NLO and NNLO Low Energy Constants for $SU(2)$ Chiral Perturbation Theory

TL;DR

This paper determines low energy constants for $SU(2)$ chiral perturbation theory at NNLO using lattice QCD data, enabling precise predictions of pion properties and scattering lengths within a specific mass range.

Contribution

It provides the first comprehensive determination of NNLO low energy constants for $SU(2)$ chiral perturbation theory from lattice data, demonstrating the expansion's robustness.

Findings

NNLO low energy constants are accurately determined from lattice data.

The $SU(2)$ expansion fits lattice data with percent-level accuracy.

Predictions for pion mass splitting and scattering lengths are made.

Abstract

We have performed global fits of $f_{\pi}$ and $m_{\pi}$, from a variety of RBC-UKQCD domain wall fermion ensembles, to $SU(2)$ partially quenched chiral perturbation theory at NNLO. We report values for 9 NLO and 8 linearly independent combinations of NNLO partially quenched low energy constants, which we compare to other lattice and phenomenological determinations. We discuss the convergence of the expansion and use our large set of low energy constants to make predictions for the pion mass splitting due to QCD isospin breaking effects and the s-wave $\pi \pi$ scattering lengths. We conclude that, for the range of pseudoscalar masses explored in this work, $115~\mathrm{MeV} \lesssim m_{\rm PS} \lesssim 430~\mathrm{MeV}$, the NNLO $SU(2)$ expansion is quite robust and can fit lattice data with percent-scale accuracy.