Next-to-leading order transverse momentum-weighted Sivers asymmetry in semi-inclusive deep inelastic scattering: the role of the three-gluon correlator

# Next-to-leading order transverse momentum-weighted Sivers asymmetry in semi-inclusive deep inelastic scattering: the role of the three-gluon correlator

Ling-Yun Dai Jefferson Lab, 12000 Jefferson Avenue, Newport News, VA 23606, USA    Zhong-Bo Kang Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA    Alexei Prokudin Division of Science, Penn State Berks, Reading, PA 19610, USA Jefferson Lab, 12000 Jefferson Avenue, Newport News, VA 23606, USA    Ivan Vitev Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
July 13, 2019
###### Abstract

We study the Sivers asymmetry in semi-inclusive hadron production in deep inelastic scattering. We concentrate on the contribution from the photon-gluon fusion channel at , where three-gluon correlation functions play a major role within the twist-3 collinear factorization formalism. We establish the correspondence between such a formalism with three-gluon correlation functions and the usual transverse momentum dependent (TMD) factorization formalism at moderate hadron transverse momenta. We derive the coefficient functions used in the usual TMD evolution formalism related to the quark Sivers function expansion in terms of the three-gluon correlation functions. We further perform the next-to-leading order calculation for the transverse-momentum-weighted spin-dependent differential cross section, and identify the off-diagonal contribution from the three-gluon correlation functions to the QCD collinear evolution of the twist-3 Qiu-Sterman function.

###### pacs:
12.38.Bx, 12.39.St, 13.85.Hd, 13.88.+e
preprint: JLAB-THY-14-1947preprint: LA-UR-14-27352

## I Introduction

In recent years, the study of single transverse-spin asymmetries (SSAs) has become a forefront of both experimental and theoretical research in QCD and hadron physics. With extensive experimentation underway and major theoretical advances, we have begun to obtain a deeper understanding of the nucleon structure and the partons’ transverse motion. A lot of progress was made in understanding the underlying QCD mechanisms that generate these asymmetries. The transverse momentum dependent (TMD) factorization scheme Ji:2004wu (); Ji:2004xq (); Collins:2011zzd () and the twist-3 collinear factorization approach Efremov:1981sh (); Efremov:1984ip (); Qiu:1991pp () were studied theoretically and applied phenomenologically to describe the SSAs in various processes, including Drell-Yan, semi-inclusive deep inelastic scattering (SIDIS), annihilation, and hadron and jet production in scattering. These two mechanisms were shown to be closely related and provide a unified picture for SSAs Ji:2006ub (); Bacchetta:2008xw ().

SIDIS is one of the key experimental tools to study the spin asymmetries and the associated nucleon structure. A particular twist-3 quark-gluon correlation function, often called Qiu-Sterman function Qiu:1991pp (), plays a crucial role in generating non-zero SSA and is related to the quark Sivers function Sivers:1989cc (); Boer:1997nt (). SSAs enabled by the Sivers function were extensively studied experimentally in the SIDIS process by HERMES Airapetian:2009ae (), COMPASS Alekseev:2008aa (); Adolph:2013stb (); Adolph:2014zba () and JLab Qian:2011py (). It was discovered theoretically that the Sivers function should change sign when measured in the Drell-Yan process with respect to the SIDIS process Brodsky:2002cx (); Collins:2002kn (); Kang:2009bp () and a number of experiments, including COMPASS, RHIC experiments, and Fermilab experiments, are planned to test this prediction experimentally. Knowledge of the evolution of the Sivers function Kang:2011mr (); Aybat:2011ge (); Echevarria:2012pw (); Sun:2013hua (); Echevarria:2014xaa () (and Qiu-Sterman function Kang:2008ey (); Zhou:2008mz (); Vogelsang:2009pj (); Braun:2009mi (); Kang:2012em (); Schafer:2012ra (); Kang:2010xv (); Schafer:2013opa ()) with the hard scale is very important for accurate phenomenological applications and, eventually, for precise extraction of these functions. Next-to-leading order (NLO) corrections involving the Qiu-Sterman function were calculated for Drell-Yan Vogelsang:2009pj () and SIDIS Kang:2012ns ().

Special three-gluon correlation functions Ji:1992eu (); Eguchi:2006qz (); Eguchi:2006mc (); Beppu:2010qn (); Kang:2008ey (); Kang:2008qh (); Kang:2008ih () become relevant at NLO () and can be studied experimentally via open charm production in SIDIS Kang:2008ih (); Beppu:2010qn (). The purpose of our current paper is to study the role of three-gluon correlation functions in SIDIS, and their connection to the quark Sivers function. Concentrating on the photon-gluon fusion channel, we first calculate the contributions of the three-gluon correlation functions to the transverse spin-dependent differential cross section within the twist-3 collinear factorization formalism. We then demonstrate that our result can be matched onto the TMD factorization formalism at moderate hadron transverse momenta, and that we can extend the unification of the two mechanisms to the case involving three-gluon correlation functions. We also derive the coefficient functions widely used in the TMD evolution formalism. This is achieved by expanding the quark Sivers function (in the Fourier transformed -space) in terms of a convolution of the coefficient functions and the three-gluon correlation functions.

In the second part of our paper, we study the NLO perturbative QCD corrections to the transverse momentum-weighted spin-dependent SIDIS cross section. Our primary focus is again on the contributions of the three-gluon correlation functions. By analyzing the collinear divergence structure, we identify the evolution kernel for the Qiu-Sterman function that includes the off-diagonal contribution from the three-gluon correlators. The hard coefficient function is evaluated at one-loop order.

## Ii Sivers asymmetry from three-gluon correlation functions

In this section we first study the contribution of the three-gluon correlation functions to the Sivers asymmetry in SIDIS within the twist-3 collinear factorization formalism. We establish the correspondence between such a formalism and the usual TMD factorization formalism at moderate hadron transverse momenta to be defined below. Coincidently, we derive the coefficient functions by expanding the quark Sivers function in the conjugate Fourier -space in terms of the three-gluon correlation functions. Such coefficient functions are a key ingredient of the usual TMD evolution formalism.

### ii.1 Three-gluon correlation functions

To define the twist-3 three-parton correlation functions, we consider a nucleon of momentum , with expressed in light-cone coordinates, where we write any four-vector with and . We also define a conjugated light-like vector , which obeys , , and . The widely studied twist-3 quark-gluon correlation function (the so-called “Qiu-Sterman” function) is defined as follows Kang:2011hk ():

 MαF,aij(x1,x2) =gs∫dy−1dy−22πeix1p+y−1ei(x2−x1)p+y−2⟨ps|¯ψj(0)Fα+a(y−2)ψi(y−1)|ps⟩ =12[Tq,F(x1,x2)γ⋅¯nϵαn¯ns2N2c−1(ta)ij+⋯], (1)

where represents the nucleon wave-function with the momentum of the nucleon given above, and spin vector . is the strong coupling, are the quark fields with color indices in the fundamental representation of the color group. We have the number of the colors, is the standard generating matrix of the group. with the standard gluon field strength. are the momentum fractions carried by the quarks (represented by and , respectively), and will be the momentum fraction for the gluon following the momentum conservation. Note that with the Levi-Civita tensor, and we use the convention here.

Classification of three-gluon correlation functions was first considered in Ji:1992eu (). These functions have been studied in the context of open charm production in Kang:2008qh (); Kang:2008ih (); Eguchi:2006qz (); Eguchi:2006mc (); Beppu:2010qn (). Generically, three-gluon correlation functions can be constructed as combinations of the gauge invariant correlation functions and , where and are the anti-symmetric and symmetric structure constants of the color group. With slightly different normalization from Ref. Beppu:2010qn (), we define the following three-gluon correlation function:

 MαβγF,abc(x1,x2) =gs∫dy−1dy−22πeix1p+y−1ei(x2−x1)p+y−21p+⟨ps|Fβ+b(0)Fγ+c(y−1)Fα+a(y−2)|ps⟩ =(C(d)g)abcOαβγ(x1,x2)−(C(f)g)abcNαβγ(x1,x2), (2)

where the gluonic color projection operators and are given by

 (C(d)g)abc =Nc(N2c−1)(N2c−4)dabc, (3) (C(f)g)abc =iNc(N2c−1)fabc. (4)

The functions and correspond to symmetric and anti-symmetric combinations of gluon field-strength tensors and read Beppu:2010qn ()

 Oαβγ(x1,x2) =gs∫dy−1dy−22πeix1p+y−1ei(x2−x1)p+y−21p+⟨ps|dbcaFβ+b(0)Fγ+c(y−1)Fαna(y−2)|ps⟩ =12[O(x1,x2)gαβ⊥ϵγn¯ns+O(x2,x2−x1)gβγ⊥ϵαn¯ns+O(x1,x1−x2)gγα⊥ϵβn¯ns], (5) Nαβγ(x1,x2) =gs∫dy−1dy−22πeix1p+y−1ei(x2−x1)p+y−21p+⟨ps|ifbcaFβ+b(0)Fγ+c(y−1)Fαna(y−2)|ps⟩ =12[N(x1,x2)gαβ⊥ϵγn¯ns−N(x2,x2−x1)gβγ⊥ϵαn¯ns−N(x1,x1−x2)gγα⊥ϵβn¯ns], (6)

where . Our definitions are related to those of Refs. Eguchi:2006qz (); Eguchi:2006mc (); Beppu:2010qn () by Koike et.al. as follows:

 O(x1,x2) =8πMO(x1,x2)|Koike, N(x1,x2) =8πMN(x1,x2)|Koike, (7)

with being the nucleon mass.

### ii.2 Spin-dependent cross section for SIDIS: three-gluon correlation functions

We now consider the contribution of the three-gluon correlation functions to the Sivers asymmetry for the SIDIS process, . Here is the transverse spin vector of the incoming nucleon with momentum , whereas and are the momenta of the lepton before and after the collision. represents the observed final-state hadron with momentum , and the exchanged virtual photon has momentum with the invariant mass . We will work in the so-called hadron frame Meng:1991da (); Koike:2003zc (); Kang:2008qh (); Beppu:2010qn (), where both the virtual photon and the incoming polarized nucleon have only the -component, i.e., vanishing transverse momentum. In this frame, the final observed hadron has transverse momentum, which will be denoted as below with its magnitude .

This process has already been studied in Beppu:2010qn (), for open charm production. We will reproduce the result here. However, the purpose of our calculation is quite different. What we investigate here is the connection between the twist-3 formalism and the TMD factorization approach, in particular, for the three-gluon correlation functions. In the course of such a study, we will further derive the contribution of three-gluon correlation functions to the evolution of Qiu-Sterman function . Finally, we will derive the so-called coefficient functions which are widely used in the TMD evolution formalism. Except for the very first result available in Beppu:2010qn (), all other calculations (matching, evolution, and coefficient functions) are performed only in the current paper. Because of the different goal in our calculations, we will study light hadron production, i.e., we consider the mass of the hadron is much smaller than the hard scale .

The differential cross section that includes the Sivers effect, i.e. the module, can be written as follows Bacchetta:2006tn (); Kang:2012xf (); Kang:2012ns ():

 dσSiversdxBdydzhd2ph⊥=σ0[FUU+sin(ϕh−ϕs)Fsin(ϕh−ϕs)UT], (8)

where and are the spin-averaged and transverse spin-dependent structure functions, respectively, and are the azimuthal angles of the final-state hadron transverse momentum and the proton spin relative to the scattering plane of the lepton. is given by

 σ0=2πα2emQ21+(1−y)2y, (9)

and , , and are the standard SIDIS kinematic variables,

 S=(p+ℓ)2,xB=Q22p⋅q,y=p⋅qp⋅ℓ=Q2xBS,zh=p⋅php⋅q. (10)

The transverse spin-dependent differential cross section , which is the main focus of this section. It can be written as Beppu:2010qn ()

 dΔσdxBdydzhd2ph⊥=α2emy32π3Q4zhLμνWμν(p,q,ph), (11)

where is the leptonic tensor and is the hadronic tensor. The hadronic tensor is related to the partonic tensor by

 Wμν(p,q,ph)=∫dzz2Dh/q(z)wμν(p,q,pc), (12)

where is the fragmentation function of a quark into a hadron , and the parton momentum . In the following (and throughout the paper) we will only consider the so-called metric contribution Graudenz:1994dq (); Daleo:2004pn (); Kang:2014ela (); Kang:2013raa (), i.e. we contract with and write below. Within the collinear factorization formalism, the transverse spin-dependent cross section is a twist-3 effect. To extract this effect, one has to perform a collinear expansion around a vanishing parton . For three-gluon correlation functions, the contribution can be written as Beppu:2010qn ()

 w(p,q,pc)=∫dx1x1dx2x2∂∂kλ⊥[Habcρδσ(p,q,pc,k⊥)pδ]k⊥→0ωραωσβωλγMαβγF,abc(x1,x2), (13)

with . A generic diagram to calculate the photon-gluon hard-part function is sketched in Fig. 1.

Using Eqs. (2), (5), and (6), we can rewrite Eq. (13) as

 w(p,q,pc)=∫dx1x1dx2x2∂∂kλ⊥[Habcρδσ(p,q,pc,k⊥)pδ]k⊥→0FρσλNO(x1,x2), (14)

where represents

 FρσλNO(x1,x2)= (C(d)g)abc(O(x1,x2)gρσ⊥ϵλn¯ns+O(x2,x2−x1)gσλ⊥ϵρn¯ns+O(x1,x1−x2)gλρ⊥ϵσn¯ns) −(C(f)g)abc(N(x1,x2)gρσ⊥ϵλn¯ns−N(x2,x2−x1)gσλ⊥ϵρn¯ns−N(x1,x1−x2)gλρ⊥ϵσn¯ns). (15)

The relevant Feynman diagrams for the transverse momentum dependent differential cross section at leading order (LO) are listed in Fig. 2. The technique to extract twist-3 contributions is well explained in the literature. The idea is that the so-called “pole-propagators” and the on-mass-shell condition for the unobserved parton in the final-state lead to kinematic -functions, which can be used to integrate out the parton momentum fractions and . These parton momentum fractions generally depend on , and, thus, are expanded with respect to . After some algebraic manipulation we have the following “master formula”:

 w(p,q,pc)= (v1−v2)λ1x2⎛⎝dFρσλNO(x,x)dx−2FρσλNO(x,x)x⎞⎠HLρσ(x,x,0)+FρσλNO(x,x)x2 ×limk⊥→0∂∂kλ⊥[HLρσ(x+(v2−v1)⋅k⊥,x+v2⋅k⊥,k⊥)−HRρσ(x,x+v1⋅k⊥,k⊥)], (16)

where are remainders of the hard parts given in Fig. 2, while are the mirror diagrams where the middle gluon is to the right of the unitary cut. The two four-vectors and are given by

 vμ1=−2x^upμc,vμ2=−2x^tpμc, (17)

with the partonic Mandelstam variables

 ^s=(xp+q)2,^t=(q−pc)2,^u=(xp−pc)2. (18)

The final result for the transverse spin-dependent differential cross section is given by

 dΔσdxBdydzhd2ph⊥= σ0(ϵαβsα⊥pβh⊥)∑qe2q(14)αs2π2∫1xBdxx∫1zhdzzDh/q(z)1zQ2δ(p2h⊥−z2hQ2(1^x−1)(1^z−1)) ×{[(dO(x,x)dx−2O(x,x)x)H1+(dO(x,0)dx−2O(x,0)x)H2+O(x,x)xH3+O(x,0)xH4] +[(dN(x,x)dx−2N(x,x)x)H1−(dN(x,0)dx−2N(x,0)x)H2+N(x,x)xH3−N(x,0)xH4]}. (19)

where is a two-dimensional anti-symmetric tensor with , and thus . Note that the integration limit for and are the standard ones, since we are considering light hadron production, as we have emphasized in the beginning of this subsection. Thus, there are no specific restrictions on the integration limit for and as pointed out in Beppu:2010qn (), which only apply when the mass of the hadron is important. The hard-part functions have the following expressions:

 H1 =^x[2^x2−2^x+(1−2^z+2^z2)]^z2(1−^z)2, (20) H2 =^x[4^x2−4^x+(1−2^z+2^z2)]^z2(1−^z)2, (21) H3 =2^x2(1−2^x)^z2(1−^z)2, (22) H4 =2^x2(1−4^x)^z2(1−^z)2, (23)

with and . Our results are consistent with those in Beppu:2010qn () 111One simply realizes that in Beppu:2010qn ()..

### ii.3 Matching onto the TMD factorization formalism

It has been demonstrated that the collinear twist-3 factorization formalism and the TMD factorization formalism are consistent with each other (so-called “matching”) for moderate hadron transverse momenta, i.e., in the kinematic region , see e.g., Ji:2006ub (); Ji:2006vf (); Ji:2006br (); Koike:2007dg (); Bacchetta:2008xw (); Yuan:2009dw (); Boer:2010ya (). However, in these earlier studies the matching was only demonstrated/shown for the quark-gluon correlation function . In this paper we generalize the known correspondence to include the three-gluon correlation functions for the first time. To demonstrate such a connection, we first study the limit of the transverse spin-dependent cross section in Eq. (19) derived from the collinear twist-3 factorization formalism when . Using Ji:2006br ()

 δ(p2h⊥−z2hQ2(1^x−1)(1^z−1))∣∣∣ph⊥≪Q=(1−^x)(1−^z)p2h⊥[δ(1−^x)(1−^z)++δ(1−^z)(1−^x)++δ(1−^x)δ(1−^z)ln(z2hQ2p2h⊥)], (24)

we find that

 dΔσdxBdydzhd2ph⊥∣∣∣ph⊥≪Q= −zhσ0(ϵαβsα⊥pβh⊥)1(p2h⊥)2∑qe2qαs2π2∫dzzDh/q(z)δ(1−^z) ×∫dxx2Pq←g(^x)(12)[O(x,x)+O(x,0)+N(x,x)−N(x,0)], (25)

where is the usual gluon-to-quark splitting kernel

 Pq←g(^x)=TR[^x2+(1−^x)2], (26)

with the color factor . It is instructive to point out that to arrive at the final result in Eq. (25) we have used integration by parts in Eq. (19) to convert all the derivative terms to non-derivative terms, as well as the fact that vanish when parton momentum fraction .

On the other hand, the TMD factorization formalism Ji:2004wu (); Ji:2004xq (); Collins:2011zzd () for the SIDIS process gives

 dΔσdxBdydzhd2ph⊥= σ0∑qe2q∫d2k⊥d2p⊥d2λ⊥δ2(zhk⊥+p⊥+λ⊥−ph⊥) ×ϵαβsα⊥kβ⊥Mf⊥q1T(xB,k2⊥)Dh/q(zh,p2⊥)S(λ⊥)H(Q2), (27)

where is the quark Sivers function, is the transverse momentum dependent fragmentation function, and denotes the soft and hard factors, respectively. Note that here both and are the so-called unsubtracted TMD functions Collins:2011zzd (). To make contact with the result from the collinear twist-3 formalism in Eq. (25), we need to compute the perturbative tail of the various factors in the TMD formalism in Eq. (27). In particular we need the expansion of the quark Sivers function in terms of the three-gluon correlation functions when . This is usually referred to as the so-called off-diagonal term, where the quark Sivers function receives contributions from the three-gluon correlation functions (quark from gluon), as opposed to the known diagonal term in Ji:2006ub (); Ji:2006vf (); Ji:2006br (); Koike:2007dg (); Bacchetta:2008xw (); Yuan:2009dw (); Boer:2010ya (), where the quark Sivers function receives contributions from the quark-gluon correlation function (quark from quark).

The relevant Feynman diagrams to compute the quark Sivers function in terms of the three-gluon correlation functions are shown in Fig. 3. This is the forward cut scattering diagram, where the left side of the cut (the magenta dashed line) is the amplitude and the right side of the cut is the conjugate to the amplitude. The upper part of this diagram represents the quark Sivers function with the momentum for the quark, where with the momentum of the nucleon. Note that the nucleon is represented by the grey blob in the bottom of the diagram. The double line represents the gauge link (eikonal line) in the definition of the quark Sivers function. In the middle part of the diagram, we have three-gluon correlation functions in the nucleon (as represented by three gluons coming out of the nucleon). In other words, such a diagram just represents the contribution of three-gluon correlation functions to the quark Sivers function, which is very similar to those contributions of the quark-gluon correlation function to the quark Sivers functions, see, e.g., Fig. 9 of Ji:2006vf ()222see also the similar Feynman diagram Fig. 8(e) for the collinear gluon distribution contribution to the unpolarized quark TMD Ji:2006vf ().. To obtain the final result, we have to perform the same collinear expansion as in Eqs. (13) and (16), and the result can be written as the following form

 1Mf⊥q1T(xB,k2⊥)=−αs2π21(k2⊥)2∫1xBdxx2Pq←g(^x)(12)[O(x,x)+O(x,0)+N(x,x)−N(x,0)], (28)

where to arrive at the above result we have again used integration by parts to convert all derivative terms to non-derivative terms.

In order to calculate the explicit -dependence generated by the TMD factorization in Eq. (27) (particularly those related to the quark Sivers function), following Ji:2006br (), we let to be of the order of and the others ( and ) much smaller: . In this case, we can neglect and compared with in the delta function and obtain

 δ2(zhk⊥+p⊥+λ⊥−ph⊥)\lx@stackrelk⊥∼ph⊥−−−−→δ2(zhk⊥−ph⊥). (29)

At the same time, the integration over the other transverse momentum yields the ordinary collinear fragmentation function,

 ∫d2p⊥Dh/q(zh,p2⊥)=Dh/q(zh), (30)

whereas the integration over leads to , for details, see Ji:2006br (); Ji:2004wu (); Ji:2004xq (). Finally, substituting the expansion of the quark Sivers function in terms of three-gluon correlation functions, Eq. (28), into the TMD factorization, Eq. (27), we find:

 dΔσdxBdydzhd2ph⊥∣∣∣ph⊥≪Q= −zhσ0(ϵαβsα⊥pβh⊥)1(p2h⊥)2∑qe2qαs2π2∫dzzDh/q(z)δ(1−^z) ×∫dxx2Pq←g(^x)(12)[O(x,x)+O(x,0)+N(x,x)−N(x,0)]. (31)

It is evident that the above result reproduces the transverse spin-dependent differential cross section in Eq. (25), the one derived from the collinear twist-3 factorization formalism. We have thus demonstrated the consistency between the collinear twist-3 factorization formalism and TMD factorization formalism for the twist-3 three-gluon correlation functions at moderate transverse momenta, .

In principle, to establish fully the connection between TMD and collinear twist-3 formalism at the first non-trivial order, one should also consider the situation where either to be of the order , or to be the order of . Both situations were studied in Ji:2006br (), and they establish the connection for the quark-gluon correlation function or the Qiu-Sterman function, which is different from what we have done here.

Another result from our calculation above will be to obtain the QCD evolution equation of the Qiu-Sterman function , specifically the contribution from the three-gluon correlation functions. To achieve this, we start from Eq. (28) and using the following identity on the left hand side Boer:2003cm (); Kang:2011hk (),

 Tq,F(xB,xB,μ2f)=−1M∫μ2fd2k⊥k2⊥f⊥q1T(xB,k2⊥)|SIDIS, (32)

we find (in the cut-off scheme)

 Tq,F(xB,xB,μ2f)=∫μ2fdk2⊥k2⊥αs2π∫1xBdxx2Pq←g(^x)(12)[O(x,x)+O(x,0)+N(x,x)−N(x,0)]. (33)

The evolution equation corresponding to the above expression is then

 ∂∂lnμ2fTq,F(xB,xB,μ2f)=αs2π∫1xBdxx2Pq←g(^x)(12)[O(x,x,μ2f)+O(x,0,μ2f)+N(x,x,μ2f)−N(x,0,μ2f)], (34)

which is exactly the same as the one derived before from different approaches Kang:2008ey (); Braun:2009mi (); Ma:2012xn (). In the next section, we will perform a complete NLO calculation for the -weighted transverse spin-dependent cross section, and re-derive this evolution equation using dimensional regularization.

### ii.4 Coefficient functions in the TMD evolution formalism

To study the QCD evolution of TMDs, one usually defines the TMDs in the Fourier conjugated 2-dimensional coordinate space - the so-called “-space”. For the quark Sivers function, the common definition in -space is the following Kang:2011mr (); Echevarria:2014xaa () 333Note the proper defined TMDs depend on two additional scales, i.e., the factorization scale and another scale associated with rapidity divergence. Here we suppress both dependences for simplicity.

 f⊥q(α)1T(xB,b)=1M∫d2k⊥e−ik⊥⋅bkα⊥f⊥q1T(xB,k2⊥). (35)

In the perturbative region , one can expand the above quark Sivers function in terms of the corresponding collinear functions, i.e. the twist-3 Qiu-Sterman function as well as the three-gluon correlation functions and . If we collectively denote them as , we can write formally

 f⊥q(α)1T(xB,b)=(ibα2)Cq←i(^x1,^x2)⊗f(3)i(x1,x2), (36)

where is the coefficient function with . The precise meaning of the convolution  will be defined below, where the inclusion of the factor will also become clear.

At leading order, one has Kang:2011mr (); Aybat:2011ge (); Echevarria:2014xaa ():

 f⊥q(α)1T(xB,b)=(ibα2)∫1xBdxxδ(1−^x)Tq,F(x,x), (37)

which tells us that the coefficient function at leading order is given by

 Cq←i=δqiδ(1−^x). (38)

Now, we will study the coefficient function from the expansion of the quark Sivers function in terms of three-gluon correlation functions. To start, we redo the calculation which leads to Eq. (28) in dimensions, and obtain the following result:

 1Mf⊥q1T(xB,k2⊥)= −αs2π2(4π2