Parton Distribution Function with Non-perturbative Renormalization from Lattice QCD

TL;DR

The paper develops a nonperturbative RI/MOM renormalization framework for lattice QCD quasi-PDFs to extract the isovector unpolarized PDF $q(x,\mu)$ from the spatially extended operator $O_{\gamma_z}(z)$. It computes the renormalization matrix $Z(z,p_z,a,\mu_R)$ and the renormalized quasi-PDF $\tilde{q}_R(x,P_z,\mu_R)$, then applies a one-loop RI/MOM-to-$\overline{\text{MS}}$ matching with mass corrections to obtain the physical PDF, while accounting for mixing with the scalar operator due to chiral-symmetry breaking. The study uses a MILC HISQ lattice with $a\approx0.12$ fm and $m_\pi\approx310$ MeV, employing momentum-smearing to access large $P_z$, and demonstrates that the RI/MOM renormalization can stabilize the extraction of $q(x)$, removing the unphysical dip at $x\approx0$ but introducing sizable uncertainties at large $|x|$ from the exponential growth of the renormalization factor. The work highlights both the feasibility and the challenges of achieving precise, first-principles PDFs from lattice QCD, pointing to improvements via finer lattices, larger volumes, and alternative renormalization schemes.

Abstract

We present lattice results for the isovector unpolarized parton distribution with nonperturbative RI/MOM-scheme renormalization on the lattice. In the framework of large-momentum effective field theory (LaMET), the full Bjorken-$x$ dependence of a momentum-dependent quasi-distribution is calculated on the lattice and matched to the ordinary lightcone parton distribution at one-loop order, with power corrections included. The important step of RI/MOM renormalization that connects the lattice and continuum matrix elements is detailed in this paper. A few consequences of the results are also addressed here.

Figures (6)

• Figure 1: Comparison between the renormalization constants obtained with the point source and the momentum source for $z\le 2$, taking the $p_z=6 \pi/L$ case as an example. The values are normalized by the central value of the renormalization constant at $z=0$ and the real parts are subtracted by unity for a better comparison. It is obvious that with the same configurations, the signal from the momentum source can be much better than that from the point source, while the values are consistent with each other.
• Figure 2: The renormalization constant of the quasi-PDF operator $O_{\gamma_z}(z)$ (red boxes) and the mixing with the scalar quasi-PDF operator $O_{\cal I}(z)$ (blue dots) with the momentum along the Wilson link being $6\pi/L=1.29$ GeV and $\mu_R^2=p^2=5.74\hbox{GeV}^2$. The size of the mixing coefficient is about an order of magnitude smaller than the renormalization factor in the large-$z$ region.
• Figure 3: The bare $\tilde{h}_{\gamma_z}(z,P_z,\mu_R)$ (blue) and renormalized $\tilde{h}_R(z,P_z,\mu_R)$ (red) for $P_z=4\pi/L$ (upper row) and $6\pi/L$ (lower row) with the renormalization scale $\mu_R=2.4$ GeV. The left and right panels show the real and imaginary parts, respectively.
• Figure 4: The renormalized unpolarized isovector quark distribution after one-loop matching and mass correction at the renormalization scale $\mu^2=5.76$ GeV$^2$ in the $\overline{\text{MS}}$ scheme, with $P_z=6\pi/L$ and three different RI/MOM renormalization scales $\mu_R$. Three distributions agree with each other within the statistical uncertainties, it shows the $\mu_R$ dependence is almost cancelled numerically. The negative-$x$ part is related to the antiquark distribution via $\bar{u}(x)-\bar{d}(x) = - u(-x)+ d(-x)$ for $x>0$.
• Figure 5: The renormalized unpolarized isovector quark distribution after one-loop matching and mass correction at the renormalization scale $\mu^2=5.76$ GeV$^2$ in the $\overline{\text{MS}}$ scheme. The negative-$x$ part is related to the antiquark distribution via $\bar{u}(x)-\bar{d}(x) = - u(-x)+ d(-x)$ for $x>0$.