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.