# Spectral functions of nucleon form factors: Three-pion continua at low energies

## TL;DR

This paper investigates the spectral functions of nucleon form factors near the 3-pion threshold using covariant baryon chiral perturbation theory, highlighting the limitations of heavy baryon approaches and the influence of resonances.

## Contribution

It provides a detailed two-loop calculation of the imaginary parts of nucleon form factors near the 3-pion threshold, incorporating resonance effects and comparing covariant and heavy baryon methods.

## Key findings

- Heavy baryon approximation overestimates 3-pion continua.
- Chiral 3π-dynamics dominate near threshold in certain windows.
- Radiative corrections from π0γ intermediate states are significant.

## Abstract

We study the imaginary parts of the isoscalar electromagnetic and isovector axial form factors of the nucleon close to the $3\pi$-threshold in covariant baryon chiral perturbation theory. At the two-loop level, the contributions arising from leading and next-to-leading order chiral $\pi N$-vertices, as well as pion-induced excitations of virtual $\Delta(1232)$-isobars, are calculated. It is found that the heavy baryon treatment overestimates substantially these $3\pi$-continua. From a phenomenological analysis, that includes the narrow $\omega(783)$-resonance or the broad $a_1$-resonance, one can recognize small windows near threshold, where chiral $3\pi$-dynamics prevails. However, in the case of the isoscalar electromagnetic form factors $G_{E,M}^s(t)$, the radiative correction provided by the $\pi^0\gamma$-intermediate state turns out to be of similar size.