Structure of parton quasi-distributions and their moments

We discuss the structure of the parton quasi-distributions (quasi-PDFs) $Q(y, P_3)$ outside the"canonical"$-1 \leq y \leq 1$ support region of the usual parton distribution functions (PDFs). Writing the $y^n$ moments of $Q(y, P_3)$ in terms of the combined $x^{n-2l} k_\perp^{2l}$-moments of the transverse momentum distribution (TMD) ${\cal F} (x,k_\perp^2)$, we establish a connection between the large-$|y|$ behavior of $Q(y,P_3)$ and large-$k_\perp^2$ behavior of ${\cal F} (x,k_\perp^2)$. In particular, we show that the $1/k_\perp^2$ hard tail of TMDs in QCD results in a slowly decreasing $\sim 1/|y|$ behavior of quasi-PDFs for large $|y|$ that produces infinite $y^n$ moments of $Q(y,P_3)$. We also relate the $\sim 1/|y|$ terms with the $\ln z_3^2$-singulariies of the Ioffe-time pseudo-distributions $\mathfrak{M} (\nu, z_3^2)$. Converting the operator product expansion for $\mathfrak{M} (\nu, z_3^2)$ into a matching relation between the quasi-PDF $Q(y,P_3)$ and the light-cone PDF $f(x, \mu^2)$, we demonstrate that there is no contradiction between the infinite values of the $y^n$ moments of $Q(y,P_3)$ and finite values of the $x^n$ moments of $f(x, \mu^2)$.


Introduction
In the original Feynman approach [1], the parton distribution functions (PDFs) f (x) were introduced as the infinite momentum P 3 → ∞ limit of distributions in the longitudinal k 3 = xP 3 momentum of partons. These functions basically coincide with the quasi-PDFs Q(x, P 3 ) introduced more recently by X. Ji [2].
In the parton model, f (x) were treated as k ⊥ -integrals of more detailed f (x, k ⊥ ) distributions that involve also the transverse momentum k ⊥ . From the start, it was understood by Feynman that the Q(x, P 3 → ∞) → f (x) limit exists only if f (x, k ⊥ ) rapidly decreases with k ⊥ , so that the integral over k ⊥ does not diverge. This happens, in particular, in the theories/models with transverse momentum cut-off k ⊥ Λ, e.g., in superrenormalizable models, but not in QED and other renormalizable field theories.
One may ask two natural questions. First, why Q(x, P 3 ) for a finite P 3 differs from f (x)? Second, how does Q(x, P 3 ) convert into f (x) when P 3 → ∞? A qualitative answer is that the parton's longitudinal momentum k 3 = yP 3 comes from two sources: from the motion of the hadron as a whole (xP 3 ) and from a Fermi motion of quarks inside the hadron, so that (y− x)P 3 ∼ 1/R hadr . As P 3 → ∞, the role of the y− x ∼ 1/PR hadr fraction decreases and Q(x, P 3 ) → f (x).
In this picture, the (y − x)P 3 part has the same physical origin as the parton's transverse momentum. Hence, one should be able to relate quasi-PDFs to the transverse momentum distributions (TMDs) and quantify the difference between Q(x, P 3 ) and f (x) in terms of TMDs f (x, k ⊥ ).
An important point is that the components of k ⊥ may take any values from −∞ to ∞, even when the distribution in k ⊥ is mostly restricted to a limited range, like in a Gaussian e −k 2 ⊥ /Λ 2 . Similarly, the (y − x)P 3 part of the k 3 -distribution may take any values. As a result, Q(y, P 3 ) formally has the −∞ < y < ∞ support region, though possibly with a rapid decrease (say, like e −y 2 P 2 3 /Λ 2 ) for large y. In other words, for a finite P 3 , there is no requirement that the fraction y is smaller than 1 or positive. Even in a fast-moving hadron, there is some probability that a parton moves in the opposite direction, and hence, that some other parton has the momentum k 3 larger than P 3 . Still, with increasing P 3 , the chances for fractions outside the [0, 1] segment decrease rapidly, reflecting the large-k ⊥ dependence of the relevant TMD f (x, k ⊥ ).
When Q(y, P 3 ) ∼ e −y 2 P 2 3 /Λ 2 , one may consider y n moments of quasi-PDFs Q(y, P 3 ) calculated over the whole −∞ < y < ∞ axis and study their relation to the x n moments of the light-cone PDFs f (x).
Still, starting with the first papers [2,3] on quasi-PDFs, it was known that the simplest perturbative calculations produce ∼ 1/|y| behavior for quasi-PDFs at large |y|. Such a behavior reflects a slow ∼ 1/k 2 ⊥ decrease of the perturbative hard tail of TMDs in renormalizable theories. Clearly, if Q(y, P 3 ) ∼ 1/|y|, then even the zeroth moment of Q(y, P 3 ) diverges, so that it apparently makes no sense to consider y n moments of Q(y, P 3 ). Since the standard procedures of extracting PDFs from the lattice [4,5,6] do not involve a calculation of the moments, the divergence of these moments did not attract much attention.
However, recently it was argued by G.C. Rossi and M. Testa [7,8] that the divergence of the y n moments of Q(y, P 3 ) poses a serious problem for extraction of PDFs from lattice QCD simulations. The basic claim is that the infinite values of y n Q quasi-PDF moments are in conflict with the finite values of the x n f moments of the usual PDFs. Irrespectively of these claims, we find that the structure of quasi-PDFs Q(y, P 3 ) outside the central |y| ≤ 1 region is an interesting problem on its own, and we analyze it in the present paper. Our study is based on the concept [9] of the Ioffe-time pseudo-distributions (pseudo-ITDs) M(ν, −z 2 ). They are basically the matrix elements M(z, p) of bilocal operators ∼ φ(0)φ(z) treated as functions of the Lorentz invariants, the Ioffe time ν = −(zp) [10,11] and the invariant interval z 2 . [Our convention is to add "pseudo" to the name of distributions defined for nonzero z 2 , and skip it for their light-cone analogs.] While M(ν, −z 2 ) does not involve momentum fraction variables like y and x, quasi-PDFs Q(y, P) and pseudo-PDFs P(x, z 2 3 ) may be obtained [9] from M(ν, −z 2 ) as Fourier transforms. The advantage of this approach is a direct use of the coordinate representation that greatly simplifies further considerations of pseudo-PDFs, TMDs and quasi-PDFs.
Furthermore, as we will show, the fact that the quasi-PDFs Q(y, P) do not vanish outside the |y| ≤ 1 region, is directly connected with the presence of a non-trivial z 2 3 -dependence in the relevant pseudo-PDFs P(x, z 2 3 ). The paper is organized as follows. In Section 2, we start with reminding the definition of the pseudo-ITDs and their relation to pseudo-PDFs, quasi-PDFs and TMDs. We write a formal 1/P 2l series expansion for the y n Q moments of the quasi-PDFs in terms of the combined x n−2l k 2l ⊥ F moments of TMDs F (x, k 2 ⊥ ). In the case of "very soft" TMDs, i.e., those vanishing faster than any inverse power of k 2 ⊥ for large k ⊥ , this expansion allows to study y n Q moments (which are finite in this case) and their relation to x n f moments of the usual PDFs. In Section 3, we study the consequences of having a hard ∼ 1/k 2 ⊥ tail of TMDs, present in renormalizable theories, including QCD. In this case, the combined x n−2l k 2l ⊥ F moments diverge. For l = 0, one has a logarithmic divergence corresponding to the usual perturbative evolution. For l ≥ 1, one faces power divergences equivalent to those discussed in Refs. [7,8]. We show that they reflect the slowly ∼ 1/|y| decreasing perturbative contributions to Q(y, P 3 ). We also show that the |y| > 1 parts of Q(y, P 3 ) are generated by the z 2 3 -dependence of the pseudo-PDFs P(x, z 2 3 ). In Section 4, we study possible forms of the z 2 3 -dependence. In Section 5, we discuss the matching relations connecting the lightcone PDFs to pseudo-ITDs and quasi-PDFs. According to the operator product expansion (OPE), the reduced pseudo-ITD M(ν, z 2 3 ) is given by the MS-ITD I(ν, µ 2 ) plus O(α s ) perturbative contribution that contains the ∼ ln z 2 3 term responsible for the slowly varying ∼ 1/|y| terms in the |y| > 1 part of the quasi-PDF Q(y, P 3 ). The latter, hence, is given by the MS-PDF f (x, µ 2 ) plus O(α s ) perturbative contribution that contains the slowly varying ∼ 1/|y| terms in the |y| > 1 part.
Vice versa, f (x, µ 2 ), is given by the difference between the lattice quasi-PDF Q L (y, P 3 ) and that O(α s ) perturbatively calculable contribution. This means that the implementation of the matching condition includes a subtraction, though not of the kind discussed by Rossi and Testa in Refs. [7,8]. The final point is that, for large P 3 , the quasi-PDF Q L (y, P 3 ) must be purely perturbative in the |y| > 1 region. Hence, the above difference vanishes outside the |y| ≤ 1 segment, and the moments of the light-cone PDF f (x, µ 2 ) extracted in this way are finite. Section 6 contains summary and conclusions.

Ioffe-time distributions and pseudo-PDFs
Defining a parton distribution either in a continuum theory or on the lattice, one starts with a matrix element p|φ(0)φ(z)|p ≡ M(z, p) of a product of two parton fields. We use here simplified scalar notations, since the details of parton spin structure are not central to the concept of parton distributions, and may be added, if needed, at later stages.
By Lorentz invariance, M(z, p) is a function of two scalars, the Ioffe time [10,11] (pz) ≡ −ν and the interval z 2 M(z, p) =M(−(pz), −z 2 ) . (2.1) As shown in Refs. [12,13], for any contributing Feynman diagram, the Fourier transform of M(ν, −z 2 ) with respect to the Ioffe time ν has the −1 ≤ x ≤ 1 support, familiar from the studies of the usual parton densities, When z is on the light cone, z 2 = 0, we deal with the ordinary (or light-cone) parton distributions Thus, P(x, 0) = f (x), and the function P(x, −z 2 ) generalizes the concept of PDFs onto the case of non-lightlike intervals z. Following Ref. [9], we will refer to it as pseudo-PDF or parton pseudo-distribution function.

Quasi-PDFs
The simplest example of a spacelike interval is obtained when just one component is nonzero, z = {0, 0, 0, z 3 }. Choosing p = (E, 0 ⊥ , P), one can define the quasi-PDF [2] as the Fourier transform of M(z 3 , P) with respect to z 3 Combining Eqs. (2.2) and (2.4) gives a relation between the quasi-PDF Q(y, P) and the pseudo-PDF P(x, z 2 3 ) corresponding to the z = z 3 separation One can see that though the pseudo-PDFs have the −1 ≤ x ≤ 1 support, the quasi-PDFs Q(y, P) are defined for all real y. Another observation is that if the pseudo-PDF does not depend on z 2 3 , i.e., if P(x, z 2 3 ) = f (x), then the quasi-PDF Q(y, P) does not depend on P, and Q(y, P) = f (y).
Thus, it is the dependence of P(x, z 2 ) (or, equivalently, of M(ν, z 2 )) on z 2 that determines the deviation of quasi-PDFs from PDFs. In particular, it generates the parts of Q(y, P) outside the PDF support region |y| ≤ 1.
In QCD and other renormalizable theories, the presence of the z 2 3 -dependence is unavoidable, because M(ν, z 2 ) has ∼ ln z 2 contributions for small z 2 . Furthermore, these terms are singular in the z 2 → 0 limit which complicates the definition of the light-cone PDFs.

Transverse Momentum Dependent PDFs
A very convenient way to parametrize the z 2 -dependence for a space-like z is provided by a description in terms of the transverse momentum dependent PDFs. Using again p = (E, 0 ⊥ , P) and choosing z that has only z − and z ⊥ = {z 1 , z 2 } components, while z + = 0, we have ν = −p + z − , z 2 = −z 2 ⊥ , and the TMD is defined by Due to the rotational invariance, this TMD depends on k 2 ⊥ only. Integrating over the angle between k ⊥ and z ⊥ gives where J 0 is the Bessel function. Now recall that P(x, −z 2 ) is a function defined in a covariant way by Eq. (2.2). This implies that this TMD representation [14] may be written for a general spacelike z. One should just change z ⊥ → √ −z 2 and k ⊥ → k in Eq. (2.7). In particular, one may take z = {0, 0, 0, z 3 }, i.e., choose z in the purely longitudinal direction, and write is a function that coincides with the TMD, one does not need to specify a "transverse" plane and treat k as the magnitude of a 2-dimensional momentum in that plane.

Support mismatch
Using the TMD parametrization (2.8) in the quasi/pseudo-PDF relation (2.5), and expanding J 0 (kz 3 ) into the Taylor series, we get a formal 1/P 2l expansion for the quasi-PDF Q(y, P) To shorten formulas, we have switched here back k → k ⊥ in the notation for the integration variable of the TMD representation (2.8), and also wrote the resulting 2πk ⊥ dk ⊥ as d 2 k ⊥ . We can do this because the TMD F (x, k 2 ⊥ ) does not depend on angles. As a matter of caution, we repeat again that k or k ⊥ should be understood simply as scalar variables of the TMD parametrization. There is no need to specify in which plane k ⊥ is.
According to Eq. (2.5), the quasi-PDF Q(y, P) has the −∞ < y < ∞ support region. However, the quasi-PDF Q(y, P) in Eq. (2.9) is given by a sum of terms involving the TMD F (y, k 2 ⊥ ) that has the −1 ≤ y ≤ 1 support. The explanation of the apparent discrepancy is that the innocently-looking derivatives of F (y, k 2 ⊥ ) in the expansion (2.9) may generate an infinite tower of singular functions like δ(y), δ(y ± 1) and their derivatives. To this end, we recollect that, even when a function f (y) has a nontrivial support Ω (say, −1 ≤ y ≤ 1), one may formally represent it by a series over the functions δ (N) (y) with an apparent support at one point y = 0 only. Here, M N are the moments of f (y), Hence, the support mismatch may be explained by the fact that the delta-function and its derivatives are integration prescriptions (mathematical distributions) rather than ordinary functions. But this also means that while the difference between Q(y, P) and f (y) is formally given by a series in powers of 1/P 2 , its coefficients are not the ordinary functions of y.

Moments of very soft quasi-PDFs
In order to get relations involving usual functions, one may wish to integrate the equations in which these distributions enter, e.g., to take moments. Indeed, the derivatives disappear if we calculate the y n moments y n Q of the quasi-PDFs where x n−2l k 2l ⊥ F are the combined moments of TMDs In the case of very soft distributions which vanish faster than any power of 1/k 2 ⊥ for large k ⊥ , all the combined moments x n−2l k 2l ⊥ F are finite and Eq. (2.12) tells us that then y n Q differs from x n f by terms having the ( k 2 ⊥ F /P 2 ) l structure. Two lowest moments n = 0 and n = 1 do not involve l ≥ 1 terms. For the normalization integral, Eq. (2.12) gives (2.14) Thus, the area under Q(y, P) does not change with P and is equal to the area under f (x), the phenomenon corresponding to the quark number conservation. Similarly, the first y-moment is given by which corresponds to the momentum conservation. These two sum rules have been originally derived in our paper [12].

Perturbative evolution
In renormalizable theories (most importantly, in QCD, but also in models with Yukawa gluons), the TMD F (x, k 2 ⊥ ) must 3 have a hard part that has the 1/k 2 ⊥ behavior for large k ⊥ . For non-singlet densities in QCD, it is given at one loop by and B(u) is the Altarelli-Parisi (AP) evolution kernel [15] If one calculates the combined moments x n−2l k 2l ⊥ F for the hard term, they diverge, starting from the lowest l = 0 moment in k 2 ⊥ . In the l = 0 case, the divergence is logarithmic. Let us see that it just reflects the fact that the quasi-PDF Q(y, P) for large P in this case has the logarithmic perturbative evolution with respect to P 2 . To begin with, we write the hard part in the coordinate representation where m is some infrared regularization scale. Rewriting the quasi-PDF definition in terms of the pseudo-ITD as As a result, the hard part of the quasi-PDF Q(y, P) has the evolution ln P 2 part Comparing with Eq. (3.1), we conclude that, calculating the evolution part, one should cut-off the k ⊥ integral at |k ⊥ | ∼ P values, so that it is given by

Two lowest moments
As we have seen, for very soft distributions, the n = 0 and n = 1 moments of quasi-PDF Q(y, P) coincide with these moments of the PDF f (x). To proceed with the hard part, we use where γ n are related to anomalous dimensions of operators with n derivatives, Thus, for the zeroth moment of Q ev (y, P), the coefficient in front of ln P 2 is proportional to the anomalous dimension γ 0 of the vector current. Since γ 0 vanishes, the area under Q(y, P) does not change with P and is equal to the area under f (x), the phenomenon corresponding to the quark number conservation. Similarly, the first y-moment of the hard part of Q(y, P) has the ln P 2 part proportional to the anomalous dimension γ 1 = 4/3 that is nonzero. This reflects the fact that the quarkgluon interactions change the momentum carried by the quarks, and only the total momentum of quarks plus gluons is conserved in the evolution process.

Higher moments and large-|y| behavior
According to the general formula (2.12), the y 2 -moment is given by (see also Ref. [16]), where When F (x, k 2 ⊥ ) vanishes faster than 1/k 4 ⊥ for large k ⊥ , the k ⊥ -integral converges. Then the difference between y 2 Q and x 2 F decreases as k 2 ⊥ F /2P 2 for large P. 1 However, for a hard ∼ 1/k 2 ⊥ TMD, the k 2 ⊥ F integral diverges quadratically. If, by analogy with Eq. (3.8), we would set the upper limit of k ⊥ integration to be proportional to P, the k 2 ⊥ -weighted integral (3.12) would be proportional to P 2 .
Because of the compensation of the initial 1/P 2 suppression factor by the P 2 factor resulting from the quadratic divergence of the k ⊥ -integral, the contribution of the k 2 ⊥ F /2P 2 term does not disappear in the P → ∞ limit. One may also argue that, on the lattice, the upper limit on the k ⊥ integral may be set by the lattice spacing. Then, a cut-off for the k ⊥ integral at the ∼ 1/a value would result in a ∼ 1/a 2 P 2 contribution.
These worries have been formulated in recent papers by G.C. Rossi and M. Testa [7,8], who warned that one might need to perform a nonperturbative subtraction of such terms in lattice calculations. The questions raised in Ref. [7] have been subsequently addressed by X. Ji et al. in Ref. [18], who stated that the extraction of PDFs does not involve taking moments of quasi-PDFs. It was also argued that the moments of quasi-PDFs do not exist because Q(y, P) decreases as 1/|y| for large y. While we agree with these statements in general, we think that the problem deserves a more detailed investigation.

Sources of z 2 dependence
As we discussed already, the |y| > 1 parts of quasi-PDFs Q(y, P) are generated by the z 2 3 -dependence of the ITD M(ν, z 2 3 ). In particular, for large z 2 3 , M(ν, z 2 3 ) has a fast decrease with z 3 . This reflects a finite size of the system. Such a behavior should appear in any reasonable theory/model used to describe hadrons.
The second type of the z 2 3 -dependence appears in renormalizable theories. As already mentioned, then P(x, −z 2 ) and M(ν, −z 2 ) contain, for small −z 2 , the terms ∼ ln(−z 2 ) corresponding to the ∼ 1/k 2 ⊥ hard tail of F (x, k 2 ⊥ ). The tail is generated by hard gluon exchanges and is proportional to a small parameter α s /π ∼ 0.1.
Finally, in QCD (and other gauge theories), there is the third source of the z 2 -dependence related to some special contributions originating from the gauge link. These contributions vanish on the light cone z 2 = 0, but do not vanish for spacelike z 2 . Moreover, they contain link-specific UV divergencies, similar to those one encounters in the heavy-quark effective theory (HQET). Let us discuss these types of z 2 -dependence.

Long-distance z 2 -dependence
To begin with, P(x, z 2 3 ) describes a finite-size system (moreover, a system of confined quarks). Hence, it should rapidly decrease for large z 3 , say, like a Gaussian ∼ e −z 2 3 /R 2 or an exponential ∼ e −z 3 /R (one may choose here one's own favorite model), where R characterizes the size of the system. A finite size of the system imposes no restrictions on the behavior of P(x, z 2 3 ) for small z 2 3 . Such a behavior is determined by the short-distance dynamics. In a model involving just soft interactions, one would expect that P(x, z 2 3 ) is finite in the z 3 → 0 limit, like in the Gaussian and exponential cases. Then one may simply take z 3 = 0 in P(x, z 2 3 ) to get f (x).

Evolution-related z 2 3 -dependence
Since the small-z 2 limit in QCD is perturbative, one would expect that the only singularities of P(x, −z 2 ) for z 2 = 0 are those generated by perturbative corrections. As already mentioned, at one loop one gets ∼ α s ln(−z 2 ) terms. Hence, it makes sense to treat P(x, −z 2 ) as a sum of a "primordial" soft part P soft (x, −z 2 ) that has a finite z 2 → 0 limit, and a logarithmically singular hard part reflecting the evolution, and generated by hard gluon corrections to the original purely soft function. The same applies to M(ν, −z 2 ).
A singularity at z 2 = 0 means that the lightcone object M(ν, −z 2 = 0) is a divergent quantity. In perturbative calculations of the lightcone matrix element, the ln(−z 2 ) singularities convert into ultraviolet logarithmic divergences. These UV divergences are then additional to the usual UV divergences related to the propagator and vertex renormalization.
Still, as far as z 2 is kept finite, one does not have these additional UV divergences, and does not need to introduce a regularization for theψ(0) . . . ψ(z) operator. One should deal with the usual UV divergences and their renormalization only. Such a renormalization (characterized by some parameter λ) would produce (in a covariant gauge, say) just a trivial Z ψ (λ/m) renormalization factor for the ψ-fields (m being an infrared cut-off, e.g., a mass of the ψ field). This factor is the same whether z is on the light cone or not.
Except for this trivial dependence on the UV cut-off λ, the pseudo-ITDs M(ν, −z 2 ) in a general renormalizable (but nongauge) theory, depend on ν and z 2 only. The ln(−z 2 ) terms are just a particular form of the z 2 -dependence, and they do not require any regularization as far as z 2 is finite, which is the case in lattice simulations.
Theoretically, one may take z on the light cone. Then one should regularize the resulting extra UV divergences in some way, e.g., by imposing a momentum cut-off or by incorporating the MS scheme, etc. The resulting lightcone ITD I(ν, µ 2 ) introduced in Ref. [11] naturally depends on the parameter µ involved in the regularization of these ultraviolet divergences generated by taking ln z 2 for z 2 = 0.

UV singular terms generated by the gauge link
Furthermore, in QCD, the gauge link factor connectingψ(0) and ψ(z) generates contributions that are absent on the light cone, and moreover, are ultraviolet divergent. These divergences may be regularized using, e.g., the Polyakov prescription [19] 1/z 2 → 1/(z 2 − a 2 ) for the gluon propagator in the coordinate space. Then one finds that, for a fixed UV cut-off a, these terms vanish in the z 2 3 → 0 limit, like |z 3 |/a for the linear UV divergence and like ln(1 + z 2 3 /a 2 ) for the logarithmic one. That is why such terms are invisible on the light cone. Hence, we must make an effort to completely exclude these terms from M(ν, z 2 3 ). We emphasize that we need to eliminate the terms invisible in the light-cone limit even if they are UV finite.
As a matter of fact, in QCD they are UV divergent, and one should add the regularization parameter (a in this case) to the argument of the regularized pseudo-ITD: M(ν, −z 2 ) → M(ν, −z 2 ; a). These UV divergences are similar to those known from the HQET studies, and are multiplicatively renormalizable [20,21,22]. Since the parameter a appears only in the combination z 3 /a, the UV-sensitive terms form a factor Z(z 2 3 /a 2 ). As discussed above, this factor is an artifact of having a nonlightlike z, and has nothing to do with the lightcone PDFs. Thus, constructing the latter, we should exclude Z(z 2 3 /a 2 ) from the pseudo-ITD M(ν, z 2 3 ; a). In other words, one should build quasi-PDFs from the modified function Z −1 (z 2 3 /a 2 )M(ν, z 2 3 ; a). By construction, Z −1 (z 2 3 /a 2 )M(ν, z 2 3 ; a) does not have a → 0 UV divergences. These UV divergences were considered as the main problem in many recent papers [21,20,22,23]. However, if the goal is just to remove the divergences, then one may use any combination of the Z −1 (1/µ 2 UV a 2 )M(ν, z 2 3 ; a) type for the renormalized ITD. But the result then will have the dependence on the renormalization scale µ UV . The renormalized ITD will also contain the z 2 3 -dependence of the Z(z 2 3 /a 2 )-factor, that should be excluded in the construction of the light-cone PDFs.
In the approaches of Refs. [21,22,23], this is done at the final stage, when the matching conditions are applied.
We argue, that it is more beneficial, from the very beginning, to remove the UV divergences together with the associated z 2 3 -dependence. This may be done by multiplying M(ν, z 2 3 ; a) with the Z −1 (z 2 3 /a 2 ) factor. To do this, one should know the Z(z 2 3 /a 2 ) factor. Another possibility, proposed in our paper [9], is to use the reduced pseudo-ITD .

(4.2)
Then the UV-sensitive factor Z(z 2 3 /a 2 ) automatically cancels in the ratio (4.2), since it is ν-independent. So, there is no need to know it explicitly. The Z ψ (λ/m) factors reflecting the anomalous dimensions of the ψ fields also cancel in the ratio (4.2). The resulting function has a finite a → 0 limit, which will be denoted by M(ν, z 2 3 ). This function does not depend on any UV cut-off or a UV renormalization scale like µ UV .
We may say that M(ν, z 2 3 ) is a physical observable, just like the deep inelastic (DIS) structure functions W(x Bj , Q 2 ). The latter depend on the external variables x Bj , Q 2 , but do not depend on any ultraviolet cut-off or a renormalization scale µ, even if they are calculated in a renormalizable theory.
A widespread statement is that W(x Bj , Q 2 ) describes the hadron on the distance scale ∼ 1/Q. In this sense, M(ν, z 2 3 ) and the pseudo-PDF P(x, z 2 3 ) literally describe a hadron on the distance z 3 by construction.
Thus, for the reduced ITD M(ν, z 2 3 ), there are just two sources of the z 2 3 -dependence: the long-distance nonperturbative dependence reflecting the finite size of the system, and the shortdistance perturbative ∼ ln z 2 3 dependence related to the usual perturbative evolution. In this respect, the reduced pseudo-ITD M(ν, z 2 3 ) in QCD has the z 2 3 -structure similar to that in nongauge renormalizable theories, in which we also have just two first types of the z 2 3 -dependence.

Matching
The relations for the moments, like the formula (3.11) for y 2 Q , and the general formula (2.12), that have been used in our preceding discussion, are based on the Taylor expansion of P(x, z 2 3 ) over z 2 3 . Rossi and Testa in Refs. [7,8] also appeal to a Taylor expansion in z 3 . While this may be reasonable in a very soft case (in which all the derivatives with respect to z 2 3 exist at z 2 3 = 0), it is clear that to use the Taylor expansion at z 2 3 = 0 for the hard logarithm ln z 2 3 amounts to just asking for trouble. The basic reason for using the Taylor expansion is that the z 3 -dependence of the matrix element is, in general, unknown. So, a natural idea is to parametrize it through the values of the matrix elements of local operators.
The advantage of the hard contribution is that its z 2 3 -dependence at small z 2 3 (unlike that of the soft contribution) is known: at one loop it is given by ln z 2 3 . Thus, if one needs to calculate a quasi-PDF corresponding to the ln z 2 3 part of the matrix element, one can do this by simply calculating the Fourier transform of ln z 2 3 dictated by the quasi-PDF definition (2.4) rather than to use a Taylor expansion for a non-expandable function.

OPE and matching conditions for ITDs
When ln(−z 2 ) terms are present, a formal light-cone limit z 2 → 0 is singular. However, the PDF community wants lattice predictions for the light cone PDFs. In the continuum, the singular nature of the z 2 → 0 limit is perceived as an ultraviolet divergence in the Feynman integrals for operators on the light cone. It is worth repeating once more that these UV divergences are a consequence of our desire to take z 2 = 0. As far as z 2 is finite, these divergences are absent.
To work at z 2 = 0, we need to arrange an UV cut-off for these hand-made divergences. Using, say, the dimensional regularization and MS scheme, one would define the light-cone ITD I(ν, µ 2 ). Its connection to the pseudo-ITD M(ν, z 2 3 ) is given by the operator product expansion. At one loop in QCD, we have [18,23,24,25] The OPE tells us that, for small z 2 3 , the dependence of M(ν, z 2 3 ) on z 2 3 must be given by the ln z 2 3 term on the right-hand side. Hence, to get the light-cone ITD I(ν, µ 2 ) from, say, lattice calculations of M(ν, z 2 3 ), one should subtract from the lattice pseudo-ITD M(ν, z 2 3 ) its perturbative ln z 2 3 part present in the r.h.s. of Eq. (5.1). For an appropriately chosen/fitted α s , the result of such a subtraction should be z 2 3 -independent. Such a procedure of extracting I(ν, µ 2 ) from the lattice data of Ref. [26] was described in our Ref. [25].

Matching conditions for quasi-PDFs
Multiplying Eq. (5.1) by Pe −iyz 3 P and integrating over z 3 , we get a relation between the (reduced pseudo-ITD based) quasi-PDF Q(y, P) and the light cone PDF f (x, µ 2 ). It has the following structure where the kernel L(y, x; P) is formally given by It involves the Fourier transform of ln z 2 3 and it is the only term that produces contributions in the |y| > 1 region. Eq. (5.2) tells us that the quasi-PDF Q(y, P) must have O(α s ) perturbative contributions in the |y| > 1 region. In actual lattice calculations it is desirable (though challenging) to try to check if the lattice quasi-PDF in the |y| > 1 region is indeed close to the convolution of the fitted PDF with the L-kernel.
For large P, the soft contributions disappear from the |y| > 1 region, and the perturbative terms are the only ones remaining for |y| > 1. This means that extracting the PDF f (y, µ 2 ) from the lattice data for Q(y, P), one deals with the combination that vanishes in the |y| > 1 region for large P (provided that we trust perturbative QCD!). We may say that the f ⊗ L contribution cancels the perturbative slow-decreasing terms of the |y| > 1 part of Q(y, P). After that, all the remaining terms in Eq. (5.2) have the |y| ≤ 1 support. In other words, the process of getting MS PDFs from quasi-PDFs involves a subtraction of the perturbative |y| > 1 contributions generated by the ln z 2 3 term.

Hard part of quasi-PDFs
An evident and important observation from the study of the hard contribution is that the quasi-PDFs do not simply convert into the usual PDFs in the large-P limit. They convert into PDFs only in the case of soft TMDs and quasi-PDFs generated from them.
When the hard part is included, Q(y, P) contains the terms that are not present in the lightcone PDFs and which are, moreover, finite (for a fixed α s ) in the P → ∞ limit. Such terms appear both in the "canonical" −1 ≤ y ≤ 1 region and, most importantly, outside it. The presence of such terms was known since the first papers on quasi-PDFs [2,3].
In the context of pseudo-PDFs, these terms are generated by the Fourier transform of the ln z 2 3 hard term. In the momentum representation, ln z 2 3 corresponds to the 1/k 2 ⊥ behavior, which needs some IR regularization. Let us choose the mass-type modification 1/k 2 ⊥ → 1/(k 2 ⊥ + m 2 ). Then ln(z 2 3 ) → −2K 0 (z 3 m), and we have (see Ref. [24]) where the kernel R(η, m 2 /P 2 ) is given by In lattice extractions, the real part of the pseudo-ITD corresponds to an even function of y, while the imaginary part corresponds to an odd function of y. Hence, in both cases, it is sufficient to consider positive y only. For η, we need then to analyze three regions, η < 0, 0 ≤ η ≤ 1 and η > 1.
In the central 0 ≤ η ≤ 1 region, the P → ∞ limit is singular, reflecting the presence of the evolution ∼ ln P 2 /m 2 term (3.7). There are also terms [24] R middle (η) = 1 + η 2 1 − η ln 4η(1 − η) that are independent of P in the P → ∞ limit. For y outside the 0 ≤ y ≤ 1 segment, we can neglect m 2 /P 2 in the P → ∞ limit and get with the kernel R(η; 0) ≡ R(η) specified by At first sight, one would expect a ∼ 1/|η| behavior for large |η| from Eq. (5.9). However, the 1/|η| term is accompanied by the integral of B(u) which vanishes because of the plus-prescription structure of B(u). This is also the reason why γ 0 in Eq. (3.10) vanishes. Hence, in the region η > 1, we can write the kernel as a series in 1/η starting with n = 1, 10) or, in a closed form [24], Similarly, for negative values, we have the expansion and a closed-form expression [24] R(η; 0)| η<0 ≡ R < (η) (5.13)

Large-|y| behavior in QCD
According to Eq. (3.10), we have γ 1 = 4/3. Thus, the asymptotic behavior for large |η| is given by 14) The ∼ sgn(η)/η 2 behavior of R(η) translates into the ∼ sgn(y)/y 2 behavior of the quasi-PDF Q(y, P) for large values of |y|. As a result, the y 0 moment of Q(y, P) converges for large |y|, while further moments involve divergences, in agreement with observations made in Sect. 3.2 . In particular, the y 2 moment involves a linear divergence. If B(u) would not have the plus-prescription property, the divergence would be quadratic. This agrees with the estimate made in Sect. 3.3.
Hence, the divergences of the y integrals correspond to the presence of the P-independent terms ∼ 1/y 2 in the hard part of the quasi-PDFs Q(y, P) outside of the 0 ≤ y ≤ 1 region.
As we discussed, the ln z 2 3 part of the pseudo-ITDs contributes slowly-decreasing (∼ 1/y or ∼ 1/y 2 ) terms into the |y| > 1 part of quasi-PDFs. It is these terms that lead to the divergence of the y n moments of the quasi-PDFs Q(y, P).

Large-P matching
These terms are not eliminated by just taking the P → ∞ limit. However, they disappear when one extracts f (y, µ 2 ) using the matching condition (5.2). Namely, we have Since both the O(1/P 2 ) soft part and the Q(y, P) combination of Eq. (5.4) vanish for |y| > 1 in the P → ∞ limit, Eq. (5.15) resolves the problem of the support mismatch between f (y, µ 2 ) and Q(y, P). As a result, one can calculate the y n moments of the light-cone PDFs f (y, µ 2 ) using Eq. (5.2) without getting divergences in its right-hand side.
If we separate quasi-PDFs corresponding to the real [Q − (y, P)] and imaginary [Q + (y, P)] parts of the ITD, it is sufficient to consider positive y only. Using the fact that perturbative part of Q(y, P) vanishes outside the |y| ≤ 1 region, we may write iterative solution of Eq. (5.15) for y > 0 as f ∓ (y, µ 2 ) = Q ∓ (y, P)θ(0 ≤ y ≤ 1) Here the function f − (y) corresponds to the real part of the ITD and is given by q(y) −q(y), while f + (y) corresponds to the imaginary part of the ITD and is given by q(y) +q(y). The kernels R > (η), R < (η) are given by Eqs. (5.11) and (5.13). The third line of Eq. (5.16) comes from R middle (η) of Eq. (5.7) and terms from Eq. (5.1). All the terms explicitly written in Eq. (5.16) involve quasi-PDFs in the y < 1 region only. The y > 1 part of Q(y, P) is included in O(1/P 2 ) term and vanishes in the P → ∞ limit. We remind that the starting point for the derivation of Eq. (5.16) is based on Eqs. (5.1) and (5.2). Hence, Eq. (5.16) applies to quasi-PDFs built from the reduced pseudo-ITDs (4.2).

Summary and conclusions
In this paper, we discussed a specific feature of the quasi-PDFs Q(y, P 3 ) in which they differ from the usual PDFs f (x), namely, the presence of terms outside the |y| ≤ 1 region.
In a model with a transverse momentum cut-off, such terms disappear in the P → ∞ limit. However, in renormalizable theories, including QCD, one has |y| > 1 terms persisting (for a fixed α s ) even in the P → ∞ limit. These terms have a perturbative origin that may be traced to the ln z 2 3 singularities of the generating matrix element p|ψ(0) . . . ψ(z 3 )|p .
Since one knows that such terms, absent in the light-cone PDFs f (x), must be present in the quasi-PDFs Q(y, P 3 ), one should just subtract them from Q(y, P 3 ) obtained on the lattice. The resulting "reduced" quasi-PDF Q R (y, P) for large P has support in the canonical region |y| ≤ 1 only. On a formal level, such a subtraction is automatically provided by implementing the matching conditions.
Eq. (5.16), that is given at the end of the paper, provides an explicit expression for the lightcone PDF f (y, µ 2 ) involving the quasi-PDF Q(y, P) in the |y| ≤ 1 region. Hence, in actual lattice PDF extractions, one may ignore the |y| > 1 region altogether and operate with Q(y, P) obtained in the |y| ≤ 1 region only.
A related practical question is if the complications with the |y| > 1 region may be avoided? Indeed, according to the OPE (5.1), the reduced pseudo-ITD M(ν, z 2 3 ), a function directly "coming out of the computer box", may be used, without intermediaries, to extract the lightcone ITDs I(ν, µ 2 ). The latter are the Fourier transforms of the lightcone PDFs f (x, µ 2 ), the functions that have the canonical |x| ≤ 1 support. Such an approach has been already applied in the exploratory lattice calculation [26] and in the construction [25] of MS ITD I(ν, µ 2 ) based on its results.