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We analyze the lattice spacing dependence for the pion unpolarized matrix element of a quark bilinear operator with a Wilson link (parton quasidistribution functions operator, quasi-PDFs) in the rest frame, using 13 lattice spacings ranging from 0.032 to 0.121 fm. We compare results for three different fermion actions with or without good chiral symmetry on dynamical gauge ensembles from three collaborations. This investigation is motivated by the fact that the gauge link generates a

Parton distribution functions (PDFs) play a key role for most processes in high energy and hadron physics, ranging, e.g., from the search for new physics to the detailed analysis of nucleon properties. Therefore, it is one of the most important tasks of lattice QCD to determine PDFs by first principle theory calculations. While the calculation of PDF Mellin moments, which can be expressed as matrix elements of local operators, has a long, high-profile history, the calculation of their full

Setup of the ensembles, including the bare coupling constant

LaMET starts from the quasi-PDF operator

Studies in the continuum

To close this gap, we study the RI/MOM renormalized pion quasi-PDF matrix element in the rest frame, for 13 lattice spacings ranging from 0.032 to 0.121 fm using different quark and gluon actions. Our results show that the cancellation deteriorates with decreasing lattice spacing and that the RI/MOM method leaves a linearly divergent residue for quasi-PDFs. We also show that in the Landau gauge the interaction between the Wilson link and the external state results in a linear divergence which depends on the discretized fermion action.

When we consider the quasi-PDF nucleon matrix element in the moving frame, the gap

Using the pion rest frame can, therefore, be the better choice as it avoids this problem. We choose

In RI/MOM renormalization, we define the renormalization constant as

We can apply

In order to check if the linear divergence is related to the fermion action, we compare results for two kinds of fermion actions in this work: the clover (CL) and overlap (OV) actions. The clover action is computationally relatively cheap and widely used in quasi-PDF calculations, while the overlap action is much more expensive but conserves chiral symmetry.

The clover fermion action is

To eliminate

We use three sets of ensembles:

The details of the ensembles we used in this work are collected in Table

We start with the pion matrix element and the MILC ensemble using the overlap action to demonstrate the elimination of excited state contaminations. As shown in Fig.

The bare ratios

The ratios between the pion matrix element

Based on a symmetry analysis and 1-loop calculation, we can decompose the amputated Green’s function

At the same time, setting

The real part of the relative mixing between

It is also popular to require all momentum components to be nonzero to suppress discretization errors and use the p-slash projection as proposed in Ref.

As for the RI/MOM renormalization constant in this work, we use the momentum

The

However, the residual

Another thing we need to mention here are the normalization conditions Eqs.

The normalization factor

We start from the overlap fermion case which is free of a possible linear divergence due to a possible mistuning of

The RI/MOM renormalized pion matrix element of the vector current, using overlap (OV) fermions on DWF sea ensembles. The data points correspond to the cases with 1-step of HYP smearing, and the bands are the results without HYP smearing of the Wilson link.

Thus, we switch to the other class of ensembles using the HISQ sea, which covers lattice spacings from 0.032 to 0.121 fm, see Fig.

The RI/MOM renormalized pion matrix element of the vector current, using overlap (OV) fermions on HISQ sea ensembles and using a Wilson link without HYP smearing (upper panel) and with 1-step of HYP smearing (lower panel). The residual linear divergences in both cases are pronounced and similar.

In the clover fermion case, the RI/MOM renormalized

The same RI/MOM renormalized pion matrix element as in Fig.

Since

The same RI/MOM renormalized pion matrix element as in Fig.

It is natural to guess that the residual linear divergence is stronger in the clover fermion case than in the overlap case because of chiral symmetry breaking effects. However, there can be contributions from various operators, e.g.,

In Ref.

According to the lattice perturbative theory calculation

We guess that the Landau gauge fixing we used for the quark matrix element introduces an additional linear divergence. As the Wilson link can have a gauge-dependent logarithmic divergence at the 1-loop level

The ratio of gauge links in Coulomb gauge and Landau gauge on MILC ensembles with 1-step of HYP smearing. These curves seem to converge at small lattice spacings, which indicates that the linear divergence of the Wilson link is independent from the gauge fixing condition, at least for the Landau or Coulomb ones.

We also checked the correlation between the Wilson line and external states by considering the following correlation:

The value of

Eventually, we consider the ratio

The pion quasi-PDF matrix element renormalized by the Coulomb gauge fixed Wilson line, using the overlap fermion on HISQ ensembles.

In this paper, we study systematically the continuum limit for quasi-PDFs with RI/MOM renormalization in the LaMET approach. We compare results for a variety of valence and sea quark fermion actions and lattice setups. We find that the traditional RI/MOM method cannot eliminate the characteristic linear divergences of quasi-PDFs completely and that the remnants of this linear divergence blow up with decreasing lattice spacing. Obviously this greatly complicates control of the continuum limit.

We make a number of observations which can prove useful to clarify the origin of these remnants and develop strategies to cope with them.

The RI/MOM renormalized pion matrix elements in the rest frame include residual linear divergences from certain higher loop effects, so higher loop calculations in lattice perturbation theory are needed.

Simulations with chiral fermions (overlap or domain wall) are much less affected than those with clover fermions, so chiral symmetry seems to play an important role.

Since the renormalization of operators should be frame independent, we believe that this conclusion should also apply to all quasi-PDF calculations in a moving frame. For a quasi-PDF calculation using an ensemble with

We believe that similar checks should be done for all LaMET calculations using bilinear operators with Wilson link, e.g., quasi-PDFs, quasi-DAs, quasi-TMDs, and so on. Comparison to calculations with large momentum for the nucleon suggests that the proposed check just requires the calculation of

The outcome of many such tests should allow one to better understand the origin of these residues and should provide clues for how to remove them systematically

The perturbative and nonperturbative study in lattice regularization beyond the 1-loop level is also essential, as a residual linear divergence is forbidden at the 1-loop level. We also confirmed the following expectations:

The linear divergences in the pion quasi-PDF matrix element with different fermion actions (overlap and clover) are the same. Combining with the consistency of the pion and nucleon matrix element shown in Ref.

The linear divergence in the Wilson link is indeed gauge independent up to certain lattice artifacts, based on our calculation in the Coulomb and Landau gauges.

There is a residual linear divergence due to the gluon exchange between the external state and the Wilson line, which is absent at the 1-loop level. Such a linear divergence is action dependent for a quark state in the Landau gauge, while it is action independent in the hadron state.

We thank the CLS, MILC, and RBC/UKQCD Collaborations for providing us their gauge configurations and Long-Cheng Gui, Xiangdong Ji, Keh-Fei Liu, Wei Wang, Jian-Hui Zhang, and Yong Zhao for useful information and discussion. The calculations were performed using the Chroma software suite

The expression for the amputated Green’s function reads

First of all, Fig.

The

Then, we turn to the p-slash term

The

The cases with

The last part is the

Similar to Fig.

Thus, using the RI/MOM renormalization constant defined in Eq.

Eventually, in Fig.

The RI/MOM renormalized pion matrix elements