Chiral Perturbation Theory Predictions for $\eta \to \pi^+ \pi^- \pi^0 \gamma$

The $\eta \to \pi^+ \pi^- \pi^0 \gamma$ decay is discussed in the general context of Chiral Perturbation Theory (ChPT), assuming that the low--energy constants (counter--terms) are saturated by vector-meson resonances. The $\eta \to \pi^+ \pi^- \pi^0 \gamma$ amplitude can be separated in two distinct pieces: the inner bremsstrahlung, $A^{(IB)}$, and the structure dependent (or direct emission), $A^{(SD)}$, amplitudes. The former -- which essentially contains the same physics as $A(\eta \to \pi^+ \pi^- \pi^0)$ -- is found to dominate over the second one -- which looks more interesting from the ChPT point of view.