production at hadron colliders

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  • 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

    I. INTRODUCTION

    Nonrelativistic QCD (NRQCD) [1] 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 [13]and D-wave [4,5] decay widths of heavy quarkonia. InRef. [6], 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 [7]. (For a review of theseissues, one can refer to Refs. [810]). 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 [1315] 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 [20]. Much work has beendone and it seems that those discrepancies could be re-solved by including NLO QCD corrections [2124] 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 [27], 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 [28]. We findthat the relativistic effect on the color-singlet J=c produc-tion at both the Tevatron and LHC is tiny and negligible,

    *ying.physics.fan@gmail.comyqma.cn@gmail.comktchao@th.phy.pku.edu.cn

    PHYSICAL REVIEW D 79, 114009 (2009)

    1550-7998=2009=79(11)=114009(8) 114009-1 2009 The American Physical Society

    http://dx.doi.org/10.1103/PhysRevD.79.114009

  • 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 [1], the inclusivecross section for the hadroproduction of J=c can be writ-ten as

    d

    dtg g! J=c g X

    n

    Fn

    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:

    d

    dtg g! J=c g F

    3S11 m2c

    h0jOJ=c 3S11 j0i

    G3S11 m4c

    h0jP J=c 3S11 j0i

    Ov4; (2)and the explicit expressions of the matrix elements are [1]

    h0jOJ=c 3S11 j0i h0jyic aycac c yij0i;

    h0jP J=c 3S11 j0i 0

    12yic ayc ac c yi

    i2D$2 H:c:

    0;

    (3)

    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:

    d

    dtg g! J=c gjpert QCD

    F3S11 m2c

    h0jOJ=c 3S11 j0i

    G3S11 m4c

    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 [29] to generate Feynman diagrams and am-plitudes, FeynCalc [30] 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)

    114009-2

  • 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 of Eq. (4) is

    d

    dtg g! J=c gjpert QCD

    116s2

    XjMg g! J=c gj2h0jOJ=c 3S11 j0i;(6)

    where h0jOJ=c 3S11 j0i is the matrix element evaluated attree level, and the summation/average of the color andpolarization degrees of freedom for the final/initial state

    has been implied by the symbol P. The amplitude for

    the color-singlet process gp1 gp2 ! J=c p3 P gp4 is

    M g g! J=c g ffiffiffiffiffiffi1

    Eq

    sTrC11Mam; (7)

    whereMam denotes the parton-level amplitude amputated

    of the heavy quark spinors. The factorffiffiffiffi1Eq

    qcomes from the

    normalization of the composite state j3S11 i [5]. Here thecovariant projection operator method [33,34] is adopted.

    For a color-singlet state, the color projector C1 ijffiffiffiffiffiNcp .

    The covariant spin-triplet projector1 in (7) is defined by

    1 Xss

    vs us1

    2; s;

    1

    2; sj1; Sz

    ; (8)

    with its explicit form

    1 1ffiffiffi2p Eq mc

    6P2 6qmc

    6P 2Eq4Eq

    6 6P2 6qmc

    ; (9)

    where the superscript (1) denotes the spin-triplet state and is the polarization vector of the spin 1 meson. The

    Lorentz-invariant Mandelstam variables are defined by

    s p1 p22 p3 p42;t p1 p32 p2 p42;u p1 p42 p2 p32;

    (10)

    and they satisfy

    s t u P2 4E2q 4m2c j ~qj2: (11)Furthermore, the covariant spinors are normalized relativ-istically as uu vv 2mc.Let M be short for the amplitude Mg g! J=c

    g in Eq. (7), and it can be expanded in powers of v orequivalently j ~qj. That is

    M M

    M0 1

    2qq

    @2M

    @q@q

    q0Oq4;

    (12)

    where high order terms in four momentum q have beenomitted. Terms of odd powers in q vanish because theheavy quark pair is in an S-wave configuration. Note thatthe polarization vector also depends on q, but it only has

    even powers of four momentum q, and their expressionsmay be found e.g. in the appendix of Ref. [35]. Thereforeexpansion on q2 of can be carried out after amplitude

    squaring. The following substitute is adopted:

    qq 13j ~qj2

    g P

    P

    P2

    1

    3j ~qj2: (13)

    This substitute should be understood to hold in the inte-gration over relative momentum ~q and in the S-wave case.Here, j ~qj2 can be identified as [33,36]

    j ~qj2 jh0jy i2D

    $2c jc 3S11 ijjh0jyc jc 3S11 ij

    h0jPJ=c 3S11 j0i

    h0jOJ=c 3S11 j0i1Ov4: (14)

    Then the amplitude squared defined in Eq. (6) up to Ov2is

    X jMj2 M0M0X 1

    6j ~qj2

    @2M

    @q@q

    q0M0

    @2M

    @q@q

    q0M0X

    Ov4: (15)The heavy quark and antiquark are taken to be on shell,which means that P q 0, and then gauge invariance ismaintained. The polarization sum in Eq. (15) is

    X

    g

    PP

    P2: (16)

    RELATIVISTIC CORRECTION TO J=c PRODUCTION . . . PHYSICAL REVIEW D 79, 114009 (2009)

    114009-3

  • It is clearly seen that the polarization sum above onlycontains even order powers of four momentum q, thereforeit will make a contribution to the relativistic correction atOv2 in the first term on the right-hand side of Eq. (15)when the contraction over indices and is carried out.However, since the second term on the right-hand side ofEq. (15) already has a term proportional to q2, i.e. j ~qj2, thefour momentum q can be set to zero throughout the index

    contraction. Then we have

    X jMj2 A Bj ~qj2 Ov4; (17)where A and B are independent of j ~qj. By comparingEqs. (4) and (6), we obtain the short-distance coefficientsshown explicitly below. The leading-order one is

    F3S11 m2c

    116s2

    1

    64

    1

    4

    1

    2Nc

    1

    3A

    116s2

    1

    64

    1

    4

    1

    2Nc

    1

    34s35120mc16s2 ts t2m4c 42s3 3ts2 3t2s 2t3m2c

    s2 ts t22=9s 4m2c2t 4m2c2s t2; (18)and the relativistic correction term is

    G3S11 m4c

    116s2

    1

    64

    1

    4

    1

    2Nc

    1

    3B

    116s2

    1

    64

    1

    4

    1

    2Nc

    1

    34s3256020483s2 2ts 3t2m10c 2565s3 2ts2 2t2s 5t3m8c

    3203s4 10ts3 10t2s2 10t3s 3t4m6c 1621s5 63ts4 88t2s3 88t3s2 63t4s 21t5m4c 47s6 18ts5 23t2s4 28t3s3 23t4s2 18t5s 7t6m2c sts ts2 ts t22=27mc4m2c s34m2c t3s t3: (19)

    Each of the factors has its own origin: 1=16s2 isproportional to the inverse square of the Mllers invariantflux factor, 1=64 and 1=4 are the color average andspin average of initial two gluons, respectively, 1=2Nccomes from the color-singlet long-distance matrix elementdefinition in Eq. (3) with Nc 3, 1=3 is the spinaverage for total spin J 1 states, and 4s3 quantifiesthe coupling in the QCD interaction vertices. Further-more the variable u has been expressed in terms of sand t through Eq. (11). To verify our results, we findthat those in Ref. [31] discussed for J=c photoproductionare consistent with ours under replacement 4e2c !4s, and the result in Ref. [37] agrees with oursat leading order after performing the polarizationsummation.

    B. Nonperturbative long-distance matrix elements

    The long-distance matrix elements may be determinedby potential model [25,36] or lattice calculations [38], andby phenomenological extraction from experimental data.Here we first extract the decay matrix elements fromexperimental data. Up to NLO QCD and v2 relativisticcorrections, decay widths of the color-singlet J=c to lighthadrons (LH) and ee can be expressed analytically asfollows [33]:

    J=c ! LH FLH3S11 m2c

    hHjOJ=c 3S11 jHi

    GLH3S11

    m4chHjP J=c 3S11 jHi;

    J=c ! ee Fee3S11

    m2chHjOJ=c 3S11 jHi

    Gee3S11

    m4chHjP J=c 3S11 jHi;

    (20)

    where the short-distance coefficients are [33]

    FLH3S11 N2c1N2c4

    N3c

    2918

    3s2mc

    19:46CF4:13CA1:161Nfs

    2e2QXNfi1

    Q2i

    2e

    113

    4CF

    s

    ;

    GLH3S11 5192132

    7293s2mc;

    Fee3S11 2e2Q

    2e

    3

    14CFs2mc

    ;

    Gee3S11 8e2Q

    2e

    9:

    (21)

    YING FAN, YAN-QING MA, AND KUANG-TA CHAO PHYSICAL REVIEW D 79, 114009 (2009)

    114009-4

  • Then, the production matrix elements can be related to thedecay matrix elements through vacuum saturation approx...

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