Chiral anomaly, pion to photon decay

[JosephButler]

Hello, I understand that the non-zero (or non-small) rate for $$\pi^0 \rightarrow \gamma\gamma$$ was historically a big motivation for the non-conservation of the axial current. I've been trying to work on problem IV.7.2 (p. 252) in Zee which asks to show that this amplitude vanishes if [tex]\partial_\mu J_5^\mu = 0[\tex] and $$m_\pi = 0$$. He suggests following the argument he used in a previous section where he motivated the pion as a goldstone boson (sec IV.2), leading up to the Goldberger-Treiman relation. <br /> <br /> I understand heuristically what he's asking: show that the rate for $$\pi^0 \rightarrow \gamma\gamma$$ is much larger than what would be expected without the chiral anomaly. However, I don't quite understand the limiting case that he's asking us to confirm in the problem. In the case $$m_\pi = 0$$, the decay is impossible kinematically. Peskin (ch 19.3, p. 675-676) does a similar thing where he takes the limit of the pion mass to be zero and then fills in factors of $$m_\pi$$ in the kinematics. But Peskin doesn't assume that the axial current is conserved and fixes terms based on the existence of the anomaly.<br /> <br /> So what I'm confused about is how to approach the problem in the 1950's point of view, the way that Zee wants. I want to assume the axial current is conserved and that the pion is a goldstone boson (massless), and I want to show that the amplitude for pion decay into photons vanishes. Is it necessary to assume that the pion has a small mass and then go to the massless limit after deriving a result? At any rate, the pion having a mass explicitly violates $$\partial_\mu J^\mu_5 = 0$$ since the amplitude is proportional to: (by Lorentz invariance)<br /> <br /> $$\langle 0| J^\mu_5 | \pi(k) \rangle = fk^\mu$$<br /> <br /> (which defines the constant $$f$$), and hence<br /> <br /> $$\langle 0| \partial_\mu J_5^\mu | \pi(k) \rangle = f m^2_\pi$$.<br /> <br /> Thus a conserved current ($$\partial_\mu J^\mu_5 = 0$$) means the pion has to be massless. <br /> <br /> I'm just not really sure what series of steps Zee wants us to take.<br /> <br /> Any tips would be greatly appreciated!<br /> Cheers,<br /> JB[/tex]

 

[TriTertButoxy]

I can say that although kinematically a massless pion decaying to two massless photons is impossible, the amplitude for the process doesn't necessarily forbid it.

Recall, that the transition operator is factored into a 4-momentum conserving delta function and the amplitude:

$$iT=(2\pi)^4\delta^{(4)}(p_1+p_2-k_1-k_2)\,i\mathcal{M}(p_1\,p_2\rightarrow k_1\,k_2)$$​

So, $\mathcal{M}(p_1\,p_2\rightarrow k_1\,k_2)$ may not be zero, but the kinematics is partly taken care of by the momentum conserving delta function.

 

[Avodyne]

The result you want is known as the Sutherland-Veltman theorem. I believe the approach is to work with a massive pion, and show that the amplitude M has a factor of m_pi^2, so that M vanishes in the massless limit.