Relativistic correction to J=c production at hadron colliders
Ying Fan,* Yan-Qing Ma, and Kuang-Ta Chao
Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China(Received 20 April 2009; published 10 June 2009)
Relativistic corrections to the color-singlet J=c hadroproduction at the Tevatron and LHC are
calculated up to Ov2 in nonrelativistic QCD (NRQCD). The short-distance coefficients are obtainedby matching full QCD with NRQCD results for the subprocess g g! J=c g. The long-distancematrix elements are extracted from observed J=c hadronic and leptonic decay widths up to Ov2. Usingthe CTEQ6 parton distribution functions, we calculate the leading-order production cross sections and
relativistic corrections for the process p pp ! J=c X at the Tevatron and LHC. We find that theenhancement of Ov2 relativistic corrections to the cross sections over a wide range of large transversemomentum pt is negligible, only at a level of about 1%. This tiny effect is due to the smallness of the
correction to short-distance coefficients and the suppression from long-distance matrix elements. These
results indicate that relativistic corrections cannot help to resolve the large discrepancy between leading-
order prediction and experimental data for J=c production at the Tevatron.
DOI: 10.1103/PhysRevD.79.114009 PACS numbers: 12.38.Bx, 12.39.St, 13.85.Ni, 14.40.Gx
Nonrelativistic QCD (NRQCD)  is an effective fieldtheory to describe production and decay of heavy quark-onium. In this formalism, inclusive production cross sec-tions and decay widths of charmonium and bottomoniumcan be factored into short-distance coefficients, indicatingthe creation or annihilation of a heavy quark pair, and long-distance matrix elements, representing the evolvement of afree quark pair into a bound state. The short-distance partcan be calculated perturbatively in powers of couplingconstant s, while the nonperturbative matrix elements,which are scaled as v, the typical velocity of heavy quarkor antiquark in the meson rest frame, can be estimated bynonperturbative methods or models, or extracted fromexperimental data.
One important aspect of NRQCD is the introduction ofthe color-octet mechanism, which allows the intermediateheavy quark pair to exist in a color-octet state at shortdistances and evolve into the color-singlet bound state atlong distances. This mechanism has been applied success-fully to absorb the infrared divergences in P-wave and D-wave [4,5] decay widths of heavy quarkonia. InRef. , the color-octet mechanism was introduced toaccount for the J=c production at the Tevatron, and thetheoretical prediction of production rate fits well withexperimental data. However, the color-octet gluon frag-mentation predicts that the J=c is transversely polarized atlarge transverse momentum pt, which is in contradictionwith the experimental data . (For a review of theseissues, one can refer to Refs. ). Moreover, inRefs. [11,12] it was pointed out that the color-octet long-
distance matrix elements of J=c production may be muchsmaller than previously expected, and accordingly this mayreduce the color-octet contributions to J=c production atthe Tevatron.In the past a couple of years, in order to resolve the large
discrepancy between the color-singlet leading-order (LO)predictions and experimental measurements  ofJ=c production at the Tevatron, the next-to-leading-order(NLO) QCD corrections to this process have been per-formed, and a large enhancement of an order of magnitudefor the cross section at large pt is found [16,17]. But thisstill cannot make up the large discrepancy between thecolor-singlet contribution and data. Similarly, the observeddouble charmonium production cross sections in eeannihilation at B factories [18,19] also significantly differfrom LO theoretical predictions . Much work has beendone and it seems that those discrepancies could be re-solved by including NLO QCD corrections  andrelativistic corrections [25,26]. One may wonder if therelativistic correction could also play a role to some extentin resolving the long standing puzzle of J=c production atthe Tevatron.In this paper we will estimate the effect of relativistic
corrections to the color-singlet J=c production based onNRQCD. The relativistic effects are characterized by therelative velocity vwith which the heavy quark or antiquarkmoves in the quarkonium rest frame. According to thevelocity scaling rules of NRQCD , the matrix elementsof operators can be organized into a hierarchy in the smallparameter v. We calculate the short-distance part pertur-batively up to Ov2. In order to avoid model dependencein determining the long-distance matrix elements, we ex-tract the matrix elements of up to dimension-8 four fermionoperators from observed decay rates of J=c . We findthat the relativistic effect on the color-singlet J=c produc-tion at both the Tevatron and LHC is tiny and negligible,
PHYSICAL REVIEW D 79, 114009 (2009)
1550-7998=2009=79(11)=114009(8) 114009-1 2009 The American Physical Society
and relativistic corrections cannot offer much help to re-solve the puzzle associated with charmonium production atthe Tevatron, and other mechanisms should be investigatedto clarify the problem.
The rest of the paper is organized as follows. In Sec. II,the NRQCD factorization formalism and matching condi-tion of full QCD and NRQCD effective field theory at longdistances are described briefly, and then detailed calcula-tions are given, including the perturbative calculation ofthe short-distance coefficient, the long-distance matrixelements extracted from experimental data, and theparton-level differential cross section convolution withthe parton distribution functions (PDF). In Sec. III, nu-merical results of differential cross sections over transversemomentum pt at the Tevatron and LHC are given anddiscussions are made for the enhancement effects of rela-tivistic corrections. Finally the summary of this work ispresented.
II. PRODUCTION CROSS SECTION IN NRQCDFACTORIZATION
According to NRQCD factorization , the inclusivecross section for the hadroproduction of J=c can be writ-ten as
dtg g! J=c g X
mdn4ch0jOJ=cn j0i: (1)
The short-distance coefficients Fn describe the production
of a heavy quark pair Q Q from the gluons, which comefrom the initial state hadrons, and are usually expressed inkinematic invariants. mc is the mass of charm quark. The
long-distance matrix elements h0jOJ=cn j0i with mass di-mension dn describe the evolution of Q Q into J=c . Thesubscript n represents the configuration in which the c cpair can be for the J=c Fock state expansion, and it is
usually denoted as n 2S1L1;8J . Here, S,L, and J standfor spin, orbital, and total angular momentum of the heavyquarkonium, respectively. Superscript 1 or 8 means thecolor-singlet or color-octet state.
For the color-singlet 3S1 c c production, there are onlytwo matrix elements contributing up toOv2: the leading-order term h0jOJ=c 3S11 j0i and the relativistic correctionterm h0jP J=c 3S11 j0i. Therefore the differential crosssection takes the following form:
dtg g! J=c g F
h0jOJ=c 3S11 j0i
h0jP J=c 3S11 j0i
Ov4; (2)and the explicit expressions of the matrix elements are 
h0jOJ=c 3S11 j0i h0jyic aycac c yij0i;
h0jP J=c 3S11 j0i 0
12yic ayc ac c yi
where c annihilates a heavy quark, creates a heavyantiquark, ayc and ac are operators creating and annihilat-
ing J=c in the final state, and D$ ~DD .
In order to determine the short-distance coefficients
F3S11 and G3S11 , the matching condition of fullQCD and NRQCD is needed:
dtg g! J=c gjpert QCD
h0jOJ=c 3S11 j0i
h0jP J=c 3S11 j0ijpert NRQCD: (4)
The differential cross section for the production of char-monium J=c on the left-hand side of Eq. (4) can becalculated in perturbative QCD. On the right-hand sidethe long-distance matrix elements are extracted from ex-perimental data. Quantities on both sides of the equationare expanded at leading order of s and next-to-leading
order of v2. Then the short-distance coefficients F3S11 and G3S11 can be obtained by comparing the terms withpowers of v2 on both sides.
A. Perturbative short-distance coefficients
We now present the calculation of relativistic correctionto the process g g! J=c g. In order to determine theOv2 contribution in Eq. (2), the differential cross sectionon the left-hand side of Eq. (4) or equivalently the QCDamplitude should be expanded up to Ov2. We useFeynArts  to generate Feynman diagrams and am-plitudes, FeynCalc  to handle amplitudes, andFORTRAN to evaluate the phase space integrations. A typi-
cal Feynman diagram for the process is shown in Fig. 1.
FIG. 1. Typical Feynman diagram for 3S11 c c hadroproductionat LO.
YING FAN, YAN-QING MA, AND KUANG-TA CHAO PHYSICAL REVIEW D 79, 114009 (2009)
The momenta of quark and antiquark in the lab frame are[26,31,32]:
12P q L12Pr qr; 12P q L12Pr qr; (5)
where L is the Lorenz boost matrix from the rest frame ofthe J=c to the frame in which it moves with four momen-
tum P. Pr 2Eq; 0, Eq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2c j ~qj2
p, and 2qr
20; ~q is the relative momentum between heavy quarkand antiquark in the J=c rest frame. The differential crosssection on the left-hand side