I'm trying to determine if the reaction
$$n\rightarrow p + \pi^-$$
is allowed. First of, this doesn't list this as one of the decay modes of the neutron, so I suspect that it should not be allowed. However, I would like to be able to argue this fact without reference to the link.
I have found that this reaction violates parity, because $P(n)=P(p)=+1$, but $P(\pi^-)=-1$, leading to the inequality $P(n) \neq P(p)P(\pi^-)$. This excludes the electromagnetic and strong force, and leaves me only with the possibility that the reaction is due to the weak force.
The reaction conserves angular momentum $\vec J$, baryonic number $B$ and electric charge $Q$, so no luck there.
I know that this reaction should respect $CP$, or at least $CPT$ invariance, but I'm having a hard time determining the effect of charge conjugation ($C$) on this reaction, because for each of the particles, applying $C$ changes the particle, and so I can't assign an eigenvalue $\pm 1$. For instance, $C\,|n\rangle = |\bar n\rangle$ is not an eigenvalue equation.
How can I determine if this reaction respects $CP$, or is there something else I should look at that I have failed to see so far? And if this reaction is indeed allowed, is the only way to confirm this to check every conservation law?