^{3}.

Perturbative expansions for short-distance quantities in QCD are factorially divergent and this deficiency can be turned into a useful tool to investigate nonperturbative corrections. In this work, we use this approach to study the structure of power corrections to parton quasidistributions and pseudodistributions which appear in lattice calculations of parton distribution functions. As the main result, we predict the functional dependence of the leading power corrections to quasi(pseudo)-distributions on the Bjorken

Lattice calculations in QCD have demonstrated the ability to complement, and in certain cases with the exceeding precision, significant amount of experimental measurements. Now, the lattice evaluation of parton distribution functions (PDFs) are coming on the agenda. New techniques are being explored aiming at the access of PDFs directly in the momentum fraction space, in addition to the standard approach that allows one to calculate first Mellin moments of PDFs. The existing actual proposals

A particularly popular suggestion

Lattice calculations of PDFs using these new methods are presently moving from exploratory stage towards precision calculations, therefore questions like whether the higher-twist (power suppressed) corrections are well under control have to be addressed. One possibility to investigate the impact of higher-twist corrections is to extract PDFs from the global analysis of many Euclidean correlation functions introducing such corrections as free parameters. This approach is showcased in

The renormalon approach to the investigation of power corrections is founded on the fact that operators of different twist mix with each other under renormalization, due to the violation of QCD scale invariance through the running of the coupling constant. In cutoff schemes, this mixing is explicit, whereas in dimensional regularization, it manifests itself in factorial divergence of the perturbative series. Independence of a physical observable on the factorization scale implies intricate cancellations between different twists—the cancellation of renormalon ambiguities. In turn, the existence of these ambiguities in the leading-twist expressions can be used to estimate the size of power-suppressed corrections. Conceptually, it is similar to the estimation of the accuracy of fixed-order perturbative results by the logarithmic scale dependence. The renormalon approach was used before for the study of

In order to explain how the concept of renormalons can be used to get insight in the structure of power corrections, let us consider the usual expression for the quasidistribution

To understand the role of renormalons, it is necessary to examine carefully the separation made in

In practice, perturbative calculations are usually done using dimensional regularization. In this case, powerlike terms as in

Returning to

In this work we calculate the function

The presentation is organized as follows. In Sec.

Let us start with the following nucleon matrix element

The operator product in Eq.

The “longitudinal” and “transverse” invariant functions in Eq.

The invariant functions

The qPDFs

In renormalization schemes with an explicit regularization scale, the Wilson line in Eq.

The general approach to collinear factorization of QCD amplitudes in the position space is provided by the light-ray OPE

The leading-twist projection of the nonlocal quark-antiquark operator is defined as the generating function of

The light-ray OPE differs from the usual short-distance Wilson expansion in local operators by imposing a different power counting. In the short-distance expansion one assumes that the distance between the quarks is small,

Note that the above power counting is applicable both in Minkowski and Euclidean space. In Minkowski space, one can go over to a different reference frame where all components of the momentum are of order

On the calculation level, the light-ray OPE provides one with a convenient framework to operate with the leading-twist projected operators

The nucleon matrix element of the leading-twist projected operator

Taking the nucleon matrix element of the operator relation in Eq.

Making the Fourier transformation

The target mass correction for the “transverse” qPDF can be calculated in a similar way, starting from the nonlocal operator with an open Lorentz index

The coefficient function

A convenient way to handle such a series is to consider the Borel transform

Naturally, a full all-order calculation cannot be performed. Instead, we employ the approximation

The singularity structure of the Borel transform can be extracted separately without explicit evaluation of the bubble-chain. It can be done by replacing the running coupling constant in the loop diagrams by its effective form,

The leading contributions to the renormalon singularities in the coefficient functions for qITDs are shown in Fig.

Bubble-chain contribution to the coefficient function. The Wilson line factor is shown by the double dotted line.

The structure of singularities of the Borel transform of the coefficient functions is illustrated in Fig.

Singularity structure of the Borel transform.

The singularity at

The leading IR renormalon singularity is at

Renormalon singularities at

A singularity on the integration path in Eq.

Considering

For the normalized qITDs defined in Eq.

In order to visualize the functional dependence of the power correction on the “Ioffe time”

In general, the qITDs

Real parts of

Making the Fourier transformation of the above results for the qITDs we obtain the qPDFs

The normalized QPDFs

For a numerical study we have used the MSTW NLO valence

The

Constructing the qPDFs from the qITDs normalized to the value at zero momentum has a large effect. This is illustrated in Fig.

Upper panels:

We see that the normalization to the qITD at zero momentum significantly reduces the power correction at moderate values of

Power corrections for the pPDF can be obtained easily from the corresponding expression for the “transverse” qITDs

Power correction to the pPDF

Note that

We have presented an analysis of power-suppressed (higher-twist) contributions to qPDFs and pPDFs based on the study of factorial divergences (renormalons) in the corresponding coefficient functions within the bubble-chain approximation. Factorial asymptotic implies that the sum of the series is only defined to a power accuracy and therefore, the QCD perturbation theory must be corrected by nonperturbative power-suppressed contributions to produce unambiguous predictions. Our results have to be considered as a “minimal model” for the higher-twist corrections that captures effects that are necessary for the self-consistency of the theory, but possibly misses other nonperturbative corrections that are, e.g., protected by symmetries and not “seen” through perturbative expansions. Our main conclusions are as follows:

Position space PDFs (qITDs) have flat power corrections at large Ioffe times. Generally, power corrections are much larger for the “transverse” projection as compared to the “longitudinal” projection.

Power corrections for qPDFs have a generic behavior

Power corrections for pPDFs have a generic behavior

This work was partially supported by the DFG Grant SFB/TRR-55 “Hadron Physics from Lattice QCD”.

In this Appendix we present some details of the calculation of coefficient function

To obtain the twist-two coefficient function in

The diagrams contributing to the

The leading

The factorial divergences

Strictly speaking, the expansion in powers of the large momentum corresponds to the classification in terms of the so-called collinear twist, see e.g.,