Quasi-PDFs, momentum distributions and pseudo-PDFs

TL;DR

The work reframes quasi-PDFs as hybrids of light-front PDFs with primordial rest-frame parton momenta, explaining their slow convergence and guiding toward an alternative pseudo-PDF framework tied to Ioffe-time distributions. By clarifying how quasi-PDFs relate to TMDs through Lorentz invariance and by isolating the $z_3^2$-dependent evolution, the authors propose using pseudo-PDFs to maintain the correct $ν$-dependence and avoid lattice gauge-link renormalization issues. They present factorized models and LO evolution for reduced Ioffe-time distributions, and outline a lattice strategy that leverages ${\mathfrak M}(ν,z_3^2)$ to extract PDFs with less dependence on large momenta. Early lattice results support the approach, showing suppressed $z_3^2$ effects in the reduced distributions and a Gaussian behavior that cancels renormalization factors in the ratio.

Abstract

We show that quasi-PDFs may be treated as hybrids of PDFs and primordial rest-frame momentum distributions of partons. This results in a complicated convolution nature of quasi-PDFs that necessitates using large $p_3 \sim 3$ GeV momenta to get reasonably close to the PDF limit. As an alternative approach, we propose to use pseudo-PDFs $P(x, z_3^2)$ that generalize the light-front PDFs onto spacelike intervals and are related to Ioffe-time distributions $M (ν, z_3^2)$, the functions of the Ioffe time $ν= p_3 z_3$ and the distance parameter $z_3^2$ with respect to which it displays perturbative evolution for small $z_3$. In this form, one may divide out the $z_3^2$ dependence coming from the primordial rest-frame distribution and from the problematic factor due to lattice renormalization of the gauge link. The $ν$-dependence remains intact and determines the shape of PDFs.

Figures (3)

• Figure 1: Momentum distributions $R(k,P)$ in the factorized Gaussian model for $P/\Lambda =1,10,50$.
• Figure 2: Evolution of quasi-PDF $Q(y,P)$ in the factorized Gaussian model for $P/\Lambda =1,10,50$.
• Figure 3: Model Ioffe-time distribution ${\cal M} (\nu,0)$ and the function $B \otimes {\cal M}$ governing its evolution.