^{1}

^{,*}

^{2,3}

^{,†}

^{1,4,5}

^{,‡}

^{6}

^{,§}

Corresponding author.

Corresponding author.

Corresponding author.

Corresponding author.

^{3}.

We complete the procedure of extracting parton distribution functions (PDFs) using large momentum effective theory at leading power accuracy in the hadron momentum. We derive a general factorization formula for the quasi-PDFs in the presence of mixing and give the corresponding hard matching kernel at

Understanding the internal structure of hadrons from quarks and gluons—the fundamental degrees of freedom of the QCD Lagrangian—has been a key goal in hadron physics. However, this is profoundly difficult because it requires solving QCD at large distance scales and thus at strong coupling. In high-energy collisions, the hadron and/or the probe moves nearly at the speed of light; the hadron structure greatly simplifies and can be characterized by certain parton observables such as the parton distribution functions (PDFs), distribution amplitudes (DAs), etc., The parton observables are defined as the expectation value of light-cone correlations in the hadron state and therefore cannot be readily computed on a Euclidean lattice. Currently, the most widely used approach to determine them is to assume a smoothly parametrized form and fit the unknown parameters to a large variety of experimental data (for a recent review, see e.g., Ref.

In the past few years, a breakthrough has been made to circumvent the above difficulty, which has now been formulated as large momentum effective theory (LaMET)

Since LaMET was proposed, much progress has been achieved both in the theoretical understanding of the formalism

So far, the lattice calculations of PDFs have been focused on the isovector quark PDFs only, which do not involve mixing with gluon PDFs and therefore are the easiest to calculate. In the past few years, there has been increasing interest in calculating flavor-singlet quark PDFs and gluon PDFs from lattice QCD. Such calculations are possible only if the renormalization and mixing pattern of gluon quasi-PDFs are fully understood. The ultraviolet (UV) structure of gluon quasi-PDFs was first studied in Refs.

In this paper, we provide all necessary inputs for extracting both the flavor-singlet quark PDF and the gluon PDF from lattice QCD, thereby completing the procedure of calculating PDFs using LaMET at leading power accuracy in the hadron momentum. We explain how to nonperturbatively renormalize the quark and gluon quasi-PDFs, and derive a general factorization formula for the renormalized quasi-PDFs in the presence of mixing, following the operator product expansion (OPE) method in Refs.

The rest of the paper is organized as follows: In Sec.

In this section, we give a brief review of the renormalization and factorization of quark and gluon quasi-PDFs in LaMET.

In high-energy collisions, the PDFs are defined as the hadron matrix elements of quark and gluon nonlocal correlators along the light cone. For example, the unpolarized quark distribution is defined as

Analogously, the unpolarized gluon distribution can be defined as

The quark and gluon PDFs defined above cannot be directly computed on the lattice due to their real-time dependence. However, according to LaMET, they can be extracted from lattice calculations of appropriately constructed quasi-PDFs via a factorization procedure. For the unpolarized quark PDF, a well-suited quasi-PDF candidate is given by

In comparison with the quark case, the most appropriate operator to define the gluon quasi-PDF is less obvious. In principle, one can use

All of the above gluon quasi-PDF operators are defined in terms of an adjoint gauge link. Alternatively, these operators can also be parametrized using gauge links in the fundamental representation

In the forthcoming subsections, we briefly review the renormalization of quasi-PDFs in the auxiliary field approach, following our earlier work in Refs.

In the auxiliary field approach

The effective Lagrangian with an auxiliary fundamental heavy quark field (denoted as

As shown in Ref.

From the discussions above, one can see that the Wilson line

In dimensional regularization (DR), the local operators

In lattice regularization, when going beyond leading-order perturbation theory, the self-energy of the heavy quark generates a linear divergence that does not show up in DR. Such a linear divergence can be absorbed into an effective mass counterterm,

Including the effective mass term Eq.

For the nonlocal gluon quasi-PDF operators, the desired auxiliary Lagrangian has exactly the same form as that for the quark, except that now the auxiliary heavy quark and the covariant derivative are defined in the adjoint representation. To distinguish from the fundamental auxiliary field used in the previous subsection, we denote the adjoint field as

With the auxiliary

The operator

The renormalization of the above three types of composite operators then takes the following form:

To extract the UV divergences, in particular, the genuine power divergences inherited from the operator

The one-loop diagrams that give rise to linearly divergent contributions to the operator

One-loop corrections with linear divergences to the

Based on the renormalization analysis above, one can derive useful building blocks for the construction of appropriate gluon quasi-PDFs. To this end, we may use one of the indices in

In a cutoff scheme like the lattice regularization, the mass term of the

With

Actually, the operators

From the discussions above, it is clear that the nonlocal operators at different

The quark and gluon quasi-PDFs can, in general, mix with each other under renormalization. In Ref.

The renormalization factors in the above mixing matrix can be determined using the following renormalization conditions:

Defining the inverse of the renormalization matrix in Eq.

Denoting the hadron matrix element of

In Ref.

The renormalized quark and gluon nonlocal operator matrix elements can be expanded in terms of gauge-invariant local operator matrix elements to the leading-twist approximation as

Let us first consider

Now let us turn to

Restoring all renormalization scales, the general factorization of the quark and gluon quasi-PDFs reads

As shown in the previous section, when the hadron momentum

Let us start with the gluon matrix element of the gluon quasi-PDF operator, which is the most complicated among all calculations. At tree level one finds

At one-loop level, the partonic quasi-PDF can be written as follows:

Before we proceed, a few general remarks on the calculation below are in order.

The above equations apply to bare operator matrix elements. One can write down similar equations for the renormalized ones. In our calculation of the matching coefficients, the PDF is renormalized in the

The off-shell gluon matrix elements of gauge-invariant operators can mix with those of gauge-variant operators. To illustrate this point, it is worthwhile to consider the UV divergence from the off-shell gluon matrix element of the local gluon operator

In pure Yang-Mills theory,

In general, one should be cautious about off-shell gluons, as calculating the matrix element of the gluon energy momentum tensor in off-shell gluon states and then taking the on-shell limit is rather tricky due to the existence of IR divergences

Now we present the one-loop results for the partonic quasi-PDF and PDF. The calculation is carried out in Landau gauge, and the steps are similar to those presented in Refs.

One-loop diagrams for the gluon quasi-PDF. The gluon self-energy diagrams are not shown.

The renormalized light-cone PDF can be calculated analogously, and it gives

The one-loop matching coefficient is given by the difference in the renormalized quasi-PDF and light-cone PDF,

This case has already been considered at one-loop level in Ref.

One-loop diagrams for the quark quasi-PDF. The quark self-energy diagrams are not shown.

Owing to the off-shellness of the external quark, the one-loop quark quasi-PDF contains two more Dirac structures apart from the tree-level one

In Ref.

The matching coefficient can then be extracted as

Now we turn to the mixing contributions. Let us first consider the quark matrix element of the gluon quasi-PDF operator, whose one-loop diagram is given in Fig.

One-loop diagram for the quark matrix element of the gluon quasi-PDF operator.

To illustrate the kinematic dependence of the mixing terms, it is useful to begin with the one-loop quark matrix element of the local operator

The renormalized mixing contribution from the light-cone gluon PDF has the following form:

For the quasi-PDF, we follow the decomposition as in the quark case:

As before, the matching coefficient can be extracted as

Now let us consider the gluon matrix element of the quark quasi-PDF operator, and the one-loop diagram is shown in Fig.

One-loop diagram for the gluon matrix element of the quark quasi-PDF operator.

For the light-cone PDF, the result of the mixing diagram in Fig.

The matching coefficient is then given by

Now we turn to the polarized case. The calculation can be done in complete analogy with that presented in the previous section. As demonstrated in Ref.

As before, we decompose the polarized quasi-PDF as

The local matrix elements have the following UV divergence structure:

The light-cone PDF yields the following real contribution:

The virtual contribution is the same as the unpolarized case, whereas the real contribution differs in the asymptotic limit

The matching kernel can be written using the matching kernel for the unpolarized gluon quasi-PDF as

For completeness, we also give the result for the polarized quark quasi-PDF and PDF defined as follows:

For the quasi-PDF, the one-loop result can be decomposed as

The matching coefficient can then be extracted as

The matrix element of the local gluon operator between the polarized quark states reads

The light-cone result for the polarized PDF is given as

The matching coefficient can be extracted as

In this case, the light-cone result is

The matching coefficient is then given by

In this paper, we have studied how to extract the flavor-singlet quark PDF and the gluon PDF from LaMET, both in the unpolarized and in the polarized case. After briefly reviewing the auxiliary heavy quark formalism used in our earlier work to prove the multiplicative renormalizability of quark and gluon quasi-PDF operators, we explained how a nonperturbative RI/MOM renormalization can be carried out for the quark and gluon quasi-PDFs on the lattice in the presence of mixing. Using OPE, we also derived the factorization formulas that connect them to the usual quark and gluon PDFs in the

It is interesting to note that the matrix elements of those nonfavorable gluon quasi-PDF operators have nontrivial momentum dependence in their asymptotic behavior at large

We thank Vladimir Braun, Jiunn-Wei Chen, Xiangdong Ji, Yi-Zhuang Liu, Yu-Sheng Liu, Jian-Wei Qiu, Anatoly Radyushkin, Andreas Schäfer, Yi-Bo Yang, Feng Yuan, and Yong Zhao for helpful discussions. This work is supported in part by National Natural Science Foundation of China under Grants No. 11575110, No. 11735010, No. 11705092, and No. 11911530088; by Natural Science Foundation of Shanghai under Grant No. 15DZ2272100; by Natural Science Foundation of Jiangsu under Grant No. BK20171471; and by the Sonderforschungsbereich/Transregio-55 grant “Hadron Physics from Lattice QCD.” The work of S. Z. is also supported by Jefferson Science Associates, LLC under U.S. DOE Contract No. DE-AC05-06OR23177 and by U.S. DOE Grant No. DE-FG02-97ER41028.

In this appendix, we present the results for the one-loop matrix elements of all gluon quasi-PDF operators in general

The quark matrix elements of the gluon quasi-PDF operators are given as follows: