Chiral anomaly, pion to photon decay

In summary, the conversation discusses the motivation for the non-conservation of the axial current and the problem of showing that the amplitude for \pi^0 \rightarrow \gamma\gamma vanishes in the case of a massless pion. It is suggested to follow the argument used in previous sections regarding the pion as a goldstone boson. There is confusion about the approach Zee wants, whether to assume a small mass for the pion or go to the massless limit after deriving a result. It is also mentioned that the Sutherland-Veltman theorem may be relevant in this case.
  • #1
JosephButler
18
0
Hello, I understand that the non-zero (or non-small) rate for [tex]\pi^0 \rightarrow \gamma\gamma[/tex] 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 [tex]m_\pi = 0[/tex]. 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.

I understand heuristically what he's asking: show that the rate for [tex]\pi^0 \rightarrow \gamma\gamma[/tex] 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 [tex]m_\pi = 0[/tex], 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 [tex]m_\pi[/tex] in the kinematics. But Peskin doesn't assume that the axial current is conserved and fixes terms based on the existence of the anomaly.

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 [tex]\partial_\mu J^\mu_5 = 0[/tex] since the amplitude is proportional to: (by Lorentz invariance)

[tex]\langle 0| J^\mu_5 | \pi(k) \rangle = fk^\mu[/tex]

(which defines the constant [tex]f[/tex]), and hence

[tex]\langle 0| \partial_\mu J_5^\mu | \pi(k) \rangle = f m^2_\pi[/tex].

Thus a conserved current ([tex]\partial_\mu J^\mu_5 = 0[/tex]) means the pion has to be massless.

I'm just not really sure what series of steps Zee wants us to take.

Any tips would be greatly appreciated!
Cheers,
JB
 
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  • #2
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:

[tex]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) [/tex]​

So, [itex]\mathcal{M}(p_1\,p_2\rightarrow k_1\,k_2)[/itex] may not be zero, but the kinematics is partly taken care of by the momentum conserving delta function.
 
Last edited:
  • #3
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.
 

1. What is chiral anomaly?

Chiral anomaly is a phenomenon in quantum field theory where the conservation of a certain symmetry (known as chiral symmetry) is violated at the quantum level. This results in the breaking of chiral symmetry, which is a fundamental concept in the Standard Model of particle physics.

2. What is pion to photon decay?

Pion to photon decay is a process in which a pion, a subatomic particle made up of a quark and an antiquark, decays into a photon, the fundamental particle of light. This process is governed by the laws of the Standard Model and can be studied in experiments to understand the behavior of subatomic particles.

3. How is chiral anomaly related to pion to photon decay?

The chiral anomaly plays a crucial role in the pion to photon decay process. The violation of chiral symmetry results in the production of more photons than predicted by the Standard Model, which can be observed in experiments. This phenomenon is known as the chiral anomaly and is an important aspect of understanding the behavior of particles at the quantum level.

4. What are the implications of the chiral anomaly in particle physics?

The chiral anomaly has significant implications in particle physics as it affects the predictions of the Standard Model. The violation of chiral symmetry can lead to unexpected results in particle interactions, which can be observed in experiments. It also plays a role in the understanding of the strong nuclear force and the behavior of quarks and gluons.

5. Can the chiral anomaly be observed in experiments?

Yes, the chiral anomaly can be observed in experiments through the study of processes such as pion to photon decay. Many experiments have been conducted to measure the effects of chiral anomaly, and the results have been consistent with the predictions of the Standard Model. However, more research is still being done to fully understand the implications of the chiral anomaly in particle physics.

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