Generalized parton distributions and pseudodistributions

We derive one-loop matching relations for the Ioffe-time distributions (ITDs) related to the pion distribution amplitude (DA) and generalized parton distributions (GPDs). They are obtained from a universal expression for the one-loop correction in an operator form, and will be used in the ongoing lattice calculations of the pion DA and GPDs within the parton pseudodistributions approach.


I. INTRODUCTION
Extraction of parton distribution functions (PDFs) from lattice simulations attracts now a considerable interest and efforts (for a recent review see Ref. [1]). An intensive development in this field has started with the paper by X. Ji [2], who proposed the concept of parton quasi-distributions (quasi-PDFs) formalized later within a general framework of the large momentum effective theory (LaMET) [3]. The basic idea of Ref. [2] (preceded in Refs. [4,5]) to study equal-time correlators is also used in the "good lattice cross sections" approach [6,7] and in the pseudo-PDF approach [8][9][10].
It should be noted that, in all the papers mentioned above, the derivation of the matching relations was based on separate calculations of the relevant one-loop Feynman diagrams. However, as pointed out in our paper [20], the one-loop correction in the coordinate-representation approach of Ref. [23] may be calculated in the operator form, i.e., without specifying the matrix element characteristic of a particular parton distribution.
The diagram by diagram results of such one-loop calculation for a nonsinglet quark operator are given in Ref. [20], and they were used there to obtain the matching relations between the nonsinglet pseudo-PDF and the corresponding PDF. It was also stated there that the same result obtained on the operator level may be used to derive matching relations for the pion distribution amplitude and the nonsinglet generalized parton distributions.
It is the goal of the present paper to describe the derivation of these matching relations. They can be used in future lattice extractions of the pion DA and nonsinglet GPDs within the pseudo-PDF approach.
To make the paper self-contained, we describe in Sec. II the derivation of the known matching relations for nonsinglet PDFs. In Sec. III, we derive matching relations for the pion distribution amplitude. The matching relations for nonsinglet GPDs are derived in Sec. IV. Section V contains the summary of the paper.

A. Operators and parton distributions
In the present paper, we will consider the valence parton distribution functions, the pion distribution amplitude (DA) and nonsinglet generalized parton distributions. They all are given by matrix elements of nonsinglet operators of a generic form O α ðzÞ ≡ψð0ÞΓ αÊ ð0; z; AÞψðzÞ; ð2:1Þ where Γ α ¼ γ α or γ α γ 5 . The factorÊð0; z; AÞ is the standard 0 → z straight-line gauge link in the quark (fundamental) representation Eð0; z; AÞ ≡ P exp igz ν Z 1 0 dtÂ ν ðtzÞ : ð2:2Þ In particular, studying the parton distribution functions, we deal with the forward matrix elements M α ðz; pÞ ≡ hpjO α ðzÞjpi ð 2:3Þ between the hadronic states jpi with momentum p. By Lorentz invariance, M α ðz; pÞ may be represented as a sum of two structures involving the amplitudes depending on two Lorentz scalars: the interval z 2 and the invariant ðpzÞ ≡ −ν, the Ioffe time [24].
The twist-2 PDF is determined by the Ioffe-time pseudodistribution Mðν; −z 2 Þ, while M z ðν; −z 2 Þ is a purely higher-twist contamination. It can be eliminated by an appropriate choice of z and p. The usual way to define twist-2 PDF is to use z in a purely "minus" light-cone direction, i.e., z ¼ z − and take α ¼ þ. To exclude M z in lattice calculations, one may use z ¼ z 3 and α ¼ 0, as suggested in Ref. [25]. We will follow this prescription for all the parton distributions that we consider in the present paper.

B. One-loop correction in the operator form
The one-loop correction to O 0 ðz 3 Þ was calculated in the operator form in Ref. [20], and is given by Here we use the notationv In what follows, we will also use the variable The plus-prescription at zero is defined by assuming that Fð0Þ is finite. In our result (2.5), we have used the dimensional regularization for collinear singularities, and applied the MS scheme subtraction with μ IR serving as the scale parameter.
The function ZðzÞ accumulates information about local corrections associated with the ultraviolet-divergent contributions. This function is also known (see Ref. [13]), but, in the pseudo-PDF approach, we do not need its explicit form. As we will see, such terms cancel when one forms the reduced Ioffe-time pseudodistributions.
In Feynman gauge, the terms δðvÞ containing δðuÞ or δðvÞ in the coefficient function of Eq. (2.5) are produced by vertex diagrams, while the "þ1" and "−1" u; v-independent additions to them come from the box diagram (see Ref. [20]). So, we will use sometimes "vertex" and "box" to refer to these two types of contributions.

C. Matching for parton distribution functions
In the PDF case, the one-loop correction to M 0 ðz 3 ; pÞ is given by the forward matrix element hpjδO 0 ðz 3 Þjpi. The right-hand side of Eq. (2.5) brings then the matrix element where ν ¼ p 3 z 3 is the Ioffe time [24]. Using translation invariance, the "vertex" terms containing δðuÞ or δðvÞ are trivially reduced to onedimensional integrals involving, say, ðū=uÞ þ M 0 ð0;ūνÞ or ðv=vÞ þ M 0 ð0;vνÞ. Changing u or v to a common variable 1 − w, we get the w-integral of 2ðw=wÞ þ M 0 ð0; wνÞ with the plus-prescription at w ¼ 1.
For the "box" terms, we get We can represent (1 − w) as the sum of the term ð1 − wÞ þ that has the plus-prescription at w ¼ 1 and the deltafunction term 1 2 δðwÞ that we add to Zðz 3 Þ, denoting the changed Z-function byZðz 3 Þ. As a result, we have where we have abbreviated M 0 ð0; νÞ to M 0 ðνÞ and similarly for M 0 ð0; wνÞ. The structure of Eq. (2.5) implies a scenario in which the z 2 3 -dependence at short distances is determined by the "hard" logarithms ln z 2 3 generated from the initially "soft" distribution M 0 ðν; z 2 3 Þ having only a polynomial dependence on z 2 3 that is negligible for small z 2 3 . For this reason, we have skipped the z 2 3 -dependence in the argument of M 0 -functions, leaving just their ν-dependence. The combination is the nonsinglet Altarelli-Parisi (AP) evolution kernel [26]. The latter is usually defined for PDFs, i.e., in the momentum-fraction space.
The next step is to introduce the reduced Ioffe-time pseudodistribution of Refs. [8][9][10]. When the momentum p is also oriented in the z 3 direction, i.e., p ¼ fE; 0 ⊥ ; p 3 g, the function Mð0; z 2 3 Þ corresponds to the "rest-frame" p 3 ¼ 0 distribution. According to Eq. (2.9), it is given by As a result, theZðz 3 Þ terms disappear from the Oðα s Þ correction to the ratio Mðν; z 2 3 Þ=Mð0; z 2 3 Þ, and we have Such a cancellation of ultraviolet terms for Mðν; z 2 3 Þ will persist in higher α s orders, reflecting the multiplicative renormalizability of the ultraviolet divergences of Mðν; z 2 3 Þ [29-31]. A similar calculation can be performed for the light-cone Ioffe-time distribution Iðν; μ 2 Þ [32] obtained by taking z 2 ¼ 0 in Mðν; −z 2 Þ and regularizing the resulting lightcone singularities using dimensional regularization and the MS subtraction specified by a factorization scale μ. The result may be symbolically written as that allows to get Iðν; μ 2 Þ from lattice data on Mðν; z 2 3 Þ. By definition [32], the light-cone ITD Iðν; μ 2 Þ is related to the PDF fðx; μ 2 Þ by Thus, fðx; μ 2 Þ is formally given by the inverse transformation However, lattice calculations provide Iðν; μ 2 Þ in a rather limited range of ν, which makes taking this Fourier transform rather tricky (see Ref. [33] for a detailed discussion). An easier way was proposed in our paper [8]. The idea is to assume some parametrization for fðx; μ 2 Þ similar to those used in global fits (see, e.g., Ref. [34]), and to incorporate Eq. (2.19) to fit its parameters using the lattice data for Iðν; μ 2 Þ.

A. Definition and general properties
The pion distribution amplitude, initially introduced in our 1977 paper (see Ref. [37]) may be defined using the matrix element where jpi is a pion state with momentum p. A similar object was introduced within the light-front quantization formalism [38] (see Ref. [39] for comparison of the two definitions).

B. Structure of contributing terms
Let us start with the evolution terms in Eq. (2.5), i.e., with those accompanied by lnðz 2 3 Þ in Eq. (2.5). Take first the "vertex" part. In this case, Using translation invariance, we see that this is equal to MðwνÞð1 þ e iwν Þ: ð3:6Þ Transforming to the f M-function using Eq. Take now the "box" part. It is given by Note that if we would calculate the correction to the function f Mðν; z 2 3 Þ ¼ e −iν=2 Mðν; z 2 3 Þ rather than to Mðν; z 2 3 Þ, the overall factor of e iν=2 in Eqs. (3.7) and (3.9) would be absent.

C. Matching
In a similar way, one can derive formulas for other terms from Eq. (2.5). As, a result, we obtain an analog of Eq. (2.9), namely ν=2 : ð3:10Þ To form the reduced pseudo-ITD, ð3:12Þ Thus, the sinðwν=2Þ=ðν=2Þ terms present in Eq. (3.10) change into sinðwν=2Þ=ðν=2Þ − 1 2 δðwÞ in the expression for the reduced pseudo-ITD. This combination does not have a plus-prescription form, i.e., it differs from ½sinðwν=2Þ=ðν=2Þ þ , in contrast to the PDF case, when ð1 − wÞ − 1 2 δðwÞ could be written as ð1 − wÞ þ . However, just like in the PDF case, the Zðz 3 Þ term drops from the Oðα s Þ correction to the reduced pseudo-ITD. As a result, the matching condition in the pion DA case is ð3:13Þ The "tilde" ITDĨðν; μ 2 Þ is related to the light-cone pion DA Φðx; μ 2 Þ bỹ Again, the simplest way to extract Φðx; μ 2 Þ is to assume some parametrization for it, like NðxxÞ a , and fit a from the lattice data onĨðν; μ 2 Þ.
Alternatively, in analogy with Eq. (2.21), one may write f Mðν; z 2 3 Þ in terms of Φðx; μ 2 Þ and fit α s and the parameters of the Φðx; μ 2 Þ model using the lattice data for f Mðν; z 2 3 Þ. The analog of Rðxν; z 2 3 μ 2 Þ of Eq. (2.21) is also straightforward-calculable as a closed-form expression.
A few more words about the lattice implementation. While the function f Mðν; z 2 3 Þ is directly given by the matrix element of the operator with the ð−z 3 =2; z 3 =2Þ endpoints, a more practical way to calculate it is to use the ð0; z 3 Þ endpoints and multiply the function Mðν; z 2 3 Þ obtained in this way by the e −iν=2 factor to get f Mðν; z 2 3 Þ. The reason is that z 3 =2 on the lattice should be an integer multiple of the lattice spacing a, say z 3 =2 ¼ na. But then z 3 ¼ 2na, i.e., the total separations z 3 given by an odd number of lattice spacings are lost if one uses the ð−z 3 =2; z 3 =2Þ endpoints.

D. Checking the ERBL kernel
While the matching formula (3.13) has a more involved form than that for PDFs, the difference is basically the presence of sines and cosines of wν=2, which are smooth functions of w.
On the other hand, it is well known that the ERBL (for Efremov-Radyushkin-Brodsky-Lepage [38,41,42]) kernel Vðx; yÞ governing the evolution of the pion DA is given by different functions for x < y and x > y, i.e., it is only piecewise smooth, with singularities like cusps, etc., for x ¼ y. So, one may wonder if Eq. (3.10) correctly reproduces the ERBL evolution equation dy Vðx; yÞΦ 0 ðyÞ þ Á Á Á :

ð3:15Þ
Let us take first the "vertex" part corresponding to Eq. (3.6) and write it in terms of the DA, Applying the Fourier transformation that converts MðνÞ into ΦðxÞ, gives which is a well-known part of the ERBL kernel Vðx; yÞ (see, e.g., Ref. [38]). As expected, it has different analytic forms in the regions x < y and x > y. For the "box" part given by Eq. (3.8), we have Applying the Fourier transform (3.17) gives the remaining part of the ERBL kernel Vðx; yÞ. As a function of x, it is given by two straight lines intersecting at x ¼ y, with a cusp at this point. Its integral over x gives 1=2, and the − 1 2 δð1 − wÞ term in Eq. (3.13) gives the contribution − 1 2 δðx − yÞ that provides the plus-prescription for the kernel of Eq. (3.20).
On the lattice, as discussed above, it is more convenient to take theψð0Þ…ψðzÞ operator. By translation invariance, To introduce pseudo-GPDs, we choose z ¼ z 3 . Then only the value of the third component of the average momentum P is essential in the scalar product ðPzÞ. So, we can take p 1 ¼ fE 1 ; Δ 1;⊥ ; P 1 g and p 2 ¼ fE 2 ; Δ 2;⊥ ; P 2 g. As a result, we have two Ioffe-time invariants ν 1 ≡ −ðp 1 zÞ ¼ P 1 z 3 and ν 2 ≡ −ðp 2 zÞ ¼ P 2 z 3 . The skewness variable ξ in this case is given by Note, that if we choose α ¼ 0, then both z α and Δ α ⊥ contributions will be absent in the parametrization of hp 2 jψð−z=2Þγ αÊ ð−z=2; z=2; AÞψðz=2Þjp 1 i. Hence, we can define the double Ioffe-time pseudodistribution Mðν 1 ; ν 2 ; t; z 2 3 Þ hp 2 jψð0Þγ 0 …ψðz 3 Þjp 1 i ¼ 2P 0 Mðν 1 ; ν 2 ; t; z 2 3 Þ: ð4:5Þ Using the ξ-definition (4.4), we may write P 1 ¼ ð1 þ ξÞP and P 2 ¼ ð1 − ξÞP, where P ≡ P 3 . Denoting we define the generalized Ioffe-time pseudodistribution (pseudo-GITD) by This formula tells us that the third momentum component of the quark at the point z 3 is ðx þ ξÞP, as expected. The inverse transformation is given by Note that originally we had two Ioffe-time parameters ν 1 and ν 2 . However, the Fourier representation (4.9) involves integration over just one ν-parameter, proportional to their sum. The difference ν 1 − ν 2 is expressed in terms of ν and the skewness ξ that plays the role of an external fixed parameter like t or z 2 3 . Just like in the pion DA case, it is convenient to introduce the "tilde" pseudo-GITD In deriving the matching relation, we will also need the representation ð4:12Þ

B. Structure of contributing terms
Let us now collect the terms resulting from taking Eq. (2.5) between the hp 2 j…jp 1 i brackets. Take first the "vertex" part. Proceeding as in the DA case, we start with Taking matrix elements we arrive at ð4:16Þ Applying the Fourier transformation (4.9) that converts M into H, we get the following representation for the "vertex" part of the GPD evolution kernel K v ðx; y; ξÞ ¼ It is easy to check that, for ξ ¼ 0, this expression gives the "vertex" part of the AP kernel, while for ξ ¼ 1 it gives the "vertex" part (3.18) of the ERBL kernel. Consider now the "box" part. Then we deal with Just like in Eq. (4.15), we have here an overall factor of e iξν , as expected. Further steps go absolutely in parallel with the derivation of the matching relation for the pion DA. Skipping these steps, we present here the final result To extract Hðx; ξ; t; μ 2 Þ, we again propose to take some parametrization for it, and then fit the parameters using the lattice data on f Mðν; ξ; t; z 2 3 Þ. Doing this, one should keep in mind that the GPD has a nontrivial polynomiality property [43][44][45]. It amounts to the requirement that, in the nonsinglet case, its x N moment should be a polynomial of the Nth degree in ξ. A possible way to satisfy it is to use the double distribution Ansatz [46]. An equivalent alternative strategy, similar to that in the PDF and DA cases, is to start with the matching relation between the reduced pseudo-GITD f Mðν; ξ; t; z 2 3 Þ and the light-cone GITDĨðν; ξ; t; μ 2 Þ written in terms of Hðx; ξ; t; μ 2 Þ through Eq. (4.21), and fit the parameters of Hðx; ξ; t; μ 2 Þ from the lattice data on f Mðν; ξ; t; z 2 3 Þ.

C. Remarks on lattice implementation
Just like in the pion DA case, on the lattice it is more practical to measure matrix elements Mðν 1 ; ν 2 ; t; z 2 3 Þ of the operators with the ð0; zÞ endpoints, and then to multiply them by e −iξν ¼ e −iðν 1 −ν 2 Þ=2 to convert the result into the f Mðν; ξ; t; z 2 3 Þ functions corresponding to the ð−z=2; z=2Þ endpoints.
Furthermore, on the lattice, the measurements will be done on a discrete set of values for coordinates z 3 ¼ n z a and longitudinal momenta P 1 ¼ 2πN 1 =L, P 2 ¼ 2πN 2 =L, where L ¼ na is the lattice size in the z 3 direction. Thus, possible values of the Ioffe-time parameters are limited to discrete sets ν 1 ¼ 2πn z N 1 =n and ν 2 ¼ 2πn z N 2 =n. Correspondingly, possible values for skewness are given by a set of rational numbers In particular, changing N 1 and N 2 from 0 to 6, gives 13 possible values for ξ ranging from 0 to 1 and rather well representing the whole 0 ≤ ξ ≤ 1 segment. However, varying the value of ξ also changes the value of the momentum transfer t. Namely, taking purely longitudinal momenta or, in the fξ; Pg variables, When M 2 =P 2 is small (this is not a very realistic situation for the nucleon, but still), we have For small ξ, we can approximate ð4:29Þ In these formulas, t 0 increases when ξ increases. In any case, this value of t is ξ-dependent, while we need to extract GPDs as functions of x for fixed ξ and t.
To solve this problem, one may add a transverse component Δ ⊥ to the momentum transfer. We propose to use p 1 ¼ fE 1 ; Δ ⊥ ; P 1 g and p 2 ¼ fE 2 ; 0 ⊥ ; P 2 g. Just like in the z 3 case, the asymmetric choice ðΔ ⊥ ; 0 ⊥ Þ on the lattice increases the number of possible discrete values for Δ ⊥ compared to the ðΔ ⊥ =2; −Δ ⊥ =2Þ choice. Then ð4:30Þ Since we will have a discrete set of possible Δ 2 ⊥ values on the lattice, it is impossible to arrange exactly the same value of t for different values of ξ. A more modest goal is to collect a set of data with close values of t, and then make interpolation to a chosen t-value.
Another strategy is to choose first some particular values of P 1 and P 2 . This fixes the value of ξ. The next step is to take several different values of Δ ⊥ to change t. That will give the t-dependence for fixed ξ and ν. After this, changing z 3 , we will change ν leaving ξ and t unchanged. Finally, using the matching conditions to convert the ν-dependence into the x-dependence, we will end up with Hðx; ξ; t; μ 2 Þ for a fixed ξ as a function of x and t.

V. SUMMARY
In this paper, we have derived the matching relations for the pion distribution amplitude and nonsinglet generalized parton distributions that connect them with their off-thelight-cone counterparts, the pseudo-DA and pseudo-GPDs. The latter may be calculated in lattice simulations, and the matching relations are crucial in converting them into the experimentally measurable (in principle) light-cone parton distributions.
We have also derived matching relations for the usual parton distribution functions. One of them, given by Eq. (2.22), allows to express the lattice-measurable reduced pseudo-ITD Mðν; z 2 3 Þ with its PDF fðx; μ 2 Þ. Similar relations may be derived for the lattice matrix elements renormalized using the RI/MOM schemes. Then one may be able to directly fit these matrix elements by a chosen model for the PDF.
The main feature of our derivations is that we start with a universal expression for the one-loop correction in an operator form. In particular, we show how this universal expression produces particular matching conditions for ITDs related to different parton distributions. In fact, these different matching relations have a rather similar structure. Also, these relations are much simpler than the matching relations for quasi-PDFs, quasi-DAs and quasi-GPDs given in Refs. [2,[11][12][13][14][15].
The matching relations for the pseudo-PDFs have been already used in lattice calculations [10,21,[47][48][49], while these for the pion DA and GPDs will be used in the ongoing lattice calculations.