Hybrid Renormalization with Gradient Flow for Baryon Quasi-Distribution Amplitudes

TL;DR

This work tackles the lattice determination of baryon light-cone distribution amplitudes (LCDAs) by employing baryon quasi-DAs defined with gradient flow. It derives a factorization relation between flowed quasi-DAs and $\overline{\mathrm{MS}}$-renormalized LCDAs, and executes a complete one-loop calculation to obtain the finite-flow-time matching kernel ${\cal C}_q(t,z_1,z_2,\mu)$ alongside the Wilson-line linear divergence, $\delta m$. Building on these results, the authors propose a gradient-flow–based hybrid renormalization scheme that replaces self-renormalization with gradient-flow matching and uses ratio renormalization to cancel large logarithms, providing a practical path from lattice matrix elements to continuum quasi-DAs. They discuss lattice implementation details, including fixed-flow-time extrapolations and scale choices to minimize logarithms, and outline how this framework can reduce renormalization uncertainties in baryon LCDA extractions. The approach offers a systematic, controlled method to connect lattice QCD calculations of baryon quasi-DAs to continuum observables with improved numerical stability and perturbative control.

Abstract

We establish a factorization relation between baryon quasi-distribution amplitudes (quasi-DAs) defined with gradient flow and their counterparts renormalized in the $\overline{MS}\,$ scheme. Working beyond the small flow-time limit, we perform a complete one-loop calculation that yields the full matching kernel and the associated Wilson-line linear divergence for flowed quasi-DAs. Building on this relation, we formulate a hybrid renormalization scheme that combines ratio scheme with gradient-flow matching. The framework enables a systematic connection between lattice QCD matrix elements and continuum baryon quasi-DAs, with reduced renormalization uncertainties and clear guidance for practical lattice analyses.

Figures (3)

• Figure 1: One-loop contributions to the equal-time correlator $\tilde{\Phi}^{R}(z_1, z_2, P^z, t)$. The double line represents the Wilson line, and the filled squares mark the flowed gauge fields $B_\mu$ at flow time $t$.
• Figure 2: (a) Variation of the matching kernel with the flow time $t$ for fixed parameters $\bar{z}_1 = 1$, $\bar{z}_2 = 2$, and $\mu = 2~\mathrm{GeV}$. The blue curve shows the full matching kernel, while the red curve indicates its small flow-time limit. (b) $\dfrac{C_q(z_1, z_2, \mu, t) - 1}{\lim_{t \to 0} C_q(z_1, z_2, \mu, t) - 1}$, showing the ratio of the one-loop correction of the full matching kernel to its small flow-time limit, with $\bar{z}_2 = 2$ and $t = 0.03~\mathrm{fm}^2$. The renormalization scale $\mu$ is chosen such that $\log\bigl(2\mathrm{e}^{2\gamma_E} \mu^2 t\bigr) = 0$.
• Figure 3: Schematic diagram of the renormalization procedure. First, both large-momentum and zero-momentum matrix elements are converted to the $\overline{\mathrm{MS}}$ scheme using the gradient-flow formalism. Then, ratios are formed using these $\overline{\mathrm{MS}}$-renormalized matrix elements: the large-momentum matrix element in the blue (hard) region is divided by the zero-momentum element in the same region; the large-momentum matrix element in the yellow (hard-soft) region is divided by the zero-momentum element on the blue-yellow boundary; and the large-momentum matrix element in the gray (soft) region is divided by the zero-momentum element at the gray-yellow intersection.