Contributions of the $C$-odd axial-vector resonances to $\eta \to {\pi }^0\gamma \gamma$ and $\gamma \gamma \to {\pi }^0{\pi }^0$

We consider the contributions of the $C$-odd axial-vector resonances ($b_1$, $h_1$) to $\eta \to {\pi }^0\gamma \gamma$, as well as those of the usual $C$-odd vector resonances ($\rho$, $\omega$) and chiral loops. Inclusion of $b_1\mathrm{}\left(1235\right)\mathrm{}$, $h_1\mathrm{}\left(1170\right)\mathrm{}$, $h_1\mathrm{}\left(1380\right)\mathrm{}$ increases the decay width of $\eta \to {\pi }^0\gamma \gamma$, improving the theoretical prediction. The measurement of the $\gamma \gamma$ spectrum can test our model. We also discuss the effects of these $C$-odd axial-vector resonances on electromagnetic interactions of neutral pions: $\gamma \gamma \to {\pi }^0{\pi }^0$ and electric and magnetic polarizabilities of ${\pi }^0$.