# Neutral pion 2 gamma decay question

1. Apr 4, 2013

I've been reading some books on how neutral pion decay proceeds through chiral anomaly recently. What I found wired is people always use the matrix element of the axial current between vacuum and a pion as granted, saying the matrix element should be a vector and the pion momentum is the only vector one can use so we have <vac|$J_{\mu5a}$|$\pi^b(p)$> = $-i p_\mu \delta_{ab} f_\pi$. But if you take the divergence of this, you get the ${m_\pi}^2 f_\pi$ which equals zero in chiral limit. This is not true in the case of pion 2 photon decay since when the current is coupled to electromagnetic fields the divergence of the axial current is non-zero and proportional to $E \bullet B$. Another question I have is why the pion decay constant from charged pion decay can be used to calculate neutral pion decay width and get the correct result. Someone help here?