Two-Loop Extraction of the Pion-Nucleon Sigma Term

TL;DR

This paper derives a two-loop chiral perturbation theory expression for the pion-nucleon sigma term, resolving previous discrepancies between lattice QCD and phenomenological results by including intermediate rescattering effects.

Contribution

It introduces a two-loop order calculation within relativistic baryon chiral perturbation theory to accurately extract the sigma term from lattice QCD data.

Findings

The two-loop calculation resolves the tension between lattice QCD and dispersive methods.

The final sigma term value is 55.9(2.5) MeV, consistent with Roy-Steiner analysis.

Intermediate $\pi\pi$ rescattering effects are crucial at two-loop order.

Abstract

The pion-nucleon sigma term, characterizing the mass component of Higgs origin related to $u$ and $d$ quarks inside the nucleon, is investigated within relativistic baryon chiral perturbation theory at leading two-loop order using the extended-on-mass-shell renormalization scheme. The two-loop representation of the sigma term is derived from the nucleon mass via the Feynman-Hellmann theorem and verified through a direct calculation of the forward isoscalar-scalar nucleon matrix element. We apply the derived chiral expression to extract the physical pion-nucleon sigma term by extrapolating $N_f=2+1$ lattice quantum chromodynamics (QCD) data at unphysical quark masses. We find that, at the two-loop level, the long-standing tension between lattice QCD and dispersive determinations can be naturally resolved, owing to the incorporation of intermediate $\pi\pi$ rescattering effects that begin to contribute at two-loop order. Our final result for the nucleon sigma term based on recent lattice QCD calculations is $\sigma_{\pi N}=55.9(2.5)$ MeV. It is compatible with the result of the Roy-Steiner equation analysis and thus provides a satisfactory resolution to the previous debate between lattice QCD and phenomenological determinations.