Quasi-PDFs, momentum distributions and pseudo-PDFs

TL;DR

This paper explores the relationship between quasi-PDFs, pseudo-PDFs, and Ioffe-time distributions, proposing pseudo-PDFs as a more effective alternative for extracting parton distributions from lattice QCD data.

Contribution

It introduces pseudo-PDFs as a generalization of light-front PDFs that are related to Ioffe-time distributions, offering a practical approach to mitigate lattice renormalization issues.

Findings

Pseudo-PDFs exhibit a convolution structure similar to quasi-PDFs.

Pseudo-PDFs allow division of $z_3^2$ dependence, simplifying analysis.

Ioffe-time distributions retain shape information of PDFs.

Abstract

We show that quasi-PDFs may be treated as hybrids of PDFs and primordial rest-frame momentum distributions of partons. This results in a complicated convolution nature of quasi-PDFs that necessitates using large $p_3 \sim 3$ GeV momenta to get reasonably close to the PDF limit. As an alternative approach, we propose to use pseudo-PDFs $P(x, z_3^2)$ that generalize the light-front PDFs onto spacelike intervals and are related to Ioffe-time distributions $M (\nu, z_3^2)$, the functions of the Ioffe time $\nu = p_3 z_3$ and the distance parameter $z_3^2$ with respect to which it displays perturbative evolution for small $z_3$. In this form, one may divide out the $z_3^2$ dependence coming from the primordial rest-frame distribution and from the problematic factor due to lattice renormalization of the gauge link. The $\nu$-dependence remains intact and determines the shape of PDFs.

Equations

$$⟨p|\varphi (0)\varphi (z)|p⟩=$$

$$\mathcal{M}(-(pz),-z^2)$$

$$\mathcal{M}(-p_{+}{z}_{-},0)={\int }_{-1}^{1}dxf(x)e^{-ix{p}_{+}{z}_{-}},$$

$$f(x)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }d\nu e^{-ix\nu }\mathcal{M}(\nu ,0)=\mathcal{P}(x,0).$$

$$\mathcal{P}(x,z_1^2+z_2^2)$$

$$\times {\int }_{-\infty }^{\infty }d{k}_2{e}^{i{k}_2{z}_2}\mathcal{F}(x,k_1^2+k_2^2).$$

$$\mathcal{M}(\nu ,z_1^2+z_2^2)$$

$$\times {\int }_{-\infty }^{\infty }d{k}_2{e}^{i{k}_2{z}_2}\mathcal{F}(x,k_1^2+k_2^2).$$

$$\mathcal{P}(x,z_{\perp }^2)$$

$$⟨p|\varphi (0)\varphi (z_3)|p⟩=$$

$$\mathcal{M}(\nu ,z_3^2)=$$

$$Q(y,P)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }d\nu e^{-iy\nu }\mathcal{M}(\nu ,{\nu }^2/P^2).$$

$$Q(y,P)/P=$$

$$\displaystyle{\cal M}^{\alpha}(z,p)\equiv\langle p|\bar{\psi}(0)\,\gamma^{\alpha}\,{\hat{E}}(0,z;A)\psi(z)|p\rangle\$$

$${\mathcal{M}}^{\alpha }(z,p)=$$

$$+z^{\alpha }{\mathcal{M}}_z(-(zp),-z^2).$$

$${\mathcal{M}}^0(z_3,p)=2p^0{\int }_{-1}^{1}dyQ(y,P)e^{iyP{z}_3}.$$

$$R(k_3,P)=$$

$$\mathcal{R}(x,k_3)\equiv$$

$$R(k_3,P=0)\equiv$$

$$\rho (z_3^2)\equiv$$

$$r_G(k_3)=$$

$$\displaystyle{\mathfrak{M}}(\nu,z_{3}^{2})\equiv\frac{{\cal M}(\nu,z_{3}^{2})}{{\cal M}(0,z_{3}^{2})}\$$

$$\frac{d}{d\ln z_3^2}\mathfrak{M}(\nu ,z_3^2)=-\frac{{\alpha }_s}{2\pi }{C}_F{\int }_0^{1}duB(u)\mathfrak{M}(u\nu ,z_3^2)$$

$$B(u)$$

Full text

I Introduction

The parton distribution functions (PDFs) $f(x)$ Feynman:1973xc are related to matrix elements of bilocal operators on the light cone $z^{2}=0$, which prevents a straightforward calculation of these functions in the lattice gauge theory formulated in Euclidean space. The usual way out is to calculate their moments. However, recently, X. Ji Ji:2013dva suggested a method allowing to calculate PDFs as functions of $x$. To this end, he proposes to use purely space-like separations $z=(0,0,0,z_{3})$. Then one deals with quasi-PDFs $Q(y,p_{3})$ describing sharing of the $p_{3}$ hadron momentum component, and tending to PDFs $f(y)$ in the $p_{3}\to\infty$ limit. The same method can be applied to distribution amplitudes (DAs). The results of lattice calculations of quasi-PDFs were reported in Refs. Lin:2014zya ; Chen:2016utp ; Alexandrou:2015rja and of the pion quasi-DA in Ref. Zhang:2017bzy .

In our recent papers Radyushkin:2016hsy ; Radyushkin:2017gjd , we have studied nonperturbative $p_{3}$-evolution of quasi-PDFs and quasi-DAs using the formalism of virtuality distribution functions Radyushkin:2014vla ; Radyushkin:2015gpa . We found that quasi-PDFs can be obtained from the transverse momentum dependent distributions (TMDs) ${\cal F}(x,k_{\perp}^{2})$. We built models for the nonperturbative evolution of quasi-PDFs using simple models for TMDs. Our results are in qualitative agreement with the $p_{3}$-evolution patterns obtained in lattice calculations.

In the present paper, our first goal is to develop a picture for quasi-PDFs as hybrids of PDFs and primordial momentum distributions of partons in a hadron at rest. As an intermediate step, we demonstrate that the connection between TMDs and quasi-PDFs Radyushkin:2016hsy is a mere consequence of Lorentz invariance. Then we show that, when a hadron is moving, the parton $k_{3}$ momentum comes from two sources. The motion of the hadron as a whole gives the $xp_{3}$ part, governed by the dependence of the TMD ${\cal F}(x,\kappa^{2})$ on its $x$ argument. The remaining part $k_{3}-xp_{3}$ is governed by the dependence of the TMD on its second argument, $\kappa^{2}$, dictating the primordial rest-frame momentum distribution. The convolution nature of quasi-PDFs results in a rather complicated pattern of their $p_{3}$-evolution, necessitating rather large values $p_{3}\sim 3$ GeV for getting close to the PDF limit.

Thus, our second goal is to propose an alternative approach for lattice PDF extraction. To this end, we introduce pseudo-PDFs ${\cal P}(x,z_{3}^{2})$ that generalize the light-cone PDFs $f(x)$ onto spacelike intervals like $z=(0,0,0,z_{3})$. The pseudo-PDFs are Fourier transforms of the Ioffe-time Ioffe:1969kf distributions Braun:1994jq ${\cal M}(\nu,z_{3}^{2})$ that are basically given by generic matrix elements like $\langle p|\phi(0)\phi(z)|p\rangle$ written as functions of $\nu=p_{3}z_{3}$ and $z_{3}^{2}$. Unlike quasi-PDFs, the pseudo-PDFs have the “canonical” $-1\leq x\leq 1$ support for all $z_{3}^{2}$. They tend to PDFs when $z_{3}\to 0$, showing in this limit a usual perturbative evolution with $1/z_{3}$ serving as an evolution parameter. Finally, we discuss how these properties of pseudo-PDFs may be used for extraction of PDFs on the lattice.

II Parton distributions

II.1 Generic matrix element and

Lorentz invariance

Historically Feynman:1973xc , PDFs were introduced to describe spin-1/2 quarks. Since complications related to spin do not affect the very concept of parton distributions, we start with a simple example of a scalar theory. In that case, information about the target is accumulated in the generic matrix element $\langle p|\phi(0)\phi(z)|p\rangle$. By Lorentz invariance, it is a function of two scalars, $(pz)\equiv-\nu$ and $z^{2}$ (or $-z^{2}$ if we want a positive value for spacelike $z$):

[TABLE]

It can be shown Radyushkin:2016hsy ; Radyushkin:1983wh that, for all contributing Feynman diagrams, its Fourier transform ${\cal P}(x,-z^{2})$ with respect to $(pz)$ has the $-1\leq x\leq 1$ support, i.e.,

[TABLE]

Note that Eq. (2) gives a covariant definition of $x$. There is no need to assume that $p^{2}=0$ or $z^{2}=0$ to define $x$.

II.2 Collinear PDFs

Choosing some special cases of $p$ and $z$, one can get expressions for various parton distributions, all in terms of the same function ${\cal M}(-(pz),-z^{2})$. In particular, taking a light-like $z$, e.g., that having the light-front minus component $z_{-}$ only, we parameterize the matrix element by the twist-2 parton distribution $f(x)$

[TABLE]

with $f(x)$ having the usual interpretation of probability that the parton carries the fraction $x$ of the target momentum component $p_{+}$. The inverse relation is given by

[TABLE]

Since $f(x)={\cal P}(x,0)$, the function ${\cal P}(x,-z^{2})$ generalizes PDFs onto non-lightlike intervals $z^{2}$, and we will call it pseudo-PDF. The variable $(pz)$ is called the Ioffe time Ioffe:1969kf , and ${\cal M}(\nu,-z^{2})$ is the Ioffe-time distribution Braun:1994jq .

Note that the definition of ${\cal P}(x,-z^{2})$ is simpler than that of $f(x)$ because it does not require taking a subtle $z^{2}\to 0$ limit. In renormalizable theories, the function ${\cal M}(\nu,z^{2})$ has $\sim\ln z^{2}$ singularities generating perturbative evolution of parton densities. Within the operator product expansion (OPE) approach, the $\ln z^{2}$ singularities are subtracted using some prescription, say, dimensional renormalization, and the resulting PDFs depend on the renormalization scale $\mu$, i.e., $f(x)\to f(x,\mu^{2})$.

II.3 Transverse momentum dependent

distributions

When $z^{2}$ is spacelike, one can treat $-z^{2}$ as the magnitude squared of a two-dimensional vector $\{z_{1},z_{2}\}$, and introduce a two-dimensional Fourier transform with respect to its components, i.e., to write

[TABLE]

Due to rotational invariance of ${\cal P}(x,z_{1}^{2}+z_{2}^{2})$ in $\{z_{1},z_{2}\}$ plane, the function ${\cal F}(x,k_{1}^{2}+k_{2}^{2})$ depends on ${k_{1},k_{2}}$ through $k_{1}^{2}+k_{2}^{2}$, the fact already reflected in the notation. Combining this representation with Eq. (2), one has

[TABLE]

A physical interpretation of ${\cal F}(x,k_{1}^{2}+k_{2}^{2})$ may be given in the frame where the target momentum $p$ is longitudinal, $p=(E,{\bf 0}_{\perp},P)$, while the vector $\{z_{1},z_{2}\}$ is in the transverse plane. Taking $z$ that has $z_{-}$ and $z_{\perp}$ components only, one can identify ${\cal F}(x,k_{\perp}^{2})$ with the *TMD * and write

[TABLE]

In this case, the pseudo-PDFs ${\cal P}(x,z_{\perp}^{2})$ coincide with the impact parameter distributions, a well-known concept actively used in TMD studies.

The $\sim\ln z_{\perp}^{2}$ terms in ${\cal M}(\nu,z_{\perp}^{2})$ are produced by the $\sim 1/k_{\perp}^{2}$ hard tail of ${\cal F}(x,k_{\perp}^{2})$. Thus, it makes sense to visualize ${\cal M}(\nu,z_{\perp}^{2})$ as a sum of a soft part ${\cal M}^{\rm soft}(\nu,z_{\perp}^{2})$, that has a finite $z_{\perp}^{2}\to 0$ limit and a hard part reflecting the evolution. For TMDs, soft part decreases faster than $1/k_{\perp}^{2}$, say, like a Gaussian $e^{-k_{\perp}^{2}/\Lambda^{2}}$. In the $z_{\perp}$ space, the distributions are then concentrated in $z_{\perp}\sim 1/\Lambda$ region.

III Quasi-Distributions

III.1 Definition and relation to TMDs

Since one cannot have light-like separations on the lattice, it was proposed Ji:2013dva to consider spacelike separations $z=(0,0,0,z_{3})$ [or, for brevity, $z=z_{3}$]. Then, in the $p=(E,0_{\perp},P)$ frame, one introduces the quasi-PDF $Q(y,P)$ through a parametrization

[TABLE]

According to this definition, the function $Q(y,p)$ characterizes the probability that the parton carries fraction $y$ of hadron’s third momentum component $P$. Viewing the matrix element as a function of the $\nu$ and $-z^{2}$ variables (they are given by $Pz_{3}$ and $z_{3}^{2}$ in this case), we have

[TABLE]

Noticing that $z_{3}^{2}=\nu^{2}/P^{2}$, we get the inverse Fourier transformation in the form

[TABLE]

It indicates that $Q(y,P)$ tends to $f(y)$ in the $P\to\infty$ limit, as far as ${\cal M}(\nu,\nu^{2}/P^{2})\to{\cal M}(\nu,0)$.

Thus, the deviation of the quasi-PDF $Q(y,P)$ from the PDF $f(y)$ is determined by the dependence of ${\cal M}(\nu,z_{3}^{2})$ with respect to its second argument. By Eq. (6), this dependence is related to the dependence of the TMD ${\cal F}(x,\kappa^{2})$ on its second argument $\kappa^{2}$. Hence, the difference between $Q(y,P)$ and $f(y)$ may be described in terms of TMDs.

To this end, we incorporate the fact that Eq. (6) is a mathematical relation between the function ${\cal M}(\nu,z_{1}^{2}+z_{2}^{2})$ and the function ${\cal F}(x,k_{1}^{2}+k_{2}^{2})$, no matter what is a physical meaning of the variables $z_{1},z_{2}$ and $k_{1},k_{2}$. Thus, we substitute Eq. (6) with $z_{1}=0$ and $z_{2}=\nu/P$ into Eq. (10) to convert it into the expression for quasi-PDFs in terms of TMDs

[TABLE]

Originally, this relation was derived in Ref. Radyushkin:2016hsy using a Nakanishi-type representation of Refs. Radyushkin:2014vla ; Radyushkin:2015gpa . Now, we see that it is a mere consequence of Lorentz invariance.

III.2 Quantum chromodynamics (QCD) case

The formulas derived above are directly applicable for non-singlet parton densities in QCD. In that case, one deals with matrix elements of

[TABLE]

type, where ${\hat{E}}(0,z;A)$ is the standard $0\to z$ straight-line gauge link in the quark (fundamental) representation. These matrix elements may be decomposed into $p^{\alpha}$ and $z^{\alpha}$ parts:

[TABLE]

The ${\cal M}_{p}(-(zp),-z^{2})$ part gives the twist-2 distribution when $z^{2}\to 0$, while ${\cal M}_{z}((zp),-z^{2})$ is a purely higher-twist contamination, and it is better to get rid of it.

If one takes $z=(z_{-},z_{\perp})$ in the $\alpha=+$ component of ${\cal M}^{\alpha}$, the $z^{\alpha}$-part drops out, and one can introduce a TMD ${\cal F}(x,k_{\perp}^{2})$ that is related to ${\cal M}_{p}(\nu,z_{\perp}^{2})$ by the scalar formula (6). For quasi-distributions, the easiest way to remove the $z^{\alpha}$ contamination is to take the time component of ${\cal M}^{\alpha}(z=z_{3},p)$ and define

[TABLE]

Then the connection between $Q(y,P)$ and ${\cal F}(x,k_{\perp}^{2})$ is given by the scalar formula (11).

One may notice that the operator defining ${\cal M}^{\alpha}(z,p)$ involves a straight-line link from [math] to $z$ rather than a stapled link usually used in the definitions of TMDs appearing in the description of Drell-Yan and semi-inclusive DIS processes. As is well-known, the stapled links reflect initial or final state interactions inherent in these processes. The “straight-link” TMDs, in this sense, describe the structure of a hadron when it is in its non-disturbed or “primordial” state. While it is unlikely that such a TMD can be measured in a scattering experiment, it is a well-defined QFT object, and one may hope that it can be measured on the lattice.

III.3 Momentum distributions

The quasi-PDFs describe the distribution in the fraction $y\equiv k_{3}/P$ of the third component $k_{3}$ of the parton momentum to that of the hadron. One can introduce distributions in $k_{3}$ itself: $R(k_{3},P)\equiv Q(k_{3}/P,P)/P$. Then we can rewrite Eq. (11) as

[TABLE]

or, switching to the linear argument $k_{3}-xP$,

[TABLE]

where

[TABLE]

is the TMD ${\cal F}(x,\kappa^{2})$ integrated over the $k_{1}$ component of the two-dimensional vector $\kappa=\{k_{1},k_{3}\}$. According to (17), ${\cal R}(x,k_{3})$ depends on $k_{3}$ through $k_{3}^{2}$.

For a hadron at rest, we have

[TABLE]

This one-dimensional distribution may be directly obtained through a parameterization of the density

[TABLE]

given by $\langle p|\phi(0)\phi(z_{3})|p\rangle|_{{\bf p}={\bf 0}}$. Thus, $r(k_{3})$ describes a primordial distribution of $k_{3}$ (or any other component of ${\bf k}$) in a rest-frame hadron.

The formula (16) has a straightforward interpretation. According to it, when the hadron is moving, the parton’s $k_{3}$ momentum has two sources.

The first part, $xP$ comes from the motion of the hadron as a whole, and the probability to get $xP$ is governed by the dependence of the TMD ${\cal F}(x,\kappa^{2})$ on its first argument, $x$.

On the other hand, the probability to get the remaining part $k_{3}-xP$ is governed by the dependence of the TMD on its second argument, $\kappa^{2}$, describing the primordial rest-frame momentum distribution.

The parameter $x$ appears in both arguments of ${\cal R}(x,k_{3}-xP)$ in Eq. (16), i.e., $R(k,P)$ is given by a convolution. In this sense, the momentum distributions $R(k,P)$ and, hence, the quasi-PDFs have a hybrid structure influenced by the shape both of PDFs and rest-frame distributions.

III.4 Factorized models.

Since the two sources of $k_{3}$ look like independent, it is natural to demonstrate the hybrid nature of momentum distributions and quasi-PDFs using a factorized model ${\cal R}(x,k_{3}-xP)=f(x)r(k_{3}-xP)$ (the $x$ integral of $f(x)$ is normalized to 1). For original ${\cal M}(\nu,-z^{2})$ function, this Ansatz corresponds to the factorization assumption ${\cal M}(\nu,-z^{2})={\cal M}(\nu,0){\cal M}(0,-z^{2})$.

For illustration, we take a Gaussian form $\rho_{G}(z_{3}^{2})=e^{-z_{3}^{2}\Lambda^{2}/4}$ for the rest-frame density. It corresponds to

[TABLE]

For $f(x)$, we take a simple PDF resembling nucleon valence densities $f(x)=4(1-x)^{3}\theta(0\leq x\leq 1)$. As one can see from Fig. 1, the curve for $R(k,P)$ changes from a Gaussian shape for small $P$ to a shape resembling stretched PDF for large $P$.

This result is in perfect compliance with a known fact that wave functions of moving hadrons are not given by a mere kinematical “boost” of the rest-frame wave functions. Indeed, with increasing $P$, the impact of the rest-frame distribution $r(k)$ is less and less visible, and eventually the shape of $R(k,P)$ is determined by a completely different function $f(k/P)$.

Rescaling to the $y=k/P$ variable gives the quasi-PDF $Q(y,P)$ shown in Fig. 2. For large $P$, it clearly tends to the $f(y)$ PDF form. In particular, using a momentum $P\sim 10\Lambda$ one gets a quasi-PDF that is rather close to the $P\to\infty$ limiting shape. Still, since $\Lambda\sim\langle k_{\perp}\rangle$, assuming the folklore value $\langle k_{\perp}\rangle\sim$ 300 MeV one translates the $P\sim 10\Lambda$ estimate into $P\sim 3$ GeV, which is uncomfortably large. Thus, a natural question is how to improve the convergence.

III.5 Pseudo-PDFs

A formal reason for the complicated structure of a quasi-PDF $Q(y,P)$ is the fact that it is obtained by the $\nu$-integral of ${\cal M}(\nu,z_{3}^{2})e^{i\nu y}$ along a non-horizontal line $z_{3}=\nu/P$ in the $(\nu,z_{3})$ plane (see Eq. (10)). With increasing $P$, its slope decreases, the line becomes more horizontal, and quasi-PDFs convert into PDFs.

In contrast, pseudo-PDFs ${\cal P}(x,z_{3}^{2})$, by definition, are given by integration of ${\cal M}(\nu,z_{3}^{2})e^{i\nu x}$ over horizontal lines $z_{3}=$ const. A very attractive feature of the pseudo-PDFs is that they have the $-1\leq x\leq 1$ support for all $z_{3}$ values. For small $z_{3}$, they convert into PDFs.

More precisely, when $z_{3}$ is small, $1/z_{3}$ is analogous to the renormalization parameter $\mu$ of scale-dependent PDFs $f(x,\mu^{2})$ of the standard OPE approach.

There is a subtlety, however, that while the $\mu^{2}$-dependence of PDFs $f(x,\mu^{2})$ comes solely from the evolution logarithms $\ln(\mu^{2}/m^{2})$, the $z_{3}^{2}$-dependence of quasi-PDFs in QCD comes both from the evolution logarithms $\ln(z_{3}^{2}m^{2})$ and from the ultraviolet logarithms $\ln(z_{3}^{2}\mu_{R}^{2})$, where $\mu_{R}$ is a cut-off parameter for divergences related to the gauge link renormalization (see Ref. Craigie:1980qs ). At the leading logarithm level, these divergences do not depend on $\nu$. As a result, the “reduced” Ioffe-time distribution

[TABLE]

satisfies, for small $z_{3}$, the leading-order evolution equation

[TABLE]

with respect to $1/z_{3}$ that is similar to the evolution equation for $f(x,\mu^{2})$ with respect to $\mu$. The leading-order evolution kernel $B(u)$ for the non-singlet quark case is given Braun:1994jq by

[TABLE]

with $[\ldots]_{+}$ denoting the standard “plus” prescription.

For the model used above (and $x\to-x$ symmetrized, as required for non-singlet PDFs), we have ${\cal M}(\nu,0)={12}\left[\nu^{2}-4\sin^{2}(\nu/2)\right]/{\nu^{4}}\ .$ The shape of this function and of the convolution integral $B\otimes{\cal M}(\nu)$ are shown in Fig. 3. As one can see, $B\otimes{\cal M}(\nu)$ vanishes for $\nu=0$, which reflects conservation of the vector current. Thus, the rest-frame density ${\mathfrak{M}}(0,z_{3}^{2})$ is not affected by perturbative evolution.

III.6 Lattice implementation

A possible way to find the Ioffe-time distributions on the lattice (suggested by K. Orginos) is to calculate ${\cal M}(Pz_{3},z_{3}^{2})$ for several values of $P$, and then to fit the results by a function of $\nu$ and $z_{3}^{2}$.

Recalling our discussion of two apparently independent sources of obtaining $k_{3}$ for a moving hadron, one may hope that ${\cal M}(\nu,z_{3}^{2})$ factorizes, i.e., ${\cal M}(\nu,z_{3}^{2})={\cal M}(\nu,0){\cal M}(0,z_{3}^{2})$. Then the reduced function ${\mathfrak{M}}(\nu,z_{3}^{2})$ defined by Eq. (21) is equal to ${\cal M}(\nu,0)$, and the goal of obtaining ${\cal M}(\nu,0)$ is reached. Formally, what remains is just to take its Fourier transform to get the PDF $f(x)$.

In fact, such a factorization has been already observed several years ago in the pioneering study Musch:2010ka of the transverse momentum distributions in lattice QCD.

A serious disadvantage of quasi-PDFs is that they have the $x$-convolution structure (11) even in a favorable situation when the TMD [and ${\cal M}(\nu,z_{3}^{2})$] factorizes. On the other hand, using pseudo-PDFs in the form of the ratio ${\mathfrak{M}}(\nu,z_{3}^{2})$, one divides out the $z_{3}^{2}$-dependence of the primordial distribution without affecting the $\nu$-dependence that dictates the shape of PDF.

A further advantage of using the ratio (pointed out by K. Orginos) is the cancellation of the $z_{3}$-dependence generated by the lattice renormalization of the gauge link ${\hat{E}}(0,z_{3};A)$. Such a renormalization is required by linear $|z_{3}|\delta m$ (where $\delta m\sim 1/a$, and $a$ is the ultraviolet cut-off) and logarithmic $\ln(z_{3}^{2}/a^{2})$ divergences Polyakov:1980ca ; Dotsenko:1979wb . Due to their local nature, they are expected to combine into a $\nu$-independent factor $Z(z_{3}^{2}/a^{2})$ that is the same in the numerator and denominator of the ratio ${\mathfrak{M}}(\nu,z_{3}^{2})$.

The multiplicative renormalizability of the linear divergences of ${\cal M}(\nu,z_{3}^{2})$ to all orders was recently argued in Refs. Ishikawa:2016znu ; Chen:2016fxx . A general proof for both linear and logarithmic divergences was claimed in Ref. Ishikawa:2017faj on the basis of a direct analysis of relevant Feynman graphs.

Another approach Ji:2017oey ; Green:2017xeu is to treat $\hat{E}(0,z;A)$ as $h(0)\bar{h}(z)$, where the auxiliary field $h(z)$ is analogous to the infinitely heavy quark field of the heavy quark effective theory (HQET). Since HQET is known to be multiplicatively renormalizable Bagan:1993zv this means that $\bar{\psi}(0)\hat{E}(0,z;A)\psi(z)$ is also multiplicatively renormalizable to all orders in perturbation theory.

In reality, ${\mathfrak{M}}(\nu,z_{3}^{2})$ will have a residual $z_{3}^{2}$-dependence. It comes both from a possible violation of factorization for the soft part (according to results of Ref. Musch:2010ka , it is expected to be rather mild) and from mandatory perturbative evolution. For a nonzero $\nu$, the latter should be visible as a $\ln(1/z_{3}^{2}\Lambda^{2})$ spike for small $z_{3}^{2}$.

Hence, a proposed strategy is to extrapolate ${\mathfrak{M}}(\nu,z_{3}^{2})$ to $z_{3}^{2}=0$ from not too small values of $z_{3}^{2}$, say, from those above 0.5 fm2. The resulting function ${\cal M}^{\rm soft}(\nu,0)$ may be treated as the Ioffe-time distribution producing the PDF $f_{0}(x)$ “at low normalization point”. The remaining $\ln(1/z_{3}^{2}\Lambda^{2})$ spikes at small $z_{3}$ will generate its evolution.

To convert ${\mathfrak{M}}(\nu,z_{3}^{2})$ into a function of $x$, one should, in principle, know ${\mathfrak{M}}(\nu,z_{3}^{2})$ for all $\nu$, which is impossible. The maximal values of $\nu$ reached in existing lattice calculations range from $3\pi$ Lin:2014zya to $5\pi$ Alexandrou:2015rja and $6\pi$ Orginos:2017kos . Taking a Fourier transform in these limited ranges produces unphysical oscillations in $x$. Thus, the idea is to avoid the Fourier transform in $\nu$, and just compare the reduced Ioffe-time distributions obtained from the lattice with those derived from experimentally known parton distributions.

Of course, an actual technical implementation of this program should be discussed when the lattice data on ${\mathfrak{M}}(\nu,z_{3}^{2})$ will become available.

IV Summary

In this paper, we showed that quasi-PDFs may be seen as hybrids of PDFs and the primordial rest-frame momentum distributions of partons. In this context, the parton’s $k_{3}$ momentum comes from the motion of the hadron as a whole and from the primordial rest-frame momentum distribution. The complicated convolution nature of quasi-PDFs necessitates using $p_{3}\gtrsim$ 3 GeV to wipe out the primordial momentum distribution effects and get reasonably close to the PDF limit.

As an alternative approach, we propose to use pseudo-PDFs ${\cal P}(x,z_{3}^{2})$ that generalize the light-front PDFs onto spacelike intervals. By a Fourier transform, they are related to the Ioffe-time distributions ${\cal M}(\nu,z_{3}^{2})$ given by generic matrix elements written as functions of $\nu=p_{3}z_{3}$ and $z_{3}^{2}$. The advantageous features of pseudo-PDFs are that they, first, have the same $-1\leq x\leq 1$ support as PDFs, and second, their $z_{3}^{2}$-dependence for small $z_{3}^{2}$ is governed by a usual evolution equation.

Forming the ratio ${\cal M}(\nu,z_{3}^{2})/{\cal M}(0,z_{3}^{2})$ of Ioffe-time distributions one divides out the bulk of $z_{3}^{2}$ dependence generated by the primordial rest-frame distribution. Furthermore, taking this ratio one can exclude the $z_{3}^{2}$-dependent factor coming from the lattice renormalization of the $\hat{E}(0,z_{3};A)$ link creating difficulties (see, e.g., Ishikawa:2016znu ) for lattice calculations of quasi-PDFs.

Testing the efficiency of using pseudo-PDFs for lattice extractions of PDFs is a challenge for future studies.

In fact, while this paper was in the review process, an actual lattice calculation Orginos:2017kos based on the ideas of the present paper was performed. It has clearly demonstrated the presence of a linear component in the $z_{3}$-dependence of the rest-frame function ${\cal M}(0,z_{3}^{2})$, that may be attributed to the $Z(z_{3}^{2}/a^{2})\sim e^{-c|z_{3}|/a}$ behavior generated by the gauge link. It was also observed that the ratio ${\cal M}(Pz_{3},z_{3}^{2})/{\cal M}(0,z_{3}^{2})$ has a Gaussian-type behavior with respect to $z_{3}$, which indicates that the $Z(z_{3}^{2}/a^{2})$ factors entering into the numerator and denominator of the ${\mathfrak{M}}(Pz_{3},z_{3}^{2})$ ratio have been canceled, as we expected.

Furthermore, it was found that when plotted as a function of $\nu$ and $z_{3}$, the data for the reduced distribution ${\mathfrak{M}}(\nu,z_{3}^{2})$ have a very mild dependence on $z_{3}^{2}$. This observation indicates that the soft part of the $z_{3}^{2}$-dependence of ${\cal M}(\nu,z_{3}^{2})$ has been canceled by the rest-frame density ${\cal M}(0,z_{3}^{2})$. This phenomenon corresponds to factorization of the $x$- and $k_{\perp}$-dependence for the soft part of the TMD ${\cal F}(x,k_{\perp}^{2})$.

It was also demonstrated that the residual $z_{3}$-dependence for small $z_{3}\leq 4\,a$, may be explained by perturbative evolution, with the $\alpha_{s}$ value corresponding to $\alpha_{s}/\pi=0.1$.

Acknowledgements.

I thank C.E. Carlson for his interest in this work, and also V. M. Braun and X. Ji for discussions and suggestions. I am especially grateful to K. Orginos for stimulating discussions and suggestions concerning the lattice implementation of the approach. This work is supported by Jefferson Science Associates, LLC under U.S. DOE Contract #DE-AC05-06OR23177 and by U.S. DOE Grant #DE-FG02-97ER41028.