Lattice QCD exploration of pseudo-PDFs

TL;DR

Direct lattice access to PDFs is hampered by light-cone physics. The authors develop a pseudo-PDF framework using equal-time Ioffe-time distributions ${\cal M}(\nu,z^2)$ and form a reduced ratio ${\mathfrak M}(\nu,z^2)={\cal M}(\nu,z^2)/{\cal M}(0,z^2)$ to cancel gauge-link renormalization and extract PDFs from the $z^2\to0$ limit. They find Gaussian-like $z_3$-dependence and near universal behavior of ${\mathfrak M}(\nu,z^2)$ across momenta, consistent with factorization of $x$- and $k_\perp$-dependence, and observe perturbative evolution at small $z_3$ with $\alpha_s/\pi\approx0.1$ that moves extracted valence distributions toward global fits after evolution. The lattice study reports a nonzero antiquark content and provides a viable path toward first-principles PDFs, with clear directions for improvements via finer lattices, dynamical fermions, and closer-to-physical pion masses.

Abstract

We demonstrate a new method of extracting parton distributions from lattice calculations. The starting idea is to treat the generic equal-time matrix element ${\cal M} (Pz_3, z_3^2)$ as a function of the Ioffe time $ν= Pz_3$ and the distance $z_3$. The next step is to divide ${\cal M} (Pz_3, z_3^2)$ by the rest-frame density ${\cal M} (0, z_3^2)$. Our lattice calculation shows a linear exponential $z_3$-dependence in the rest-frame function, expected from the $Z(z_3^2)$ factor generated by the gauge link. Still, we observe that the ratio ${\cal M} (Pz_3 , z_3^2)/{\cal M} (0, z_3^2)$ has a Gaussian-type behavior with respect to $z_3$ for 6 values of $P$ used in the calculation. This means that $Z(z_3^2)$ factor was canceled in the ratio. When plotted as a function of $ν$ and $z_3$, the data are very close to $z_3$-independent functions. This phenomenon corresponds to factorization of the $x$- and $k_\perp$-dependence for the TMD ${\cal F} (x, k_\perp^2)$. For small $z_3 \leq 4a$, the residual $z_3$-dependence is explained by perturbative evolution, with $α_s/π=0.1$.

Figures (18)

• Figure 1: Evolution of quasi-PDF $Q(y,P)$ in the factorized Gaussian model for $P/\Lambda =1, 5, 10, 50$.
• Figure 2: Real part of model distribution ${\cal M} (\nu)$ and the function $-B \otimes {\rm Re} \, {\cal M}$ that governs its evolution (the minus sign here is for convenience of placing two curves on one figure).
• Figure 3: Imaginary part of model Ioffe-time distribution ${\cal M} (\nu)$ and the function $B \otimes {\rm Im} \, {\cal M}$ that governs its evolution.
• Figure 4: Nucleon dispersion relation. Energies and momenta are in lattice units. The solid line is the continuum dispersion relation (not a fit) while the errorband is an indication of the statistical error of the lattice nucleon energies.
• Figure 5: Typical fits used to extract the reduced matrix element. The upper panel corresponds to $p=2\pi/L \cdot 2$ and $z=4$ and the lower panel to $p=2\pi/L \cdot 3$ and $z=8$, where momentum and position are in lattice units.