Leading and higher twist transverse momentum dependent parton distribution functions in the spectator model

Transverse momentum dependent parton distribution functions, also abbreviated as TMDs, offer a three-dimensional picture of hadrons by taking the intrinsic transverse momentum of the parton into consideration. Hence, they are very important for us to understand the structure of hadrons. In this article, we calculate and summarize all TMDs of quark through the spectator model, from twist-2 to twist-4. Especially, we give complete results of TMDs at twist-4. We adopt a general analytical framework to calculate TMDs, with both scalar and axial-vector spectators being considered. All TMDs are calculated analytically in the light-cone coordinate, and single gluon rescattering is considered to generate T-odd TMDs. T-even TMDs are also calculated to this level, maybe for the first time. Different from the traditional point of view, the twist-4 TMDs can contribute to some physical observables like azimuthal asymmetries. An approximate formula of the Sivers asymmetry, including twist-4 TMDs, is given.


I Introduction
It is well known that the parton distribution function (PDF) in high energy particle physics is an important and useful tool to study the properties of hadrons. PDFs can be extracted from experiments, and they provide information about deep structures of hadrons. There are three PDFs -unpolarized distribution function f 1 (x) (in the Amsterdam notation [1], the same below), helicity distribution function g 1 (x) and transversity distribution function h 1 (x), and they have been extensively studied. These three distributions exhaust the information on the internal dynamics of the nucleon, if the partons are assumed to be collinear with the parent nucleon. Furthermore, if we take the transverse motion of partons into consideration, we can get transverse momentum dependent parton distribution functions (TMDs). TMDs -also called three-dimensional (3D) PDFs -describe the probability density to find a parton in a hadron with longitudinal momentum fraction x and transverse momentum p T with respect to the parent hadron momentum [2]. Unlike one-dimensional PDFs, TMDs give a three-dimensional picture of the parton distribution in momentum space.
There is an increasing attention to TMDs in recent years, and a lot of theoretical and experimental researches have been performed. For instance, TMDs can offer some contributions in several azimuthal spin asymmetries which were measured in semi-inclusive deep inelastic scattering (SIDIS) and elsewhere [3][4][5][6]. So, it is necessary to build a framework to calculate these TMDs. In this article, we apply an analytical model -the spectator diquark model -to study all the TMDs [7]. As its name suggests, the basic idea of the spectator diquark model [8] is to treat the intermediate states which are not collided directly by the incoming particles (real or virtual) as spectator diquark states with effective masses. This model was originally proposed to study deep inelastic lepton nucleon scattering [9,10], based on the quark-parton model picture [11,12]. For a proton target, the diquark can only be in spin 0 (scalar) state or spin 1 (axial-vector) state, due to the conservation of parity (the parities of the proton and the quark are both +1, so the parity of the diquark must be +1). From a flavor point of view, the diquark could be an isoscalar (ud) or an isovector (uu). This idea is naive and intuitive but useful, and it can help us to understand some essential features of nucleon physical quantities including TMDs.
In theoretical descriptions of high energy SIDIS processes involving hadrons, the cross sections are usually expanded in power of 1/Q, where Q is the large momentum transfer of the collision. The cross sections in leading power can be expressed as a convolution of the leading-twist (twist-2) distributions/fragmentation functions and the hard scattering coefficients [1]. In the subleading power, the twist-3 functions also contribute to the cross sections, and so on -though one needs to consider the possibility that expressing cross sections in terms of parton distribution functions rests on factorization. However, it is worth mentioning that, there are some attempts to improve the availability of TMD factorization to subleading twist, such as the work of Bacchetta et al. in 2019 [28], and a latest work in [29].
In this article, we calculate and summarize all the TMDs from twist-2 to twist-4. Certainly, we do all the calculations in the light-cone coordinate, which is a very natural framework in high energy physics. To obtain nonzero results of T-odd TMDs, one has to consider the effect of the gauge links or the rescattering between the struck quark and the spectator diquark [30][31][32]. Partial results of these TMDs at twist-2 and twist-3 have been given in earlier papers [3,7,[33][34][35][36][37], we summarize them in a unified form. Other studies of twist-3 TMDs based on, e.g., light-front wave functions (LFWFs) can be found in Refs. [38][39][40]. Notably, studies of twist-2 TMDs as overlaps of LFWFs have already been presented in detail in Ref. [33]. The contribution of gauge link to T-even TMDs is also considered, and the corresponding analytical results are calculated maybe for the first time. We also give the results of TMDs at twist-4, which have not been completely calculated yet. Some studies involving unpolarized TMDs at twist-4 can be found in, e.g., Refs. [41][42][43].
The twist-4 TMDs are used to be regarded as pure theoretical quantities, and they are only of academic interest because of the unavailable factorization and the lack of relations with observables. But, as pointed out by Zuotang Liang et al. in Ref. [44], one can obtain some concrete formulas describing the contribution of twist-4 TMDs to several azimuthal asymmetries, if the cross section is expanded up to twist-4 in the framework of collinear expansion [45][46][47][48]. Based on these formulas, we get an approximate expression for Sivers asymmetry including the contribution of twist-4 TMDs.
This article is organized as follows. In Sec. II, we adopt an analytical framework to calculate TMDs. We obtain two useful formulas for T-even and T-odd TMDs respectively. Then in Sec. III, we give the complete results of scalar-diqurak TMDs as well as axial-vector-diquark TMDs from twist-2 to twist-4 in the light-cone transversely polarized diquark states. Some phenomenological discussions are also given there. In Sec. IV we give a twist-4 modified expression for the Sivers asymmetry. We put the results of TMDs under another type of polarized diquark state and the results of T-even TMDs at one-loop level in the Appendix.

II Analytical framework
In this section, we try to establish a framework, inspired by Ref. [33], to calculate TMDs analytically from the perspective of correlators, i.e., via the Dirac decomposition of the correlation function. We put the results of these TMDs in a unified form in the next section and the Appendix.
As is showed in Ref. [33], the quark-quark correlation function in the quark-diquark model can be expressed as where p is the four-momentum of the active quark, p T is its transverse momentum, m is its mass, and x = p + /P + is the light-cone momentum fraction of the active quark to the hadron. The symbols M and M X are the masses of the hadron and the spectator diquark, respectively. This formula is also called the spectator approximation. The on-shell condition (P − p) 2 = M 2 X for the spectator diquark suggests that the quark is off shell: Usually, we choose the frame where the hadron momentum P has no transverse components [4,33]. Thus, some four-vectors used in this article can be written as (λ N -we use λ later for simplicity-is the helicity of the hadron), The symbol M (0) here is the nucleon-quark-diquark scattering amplitude at tree level (see the lower-left part of Fig. 1) as follows: and M (0) is its Hermitian conjugation multiplied by γ 0 . Here ε(P − p, λ a ) is the polarization vector of the axialvector diquark with momentum P − p and helicity λ a . There are several choices used when summing over all polarization states [33], and we set d µν = λ a ε * µ ε ν , According to Ref. [5], only the first and the last choices of d µν can give convergent results of some TMDs, like g ⊥ , even if the dipolar coupling factor (see below) is applied. In this article, we give the results of all TMDs under the first and the last choice of d µν from twist-2 to twist-4. The results at twist-2 can also be found in Refs. [7,33,35]. The results of the other choices of d µν can also be calculated through this analytical framework.
In the above expression of nucleon-quark-diquark scattering amplitude, the symbol Υ s/a denotes the nucleonquark-diquark vertex (s for the scalar diquark and a for the axial-vector diquark, the same below), and we choose the simplest forms of them [7] (other forms can be found in, e.g., Refs. [7,50]), here g X (p 2 ) is the coupling factor of the nucleon-quark-diquark vertex that takes into account the composite structure of the nucleon and the diquark spectator. The most general structure of the vertices can be found in Ref. [52]. Usually, there are three common choices of g X (p 2 ), where Λ X is the cutoff parameter, and g X is the coupling constant. In order to regularize the light-cone divergence appearing in the calculations of T-odd TMDs [53,54] and avoid the end-points singularities [55], the second choice, i.e., the dipolar form for g X (p 2 ) is usually used [3,33], and it can be rewritten as follows: where L 2 X (Λ 2 X ) (L 2 X for simplicity) has the same form with Eq. (2) by replacing m 2 with Λ 2 X . In this article, we give two formulas to calculate TMDs analytically under a general form of the coupling factor g X (p 2 ). Then, the concrete results of TMDs under the dipolar coupling of g X (p 2 ) are shown. Besides, results of twist-2 TMDs under all these choices are already given in Ref. [33]. (1) . The Hermitian conjugate diagram is not shown.
In order to obtain the results of T-odd functions, which vanish in the lowest order, we need to take the interference between the lowest-order amplitude M (0) and the one-loop-order amplitude M (1) into consideration [30][31][32]. The form of M (1) is as follows (see the lower-left part of Fig. 2, and eikonal approximation [2,56] is already used): where D i with i = 1, 2, 3, 4 in the denominator are propagators: X , and the vertex between the gluon and the scalar (Γ s ) or axial-vector (Γ a ) diquark has the following form: here e q and e s/a denote the color charge of the quark and the scalar/axial-vector diquark. The symbol κ here is the diquark anomalous chromomagnetic moment [33], and κ = 1 corresponds to the standard pointlike photon-W coupling [57]. As we can see, it does not show up in the final results under the first choice of d µν . However, it shows up in the results of T-odd TMDs under the last choice of d µν (see Appendix A). Besides, if we want to get the QCD color interaction, we shall apply the replacement: e q e X −→ 4πC F α s [30,58]. We can obtain TMDs by taking traces between the correlator and some γ matrices [4]. Here we introduce a symbol where Γ denotes any possible combination of gamma matrices. We take the Jaffe-Ji-Mulders convention to label TMDs [1,42,59]. For the twist-2 case, there are eight TMDs, six for T-evens and two for T-odds. The decomposition expressions of these functions can be found in e.g. Refs. [1,59]. For the twist-3 case, there are sixteen TMDs, eight for T-evens and eight for T-odds [4,60]. As for the twist-4 case, it has similar forms as the twist-2 case [60] (notice : By using the definition Φ(S) [Γ ] = 1 2 Tr[Φ(S)Γ ] and Eq. (1), we obtain here the symbol "· · · " denotes the expression inside the square brackets in M (0) M (0) . Under the dipolar form of Similarly, for the T-odd case, we can get where It is a little more complicated because of the loop integral. Firstly, we need to use Φ(S) [Γ ] = 1 4 Tr[(Φ(S) ± Φ(−S))Γ ] + H.c. (abbreviation for Hermitian conjugate) instead to get our results, and the sign between Φ(S) and Φ(−S) depends on the concrete TMD. The total calculation includes a trace part multiplied with a diagram loop integral part. The trace results of T-odd TMDs are imaginary, so the cut diagram plus its conjugate diagram lead us to take the imaginary part of the loop integral. Then we can use the Cutkosky rule [61] to obtain the imaginary part of the loop integral by putting its intermediate states into on shell: 1 D±iε =⇒ ∓2πiδ(D)θ(D + ). Hence, we can get the four-vector of the gluon and, similarly, we obtain here the symbol "· · · " denotes the expression inside the square brackets in ( Up to now, we establish a framework to calculate all TMDs analytically, and we get the two core formulas, Eq. (15) and Eq. (19). Then we directly give the results of TMDs, as well as corresponding phenomenological discussions in the following part. For a systematic model calculation, we should also consider the contribution of the one-gluon-exchange finalstate interaction for T-even TMDs. In contrast to T-odd TMDs, we should take the real part, i.e., the principal value term of the loop integral to obtain the results for T-even TMDs, because their trace results are real. We list these analytical results in the Appendix.

III Analytical results
In this section, we give all the results of TMDs for quark in this spectator model under the light-cone transversely polarized diquark from twist-2 to twist-4. Namely, we take 1 Twist-2 The results of TMDs at twist-2 are the same with the results in Ref. [33]. It is noted that, under the parametrization of Ref. [33], we can find that the Sivers function is negative for u quark and positive for d quark, which is consistent with extractions from experimental data [62][63][64], as well as calculations on the lattice [65].
We introduce some symbols to simplify our expressions ("2" for twist-2, "e" for even, "o" for odd, the same below), as For the T-even case, we have We notice that g s 1T = −h ⊥s 1L = 2M m+xM h s 1 , and g a 1T = xh ⊥a 1L . Also, we have the following expressions for the T-odd functions: We have the relations h ⊥s
Our analytical results of twist-3 TMDs are consistent with those in past literatures. Similarly, we introduce the following symbols for simplicity: e q e a 2(2π) 4 (1 − x) So, for the T-even functions, we have While for the T-odd ones, we obtain h a = 0.
Some T-odd TMDs are found to vanish in our model calculation, and most of them can be considered to be relevant with QCD constraints. Due to the time-reversal invariance of QCD [73], all T-odd TMDs must vanish when integrated upon p T , and this implies the following constraints [4,26,60]: Although our calculation is model dependent, we can easily find that our results satisfy these constraints. As mentioned, some of these TMDs can offer contributions to azimuthal spin asymmetries by contributing to the structure functions [3][4][5][6]. The relations between analytical results of TMDs and structure functions can be found in e.g., Ref. [4], and the structure functions can be expressed as the combination of TMDs and fragmentation functions. Numerical results based on these analytical results of TMDs can be found in e.g., Refs. [5,6,[67][68][69], and relevant physical observables are studied.

Twist-4
For simplicity, similarly, we introduce these following symbols: For the T-even case, we have Now we find g s 3T = h ⊥s 3L . For the T-odd case, we have Hence, we find f ⊥s 3T = h ⊥s 3 . TMDs at twist-4 have not been completely calculated yet in earlier papers, and we try to obtain them through our framework. Although there is still a lack of factorization proofs, the twist-4 TMDs can offer contributions to some physical observables like azimuthal asymmetries, if we expand the cross sections or structure functions of some process like SIDIS to twist-4 under the framework of collinear expansion. An example for the Sivers asymmetry is showed in next section. Besides, twist-4 contributions to some physical processes like the SIDIS process [44,74] and semi-inclusive e + e − annihilation process [75] are important for further understanding of the feature of hadrons, such as distribution functions and fragmentation functions.
In general, all these leading and higher twist TMDs offer us an important opportunity to examine the applicability and effectiveness of TMD factorization [2,76,77] together with potential extension of factorization proofs to higher twist levels. We can also get deeper understanding of the nonperturbative properties of QCD, as well as the nontrivial internal structure of hadrons through these TMDs.

IV Modification on Sivers asymmetry
In the SIDIS process like (l) + N (P ) → (l ) + h(P h ) + X, the Sivers asymmetry A Siv (also denoted as A is the amplitude of the sin(φ h − φ S ) modulation in the distribution of the produced hadrons. Here φ h and φ S are the azimuthal angles of hadron transverse momentum and the nucleon spin vector, respectively, in a reference system in which the z axis is the virtual photon direction and the x-z plane is the lepton scattering plane (see Fig. 3), in agreement with the Trento conventions [78]. Under this convention, at leading twist A Siv can be written as [4,64,79] (notice the sign difference with, e.g., Refs. [80,81]) Here the sum is over all (anti)quark flavors, e q is the quark charge, D q 1 (z, k 2 T ) is the transverse momentum dependent unpolarized fragmentation function, and the symbol C denotes the transverse momentum convolution, where k T is the transverse momentum of the hadron with respect to the direction of the fragmenting quark, P h⊥ is the measurable transverse momentum of the produced hadron, and z = P − h /k − is the longitudinal momentum fraction. The denominator of Eq. (84) can be recast as the collinear expression Previous studies on Sivers asymmetry only consider the contribution from the leading twist TMD, i.e., the Sivers function f ⊥ 1T . However, as indicated by Ref. [44], twist-4 TMDs can also participate in this asymmetry provided the cross section or the structure function is expanded up to twist-4. To utilize our analytical results of twist-4 TMDs, we make an approximation of the Sivers asymmetry formula by neglecting contributions from the two gluon scattering. Thus, we obtain where f (y) = 2(1 − y)/[1 + (1 − y) 2 ] with y the fractional energy of the virtual photon [4]. This modified formula for the Sivers asymmetry can show us the contribution of twist-4 TMDs intuitively and provide us a tool to further study the hadronic structures and high twist effects.

V Conclusions
As is shown, we adopt an analytical framework for the calculations of TMDs under the quark-diquark spectator model. Through this framework, we obtain all TMDs from twist-2 to twist-4 under two types of polarized diquark states. T-even TMDs at one-loop level are also analytically calculated. Besides, we can also calculate the transverse momentum dependent fragmentation functions in a similar way [3]. The contribution of twist-4 TMDs in the Sivers asymmetry is shown, and a modified formula for this asymmetry is obtained. Those formulas with model input of various coupling factors can be used to calculate physical observables in an explicit physical process, so that we can confront theoretical predictions with experimental data for a better understanding of the three-dimensional picture of hadrons.

Appendix A TMDs in another type of polarized diquark
Here are the results of TMDs from twist-2 to twist-4 under the last choice of d µν in Eq. (4), i.e., We make use of the A symbols above to simplify our results. It is noted that the diquark anomalous chromomagnetic moment κ appears in T-odd TMDs under this kind of d µν .
1 Twist-2 The results of TMDs at twist-2 are the same with those in Ref. [33]. For the T-even TMDs, we have For the T-odd TMDs, we have 2 Twist-3 The results for the T-even case are as follows: Here are the T-odd TMDs, h a = 0.
3 Twist-4 Similarly, for the T-even functions, we have For the T-odd functions, we have

Appendix B T-even TMDs at one-loop level
After taking the principal value terms, the loop integral can be rewritten as follows: where p.v.( 1 D ) denotes the Cauchy principal value of 1 D . The Cauchy principal value can be calculated through residue theorem [82], Thus, we can also get the constraints on l, namely, Eq. (18) still remains. The results of T-even TMDs here only come from the contribution of the one-gluon-exchange final-state interaction, i.e., the gauge link. The whole results of T-even TMDs from tree level to one-loop level would be the sum of the results above and the results here.

Twist-2
As usual, we introduce some symbols to simplify our expressions, Then we have 2 Twist-3 Two symbols for simplicity are as follows: The TMDs are 3 Twist-4 Similarly, two symbols for simplicity are Then, the TMDs are