Theory of two-meson photo- and electroproduction off the nucleon
Using the reaction γN → ππN as the paradigmatic topological framework, a field-theoretical description of the electromagnetic production of two pseudoscalar mesons off the nucleon is derived and applied to photo- and electroproduction processes, assuming only one-photon exchange for the latter. The dynamics of all explicit three-point interaction mechanisms of the ππN system is accounted for by the Faddeev-type ordering structure of the Alt-Grassberger-Sandhas equations. The modifications necessary for incorporating n-meson vertices for n ≥ 4 are discussed. The formulation is valid for hadronic two-point and three-point functions dressed by arbitrary (even nonlinear) internal mechanisms provided all associated electromagnetic currents are constructed to satisfy their respective (generalized) Ward-Takahashi identities. Coupling the photon to the Faddeev structure of the underlying hadronic two-pion production mechanisms results in a natural expansion of the full two-pion photoproduction current $M$$^{μ}_{ππ}$ in terms of multiple dressed loops involving two-body subsystem scattering amplitudes of the ππN system that preserves gauge invariance as a matter of course order by order in the number of (dressed) loops. A closed-form expression is presented for the entire gauge-invariant current $M$$^{μ}_{ππ}$ with complete three-body dynamics. Individually gauge-invariant truncations of the full dynamics most relevant for practical applications at the no-loop, one-loop, and two-loop levels are discussed in detail. An approximation scheme to the full two-meson amplitude for calculational purposes is also presented. It approximates, systematically, the full amplitude to any desired order of expansion in the underlying hadronic two-body amplitude. Furthermore, it allows for the approximate incorporation of all neglected higher-order mechanisms in terms of a locally gauge-invariant phenomenological remainder current.