C-parity of π+ π- π0

[ln(]

TL;DR Summary

C-parity of π+ π- π0 compared to π+ π-

The C-parity of π+ π- alone is simply (-1)^L, where L is the angular momentum quantum number for the system. But then what is C-parity of π+ π- π0? Is it simply (-1)^L (+1), where L is the angular momentum quantum number for the π+ π- subsystem (which isn't necessarily the angular momentum of the whole three pion system)?

 

[SBoh]

π+ π- π0's CP is(-1)(-1)^(L). So, two pions and three pions have opposite CP eigenvalues. Therefore, observation of both K_{0 L} to two pions and three pions decays is clue of CP violation.

 

[ln(]

π+ π- π0's CP is(-1)(-1)^(L). So, two pions and three pions have opposite CP eigenvalues. Therefore, observation of both K_{0 L} to two pions and three pions decays is clue of CP violation.

This implies C = +1 which seems right. But how do you calculate C eigenvalue directly?

 

[SBoh]

C is multiplicative quantum number. So, once you know C of a single pion, you just need to perform C^n for n pions system. It is same for P. So, C(pion)^n P(pion)^n (-1)^L is total CP eigenvalue.
If you ask how we can know C or P eigenvalue of single particle such as pion, I would answer that we need to set C and P eigenvalues to several basic particles such as proton and electron. Then, we can study C and P quantum number of other particles (i.e. pions) using physical process such as pion + deutron to two neutrons.