^{1}

^{2}

^{3}

^{3}.

We show that the correct way to extract parton distribution functions from the reduced Ioffe-time distribution, a ratio of the Ioffe-time distribution for a moving hadron and a hadron at rest, is through a factorization formula. This factorization exists because, at small distances, forming the ratio does not change the infrared behavior of the numerator, which is factorizable. We illustrate the effect of such a factorization by applying it to results in the literature.

One of the most important goals of QCD is to understand the structure of hadrons in terms of their fundamental constituents—quarks and gluons. This is a profoundly difficult task because it necessarily requires studies in the nonperturbative regime. Currently, the most reliable tool for such nonperturbative studies is lattice QCD, which has been widely used to study the static property of hadrons (see e.g. Refs.

Recently, a new approach has been proposed to study parton physics from lattice QCD

Alternative, but related, approaches have also been proposed: using lattice cross sections

In Refs.

In this paper, we show that the correct way to take advantage of the RITD is through a factorization formula that connects the RITD to the PDFs. Although logarithmic evolution relates the RITD at different

The paper is organized as follows. In Sec.

The RITD introduced in Ref.

The quasi- and the pseudodistributions are defined as two different Fourier transforms of the Ioffe-time distribution

So far, we have considered the Ioffe-time distribution only. Now, let us turn to the reduced distribution. The matrix element

From Eq.

These results suggest the correct way to relate the PDF and the RITD. That is, we can form a factorization not for the Ioffe-time distribution but for the reduced distribution. This is possible because the RITD is constructed from a ratio of two matrix elements,

By taking the Fourier transform of Eq.

In the second case, we have

We can take advantage of the RITD either by working directly in coordinate space, converting it to

The authors of Refs.

In the following, we apply the factorization formula in Eq.

From Eq.

Valence distribution

With the choice

The reduced Ioffe-time distribution has the advantage that the nonperturbative renormalization effects associated with the Ioffe-time distribution itself can be avoided. In this paper, we have shown that the light-front PDFs are related to the RITD through a factorization relation. This factorization exists because, at small distances, forming the ratio in the RITD does not change the IR behavior of the numerator, which is factorizable. We then illustrate the effect of such a factorization by applying it to the data used in Ref.

This work was partially supported by the U.S. Department of Energy via Grant No. DE-FG02-00ER41132, the SFB/TRR-55 grant “Hadron Physics from Lattice QCD,” a grant from National Science Foundation of China (Grant No. 11405104), the Ministry of Science and Technology, Taiwan under Grant No. 105-2112-M-002-017-MY3, and the Kenda Foundation. It stemmed from discussions during the “Workshop of Recent Developments in QCD and Quantum Field Theories” at National Taiwan University in November, 2017. We would like to thank K. Orginos for useful comments and V. Braun and A. Vladimirov for useful discussions.

Recently, a preprint by A. Radyushkin