In this talk, I briefly review several models of the heavy quarkonium
production at collider energies, and discuss the status of QCD factorization
for these production models.Comment: 7 pages, 12 figures, Talk presented at the Conference on Quark
  Confinement and Hadron Spectrum VII, Ponta Delgada, Portugal, 2-7 September,
  200

Qiu, Jian-Wei

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Jian-Wei Qiu

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QCD Factorization for heavy quarkonium production at collider energies

In this talk, I briefly review several models of the heavy quarkonium production at collider energies, and discuss the status of QCD factorization for these production models

QIU, J.-W.

UNT (University of North Texas) Digital Library

BROO EN NATI 0 M#;r( L. LAB 0 R A T 0  RY BNL-77186-2006-CP QCD Factorization for heavy quarkonium productiolz at collider energies Jian-Wei Qiu'92 'Department of Physics and Astronomy, Iowa State University, Ames, Iowa 5001 1 2Physics Dept., Brookhaven National Laboratory, Upton, NY 1 1973 USA Presented at the International Conference on Quark Confinement and Hadron Spectrum VII Ponta Delgada, Portugal September 2 to 7,2006 Physics Department Nuclear Theory Group Brookhaven National Laboratory Upton, NY 11 973-5000 www. bnl.gov P.O. Box'5000 Notice: This manuscript has been authored by employees of Brookhaven Science Associates, LLC under Contract No. DE-ACOZ 98CH10886 with the U.S. Department of Energy. 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The views and opinions of authors expressed herein do not. necessarily state or reflect those of the United States Government or any agency thereof ' I QCD Factorization for heavy quarkonium production at collider energies Jian-Wei Qiu Department of Physics and Astronomy, Iowa State University Ames, Iowa 50011, U S A .  Physics Department, Brookhaven National Laboratory Upton, New York 1 1973-5000, U.S.A. Abstract. In this talk, I briefly review several models of the.heavy quarkonium .production at collider energies, and discuss the status of QCD factorization for these production models. Keywords: Heavy quarkonium, factorization PACS: 12.38.Bx, 12.393, 13.87.Fh, 14.40Gx INTRODUCTION The production of bound states of heavy quark pairs has been the subject of a vast the- oretical -literature and of intensive experimental study, and offers unique perspectives into the formation of QCD bound states,[l]. The first step inquarkonium production, the inclusive creation of a pair of heavy quarks, is an essentially perturbative process, and takes placetat a distance scale much smaller than the physical size of a quarkonium. Therefore, the transition‘from8the produced heavy quark pairs to bound mesons is un- likely to be instantaneous. The “long” lifetime of the produced pairs allows the dynamics of QCD confinement to evolve, and provides us with a window of opportunities to probe the formation of QCD bound states via the interaction between the pairs and the medium where they.were produced. Much of the predictive content of QCD perturbation theory is contained in factoriza- tion theorems [2]. They allow us to separate long-distance physics from short-distance interactions in hadronic cross sections, and to provide physical content for the uncalcu- lable long-distance quantities, so that they can be measured independently or calculated numerically. In this talk, I first briefly review several models of heavy quarkonium pro- duction, and then, discuss the status of QCD factorization for these production models, and finally, give a brief summary and outlook. PRODUCTION MODELS In order to produce a’heavy quarkonium in hadronic collisions, the energy exchange in the collisions has to be. larger. than the invariant mass of the produced quark -pair (2 2mQ with heavy quark mass m~). ,The pairs should be produced at a distance scale Ar 5 1/2mQ 5 0.1 fm for, charmonia.or 0.025 fm for bottomonia. Since the binding energy of a heavy quarkonium of mass M is much less than heavy quark mass, (M2 - FIGURE 1. Charmonium cross section as a function of p r  along with the CDF data points [7] and the theory curves from NRQCD model from Ref. [6]  (a), and CEM from Ref. [9] (b). 4m;)/4m$ << 1,  the transition from the pair to a meson is sensitive to soft physics. The quantum interference between the production of the heavy quark pairs and the transition process is powerly suppressed by the heavy quark mass, and the production rate f0r.a heavy quarkonium state, H ,  up to corrections in powers of l / m ~ ,  can be factorized as, OA+B+H+X S d r Q Q  OA+B+QQ[n]+X (rQQ> FQQ[n]+H(rQ,> ( 1 )  with a sum over possible Q & z ]  states. and- an integration over available, QQ phase space drQ@ Neglecting the “high twist” interaction between incoming, hadrons and quarkonium formation, the nonperturbative transition probability F for a pair of off- I shell heavy quark (w)  and antiquark (x) to a quarkonium state H is proportional to the Fourier transform of following matrix elements (2) (01%’ (Y 1) %a IY(Y2) IH + N )  ( H  + NI w’ (72 )  Z’X(7l 1 IO) 7 N where yi(j$ are coordinates, and Xn and Xd are local combinations of color and spin I matrices for the QQgstate n. A properlinsertion of Wilson lines to make the operators in Eq. (2) gauge invariant is implicit [3]. In Eq: (2), OA+B,QQ[n]+x represents the production of a pair of on-shell heavy quarks and is calculable in perturbative QCD [4]. The debate on the production mechanism has 4been focusing on the transition from the pair to the meson. Color-singlet model The color-singlet model (CSM) assumes that only a color singlet heavy quark pair with the right quantum number can become a quarkonium of the same quantum number and the transition from the pair to a meson is given by the quarkonium wave function [5]. By neglecting the,dependence on the pair’s relative momentum in OA+B,QQ[n]+x, JdrQQFQ@+H in Eq. (1) is set to equal the matrix element in Eq. (2) evaluated at y i ( 7 i )  = 0, which is proportional to the square of coordinate-space quarko- nium wave function at the origin, lR~(O)12 [ l ] ,  and therefore; I 3 ._ ::: 1 0.2 + a '  0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.6 -0.8 -0.8 -1  - 1  0 5 10 15 20 25 p , ( ~ e ~ )  0 5 10 15 20 25 p , ( ~ e ~ )  (a) (b) FIGURE 2. From Ref. [6] ,  NRQCD predictions of charmonium polarizations are compared with the CDE data [ 111. The same-wave function appears in both production and decay, and the model provides absolutely normalized predictions. It works well for J/v -production in deep inelastic scattering, photon production, and some low energy experiments [l], but fails to predict the CDF data, as demonstrated by the dotted lines in Fig. l(a) [6]. Color evaporation model The color evaporation model (CEM) takes a very different approach to. handle the nonperturbative transition. It assumes that all QQ pairs with invdriant mass less than,the threshold of producing a pair of. open-flavor heavy mesons, regardless their color, spin, and invariant mass, have the same probability to become a quarkonium [8]. That is,-the FQein]+H in Eq. (1) is a constant for a given quarkonium state, H ,  and therefore, with open-flavor heavy meson mass MQ and a constant f H  [8]. With a proper choice of mQ and fH, the model gives a reasonable description of almost all data including the CDF data, as shown in Fig:l(b). Nonrelativistic QCD model The Nonrelativistic QCD (NRQCD) model is based on the fact that the typical heavy quark rest-frame kinetic energy and binding energy, mQv2, in a heavy quarkonium is much smaller than the heavy quark mass. The model separates the physics at scales of order mQ and higher from the dynamics of the binding by using NRQCD, an effective field theory [lo]. It provides a systematic prescription to calculate the physics at mQ order-by-order in powers of a,, and expands the nonperturbative dynamicsin terms of local matrix elements ordered in power series of heavy quark (a). (b) FIGURE 3. Sketch for process A + B H ( p ) . +  X: (a) sample scattering amplitude with possible factorization breaking soft interactions indicated by dashed lines, and (b) factorization with Wilson lines indicated by thin double lines. where a; is the creation operator for H ,  x (w) are two component Dirac spinors, and Xn.and 3; are defined in Eq. (2) and.can also involve covariant derivatives. .At higher orders in v, the.operator. @': can have additional dependence on field strength as well as more fermion4ields. The factorization in Eq. (5) could be understood from Eq. (1) by expanding the OA+B+Q&]+x at heavy quark relative momentum, q = ( p ~  - p ~ ) / 2  = 0. The moments, J d r Q g  8 the probability: is determined by corresponding nonperturbative matrix elements* (@f). It in principle includes thesphysics of CSM. Its octet contribution is the most important onefor high p~ quarkonium production at collider energies [6]. The NRQCD model has been most successful in interpreting data [l, 61, as shown in Fig. l(a). Quarkonium polarization and other models The key difference between the NRQCD model and the CEM is the prediction on quarkonium polarization. Once the matrix el- ements (0:) are determined; the,NRQCD model can systematically calculate the PO- larization-of produced heavy quarkonia. On the other hand, CEM does not provide a systematic prescription to calculate the polarization because of the uncontrolled radi- ations from the quark pair. The polarization of quarkonia at large p~ was considered a definite test of the NRQCD-model [6]. But, the model has failed the test, as seen in Fig. 2, if the CDF data hold [ 111. Several improved and new models have been proposed to address the issues of polarization [12]. lead to local matrix elements with high powers of v.  The NRQCD model allows every QG[n] state to become a bound quarkonium, while - FACTORIZATION The heavy quarkonium production in hadronic collisions involves both perturbative.and nonperturbative scales. Theaonperturbative physics appears not only in the transition from.the heavy quark pair to a bound state but also in incoming hadron wave functions. A typical scattering' amplitude for quarkonium production, as sketched in Fig. 3(a), can have soft and nonperturbative interactions between incoming hadrons as well as between the spectators and-the formation process. These soft interactions may introduce process dependence to the nonperturbative matrix elements, and consequently, spoil the predictive powers of Eqs. (4) and (5). A proof of the‘factorization needs to: 1) show that the square of the scattering- amplitude in Fig. 3(a), after summing over all amplitudes with the same initialsand final states, can.be expressed as a convolution of the probabilities, as sketched in Fig. 3(b); each‘ probability -represents a] square of sub-amplitudes and is evaluated at its own momentum scale@); 2) show that the piece evaluated at perturbative scale(s) is infrared safe and those evaluated at,nonperturbative scales are universal. As argued in Ref. [4], cross sections for producing on-shell heavy quark pairs can be computed’in terms of QCD factorization..Therefore,, the right-hand-side of the CEM formalism in Eq. (4) can be factorized into a convolution of a calculable partonic cross section of producing the heavy quark pair and two universal parton .distributions from ‘respective hadrons. However, the factorization. statement here does not provide justification. that the FQ&++H in Eq. (1) is independent, of. the pair’s invariant mass me&, spin, and other-quantum numbers. For-quarkonium production at a large p ~ ,  the f H  should be universal within the model because soft interactions in Fig. 3(a) are suppressed by powers of , 1 / p ~ .  When p~ << m ~ ,  the f H  may not be universal. Fully .convincing arguments have not yet been given .for NRQCD factorization for- malism in Eq. (5) [6,.10]. Since the interaction between the beam jet and the jet of heavy quark pair. should be- suppressed *by powers of l / p ~ ,  one might expect the NRQCD factorization lformalism to work .at large p ~ :  When p~ >> ITZQ, the- heavy quarkonium production is similar to the single light hadron production, and is dominated byJpar- ton -fragmentation. The cross section is proportional to the universal parton-to-hadron fragmentation functions [3], oA+B+H+X(PT) aA+B+i+X(PT/Z,p)  @ D H / i ( Z , m Q t Y >  f @(mi/&’) - (7) i Here, @ a  represents- a. convolution in the momentum fraction z. The cross section aA+B+i+X includes all information on the incoming state, including convolutions with parton distributions of hadrons A and B at factorizationdscale p, as sketched in Fig. 3(b). As a necessary condition for NRQCD factorization in EQ. (5), the following factorization .. is required to be’valid to all orders in a; and all powers in v-expansion for all parton- to-quarkonium fragmentation%functions [3]. In Eq. (8), diLQ~[nl describes the evolution of an off-shell *parton into a heavy quark pair in state [n], including logarithms of p/mQ,. and should be infrared safe [3]. The factorization relation in Eq. (8) was tested .up to next-to-next-to-leading order ( N O )  in a, at v2 order in Ref. [3], as well as at finite v in Ref. [13]. Consider representative NNLO contributions to the fragmentation process of transforming a color octet heavy quark pair. to. a singlet, as sketched in Fig. 4. The individual classes of diagrams in Fig. 4(I) and (II),,for which% two gluons are exchanged between the quarks and the Wilson line, satisfy the infrared cancellation conjecture of Ref. [lo], by summing over the possible cuts and.connections.to quarkqand antiquark lines, as do diagrams that have three gluon-eikonal vertices on the quarkpair and one on the Wilson line [3]. For Fig. ,4(III) type of diagrams, however, with a three-gluon interaction, this cancellation (1) (11) (In) FIGURE 4. Representative NNLO contributions to g --f QQ fragmentation in eikonal approximation, see Ref. [3] for the details. fails. By summing over all contributions.to second order in the relative momentum q, it was found, that the fragmentation function has a noncanceling real pole [3] (9). From Fig: 4(1II), this infrared divergence is not topologically factorizable and can. not be absorbed into the matrix elements (8';). As demonstrated in Ref. [3], as a necessary condition for-restoring the NRQCD factorization at NNLO, the conventional @(v2) octet NRQCD production matrix elements (8';) must be modified by incorporating Wilson lines thatlmake them manifestly gauge invariant, so that the infrared divergence in Eq. (9) can be absorbed into the gauge-completed matrix elements. For quarkonium production, all  heavy quark pairs with invariant. mass less than a pair of open-flavor- heavy mesons, could become la  bound quarkonium. Therefore, the effective. velocity of. heavyd quark' pair in the production could be much larger than that in decay. For charmonium production, charm quark velocity, vc - l&l/mc 5 (4M2 - 4m;)/(4mc) -0.88, is not. small; and therefore, the velocity expansion for charmonium production may not be a good approximation, unless one can identify and resum .large:contributions to allsorder in v or have a factorized formalism -at finite-v. It was found in Ref. [13] that with the gauge-completed matrix elements, infrared, sin- gularities in.the fragmentation function for a color octet pair to a.singlet at NNLO are consistent-with.NRQCD factorization<to all orders in v or to a finite v, 7where f(v) = 2v/( l.+ v2). The.result in finite v is remarkably compact and intriguing, and should encourage further work on the factorization theorem. SUMMARY AND OUTLOOK After more than 30 years since the discovery of J/y, we still have not been able to fully understand the production. mechanism of 'heavy quarkonium in higli energy collisions, inparticular, the transition from the heavy quark pair to a bound quarkonium. Although t P there are good reasons for- each production model, none of the factorized production formalism, including that of the NRQCD model, has been proved *theoretically. Further work on the’factorization theorem-is critical. The heavy quarkonium: production has a “long” transition time from the produced pairs to bound mesons, and can.be a good probe for the properties of newly formed hot and dense matter of quarks and gluons, or. the quark-gluon plasma, in ultrarelativistic heavy ion collisions [14; 151, if we can calibrate the production. Since the medium in- teraction is sensitive to the quantum state of the proporgating heavy quark pair, nuclear matter could be an effective filter to distinguish the production mechanism [16]: De- tailed studies of nuclear dependence of heavy quarkonium production in hadron-nucleus collisions should provide invaluable information on the formation of heavy quarkonia in hadronic collisions. ACKNOWLEDGMENTS The,author thanks G. T. Bodwin, G. Nayak, and G. Sterman for discussions, and nuclear theory group’at Brookhaven National Laboratory for its hospitality during the %writing of this contribution. This work is supported in part by the U.S. Department of Energy, Grant No. DE-FG02-87ER40371, and Contract No. DE-AC02-98CH10886. REFERENCES 1. N. Brambilla et al. [Quarkonium Working Group], arXiv:hep-ph/0412158. 2. J. C. Collins, D. E. Soper and G. Sterman, Adv. Ser: Direct. High Energy Phys. 5 , l  (1988). 3. G. C. Nayak; J. W. Qiu and G. Sterman, Phys. Lett:B 613,45 (2005); Phys. Rev. D 72,114012 (2005). 4. J. C. Collins, D. E. Soper and G. Sterman, Nucl. Phys. B 263,37 (1986). 5. C. H. Chang, Nucl. Phys. B 172,425 (1980); R. Baier and R. Ruckl, Phys. Lett. B 102, 364 (1981); E. L. Berger andfD. L;. Jones, Phys. Rev. D 23,1521 (1981). 6. E. Braaten, S. Fleming and T. C. *Yuan, Ann. Rev. Nucl. Part. Sci. 46, 197 (1996); M. Kramer, Prog. Part. Nucl. Phys. 47, 141 (2001); G. T. Bodwin, Int. J. Mod. Phys. A 21,785 (2006). 7. F. 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