Relativistic effect of $J/\psi$ hadroproduction in large $p_T$ region

By combining NRQCD factorization and collinear factorization, we compute a series of relativistic corrections for $J/\psi$ hadroproduction to all orders in $v^2$ at large $p_T$ limit. The $v^2$ expansion converges well for all channels. We find that the ratio of relativistic correction term to the corresponding leading term is independent of kinematic variables for any channel, which generalizes the proportional relations found in previous works to all orders.


Introduction
The study of heavy quarkonium production is important to understand hadronization physics of QCD. Due to the successful performance of the Tevatron and the LHC, the J/ψ hadroproduction receives especial attention recently. The most widely used theory to describe heavy quarkonium production in the past two decades is the non-relativistic QCD (NRQCD) factorization [1]. However, based on it, prediction of prompt J/ψ hadroproduction at leading order (LO) in α s is significantly transversely polarized [2] which contradicts with the measurement by the CDF Collaboration [3,4], where one found that the produced J/ψ is almost unpolarized and even slightly longitudinally polarized as p T increases.
To solve the polarization puzzle, the complete next-to-leading order (NLO) in α s corrections for all important channels ( 3 S [1] 1 , 1 S [8] 0 , 3 S [8] 1 and 3 P [8] J ) have been calculated [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Full NLO prediction can indeed describes both yield and polarization of prompt J/ψ hadroproduction consistently [14,24], but the corresponding color-octet (CO) long-distance matrix elements (LDMEs) are much larger than that needed in J/ψ production at B factories [25][26][27], which challenges the universality of LDMEs. It seems like that still some other important contributions have not been included yet in J/ψ hadroproduction. A possible source may be relativistic effects, as the relative momentum between cc pair is not very small in J/ψ system (v 2 ≈ 0.3). Along this line, a soft gluon factorization was proposed to resum relativistic corrections to all orders [28]. In this paper, we will concentrate on relativistic corrections within NRQCD factorization. For J/ψ hadroproduction, NLO relativistic corrections are calculated in Refs. [29,30], where it was found that, although corrections for 3 S [1] 1 channel are small [29], corrections for CO channels are non-ignorable [30]. As NLO relativistic corrections are important, it is also needed to study the importance of even higher order relativistic corrections to explore all important contributions for J/ψ hadroproduction and to test the convergence of relativistic expansion. Another interesting finding in Refs. [29,30] is that, in large transverse momentum p T limit, the ratio of relativistic correction term to the corresponding leading term is a constant number for each of the four channels. The proportional relation for 3 S [1] 1 channel is well understood [31], but not for other three channels. An understanding of these proportional relations is helpful to illustrate the structure of relativistic effects.
In high p T region, it is more convenient to use collinear factorization framework [32][33][34][35][36][37][38], which has been rigorously proven to leading power (LP) and next-to-leading power (NLP) in p T expansion [37]. As contributions beyond NLP are surely negligible when p T is sufficiently large, this collinear factorization method can capture the main feature in this region. For example, using the collinear factorization, a simple LO calculation can already reproduce NLO NRQCD calculations at large p T [39]. As the proportional relations mentioned in the last paragraph are large p T behavior, they should also be reproducible in the collinear factorization. In fact, we will see that, in large p T limit, relativistic corrections for all the three CO channels can be easily calculated to all orders in v 2 in the framework of collinear factorization and the proportional relations hold to all orders.
The rest of the paper is organized as follows. In Sec. 2, we review the NRQCD factorization and give the explicit large p T relations that were found in Refs. [29,30]. We then review the collinear factorization for heavy quarkonium production in Sec. 3. The combination of NRQCD factorization and collinear factorization is also discussed. Based on these factorizations, we study the relativistic effects of CO J/ψ production via 3 S [8] 1 , 1 S [8] 0 and 3 P [8] channels in Sec. 4 one by one. We summarize our study in Sec. 5. Finally, we provide formulas to project short distance coefficients to definite orbital angular momentum state in Appendix A.

Relativistic corrections within NRQCD factorization
In NRQCD factorization, differential cross section of the J/ψ production at hadron colliders is factorized as [1] where ψ (χ) is field of heavy (anti-)quark in NRQCD effective field theory [40], K n and K † n contain a color matrix, a spin matrix and a polynomial of covariant differential operator D, respectively and P J/ψ is the projection operator of J/ψ with the form where X S includes all soft hadrons. In principle, one needs infinite number of LDMEs to reproduce QCD result. However, as each LDME has a definite power counting in v, not all of them are relevant to finite accuracy [1]. The most relevant LDMEs for J/ψ production . We will also use the following definition in this paper (2.5) To consider the first order relativistic corrections, one needs the following relatively v 2 suppressed LDMEs [29,30], To use NRQCD factorization, one needs to calculate short-distance coefficients dσ i+j→cc[n]+X H in Eq. (2.1). We denote the short-distance coefficient as F (n) and G(n) if the corresponding LDME is O J/ψ (n) and P J/ψ (n) , respectively.
In the rest frame of the cc pair 1 , we denote their momenta in the amplitude as where q is half of the relative momentum between cc pair, E = m 2 + | q| 2 is half of the invariant mass of cc pair. In the complex conjugated amplitude, the relative momentum q can be in principle different from q. Momentum conservation gives p c + p c = p c + p c , which results in E = E and | q | = | q|. Therefore, we do not distinguish E and | q | from E and | q| in the following. It is convenient to define a ratio which is also the same in amplitude and complex conjugated amplitude. In arbitrary frame, with P is the total momentum of cc pair and q is boosted from (0, q) in the rest frame.
To produce a cc[n] state, one should project both color and spin of cc pair to this definite state. There are two kinds of color states with projection operators for singlet and octet respectively. For spin singlet or triplet, one needs the following projection operators Denoting momenta of light partons in initial state and final state as k 1 , k 2 and k 3 respectively, one has Lorentz invariant Mandelstam variables (2.12) 1 We call it rest frame in the rest of this paper for simplicity. with s + t + u = P 2 = 4E 2 . In principle, both F (n) and G(n) are complicated functions of s, t, u, and m. However, in Refs. [29,30], it was found that there are very simple relations in large p T limit (i.e. in the limit that p 2 T ∼ s, t, u m 2 ), where the superscript "(1)" means the ratio of the first order of relativistic corrections to the lowest order results. Among them, relation for 3 S 1 channel is understood in Ref. [31]. In the rest of this paper, we will be devoted to understand the proportional relations for CO channels and generalize them to higher orders.

Collinear factorization for J/ψ production
Production cross section of the J/ψ in the collinear factorization is given by [37,38] where p is the momentum of the observed J/ψ in the final states. The first (second) term on the right-hand side gives the contribution of LP (NLP) in m 2 /p 2 T expansion. D ψ/f (z, m) is the fragmentation function (FF) for J/ψ from a single parton f with momentum fraction z and D ψ/[cc(κ)] (z, ζ, ζ , m) is the charm quark-pair FF. Operator definition of single parton FF can be found in Refs. [32,33,41], and that of double parton FF can be found in Refs. [37,38]. The hard-scattering function dσ i+j→f +X (p f =P /z) (dσ i+j→[cc(κ)]+X (P (1 ± ζ)/2z,P (1±ζ )/2z) ) describes the production of an on-shell light parton (collinear cc pair).
The pair [cc(κ)] in fragmentation function is massive and, in general, is off shell, thus gauge link is needed to ensure gauge invariance. The charm quark pair [cc(κ)] in hard part are massless and moving in the "+z" direction with light-cone momentum componentŝ P µ /z = (P + /z, 0, 0 ⊥ ), whereP µ is a light like momentum whose plus component equals to the J/ψ's plus componentP + = P + J/ψ = n · P J/ψ . κ represents the pair's color and spin, which has two kinds of color states with the projection operators defined in Eq. (2.10) for both fragmentation and hard part, and three kinds of spin states, for effective axial vector (a), vector (v) and tensor (t) "currents", described by relativistic Dirac spin projection operators [37,38] for FFs and for hard parts. The momentum fractions z, ζ and ζ are defined aŝ z measures the fractional momentum of the collinear cc pair carried by J/ψ in this leading region, which is the same on both amplitude side and its complex conjugate side. Both single parton and double parton FFs can be in principle determined by fitting experimental data. However, it will be much powerful if we combine this collinear factorization with NRQCD factorization. Applying NRQCD factorization to J/ψ FFs, all FFs can be expressed in terms of some unknown LDMEs Note the difference that [cc(κ)] is a perturbative QCD state, while cc[n] is a NRQCD state. What was shown above is a two steps factorization: first, using collinear factorization to express the cross section in terms of hard parts and FFs of a heavy quarkonium; then using NRQCD factorization to express these FFs in terms of LDMEs. However, it is more convenient for us to reorganize the factorization as: first, using NRQCD factorization to express the cross section in terms of short-distance coefficients and LDMEs; then using collinear factorization to express the short-distance coefficients in terms of multiplication of hard parts with FFs of cc[n]. Based on the later factorization steps, we can study relations between short-distance coefficients of J/ψ production at large p T limit for each channel directly. Single parton fragmentation functions have been studied extensively . Double parton fragmentation functions have also been calculated to order α s v 0 [67][68][69].
Recall that we are interested in J/ψ production in 3 S 1 channel is dominated by gluon fragmentation and dσ/dp 2 T behaves as p −4 T in large p T limit. 1 S [8] 0 and 3 P [8] channels do not have single particle fragmentation contributions at this order, and their leading contributions come from cc pair fragmentation, which behaves as 1 channel behaves as p −8 T [31], thus it cannot be interpreted in terms of the above formula. In the following sections, we will discuss the three CO channels one by one.
4 Relativistic effect of color-octet J/ψ production The short-distance coefficient for producing a cc[ 3 S [8] 1 ] in large p T limit can be factorized as where i, j and k denote various light partons. At LO in α s , we can denote D cc .

(4.2)
Then we get the ratio where the superscript "(0)" denotes keeping the relative momentum to the lowest order. Therefore, to calculate R( 3 S . (4.4) The cut diagram of gluon fragmentation to cc[ 3 S [8] 1 ] is shown in Fig. 1. The amplitude is where α (h) is the polarization (color) index of cc[ 3 S [8] 1 ]. Considering that P µ and P α can be set to zero here, we have Then using Eq. (A11) to project the above amplitude to S-wave by the replacement q α q µ → Π αµ , we get FF is achieved by squaring the amplitude and summing/averaging over quantum numbers with ∆ = E/m, P αβ = −g αβ + P α P β P 2 and d αβ = −g αβ + P α n β +n α P β P ·n − P 2 n α n β (P ·n) 2 . This result agrees with the calculation in Ref. [28]. Therefore, we get The expansion of R( 3 S where the second term indeed reproduces the result in Eq. (2.13c) of NLO relativistic correction in NRQCD framework. This result tells us that the proportional relation for 3 S 1 channel holds to all orders in velocity expansion. We will see that this property holds also for the other two channels.
To give an estimation of the convergence of velocity expansion in this part, we choose an average value for δ in Eq.(4.10). The expansions of R( 3 S [8] 1 ) are listed in Tab. 1, where δ is chosen as 0.3 for J/ψ and 0.1 for Υ. From Tab. 1, we find that after a few corrections, the expansion can converge to its exact result soon. Obviously, the convergence of Υ system is much faster than that of the J/ψ system. 1 ). δ = | q| 2 /m 2 is chosen to be 0.3 and 0.1 in the second and third row, respectively. [8] 0 channel A cc pair which fragments to cc[ 1 S [8] 0 ] at LO in α s must be in CO. Then, there are three possible channels, a [8] , v [8] and t [8] . The hard part i + j → [cc(t [8] )] + k at LO in α s are helicity suppressed, whose contributions are at higher power in m 2 /p 2 T , thus t [8] channel can be neglected. D cc[ 1 S [8] 0 ]/[cc(v [8] )] vanishes at LO in α s because there are only three independent vectors but a γ 5 exists in the trace of gamma matrices. All in all, the short-distance coefficient for producing a cc[ 1 S [8] 0 ] in large p T limit can be factorized as dσ i+j→cc[ 1 S [8] 0 ]+k = dzdζdζ dσ i+j→[cc(a [8] )]+k (z, ζ, ζ )D cc[ 1 S [8] 0 ]/[cc(a [8] )] (z, ζ, ζ , m) . Let's first calculate the fragmentation function D cc[ 1 S [8] 0 ]/[cc(a [8] )] . The cut diagram for a perturbative QCD cc pair fragments to a NRQCD cc pair is shown in Fig. 2. The amplitude for D cc[ 1 S [8] 0 ]/[cc(a [8] )] is

Relativistic effect in 1 S
where the relation P c = P guaranteed by momentum conservation has been used. It is straightforward to get (4.13) Squaring the amplitude and summing/averaging over quantum numbers, we get D cc[ 1 S [8] 0 ]/[cc(a [8] )] (z, ζ, ζ , m) = z) . (4.14) It was found in Ref. [38] that hard parts have the behavior dσ i+j→[cc(a [8] )]+k (z, ζ, ζ ) = C ijk a + (terms odd in ζ or ζ ), where C ijk a is independent of ζ and ζ . Thus the Eq. (4.11) becomes dσ i+j→cc[ 1 S [8] 0 ]+k = dzdζdζ (C ijk a + terms odd in ζ or ζ ) where in the last equation we suppress terms odd in ζ (or ζ ) which do not contribute to S-wave. Keep to lowest order, one gets and

Axial-vector channel
The amplitude for D cc[ 3 P [8] ]/[cc(a [8]  Squaring the amplitude and summing/averaging over quantum numbers, we get D cc[ 3 P [8] ]/[cc(a [8] )] (z, ζ, ζ , m) = 1 9 which is odd in q and q if we do not consider delta functions. As only terms with odd number of q and q contribute to P-wave and hard parts have the behavior dσ i+j→[cc(a [8] )]+k (z, ζ, ζ ) = C ijk a + (terms odd in ζ or ζ ), we have Using Eq. (A32) to project the above expression to P-wave, we get and (4.24) Therefore, (4.25)

Summation of the two channels
Finally, we get (4.33) In Ref. [38], it is interesting to find that C ijk a = C ijk v , which results in where the second term in the last line indeed reproduces the result in Eq. (2.13d) of NLO relativistic correction for 3 P [8] channel in NRQCD framework, while higher order terms here are new. As an estimate of convergence of perturbation expansion, choosing δ = 0.3 and δ = 0.1, the expansions of R( 3 P [8] ) are listed in Tab. 3, where the convergence is good.  Table 3. Perturbative expansion of R( 3 P [8] ). δ = | q| 2 /m 2 is chosen to be 0.3 and 0.1 in second and third row, respectively.

Summary and outlook
By combining NRQCD factorization and collinear factorization for heavy quarkonium production, we calculate the relativistic correction for J/ψ hadron production to all orders in v 2 at large p T limit. As large p T data are very important to determine CO LDMEs in NRQCD, our calculation should be useful for this purpose. It is interesting to find that the ratio of relativistic correction contribution to the leading contribution R(n) is independent of kinematic variables for all production channels, which generalizes the finding in Ref. [30] to all orders in v 2 expansion. Specifically, for 3 S [8] 1 channel, relative momentum does not flow into hard part after the factorization in Eq. (4.1), thus kinematics dependence in R( 3 S [8] 1 ) cancels between denominator and numerator. For 1 S [8] 0 channel, relative momentum still flows into hard part in terms of ζ and ζ even after the factorization in Eq. (4.11). However, to produce a S-wave cc pair, only a relative momentum independent kinematic configuration C ijk a in hard part contributes, which cancels between denominator and numerator of R( 1 S [8] 0 ). The 3 P [8] channel is even complicated because two terms contribute in the factorization formula in Eq. (4.19). Fortunately, kinematic configurations for the two terms (C ijk a and C ijk v ) are identical, thus they can be canceled in R( 3 P [8] ). By expanding R(n) according to powers of δ = | q| 2 /m 2 , we find O(δ) terms reproduce large p T results in Ref. [30]. By setting δ = 0.3 or δ = 0.1, we find a good convergence for R(n) expansion.
Note that, the calculation in this paper only covers one series of relativistic correction terms. Another series of relativistic correction terms caused by soft gluons emission are also important for J/ψ hadroproduction, which can be taken into account by the soft gluon factorization [28]. Our calculation is also a start point for the use of soft gluon factorization framework.
amplitude to state with orbital angular momentum l and polarization m. In the rest frame of the intermediate cc pair and parameterizing the relative momentum as q µ = (0, q) = | q|(0, sin θ cos φ, sin θ sin φ, cos θ), F lm acts on amplitude A( q) gives where Y m l (θ, φ) are spherical harmonics functions. If there is no L-S coupling, we can use the following projection operator to get unpolarized differential cross section F l (M ( q, q )) = m=±l,±(l−1),··· ,0 which acts on the squared amplitudes. In this work we consider only S-wave (l = 0) and P-wave (l=1), explicit spherical harmonics functions of which are Once the projected results in the rest frame are calculated, results in arbitrary frame can be easily obtained using Lorentz covariance. Specifically, one needs to do the following replacement where k 0 is the "0"-component of a momentum k.

A.1 S-wave
For S-wave, projection operator gives which satisfies Because of parity symmetry, F 0 acts on product of odd number of q vanishes F 0 acts on two q gives dφ (k 1x sin θ cos φ + k 1y sin θ sin φ + k 1z cos θ) × (k 2x sin θ cos φ + k 2y sin θ sin φ + k 2z cos θ) which means In arbitrary frame, it gives Expression in Eqs. (A10) and (A11) can be generalized to any even number of q. Notice that there is no vector in F 0 ( 2N j=1 q i j ) after integration, thus its result must be a symmetric tensor containing only terms like δ ij . Therefore we have where {[δ] N } i 1 ···i 2N means a tensor combination symmetric under i 1 , · · · , i 2N , and each term in the combination has N delta functions. For example, It is easy to find that {[δ] N } i 1 ···i 2N contains (2N − 1)!! terms. Contract both sides of Eq. (A12) with δ i 1 i 2 δ i 3 i 4 · · · δ i 2N −1 i 2N , one can determine its proportionality factor to be | q| 2N (2N +1)!! . Thus we get and In this work, we also need to calculate F 0 ( 1 1−ζ 2 ) with ζ = 2q·n P ·n . It is convenient to calculate it in the rest frame and choose n to only have t and z directions. We get where β = | q| E . (A16)
Generalizing Eqs. (A23), (A30) and (A31) to arbitrary frame and contracting indexes, we get the following explicit expressions that are used in this work,