Baryon Axial Charge in a Finite Volume

TL;DR

The paper addresses how finite-volume effects in lattice QCD affect the nucleon axial charge $g_A$ and related Delta couplings. It develops the leading finite-volume corrections within heavy-baryon chiral perturbation theory with explicit Delta degrees of freedom and expresses the volume dependence through functions $F_1$–$F_4$ that weight the infrared axial couplings and the Delta-nucleon mass splitting. By analyzing the volume dependence across multiple lattice sizes, the authors show that one can extract the chiral-limit axial-vector couplings $g_A$, $g_{\Delta N}$, and $g_{\Delta\Delta}$ and even determine the chiral-multiplet mixing angle $\psi$. This approach provides a practical pathway to connect lattice results with physical axial properties, especially in the regime $m_\pi<\Delta$. Overall, the work offers a method for using finite-volume effects to constrain fundamental hadronic axial properties from lattice QCD.

Abstract

We compute finite-volume corrections to nucleon matrix elements of the axial-vector current. We show that knowledge of this finite-volume dependence --as well as that of the nucleon mass-- obtained using lattice QCD will allow a clean determination of the chiral-limit values of the nucleon and Delta-resonance axial-vector couplings.

Figures (4)

• Figure 1: One-loop graphs that contribute to the matrix elements of the axial current in the nucleon. A solid, thick-solid and dashed line denote a nucleon, a $\Delta$-resonance, and a pion, respectively. The solid-squares denote an axial coupling given in eq.(\ref{['eq:intsQCD']}), while the crossed circle denotes an insertion of the axial-vector current operator. Diagrams (a) to (e) are vertex corrections, while diagrams (f) and (g) give rise to wavefunction renormalization.
• Figure 2: Plot of ${\bf F_1}$ and the ratios ${\bf F_2}/{\bf F_1}$, ${\bf F_3}/{\bf F_1}$ and ${\bf F_4}/{\bf F_1}$ vs. $L$. The solid and dashed lines correspond to $m_\pi=139~{\rm MeV}$ and $300~{\rm MeV}$, respectively, for the physical $\Delta$-nucleon mass splitting, $\Delta=293~{\rm MeV}$.
• Figure 3: The volume dependence of $g_A$ for chiral-multiplet mixing-angles $\psi=\pi/4$ and $\psi=0$. The left panel shows $\delta g_A$ vs. $L$ with $\psi=\pi/4$, where the solid, dotted and dashed lines correspond to $m_\pi=139~{\rm MeV}$, $200~{\rm MeV}$ and $300~{\rm MeV}$, respectively. The right panel shows $\delta g_A$ vs. $L$ with $\psi=0$ (spin-flavor $SU(4)$ values of axial-vector couplings). The physical $\Delta$-nucleon mass splitting, $\Delta=293~{\rm MeV}$, is used for both panels.
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