Structure analysis of the generalized correlator of quark and gluon for a spin-1/2 target

We analyze the structure of generalized off-diagonal and transverse-momentum dependent quark-quark and gluon-gluon correlators for a spin-1/2 hadron. Using the light-front formalism, we provide a parametrization in terms of the parton generalized transverse-momentum dependent distributions that emphasizes the multipole structure of the correlator. The results for the quark-quark correlation functions are consistent with an alternative parametrization given in terms of Lorentz covariant structures. The parametrization for the gluon-gluon generalized correlator is presented for the first time and allows one to introduce new correlation functions which can be relevant for phenomenological applications.


Introduction
Both the generalized parton distributions (GPDs) [1][2][3][4][5][6][7], appearing in the description of hard exclusive reactions like deeply virtual Compton scattering, and the transversemomentum dependent parton distributions (TMDs) [8][9][10][11], appearing in the description of semi-inclusive reactions like semi-inclusive deep inelastic scattering and Drell-Yan process, have been intensively studied in the last two decades. These distributions provide us with essential information about the distribution and the orbital motion of partons inside hadrons, and allow us to draw three-dimensional pictures of the nucleon, either in mixed position-momentum space or in pure momentum space [12].
Despite numerous suggestions in the literature [13][14][15][16][17][18][19], no nontrivial model-independent relations between GPDs and TMDs have been found [20,21]. However, both the GPDs and the TMDs appear to be two different limits of more general correlation functions called generalized TMDs 1 (GTMDs) [20,21] which can show up in the description of hard QCD processes [22,23]. They depend on the 3-momentum of the partons and, in addition, contain information on the momentum transfer to the hadron. The quark GTMDs typically appear at subleading twist and in situations where the standard collinear factorization cannot be applied, see e.g. refs. [24][25][26]. On the other hand, gluon GTMDs have been extensively used in the description of high-energy processes like e.g. diffractive vector meson production [27] and Higgs production at the Tevatron and the LHC [28][29][30] using the k T -factorization framework. In ref. [31] is also suggested an approximate method for constraining the unpolarized gluon GTMD. The GTMDs have a direct connection with Wigner distributions of the parton-hadron system [4,[32][33][34] which represent the quantummechanical analogues of the classical phase-space distributions and have recently been discussed to access the orbital angular momentum structure of partons in hadrons [35][36][37][38].
The parametrization of the generalized (off-diagonal) quark-quark correlation functions for a spin-0 and spin-1/2 hadron has been given for the first time in refs. [20,21]. Here, we want to extend this study to the generalized gluon-gluon correlator, proposing a convenient formalism which allows us to discuss in a unified framework also the quark-quark correlator. Such a formalism is based on the light-front quantization and on the analysis of the multipole pattern given by the parton operators entering the two-parton generalized correlators at different twists. We first identify the spin-flip number of each parton operator, defined in terms of the helicity and orbital angular momentum transferred to the parton. To each spin-flip number we can then associate a well-defined multipole structure that can be represented in terms of the four-vectors at our disposal, multiplied by Lorentz scalar functions representing the parton GTMDs.
The various step of this derivation are presented as follows. In the next section we introduce the definition of the two-parton generalized correlator. In section 3, we describe the derivation of the parametrization of the generalized correlators in terms of GTMDs. In particular, we discuss the angular momentum structure and multipole pattern of the twoparton correlators at different twists. Taking into account also the constraints of discrete symmetries and hermiticity, we obtain a basis to parametrize both the quark-quark and the gluon-gluon correlation functions. The results in the gluon sector are given here for the first time, while the parametrization in the quark sector is alternative, but equivalent, to that one given in terms of Lorentz covariant structures in ref. [20]. The relations between these two sets of quark GTMDs are given in the appendix. At leading twist, we also present the results for the light-front helicity amplitudes, discussing the physical interpretation of the twist-2 GTMDs in terms of nucleon and parton polarizations. In section 4, we discuss the TMD limit and the GPD limit of the GTMDs, and provide the dictionary to relate them with other existing parametrizations of the gluon and quark distribution functions. In the last section we draw our conclusions.

Generalized parton correlators
The maximum amount of information on the parton distributions inside the nucleon is contained in the fully-unintegrated two-parton correlator W for a spin-1/2 hadron. The general quark-quark correlator is defined as 2 [4,20,32,33] W [Γ] Λ ′ Λ (P, k, ∆, N ; η) = 1 2 and the general gluon-gluon correlator can be defined likewise (2.2) These correlators are functions of the initial (final) hadron light-front helicity Λ (Λ ′ ), the average hadron four-momentum P = (p ′ + p)/2, the average parton four-momentum k, and the four-momentum transfer to the hadron ∆ = p ′ − p. The superscript Γ in eq. (2.1) stands for any element of the basis {1, γ 5 , γ µ , γ µ γ 5 , iσ µν γ 5 } in Dirac space. The Wilson lines W ≡ W(− z 2 , z 2 |ηn) and W ′ ≡ W( z 2 , − z 2 |η ′ n) ensure the color gauge invariance of the correlators [39], connecting the points − z 2 and z 2 via the intermediary points − z 2 + η ′ ( ) ∞ · n and z 2 + η ′ ( ) ∞ · n by straight lines 3 , where n is a lightlike vector n 2 = 0. Since any rescaled four-vector αn with some positive parameter α could be used to specify the Wilson lines, the correlators actually depend on the four-vector The parameters η ′ ( ) indicate whether the Wilson lines are future-pointing (η ′ ( ) = +1) or past-pointing (η ′ ( ) = −1). For convenience, we choose the spatial axes such that n ∝ e z and work in a symmetric frame, see figure 1. The two-parton correlators defining TMDs, GPDs, PDFs, FFs and charges are obtained by considering specific limits or projections of eqs. (2.1) and (2.2). These correlators have in common the fact that the parton fields are taken at the same light-front time z + = 0. We then focus our attention on the k − -integrated version of eqs. (2.1) and (2.2) where we used for a generic four-vector a µ = [a + , a − , a T ] the light-front components a ± = (a 0 ± a 3 )/ √ 2 and the transverse components a T = (a 1 , a 2 ), and where x = k + /P + is the fraction of average longitudinal momentum and k T is the average transverse momentum of the parton. These correlators are parametrized in terms of the so-called GTMDs, which can be considered as the mother distributions of GPDs and TMDs. A complete parametrization of the quark-quark correlator (2.4) in terms of GTMDs has been given in ref. [20]. In the present work, we give for the first time a complete parametrization of the gluon-gluon correlator (2.5), and provide the dictionary between the corresponding daughter functions (GPDs, TMDs, PDFs) and other partial parametrizations given in the literature. Moreover, we present an alternative (but equivalent) parametrization of the quark-quark correlator (2.4) which emphasizes better the underlying multipole pattern.

Parametrization
The correlators (2.4) and (2.5) can generally be written as where O(z) stands for the relevant quark or gluon operator, and M O is a matrix in Dirac space, with O = [Γ] in the quark sector and O = µν; ρσ in the gluon sector. A general, model-independent parametrization of these objects is obtained by giving an explicit form of M O in terms of the four-vectors at our disposal (P, k, ∆, N ), of the Dirac matrices (1, γ 5 , γ µ , · · · ), of the invariant tensors g µν and ǫ µνρσ , and of Lorentz scalar functions where, for convenience, we denoted the set of all parameters η simply by η i .
Traditionally, one writes down all the possible structures compatible with the Lorentz covariance, the discrete symmetry and the hermiticity constraints. All the allowed structures are usually not independent. Using on-shell relations like e.g. the Gordon identities, one can eventually extract an independent subset. Such an independent subset can be thought of as a basis for the parametrization of the correlators. Note however that because of the on-shell identities, one has a certain freedom in choosing the actual basis. Most of the time, the basis with the simplest structures is chosen. However, such a choice will generally not display the underlying twist and multipole patterns. As a result, the corresponding Lorentz scalar functions have often no simple physical interpretation.
Alternatively, one can use the light-front formalism. It has the advantage of unravelling the underlying twist and multipole patterns. Another advantage is that it is also much easier in practice, especially when there are many four-vectors at our disposal. The two methods are of course equivalent. They lead at the end to the same number of independent structures and can be translated into each other.

Angular momentum and multipole pattern
The quark spinors ψ(k, λ) and gluon polarization four-vectors ε µ (k, λ) have definite lightfront helicity λ corresponding to the eigenvalue ofĴ z =Ŝ z +L z , whereŜ z is the standard spin operator andL z is the orbital angular momentum (OAM) operator given in momentum space byL When discussing the angular momentum along the z direction, it is convenient to use the polar combinations a R,L = a 1 ± ia 2 for the transverse indices. It turns out to be particularly convenient to work with a complete set of partonic operators having a well-defined spin-flip number defined as ∆S z = λ ′ − λ + ∆L z , where λ (λ ′ ) is the initial (final) parton light-front helicity and ∆L z is the eigenvalue of the operator where k = (k f + k i )/2 and ∆ = k f − k i with k i (k f ) the initial (final) parton momentum. For example, one can easily see that the generic structure k m 1 R k m 2 L ∆ m 3 R ∆ m 4 L carries m 1 − m 2 + m 3 − m 4 units of OAM. For the quark operators, we have where the scalar, pseudoscalar, vector, axial-vector and pseudotensor quark bilinears are respectively given by

9)
A µ = ψγ µ γ 5 ψ, (3.10) T µν = ψiσ µν γ 5 ψ. (3.11) For the gluon operators, we have 14) where i, j = 1, 2 are transverse indices, ǫ 12 T ≡ ǫ −+12 = +1, and where we have defined Interestingly, the twist-2 partonic operators have ∆L z = 0 leading therefore to a simple interpretation in terms of light-front helicities ∆S z = λ ′ − λ. For the higher-twist partonic operators, a simple interpretation does not exist since the light-front helicities are usually mixed with the OAM. Just like the quark spinors and the gluon polarization four-vectors, the nucleon states |p, Λ have definite light-front helicity Λ corresponding to the eigenvalue ofĴ z =Ŝ z +L z . By conservation of angular momentum, the amplitude W O Λ ′ Λ is associated with the change of OAM ∆ℓ z = ∆S z − (Λ ′ − Λ). Since k T and ∆ T are the only possible transverse vectors available 4 , ∆ℓ z has to coincide with the eigenvalue obtained by applying the OAM operator (3.3) to the amplitude W O Λ ′ Λ . Therefore, the general structure of the amplitude W O Λ ′ Λ is given in terms of explicit global powers of k T and ∆ T , accounting for the change of OAM, multiplied by a Lorentz scalar function X(x, ξ, k 2 T , k T · ∆ T , ∆ 2 T ; η i ). Since any structure of the form can be reabsorbed in the definition of the Lorentz scalar functions X(x, ξ, k 2 T , k T · ∆ T , ∆ 2 T ; η i ), there can only be 2 independent structures for each value of ∆ℓ z . For ∆ℓ z = 0, we choose 1 and i( k T × ∆ T )z to choose a light-front frame such that PT = 0T . This is related to the fact that, thanks to translation invariance, a parametrization does not actually depend on P apart from a trivial global factor. the following forms . . . . . .
where a, b simply label the Lorentz scalar functions associated with the two independent structures for a given ∆ℓ z .

Discrete symmetry and hermiticity constraints
The hermiticity constraint relates amplitudes with initial and final light-front helicities interchanged, and changes the sign of the momentum transfer where a * is the complex conjugate of a, and O H is given by for quarks and by (µν; ρσ) H = (ρσ; µν) * for gluons. For later convenience, we will use the notation a ∆ to indicate that the sign of ∆ has been changed in the function a.
For the discrete symmetries, it is convenient to use the ones adapted to the light-front coordinates [43][44][45]. The light-front parity changes the sign of the a 1 component of any four-vector a and flips the light-front helicities for quarks and by (µν; ρσ) P =μν;ρσ for gluons. Finally, under light-front time-reversal any four-momentum transforms as q →q, while any position four-vector transforms as x → −x. As a result, invariance under light-front time-reversal implies for quarks and by (µν; ρσ) T = (μν;ρσ) * for gluons. In the symmetric frame one has naturallyP = P . The momentum arguments of the Lorentz scalar functions X(x, ξ, k 2 T , k T · ∆ T , ∆ 2 T ; η i ) are invariant under light-front parity and time-reversal transformations. For later conve-nience, we then introduce the following notations: To each partonic operator in eqs. (3.4)-(3.16), we associate c H , c P and c T coefficients determining their properties under hermiticity, light-front parity and light-front time-reversal transformation, respectively where the replacement rule affects only the uncontracted transverse indices. An explicit pair of indices O LR has to be considered as contracted since it can be rewritten in terms of δ ij T O ij and −iǫ ij T O ij . We chose the factors of i in the partonic operators (3.4)-(3.16) such that c T = +1. For the quark operators, we have and for the gluon operators, we have

Quark and gluon GTMDs
For a given partonic operator O, the amplitude W O Λ ′ Λ can conveniently be represented as a 2 × 2 matrix in the proton light-front helicity basis. The amplitudes with ∆S z = 0 and parity c P = ±1 have the following generic structure where the row entries correspond to Λ ′ = 1 2 , − 1 2 and the column entries are likewise Λ = 1 2 , − 1 2 . Furthermore, the hermiticity constraint imposes the following relations Similarly, we have the following generic structure for ∆S z = ±1 where the hermiticity constraint imposes Finally, we have the following generic structure for ∆S z = ±2 where the hermiticity constraint imposes The 2 × 2 matrices in eqs. (3.46), (3.48) and (3.50) can be expressed in the more conventional bilinear form where M ∆Sz is a Dirac matrix. The general structure of these matrices can be written in the following form: where t + 1 is the twist of the partonic operator, and we used the notations ǫ ab The general parametrization of the GTMD correlators (2.4) and (2.5) is given by eqs. (3.52)-(3.57) and is determined by the twist t + 1, the spin-flip ∆S z and the parity coefficient c P of the partonic operator summarized in tables 1 and 2. The relations between the quark GTMDs in eqs. (3.53)-(3.56) and the nomenclature introduced in ref. [20] are given in appendix A.

Quark and gluon light-front helicity amplitudes
For the two-parton correlators at leading twist, it is also convenient to represent them in terms of helicity amplitudes. We will restrict ourselves to the region x > ξ where the GTMDs describe the emission of a parton with momentum k i and helicity λ from the nucleon, and its reabsorption with momentum k f and helicity λ ′ . Any parton operator O occurring in the definition of the parton correlators (2.4) can be decomposed in the parton light-front helicity basis as follows O = λ ′ ,λ c λ ′ λ O λ ′ λ . The light-front helicity amplitudes are then defined as the matrix elements of O λ ′ λ in the states of definite hadron light-front helicities [46] H and depend in general on all the four-vectors at our disposal. At leading twist, the spin-flip ∆S z associated with the partonic operator can be identified with the difference of light-front helicities of the parton between the final and initial states, i.e. ∆S z = λ ′ −λ. Then, by conservation of the total angular momentum, the orbital angular momentum transfer to the parton is simply given by ∆ℓ z = (Λ − λ) − (Λ ′ − λ ′ ). As a result, to each value of the spin-flip ∆S z one can associate at leading twist a well-defined state of polarization for the active parton [6]. In the quark sector, 1 2 V + corresponds to the unpolarized quark operator, 1 2 A + corresponds to the longitudinally polarized quark operator, and 1 2 T R(L)+ correspond to the transversely polarized quark operators Similarly, in the gluon sector, δ ij T Γ +i;+j corresponds to the unpolarized gluon operator, −iǫ ij T Γ +i;+j corresponds to the longitudinally polarized gluon operator, and −Γ +R(L);+R(L) correspond to the transversely polarized gluon operators where for the gluon polarization vectors we used Denoting the matrix elements of these leading-twist operators as follows we obtain the following matrix representation for the light-front helicity amplitudes Using the discrete symmetry and hermiticity constraints discussed in section 3.2, one obtains the following properties for the helicity amplitudes:

TMD limit
The forward limit ∆ = 0 of the correlators W , denoted as Φ, gives the quark-quark and gluon-gluon correlators which are parametrized in terms of quark and gluon TMDs, respectively. These TMDs can be seen as the forward limit of the GTMDs. For ∆ = 0, the imaginary part of the GTMDs belonging to the class X + and the real part of the GTMDs belonging to the class X − vanish because they are odd under a sign change of ∆, see eqs. (3.60) and (3.61). In addition, the functions in eqs. (3.53)-(3.57) which are multiplied by a coefficient proportional to ∆, i.e. those labeled by b, do not appear in the correlator Φ any longer.
In the gluon sector, we find 8 TMDs at twist 2, 16 TMDs at twist 3, 24 TMDs at twist 4. The correlators at twist 5 and twist 6 are copies of the correlators at twist 3 and 2, respectively. At each twist, half of the TMDs are T-odd functions and half are T-even functions. We will discuss explicitly the parametrizations for the gluon correlators at twist 2 and at twist 3, comparing with the results derived in refs. [47,49].

GPD limit
Integrating the correlator W over k T , one obtains the parton correlators denoted as F The integration over k T removes the dependence on η i , and we are left with a Wilson line connecting directly the points − z − 2 and z − 2 by a straight line. As a consequence, all the T-odd contributions given by the imaginary part of the GTMDs disappear, and the generic structures parametrizing the correlators (4.55)-(4.56) can be obtained from eqs. (3.20)-(3.23) as We refer to [20] for the complete list of quark GPDs up to twist 4, where the results at twist 3 in the chiral-odd sector and at twist 4 have been derived for the first time, the results at twist 2 follow the common definitions [6], and the definitions at twist 3 in the chiral-even sector can easily be related to the set of GPDs introduced in ref. [50]. The relations between the standard GPDs and the GPD limit of our GTMDs read: -at twist 2, in the chiral-even sector -at twist 2, in the chiral-odd sector -at twist 4, in the chiral-odd sector (4.84) Using the results in appendix A to relate the quark GTMDs introduced in this work and the ones of ref. [20], we reproduce the GPD limit of the quark GTMDs given in eqs. (4.47)-(4.78) of ref. [20]. Using the hermiticity constraint (3.59) for the GTMDs, one derives the symmetry behavior of the GPDs under the transformation ξ → −ξ. In the quark sector, the 10 GPDsẼ q T ,H q 2 , H ′q 2 , E ′q 2 ,Ẽ ′q 2 , H q 2T , E q 2T ,H q 2T ,Ẽ ′q 2T andẼ q 3T are odd functions in ξ, while all the 22 other ones are even in ξ.
At twist 2, the gluon generalized correlators in the GPD limit are parametrized as [18] δ ij T F +i;+j The relations between these twist-2 gluon GPDs and the GPD limit of our GTMDs read: -in the chiral-even sector + 2ξ 2 P 0,+;g 1,1 , E g = 2 1 − ξ 2 P 0,+;g 1,1 , (4.88) . (4.93) The gluon GPDs at twist 3 are introduced here for the first time. For the gluon correlators at twist 3, we follow the conventions of the corresponding quark correlators at the same twist order and with the same values of ∆S z and c P (see tables 1 and 2). Explicitly, the gluon GPDs at twist 3 can be defined according to 2T (x, ξ, t) u(p, Λ). (4.97) All the gluon GPDs at twist 3 are chiral-odd functions. The relations between these GPDs and the GPD limit of our GTMDs read: , (4.103) . (4.113) From the hermiticity constraint (3.59), one finds that the 9 gluon GPDsẼ g 2T are odd functions in ξ, while all the 15 other ones are even in ξ.

Conclusions
We discussed the parametrization of the generalized off-diagonal two-parton correlators in terms of generalized transverse-momentum dependent parton distributions. Such distributions contain the most general information on the two-parton structure of hadrons and reduce in specific limits or projections to the GPDs, TMDs and PDFs, and form factors accessible in various inclusive, semi-inclusive, exclusive, and elastic scattering processes.
The structure of the generalized two-parton correlator has been analyzed by proposing a new method which can be applied in general to any matrix element of partonic operators and allows one to unravel the underlying spin and orbital angular momentum content. Such a method is based on the light-front formalism which provides the most natural and practical tools when dealing with distribution of partons in a fast moving hadron. We first give the classification of the parton operators in terms of i) the spin-flip number, defined in terms of the change of the light-front helicity and orbital angular momentum of the partons between the initial and final states, ii) the properties under transformation by discrete symmetries, such as light-front parity and time-reversal, and iii) the constraints from hermiticity. When calculating the off-diagonal matrix element of the parton operators between hadron states with given values of the light-front helicities and four-momentum, we can associate to each correlation function a unique multipole structure, related to the orbital angular momentum transferred to the hadrons. Such multipoles are then expressed in terms of powers of the average transverse momentum of the partons and the transverse momentum transferred to the hadrons, multiplied by Lorentz scalar functions representing the GTMDs.
The method is applied simultaneously to the quark-quark and gluon-gluon correlation functions. In the quark sector, we obtain an alternative, but equivalent, parametrization to the one proposed in ref. [20] in terms of Lorentz covariant structures. The results for the gluon sector are presented here for the first time. We also discussed the GPD and TMD limit of the GTMDs, providing the relations with other existing parametrizations up to twist 3. The main advantage of the new nomenclature we propose is to have a transparent and direct interpretation in terms of the spin and orbital angular momentum correlations encoded in each functions. This becomes particularly evident at leading twist, where the spin-flip number of the partonic operator can be identified with the difference of light-front helicities of the parton between the final and initial states, and therefore can be directly associated with a well-defined state of polarization of the parton. As outlined before, the proposed framework can be systematically used for any matrix element of partonic operator and therefore provides a useful framework for the definition of new correlation functions that can be relevant for future phenomenological applications.

A Relations between different definitions of quark GTMDs
In this appendix, we list the relations between the quark GTMDs introduced in ref. [20] and the nomenclature adopted in this work.
The quark correlator at twist-3 with V R(L) is the only one with ∆S z = ±1 and c P = +1 (fifth column of table 1). The relations between the two sets of GTMDs read