Structure of parton quasi-distributions and their moments

TL;DR

This paper analyzes the structure of parton quasi-distributions outside the usual support region, linking their large-y behavior to the transverse momentum distribution's tail, and clarifies the relationship between quasi-PDF moments and light-cone PDFs.

Contribution

It establishes a connection between the large-|y| behavior of quasi-PDFs and the high-k_perp tail of TMDs, and clarifies the relation between their moments and those of light-cone PDFs.

Findings

Large-|y| quasi-PDFs decay as 1/|y| due to TMD tails.

Infinite moments of quasi-PDFs are consistent with finite moments of light-cone PDFs.

No contradiction exists between the divergent quasi-PDF moments and finite light-cone PDF moments.

Abstract

We discuss the structure of the parton quasi-distributions (quasi-PDFs) $Q(y, P_3)$ outside the "canonical" $-1 \leq y \leq 1$ support region of the usual parton distribution functions (PDFs). Writing the $y^n$ moments of $Q(y, P_3)$ in terms of the combined $x^{n-2l} k_\perp^{2l}$-moments of the transverse momentum distribution (TMD) ${\cal F} (x,k_\perp^2)$, we establish a connection between the large-$|y|$ behavior of $Q(y,P_3)$ and large-$k_\perp^2$ behavior of ${\cal F} (x,k_\perp^2)$. In particular, we show that the $1/k_\perp^2$ hard tail of TMDs in QCD results in a slowly decreasing $\sim 1/|y|$ behavior of quasi-PDFs for large $|y|$ that produces infinite $y^n$ moments of $Q(y,P_3)$. We also relate the $\sim 1/|y|$ terms with the $\ln z_3^2$-singulariies of the Ioffe-time pseudo-distributions $\mathfrak{M} (\nu, z_3^2)$. Converting the operator product expansion for $\mathfrak{M} (\nu, z_3^2)$ into a matching relation between the quasi-PDF $Q(y,P_3)$ and the light-cone PDF $f(x, \mu^2)$, we demonstrate that there is no contradiction between the infinite values of the $y^n$ moments of $Q(y,P_3)$ and finite values of the $x^n$ moments of $f(x, \mu^2)$.