# Higher-order relativistic corrections to gluon fragmentation into spin-triplet -wave quarkonium

###### Abstract

We compute the relative-order- contribution to gluon fragmentation into quarkonium in the color-singlet channel, using the nonrelativistic QCD (NRQCD) factorization approach. The QCD fragmentation process contains infrared divergences that produce single and double poles in in dimensions. We devise subtractions that isolate the pole contributions, which ultimately are absorbed into long-distance NRQCD matrix elements in the NRQCD matching procedure. The matching procedure involves two-loop renormalizations of the NRQCD operators. The subtractions are integrated over the phase space analytically in dimensions, and the remainder is integrated over the phase-space numerically. We find that the order- contribution is enhanced relative to the order- contribution. However, the order- contribution is not important numerically at the current level of precision of quarkonium-hadroproduction phenomenology. We also estimate the contribution to hadroproduction from gluon fragmentation into quarkonium in the color-octet channel and find that it is significant in comparison to the complete next-to-leading-order-in- contribution in that channel.

###### Keywords:

quarkonium, fragmentation, NRQCD, relativistic corrections1208.5301 \preprintANL-HEP-PR-12-65

## 1 Introduction

Heavy-quarkonium production in hard-scattering collisions has a long and rich history of experimental measurements and theoretical calculations Brambilla:2010cs . Intense efforts in this area are expected to continue as the Large Hadron Collider (LHC) makes available data with unprecedented momentum transfers and statistics.

In recent years, a great deal of theoretical effort has been focused on the nonrelativistic QCD (NRQCD) factorization approach BBL to calculations of quarkonium production rates. In this approach, it is conjectured that the inclusive quarkonium production cross section at large transverse momentum () can be written as a sum of products of short-distance coefficients and long-distance matrix elements (LDMEs):

(1) |

Here, is the factorization scale, which is the cutoff of the effective field theory NRQCD. A short-distance coefficient is, in essence, the partonic cross section to produce a heavy-quark-antiquark () pair with certain quantum numbers, convolved with parton distribution functions. The short-distance coefficients can be calculated as perturbation series in the strong-coupling constant . A production LDME is the probability for a pair with certain quantum numbers to evolve into a particular heavy-quarkonium state. It is expressed as the vacuum expectation value of a four-fermion operator

(2) |

where and are two-component (Pauli) fields that
create a heavy quark and a heavy antiquark, respectively, and
and are combinations of Pauli and color matrices.^{1}^{1}1
It was pointed out by Nayak, Qiu, and Sterman that gauge invariance
requires that the definitions of the NRQCD LDMEs include
Wilson lines that run from the quark and antiquark fields to infinity
Nayak:2005rw ; Nayak:2005rt . For simplicity, we have omitted these
Wilson lines here.

(3) |

is a projection onto a state consisting of a quarkonium , with four-momentum , plus anything. contains a sum over any quarkonium polarization quantum numbers that are not specified explicitly. The NRQCD LDMEs are evaluated in the rest frame of the quarkonium, in which , where is the quarkonium mass. In the remainder of this paper, we suppress the momentum argument of in NRQCD LDMEs.

The production LDMEs for the evolution of color-singlet pairs into a quarkonium state are related to the color-singlet quarkonium decay LDMEs. These color-singlet production LDMEs can be determined from comparison of theory with quarkonium production or decay data or from lattice QCD calculations. However, the production LDMEs for the evolution of color-octet pairs into a quarkonium state can be determined, at least at present, only through comparison of theory with experimental quarkonium-production data.

Complete calculations of short-distance coefficients in the NRQCD factorization approach now exist through next-to-leading order (NLO) in for production of the and the in collisions, in collisions, and in and collisions Brambilla:2010cs ; Butenschoen:2009zy ; Ma:2010yw ; Butenschoen:2010rq ; Butenschoen:2010px ; Ma:2010jj ; Butenschoen:2011yh ; Butenschoen:2011ks ; Butenschoen:2012px ; Chao:2012iv ; Butenschoen:2012qh ; Gong:2012ug . These calculations include the contributions from all of the color-octet channels through relative order , as well as the contribution of the color-singlet channel at leading order (LO) in . Specifically, the calculations include the contributions of the , and color-octet channels and the color-singlet channel at the leading nontrivial order in in each channel. Here, is half the relative velocity of the heavy quark and the heavy antiquark in the quarkonium rest frame. for the , and for the . These theoretical results are generally compatible with experimental measurements of quarkonium production cross sections. However, significant discrepancies remain between theoretical predictions for quarkonium polarization and experimental measurements Chao:2012iv ; Butenschoen:2012qh ; Gong:2012ug . These discrepancies might point to as-yet-uncalculated theoretical contributions, to experimental difficulties, to a failure of convergence of the NRQCD series in or , or to a failure of the NRQCD factorization conjecture itself.

The calculations at NLO in have revealed very large corrections in that order to the color-octet channel and the color-singlet channel. Large corrections have also been found in a calculation of the real-emission contributions to hadroproduction at next-to-next-to-leading order in Artoisenet:2008fc . The large corrections are the result of kinematic enhancements of the higher-order cross sections at high relative to the LO cross sections. The sizes of these corrections have cast some doubt on the convergence of the perturbation series.

It has been suggested recently that the large higher-order corrections to quarkonium production can be brought under control by re-organizing the perturbation series according to the behavior of the various contributions Kang:2011zza . In this approach, the cross section can be shown to factorize into convolutions of hard-scattering cross sections with fragmentation functions. The factorization holds up to corrections of relative order , where is the charm-quark mass. The factorized cross section consists of a leading contribution, which arises from single-particle fragmentation into a quarkonium and falls as in the partonic cross section and a first subleading contribution, which arises from two-particle fragmentation into a quarkonium and falls as in the partonic cross section. If NRQCD factorization holds, then the various fragmentation functions can be expressed in terms of a sum of products of short-distance coefficients and NRQCD LDMEs. This picture has been shown to account for the large corrections at NLO in in the color-singlet channel Kang:2011mg .

In this paper, we compute the NRQCD short-distance coefficient for gluon fragmentation into a color-singlet pair in relative order . The short-distance coefficients for gluon fragmentation in this channel have been computed in relative order Braaten:1993rw ; Braaten:1995cj and relative order Bodwin:2003wh . In both cases, the contributions are not important phenomenologically. Nevertheless, it is worthwhile to consider the order- contribution for two reasons. First, this contribution is interesting theoretically because it is in order that the color-singlet fragmentation channel first develops soft divergences in full QCD. As we shall see, these soft divergences in full QCD correspond to soft divergences in the LDMEs for the and color-octet channels and cancel in the short-distance coefficients, as is required by NRQCD factorization. A second motivation for examining the order- contribution is that it is potentially large. Contributions from gluon fragmentation into the and color-octet channels are known to be important phenomenologically. The color-singlet channel mixes with these channels in order , and the partitioning of the various contributions is controlled by single and double logarithms of the factorization scale. Therefore, it is plausible that the order- contributions to the color-singlet channel could be large.

Our method of calculation is based on the Collins-Soper definition Collins:1981uw of the fragmentation function for a gluon fragmenting into a quarkonium. We assume that NRQCD factorization holds, that is, that the fragmentation function can be decomposed into a sum of products of short-distance coefficients and NRQCD LDMEs. We then compute the full-QCD fragmentation functions for a gluon fragmenting into free states with various quantum numbers. The ultimate aim is to match these full-QCD fragmentation functions to the corresponding NRQCD fragmentation functions in order to determine the NRQCD short-distance coefficients. Some additional details of this approach can be found in Ref. Bodwin:2003wh .

This paper is organized as follows. We give the Collins-Soper definition of the fragmentation function in Sec. 2. Section 3 contains the NRQCD factorization formula for the fragmentation function and also contains a discussion of the NRQCD LDMEs and short-distance coefficients that are relevant through relative order . The kinematics and variables that we use in our calculation are described in Sec. 4. In Sec. 5, we discuss the calculation of the fragmentation processes in full QCD. As we have mentioned, an important feature of the present calculation is that soft divergences arise in the color-singlet channel in both full QCD and NRQCD. These divergences ultimately cancel in the short-distance coefficients when we carry out the matching between full QCD and NRQCD. In both full QCD and NRQCD, we regulate the divergences dimensionally. In the case of full QCD, we devise subtractions that remove the divergent terms from the integrand, and we compute the subtraction contributions analytically. This computation is described in Sec. 5.3. After we remove the subtraction terms, we calculate the remainder of the full-QCD contribution in four dimensions, carrying out the integration numerically. We compute the relevant NRQCD LDMEs for free states analytically in dimensional regularization. These calculations are described in Sec. 6. We also determine the evolution equations for the LDMEs and find a discrepancy with a result in Ref. Gremm:1997dq . In Sec. 7, we match the NRQCD and full-QCD fragmentation functions to obtain the short-distance coefficients, and we present numerical results for them in Sec. 8. Finally, in Sec. 9, we summarize our results.

## 2 Collins-Soper definition of the fragmentation function

Here, and throughout this paper, we use the following light-cone coordinates for a four-vector :

(4a) | |||||

(4b) | |||||

(4c) |

The scalar product of two four-vectors and is then

(5) |

The Collins-Soper definition for the fragmentation function for a gluon fragmenting into a hadron (quarkonium) Collins:1981uw is

Here, is the fraction of the gluon’s component of momentum that is carried by the hadron, is the gluon field-strength operator, is the momentum of the field-strength operator, is the factorization scale, and is the number of space-time dimensions. There is an implicit average over the color and polarization states of the initial gluon. The projection is given in Eq. (3). The fragmentation function is evaluated in the frame in which the hadron has zero transverse momentum: . The operator is a path-ordered exponential of the gluon field:

(7) |

where is the QCD coupling constant and is the gluon field. Both and are SU(3) matrices in the adjoint representation. The expression (LABEL:eq:D-def) is manifestly gauge invariant. We use the Feynman gauge in our calculation.

The Feynman rules for the perturbative expansion of Eq. (LABEL:eq:D-def) are given in Ref. Collins:1981uw . The quantity appears in the Feynman rules as an eikonal line. Owing to the charge-conjugation properties of the states that we consider and the Landau-Yang theorem landau-thm ; Yang:1950rg , gluon attachments to the eikonal lines from do not appear in our calculation. Hence, we need only the standard QCD Feynman rules, an overall factor

(8) |

from Eq. (LABEL:eq:D-def), and the special Feynman rule for the vertex that creates a gluon and an eikonal line. That vertex is shown in Fig. 1. Its Feynman rule, in momentum space, is a factor

(9) |

where is the sum of the momenta of the gluon and the eikonal line, is the momentum of the gluon, is the polarization index of the gluon, and and are the color indices, respectively, of the gluon and the eikonal line. In the absence of interactions with the eikonal lines, . is a light-like vector whose components are given by .

The final-state phase space that is implied by Eq. (LABEL:eq:D-def) is

(10) |

where is the statistical factor for identical particles in the final state, is the momentum of the th final-state particle, and the product is over all of the final-state particles except . We use nonrelativistic normalization for the state , and so a factor appears in the phase space in order to cancel the relativistic normalization of in the definition (LABEL:eq:D-def). We use relativistic normalization for all particles other than .

## 3 NRQCD factorization

We assume that the fragmentation function for a gluon fragmenting into a quarkonium satisfies NRQCD factorization. Then, in analogy with Eq. (1), we have

(11) |

where the are NRQCD LDMEs and the are the fragmentation short-distance coefficients. We have suppressed the dependences of and on the factorization scale . In discussing specific cases, we use the notation for the LDMEs and the notation for the short-distance coefficients, where is the standard spectroscopic notation for the angular-momentum quantum numbers of the corresponding NRQCD operator, and is the color quantum number of the NRQCD operator ( or ). is the order in , relative to the leading order, of the field operators and derivatives in , excluding factors of from the projection onto the final state . is an integer that is used to distinguish operators that have the same quantum numbers and order in . We denote the contribution of order to by .

We can write the fragmentation functions for gluon fragmentation into free states as

(12) |

Since the short-distance coefficients are independent of the specifics of the hadronic states, the in Eq. (12) are identical to the in Eq. (11). We determine the by computing the left side of Eq. (12) in full QCD and comparing it with the right side, in which the free LDMEs are computed in NRQCD. Since we choose a factorization scale of order the heavy-quark mass , we can carry out this computation in perturbation theory. We will denote the contribution of order to by .

If is a quarkonium state, such as the , then, in LO in , we must consider the LDME

(13) |

In relative order , we must consider the LDME

(14) |

In relative order , we must consider the LDME

(15) |

In relative order , we must consider the LDMEs

(16a) | |||||

(16b) | |||||

(16c) | |||||

(16d) | |||||

(16e) | |||||

(16f) | |||||

(16g) |

Here, the symmetric traceless product is defined by

(17) |

and the antisymmetric product is defined by

(18) |

For purposes of our calculation, it is also useful to define

(19) | |||||

It was shown in Ref. Bodwin:2002hg that, by making use of the NRQCD equations of motion, one can express the LDME in terms of the LDMEs and . (Equivalently, one can eliminate the LDME by making use of a field redefinition Brambilla:2008zg .) Hence, we need not consider in our analysis.

In the LDMEs and , one can replace in the projector with (vacuum-saturation approximation), making an error of relative order . If one takes this approximation and evaluates the LDMEs in dimensional regularization in a potential model, then they are equal Bodwin:2006dn . Since the static potential model is valid up to corrections of order , we have

(20) |

Hence, up to corrections of relative order , only the sum of short-distance coefficients appears in the fragmentation function.

### 3.1 NRQCD factorization formulas for through order

In summary, we have the following NRQCD factorization formulas for gluon fragmentation into through relative order .

In relative order we have

(21) |

The short-distance coefficient was calculated in Refs. Braaten:1993rw ; Braaten:1995cj .

In relative order we have

(22) |

The short-distance coefficient was calculated in Ref. Bodwin:2003wh .

In relative order we have

(23) |

The short-distance coefficient was calculated in Ref. Braaten:1996rp and differs from the short-distance coefficient in Eq. (23) only by a color factor, which we provide in Sec. 7.

In relative order we have

(24) | |||||

The short-distance coefficient was calculated at LO in in Refs. Braaten:1996rp ; Bodwin:2003wh and at NLO in in Refs. BL:gfrag-NLO ; Lee:2005jw . We verify the LO calculations in Refs. Braaten:1996rp ; Bodwin:2003wh in the present paper, giving our result in Sec. 7. We compute in this paper, giving the result in Sec. 7. The short-distance coefficient was calculated in Refs. Braaten:1996rp and differs from the short-distance coefficient in Eq. (24) only by a color factor. The computation of the combination of short-distance coefficients is the main goal of this paper. The result of that computation is given in Sec. 7.

## 4 Kinematics

In the calculations to follow, in both full QCD and NRQCD, we employ the following kinematics.

We take the and the to be free (on-shell) states with momenta

(25a) | |||||

(25b) |

respectively. The heavy quark has three-momentum in the rest frame, and, so, the invariant mass of the state is

(26) |

where

(27) |

We work in the frame in which the transverse momentum of the pair vanishes. In this frame, the initial-state gluon, the final-state pair and the final-state gluons, respectively, have the momenta

(28a) | |||||

(28b) | |||||

(28c) | |||||

(28d) |

where we have introduced the longitudinal momentum fractions

(29a) | |||||

(29b) | |||||

(29c) |

Because of the conservation of four-momentum, , the momenta , and depend implicitly on , and, therefore, on . We can make the dependence on explicit by writing quantities in terms of dimensionless momenta

(30a) | |||||

(30b) | |||||

(30c) | |||||

(30d) |

It is also useful to express the Lorentz invariants in terms of the following dimensionless variables:

(31a) | |||||

(31b) | |||||

(31c) |

where is the unit vector that is parallel to the three-vector in the rest frame.

The phase space in Eq. (10) can be expressed in terms of the dimensionless variables as

(32a) | |||||

(32b) | |||||

(32c) |

We have not replaced the overall factor in Eq. (32) with , because it ultimately will be cancelled by the factor in in Eq. (8). The ranges of the variables , and are completely determined by the and functions. When we expand the fragmentation function in powers of , it is convenient to make use of the phase space at LO in ,

(33) |

where means that, in the phase space in Eq. (32), we replace with . Then can be expressed in terms of as follows:

(34a) | |||||

(34b) | |||||

(34c) |

We express our results for the fragmentation contributions in terms of integrals over the phase spaces :

(35) |

The factors of in Eq. (34) are then an additional source of relativistic corrections.

## 5 Full-QCD calculations

In this section, we compute the relevant fragmentation functions for free states in full QCD. We have carried out the calculations by writing independent codes using reduce REDUCE and using the feyncalc package Mertig:an in mathematica MATH . At each stage of the calculations we have checked that the independent codes give identical results.

The computations are carried out in dimensions with dimensional-regulariza-tion scale . We use the modified-minimal-subtraction () scheme throughout. Then, in dimensions, there is a factor that is associated with each factor of the strong coupling , where is the Euler-Mascheroni constant.

In computing the fragmentation functions, it is convenient to make use of projection operators for the spin and color states of the pair. The projection operators for a pair in the color-singlet and color-octet configurations are

(36a) | |||||

(36b) |

where and are the identity matrix and the generator of the fundamental (triplet) representation of SU(3), is the adjoint-representation color index (), and . Spin-projection operators at LO in were first given in Refs. Barbieri:1975am ; Barbieri:1976fp ; Chang:1979nn ; Guberina:1980dc ; Berger:1980ni . Projectors accurate to all orders in were given in Ref. Bodwin:2002hg . For the spin-triplet state, the projection operator, correct to all orders in , is

(37) |

where is the spin polarization of the pair, and . Note that we use nonrelativistic normalization for the heavy-quark spinors.

The use of the spin projection (37) in dimensions
requires some justification. It accounts for only the vector
polarization states, which, in the rest frame, correspond to
the Pauli matrices . In general, in dimensions, one
must consider states that correspond to products of the that
are linearly independent of the Braaten:1996rp . These
additional states vanish as goes to zero. Hence, they can
contribute only in conjunction with a pole in . The poles in
in our calculation correspond to soft divergences. The
divergent parts of soft interactions arise from the convection current
on fermions lines, and, hence, do not change the fermion spin.
Therefore, the additional states that correspond to products of the
never mix in our calculation with the vector states that
correspond to the . Consequently, we need consider only the
vector states in our calculation.^{2}^{2}2Some elements of this
argument were presented in Ref. Petrelli:1997ge .

The spin-triplet, color-singlet part of an amplitude is

(38) |

and the spin-triplet, color-octet part of an amplitude is

(39) |

where the traces are over the Dirac and color indices. The amplitude includes the propagator of the initial gluon, as well as the associated polarization factor in Eq. (9). In our calculation, the amplitudes , , and are all expressed in terms of the dimensionless variables in Eq. (30) or invariants that are formed from them, and so the dependence on is explicit.

The -wave part of (with color index suppressed in the color-octet case) can be written as an expansion in powers of :

(40) |

where

(41a) | |||||

(41b) | |||||

(41c) |

and

(42a) | |||||

(42b) |

In order to project out the -wave part of the amplitude ,
we multiply by the -wave orbital-angular-momentum state
and average over the direction of .^{3}^{3}3
In some calculations in
NRQCD, the -wave orbital-angular-momentum state
is normalized as .
Here, is the polarization vector for the
orbital-angular-momentum state, and
in the rest frame of the
pair. Then, the -wave part of the amplitude is

(43) |

where

(44) |

We define squared amplitudes for the color-singlet and color-octet states as

(45a) | |||||

(45b) |

where is given in Eq. (8), and it is implicit that there are sums over the spin and orbital-angular-momentum polarizations of the states and sums over the polarizations of the initial and final gluons. Note that

(46) |

We denote the order- contribution to by .