# Transverse-momentum-dependent gluon distributions from JIMWLK evolution

###### Abstract

Transverse-momentum-dependent (TMD) gluon distributions have different operator definitions, depending on the process under consideration. We study that aspect of TMD factorization in the small- limit, for the various unpolarized TMD gluon distributions encountered in the literature. To do this, we consider di-jet production in hadronic collisions, since this process allows to be exhaustive with respect to the possible operator definitions, and is suitable to be investigated at small . Indeed, for forward and nearly back-to-back jets, one can apply both the TMD factorization and Color Glass Condensate (CGC) approaches to compute the di-jet cross-section, and compare the results. Doing so, we show that both descriptions coincide, and we show how to express the various TMD gluon distributions in terms of CGC correlators of Wilson lines, while keeping finite. We then proceed to evaluate them by solving the JIMWLK equation numerically. We obtain that at large transverse momentum, the process dependence essentially disappears, while at small transverse momentum, non-linear saturation effects impact the various TMD gluon distributions in very different ways. We notice the presence of a geometric scaling regime for all the TMD gluon distributions studied: the ”dipole” one, the Weizsäcker-Williams one, and the six others involved in forward di-jet production.

## I Introduction

In hadronic collisions that feature a large transfer of momentum, the standard perturbative QCD framework of collinear factorization is appropriate to calculate scattering cross sections, which are measurable in particular at the Large Hadron Collider. However, some hadronic processes involve, in addition, smaller momentum scales, and for those one needs to resort to a more involved QCD framework, using the concept of transverse-momentum-dependent (TMD) parton distributions, or in short, TMDs. This relates to a large number of observables such as the production of heavy bosons at small transverse momentum Collins:1984kg ; Qiu:2000ga , transverse spin asymmetries measured in high-energy collisions with polarized beams Ralston:1979ys ; Boer:2003cm , or in general hadronic scattering in the high-energy limit Lipatov:1985uk ; Catani:1990eg .

One of the main theoretical obstacles has been the fact that, even in cases for which TMD factorization could be established, the precise operator definition of the parton distributions is dependent on the process under consideration Mulders:2000sh ; Belitsky:2002sm , implying a loss of universality. In this paper, our goal is to study that aspect of TMDs, in the limit of small longitudinal momentum fraction , where the parton transverse momentum generically plays a central role. Restricting ourselves to unpolarized gluon TMDs, we investigate what happens when the large gluon density reaches the saturation regime, and how the different gluon TMDs are affected by non-linear effects when becomes of the order of the saturation scale , or below. To do so, we shall use the Color Glass Condensate (CGC) framework, an effective theory of QCD which encompasses its small- dynamics, both in the linear and non-linear regimes Gelis:2010nm .

In order to perform our study of the various unpolarized gluon TMDs for protons and nuclei in the small- regime, we choose to consider the process of forward di-jet production in proton-proton (p+p) and proton-nucleus (p+A) collisions, respectively. On the one hand, di-jets, when produced nearly back-to-back, provide the two necessary transverse momentum scales, and the strong ordering needed between them, for TMDs to be relevant: the hard scale is the typical single-jet transverse momentum while the softer scale is the total transverse momentum of the jet pair , and TMD factorization applies when Vogelsang:2007jk . On the other hand, the production at forward rapidities probes small values of : for kinematical reasons, only high-momentum partons from the ”projectile” hadron contribute, while on the ”target” side, it is mainly small- gluons that are involved Marquet:2007vb . The forward di-jet process is therefore an ideal playground to apply both the TMD and CGC frameworks and to compare them.

Note that the asymmetry of the problem, and , implies that gluons from the target have a much bigger average transverse momentum (of the order of ) compared to that of the partons from the projectile (which is of the order of ). Therefore we shall always neglect the transverse momentum of the high- partons from the projectile compared to that of the low- gluons from the target. As a result, the parton content of the projectile hadron will be described by regular parton distributions and TMDs will be involved only on the target side, with the transverse momentum of those small- gluons being equal to the transverse momentum of jet pair . This simplification is actually needed in order to apply TMD factorization for the di-jet process, since for this final state, there is no such factorization with TMDs for both incoming hadrons Collins:2007nk ; Rogers:2010dm .

In order to compare the TMD and CGC approaches in their overlapping domain of validity, we could have considered a simpler process where this issue does not arise, such as for instance semi-inclusive deep-inelastic scattering in electron-proton or electron-nucleus collisions Ji:2004wu ; Marquet:2009ca . However, with simpler processes, one encounters only small sub-sets of all the possible operator definitions for the gluon TMDs. The advantage of the di-jet process in p+p or p+A collisions is that it involves all the possible gluon TMDs encountered so far in the literature Bomhof:2006dp , and therefore it allows us to be comprehensive and study the specifics of the process dependence of gluon TMDs at small- in an exhaustive way. All our findings, such as the geometric scaling of all the gluon TMDs, will naturally carry over to other processes for which only one or a few of them play a role, like di-jet or heavy-quark production in deep-inelastic scattering and Drell-Yan or photon-jet in p+p and p+A collisions, for instance Dominguez:2010xd .

The TMD description of the forward di-jet process, valid in the limit but with no (other than kinematical) constraints on the value of , calls for the use of eight different operator definitions for the gluon TMDs Kotko:2015ura . They all involve a correlator of two field strength operators, but they differ from each other in their gauge link content. Each of the gluon TMDs is also associated to a different hard factor, made of a sub-set of the possible diagrams. We show that in the small- limit, all the gluon TMDs can be simplified and expressed as Fourier transforms of Wilson-line correlators, made either of two, four, six or eight Wilson lines, but with only two different transverse positions whose difference is conjugate to the transverse momentum .

The CGC description of the forward di-jet process, valid in the small- limit but with no (other than kinematical) constraints on the values of the transverse momenta of the jets, involves correlators of up to eight Wilson lines, all of which sit at different transverse positions Marquet:2007vb ; Dominguez:2011wm . We show that in the limit, the CGC formula coincides with the small- limit of the TMD formula. In particular, we show how the various gluon TMDs emerge from the framework, how their different operator definitions correspond to different Wilson lines structure of the CGC correlators. We obtain full agreement in the overlapping domain of validity, hereby extending the results of Dominguez:2011wm to the case of finite .

It is important to note that in the limit, saturation effects do not disappear. Indeed, even though the hard scale is much bigger than the saturation scale, the transverse momentum of jet pair may be of the order of , and formally all powers of may still be included in the definition of the gluon TMDs. They are all contained if the Wilson-line correlators are properly evaluated in the CGC. In particular, the non-linear QCD evolution of all the gluon TMDs can be obtained from the Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner (JIMWLK) JalilianMarian:1997jx ; JalilianMarian:1997dw ; Iancu:2000hn ; Ferreiro:2001qy ; Weigert:2000gi equation, in the leading approximation. Using a numerical simulation of the JIMWLK equation on a discretized lattice, we are able to extract their dependence and evolution towards small values of , as well as some important properties: all the gluon TMDs feature geometric scaling (i.e. they are functions of only, as opposed to and separately) in the saturation region , and they either vanish or coincide for . Finally, having understood how their process dependence manifests itself in the CGC allows us to restore universality: potential information extracted from a particular process, for one gluon TMD, can be consistently fed into the others.

The paper is organized as follows. In section II, we recall the TMD factorization formula for forward di-jets as well as the operator definitions of the eight gluon TMDs involved, and we explain the simplifications obtained in the small- limit. In section III, we recall the CGC formula for forward di-jets, take the limit, and show that the result coincides with the one obtained in the TMD framework, in this overlapping domain of validity. In section IV, we describe the numerical method used in order to solve the JIMWLK equation: a lattice implementation of the Langevin formulation of the equation. In section V, we present numerical results for the various gluon TMDs and discuss several properties of their small- evolution such as geometric scaling. Finally, section VI is devoted to conclusions and outlook.

## Ii Small-x limit of the TMD factorization framework

We consider the process of inclusive di-jet production in the forward region, in collisions of dilute and dense systems

(1) |

The process is shown schematically in Fig 1. The four-momenta of the projectile and the target are massless and purely longitudinal. In terms of the light cone variables, , they take the simple form

(2) |

where is the squared center of mass energy of the p+A system. The energy (or longitudinal momenta) fractions of the incoming parton (either a quark or gluon) from the projectile, , and the gluon from the target, , can be expressed in terms of the rapidities and transverse momenta of the produced jets as

(3) |

where , are transverse Euclidean two-vectors. By looking at jets produced in the forward direction, we effectively select those fractions to be and . Since the target A is probed at low , the dominant contributions come from the subprocesses in which the incoming parton on the target side is a gluon

(4) |

Moreover, the large- partons of the dilute projectile are described in terms of the usual parton distribution functions of collinear factorization and , with a scale dependence given by DGLAP evolution equations, while the small- gluons of the dense target are described by several transverse-momentum-dependent (TMD) distributions, which evolve towards small values of according to non-linear equations. Indeed, besides its longitudinal component , the momentum of the incoming gluon from the target has in general a non-zero transverse component

(5) |

which leads to imbalance of transverse momentum of the produced jets: . The Mandelstam variables of the process are:

(6) | |||||

(7) | |||||

(8) |

with

(9) |

They sum up to .

### ii.1 The TMD factorization formula for forward di-jets

Just as collinear factorization, the TMD factorization framework is a ”leading-twist” framework valid to leading power of the hard scale, but it can only be established for a subset of hard processes, compared to collinear factorization. In particular, there exists no general TMD factorization theorem for jet production in hadron-hadron collisions. However, such a factorization can be established in the asymmetric “dilute-dense” situation considered here, where only one of the colliding hadrons is described by a transverse momentum dependent (TMD) gluon distribution. Again, selecting di-jet systems produced in the forward direction implies and , which in turn allows us to make that assumption.

In this context, the validity domain of the TMD factorization formula is

(10) |

This means that the transverse momentum imbalance between the outgoing particles, Eq. (5), must be much smaller than their individual transverse momenta, which corresponds to the situation of nearly back-to-back di-jets. The jet momenta must also be much bigger than the other momentum scale in the problem, the saturation scale of the dense target , and in practice this is always the case.

The TMD factorization formula reads Dominguez:2011wm ; Kotko:2015ura :

(11) |

where denotes several distinct TMD gluon distributions, with different operator definitions. Each of them is accompanied by its own hard factor . These were calculated in Dominguez:2011wm and expressed in terms of the Mandelstam variables (6). Because of the condition , those hard factors are on-shell (i.e. ), and the dependence of the cross-section comes from the gluon distributions only.

It was shown in Kotko:2015ura that the offshellness of the small- gluon can be restored in the hard factors (i.e. ) in order to extend the validity of formula (11) to a wider kinematical range: without any condition on the magnitude of . But in this work, we stick to the strict TMD limit. Explicitly, the three channels read (in (12) denotes the momentum of the final-state gluon):

(12) |

(13) |

(14) |

with

(15) |

Several gluon distributions , with different operator definition, are involved here. Indeed, a generic unintegrated gluon distribution of the form Mulders:2000sh

(16) |

where are components of the gluon field strength tensor, must be also supplemented with gauge links, in order to render such a bi-local product of field operators gauge invariant Belitsky:2002sm . The gauge links are path-ordered exponentials, with the integration path being fixed by the hard part of the process under consideration. In the following, we shall encounter two gauge links and , as well as loops and . The various gluon distributions needed for the di-jet process are given by Bomhof:2006dp ; Kotko:2015ura :

(17) | |||||

(18) | |||||

(19) | |||||

(20) | |||||

(21) | |||||

(22) | |||||

(23) | |||||

(24) |

The gauge links are composed of Wilson lines, their simplest expression is obtained in the gauge (but the expressions above are gauge-invariant):

(25) |

where are the generators of the fundamental representation of . The gluon TMDs are normalized such that , except for which vanishes when integrated.

### ii.2 Taking the small-x limit

In (16), the matrix element is calculated for a hadronic/nuclear state with a fixed given momentum , normalized such that . Therefore, using translational invariance, we may write

(26) |

In the small limit, we set

(27) |

Then, for instance, we can write for (see (17)):

(28) |

More details on their dependence are given below, but first let us simplify further their expressions. From Eq. (28) we have:

(29) | |||||

Using the formula for the derivative of the Wilson lines

(30) |

with in the gauge (otherwise, the other piece of the field strength tensor comes additional transverse gauge links in ), we obtain

(31) |

where

(32) |

Due to its simple Wilson line structure, has been dubbed the ”dipole” gluon distribution.

To give a second example which leads to a more complicated Wilson line structure, let us also simplify the so-called Weizsäcker-Williams gluon distribution :

(33) | |||||

Following the same lines, we obtain for the other gluon TMDs:

(34) | |||||

(35) | |||||

(36) | |||||

(37) | |||||

(38) | |||||

(39) |

The CGC averages are averages over color field configurations in the dense target. They may be written

(40) |

where represents the probability of a given field configuration. The CGC wavefunction effectively describes, in terms of strong classical fields, the dense parton content of a hadronic/nuclear wave function, at small longitudinal momentum fraction . In the leading-logarithmic approximation, the evolution of with decreasing is obtained from the JIMWLK equation:

(41) | |||||

(42) |

where the functional derivatives act at the largest value of :

(43) |

and where

(44) |

with denoting the generators of the adjoint representation of . After integrating by parts, the evolution of any CGC average may be written:

(45) |

We note that recently, more general evolution equations have been derived for the gluon TMDs Balitsky:2015qba ; Zhou:2016tfe and Balitsky:2016dgz . These equations contain JIMWLK evolution in the small- limit (or Balitsky-Fadin-Kuraev-Lipatov (BFKL) evolution Lipatov:1976zz ; Kuraev:1976ge ; Balitsky:1978ic in the linear approximation), and in addition for generic values of , the hard scale dependence, which at small boils down to Sudakov factors Mueller:2012uf ; Mueller:2013wwa . Presumably, similar equations could also be obtained for the other gluon TMDs. It is not known how to solve them however, and in this work we stick to the small- JIMWLK evolution.

## Iii Leading power of the CGC framework

In this section, our starting point is CGC formalism for double-inclusive particle production in dilute-dense collisions. We shall extract the leading power in and show that the result coincides with the small- limit of TMD factorization formula for forward di-jets (11), namely Eq. (12)-(14) with the gluon TMDs given by (31), (33) and (34)-(39). In the large- limit, this has already been demonstrated in Dominguez:2011wm for all three channels (see (4)), in Akcakaya:2012si for the case and in Iancu:2013dta for the subprocess.

In the present work, we show the equivalence between the CGC and TMD framework, in their overlapping domain of availability, while keeping finite. We start with the quark initiated channel, for which we shall explain the derivation in details, and then we deal with the gluon initiated channels.

### iii.1 The quark initiated channel

The amplitude for quark-gluon production is schematically presented in Fig. 2 as in Ref. Marquet:2007vb . In the CGC formalism, the scattering of the partons from the dilute projectile with the dense target is described by Wilson lines that resum multi-gluon exchanges; fundamental Wilson lines for quarks and adjoint Wilson lines for gluons. As a result, the cross section involves multipoint correlators of Wilson lines. In particular, the square of the amplitude from Fig. 2 contains four terms: a correlator of four Wilson lines, , corresponding to interactions happening after the emission of the gluon, both in the amplitude and the complex conjugate, then a correlator of two Wilson lines, , representing the case when interactions with the target take place before the radiation of the gluon in both amplitude and complex conjugate, and two correlators of three Wilson lines, , for the cross terms.

Denoting, as in the previous section, the momentum of the outgoing gluon and the momentum of the outgoing quark, the cross-section reads Marquet:2007vb :

(46) |

where

(47) |

denote the transverse positions of the final-state quark in the amplitude and the conjugate amplitude, respectively, and

(48) |

denote the transverse positions of the final-state gluon in the amplitude and the conjugate amplitude, respectively. is conjugate to , and is conjugate to the total transverse momentum of the produced particles .

The Wilson line correlators are given by:

(49) | |||||

(50) | |||||

(51) |

and the functions denote the splitting wave functions. In the limit of massless quarks, the wave function overlap is simply given by:

(52) |

### iii.2 Extracting the leading power

In the limit, the integrals in (46) are controlled by configurations where and are small compared to the other transverse-size variables, and the leading power of this expression can be extracted by expanding around and . To do this, let us first rewrite all the Wilson line correlators in terms of fundamental Wilson lines only:

(53) | |||||