Why does $C$-parity forbid odd number of photons in pion decay, but not in positronium decay? (Martin-Shaw)

Question

I'm reading Martin and Shaw's "Particle Physics."

In Section 5.4.1 they show how $C$-parity restricts the number of photons in pion decay to an even number of photons:

$$ \pi^0=u\bar u\textrm{ (or }d\bar d) \rightarrow 2n\gamma,~n\in\mathcal I$$

And in Section 1.3.3, they mention that positronium decay can give both even and odd number of photons:

$$ e^-+e^+\rightarrow n\gamma,~n\in\mathcal I $$

In both cases a particle-antiparticle pair annihilates, so why do we not get odd suppression in both cases?

Is there perhaps an unstated assumption that for the pion we look at the ground state, while for positronium we also allow excited states?

Answer 1

For a fermion-antifermion pair, you can easily see that $$ C= (-)^{L+S}, $$ so, e.g., $$ C(^1S_0)=+ , ~~~\leadsto ~~~~ \to 2\gamma,\\ C(^3S_1)=- , ~~~\leadsto ~~~~ \to 3\gamma, $$ the photons being odd under C.

The first case covers the lowest-lying pseudoscalars, like the pion, and parapositronium; while the second covers the ρ, ψ, ..., and orthopositronium.