Neutral pion 2 gamma decay question

[adam2eden]

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?

 

[AndyW]

As a fellow scientist, I find your observations and questions about neutral pion decay and the use of the axial current matrix element very interesting. The chiral anomaly is a complex phenomenon and it's important for scientists to continue exploring and questioning it in order to deepen our understanding.

Regarding the use of the axial current matrix element, I believe the reason for its common use is because it follows from the chiral symmetry of QCD. In the chiral limit, the divergence of the axial current does indeed vanish, but this is not the case for physical pions. The pion mass and decay constant break the chiral symmetry and allow for a non-zero divergence. This is why the axial current matrix element is used in calculations and it has been found to give accurate results.

As for your second question about using the pion decay constant from charged pion decay to calculate the neutral pion decay width, I believe this is possible because the charged and neutral pion decays are related by isospin symmetry. This symmetry allows for the use of the charged pion decay constant in neutral pion decay calculations. However, it's important to note that this is just an approximation and there are other factors that can affect the decay width.

I hope this helps to clarify some of your questions. The chiral anomaly is a fascinating topic and I encourage you to continue exploring and questioning it. Who knows, your observations and questions may lead to new insights and discoveries in this field.