The Mesonic Chiral Lagrangian of Order $p^6$

TL;DR

The paper delivers the most general mesonic chiral Lagrangian at order $O(p^6)$ in CHPT with external fields, rigorously establishing a minimal, symmetry-consistent basis by using partial integration, equations of motion, Bianchi identities, and Cayley–Hamilton relations. It proves the equivalence of EOM and field redefinitions for removing spurious terms and provides detailed reductions from a general $SU(n)$ basis to smaller, phenomenologically relevant cases: $n=3$ with 90 invariants plus 4 contact terms and $n=2$ with 53 invariants plus 4 contact terms. A guided tour highlights that only a subset of terms are needed for common processes, exemplified by a 16-term sector in the $n=2$ limit relevant to four-pion and current interactions. The work establishes a framework essential for two-loop renormalization and for connecting low-energy constants across processes through theoretical input beyond symmetry alone.

Abstract

We construct the effective chiral Lagrangian for chiral perturbation theory in the mesonic even-intrinsic-parity sector at order $p^6$. The Lagrangian contains 112 in principle measurable + 3 contact terms for the general case of $n$ light flavours, 90+4 for three and 53+4 for two flavours. The equivalence between equations of motion and field redefinitions to remove spurious terms in the Lagrangians is shown to all orders in the chiral expansion. We also discuss and implement other methods for reducing the number of terms to a minimal set.