Chiral perturbation theory at non-zero lattice spacing

TL;DR

This review presents a systematic framework for chiral perturbation theory applied to lattice QCD at non-zero lattice spacing, built via a two-step matching: (1) Symanzik’s effective theory organizes discretization effects in powers of $a$, and (2) a chiral Lagrangian with spurion-induced symmetry breaking yields a lattice-aware expansion. It covers Wilson, twisted-mass, partially quenched, mixed-fermion, and staggered fermions, including heavy-light extensions, and highlights how lattice spacing modifies chiral logs, mass relations, and phase structure (e.g., the Aoki phase) while detailing specific one-loop results and density of low-energy constants. The review emphasizes practical implications for lattice data analysis: lattice artifacts can significantly alter extrapolations, and data fits must employ lattice-aware χPT forms (often beyond continuum χPT), with mixed results across actions (e.g., Wilson vs. staggered) and quark masses. Overall, χPT at non-zero lattice spacing provides essential analytical control over discretization effects, guiding reliable extrapolations and error budgeting in lattice QCD simulations.

Abstract

A review of chiral perturbation theory for lattice QCD at non-zero lattice spacing is given.

Figures (2)

• Figure 1: Fit result for $M^{2}_{\pi}/2m_{\rm AWI}$ using resummed Wilson $\chi$PT. From Ref. Namekawa:2004bi.
• Figure 2: Data for $M_{\pi}^{2}/2m_{\rm AWI}$ as a function of $m_{\rm AWI}$, normalized by the values at the heaviest quark mass (see text). From Ref. Farchioni:2004tv.