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We present the first direct lattice-QCD calculation of the

The pion plays a fundamental role in QCD. As the lightest meson and the Goldstone boson associated with dynamical chiral symmetry breaking, it provides an important testing ground for our understanding of nonperturbative QCD. Currently, our experimental knowledge of the pion structure comes primarily from the Drell-Yan data for pion-nucleon/pion-nucleus scattering

This became possible recently due to the breakthrough made by large-momentum effective theory (LaMET) in direct lattice calculation of the

LaMET has been applied to compute various nucleon PDFs

In this paper, we carry out the first direct lattice calculation for the valence quark distribution of the pion using the LaMET approach. The calculation is done using clover valence fermions on an ensemble of gauge configurations with

The quark PDF in the pion is defined as

The quark quasi-PDF can be defined in a similar way to Eq.

The factorization connecting the quasi-PDF and the PDF was first presented in Refs.

In the RI/MOM scheme, the bare coordinate-space matrix element

Then the nonperturbatively renormalized quasi-PDF can be matched to the PDF in the

For power corrections, the meson mass correction associated with the choice of Dirac matrix

In addition to the setup described in the introduction, the gauge links are one step hypercubic smeared

It is worthwhile to point out that for the valence quark PDF considered in the present paper, the imaginary part vanishes. The reason is that the imaginary part yields

To reach larger pion momentum, the optimal Gaussian smearing parameter was chosen by varying the parameters. Then

An example plot for

Example plot of the three- to two-point function ratios vs the insertion time

In Fig.

Comparison between one-state and two-state fits. The data points from left to right indicate the single-state fits [i.e., keeping only the

In Fig.

The real and imaginary part of the

The consistency of the one- and two-state fits with multiple

Now we present our numerical results for the valence quark distribution in the pion and discuss their physical implications. We first Fourier transform the renormalized lattice data to momentum space,

The pion valence quark PDF result from the Fourier transform in Eq.

In our earlier work

Comparison between the results with (red) and without (blue) using the derivative method, for the momentum

In Fig.

The pion momentum dependence of the results in the Wilson line renormalization scheme (WL ren.) and RI/MOM scheme. We have chosen

It is worthwhile to stress that matching is a necessary step in converting the quasi-PDF to PDF. It yields sizeable contributions and changes, in particular, for the distribution in the unphysical region. In Ref.

In Fig.

Our pion valence quark PDF result at the scale

We point out several potential sources of uncertainty or artifact in the above analysis, which we aim to improve in the future. First, the contribution at large

We thank the MILC Collaboration for sharing the lattices used to perform this study. We also thank Yu-Sheng Liu and Yi-Bo Yang for providing part of the data/plots and useful discussions. The LQCD calculations were performed using the Chroma software suite

In this scheme, the nonperturbative renormalization reads

The matching kernel