Extracting Mellin moments of double parton distributions from lattice data

TL;DR

This work tackles the challenge of reconstructing Mellin moments of double parton distributions (DPDs) from Euclidean lattice data by introducing a Fourier-conjugate skewness parameter $\zeta$ and studying its impact on moment extraction via the Ioffe-time variable $\omega$. The authors develop several physically motivated models for the $\zeta$-dependence of the lowest DPD Mellin moment, fit them to lattice data for the flavor combination $u\,d$, and assess whether the $\zeta$- and $y$-dependencies factorize. They find that while polynomial and power-law forms can describe the data, they lead to large uncertainties for $\zeta=0$ (the directly relevant moment for double parton scattering) unless additional theory priors (e.g., sum rules) are imposed, and that nonzero $\zeta$ moments remain informative about parton correlations. The study suggests that higher-precision lattice data at larger Ioffe times $|\omega|$ and a more robust treatment of endpoint behavior are needed to reliably determine the $\zeta=0$ Mellin moment, with implications for DPD phenomenology and possible extensions to quasi- or pseudo-distributions in lattice QCD.

Abstract

Reconstructing Mellin moments of double parton distributions from calculations on a Euclidean lattice requires taking an integral over a variable that may be regarded as a Ioffe time. The Fourier conjugate of this variable plays the role of a kinematic skewness in the double parton distributions. We discuss the skewness dependence of the relevant hadronic correlation functions. Using several models, we study the impact of this dependence on extracting moments of double parton distributions from existing lattice data.

Figures (13)

• Figure 1: Illustration of the skewed DPD $F_{u d}\left(x_1, x_2, \zeta, y^2\right)$ for the cases where (a) all fractions $x_i \pm \zeta / 2$ or (b) all fractions $\zeta / 2 \pm x_i$ are positive. Below the vertical dashed line we give the total momentum fraction carried by the spectator partons.
• Figure 2: Support regions of $F_{u d}\left(x_1, x_2, \zeta, y^2\right)$ in the $(x_1, x_2)$-plane for negative (left) and positive (right) values of $\zeta$. For each region we indicate the (anti-)quark content of the wave function and its complex conjugate. The notation $u | \bar{d} d u$ means that we have a $u$-quark in the proton wave function and $\bar{d} d u$ in its complex conjugate.
• Figure 3: Graphs for the perturbative splitting mechanism of DPDs at leading (a) or next-to-leading (b, c) order in $\alpha_s$. Graph (c) only contributes for $\zeta \neq 0$.
• Figure 4: Illustration of the five kinds of Wick contractions (graphs) contributing to a four-point function of a baryon. The explicit contributions for the graphs $C_1$, $C_2$ and $S_1$ depend on the quark flavor of the current insertions (red points). In the case where all quark flavors have the same mass, $C_2$ only depends on the flavors of the two propagators connected to the source or the sink. These flavors have to be the same for proton-proton matrix elements. For the graphs $S_1$, $S_2$ and $D$ we also indicate the parts connected to the proton source and sink, i.e. $G_{3\mathrm{pt}}$ and $G_{2\mathrm{pt}}$ (blue), as well as the disconnected loops $L_1$ and $L_2$ (orange). The indices $i$ and $j$ specify the currents and are both equal to $V$ in our context. [Figure taken from Ref. Bali:2021gel]
• Figure 5: Lattice data of the invariant function $A_{u d}$ for $\omega = 0$, together with the fit \ref{['y-ansatz']}. The shaded region $y \le 0.11 \operatorname{fm}$ is not used in our analysis.