Matching the Quasi Parton Distribution in a Momentum Subtraction Scheme

TL;DR

The paper addresses extracting collinear PDFs from lattice QCD by using quasi-PDFs. It develops a nonperturbative RI/MOM renormalization for the quasi-PDF, then performs a one-loop perturbative matching to the $ ext{MS}$ PDF, yielding the coefficient $C^{ m OM}$. The analysis shows IR-regulator independence, improved UV behavior compared to transverse-cutoff schemes, and good numerical agreement after matching, indicating enhanced accuracy for lattice-to-PDF conversions. The results support RI/MOM as a favorable path toward precise, first-principles determinations of parton distributions from lattice QCD, with guidance for gauge choices and future higher-order work.

Abstract

The quasi parton distribution is a spatial correlation of quarks or gluons along the $z$ direction in a moving nucleon which enables direct lattice calculations of parton distribution functions. It can be defined with a nonperturbative renormalization in a regularization independent momentum subtraction scheme (RI/MOM), which can then be perturbatively related to the collinear parton distribution in the $\overline{\text{MS}}$ scheme. Here we carry out a direct matching from the RI/MOM scheme for the quasi-PDF to the $\overline{\text{MS}}$ PDF, determining the non-singlet quark matching coefficient at next-to-leading order in perturbation theory. We find that the RI/MOM matching coefficient is insensitive to the ultraviolet region of convolution integral, exhibits improved perturbative convergence when converting between the quasi-PDF and PDF, and is consistent with a quasi-PDF that vanishes in the unphysical region as the proton momentum $P^z\to \infty$, unlike other schemes. This direct approach therefore has the potential to improve the accuracy for converting quasi-distribution lattice calculations to collinear distributions.

Figures (7)

• Figure 1: One-loop Feynman diagrams for the quasi-PDF. The standard quark self energy wavefunction renormalization is also included, and denoted $\tilde{q}_{\rm w.fn.}^{(1)}(z)$.
• Figure 2: Comparison between the PDF $xf_{u-d}$ and the quasi-PDF result obtained from $x(C^{\text{OM}}\otimes f_{u-d})$ in Feynman gauge. The orange and blue bands indicate the results from varying the factorization scale $\mu$ by a factor of two. Left: $x(C^{\text{OM}}\otimes f_{u-d})$ and $xf_{u-d}$. Right: differences when taking $x(C^{\text{OM}}\otimes f_{u-d})$ or $xf_{u-d}$, and subtracting $xf_{u-d}(x,3\text{ GeV})$.
• Figure 3: Comparison between the PDF $xf_{u-d}$ and the quasi-PDF obtained from $x(C^{\text{OM}}\otimes f_{u-d})$ in the Landau gauge. The orange, blue, and green bands indicate the results from varying the factorization scale $\mu$ by a factor of two.
• Figure 4: Left panel: Comparison between the PDF $xf_{u-d}$ and the quasi-PDF from $x(C^{\text{OM}}\otimes f_{u-d})$ determined at different $\mu_R$s. Right panel: The $p_R^z$ dependence of the quasi-PDF $x(C^{\text{OM}}\otimes f_{u-d})$, compared to the PDF $xf_{u-d}$ which is independent of $p_R^z$. In both panels the blue band indicates the $\mu$ renormalization scale dependence of the PDF from variation by a factor of two.
• Figure 5: Comparison between the PDF $xf_{u-d}$ and the quasi-PDF from $x(C^{\text{OM}}\otimes f_{u-d})$ determined at different $P^z$s. The blue band indicates the $\mu$ renormalization scale dependence of the PDF from variation by a factor of two.