# Gluon Quasi-PDF From Lattice QCD

###### Abstract

We present the first attempt to access the -dependence of the gluon unpolarized parton distribution function (PDF), based on lattice simulations using the large-momentum effective theory (LaMET) approach. The lattice calculation is carried out with pion masses of 340 and 678 MeV on a 2+1-flavor DWF configuration with lattice spacing fm, for the gluon quasi-PDF matrix element with the nucleon momentum up to 0.93 GeV. Taking the normalization from similar matrix elements in the rest frame of the nucleon and pion, our results for these matrix elements are consistent with the Fourier transform of the global fit CT14 and PDF4LHC15 NNLO of the gluon PDF, within statistical uncertainty and the systematic one up to power corrections, perturbative matching and the mixing from the quark PDFs.

###### pacs:

12.38.Mh, 12.39.-x, 25.75.Nq^{†}

^{†}preprint: MSUHEP-18-014

## I Introduction

The unpolarized parton distribution function (PDF) is the probability density for finding the corresponding parton with a certain longitudinal momentum fraction at renormalization scale . In the leading-twist collinear factorization, PDFs are process-independent and satisfy the hadron momentum sum rules,

(1) |

where and are the unpolarized gluon and quark PDF respectively. Although contributes at next-to-leading order to the deep inelastic scattering (DIS) cross section, it enters at leading order in jet production. Top-quark pair production at the LHC can provide significant constraints to the global fit of at region Czakon et al. (2013), and small- () region of is strongly constrained by charm production at high energies Gauld et al. (2015). Thus, many fits have been done to constrain by combining data from DIS and jet-production cross sections. With more experimental data and better fit approaches, our understanding of from experiments continues to improve.

On the theoretical side, the unpolarized gluon PDF is defined by the Fourier transform (FT) of the lightcone correlation in the hadron,

(2) |

where is the spacetime coordinate along the lightcone direction, the hadron momentum , is the hadron state with momentum with the normalization , is the renormalization scale of the glue operator, is the lightcone Wilson link from to 0 with being the gauge potential in the adjoint representation, and is the gauge field tensor. Based on such a definition, all the odd moments vanish due to the parity of the glue matrix elements, while the even ones survive.

Even though the definition in Eq. I involves a Minkowski spacetime correlation and is infeasible to construct in a Euclidean lattice simulation, its second moment is calculable in Euclidean space as the matrix elements of local operators:

(3) | ||||

(4) |

where the gauge energy-momentum tensor . Note that all the definitions in Eq. (4) are frame-independent and can be calculated in a frame far from the infinite momentum. Moreover, the latter two definitions can be used to carry out the calculation in the rest frame of the hadron. Lattice calculations of in the nucleon Horsley et al. (2012); Deka et al. (2015); Alexandrou et al. (2017a, b); Yang et al. (2018) have been significantly refined in the last decade, while calculations of moments beyond the second moment are still absent.

Based on the large-momentum effective theory (LaMET) Ji (2013, 2014) approach, a proper definition of the gluon quasi-PDF inspired by the last right-hand side of Eq. (4) is

(5) |

where is the gluon quasi-PDF matrix element

(6) | ||||

renormalized at the scale with . When , is a local operator and equals to . In the large momentum limit, only the leading twist contribution in survives, and then can be factorized into the the gluon PDF and a perturbative calculable kernel , up to mixing with the quark PDF and the higher-twist corrections .

More precisely, the renormalized can be factorized into the perturbatively calculable matching kernel and the lightcone PDF through the factorization theorem Ma and Qiu (2018); Radyushkin (2018); Zhang et al. (2018); Izubuchi et al. (2018) as

(7) |

up to higher-twist corrections and mixing with the quark PDF at , where with . When , the above equation reduces to

Thus if the multiplicative renormalization of gluon quasi-PDF matrix element can be proven like the quark case, then the ratio is free of renormalization of the operator and also its linear divergence Wang et al. (2018); Wang and Zhao (2018) under lattice regularization, and can be connected to through

Since the is a matrix element with the local operator which can be trivially renormalized in the scheme, and it exactly equals to at the scale , so the “ratio renormalized” gluon quasi-PDF matrix element

(8) |

can also be factorized by

(9) |

with the assumption of multiplicative renormalization of .

The choice for the quasi-PDF operator is not unique. Any operator that approaches the lightcone one in the large-momentum limit is a candidate, such as the other choices inspired by Eq. (4)

(10) |

as well as

(11) |

proposed in Ref. Ji (2013). As we will address in the latter part of this work, the quasi-PDF using has larger higher-twist corrections and/or statistical uncertainty compared to that from using .

## Ii Numerical setup

The lattice calculation is carried out with valence overlap fermions on 203 configurations of the -flavor domain-wall fermion gauge ensemble “24I” Aoki et al. (2011) with , fm, and =330 MeV. For the nucleon two-point function, we calculate with the overlap fermion and loop over all timeslices with a 2-2-2 grid source and low-mode substitution Li et al. (2010); Gong et al. (2013), and set the valence-quark mass to be roughly the same as the sea and strange-quark masses (the corresponding pion masses are 340 and 678 MeV, respectively). Counting independent smeared-point sources, the statistics of the two-point functions are , where the last factor of 2 coming from the averaging between the forward and backward nucleon propagators.

On the lattice, is defined by

(12) |

where is defined in the Euclidean space with the gauge link in the fundamental representation, and the clover definition of the field tensor is the same as that used in our previous calculation of the glue momentum fraction Yang et al. (2018). The alternative operators can be defined on the lattice similarly.

The bare glue matrix element with the Wilson link length and nucleon momentum can be obtained from the derivative of the summed ratio following the recent high-precision calculation of nucleon matrix elements Berkowitz et al. (2017),

(13) |

where

(14) |

and . To further improve the signal of , we applied up to 5 steps of HYP smearing on the glue operators.

## Iii Results

As illustrated in Fig. 1 for with 5 HYP smearing steps, the value of saturates after and a constant fit can provide the same result as what can be obtained from the two-state fit of with larger . In the limit, both and saturate to the same as in the figure, while such a limit can be reached with smaller in the case. Using the renormalization constant of in at 2 GeV with 5 steps of the HYP smearing calculated in Ref. Yang et al. (2018) of 0.90(10) and ignoring mixing from the quark momentum fraction, the renormalized agrees with the phenomenological determination 0.42(2) Dulat et al. (2016) within uncertainties.

Due to its linear divergence Wang and Zhao (2018), the bare decays exponentially as increases. Fig. 2 shows the dependence of with GeV and 1, 3 and 5 HYP smearing steps. It is obvious to see that the decay rates decreases when more steps of smearing are applied, since the corresponding linear divergence becomes smaller. Note that is purely real and symmetric with respect to ; thus, we just plot the real part in the positive- region. The “ratio renormalized” matrix elements with different HYP smearing steps are consistent with each other, as shown in Fig. 2, while more HYP smearing can reduce the statistical uncertainties significantly.

Then, we plot the “ratio renormalized” using for the glue operator with 5 HYP smearing steps and , 0.46, 0.92 GeV in the top panel of Fig. 3. All the cases with provide consistent results, except which suffers from large mixing with the higher-twist operator . With larger , the value of becomes less negative as higher-twist contamination becomes smaller.

The lower panel of Fig. 3 shows with different operators and , 0.46, 0.92 GeV. The case also suffers from large higher-twist contamination like the case; the results with seem to be slightly different from each other at GeV, while the consistency at GeV is much better. Since the operators can provide consistent results but the uncertainty using is slightly smaller than the other two cases, we will concentrate on this case in the following discussion.

Finally, the coordinate-space gluon quasi-PDF matrix element ratios are plotted in Fig. 4, compared with the corresponding FT of the gluon PDF, H(z, =2 GeV), based on the global fits from CT14 Dulat et al. (2016) and PDF4LHC15 NNLO Butterworth et al. (2016). Since the uncertainties increases exponentially at larger , our present lattice data with good signals are limited to the range 2 or so, and the values at different are consistent with each other. At the same time, doesn’t changes much either in this region as in Fig. 4. Up to perturbative matching and power correction at , they should be the same, and our simulation results are within the statistical uncertainty at large . The results at the lighter pion mass (at the unitary point) of 340 MeV is also shown in Fig. 4, which is consistent with those from the strange quark mass case but with larger uncertainties. We also study the pion gluon quasi-PDF (see Fig. 5) and similar features are obsevered.

If keeps flat outside the region where we have good signal, the gluon quasi-PDF will be a delta function at through FT. On the other hand, the width of will be in if we suppose for all the 3. We conclude the FT of our present results of cannot provide any meaningful constraint on the gluon PDF .

## Iv Summary and outlook

In summary, we present the first gluon quasi-PDF result for the nucleon and pion with multiple hadron boost momenta and explore different choices of the operators. With proper renormalization, the quasi-PDF matrix elements we obtain agree with the FT of the global-fit PDF within statistical uncertainty, up to mixing from the quark PDF, perturbative matching and higher-twist correction .

Since global fitting results shows that most of the contribution of comes from the region, the width of its FT, , is pretty large as the becomes half of of its maximum value (at =0) at . At the same time, the signal of the lattice simulation and also the validity of the factorization limit us to the small region. Thus to discern the width of gluon PDF, the lattice simulation with much larger nucleon momentum , such as 2-3 GeV, is needed. To archive a good signal with such a large , the momentum smearing Bali et al. (2016) and cluster decomposition error reduction Liu et al. (2018) should be helpful.

## Acknowledgments

We thank J.W. Chen, L. Jin and J.H. Zhang for useful discussions, and the RBC and UKQCD collaborations for providing us their DWF gauge configurations. ZF, HL and YY are supported by the US National Science Foundation under grant PHY 1653405 “CAREER: Constraining Parton Distribution Functions for New-Physics Searches”. KL is partially supported by DOE grant DE-SC0013065 and DOE TMD topical collaboration. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231 through ALCC and ERCAP; facilities of the USQCD Collaboration, which are funded by the Office of Science of the U.S. Department of Energy, and supported in part by Michigan State University through computational resources provided by the Institute for Cyber-Enabled Research.

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