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Corresponding author.

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Gluon parton distribution functions (PDFs) in the proton can be calculated directly on Euclidean lattices using large momentum effective theory (LaMET). To realize this goal, one has to find renormalized gluon quasi-PDFs in which power divergences and operator mixing are thoroughly understood. For the unpolarized distribution, we identify four independent quasi-PDF correlators that can be multiplicatively renormalized on the lattice. Similarly, the helicity distribution can be derived from three independent multiplicatively renormalizable quasi-PDFs. We provide a LaMET factorization formula for these renormalized quasi-PDFs from which one can extract the gluon PDFs.

Parton distribution functions (PDFs) are crucial quantities characterizing the structure of hadrons at high energy. They are defined as momentum distributions of quarks and gluons in an infinite-momentum hadron, and provide a necessary input for the description of experimental data from hadron-hadron or lepton-hadron colliders. However, calculating PDFs from first principles has been a long-standing problem in hadron physics, since PDFs are low-energy properties of a hadron defined in terms of lightcone correlations. Traditional lattice QCD methods focus on the calculation of their moments, but can only determine the first few moments due to the power divergent mixing between different moment operators

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

Lattice calculations of gluon parton physics are even more crucial because gluons play an extremely important role in experiments performed at the Large Hadron Collider (LHC) and the future Electron-Ion Collider (EIC) in the U.S. However, in contrast to the intensive work on extracting quark PDFs using LaMET, much less effort has been devoted to the study of gluon quasi-PDFs. In Refs.

In this Letter, we perform a systematic study of the renormalization of gauge-invariant nonlocal operators defining the gluon quasi-PDFs, focusing on their power divergence structure. Following an earlier work involving two of us

Let us start with the gluon quasi-PDF defined in Ref.

Alternatively, the gluon quasi-PDF can be defined using fundamental gauge links

To renormalize

With the auxiliary heavy quark

The renormalization of the effective Lagrangian Eq.

In DR, the operators allowed by BRST symmetry to mix with

In the presence of mixing, the renormalized operators can be written in a triangular mixing form

Our calculations show that no linear divergences occur in the one-loop correction to

In Eq.

In Ref.

Now, we are ready to construct an appropriate gluon quasi-PDF operator. To this end, we need to identify one of the indices in

From the discussions above, we identify the following building blocks,

In the large momentum limit, the operators

In the above discussion, we considered the gluon quasi-PDF only. If we insert the gluon quasi-PDF operator into a quark state, it gives rise to UV finite contributions, provided that all subdivergences have been renormalized. The reason is that the quasi-PDF is defined at spacelike separations; therefore there is no nonlocal UV divergence apart from the exponential mass renormalization. It indicates that the gluon quasi-PDF does not mix with the quark ones under renormalization in the above sense, while the mixing occurs at the factorization stage. This has been confirmed by the one-loop calculations in Ref.

Following the same procedure, we can define three operators that can be used to calculate the gluon helicity distribution

To define a renormalized gluon quasi-PDF on the lattice, one needs to determine the renormalization factors

In the RI/MOM scheme, the renormalization factor of the gluon quasi-PDF can be determined by calculating the amputated Green’s function of the corresponding operator in a far-off-shell gluon state and requiring that the counterterm cancels all loop contributions to this Green’s function at a specific gluon momentum. Schematically, one can write (a detailed investigation of the RI/MOM renormalization will be given elsewhere

After the nonperturbative renormalization, we can write down the factorization for the renormalized gluon or quark quasi-PDFs

In summary, we have performed a systematic study of the renormalization property of the gluon quasi-PDF defined in LaMET, using an auxiliary adjoint heavy quark Lagrangian. We have shown that all power divergences in the gluon quasi-PDF arise from the Wilson line self energy, and can be removed by a mass renormalization. We have also identified a set of multiplicatively renormalizable gluon quasi-PDF operators appropriate for lattice simulations. Our findings provide a theoretical basis for directly extracting the gluon PDFs from lattice simulations.

We thank V. Braun, J.-W. Chen, M. Göckeler, Y.-S. Liu, Y.-B. Yang, F. Yuan, Y. Zhao, and R.-L. Zhu for helpful discussions. This work was partially supported by the U.S. Department of Energy Office of Science, Office of Nuclear Physics under Awards No. r DE-FG02-93ER-40762 and No. DE-SC0011090 (X. J.), and, also, within the framework of the TMD Topical Collaboration (X. J. and A. S.). X. J. acknowledges support from the Alexander von Humboldt Foundation. J. H. Z. and A. S. are supported by the SFB/TRR-55 grant “Hadron Physics from Lattice QCD.” This work is also supported in part by National Natural Science Foundation of China under Grants No. 11575110, No. 11655002, No. 11735010, and by Natural Science Foundation of Shanghai under Grant No. 15DZ2272100.