On the role of isospin violation in the pion-nucleon $\sigma$-term

TL;DR

This paper examines how isospin violation affects the pion-nucleon sigma term, resolving discrepancies between phenomenological and lattice QCD results by applying a correction based on pion mass conventions.

Contribution

It identifies and quantifies the impact of inconsistent conventions in defining the isospin limit, providing a correction that reconciles phenomenological and lattice QCD determinations of the sigma term.

Findings

The correction reduces the tension between phenomenological and lattice QCD results.

Updated sigma term value from pionic atoms: 59.0(3.5) MeV.

The correction amounts to approximately 3.1 MeV.

Abstract

In recent years, a persistent tension between phenomenological and lattice QCD determinations of the pion-nucleon $\sigma$-term $\sigma_{\pi N}$ has developed. In particular, lattice-QCD calculations have matured to the point that isospin-violating effects need to be included. Here, we point out that the standard conventions adopted in both fields are incompatible, with the data-driven extraction based on the charged-pion mass, but lattice-QCD conventions relying on the mass of the neutral pion to define the isospin limit. The corresponding correction amounts to $\Delta \sigma_{\pi N}=3.1(5)\,\text{MeV}$ when evaluated in chiral perturbation theory with low-energy constants determined from a Roy-Steiner analysis of pion-nucleon scattering as well as $\sigma_{\pi N}$ itself. It reduces the tension with lattice QCD, and should be included in the comparison to phenomenological determinations. We also update the extraction from pionic atoms accounting for the latest measurement of the width of pionic hydrogen, $\sigma_{\pi N}=59.0(3.5)\,\text{MeV}$, and provide the corresponding set of scalar couplings of the nucleon.