Gluon Quasi-Parton-Distribution Functions from Lattice QCD

We present the first attempt to access the $x$ dependence of the gluon unpolarized parton-distribution function (PDF), based on lattice simulations using the large-momentum effective theory approach. The lattice calculation is carried out with pion masses of 340 and 678 MeV on a ($2+1$)-flavor domain-wall fermion configuration with lattice spacing $a=0.111\mathrm{fm}$, 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 $\mathcal{O}({\alpha }_s)$ matching and the mixing from the quark PDFs.

Figure 1

The ratio $R(t_{\mathrm{sep}},t)$ for ${\tilde{H}}_0(0,0)$ at different $t_{\mathrm{sep}}$ as a function of operator insertion time $t$ (left panel), and the ratio $\tilde{R}(t_{\mathrm{sep}})$ as a function of source-sink seperation $t_{\mathrm{sep}}$ (right panel). Four colored points in the right panel corresponds to the $\tilde{R}$ at the separations plotted in the left panel.

Figure 2

The bare $\tilde{H}(z,P_z=0.46\mathrm{GeV})$ and the renormalized one ${\tilde{H}}^{\text{Ra}}$ at 2 GeV with 1,3,5 HYP smearing steps, as functions of $z$. In ${\tilde{H}}^{\text{Ra}}$, the exponential falloff in the bare $\tilde{H}$ due to the linear divergence is obviously removed by the ratio renormalization factor $Z(\mu ,z)\equiv H_0^{\overline{\mathrm{MS}}}(0,0,\mu )/{\tilde{H}}_0(z,0)$. Some data using the same HYP smearing steps are shifted horizontally to enhance the legibility.

Figure 3

The renormalized ${\tilde{H}}_{i=0,1,2,3}^{\text{Ra}}(z,P_z)$ as a functions of $P_z$ at $z=0$ (top) and 3 (bottom). Some data with the same $P_z$ are shifted horizontally to enhance the legibility. The case with ${\mathcal{O}}_{i=3}$ suffers from a large contamination from higher-twist distributions, while the results with ${\mathcal{O}}_{i=0,1,2}$ are consistent with each other, especially at larger $P_z$.

Figure 4

The final results of ${\tilde{H}}_0^{\text{Ra}}(z,P_z)$ at 678 (top) and 340 MeV (bottom) pion mass as a functions of $z{P}_z$, in comparison with the FT of the gluon PDF from the global fits CT14 [24] and PDF4LHC15 NNLO [25]. The data with $P_z=0.92\mathrm{GeV}$ with each other within the uncertainty.

Figure 5

The similar figure for the pion gluon quasi-PDF matrix elements with $M_{\pi }=678\mathrm{MeV}$. The shape is quite similar to the case in Fig. 4.