References
###### Abstract

We study the one-loop correction in Transverse-Momentum-Dependent(TMD) factorization for Drell-Yan processes at small transverse momentum of the lepton pair. We adopt the so-called subtractive approach, in which one can systematically construct contributions for subtracting long-distance effects represented by diagrams. The perturbative parts are obtained after the subtraction. We find that the perturbative coefficients of all structure functions in TMD factorization at leading twist are the same. The perturbative parts can also be studied with scattering of partons instead of hadrons. In this way, the factorization of many structure functions can only be examined by studying the scattering of multi-parton states, where there are many diagrams. These diagrams have no similarities to those treated in the subtractive approach. As an example, we use existing results of one structure function responsible for Single-Spin-Asymmetry, to show that these diagrams in the scattering of multi-parton states are equivalent to those treated in the subtractive approach after using Ward identity.

QCD Corrections of All Structure Functions in Transverse Momentum Dependent Factorization for Drell-Yan Processes

J.P. Ma and G.P. Zhang

Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing 100190, China

Center for High-Energy Physics, Peking University, Beijing 100871, China

1. Introduction

QCD factorization is an important concept for studying high energy scattering, in which both long-distance- and short-distance effects exist. With a proven factorization one can consistently separate short-distance effects from long-distance effects. The separated short-distance effects can be safely calculated with perturbative QCD. The long-distance effects can be represented by matrix elements, which are consistently defined with QCD operators[1]. This fact allows us not only to make predictions, but also to explore the inner structure of hadrons through determining these matrix elements from experiment.

There are two types of QCD factorizations for inclusive processes. One is of collinear factorization, in which one neglects the transverse motion of partons inside hadrons at the leading twist or at the leading power. Another one is of Transverse-Momentum-Dependent(TMD) factorization, where one takes the transverse momenta of partons into account at the leading power. Using this factorization allows one to study the transverse motion of partons in hadrons and hence to obtain 3-dimensional information about the inner structure of hadrons. TMD factorization is applicable for processes in certain kinematical regions. E.g., in Drell-Yan processes, the TMD factorization can only be used if the lepton pair has a small transverse momentum which is much less than the invariant mass of the lepton pair. In this work we focus on one-loop correction in TMD factorization for Drell-Yan processes.

TMD factorization has been first studied in the case where a nearly back-to-back hadron pair is produced in -annihilations[2]. A factorization theorem in this case is established. Later, such a factorization has been established or examined in Drell-Yan processes[3], Semi-Inclusive Deeply Inelastic Scattering(SIDIS)[4, 5], and has been extended to the polarized case[6]. The established TMD factorization in SIDIS and Drell-Yan processes only involves TMD quark distributions at the order of leading power . There exist TMD gluon distributions. These distributions can be extracted from inclusive processes in hadron collision like Higgs-production[7, 8], quarkonium production [9, 10] and two-photon production[11]. It should be noted that studies of TMD factorization will not only help to explore the inner structure of hadrons, but also provide a framework for resummtion of large log terms with in perturbative expansion with . The classical example is for Drell-Yan processes studied in [3].

Unlike parton distributions in collinear factorization at leading twist, there are many TMD parton distributions at leading twist. Structure functions, e.g., in Drell-Yan processes, are factorized with these distributions. The perturbative coefficients at tree level in TMD factorization can be easily derived. However, for reliable predictions one needs to know higher-oder corrections in the factorization. This is also important for giving physical predictions of experiments performed at different energy scales, since the dependence on the scales of perturbative coefficients appears beyond tree-level.

In Drell-Yan processes, one-loop correction of some structure functions can be obtained by studying partonic scattering and TMD parton distributions of a single parton, where one replaces each hadron with a single parton, i.e., the scattering with or as a single parton state. The one-loop corrections of the studied structure functions in [6] are in fact obtained in this way. But this approach for obtaining higher-order corrections does not work for many other structure functions, e.g., the structure function for Single transverse-Spin Asymmetry(SSA) in the case that one initial hadron is transversely polarized. This structure function is factorized with the TMD parton distribution, called Sivers function[12]. If one replaces the transversely polarized hadron with a transversely polarized quark, one will always have zero results for the structure function and the Sivers function, because the chirality of a massless quark is conserved in perturbative QCD. Therefore, one needs to use multi-parton state instead of a single parton state to study those structure functions. Such a study for SSA has been done mainly in the framework of collinear factorization in [13, 14, 15]. The approach with multi-parton states has provided a different way to solve some discrepancies in collinear factorization of SSA[15, 16].

In principle one can use these multi-parton states to study higher-order corrections in TMD factorization. Since scattering with multi-parton states is more complicated, the one-loop correction is difficult to obtain, because too many diagrams are involved. In this work we use the so-called subtractive approach to study the problem. The approach is based on diagram expansion and explained in [17, 18, 19]. In general, it is relative easy to find the leading order contribution to structure functions in the factorized form. At the next-to-leading or one-loop order, the diagrams, which give possible contributions, contain in general contributions from TMD parton distributions, which are of long-distance effects and need to be subtracted for obtaining the one-loop perturbative coefficients. In the subtractive approach one can systematically construct such subtractions in terms of diagrams. A comparison of the two approaches can be noticed in the following: In the approach with multi-parton states one explicitly calculates in detail the contributions of structure functions and the correspond contributions of TMD parton distributions for the subtraction. In the subtractive approach one only calculates in detail the contributions to the structure functions subtracted with the contributions of TMD parton distributions. At leading twist of TMD factorization, the symmetric part of the hadronic tensor has 24 structure functions[20, 21]. With the work presented here, it turns out that the one-loop correction is the same for all structure functions. This result can be generalized beyond one-loop order.

It may be difficult to understand why the one-loop correction is the same for all structure functions. Taking SSA factorized with Sivers function as an example, the diagrams treated in the subtractive approach have no similarity to those diagrams treated with multi-parton states. We will make a comparison for a part of existing results for SSA to show that the contribution of the studied part is the same obtained from diagrams in the subtractive approach. This is in fact a consequence of Ward identity.

In Drell-Yan processes, the interpretation of the small is that it is partly generated with the transverse momentum of incoming partons from hadrons. In TMD factorization, as we will see in the subtractive approach, one momentum-component of partons is set to be zero as an approximation. This may give the impression that one deals here with scattering of off-shell partons and hence brings up the question if the TMD factorization is gauge-invariant. We will discuss this problem and show that the factorization is gauge-invariant.

Our work is organized as in the following: In Sect. 2. we give our notation and derive the tree-level result. In Sect. 3. we discuss the issue of gauge invariance mentioned in the above. In Sect. 4. and Sect. 5. we analyse the one-loop contributions in the factorization with the subtractive approach and give our main results. In Sect. 6. we make a comparison with a part of results derived with the subtractive approach and the existing results calculated with multi-parton states. Sect.7. is our summary. Detailed results for all factorized 24 structure functions are given in the Appendix.

2. Notations and Tree-Level Results

We consider the Drell-Yan process:

 hA(PA)+hB(PB)→γ∗(q)+X→ℓ−+ℓ++X. (1)

We will use the light-cone coordinate system, in which a vector is expressed as and . We take a light-cone coordinate system in which:

 PμA≈(P+A,0,0,0),    PμB≈(0,P−B,0,0). (2)

moves in the -direction, i.e., is the large component. The lepton pair or the virtual photon carries the momentum with . We will study the case with and that the two initial hadrons are of spin-1/2. The two hadrons are polarized. The polarization of hadron can be described by the helicity and a transverse spin vector . The polarization of hadron is described by and . For convenience we also introduce two light-cone vectors: and , and two transverse tensors:

 gμν⊥=gμν−nμlν−nνlμ,      ϵμν⊥=ϵαβμνlαnβ, (3)

The relevant hadronic tensor for Drell-Yan processes is defined as:

 Wμν=∑X∫d4x(2π)4eiq⋅x⟨hA(PA),hB(PB)|¯q(0)γνq(0)|X⟩⟨X|¯q(x)γμq(x)|hB(PB),hA(PA)⟩. (4)

The tensor can be decomposed into various structure functions. In this work we will only give results for those structure functions which receive leading-twist contributions in TMD factorization.

Taking hadrons as bound states of partons, i.e., quarks and gluons, the scattering of hadrons, hence the hadronic tensor can be represented by Feynman diagrams. Regardless how these diagrams are complicated, one can always divide each diagram into three parts: One part contains the insertion of two electromagnetic currents as indicated in Eq.(4). The other two parts are related to the hadron or . The three parts are connected with parton lines. An example is given in Fig.1a. In Fig.1a, the middle part contains the two electromagnetic vertices, the lower part is associated with and the upper part is associated with . Two quark lines connect the middle part with the part of and two antiquark lines connect the middle part with the part of . The parton lines from the part of only denote the contraction of Dirac- and color indices with the middle part, and the momentum flow into the middle part. The propagators associated with the parton lines are in the part of . The same is also for the part of . The middle part can be classified with the order of . E.g., at tree-level the middle part of Fig.1a is given by Fig.1b. Hereafter, we denote the part of or as sum of all possible diagrams for a given middle part. In this work we use Feynman gauge.

The part of and of in Fig.1a can be identified as:

 Γji(PA,kA) = ∫d4ξ(2π)4e−iξ⋅kA⟨hA(PA)|[¯q(ξ)]i[q(0)]j|hA(PA)⟩, ¯Γij(PB,kB) = ∫d4ξ(2π)4e−iξ⋅kB⟨hB(PB)|[q(ξ)]i[¯q(0)]j|hB(PB)⟩, (5)

where stand for Dirac- and color indices. We denote the middle part in Fig.1a as , the contribution of Fig.1a as:

To factorize the contribution from Fig.1a with and , especially at tree level, certain approximations can be made. Because we are interested in the kinematical region of , the momenta of and can not be arbitrarily large. They are restricted as . Here a detailed discussion is needed to clarify what is in fact included in the hadronic parts in Fig.1b. This is also important for the comparison with the detailed calculation in Sect. 6. For this we take as an example. In principle there can be the case that receives contributions from large and also large in Fig.1a. But these contributions can be calculated with perturbation theory. They need to be factorized out from and will in general give power suppressed contributions to beyond tree-level. Therefore, the dominant contributions only come from the case that is only characterized with the energy scale , i.e., . Hence one has and . In other word, in Fig.1b is the sum of all diagrams with with . Similarly, one can also find that for in Fig.1b one has .

With the above discussion one can find the space-time picture of the hadronic matrix element in . The -dependence is characterized by the small scale . The -dependence is characterized by the scale , and the -dependence is characterized by the scale . Therefore, we can first neglect the -dependence. This is equivalent to take the leading result by expanding in . In we neglect the -dependence of the hadronic matrix element. Another approximation can be made is that the leading contributions are only given by the matrix elements containing the good component of quark fields. One can always decompose a quark field as:

 q(x)=12γ+γ−q(x)+12γ−γ+q(x). (7)

For the first term is the good component, the second term can be solved with equation of motion and gives a power-suppressed contribution. For the second term is the good component. After making these approximations, we can write the two parts as:

 Γij(PA,kA) ≈ δ(k−A)Mij(x,kA⊥)+⋯,¯Γij(PB,kB)≈δ(k+B)¯Mij(x,kB⊥)+⋯, Mij(x,kA⊥) = ∫dξ−d2ξ⊥(2π)3e−iξ⋅~kA⟨hA(PA)|[¯q(ξ)]j[q(0)]i|hA(PA)⟩∣∣∣ξ+=0, ¯Mij(x,kB⊥) = ∫dξ+d2ξ⊥(2π)3e−iξ⋅~kB⟨hB(PB)|[q(ξ)]i[¯q(0)]j|hB(PB)⟩∣∣∣ξ−=0, (8)

with

 ~kμA=(xP+A,0,k1A,k2A),~kμB=(0,xP−B,k1B,k2B). (9)

In Eq.(8), the stand for higher-twist- or power-suppressed contributions. and are of leading-twist- or leading power. The quark fields in or are correspondingly good components. Therefore, we always have:

 γ+¯M=¯Mγ+=0,γ−M=Mγ−=0. (10)

This property will help us to extract the contributions of TMD parton distributions as we will see later. The approximation made in the above is valid for the case that the transverse momentum of the lepton pair is at order . The correction of the approximation to the hadronic tensor is at the order or relative to the leading order. For one can make a further approximation by neglecting or expanding the -dependence in hadron matrix elements in or . This will lead to collinear factorizations.

We first consider the leading order given by Fig.1b. The middle part can be then given explicitly:

 Hμνij,lk(kA,kB,q)=δ4(kA+kB−q)[γμ]lj[γν]ik. (11)

Using the above approximated results for and , one obtains the hadronic tensor at leading order of as

 Wμν≈∫d2kA⊥d2kB⊥Tr[γμM(x,kA⊥)γν¯M(y,kB⊥)]δ2(kA⊥+kB⊥−q⊥), (12)

with and . According to the notation in [17], we denote the approximations made for Fig.1b to derive the above result represented by Fig.1c, it is just Fig.1b with the hooked lines. The hooked line in the lower part denotes the approximations made for and the hooked line in the upper part denotes the approximations made for , as indicated in Eq.(8). In the above we have worked out the contribution with a quark from and an antiquark from . Similarly, one can work out the case where a quark is from and an antiquark is from .

The density matrix can be decomposed into various TMD parton distributions. The decomposition has been studied in [22, 23, 24, 25]. At leading twist, the decomposition is:

 Mij(x,k⊥) = 12Nc[f1(x,k⊥)γ−−f⊥1T(x,k⊥)γ−ϵμν⊥k⊥μSAν1MA (13) +g1L(x,k⊥)λAγ5γ−−g1T(x,k⊥)γ5γ−k⊥⋅SA1MA +h1T(x,k⊥)iσ−μγ5SAμ+h⊥1(x,k⊥)σμ−k⊥μ1MA+h⊥1L(x,k⊥)λAiσ−μγ5k⊥μ1MA −h⊥1T(x,k⊥)iσ−μγ5k⊥μk⊥⋅SA1M2A]ij.

There are 8 TMD parton distributions at leading twist. is the mass of . In the above and are odd under naive time-reversal transformation. is the Sivers function, is called as Boer-Mulders function. For the decomposition we have implicitly assumed that the gauge links discussed in the next section is added in .

Similarly one has the decomposition for as:

 ¯Mij(x,k⊥) = 12Nc[¯f1(x,k⊥)γ++¯f⊥1T(x,k⊥)γ+ϵμν⊥k⊥μSBν1MB (14) −¯g1L(x,k⊥)λBγ5γ++¯g1T(x,k⊥)γ5γ+k⊥⋅SB1MB +¯h1T(x,k⊥)iσ+μγ5SBμ+¯h⊥1(x,k⊥)σμ+k⊥μ1MB+¯h⊥1L(x,k⊥)λBiσ+μγ5k⊥μ1MB −¯h⊥1T(x,k⊥)iσ+μγ5k⊥μk⊥⋅SB1M2B]ij.

It should be noted that and are diagonal in colour space. With these decomposition one can work out the hadronic tensor at leading order. The results for the tensor can be represented with structure functions and each structure function is factorized with corresponding TMD parton distributions.

3. Gauge Invariance and Gauge Links

In the last section we have worked out the tree-level TMD factorization by considering the diagram Fig.1b. In this diagram or Fig.1a, there are only two parton lines connecting the middle part with the part of , and two parton lines connecting the middle part with the part of . From argument of power-counting, if the connection is made by more parton lines in Fig.1a or 1b, the contributions are power-suppressed but with exceptions. The exceptions are well-known. If there are gluon lines connecting the middle part with the part of , and those lines are for -gluons collinear to , the resulted contributions are not power-suppressed. The tree-level diagram with many collinear gluons is illustrated in Fig.2a.

It is well-known how those diagrams with many collinear gluons can be summed. The summation is achieved by introducing gauge links. In this work we follow [4] by using the gauge link along the direction with :

 Lu(ξ,−∞)=Pexp(−igs∫0−∞dλu⋅G(λu+ξ)). (15)

The diagrams like that given in Fig.2a can be summed by inserting in the product . Similarly, there can be diagrams with many collinear -gluons emitted by the part of . These diagrams can also be summed by introducing the gauge link along the direction with . Finally, the summation can be represented by Fig.2b. The summation is made by re-defining:

 M(x,k⊥) = ∫dξ−d2ξ⊥(2π)3e−ixξ−P+A−iξ⊥⋅k⊥⟨hA(PA)|¯q(ξ)Lu(ξ,−∞)L†u(0,−∞)q(0)|hA(PA)⟩∣∣∣ξ+=0, ¯M(x,k⊥) = −∫dξ−d2ξ⊥(2π)3e−ixξ+P−B−iξ⊥⋅k⊥⟨hB(PB)|¯q(0)Lv(0,−∞)L†v(ξ,−∞)q(ξ)|hB(PB)⟩∣∣∣ξ−=0,

these matrices are diagonal in color-space and -matrices in Dirac space, and . With these gauge links, TMD parton distributions will not only depend on and the renormalization scale but also depend on those parameters:

 ζ2u=(2PA⋅u)2u2,ζ2v=(2PB⋅v)2v2. (17)

The dependence on these parameters is controlled by Collins-Soper equation[2]. The Collins-Soper equations of the introduced TMD parton distributions can be found in [2, 26]. In general one needs to add in Eq.(LABEL:DENMG) gauge links along transverse direction at infinite space-time to make density matrices gauge invariant as shown in [27]. In this work, we will take a non-singular gauge, i.e., Feynman gauge. In a non-singular gauge gauge links at infinite space-time vanish.

With the added gauge links the TMD parton distributions are gauge invariant. But there seems another problem of gauge invariance related to the tree-level result in Eq.(12). The result at the leading order seems that one can interpret the process as: One quark with the momentum from annihilates an antiquark with the momentum from into the virtual photon. From the momenta one can realize that the quark and the antiquark are off-shell because of and . One may conclude that the result is not gauge invariant because the perturbative coefficients are extracted from scattering amplitudes of off-shell partons. However it can be shown as in the following that this is not the case.

We introduce two momenta which are:

 ¯kμA=(k+A,0,0,0),¯kμB=(0,k−B,0,0). (18)

These are momenta of on-shell partons. Now using the properties in Eq.(10) we can derive:

 Tr[γμMγν¯M] = 116(k+Ak−B)2∑s1,s2,s3,s4[¯v(¯kB,s1)γμu(¯kA,s2)]¯u(¯kA,s2)γ+Mγ+u(¯kA,s3) (19) ⋅[¯u(¯kA,s3)γνv(¯kB,s4)]¯v(¯kB,s4)γ−Mγ−v(¯kB,s1),

from this one can see that the perturbative coefficients are in fact extracted from scattering amplitudes of on-shell partons, i.e., from the annihilation amplitude with , indicated by the two terms in the two terms . The effect of transverse momenta of partons are only taken into account in the momentum conservation, i.e., in the -function . Therefore, the tree-level result in Eq.(12) are gauge invariant.

It is rather obscure to see if the tree-level result in Eq.(12) is gauge invariant from momenta carried by patrons, because the amplitudes there are constant. If we go beyond tree-level, we can see this more clearly. We consider a class of diagrams in which there is no parton crossing the cut. This case is represented by Fig.3a. After making the approximations indicated with Eq.(8) the contribution from Fig.3a can be in general written as:

 Wμν∼Tr[HL(~kA,~kB,q)M(~kA)HR(~kA,~kB,q)¯M(~kB)]δ2(kA⊥+kB⊥−q⊥). (20)

The contribution to perturbative coefficients from Fig.3a is obtained by subtracting the corresponding contribution of TMD parton distributions from the above contribution. Since the TMD factorization is for the leading contribution in , one should find the leading contribution from Fig.3a. In its contribution the -function already gives the leading contribution. Hence we need to expand in all transverse momenta. It is easy to find the leading contribution:

 (21)

where stand for higher order contributions in . It is clear now in the momenta of incoming partons are of on-shell. Therefore, the contribution from diagrams like that in Fig.3a is gauge invariant. This also tells us that for leading power contribution one only needs to calculate with on-shell momenta. The collinear- and infrared singularities are then regularized with dimensional regularization.

We notice here that it is important and crucial to use dimensional regularization for TMD factorization here for collinear- and infrared singularities. In other word, one should set in before performing integrations of loop momenta. One may think that one can keep a nonzero but infinite small and to regularize collinear- and infrared singularities. Then these singularities will appear in as terms with and . After subtraction of contributions from TMD parton distributions, perturbative coefficients do not contain such terms. In fact, this is not the case. This can be seen by calculating one-loop contribution of with nonzero transverse momenta of partons. The contribution will contain terms of . The reason for existence of such terms is that the one-loop contribution always contain double log terms. Such terms can never be subtracted with the contributions of TMD parton distributions, because the contributions of TMD parton distributions at one-loop do not have such terms. Therefore, with nonzero and in TMD factorization can not be made, or the TMD factorization is gauge dependent with perturbative coefficients depending on and .

The above discussion is for diagrams without any parton crossing the cut. There are diagrams with partons crossing the cut like the one given in Fig.3b. For these diagrams there is no reason to set the transverse momenta of incoming partons to be zero as the leading approximation. These diagrams may have problems with gauge invariance. But, in TMD factorization, the contributions from diagrams like Fig.3b will be totally subtracted, as we will explicitly show at one-loop, and will not contribute to perturbative coefficients. Also, the subtracted contributions are from TMD parton distributions which are now gauge invariant, and from a gauge-invariant soft factor which will be introduced later. Therefore, there is no problem of gauge invariance with contributions from Fig.3b.

Before ending this section, we briefly explain the approximation denoted by hooked lines introduced in [17], or the rule of theses lines here for TMD factorization. The approximation can be called as parton model approximation. We consider an example with a quark, which originally comes from and enters the annihilation into the virtual photon, as given in Fig.4. In Fig.4 the lower part is associated with , the upper part contains the annihilation with partons from . is the Dirac index. The contribution from Fig.4 without the hooked line can be generically written as

 Γ=∫d4kAUi(kA)Li(kA)=∫d4kAUi(kA)12(γ−γ++γ+γ−)ijLj(kA), (22)

with the hooked line it means that we take the approximation for the expression as

 Γ=∫d2kA⊥dk+A12Ui(~kA)(γ−γ+)ij(∫dk−ALj(kA))+⋯,    ~kμA=(k+A,0,k1A,k2A), (23)

where stand for contributions which will give power-suppressed contributions to the hadronic tensor and are neglected. Similarly, one can also find the rule of hooked lines for parton lines coming from the part of orginally.

4. One-Loop Real Correction

In this section we study how the real part of one-loop correction is factorized. This part corresponds to Fig.3b. We first consider Fig.5a. We will use this example to explain the subtractive approach in detail.

From the topology of the diagram, we can apply the parton model approximation in different places in the lower part of the diagram, e.g., one can apply the approximation for the parton lines connecting to the electromagnetic vertices directly, as given in Fig.5b, or one can also apply the approximation as given in Fig.5c and Fig.5d. It is clearly that the contribution from Fig. 5b is already included in Fig.1c, i.e., in the tree-level contribution. Therefore, in order to avoid double -counting and to obtain true one-loop contribution, one needs first to subtract the contribution from Fig.5b from Fig.5a, the remaining part will give the true one-loop correction after making the parton-model approximation for the two parton lines from the lower bubble of . With the approximation Fig.5a becomes Fig.5c and Fig.5b becomes Fig.5d. The true one-loop correction is then given by the contribution from Fig.5c subtracted with the contribution from Fig.5d. In the sense of the subtraction, the part of Fig.5d below the middle hooked line can be interpreted as a contribution from TMD parton distributions, because in this part all transverse momenta are at order of . Corresponding to the discussion for after Eq.(6), this part is already included in or .

We denote the momentum carried by the gluon in Fig.5 as , and the gluon is with the polarization index . is the momentum entering the left electromagnetic vertex along the quark line from . The contribution from Fig.5c can be easily found as:

Since we are interested in the kinematic region of , we need to find the leading contribution in the limit. For this diagram, it is easy to find the leading contribution appears if the exchange gluon is collinear to , i.e.:

 (~kA−k)μ∼O(1,λ2,λ,λ),    λ2=q2⊥Q2. (25)

This also implies that is collinear to because . Using the property in Eq.(10) we obtain the leading contribution in the limit :

i.e., the leading contribution has the indices as transverse. The summed index is always transverse. Again one can use the property in Eq.(10) to re-write the leading contribution in the form:

where stand for power-suppressed contributions which are neglected. Some integrations have been done with -functions. It gives and . Now it is interesting to note that the part in is just the expression for the part in Fig.5d between the two hooked lines below the electromagnetic vertices. This part can be identified as the correction of TMD parton distributions. This fact tells that the leading contribution from Fig.5c is the same as the contribution of Fg.5d:

 Wμν∣∣∣5c≈Wμν∣∣∣5d. (28)

Therefore, after the subtraction Fig.5a will not contribute to perturbative coefficients in TMD factorization at one-loop.

Now we consider the contribution from Fig.6a, which also gives possible contribution at one-loop. By applying the two hooked lines in Fig.6a to parton lines nearest to the bubbles for hadrons, we obtain the contribution of Fig.6a as:

here is the momentum of the exchanged gluon. It is noted that in this case the index can be any of . Again we need to find the leading contributions from Fig.6a in . These leading contributions appear in the case that is collinear to or to , and is soft. We first consider the case that the exchanged gluon is collinear to , i.e., . The leading contribution in this case is given by:

Here we use the subscript to denote the leading contribution from the momentum region collinear to . It is well-known that there will be a light-cone singularity with the light-cone vector . To avoid this we can replace the vector with introduced before. We can use the property in Eq.(10) to re-write the above result as:

where and . With the result given in the form in the above, one easily finds that this contribution is the same as that from Fig.6b. The contribution from Fig.6b is already contained in Fig.1c and should be subtracted from Fig.6a. Hence, the considered contribution in Eq.(31) is exactly subtracted and gives no contribution to the factorization at one-loop. Similarly, one also finds that the leading contribution from Fig.6a in the region where the exchanged gluon is collinear to is also exactly subtracted.

The interesting part is from the region where the gluon is soft, i.e., . To analyze this part we make the substitution , i.e., now the gluon carries the momentum . The leading contribution from this region gives:

 Wμν∣∣∣6aS ≈ g2s∫d2~kA⊥d2~kB⊥d4k(2π)42πδ(k2)δ2(kB⊥+kA⊥−k⊥−q⊥) (32) Tr[¯M(~kB)γμTaM(~kA)γνTa]n⋅l(l⋅k−iε)(n⋅k+iε)+⋯,

where we use the subscript to denote the contribution from the soft gluon. We have here and . We replace the vector with and with . The color trace can be taken out by noting that the density matrices are diagonal in color space. Therefore,

 Wμν∣∣∣6aS ≈ g2s∫d2~kA⊥d2~kB⊥d4k(2π)42πδ(k2)δ2(kB⊥+kA⊥−k⊥−q⊥) (33) Tr[¯M(~kB)γμM(~kA)γν]1NcTr(TaTa)u⋅v(v⋅k−iε)(u⋅k+iε)+⋯.

This contribution can be represented with Fig.6c. We note here that this contribution is not contained in Fig.1c. It can be subtracted with a soft factor as shown in [3, 4]. The need of such a soft factor is also necessary for one-loop virtual correction as shown later.

If we want to subtract the soft region of the gluon from the contribution from Fig.6a, we should notice that the collinear contribution in Eq.(31) also contains the contribution from the soft region. One can easily find that the soft contribution in Eq.(31) is exactly the same as the soft contribution from Fig.6a given in Eq.(33). We can re-write the contribution from Fig.6a as:

 Wμν∣∣∣6a=(Wμν∣∣∣6a−Wμν∣∣∣6aA−Wμν∣∣∣6aB+Wμν∣∣∣6aS)+(Wμν∣∣∣6aA+Wμν∣∣∣6aB−Wμν∣∣∣6aS), (34)

where is the contribution from the region in which the gluon is collinear to . This contribution contains the same soft contribution as given by Fig.6c. In the first of Eq.(34) there are no soft- and collinear contributions. In fact the leading contribution from the first is zero as discussed before. In the second the collinear contributions are already included in Fig.1c, hence give no contribution to one-loop factorization.

We define the soft factor for the subtraction of soft gluons as:

 ~S(k⊥)=∫d2ξ⊥(2π)2e−iξ⊥⋅k⊥Nc⟨0|Tr[L†v(ξ⊥,−∞)Lu(ξ⊥,−∞)L†u(0,−∞)Lv(0,−∞)]|0⟩. (35)

At tree-level one has:

 ~S(0)(k⊥)=δ2(k⊥). (36)

At one-loop it receives contributions from Fig.7a and Fig.7b. The total one-loop correction is the sum of the two diagrams, their conjugated diagrams and those diagrams for self-energy of gauge links and for one gluon exchange between