Closed-Fermion-Loop Calculations in Chiral Models of Electromagnetic Processes

The amplitudes involving two external $\gamma \text{'}\mathrm{s}$ and an arbitrary number of soft $\pi$ mesons emitted from a closed nucleon loop have been calculated in a model with a formally conserved (neutral) axial-vector current. The results violate the formal chiral symmetry of the underlying Lagrangian, but in a way that has the following simple group-theoretic interpretation stated in terms of an effective Lagrangian ${\mathcal{L}}_{\mathrm{eff}}$: Instead of finding $\delta {\mathcal{L}}_{\mathrm{eff}}=0\mathrm{}$ under those chiral variations which leave the electromagnetic couplings invariant, we find $\delta {\mathcal{L}}_{\mathrm{eff}}$ to be independent of the chirally transforming fields. If the group-theoretic principle is abstracted from the detailed calculation it leads to predictions for the processes $\gamma +\gamma \to \left(\mathrm{an}\mathrm{odd}\mathrm{number}\mathrm{of}\mathrm{pions}\right)$ which should hold true in a wide class of chiral models. The generalization is given to a quark-loop calculation in a theory invariant under the neutral transformations from $\mathrm{SU}\left(3\right)\mathrm{}\otimes \mathrm{SU}\left(3\right)\mathrm{}\otimes U\left(1\right)\mathrm{}$, and the results for $\gamma \gamma \to \mathrm{pions}$ are unchanged.