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

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.