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Corresponding author.

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We present a detailed lattice QCD study of the unpolarized isovector quark parton distribution function (PDF) using a large-momentum effective theory framework. We choose a quasi-PDF defined by a spatial correlator which is free from mixing with other operators of the same dimension. In the lattice simulation, we use a Gaussian-momentum-smeared source at

Parton distribution functions (PDFs) of nucleons are not only important quantities characterizing the internal hadron structures but are also key ingredients to make predictions for high-energy scattering processes

Within QCD factorization

PDFs are defined with light-cone coordinates, but the lattice simulation can only be conducted in Euclidean space with no proper treatment for light-cone quantities, which involves real time. Thus simulating PDFs on a Euclidean lattice is an extremely difficult task. Early studies based on operator product expansion were only able to derive the lowest few moments of the PDFs

Recently, a novel approach that allows to directly access the

To calculate the quark PDF in LaMET, one starts with a “quasi-PDF” which is defined as an equal-time correlation of quarks along the

Since the proposal of LaMET, remarkable progress has been made in both theoretical aspect and lattice calculations. It should be pointed out that these developments are achieved in an interactive way. The LaMET was first used to calculate the proton isovector quark distribution

In addition to LaMET, other interesting approaches have been proposed in recent years to calculate the PDFs from lattice QCD. For example, one can extract the PDFs from a class of “lattice cross sections"

It was argued that the power divergent mixing between local moment operators may spoil the renormalization of quasi-PDFs

Most of the available lattice calculations have used

In this work, we will carry out a lattice calculation of the unpolarized isovector quark distribution from the quasi-PDF with

The rest of this paper is organized as follows. In Sec.

To recover the continuum limit of a quasi-PDF matrix element, nonperturbative renormalization on the lattice is required to deal with linear and logarithmic UV divergences. In this work, we follow the RI/MOM scheme elaborated in Refs.

The spatial correlator

For each value of

Based on the symmetry of

The bare nucleon matrix element from a lattice calculation in coordinate space

The quasi-PDFs will eventually be matched to the same

Using the same logic one reaches the conclusion that the quasi-PDF’s dependence on the RI/MOM scales

To obtain the matching coefficient between the quasi-PDF

The lowest-order quark quasi-PDF is

For the light-cone PDF with the same off-shell IR regulation in Landau gauge, the tree-level contribution is

To match the quasi-PDF to the light-cone PDF, one needs to take the on-shell limit (

For the light-cone PDF, the coefficient of

In RI/MOM, the quasi-PDF is renormalized with an additional counterterm. We find that in the

For

In this section, we give the results of a lattice-QCD calculation using clover valence fermions on the CLS

First of all, we will explore the nonperturbative renormalization in the RI/MOM scheme. Following Ref.

Momentum modes used in the NPR analysis. The four digits (in units of

As shown in Eq.

(Top panel) The NPR

In Fig.

The inverse of the minimum projection renormalization factor

In the following we will take

In the calculation of the nucleon matrix element, we use Gaussian momentum smearing

Such a momentum source is designed to increase the overlap with nucleons of the desired boost momentum and we are able to reach higher-boosted momentum for the nucleon states than in the previous work

On the lattice, we calculate the time-independent and nonlocal in the

As the nucleon boost momentum increases, one anticipates that excited-state contributions are more severe; therefore, a careful study of the excited-state contamination is necessary. To do so, we calculate the nucleon matrix element

Eventually we apply the joint fit with the 3pt functions at several

In Fig.

The real (left panels) and imaginary (right panels) parts of the isovector nucleon matrix elements for unpolarized PDFs as functions of

For a comparison between data and the fit, we show our results at large

The real (left panels) and imaginary (right panels) parts of the renormalized isovector nucleon matrix elements for unpolarized PDFs with

The renormalized quasi-PDF matrix elements with two values of

The renormalized quasi-PDF matrix elements with

In this section, we will consider four systematic uncertainties, from Fourier transformation (FT), unphysical scales

In the following, we explain the details to include these systematic uncertainties.

(1) Fourier transformation. As shown in Fig.

The quasi-PDF (dashed-red lines) with nucleon boost momentum 2.3 GeV and matched PDF (solid-black lines) at

Different contributions to the systematic errors. See the text for detailed information.

Extending the lattice calculations restricted to a finite number of determinations of matrix elements to recapture all information on the PDF is a sophisticated problem. Thus it is necessary to point out that the derivative method we adopted in Eq.

(2) Unphysical scales

(3) Dependencies on the projection. There are two projections discussed in this work: the minimum and “

(4) Inversion of matching. To extract the PDF from the quasi-PDF, one needs to invert the factorization formula Eq.

Effects of the inversion matching formula using minimal projection. The solid-black, dotted-red, dotted-blue, and dot-dashed-green lines represent the CT14nnlo PDF, applying inverse matching from the CT14nnlo PDF

There are more sophisticated methods to invert the factorization, such as using a recursion procedure. However, as we can see in Fig.

As shown in Fig.

With the derivative method of FT and the matching using

Nucleon boost momentum dependence of the matched unpolarized isovector PDFs: the dotted-green and solid-blue lines correspond to the nucleon momentum

Finally, we show our results for the PDF and a comparison with global analysis

Results for PDF at

In this paper, we have studied the quasi-PDF defined with

We have found that the systematic uncertainties from the FT truncation method and also the

Controlling systematic uncertainty from the excited state is very challenging since the relative uncertainty grows very fast when either source-sink separation

Besides the uncertainties that we have studied, in the future we plan to investigate other systematics such as lattice discretization and finite volume effects

Our final result for light-cone PDF agrees with the global analysis in the large-

We thank the CLS Collaboration for sharing the lattices used to perform this study. The lattice QCD calculations were performed using the

The gluon propagator in the general covariant gauge is

For