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In the large-momentum effective field theory approach to parton physics, the matrix elements of nonlocal operators of quark and gluon fields, linked by straight Wilson lines in a spatial direction, are calculated in lattice quantum chromodynamics as a function of hadron momentum. Using the heavy-quark effective theory formalism, we show a multiplicative renormalization of these operators at all orders in perturbation theory, both in dimensional and lattice regularizations. The result provides a theoretical basis for extracting parton properties through properly renormalized observables in Monte Carlo simulations.

One of the most important goals of quantum chromodynamics (QCD) is to understand the hadron structure from its fundamental degrees of freedom—quarks and gluons. A powerful tool of obtaining such results is lattice QCD, which has been used to study the static properties of the hadron, such as mass, charge radius, etc., to good accuracy (see, e.g., Refs.

In the past few years, a new approach has been proposed to study parton physics from lattice QCD, which is now known as the large-momentum effective theory (LMET)

Renormalization of nonlocal operators in QCD has not been well studied in the literature. A distinct feature of the LMET nonlocal operators is the appearance of Wilson lines connecting the parton fields at different spacetime points to ensure gauge invariance. The renormalization of an open Wilson line as well as a closed Wilson loop was studied decades ago

In this Letter, we perform a systematic study of the renormalization of gauge-invariant nonlocal operators for the quasi-PDF. We first present a general framework by introducing an auxiliary “heavy-quark” field into the QCD Lagrangian, where the Wilson line can be replaced by a product of the auxiliary fields. In analogy to heavy-quark effective theory (HQET), we argue that this theory is renormalizable to all orders in perturbation theory. By taking the unpolarized quark quasi-PDF as an example, we then show that its renormalization reduces to that of two heavy-light quark currents, which can also be done to all orders in perturbation theory. This removes the obstacle to define a continuous quasi-PDF through lattice simulations. The same conclusion is not limited to the quark quasi-PDF, but can also be generalized to the gluon ones, as well as other nonlocal correlators, such as the Euclidean observables defined in the LMET framework to extract the generalized parton distributions, transverse-momentum-dependent distributions, hadron distribution amplitudes, etc.

To be concrete, consider the following nonlocal operator:

To study the renormalization property of the nonlocal operator in Eq.

In the above theory, we can replace the bilocal operator

Let us consider renormalization of the above effective theory with a heavy quark. We will show that such a theory can be renormalized perturbatively to all orders in perturbation theory, first in dimensional regularization where power divergences do not exist. The basic argument is the same as the all-order proof of the renormalization for HQET, first presented in Ref.

The standard proof of the renormalization of QCD chooses the covariant gauge

In HQET where the Lagrangian has a similar form as Eq.

Note that the above arguments are valid independent of whether the heavy-quark field in the HQET Lagrangian is defined by a timelike or spacelike vector. Therefore, the statement about renormalizability of the HQET in Ref.

Local composite operators in this effective theory can be renormalized using the standard approach. In particular, the heavy-light composite operators, such as

Now let us consider the renormalization of the nonlocal operator, such as in Eq.

Thus, the renormalized operator becomes

The HQET Lagrangian in Eq.

However, in UV cutoff regularizations such as the lattice regularization, when going beyond leading-order perturbation theory, the self-energy of the heavy quark introduces a linear divergence which has to be absorbed into an effective mass counterterm,

Thus, the total heavy-quark Lagrangian now becomes

It is worthwhile to point out that the explicit one-loop calculation

The renormalization presented above works the same when

In this Letter, we have shown the nonlocal operator involving a spacelike Wilson line calculated in LMET can be renormalized multiplicatively to all orders in perturbation theory. Apart from the mass counterterm which yields a

This work was partially supported by the U.S. Department of Energy Office of Science, Office of Nuclear Physics under Awards No. DE-FG02-93ER-40762 and No. DE-SC0011090, a grant from the National Science Foundation of China (No. 11405104), and the SFB/TRR-55 grant “Hadron Physics from Lattice QCD.” The work of X. J. and Y. Z. was also supported in part by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, within the framework of the TMD Topical Collaboration.

Recently, we learned that a similar consideration has been made by the DESY group