Positronium allowed decays (Peskin)

[Hao]

Homework Statement

(This is not homework)
This refers to question 3.8 in Peskin's QFT

Using the fact that the electromagnetic interaction term in the Dirac + EM lagrangian is invariant under Parity (P) and Charge conjugation (C), and that spin 0 and spin 1 states are odd and even under exchange of spins,

show that the
1) spin 0 positronium ground state (S wavefunction) decays into 2 photons, and that the
2) spin 1 positronium ground state (S wavefunction) must decay into 3 photons
3) The above for P, D states.

Homework Equations

EM coupling
$$\Delta H=\int A_{\mu}j^{\mu}d^{3}x$$

We know that under parity, $$j^{\mu}\rightarrow (j^{0},-j^{1},-j^{2},-j^{3})$$
We know that under parity, $$j^{\mu}\rightarrow -j^{\mu}$$

The Attempt at a Solution

By handwaving, we can say that these transitions occur due to conservation of angular momentum as a photon has a spin of 1.

However, how would these transitions be derived on the basis of C and P symmetries alone?

One could probably consider the interaction matrix term:

$$\left\langle photons\right|\Delta H\left|positronium\right\rangle$$

And determine how it transforms under C and P

The problem I have is in evaluating the P and C eigenvalues of states that contain only photons.

For a state involving a fermion and antifermion (eg. positronium), and with orbital angular momentum L, P|state> = (-1)^L+1|state>. The extra factor of +1 is due to the anticommutativity of spin 1/2 creation operators.

Thanks

 

[Hao]

For those who are interested, I've found a solution by somebody else.

http://www.maths.tcd.ie/~powersr/New/