Gauge-invariant theory of two-pion photo- and electro-production off the   nucleon
  A field-theoretical description of the photoproduction of two pions off the nucleon is presented that applies to real as well as virtual photons in the one-photon approximation. The Lorentz-covariant theory is complete at the level of all explicit Faddeev-type three-body final-state mechanisms of dressed interacting hadrons, including those of the nonlinear Dyson-Schwinger type. All electromagnetic currents are constructed to satisfy their respective (generalized) Ward-Takahashi identities and thus satisfy local gauge invariance as a matter of course. The Faddeev-type ordering structure results in a natural expansion of the full two-pion photoproduction current $\Mpp^\mu$ in terms of multiple loops that preserve gauge invariance order by order in the number of loops, which in turn lends itself naturally to practical applications of increasing sophistication with increasing number of loops. 
Helmut Haberzettl, Kanzo Nakayama, and Yongseok Oh
nucl-th hep-ph