# Introduction to the Transverse-Momentum-Weighted Technique in the Twist-3 Collinear Factorization Approach

Authors: Hongxi Xing; Shinsuke Yoshida
Journal: Advances in High Energy Physics
 (2019)

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## Abstract

The twist-3 collinear factorization framework has drawn much attention in recent decades as a successful approach in describing the data for single spin asymmetries (SSAs). Many SSAs data have been experimentally accumulated in a variety of energies since the first measurement was done in the late 1970s and it is expected that the future experiments like Electron-Ion-Collider will provide us with more data. In order to perform a consistent and precise description of the data taken in different kinematic regimes, the scale evolution of the collinear twist-3 functions and the perturbative higher-order hard part coefficients are mandatory. In this paper, we introduce the techniques for next-to-leading order (NLO) calculation of transverse-momentum-weighted SSAs, which can be served as a useful tool to derive the QCD evolution equation for twist-3 functions and to verify the QCD collinear factorization for twist-3 observables at NLO, as well as obtain the finite NLO hard part coefficients.


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## Body

## 1. Introduction

The large single transverse spin asymmetries (SSAs) have been a longstanding problem over 40 years since it was turned out that the conventional perturbative calculation based on the parton model picture failed to describe the large SSAs which were experimentally observed in pion and polarized hyperon productions [1, 2]. In the recent several years, two QCD factorization frameworks have been proposed to study phenomenologically the observed SSAs: the transverse momentum dependent factorization approach [3–15] and the twist-3 collinear factorization approach [16–38]. These two frameworks are shown to be equivalent in the common applied kinematic region [39–46].The twist-3 collinear factorization framework is a natural extension of the conventional perturbative QCD framework and it could give a reasonable description of the large SSAs. Measurements of SSAs at Relativistic-Heavy-Ion-Collider (RHIC) [47–51] have greatly motivated the theoretical work on developing the twist-3 framework, because it is a unique applicable framework for single hadron productions in proton-proton collision. A series of important works have been done in the past a few decades and the SSAs for the hadron production were completed at leading order (LO) with respect to the QCD strong coupling constant αs[16–38]. Recent numerical simulations based on the complete LO result confirmed that the twist-3 approach gives a reasonable description of the SSA data provided by RHIC [52, 53].Electron-Ion-Collider (EIC) is a next-generation hadron collider expected to provide more data in different kinematic regimes for SSAs. In order to extract the fundamental structure of the nucleon from the measurements at a future EIC, comprehensive and precise calculations for SSAs in transversely polarized lepton-proton collision are highly demanded. It is well known that nonperturbative functions in the perturbative QCD calculation, in general, receive logarithmic radiative corrections and the evolution equation with respect to this logarithmic scale is necessary for a systematic treatment of the cross sections in wide range of energies. Most famous example is the DGLAP evolution equation of the twist-2 parton distribution functions (PDFs). Correct description of the small scale violation of the structure function controlled by the DGLAP equation was an important success of the QCD phenomenology in the early days. The twist-3 function is expected to have similar logarithmic dependence and its evolution equation will play an important role in global fitting of the SSA data accumulated in different energies. Consistent description of the data will be a good evidence that the twist-3 framework, one of major fundamental developments in recent QCD phenomenology, is a feasible theory to solve the 40-year mystery in high energy physics. The evolution equations for the twist-3 functions have been derived in two different methods. The first method is a calculation of the higher-order corrections to the nonperturbative function itself [54–61]. Since the nonperturbative function has the operator definition, we can investigate the infrared singularity of the operator through higher-order perturbative calculation. We can read the evolution equation from the infrared structure of the function. This is a standard technique and the evolution equations have been derived for the twist-3 distribution functions for initial state proton [54–59] and the twist-3 fragmentation functions for final state hadron [60, 61]. The second method which we will review in this paper is a transverse-momentum-weighted technique for the SSAs [62–67]. Except for deriving the QCD evolution equation for twist-3 nonperturbative functions, the transverse-momentum-weighted technique can be also used as a tool to verify the twist-3 collinear factorization at higher orders in strong coupling constant αs. There is also phenomenological interest related to this technique. One can use the standard dimensional regularization method to derive the NLO hard part coefficient for transverse-momentum-weighted SSAs, which can be used for high precision extraction of twist-3 functions from the relevant experimental data. The recent measurement of the transverse-momentum-weighted SSAs at COMPASS [68] strongly motivates the phenomenological application of the results reviewed in this paper. We expect more data will be produced in future COMPASS and EIC measurements.The rest of the paper is organized as follows. In Section2 we present the notation and the calculation of transverse-momentum-weighted SSAs at leading order for semi-inclusive deep inelastic scattering (SIDIS). In Section 3 we present the detail of NLO calculation for both real and virtual corrections, we show the cancelation of soft divergence in the sum of real and virtual corrections, and the collinear divergences can be absorbed into the redefinition of twist-3 Qiu-Sterman function and unpolarized leading twist fragmentation function. In Section 4 we review the application of the transverse-momentum-weighted technique to other processes that have been done in recent years. We conclude our paper in Section 5.


## 2. Transverse-Momentum-Weighted SSA at Leading Order

In this paper, we take the process of SIDIS as an example to show the techniques of perturbative calculation for transverse-momentum-weighted differential cross section at twist-3. We start this section by specifying our notation and the kinematics of SIDIS and present the calculation for transverse-momentum-weighted SSA at leading order (LO).


### 2.1. Notation

We consider the scattering of an unpolarized lepton with momentuml on a transversely polarized proton with momentum p and transverse spin S⊥ and observe the final state hadron production with momentum Ph,(1)el+p↑p,S⊥→el′+hPh+X.We focus on one-photon exchange process with the momentum of the virtual photon given by q=l-l′ and its invariant mass Q2=-q2. We define all vectors in the so-called hadron frame. We define pc=Ph/z to be the momentum for the parton that fragments into the final state hadron. The conventional Lorentz invariant variables in SIDIS are defined as(2)Sep=p+l2,xB=Q22p·q,zh=p·Php·q,y=p·qp·l.For clear understanding, we start with the Ph⊥-integrated cross section at leading twist in unpolarized lepton-proton scattering(3)dσdxBdydzh=∫d2Ph⊥dσdxBdydzhd2Ph⊥.There is only one hard scale Q2 in this case; therefore the differential cross section shown above can be reliably computed by using the standard collinear factorization formalism. The LO contribution is given by 2→1 scattering amplitude γ∗+q→q. It is trivial that the LO cross section is proportional to the unpolarized PDFs and the unpolarized fragmentation functions (FFs),(4)dσLOdxBdydzh=σ0∑qfq/pxB,μ2Dq→hzh,μ2,where σ0 is the LO Born cross section σ0=2παem2/Q21+(1-y)2/y with αem=e2/4π being the QED coupling constant. The bare results at O(αs) contain infrared divergences which represent the long-range interaction in hadronic collision process. These divergences are canceled by the renormalization of the PDFs and FFs and the DGLAP evolution equations are derived as the renormalization group equations. The final result at NLO can be written as the convolution of finite hard part coefficient H and nonperturbative functions (PDFs and FFs) [63](5)dσNLOdxBdydzh=∑i,jfi/p⊗Hγ∗+i→j+kNLO⊗Dj→h.where ⊗ represents convolution.The concept of the transverse-momentum-weighted technique is mostly the same with the twist-2 case. Notice that directPh⊥-integration of the cross section for unpolarized lepton scattering off transversely polarized proton vanishes due to the linear dependence of Ph. Realize that the SSA is characterized in terms of three vectors: the momentum of the final state hadron, the momentum, and the spin of the initial state proton, which can be combined as(6)ϵαβρσPhαS⊥βpρnσ=ϵijPh⊥iS⊥j≡ϵPh⊥S⊥pn,where ϵij is a two-dimensional antisymmetric tensor with ϵ12=1; n is an arbitrary vector which satisfies p·n=1 and n2=0. We introduce a weight factor ϵPh⊥S⊥pn and consider the following transverse-momentum-weighted differential cross section(7)dPh⊥ΔσdxBdydzh≡∫d2Ph⊥ϵPh⊥S⊥pndΔσdxBdydzhd2Ph⊥,which is well defined after Ph⊥-integration. Since the virtuality Q2 is the only hard scale after the Ph⊥-integration, one can safely use the collinear twist-3 factorization formalism, and the technique in performing NLO calculation will follow those used at leading twist. Same technique has been applied to Drell-Yan dilepton production in proton-proton collisions [62, 66] and can be extended to polarized electron-positron collisions.We recall the cross section for SIDIS presented in [64],(8)dΔσdxBdydzhd2Ph⊥=αem2128π4zhxB2Sep2Q2LμνWμν,where Lμν=2(lμlν′+lνlμ′)-Q2gμν is the leptonic tensor. We focus on the metric contribution Lμν→-Q2gμν and the SSA generated by initial state twist-3 distribution functions of the transversely polarized proton. Then we can factorize the nonperturbative part by introducing the usual twist-2 unpolarized fragmentation function(9)Wμν=∑i∫dzz2wiμνDi→hz.The hadronic tensor wiμν describes a scattering of the virtual photon and the transversely polarized proton. We will make the subscript i implicit in the rest part of this paper for simplicity.


### 2.2. Leading Order

We demonstrate how to derive the LO cross section for the transverse-momentum-weighted SSA based on the collinear twist-3 framework and show that the LO cross section is proportional to the first moment of the TMD Sivers function. The twist-3 calculation is well formulated in the diagrammatic method. We consider a set of the general diagrams shown in Figure1 and extract twist-3 contributions from these diagrams. We start from the first diagram in Figure 1, which can be expressed as(10)w1μν=∫d4ξ∫d4k2π4eik·ξ·pS⊥ψ¯j0ψiξpS⊥·Hjiμνkδ2k⊥-pc⊥,where k and pc are the momenta of the parton from initial state proton and that fragments to the final state observed hadron, respectively, pc⊥=Ph⊥/z. The hard part at LO is given by(11)Hjiμνk=γνk+qγμji2π4δk++q+-pc+δk-+q--pc-.We perform Ph⊥-integration before the collinear expansion,(12)∫d2Ph⊥ϵPh⊥S⊥pnδ2k⊥-pc⊥=z3ϵαS⊥pnk⊥α.Because the Ph⊥-integration gives O(k⊥) factor, we can identify the leading term in the collinear expansion as a twist-3 contribution. We perform the collinear expansion of the hard part around kμ=(k·n)pμ≡kpμ(13)Hμνk=Hμνkp+Ok⊥,and substitute the above expansion into the hadronic tensor as shown in (10)(14)∫d2Ph⊥ϵPh⊥S⊥pnw1μν=z3ϵαS⊥pn∫d4ξ∫d4k2π4·eik·ξpS⊥ψ¯j0ψiξpS⊥k⊥αHjiμνkp=iz3ϵαS⊥pn∫dx∫dλ2π·eiλxpS⊥ψ¯j0∂αψiλpS⊥Hjiμνxp.where λ=p+ξ-. Now we turn to the second and the third diagrams in Figure 1. These two diagrams can be expressed as(15)w2μν=∫d4ξ1∫d4ξ2∫d4k12π4∫d4k22π4·eik1·ξ1eik2-k1·ξ2pS⊥ψ¯j0gAρξ2ψiξ1pS⊥×HLρjiμνk1,k2δ2k2⊥-pc⊥+HRρjiμνk1,k2·δ2k1⊥-pc⊥HLρjiμν,where the hard parts are given by(16)HLρjiμνk1,k2=-γνk2+qγρk1+qγμji1q+k12+iϵ2π4δk2++q+-pc+δk2-+q--pc-,HRρjiμνk1,k2=-γνk2+qγρk1+qγμji1q+k22-iϵ2π4δk1++q+-pc+δk1-+q--pc-.The Ph⊥-integration gives O(k1,2⊥) and then the leading term in collinear expansion gives twist-3 contribution again,(17)HLRρjiμνk1,k2=HLRρjiμνk1p,k2p+Ok1,2⊥.For the matrix element, we have to separate the components of the gluon field Aρ into “longitudinal" and “transverse" part as(18)Aρ=Anpρ+Aρ-Anpρ.The longitudinal part Anpρ gives the leading contribution. It is straightforward to derive the Ward-Takahashi identities (WTIs) for the hard parts,(19)pρHLρjiμνk1,k2=γνx2p+qγμji1x2-x1-iϵ2π42xBQ2δx2-xBδ1-z^=1x2-x1-iϵHjiμνx2p,pρHRρjiμνk1,k2=-γνx1p+qγμji1x2-x1-iϵ2π42xBQ2δx1-xBδ1-z^=-1x2-x1-iϵHjiμνx1p.Finally the hadronic tensor shown in (15) can be expressed as(20)∫d2Ph⊥ϵPh⊥S⊥pnw2μν=z3ϵαS⊥pn∫d4ξ1∫d4ξ2·∫d4k12π4∫d4k22π4·eik1·ξ1eik2-k1·ξ2·pS⊥ψ¯j0gAnξ2ψiξ1pS⊥×1x2-x1-iϵ·k2⊥αHjiμνx2p-k1⊥αHjiμνx1p=z3ϵαS⊥pn∫d4ξ1·∫d4ξ2∫d4k12π4·∫d4k22π4eik1·ξ1eik2-k1·ξ2·pS⊥ψ¯j0gAnξ2ψiξ1pS⊥×1x2-x1-iϵHjiμνx2p-Hjiμνx1pk2⊥α-k1⊥α·Hjiμνx2p+k1⊥αHjiμνx2p-Hjiμνx1p=iz3ϵαS⊥pn∫dx∫dλ2πeiλx·pS⊥ψ¯j0ig∫λ∞dλ′∂αAnλ′n-∂nAαλ′nψiλnpS⊥·Hjiμνxp+iz3ϵαS⊥pn·∫dx∫dλ2πeiλx·pS⊥ψ¯j0ig∫λ0dλ′Anλ′n∂αψiλnpS⊥·Hjiμνxp-iz3ϵαS⊥pn·∫dx∫dλ2πeiλx·pS⊥ψ¯j0igAαλnψiλnpS⊥·Hjiμνxp.Combining (14) and (20), we can obtain the result(21)∫d2Ph⊥ϵPh⊥S⊥pnwμν=iz3ϵαS⊥pn·∫dx∫dλ2πeiλx·pS⊥ψ¯j0Dαλnψiλn+ig∫λ0dλ′Anλ′n∂αψiλnpS⊥·Hjiμνxp+iz3ϵαS⊥pn·∫dx∫dλ2πeiλx·pS⊥ψ¯j0ig∫λ∞dλ′∂αAnλ′n-∂nAαλ′nψiλnpS⊥·Hjiμνxp.We can find that this matrix element corresponds to O(g) term in the first moment of the TMD correlator(22)∫d2pTpTα∫dλ2π∫dxT2πeiλxeixT·pT·pS⊥ψ¯j00,∞n·∞n,∞n+xT∞n+xT,λn+xT·ψiλn+xTpS⊥dλ2π=i∫dλ2πeiλx·pS⊥ψ¯j00,λnDαλnψiλnpS⊥+i∫dλ2πeiλx∫λ∞dλ′·pS⊥ψ¯j00,λ′nigFαnλ′nλ′n,λnψiλnpS⊥=-πMN4ϵαpnS⊥Gq,Fx,x+⋯,where [⋯] represents the Wilson line, MN is the nucleon mass, and we used the fact that the first moment of the TMD Sivers function gives the Qiu-Sterman function Gq,F(x,x). The nonlinear term in the field strength tensor Fα+ and the higher-order terms in the Wilson lines which have to be added to (21) come from the more gluon-linked diagrams in Figure 2. Finally we can derive the LO cross section formula as(23)dPh⊥ΔσLOdxBdydzh=σ~0∑qeq2Gq,FxB,xBDq→hzh.where σ~0=-πMNσ0. As we demonstrated here, the LO cross section for the weighted SSA is proportional to the first moment of the TMD function (also known as the kinematical twist-3 function). We can expect the NLO contribution gives the evolution equation to the twist-3 Qiu-Sterman function, Gq,F(x,x) in this case. This technique is quite general so that we can apply the same technique to other TMD functions. When we focus on the twist-3 fragmentation effect, we can derive the evolution equation for the first Moment of the Collins function. When we consider the double spin asymmetry ALT and change the weight factor ϵijPh⊥iS⊥j→(Ph·S⊥), we can investigate other TMD distribution functions like the Worm-Gear and the Pretzelosity.Figure 1
A series of LO diagrams in the diagrammatic method.Figure 2
The NLO virtual correction diagrams in SIDIS.


## 2.1. Notation

We consider the scattering of an unpolarized lepton with momentuml on a transversely polarized proton with momentum p and transverse spin S⊥ and observe the final state hadron production with momentum Ph,(1)el+p↑p,S⊥→el′+hPh+X.We focus on one-photon exchange process with the momentum of the virtual photon given by q=l-l′ and its invariant mass Q2=-q2. We define all vectors in the so-called hadron frame. We define pc=Ph/z to be the momentum for the parton that fragments into the final state hadron. The conventional Lorentz invariant variables in SIDIS are defined as(2)Sep=p+l2,xB=Q22p·q,zh=p·Php·q,y=p·qp·l.For clear understanding, we start with the Ph⊥-integrated cross section at leading twist in unpolarized lepton-proton scattering(3)dσdxBdydzh=∫d2Ph⊥dσdxBdydzhd2Ph⊥.There is only one hard scale Q2 in this case; therefore the differential cross section shown above can be reliably computed by using the standard collinear factorization formalism. The LO contribution is given by 2→1 scattering amplitude γ∗+q→q. It is trivial that the LO cross section is proportional to the unpolarized PDFs and the unpolarized fragmentation functions (FFs),(4)dσLOdxBdydzh=σ0∑qfq/pxB,μ2Dq→hzh,μ2,where σ0 is the LO Born cross section σ0=2παem2/Q21+(1-y)2/y with αem=e2/4π being the QED coupling constant. The bare results at O(αs) contain infrared divergences which represent the long-range interaction in hadronic collision process. These divergences are canceled by the renormalization of the PDFs and FFs and the DGLAP evolution equations are derived as the renormalization group equations. The final result at NLO can be written as the convolution of finite hard part coefficient H and nonperturbative functions (PDFs and FFs) [63](5)dσNLOdxBdydzh=∑i,jfi/p⊗Hγ∗+i→j+kNLO⊗Dj→h.where ⊗ represents convolution.The concept of the transverse-momentum-weighted technique is mostly the same with the twist-2 case. Notice that directPh⊥-integration of the cross section for unpolarized lepton scattering off transversely polarized proton vanishes due to the linear dependence of Ph. Realize that the SSA is characterized in terms of three vectors: the momentum of the final state hadron, the momentum, and the spin of the initial state proton, which can be combined as(6)ϵαβρσPhαS⊥βpρnσ=ϵijPh⊥iS⊥j≡ϵPh⊥S⊥pn,where ϵij is a two-dimensional antisymmetric tensor with ϵ12=1; n is an arbitrary vector which satisfies p·n=1 and n2=0. We introduce a weight factor ϵPh⊥S⊥pn and consider the following transverse-momentum-weighted differential cross section(7)dPh⊥ΔσdxBdydzh≡∫d2Ph⊥ϵPh⊥S⊥pndΔσdxBdydzhd2Ph⊥,which is well defined after Ph⊥-integration. Since the virtuality Q2 is the only hard scale after the Ph⊥-integration, one can safely use the collinear twist-3 factorization formalism, and the technique in performing NLO calculation will follow those used at leading twist. Same technique has been applied to Drell-Yan dilepton production in proton-proton collisions [62, 66] and can be extended to polarized electron-positron collisions.We recall the cross section for SIDIS presented in [64],(8)dΔσdxBdydzhd2Ph⊥=αem2128π4zhxB2Sep2Q2LμνWμν,where Lμν=2(lμlν′+lνlμ′)-Q2gμν is the leptonic tensor. We focus on the metric contribution Lμν→-Q2gμν and the SSA generated by initial state twist-3 distribution functions of the transversely polarized proton. Then we can factorize the nonperturbative part by introducing the usual twist-2 unpolarized fragmentation function(9)Wμν=∑i∫dzz2wiμνDi→hz.The hadronic tensor wiμν describes a scattering of the virtual photon and the transversely polarized proton. We will make the subscript i implicit in the rest part of this paper for simplicity.


## 2.2. Leading Order

We demonstrate how to derive the LO cross section for the transverse-momentum-weighted SSA based on the collinear twist-3 framework and show that the LO cross section is proportional to the first moment of the TMD Sivers function. The twist-3 calculation is well formulated in the diagrammatic method. We consider a set of the general diagrams shown in Figure1 and extract twist-3 contributions from these diagrams. We start from the first diagram in Figure 1, which can be expressed as(10)w1μν=∫d4ξ∫d4k2π4eik·ξ·pS⊥ψ¯j0ψiξpS⊥·Hjiμνkδ2k⊥-pc⊥,where k and pc are the momenta of the parton from initial state proton and that fragments to the final state observed hadron, respectively, pc⊥=Ph⊥/z. The hard part at LO is given by(11)Hjiμνk=γνk+qγμji2π4δk++q+-pc+δk-+q--pc-.We perform Ph⊥-integration before the collinear expansion,(12)∫d2Ph⊥ϵPh⊥S⊥pnδ2k⊥-pc⊥=z3ϵαS⊥pnk⊥α.Because the Ph⊥-integration gives O(k⊥) factor, we can identify the leading term in the collinear expansion as a twist-3 contribution. We perform the collinear expansion of the hard part around kμ=(k·n)pμ≡kpμ(13)Hμνk=Hμνkp+Ok⊥,and substitute the above expansion into the hadronic tensor as shown in (10)(14)∫d2Ph⊥ϵPh⊥S⊥pnw1μν=z3ϵαS⊥pn∫d4ξ∫d4k2π4·eik·ξpS⊥ψ¯j0ψiξpS⊥k⊥αHjiμνkp=iz3ϵαS⊥pn∫dx∫dλ2π·eiλxpS⊥ψ¯j0∂αψiλpS⊥Hjiμνxp.where λ=p+ξ-. Now we turn to the second and the third diagrams in Figure 1. These two diagrams can be expressed as(15)w2μν=∫d4ξ1∫d4ξ2∫d4k12π4∫d4k22π4·eik1·ξ1eik2-k1·ξ2pS⊥ψ¯j0gAρξ2ψiξ1pS⊥×HLρjiμνk1,k2δ2k2⊥-pc⊥+HRρjiμνk1,k2·δ2k1⊥-pc⊥HLρjiμν,where the hard parts are given by(16)HLρjiμνk1,k2=-γνk2+qγρk1+qγμji1q+k12+iϵ2π4δk2++q+-pc+δk2-+q--pc-,HRρjiμνk1,k2=-γνk2+qγρk1+qγμji1q+k22-iϵ2π4δk1++q+-pc+δk1-+q--pc-.The Ph⊥-integration gives O(k1,2⊥) and then the leading term in collinear expansion gives twist-3 contribution again,(17)HLRρjiμνk1,k2=HLRρjiμνk1p,k2p+Ok1,2⊥.For the matrix element, we have to separate the components of the gluon field Aρ into “longitudinal" and “transverse" part as(18)Aρ=Anpρ+Aρ-Anpρ.The longitudinal part Anpρ gives the leading contribution. It is straightforward to derive the Ward-Takahashi identities (WTIs) for the hard parts,(19)pρHLρjiμνk1,k2=γνx2p+qγμji1x2-x1-iϵ2π42xBQ2δx2-xBδ1-z^=1x2-x1-iϵHjiμνx2p,pρHRρjiμνk1,k2=-γνx1p+qγμji1x2-x1-iϵ2π42xBQ2δx1-xBδ1-z^=-1x2-x1-iϵHjiμνx1p.Finally the hadronic tensor shown in (15) can be expressed as(20)∫d2Ph⊥ϵPh⊥S⊥pnw2μν=z3ϵαS⊥pn∫d4ξ1∫d4ξ2·∫d4k12π4∫d4k22π4·eik1·ξ1eik2-k1·ξ2·pS⊥ψ¯j0gAnξ2ψiξ1pS⊥×1x2-x1-iϵ·k2⊥αHjiμνx2p-k1⊥αHjiμνx1p=z3ϵαS⊥pn∫d4ξ1·∫d4ξ2∫d4k12π4·∫d4k22π4eik1·ξ1eik2-k1·ξ2·pS⊥ψ¯j0gAnξ2ψiξ1pS⊥×1x2-x1-iϵHjiμνx2p-Hjiμνx1pk2⊥α-k1⊥α·Hjiμνx2p+k1⊥αHjiμνx2p-Hjiμνx1p=iz3ϵαS⊥pn∫dx∫dλ2πeiλx·pS⊥ψ¯j0ig∫λ∞dλ′∂αAnλ′n-∂nAαλ′nψiλnpS⊥·Hjiμνxp+iz3ϵαS⊥pn·∫dx∫dλ2πeiλx·pS⊥ψ¯j0ig∫λ0dλ′Anλ′n∂αψiλnpS⊥·Hjiμνxp-iz3ϵαS⊥pn·∫dx∫dλ2πeiλx·pS⊥ψ¯j0igAαλnψiλnpS⊥·Hjiμνxp.Combining (14) and (20), we can obtain the result(21)∫d2Ph⊥ϵPh⊥S⊥pnwμν=iz3ϵαS⊥pn·∫dx∫dλ2πeiλx·pS⊥ψ¯j0Dαλnψiλn+ig∫λ0dλ′Anλ′n∂αψiλnpS⊥·Hjiμνxp+iz3ϵαS⊥pn·∫dx∫dλ2πeiλx·pS⊥ψ¯j0ig∫λ∞dλ′∂αAnλ′n-∂nAαλ′nψiλnpS⊥·Hjiμνxp.We can find that this matrix element corresponds to O(g) term in the first moment of the TMD correlator(22)∫d2pTpTα∫dλ2π∫dxT2πeiλxeixT·pT·pS⊥ψ¯j00,∞n·∞n,∞n+xT∞n+xT,λn+xT·ψiλn+xTpS⊥dλ2π=i∫dλ2πeiλx·pS⊥ψ¯j00,λnDαλnψiλnpS⊥+i∫dλ2πeiλx∫λ∞dλ′·pS⊥ψ¯j00,λ′nigFαnλ′nλ′n,λnψiλnpS⊥=-πMN4ϵαpnS⊥Gq,Fx,x+⋯,where [⋯] represents the Wilson line, MN is the nucleon mass, and we used the fact that the first moment of the TMD Sivers function gives the Qiu-Sterman function Gq,F(x,x). The nonlinear term in the field strength tensor Fα+ and the higher-order terms in the Wilson lines which have to be added to (21) come from the more gluon-linked diagrams in Figure 2. Finally we can derive the LO cross section formula as(23)dPh⊥ΔσLOdxBdydzh=σ~0∑qeq2Gq,FxB,xBDq→hzh.where σ~0=-πMNσ0. As we demonstrated here, the LO cross section for the weighted SSA is proportional to the first moment of the TMD function (also known as the kinematical twist-3 function). We can expect the NLO contribution gives the evolution equation to the twist-3 Qiu-Sterman function, Gq,F(x,x) in this case. This technique is quite general so that we can apply the same technique to other TMD functions. When we focus on the twist-3 fragmentation effect, we can derive the evolution equation for the first Moment of the Collins function. When we consider the double spin asymmetry ALT and change the weight factor ϵijPh⊥iS⊥j→(Ph·S⊥), we can investigate other TMD distribution functions like the Worm-Gear and the Pretzelosity.Figure 1
A series of LO diagrams in the diagrammatic method.Figure 2
The NLO virtual correction diagrams in SIDIS.


## 3. Transverse-Momentum-Weighted SSA at NLO

In this section, we review the calculation for transverse-momentum-weighted SSA at NLO including both real and virtual corrections.


### 3.1. Virtual Correction

We first consider the NLO contribution from the virtual correction which is given by the2→1 scattering amplitude with one gluon loop. When we adopt the dimensional regularization scheme, the gauge-invariance of the cross section is maintained for the loop diagram. Then we can derive the same WTI as shown in (19) which is the consequence of gauge-invariance of the diagrams. Taking advantage of this fact, we just need to calculate simple diagrams shown in Figure 2 for the virtual correction. The calculation for this kind of one-loop diagrams has been well established, which is exactly the same as the vertex correction at leading twist. We follow the conventional technique here. All ultraviolet divergences can be canceled by the renormalization of the QCD Lagrangian. Then we can set the ultraviolet and infrared divergences are the same with each other in dimensions regularization approach, ϵUV=ϵIR, and identify all divergences as infrared. In this definition, we do not have to think about the first and the third amplitudes in Figure 2 because these are exactly zero in the mass case as we considered here. The hard partonic cross section with the second amplitude is given by(24)-11-ϵgμνCFg2μ2ϵ∫dDl2πDiTrxpγνpcγρpc-lγμxp-lγρ1l2pc-l2xp-l2=CFg2μ2ϵ∫dDl2πDi-41pc-l2xp-l232+ϵQ2+4Q2l2pc-l2xp-l2,where ϵ=(4-D)/2 in D-dimension, we made a change gμν→(1/(1-ϵ))gμν for D-dimensional calculation, and we used the fact that(25)∫dDl2πDi1l2=∫dDl2πDi1xp-l2=∫dDl2πDi1l2pc-l2=∫dDl2πDi1l2xp-l2=0.We perform the basic D-dimensional calculation for each integration,(26)∫dDl2πDi1pc-l2xp-l2=116π24πQ2ϵ1Γ1-ϵΓ1-ϵΓϵB1-ϵ,1-ϵ=116π24πQ2ϵ1Γ1-ϵ1ϵ+2+Oϵ,(27)∫dDl2πDi1l2pc-l2xp-l2=-116π2Q24πQ2ϵ1Γ1-ϵΓ1-ϵΓ1+ϵB1,-ϵB-ϵ,1-ϵ=-116π2Q24πQ2ϵ1Γ1-ϵ1ϵ2+Oϵ.The complex-conjugate diagram gives the same contribution. Then we can show the cross section for the NLO virtual correction as(28)dPh⊥ΔσvirtualdxBdydzh=σ~0αs2π∑qeq2Gq,FxB,xBDq→hzhCF4πμ2Q2ϵ1Γ1-ϵ-2ϵ2-3ϵ-8,this is exactly the same as the virtual correction at leading twist. The strategy in the virtual correction calculation presented here is different with that shown in [63], in which the authors did not use the WTI shown in (19) and directly calculated HL(R)ρjiμν(k1,k2) which has one more external gluon line with the momentum k2-k1 in Figure 2. The authors obtained a consistent result with (28), which demonstrated the validation of WTI through explicit calculation by including all virtual diagrams shown in Figures 3 and 4 in [63]. We would like to comment that the WTI reduces much calculational cost. The direct calculation of HL(R)ρjiμν(k1,k2) takes tremendous time as they contain significant amount of tensor reduction and integration. These two calculations should be conceptually the same with each other as long as we correctly keep track of all imaginary contributions. We confirmed in this section the consistency mathematically in 2→1-scattering case. The consistency check in a more general way will be a future task in the collinear twist-3 factorization approach.


### 3.2. Real Correction

We now complete the NLO calculation by adding the real emission contribution represented by2→2 partonic scattering process. The calculation for 2→2 scattering diagrams has been well studied in Ph⊥-unintegrated case. We just have to repeat the same calculation but in D-dimension. We adopt the conventional technique by separating the propagator into the principle value part and imaginary part [18–33],(29)1k2+iϵ→P1k2-iπδk2,and we focus on the pole contribution -iδ(k2) which is required to generate the phase space for SSA. The derivation of the cross section for the pole contribution has been well developed so far based on the diagrammatic method we reviewed in Section 2. Here we recall the result derived in [23–28] as(30)wμν=i∫dx1∫dx2MijFαx1,x2pβ∂∂k2αHLjiβpoleμνk1,k2+HRjiβpoleμνk1,k2ki=xip,where the matrix element MijFα is given by(31)MijFαx1,x2=∫dλ2π∫dλ′2π·eiλx1eiλ′x2-x1·pS⊥ψ¯j0gFαnλ′nψiλnpS⊥=MN4ϵαpnS⊥pijGq,Fx1,x2+iS⊥αγ5pijG~q,Fx1,x2+⋯.We can construct the gauge-invariant expression (30) before performing the ph-integration. There are three types of pole contributions in SIDIS which are, respectively, known as soft-gluon pole contribution (SGP, x1=x2=x), hard pole contribution (HP, x1=x,x2=xB or x1=xB,x2=x), and another hard pole contribution (HP2, x1=xB,x2=xB-x or x2=xB,x1=xB-x) [23–28, 64], and the corresponding hard parts are given by(32)HLjiβpoleμν=HLjiβSGPμνk1,k2-iπδpc-k2-k122πδk2+q-pc2+HLjiβHPμνk1,k2-iπδk1+q22πδk2+q-pc2+HLjiβHP2μνk1,k2iπδk2+q22πδk2-k1+q-pc2,HRjiβpoleμν=HRjiβSGPμνk1,k2iπδpc+k2-k122πδk1+q-pc2+HRjiβHPμνk1,k2iπδk2+q22πδk1+q-pc2+HRjiβHP2μνk1,k2-iπδk1+q22πδk1-k2+q-pc2.Typical diagrams for each pole contribution are shown in Figure 3 (full diagrams can be found in [64]). We write down the explicit form of each diagram in Figure 3 in order to help readers to follow our calculation,(33)HLjiβSGPμνk1,k2Figure3=-γσpc-qγνpcγβpc-k2+k1γμpc-k2+k1-qγρji1pc-q21pc-k2+k1-q2-g⊥ρσk2+q-pc,HLjiβHPμνk1,k2Figure3=-γσpc-qγνpcγβpc-k2+k1γρk1+qγμji1pc-q21pc-k2+k12-g⊥ρσk2+q-pc,HLjiβHP2μνk1,k2Figure3=γνk2+qγρpcγβpc-k2+k1γμpc-k2+k1-qγρji1pc-k2-q21pc-k2+k12,where g⊥ρσ is the sum of the polarization vector ∑rϵrρ(k)ϵrσ(k)=-g⊥ρσ(k). We can show the WTI for the pole diagrams(34)k2-k1βHLRjiβpoleμνk1,k2=0,and it gives a useful relation(35)pβ∂∂k2αHLRjiβpoleμνk1,k2ki=xip=-1x2-x1-iϵHLRjiαpoleμνk1,k2.For HP and HP2 contributions, we can use (35) and do not have to perform k2-derivative directly as in (30). However, we cannot use this relation for SGP contribution because it contains a delta function δ(x1-x2). In [69], the authors found a reduction formula for SGP contribution as(36)pβ∂∂k2αHLjiβSGPμνk1,k2+HRjiβSGPμνk1,k2=12NCF1x2-x1-iϵ∂∂pcα-pcαpμpc·p∂∂pcμHxp,where H(xp) is the 2→2-scattering cross section without the external gluon line with momentum (x2-x1)p. Substituting (31) into (30) and using ((35), (36)), we can derive the following result:(37)wμν=MNπ22∫dxxδxp+q-pc2-2s^+Q2ϵpcpnS⊥+u^ϵqpnS⊥ddxGq,Fx,x1t^u^TrxpHxp-2s^+Q2ϵpcpnS⊥+u^ϵqpnS⊥Gq,Fx,x1t^u^Q2∂∂s^-∂∂Q2TrxpHxp-TrxpHxp+Gq,Fx,xB1x^-1x^Q2ϵαpnS⊥TrxpHLHPαxBp,xp+TrxpHRHPαxp,xBp-G~q,Fx,xB1x^-1x^Q2iS⊥αTrγ5xpHLHPαxBp,xp-Trγ5xpHRHPαxp,xBp+Gq,FxB,xB-xx^Q2ϵαpnS⊥TrxpHLHP2αxB-xp,xBp+TrxpHRHP2αxBp,xB-xp-G~q,FxB,xB-xx^Q2iS⊥αTrγ5xpHLHP2αxB-xp,xBp-Trγ5xpHRHP2αxBp,xB-xp,where we used the Mandelstam variables(38)s^=xp+q2=1-x^x^Q2,(39)t^=pc-q2=-1-z^x^Q2,(40)u^=xp-pc2=-z^x^Q2,with x^=xB/x, z^=zh/z. Then the cross section can be written as(41)d4Ph⊥ΔσrealdxBdydzh~μ2ϵ∑qeq2∫dzDq→hz∫d2-2ϵpc⊥2π2-2ϵ∫dxxδpc⊥2-1-x^1-z^z^x^Q2×11-ϵddxGq,Fx,xHD+Gq,Fx,xHND+Gq,Fx,xBHHP+G~q,Fx,xBHHPT+Gq,FxB,xB-xHHP2+G~q,FxB,xB-xHHPT2,where we used the symmetry of the Ph⊥-integration as(42)∫d2-2ϵPh⊥Ph⊥αPh⊥βϵραpnϵβpnσ=-∫d2-2ϵPh⊥121-ϵPh⊥2g⊥αβϵραpnϵβpnσ.The authors of [63] found that the factor 1-ϵ in the denominator is essential to derive the correct evolution function of Gq,F(x,x). They calculated all SGP and HP contributions associated with Gq,F. After that, the HP2 contribution and all G~q,F contributions were calculated in [64]. The results of all hard cross sections are listed in [64]. We perform the pc⊥-integration,(43)∫d2-2ϵpc⊥2π2-2ϵδpc⊥2-1-x^1-z^z^x^Q2=12π2-2ϵ∫dpc⊥∫dΩ2-2ϵpc⊥1-2ϵδpc⊥2-1-x^1-z^z^x^Q2=14π4πQ2ϵ1Γ1-ϵ1-x^1-z^z^x^-ϵ,where Ω2-2ϵ is the solid angle and it can be integrated out as(44)∫dΩ2-2ϵ=2π1-ϵΓ1-ϵ.We carry out the ϵ-expansion as follows:(45)z^-ϵ≃1-ϵln⁡z^,(46)x^ϵ≃1+ϵln⁡x^,(47)1-z^-1-ϵ≃-1ϵδ1-z^+11-z^+-ϵln⁡1-z^1-z^+,(48)1-x^-1-ϵ≃-1ϵδ1-x^+11-x^+-ϵln⁡1-x^1-x^+.Then the cross section can be derived as follows:(49)dPh⊥ΔσrealdxBdydzh=σ~0αs2π4πμ2Q2ϵ1Γ1-ϵ·∑qeq2CF2ϵ2Gq,FxB,xBDq→hzh+-1ϵ1+x^21-x^+Dq→hzh1+x^21-x^+∫xB1dxxCF1+x^21-x^+Gq,Fx,x+N21+x^Gq,FxB,x-1+x^2Gq,Fx,x1-x^++G~q,FxB,x1+x^Gq,FxB,x-1+x^2Gq,Fx,x1-x^+-NGq,FxB,xB+12N∫xB1dxx1-2x^Gq,FxB,xB-x+G~q,FxB,xB-x1+x^21-x^++Gq,FxB,xBCF∫zh1dzz1+z^21-z^+Dq→hz1+x^21-x^++finiteterms,where the derivative term was converted to the nonderivative term through the partial integral,(50)∫xB1dxddxGq,Fx,x1+x^2=∫xB1dxxGq,Fx,x2x^2-2δ1-x^.Combine the real (see (28)) and virtual corrections (see (49)), we find that the double-pole 1/ϵ2 term, which represents a soft-collinear divergence, cancels out. The remaining collinear divergences, represented by the single pole 1/ϵ, can be eliminated by the redefinition of Qiu-Sterman function and fragmentation function,(51)Gq,FxB,xB=Gq,F0xB,xB+αs2π-1ϵ^∫xB1dxxPqqx^Gq,Fx,x+N21+x^Gq,FxB,x-1+x^2Gq,Fx,x1-x^++G~q,FxB,x-NGq,FxB,xB+12N∫xB1dxx1-2x^Gq,FxB,xB-x+G~q,FxB,xB-x,(52)Dq→hzh=Dq→h0zh+αs2π-1ϵ^∫zh1dzzPqqz^Dq→hz,where Pqq(x) is well known q→q splitting function(53)Pqqx=CF1+x21-x++32δ1-x,and we adopted the MS¯-scheme(54)1ϵ^=1ϵ-γE+ln⁡4π.This collinear singularity is consistent with that of Gq,F(x,x) explored in [54–59]. After the renormalization, the final result for the transverse-momentum-weighted SSAs at NLO is given by(55)
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x
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x
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·
1
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x
^
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x
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z
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s
2
.Note that the cross section above does not include the contribution from the gluon fragmentation channel. From the requirement that the physical cross section does not depend on the factorization scale μ,(56)∂∂ln⁡μ2dPh⊥ΔσLO+NLOdxBdydzh=0,we can derive the LO evolution equation for Gq,F(x,x),(57)∂∂ln⁡μ2Gq,FxB,xB,μ2=αs2π∫xB1dxx1+x^Gq,FxB,x,μ2-1+x^2Gq,Fx,x,μ21-x^+Pqqx^Gq,Fx,x,μ2+N2·1+x^Gq,FxB,x,μ2-1+x^2Gq,Fx,x,μ21-x^++G~q,FxB,x,μ21+x^Gq,FxB,x,μ2-1+x^2Gq,Fx,x,μ21-x^+-NGq,FxB,xB,μ2+12N∫xB1dxx1-2x^Gq,FxB,xB-x,μ2+G~q,FxB,xB-x,μ21+x^Gq,FxB,x,μ2-1+x^2Gq,Fx,x,μ21-x^+.This evolution equation based on the transverse-momentum-weighted technique was first discussed in Drell-Yan process [62], in which the authors succeeded in deriving the Gq,F terms in the parenthesis [⋯] in the evolution equation. After that, the authors of [63] pointed out that an extra term -NGq,F(xB,xB,μ2) also contributes to the evolution equation. The HP2 pole contribution and all G~q,F terms were obtained in [64] within the method of transverse momentum weighting.Figure 3
Typical diagrams for soft-gluon pole (left), hard pole (middle), and another hard pole (right). The red barred propagator gives the pole term.


## 3.1. Virtual Correction

We first consider the NLO contribution from the virtual correction which is given by the2→1 scattering amplitude with one gluon loop. When we adopt the dimensional regularization scheme, the gauge-invariance of the cross section is maintained for the loop diagram. Then we can derive the same WTI as shown in (19) which is the consequence of gauge-invariance of the diagrams. Taking advantage of this fact, we just need to calculate simple diagrams shown in Figure 2 for the virtual correction. The calculation for this kind of one-loop diagrams has been well established, which is exactly the same as the vertex correction at leading twist. We follow the conventional technique here. All ultraviolet divergences can be canceled by the renormalization of the QCD Lagrangian. Then we can set the ultraviolet and infrared divergences are the same with each other in dimensions regularization approach, ϵUV=ϵIR, and identify all divergences as infrared. In this definition, we do not have to think about the first and the third amplitudes in Figure 2 because these are exactly zero in the mass case as we considered here. The hard partonic cross section with the second amplitude is given by(24)-11-ϵgμνCFg2μ2ϵ∫dDl2πDiTrxpγνpcγρpc-lγμxp-lγρ1l2pc-l2xp-l2=CFg2μ2ϵ∫dDl2πDi-41pc-l2xp-l232+ϵQ2+4Q2l2pc-l2xp-l2,where ϵ=(4-D)/2 in D-dimension, we made a change gμν→(1/(1-ϵ))gμν for D-dimensional calculation, and we used the fact that(25)∫dDl2πDi1l2=∫dDl2πDi1xp-l2=∫dDl2πDi1l2pc-l2=∫dDl2πDi1l2xp-l2=0.We perform the basic D-dimensional calculation for each integration,(26)∫dDl2πDi1pc-l2xp-l2=116π24πQ2ϵ1Γ1-ϵΓ1-ϵΓϵB1-ϵ,1-ϵ=116π24πQ2ϵ1Γ1-ϵ1ϵ+2+Oϵ,(27)∫dDl2πDi1l2pc-l2xp-l2=-116π2Q24πQ2ϵ1Γ1-ϵΓ1-ϵΓ1+ϵB1,-ϵB-ϵ,1-ϵ=-116π2Q24πQ2ϵ1Γ1-ϵ1ϵ2+Oϵ.The complex-conjugate diagram gives the same contribution. Then we can show the cross section for the NLO virtual correction as(28)dPh⊥ΔσvirtualdxBdydzh=σ~0αs2π∑qeq2Gq,FxB,xBDq→hzhCF4πμ2Q2ϵ1Γ1-ϵ-2ϵ2-3ϵ-8,this is exactly the same as the virtual correction at leading twist. The strategy in the virtual correction calculation presented here is different with that shown in [63], in which the authors did not use the WTI shown in (19) and directly calculated HL(R)ρjiμν(k1,k2) which has one more external gluon line with the momentum k2-k1 in Figure 2. The authors obtained a consistent result with (28), which demonstrated the validation of WTI through explicit calculation by including all virtual diagrams shown in Figures 3 and 4 in [63]. We would like to comment that the WTI reduces much calculational cost. The direct calculation of HL(R)ρjiμν(k1,k2) takes tremendous time as they contain significant amount of tensor reduction and integration. These two calculations should be conceptually the same with each other as long as we correctly keep track of all imaginary contributions. We confirmed in this section the consistency mathematically in 2→1-scattering case. The consistency check in a more general way will be a future task in the collinear twist-3 factorization approach.


## 3.2. Real Correction

We now complete the NLO calculation by adding the real emission contribution represented by2→2 partonic scattering process. The calculation for 2→2 scattering diagrams has been well studied in Ph⊥-unintegrated case. We just have to repeat the same calculation but in D-dimension. We adopt the conventional technique by separating the propagator into the principle value part and imaginary part [18–33],(29)1k2+iϵ→P1k2-iπδk2,and we focus on the pole contribution -iδ(k2) which is required to generate the phase space for SSA. The derivation of the cross section for the pole contribution has been well developed so far based on the diagrammatic method we reviewed in Section 2. Here we recall the result derived in [23–28] as(30)wμν=i∫dx1∫dx2MijFαx1,x2pβ∂∂k2αHLjiβpoleμνk1,k2+HRjiβpoleμνk1,k2ki=xip,where the matrix element MijFα is given by(31)MijFαx1,x2=∫dλ2π∫dλ′2π·eiλx1eiλ′x2-x1·pS⊥ψ¯j0gFαnλ′nψiλnpS⊥=MN4ϵαpnS⊥pijGq,Fx1,x2+iS⊥αγ5pijG~q,Fx1,x2+⋯.We can construct the gauge-invariant expression (30) before performing the ph-integration. There are three types of pole contributions in SIDIS which are, respectively, known as soft-gluon pole contribution (SGP, x1=x2=x), hard pole contribution (HP, x1=x,x2=xB or x1=xB,x2=x), and another hard pole contribution (HP2, x1=xB,x2=xB-x or x2=xB,x1=xB-x) [23–28, 64], and the corresponding hard parts are given by(32)HLjiβpoleμν=HLjiβSGPμνk1,k2-iπδpc-k2-k122πδk2+q-pc2+HLjiβHPμνk1,k2-iπδk1+q22πδk2+q-pc2+HLjiβHP2μνk1,k2iπδk2+q22πδk2-k1+q-pc2,HRjiβpoleμν=HRjiβSGPμνk1,k2iπδpc+k2-k122πδk1+q-pc2+HRjiβHPμνk1,k2iπδk2+q22πδk1+q-pc2+HRjiβHP2μνk1,k2-iπδk1+q22πδk1-k2+q-pc2.Typical diagrams for each pole contribution are shown in Figure 3 (full diagrams can be found in [64]). We write down the explicit form of each diagram in Figure 3 in order to help readers to follow our calculation,(33)HLjiβSGPμνk1,k2Figure3=-γσpc-qγνpcγβpc-k2+k1γμpc-k2+k1-qγρji1pc-q21pc-k2+k1-q2-g⊥ρσk2+q-pc,HLjiβHPμνk1,k2Figure3=-γσpc-qγνpcγβpc-k2+k1γρk1+qγμji1pc-q21pc-k2+k12-g⊥ρσk2+q-pc,HLjiβHP2μνk1,k2Figure3=γνk2+qγρpcγβpc-k2+k1γμpc-k2+k1-qγρji1pc-k2-q21pc-k2+k12,where g⊥ρσ is the sum of the polarization vector ∑rϵrρ(k)ϵrσ(k)=-g⊥ρσ(k). We can show the WTI for the pole diagrams(34)k2-k1βHLRjiβpoleμνk1,k2=0,and it gives a useful relation(35)pβ∂∂k2αHLRjiβpoleμνk1,k2ki=xip=-1x2-x1-iϵHLRjiαpoleμνk1,k2.For HP and HP2 contributions, we can use (35) and do not have to perform k2-derivative directly as in (30). However, we cannot use this relation for SGP contribution because it contains a delta function δ(x1-x2). In [69], the authors found a reduction formula for SGP contribution as(36)pβ∂∂k2αHLjiβSGPμνk1,k2+HRjiβSGPμνk1,k2=12NCF1x2-x1-iϵ∂∂pcα-pcαpμpc·p∂∂pcμHxp,where H(xp) is the 2→2-scattering cross section without the external gluon line with momentum (x2-x1)p. Substituting (31) into (30) and using ((35), (36)), we can derive the following result:(37)wμν=MNπ22∫dxxδxp+q-pc2-2s^+Q2ϵpcpnS⊥+u^ϵqpnS⊥ddxGq,Fx,x1t^u^TrxpHxp-2s^+Q2ϵpcpnS⊥+u^ϵqpnS⊥Gq,Fx,x1t^u^Q2∂∂s^-∂∂Q2TrxpHxp-TrxpHxp+Gq,Fx,xB1x^-1x^Q2ϵαpnS⊥TrxpHLHPαxBp,xp+TrxpHRHPαxp,xBp-G~q,Fx,xB1x^-1x^Q2iS⊥αTrγ5xpHLHPαxBp,xp-Trγ5xpHRHPαxp,xBp+Gq,FxB,xB-xx^Q2ϵαpnS⊥TrxpHLHP2αxB-xp,xBp+TrxpHRHP2αxBp,xB-xp-G~q,FxB,xB-xx^Q2iS⊥αTrγ5xpHLHP2αxB-xp,xBp-Trγ5xpHRHP2αxBp,xB-xp,where we used the Mandelstam variables(38)s^=xp+q2=1-x^x^Q2,(39)t^=pc-q2=-1-z^x^Q2,(40)u^=xp-pc2=-z^x^Q2,with x^=xB/x, z^=zh/z. Then the cross section can be written as(41)d4Ph⊥ΔσrealdxBdydzh~μ2ϵ∑qeq2∫dzDq→hz∫d2-2ϵpc⊥2π2-2ϵ∫dxxδpc⊥2-1-x^1-z^z^x^Q2×11-ϵddxGq,Fx,xHD+Gq,Fx,xHND+Gq,Fx,xBHHP+G~q,Fx,xBHHPT+Gq,FxB,xB-xHHP2+G~q,FxB,xB-xHHPT2,where we used the symmetry of the Ph⊥-integration as(42)∫d2-2ϵPh⊥Ph⊥αPh⊥βϵραpnϵβpnσ=-∫d2-2ϵPh⊥121-ϵPh⊥2g⊥αβϵραpnϵβpnσ.The authors of [63] found that the factor 1-ϵ in the denominator is essential to derive the correct evolution function of Gq,F(x,x). They calculated all SGP and HP contributions associated with Gq,F. After that, the HP2 contribution and all G~q,F contributions were calculated in [64]. The results of all hard cross sections are listed in [64]. We perform the pc⊥-integration,(43)∫d2-2ϵpc⊥2π2-2ϵδpc⊥2-1-x^1-z^z^x^Q2=12π2-2ϵ∫dpc⊥∫dΩ2-2ϵpc⊥1-2ϵδpc⊥2-1-x^1-z^z^x^Q2=14π4πQ2ϵ1Γ1-ϵ1-x^1-z^z^x^-ϵ,where Ω2-2ϵ is the solid angle and it can be integrated out as(44)∫dΩ2-2ϵ=2π1-ϵΓ1-ϵ.We carry out the ϵ-expansion as follows:(45)z^-ϵ≃1-ϵln⁡z^,(46)x^ϵ≃1+ϵln⁡x^,(47)1-z^-1-ϵ≃-1ϵδ1-z^+11-z^+-ϵln⁡1-z^1-z^+,(48)1-x^-1-ϵ≃-1ϵδ1-x^+11-x^+-ϵln⁡1-x^1-x^+.Then the cross section can be derived as follows:(49)dPh⊥ΔσrealdxBdydzh=σ~0αs2π4πμ2Q2ϵ1Γ1-ϵ·∑qeq2CF2ϵ2Gq,FxB,xBDq→hzh+-1ϵ1+x^21-x^+Dq→hzh1+x^21-x^+∫xB1dxxCF1+x^21-x^+Gq,Fx,x+N21+x^Gq,FxB,x-1+x^2Gq,Fx,x1-x^++G~q,FxB,x1+x^Gq,FxB,x-1+x^2Gq,Fx,x1-x^+-NGq,FxB,xB+12N∫xB1dxx1-2x^Gq,FxB,xB-x+G~q,FxB,xB-x1+x^21-x^++Gq,FxB,xBCF∫zh1dzz1+z^21-z^+Dq→hz1+x^21-x^++finiteterms,where the derivative term was converted to the nonderivative term through the partial integral,(50)∫xB1dxddxGq,Fx,x1+x^2=∫xB1dxxGq,Fx,x2x^2-2δ1-x^.Combine the real (see (28)) and virtual corrections (see (49)), we find that the double-pole 1/ϵ2 term, which represents a soft-collinear divergence, cancels out. The remaining collinear divergences, represented by the single pole 1/ϵ, can be eliminated by the redefinition of Qiu-Sterman function and fragmentation function,(51)Gq,FxB,xB=Gq,F0xB,xB+αs2π-1ϵ^∫xB1dxxPqqx^Gq,Fx,x+N21+x^Gq,FxB,x-1+x^2Gq,Fx,x1-x^++G~q,FxB,x-NGq,FxB,xB+12N∫xB1dxx1-2x^Gq,FxB,xB-x+G~q,FxB,xB-x,(52)Dq→hzh=Dq→h0zh+αs2π-1ϵ^∫zh1dzzPqqz^Dq→hz,where Pqq(x) is well known q→q splitting function(53)Pqqx=CF1+x21-x++32δ1-x,and we adopted the MS¯-scheme(54)1ϵ^=1ϵ-γE+ln⁡4π.This collinear singularity is consistent with that of Gq,F(x,x) explored in [54–59]. After the renormalization, the final result for the transverse-momentum-weighted SSAs at NLO is given by(55)
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.Note that the cross section above does not include the contribution from the gluon fragmentation channel. From the requirement that the physical cross section does not depend on the factorization scale μ,(56)∂∂ln⁡μ2dPh⊥ΔσLO+NLOdxBdydzh=0,we can derive the LO evolution equation for Gq,F(x,x),(57)∂∂ln⁡μ2Gq,FxB,xB,μ2=αs2π∫xB1dxx1+x^Gq,FxB,x,μ2-1+x^2Gq,Fx,x,μ21-x^+Pqqx^Gq,Fx,x,μ2+N2·1+x^Gq,FxB,x,μ2-1+x^2Gq,Fx,x,μ21-x^++G~q,FxB,x,μ21+x^Gq,FxB,x,μ2-1+x^2Gq,Fx,x,μ21-x^+-NGq,FxB,xB,μ2+12N∫xB1dxx1-2x^Gq,FxB,xB-x,μ2+G~q,FxB,xB-x,μ21+x^Gq,FxB,x,μ2-1+x^2Gq,Fx,x,μ21-x^+.This evolution equation based on the transverse-momentum-weighted technique was first discussed in Drell-Yan process [62], in which the authors succeeded in deriving the Gq,F terms in the parenthesis [⋯] in the evolution equation. After that, the authors of [63] pointed out that an extra term -NGq,F(xB,xB,μ2) also contributes to the evolution equation. The HP2 pole contribution and all G~q,F terms were obtained in [64] within the method of transverse momentum weighting.Figure 3
Typical diagrams for soft-gluon pole (left), hard pole (middle), and another hard pole (right). The red barred propagator gives the pole term.


## 4. Application to Other Processes

A lot of works on the transverse-momentum-weighted SSA have been done in recent years. We briefly summarize all related work in this section. The evolution equation of the Qiu-Sterman function (57) is still missing the gluon mixing contribution associated with the 3-gluon distribution functions defined by(58)∫dλ2π∫dλ′2πeiλx1eiλ′x2-x1pS⊥dbcaFbβn0gFcγnλ′nFaαnλnpS⊥=2MNOx1,x2gαβϵγpnS⊥+Ox2,x2-x1gβγϵαpnS⊥+Ox1,x1-x2gαγϵβpnS⊥+⋯,(59)∫dλ2π∫dλ′2πeiλx1eiλ′x2-x1pS⊥ifbcaFbβn0gFcγnλ′nFaαnλnpS⊥=2MNNx1,x2gαβϵγpnS⊥-Nx2,x2-x1gβγϵαpnS⊥-Nx1,x1-x2gαγϵβpnS⊥+⋯,where dbca and ifbca are the structure constants of SU(N) group. Figure 4 shows typical diagrams which give the gluon mixing contribution in SIDIS and Drell-Yan.Figure 4
The diagrams which give the gluon mixing contribution to the evolution equation ofGq,F(x,x) in SIDIS (left) and Drell-Yan (right).The calculation technique for these diagrams was developed in [29–33]. There is only the soft-gluon pole (x1=x2) contribution due to the interchange symmetry of the external gluon lines and then the cross section is expressed by four independent functions O(x,x), O(x,0), N(x,x), and N(x,0). The gluon mixing term in the flavor singlet evolution was discussed in both SIDIS [65, 67] and Drell-Yan [66] and the cross section up to the finite term is derived as(60)dPh⊥ΔσSIDISdxBdydzhdϕgluon~Gq,FxB,xB,μDq→hzh,μ+αs2πln⁡Q2μ2Dq→hzh,μFg⊗TG++finiteterms,dq⊥ΔσDYdydQ2gluon~Gq,Fxa,xa,μfq¯/pxb,μ+αs2πln⁡Q2μ2fq¯/pxa,μFg⊗TG++finiteterms,where xa=Q/seη, xb=Q/se-η with the center of mass energy s and the rapidity η, TG+ is given by a linear combination of the 3-gluon distribution functions, and fq¯/p is the antiquark PDF. In Drell-Yan process, we use the transverse momentum of the virtual photon q⊥ for the weighted cross section. Using the condition as in (56), we can derive the gluon mixing term of Gq,F. The explicit form of the evolution kernel Fg and the finite terms are shown in [66, 67]. Adding the mixing term to (57), the evolution equation for the Qiu-Sterman function, the first moment of the TMD Sivers function, was completed at LO with respect to QCD coupling constant αs. As a by-product of the work on Drell-Yan, the NLO cross section related to the first momentum of the TMD Boer-Mulders function was also derived as(61)dq⊥ΔσDYdydQ2BM~Tq,Fρxa,xa,μh1q¯xb,μ+αs2πln⁡Q2μ2h1q¯xa,μFσ⊗Tq,Fρ+finiteterms,where h1 is chiral-odd transversity distribution and the twist-3 function TF(ρ)(x,x,μ) corresponds to the first moment of the Boer-Mulders function and its exact definition can be found in [66]. The evolution kernel Fσ and the finite terms of the cross section are shown in [66]. The authors in [67] also discussed the evolution equation for the first moment of the TMD Collins fragmentation function. The calculation of the twist-3 fragmentation contribution is different from the distribution case because the cross section only receives the nonpole contribution of the hard scattering. The calculation technique for the nonpole contribution has been well developed in [26, 34–36]. It is not difficult to extend it to the D-dimensional calculation. The NLO cross section was derived in [67] as(62)dPh⊥ΔσSIDISdxBdydzhdϕCollins~h1qxB,μe^∂qzh,μ+α2πln⁡Q2μ2h1qxB,μF∂⊗e^∂q+FF⊗E^Fq+FG⊗E^Gq+finiteterms,where e^∂, E^, and E^G are the twist-3 fragmentation functions for spin-0 hadron (different definition is shown in [70]). All evolution kernels and finite terms are shown in [67].The NLO transverse-momentum-weighted cross section has been completed for single inclusive hadron production in SIDIS and Drell-Yan process in proton-proton collisions by a series of work presented here. These results are useful not only for the derivation of the evolution equations, but also for the verification of twist-3 collinear factorization feasibility, as well as for the global analysis of the experimental data. Measurement of the weighted SSAs just began very recently [68] and more data will be provided by future experiments. The analysis of the data based on the NLO result will lead to better understanding of the origin of the SSAs. There are still some TMD functions which have not been discussed yet; we hope the techniques presented in this paper can help extending the application of the transverse-momentum-weighted technique to resolve all these open questions.The transverse-momentum-weighted technique has also been extended to study the transverse momentum broadening effect for semi-inclusive hadron production in lepton-nucleus scattering [71] and Drell-Yan dilepton production in proton-nucleus scattering [72]. In these studies, the QCD evolution equation for twist-4 quark-gluon correlation function was derived for the first time, and the twist-4 (double scattering) collinear factorization at NLO was confirmed through explicit calculations [73]. The finite NLO hard parts were obtained for the transverse momentum broadening effect, which can be used in global analysis of world data to extract precisely the medium properties characterized by the twist-4 matrix elements.


## 5. Summary

We reviewed the transverse-momentum-weighted technique as a useful tool to derive the scale evolution equation for the twist-3 collinear function which is expressed by the first moment of the TMD function. We first demonstrated the calculation of the LO cross section formula in a pedagogical way. Then we showed the basic techniques for the NLO calculation for both the virtual correction and real emission contributions. A lot of work have been done on the Qiu-Sterman function [62–65] and recently the application of this technique to other twist-3 functions was also discussed [66, 67]. There is still room of the application to many other TMD functions by considering appropriate twist-3 observables. We hope our review paper will provide basic knowledge needed to work on this subject.In the end, we would like to point out the importance of thePh⊥-weighted SSA from the phenomenological point of view. We introduced this observable as a tool to derive the scale evolution equation by focusing on the 1/ϵ-term in the cross section. However, the finite term is also important when we evaluate the cross section in order to compare it with the experimental data. The COMPASS experiment reported the data of the Ph⊥-weighted SSA very recently [68]. We expect that the data will be accumulated in the future experiments at COMPASS, JLab, and EIC and then the exact NLO cross section including the finite contribution will play an important role in the analysis of those weighted SSA data.



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