derivative terms, and theup and down quark contributions.In summary, the SSA in ep collisions vanishes in the color-single model, but survives inthe color-octet model.
3. SSA in pp collisions. Now, we turn to the SSA in heavy quarkonium production in pp collisions. In this process, we have both initial and final state interactions. However, aswe showed above, when the heavy quark pair are produced in a color-singlet configuration,there is no final state interaction, and we only have initial state interaction contribution.We show a typical diagram in Fig. 4(a) in the gluon-gluon fusion subprocess. This diagram7 , b [ QQ ] (1) ( a ) a k , b [ QQ ] (8) c ( b ) a k , b [ QQ ] (8) c ( c ) ad d d FIG. 4:
Only initial state interactions contribute to the SSA in hadron-production process in thecolor-singlet model (a). On the other hand, the SSA vanishes in the color-octet model, because thecancelation between initial (b) and final (c) state interactions. is in particular the dominant channel for χ c production in the color-singlet model [17]. Theinitial state interactions for this diagram can be analyzed, and we find the SSA will dependon the gluon Sivers function defined in Eq. (7), however with a gauge link going to −∞ , L ξ − ,ξ ⊥ → L −∞ ξ = P exp (cid:18) − ig Z −∞ dζ − A + ( ζ − + ξ ) (cid:19) , (10)and the associated SSA will be opposite.However, if the quark pair are produced in a color-octet configuration, we will haveboth initial and final state interactions. After taking into account both contributions, theassociated gauge link becomes, L ξ − ,ξ ⊥ → L −∞ ξ L ∞ ξ , which is responsible for the SSA. However,the contributions from the two gauge links cancel out each other completely. This is becausethe combined gauge link is invariant under time-reversal transformation: the two gauge linkstransform into each other and the combined one remains the same. We have also checkedthis cancelation by an explicit calculation up to two gluon exchange contributions. Thisresult indicates that a standard TMD factorization breaks down for this case, similar torecent discussions on the dijet-correlation in hadronic collisions [26].
4. Summary.
In this paper, we have formulated the single spin asymmetry in heavyquarkonium production in high energy scattering in the non-relativistic limit which shouldbe valid in the limit M Q → ∞ . Very interesting observations were found. The SSA vanishesin ep collision when the pair are produced in a color-singlet configuration, and a nonzeroSSA arises from the color-octet contribution from the gluon Sivers function with the gaugelink pointing to + ∞ . On the other hand, the SSA in pp collisions in the color-single modeldepends on the initial state interactions leading to a gluon Sivers function with gauge link8ointing to −∞ , and the initial and final state interactions cancel out each other in thecolor-octet model and the SSA vanishes to all orders.In our discussions, we only considered the gluon-fusion contributions. The quark channelis also important for heavy quarkonium production, especially in pp collisions at relative lowenergies. In this case, the SSA will depend on the quark Sivers function from the initialstate interaction in the color-single model, which however is opposite to the Drell-Yan SSAbecause of different color-factor, with a relative factor ( − / N c ) /C F . In the color-octetmodel, both initial and final state interactions contribute. At one-gluon exchange order, thefinal state interaction dominate over the initial one, and their total contribution carries anoverall factor − ( N c / / N c ) /C F relative to that in the Drell-Yan process for each quarkflavor contribution. It will be interesting to compare the SSA in these two processes, andscan the energy dependence [14, 27].We thank Xiangdong Ji, Min Liu, Jen-Chiel Peng, Ernst Sichtermann, and Werner Vo-gelsang for useful discussions and encouragements. We especially thank Jian-Ping Ma andJianwei Qiu for their valuable comments and discussions. This work was supported in partby the U.S. Department of Energy under contract DE-AC02-05CH11231. We are gratefulto RIKEN, Brookhaven National Laboratory and the U.S. Department of Energy (contractnumber DE-AC02-98CH10886) for providing the facilities essential for the completion of thiswork. [1] M. Anselmino, A. Efremov and E. Leader, Phys. Rept. , 1 (1995) [Erratum-ibid. , 399(1997)]; Z. t. Liang and C. Boros, Int. J. Mod. Phys. A , 927 (2000); V. Barone, A. Dragoand P. G. Ratcliffe, Phys. Rept. , 1 (2002).[2] J. w. Qiu and G. Sterman, Phys. Rev. Lett. , 2264 (1991); Nucl. Phys. B , 52 (1992).[3] S. J. Brodsky, D. S. Hwang and I. Schmidt, Phys. Lett. B , 99 (2002); Nucl. Phys. B ,344 (2002).[4] J. C. Collins, Phys. Lett. B , 43 (2002).[5] X. Ji and F. Yuan, Phys. Lett. B , 66 (2002); A. V. Belitsky, X. Ji and F. Yuan, Nucl.Phys. B , 165 (2003).[6] D. Boer, P. J. Mulders and F. Pijlman, Nucl. Phys. B , 201 (2003).
7] D. W. Sivers, Phys. Rev. D , 261 (1991).[8] D. Boer and W. Vogelsang, Phys. Rev. D , 094025 (2004).[9] M. Burkardt, Phys. Rev. D , 091501 (2004).[10] M. Anselmino, M. Boglione, U. D’Alesio, E. Leader and F. Murgia, Phys. Rev. D , 074025(2004).[11] M. Anselmino, U. D’Alesio, S. Melis and F. Murgia, Phys. Rev. D , 094011 (2006).[12] S. J. Brodsky and S. Gardner, Phys. Lett. B , 22 (2006).[13] C. J. Bomhof and P. J. Mulders, JHEP , 029 (2007).[14] M. Liu, talk at SPIN 2006 Symposium, Kyoto, 2006.[15] E. L. Berger, J. w. Qiu and Y. l. Wang, Phys. Rev. D , 034007 (2005).[16] F. Abe et al. [CDF Collaboration], Phys. Rev. Lett. , 3704 (1992).[17] G. T. Bodwin, E. Braaten and G. P. Lepage, Phys. Rev. D , 1125 (1995) [Erratum-ibid. D , 5853 (1997)].[18] N. Brambilla et al. [Quarkonium Working Group], arXiv:hep-ph/0412158.[19] G. C. Nayak, J. W. Qiu and G. Sterman, Phys. Lett. B , 45 (2005); Phys. Rev. D ,114012 (2005); Phys. Rev. D , 074007 (2006); arXiv:0707.2973 [hep-ph]; arXiv:0711.3476[hep-ph].[20] Y. J. Zhang, Y. j. Gao and K. T. Chao, Phys. Rev. Lett. , 092001 (2006); Y. J. Zhang andK. T. Chao, Phys. Rev. Lett. , 092003 (2007).[21] J. Campbell, F. Maltoni and F. Tramontano, Phys. Rev. Lett. , 252002 (2007); P. Ar-toisenet, J. P. Lansberg and F. Maltoni, Phys. Lett. B , 60 (2007).[22] We thank J.W. Qiu for bringing us attention on this issue.[23] Y. Koike, W. Vogelsang and F. Yuan, arXiv:0711.0636 [hep-ph].[24] X. Ji, J. W. Qiu, W. Vogelsang and F. Yuan, Phys. Rev. Lett. , 082002 (2006); Phys. Rev.D , 094017 (2006); Phys. Lett. B , 178 (2006).[25] X. Ji, J. P. Ma and F. Yuan, JHEP , 020 (2005).[26] J. Collins and J. W. Qiu, Phys. Rev. D , 114014 (2007); J. Collins, arXiv:0708.4410 [hep-ph];W. Vogelsang and F. Yuan, Phys. Rev. D , 094013 (2007); C. J. Bomhof and P. J. Mulders,arXiv:0709.1390 [hep-ph].[27] M. Bai, et al. , A Proposal submitted to J-PARC experiment., A Proposal submitted to J-PARC experiment.