S-wave pion-pion scattering lengths from nucleon-meson fluctuations

TL;DR

This paper calculates S-wave pion-pion scattering lengths using a nucleon-meson model with parity doubling, employing the functional renormalization group to include loop corrections, and finds good agreement with experimental data.

Contribution

It introduces a novel approach combining nucleon-meson models with FRG to accurately compute pion scattering lengths and dynamically generate the sigma mass.

Findings

Agreement with experimental scattering lengths

Dynamically generated sigma mass around 500 MeV

Chiral symmetry breaking scale near 4πf_π

Abstract

We present calculations of the $S$-wave isospin-zero and isospin-two pion-pion scattering lengths within a nucleon-meson model with parity doubling. Both scattering lengths are computed in various approximations, ranging from a mean-field (MF) calculation towards the inclusion of loop corrections by means of the functional renormalization group (FRG). The bosonic part of the investigated nucleon-meson model is formulated in terms of stereographic projections as a "natural" set of coordinates on the respective vacuum manifold. We thereby elucidate subtleties concerning the truncation of the effective action w.r.t. higher-derivative pion interactions and the "successful" computation of the scattering lengths. As the main result, we find simultaneous agreement for the isospin-zero and isospin-two scattering lengths with experimental data within the $\mathrm{LPA}^{\prime}$-truncation of the FRG, together with chiral symmetry breaking (roughly) occurring at the characteristic scale of $4\pi f_{\pi}$. The isoscalar $\sigma$-mass is dynamically generated by the FRG integration of momentum modes, and is a prediction of the model. It ends being of the order of $500\ \mathrm{MeV}$, i.e., much lower than the value ($> 1\ \mathrm{GeV}$) found in MF or one-loop treatment of this or related models. Finally, the convergence of the corresponding low-energy expansion of the quantum effective action in terms of pion momenta is discussed.