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We present lattice results for the isovector unpolarized parton distribution with nonperturbative regularization-invariant momentum-subtraction scheme (RI/MOM) renormalization on the lattice. In the framework of large-momentum effective field theory (LaMET), the full Bjorken-

Parton distribution functions (PDFs) are probability densities of quarks and gluons seen by an observer moving at the speed of light relative to the hadron. They are universal nonperturbative properties of the hadron. In a global analysis the hard-scattering cross sections can be factorized into the PDFs and the short-distance matrix elements calculable in perturbation theory. Once known, the PDFs can be used as inputs to predict cross sections in high-energy scattering experiments, one of the major successes of QCD. Today, multiple collaborations provide regular updates concerning the phenomenological determination of the PDFs

In this work, we continue the first-principles calculation of the PDFs using lattice QCD. In parton physics, the PDFs are defined as the nucleon matrix elements of quark or gluon correlation operators along the light cone direction. For example, the unpolarized quark distribution is defined as

Recently, Ji

There has been rapid development following Ji’s proposal. Lattice-QCD calculations of the proton isovector quark distribution

Despite many promising features in the previous lattice calculation of PDFs

The renormalization of the quasi-PDF has been closely studied from the perturbative point of view

This property makes it possible to carry out a nonperturbative renormalization of the quasi-PDF in the regularization-invariant momentum-subtraction scheme (RI/MOM)

In Ref.

In this work, we present lattice results for the nonperturbatively renormalized quasi-PDF in RI/MOM scheme for the isovector unpolarized case,

While this paper was being finalized, another paper

For continuum QCD, the renormalization of nonlocal quark bilinear operators has been discussed since the 1980s

On the Euclidean lattice, QCD is invariant under discrete symmetries, which include parity

The same discussion can also be applied to pseudoscalar, axial vector, and tensor operators

Therefore, the renormalization of the nonlocal vector operators for lattice fermions without chiral symmetry can be schematically presented as

In the past,

The renormalization matrix elements of Eq.

By imposing the RI/MOM renormalization condition,

Next, the renormalized proton matrix element of

In the next section, we apply the RI/MOM scheme to the renormalization of quasi-PDF on the lattice, and eventually extract the

The results of our lattice calculations are presented in two parts: The first part is the nonperturbative renormalization constants in RI/MOM scheme, the second part is the result of the isovector unpolarized PDF. The bare quasi-PDF is renormalized using the renormalization in the first part, and then matched to the PDF using the one-loop matching formula after the power corrections in

For the renormalization calculation, we used 33 configurations on the

The

A comparison of the signals between the point source and the momentum source for the

Comparison between the renormalization constants obtained with the point source and the momentum source for

Figure

The renormalization constant of the quasi-PDF operator

In order to check the

In this subsection, we present our results for the unpolarized isovector quark distribution. We first calculate the time-independent, nonlocal correlator of a nucleon with finite

The bare

Next, we Fourier transform Eq.

The

The renormalized unpolarized isovector quark distribution after one-loop matching and mass correction at the renormalization scale

The final results are shown in Fig.

The renormalized unpolarized isovector quark distribution after one-loop matching and mass correction at the renormalization scale

Another observation concerning our renormalized distribution is an oscillating behavior in negative-

Comparison of the renormalized function

Finally, we have several comments regarding our RI/MOM treatment. The first one is the possible gauge dependence induced by taking the external quarks off-shell in the nonperturbative renormalization. The gauge dependence should be canceled by the matching kernel, but the cancellation is not complete, since the kernel is only computed at one loop. It is encouraging that the one-loop matching effect is numerically small in Landau gauge that we employed. Whether the higher-loop contributions will remain small requires further study. The second one is treating

We have carried out a nonperturbative renormalization of the quasi-PDF in the RI/MOM scheme in lattice QCD. Based on the renormalized quasi-PDF, we have updated the lattice result of the unpolarized isovector quark distribution from previous studies by some of the authors. The RI/MOM renormalization of the quasi-PDF is performed for each individual

It is possible that an alternative process is needed in the large

Compared to the previous results on bare PDFs, our present result is free of the unphysical dip at

We thank the MILC Collaboration for sharing the lattices used to perform this study, the generation of those lattices used resources of Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program by USQCD. The LQCD calculations were performed using the Chroma software suite