Calculation of theη→ππγDecay Rate with Finite Dispersion Relations

Finite-dispersion-relation techniques as developed by Aviv and Nussinov are employed to calculate the decay rate for $\eta \to {\pi }^{+}{\pi }^{-}\gamma$ in an essentially parameter-free manner. The amplitude for this decay is obtained by crossing from the scattering process $\eta \pi \to \pi \gamma$ which has the $A_2$ resonance in the $s$ and $u$ channels and the $\rho$ trajectory in the $t$ channel. A single Breit-Wigner resonance of width 85 MeV for the $A_2$ meson directly leads to a prediction for the rate and photon momentum distribution in good agreement with experimental measurements. The so-called broad-narrow model can give agreement if couplings of each component to $\eta \pi$ and $\pi \gamma$ are properly chosen, but all pictures of the $A_2$ with two narrow components give inadequate results.