Double Distributions and Pseudo-Distributions

We describe the approach to lattice extraction of Generalized Parton Distributions (GPDs) that is based on the use of the double distributions (DDs) formalism within the pseudo-distribution framework. The advantage of using DDs is that GPDs obtained in this way have the mandatory polynomiality property, a non-trivial correlation between $x$- and $\xi$-dependences of GPDs. Another advantage of using DDs is that the $D$-term appears as an independent entity in the DD formalism rather than a part of GPDs $H$ and $E$. We relate the $\xi$-dependence of GPDs to the width of the $\alpha$-profiles of the corresponding DDs, and discuss strategies for fitting lattice-extracted pseudo-distributions by DDs.


INTRODUCTION
Generalized Parton Distributions (GPDs) [1][2][3][4][5][6] (for reviews see [7][8][9]) are a major object of study at future Electron-Ion Collider and existing facilities at Jefferson Lab and CERN.They provide a detailed information about hadronic structure.Being functions H(x, ξ, t) of 3 kinematic variables (while there are other GPDs: E, H, Ẽ, etc., we will use H as a generic notation), they combine properties of usual parton distributions f (x), hadronic form factors F (t) and, in the central region |x| < ξ, of the distribution amplitudes φ(x/ξ).
However, this multi-dimensional nature of GPDs highly complicates their extraction from experimental data.In particular, deeply virtual Compton scattering (DVCS), which is the main tool for obtaining information about GPDs, gives information about GPDs on the lines x = ±ξ or through the Compton form factors that are x-integrals of GPDs with the 1/(x − ξ) weight.
More complicated processes like double DVCS or recently proposed single diffractive hard exclusive photoproduction [10] may provide information about GPDs off the x = ±ξ diagonals.The study of such processes is in its early stage.
During the last decade, starting with the pioneering paper of X. Ji [11] that introduced the quasi-distribution approach (see also Ref. [12] for "lattice cross sections" approach), strong efforts have been made to calculate parton distributions on the lattice (for reviews see Refs.[13][14][15][16]).In particular, matching conditions for GPDs in the quasi-distribution approach were discussed in Refs.[17][18][19].For a review of recent lattice calculations of GPDs see Refs.[20,21].
In our paper [22], general aspects of lattice QCD extraction of GPDs have been discussed in the framework of the pseudo-distribution approach [23,24].The advantage of lattice calculations is that matrix elements M (ν, ξ, t) ("Ioffe-time distributions" (ITDs)) of nonlocal operators measured on the lattice are related to Fourier transforms of GPDs H(x, ξ, t), which may be inverted using various technics to produce GPDs as functions of x for fixed values of skewness ξ and invariant momentum transfer t.
An important property of GPDs is polynomiality [7], which states that x N moment of H(x, ξ, t) must be a polynomial of ξ of not larger than (N + 1)th power.This nontrivial correlation between x-and ξ-dependences of H(x, ξ, t) is automatically satisfied when GPDs are obtained from double distributions F (β, α, t) [1,3,4,25,26].
The goal of the present work is to outline the approach of lattice extraction of double distributions from lattice calculations.The paper organized as follows.To make it self-contained, in Sec.II we formulate the definitions of usual (light-cone) GPDs, DDs and discuss their relationship.Some basic properties of GPDs are discussed in Sec III.There we also introduce Ioffe-time distributions.Pseudo-distributions, as generalizations of the ITDs onto correlators off the light cone are introduced in Sec.IV.Some strategies for fitting lattice-extracted pseudo-distributions by DDs are discussed in Sec.V. Finally, in Section VI, we summarize our discussion.

A. Definition of GPD
In the GPD description of a nonforward kinematics proposed by X. Ji [2], the plus-components of the initial p and final p ′ hadron momenta are given by (1 + ξ)P + and (1−ξ)P + , respectively, with P being the average momentum P = (p+p ′ )/2, while the partons have (x+ξ)P + and (x − ξ)P + as the plus-components of their momenta, see Fig. 1.
The invariant momentum transfer is given by t = (p − p ′ ) 2 .In principle, the r.h.s. of Eq. (2.1) has also the r λ term, where r = p − p ′ is the momentum transfer.However, the GPD convention is to write r + = 2ξP + , where ξ is the skewness variable, and the two terms are combined in one GPD H(x, ξ, t).
One may re-write these definitions in a more covariant form that uses Lorentz invariants (Pz) and (rz) only.For pion, we have . (2.3) For nucleons, we have two GPDs . (2.4)

B. Double distribution description
An alternative approach to describe nonforward matrix elements is based on DD formalism [1,3,4,25,26].Its guiding idea is to treat P + and r + as independent variables and organize the plus-momentum flux as a "superposition" of P + and r + momentum flows.
The parton momentum in this picture is written as k + = βP + + (1 + α)r + /2, i.e., as a sum of the component βP + due to the average hadron momentum P (flowing in the s-channel) and the component (1 +α)r + /2 due to the t-channel momentum r, see Fig. 2.

Pion
In terms of DDs, the matrix element (2.3) is written as [1,3,26,27] where Ω is the DD support region, i.e., a rhombus in the (αβ)-plane defined by |α| + |β| ≤ 1.The time reversal invariance requires that F (β, α, t) is an even function of α, while G(β, α, t) is odd in α.Expanding e −iβ(Pz)−iα(rz)/2 in powers of (Pz) and (rz), one observes that the generic term (Pz) N −k (rz) k may be obtained both from F -and G-parts [28], with two exceptions.Namely, one cannot obtain the (Pz) N term from the G-part, and one cannot obtain the (rz) N term from the F -part.The usual convention is to absorb all the (Pz) N −k (rz) k terms with k < N into the F -function, leaving the (rz) N terms in the G-function [27].As a result, the G-part would not depend on (Pz), and one can write where D(α, t) is the D-term function introduced in Ref. [27].It is odd in α.

C. Fixed parity cases
Usually we are interested in the functions corresponding to operators that are symmetric or antisymmetric with respect to the inversion of z.These combinations appear when we consider "nonsinglet" q − q or "singlet" q + q parton distributions, respectively.Since the D-term contribution (without the overall (rz) factor) is odd in z, it appears in the "singlet" case only.However, the H + E sum does not contain the D-term even in the singlet case.
In fact, it is sufficient to consider matrix element of the original O λ (z) operator.The real part of this matrix element is even in z while its imaginary part is odd in z.

III. SOME PROPERTIES OF GPDS AND DDS A. DD-parts of GPDs
In this section, we consider the relations between the DDs and the "DD parts" of GPDs which they generate, thus ignoring for a while the D-term contributions to GPDs.The D-term will be discussed later in the paper.For definiteness, we will have in mind relations between the DD part of the pion GPD and its DD.All the relations are equally applicable to the DD parts of the nucleon GPDs.3. DD support rhombus and integration lines producing the DD parts of H(ξ, ξ), H(−ξ, ξ), H(x, ξ = 0) = f (x) and H(x, ξ) in (|x| > ξ) and (|x| < ξ) regions.

B. ξ = 0 limit
Taking ξ = 0, we have This means that integrating F (β, α, t) over vertical lines β = x gives the ξ = 0 ("non-skewed") GPD H (x, ξ = 0, t), which we will also denote as f (x, t).It is the simplest GPD, that was called "nonforward parton density" in the paper [30], where it has been introduced.It differs from the forward PDF f (x) by the presence of t-dependence and satisfies f (x, t = 0) = f (x).

C. Polynomiality
The DD representation automatically produces a GPD satisfying the polynomiality property.Indeed, i.e., the n th x-moment of H DD (x, ξ, t) is a polynomial in ξ of the order not exceeding n.
Note that, since F (β, α, t) is even in α, the summation over k involves even k only, i.e. (3.1) is in fact an expansion in powers of ξ 2 .

D. Ioffe-time distributions
By Lorentz invariance, the matrix element (2.3) defining GPD is a function of the scalars (pz) ≡ −ν 1 and (p ′ z) ≡ −ν 2 , two Ioffe-time parameters, so we may write where I(ν 1 , ν 2 , t) is the double Ioffe-time distribution.Since z = z − is assumed, only the values of the plus components of momenta are essential in the scalar products (p 1 z) and (p 2 z).The skewness variable ξ in this case is given by We have introduced here the variable ν = (ν 1 + ν 2 )/2.Treating ν and ξ as independent variables, we define the generalized Ioffe-time distribution (GITD) as According to (2.1), it is a Fourier transform of the GPD Using Eq. (2.5), we can write the DD part of GITD in terms of DD dα e iναξ F (β, α, t) . (3.7)

E. DD profile and ξ-dependence
If F (β, α, t) has an infinitely narrow profile in the α-direction, i.e. if F (β, α, t) = f (β, t)δ(α), then the ξ-dependence disappears, and we deal with the simplest GPD f (x, t).A nontrivial dependence on the skewness ξ is obtained if DD has a nonzero-width profile in the α-direction.
Using the DD representation (3.7) for the GITD and expanding e iναξ into Taylor series, we get the following expansion in powers of ξ 2 (odd powers of ξ do not appear because F (β, α, t) is even in α).By analogy with (3.1), we will use notation f 2 (β, t) for the second α-moment of F (β, α, t) As a result, we write

A. Definitions
On the lattice, we choose z = z 3 , and introduce pseudo-GPDs H x, ξ, t; z 2 3 (and also E x, ξ, t; z 2 3 in the nucleon case).
The two Ioffe-time parameters are given now by ν 1 = p (3) z 3 ≡ P 1 z 3 and ν 1 = p ′(3) z 3 ≡ P 2 z 3 .In terms of momenta P 1,2 , the skewness ξ is given by The pseudo-GITD will be denoted as M(ν, ξ, t; z 2 3 ), e.g., the inverse transformation for H is written as Similarly, to denote pseudo-DDs, we will just add z 2 3 to their arguments.

B. Contaminations
On the lattice, we have z 2 ̸ = 0, and we need to add extra z-dependent structures to the original twist-2 parameterization where z 2 = 0 and the ITDs H DD , E DD and D are functions of ν, ξ and t.
A classification of additional tensor structures that appear in parametrizations of M λ off the light cone was done in Ref. [31], where eight independent structures have been identified.
However, there is some subtlety related to the fact that, for lattice extractions, we need to parametrize the "non-contracted" matrix element M λ .
In this case, the index λ in local operators ψγ λ (zD) N ψ is not symmetrized with the indices D µ1 . . .D µ N in covariant derivatives, which is necessary for building twist-2 local operators.
A way to perform symmetrization for bilocal operators was indicated in Ref. [32].Further studies of parameterizations for matrix elements with an open index have been done in Refs.[33][34][35][36][37][38].An important observation made there is that M λ should contain terms that vanish when contracted with z λ , such as r λ (Pz) − P λ (rz).One can see that r λ − P λ (rz)/(Pz) ≡ ∆ λ ⊥ is the part of the momentum transfer r that is transverse to z.
As shown in these papers, one should add Wandzura-Wilczek-type (WW) terms [39] to parametrizations of GPDs to secure electromagnetic gauge invariance of the DVCS amplitude [40] with O(∆ ⊥ ) accuracy.While the WW terms are "kinematical twist-3" contributions built from twist-2 GPDs, one cannot exclude non-perturbative (dynamical) twist-3 terms accompanied by the ∆ λ ⊥ factor.
Among additional structures listed in Ref. [31], one can see the structure (ū ′ iσ λz u) that also vanishes when multiplied by z λ , and thus should be treated as a "highertwist" term.
On the other hand, two other additional structures, (ū ′ iσ zr P λ u) and (ū ′ iσ zr r λ u), after contraction with z λ , produce the same "twist-2" structure ∼ σ zr that accompanies the E DD contribution.In this sense, the invariant amplitudes accompanying these structures, have a twist-2 component.
Note, however, that combinations σ zr P λ − σ λr (Pz) and σ zr r λ − σ λr (rz) vanish after contraction with z λ .So, we propose to use these "subtracted" forms in building the basis of additional terms, rather than just σ zr P λ and σ zr r λ .Since the "subtracted" structures do not contribute to the twist-2 parameterization (4.13), the DDs associated with them should be classified as "highertwist" ones.
For this reason, we construct a parameterization for M λ in which "twist-2" and "higher-twist" terms are explicitly separated, We have here Y and X i terms whose contribution vanishes when contracted with z λ , and Z i terms that produce z 2 factor after contraction with z λ .
Formally, the r λ (Pz) − P λ (rz) combination does not contain new structures that are independent from those present in the "twist-2" line.However, the corresponding invariant amplitude, which is denoted as Y , is generated by a new DD.This "higher-twist" DD is different from the twist-2 DDs h(β, α), e(β, α) and the D-term, which are also associated with the ∼ ū′ u structures.
Of course, using Gordon decomposition we can re-write (4.14) in a form explicitly having just eight structures like in Ref. [31].To establish a direct correspondence, we note that Ref. [31] uses a basis in which (ū ′ γ λ u) is substituted by two other structures that appear in the Gordon decomposition (4.15).Also, all the terms containing (ū ′ iσ λr u) are combined in one contribution.Using this basis, we have Comparing Eq. (4.17) with the coefficients A i in Eq. ( 35) of Ref. [31], we establish the correspondence The main difference is that H DD and D contributions in Eq. (4.17) come with the contamination from the Y -function, the 9th "higher-twist" ITD.Also, the (ū ′ iσ λr u) structure is accompanied by a factor in which the Y term is absent, but there are contaminations from X 1 and X 2 contributions.

V. FITTING PSEUDODISTRIBUTIONS
A. Nonforward parton pseudo-density f (β, t, z 2  3 ) Taking ξ = 0 we have where ν = P 1 z 3 = P 2 z 3 .An important point is that ξ = 0 may be achieved for different pairs of equal initial and final momenta P 1 = P 2 ≡ P .One should check that lattice gives the same curve for different P 's, up to evolution-type dependence on z 2 3 .One can use relation (5.1) to fit f (β, t, z 2  3 ).First, taking t = 0, we fit the forward pseudodistribution f (β, z 2 3 ), just as a pseudo-PDF.After that, one can vary t, by changing the transverse components ∆ 1,2  ⊥ , for several fixed ν.In this way, one can study what kind of dependence on t we have (dipole, monopole, etc.), and how it changes with ν.
3 ) The next step is to check if the ξ-dependence of the lattice data for M(ν, ξ, t; z 2 3 ) agrees with the form and extract f 2 (β, t, z 2 3 ) using dβ e iνβ f 2 (β, t; z 2 3 ) . (5. 3) The α-dependence of the DD F (β, α) describes the distribution of the momentum transfer r = P 1 − P 2 between the initial and final quarks.It is expected that it has a shape similar to those of parton distribution amplitudes.

C. Factorized DD Ansatz
A nonzero-width profile of DD in the α-direction may be modeled by using the Factorized DD Ansatz [25,26] with N being the parameter governing the width of the α-profile of the model DD F N (β, α, t).The α-integral of F N (β, α, t) gives the nonforward parton density f (β, t).

F. D-term
When we take the z-odd part O λ − of the operator O λ (z), its parametrization contains a nonzero D-term.
In GPD description, it appears in a mixture with H DD (and also E DD in the nucleon case) GPDs.However, using all possible helicity states for nucleons and various values of λ, one can construct sufficient number of linearly independent relations.To separate the DDs that appear in the parametrization of Eq. (4.16) one can use, e.g., singular value decomposition technique.Unfortunately, as seen from Eq. (4.16), the D-term obtained in this way comes together with the Y -contamination.
Another way is to eliminate H DD , E DD , etc. contributions from the matrix element of O λ − by taking kinematics in which (Pz) = 0.As a result, α-even DD h(β, α) will be integrated with the α-odd function sin(α(rz)), etc., so that we will have On the lattice, choosing z = z 3 , we can arrange (Pz) = 0, i.e.P 3 = 0, by taking p 1 and p 2 with opposite components in z-direction, namely p = (E 1 , p 1T , P ) and p ′ = (E 2 , p 2T , −P ).Introducing the relevant Ioffe time (5.18) However, if we choose λ = 0, we get r 0 = E 1 − E 2 as the accompanying factor.It vanishes for purely longitudinal momenta p = (E, 0 T , P ), p ′ = (E 2 , 0 T , −P ), and remains rather small when one takes non-equal transverse momenta p 1T , p 2T .
Another choice is to take λ = 3.In this case, we have ∼ As we see, for a fixed ν, the contamination term decreases with P .In principle, one may try to extract I D (ν D , t) by fitting the P -dependence of the matrix element.

VI. SUMMARY
In the present paper, we have outlined the approach of lattice extraction of GPDs based on a combined use of the double distributions formalism and pseudo-PDF framework.The use of DDs guarantees that GPDs obtained from them have the required polynomiality property that imposes a non-trivial correlation between xand ξ-dependences of GPDs.
We have introduced Ioffe-time distributions writing these directly in terms of DDs, and generalized them onto correlators off the light cone.An important advantage of using DDs is that the D-term appears then as an independent quantity rather than a non-separable part of GPDs H and E.
We have discussed the relation of the ξ-dependence of GPDs with the width of the α-profiles of the corresponding DDs, and proposed strategies for fitting latticeextracted pseudo-distributions by DDs.The approach described in the present paper is already used in ongoing lattice extractions of GPDs by HadStruc collaboration.

FIG. 1 .
FIG. 1. Flux of the momentum plus-components in terms of GPD variables.

FIG. 2 .
FIG. 2. Flux of the momentum plus-components in terms of DD variables.