Chiral perturbation theory for the Wilson lattice action

TL;DR

This work develops Wilson chiral perturbation theory (WχPT) by incorporating linear lattice-spacing effects $a$ into the chiral Lagrangian, yielding a double expansion in the light-quark mass $m_q$ and $a$. The effective theory uses spurion methods with $\chi=2B_0 m_q$ and $\rho=2W_0 a c_{SW}$, so the LO Lagrangian emerges from the usual χPT by $\chi\to\chi+\rho$, and the NLO Lagrangian contains both Gasser–Leutwyler and Wilson-specific low-energy constants $L_i$ and $W_i$. They compute flavor-non-singlet meson masses and decay constants to ${\mathcal{O}}(a)$ and ${\mathcal{O}}(m_q^2)$, including loop-induced logs that couple $a$ and $m_q$, with a simple relation to ordinary χPT via the substitution $\chi\to\chi+\rho$. The framework also addresses ${\mathcal{O}}(a^2)$ effects, improvement schemes, and the partially quenched extension, providing a practical path to extracting continuum GL coefficients from finite-$a$ lattice data and guiding improved lattice simulations.

Abstract

We extend chiral perturbation theory to include linear dependence on the lattice spacing $a$ for the Wilson action. The perturbation theory is written as a double expansion in the small quark mass $m_q$ and lattice spacing $a$. We present formulae for the mass and decay constant of a flavor-non-singlet meson in this scheme to order $a$ and $m_q^2$. The extension to the partially quenched theory is also described.