Why is the $\rho^0 \rightarrow \pi^0 + \pi^0$ decay not allowed? I have seen this question but I am not satisfied with the answers. The $J^{PC}$ of the $\rho$ and $\pi$ are $1^{--}$ and $0^{-+}$ respectively. Taking the orbital angular momentum of the final two pions state to be $1$ seems to conserve everything ($J$, $P$, $C$, etc.) but is apparently not allowed by the spin-statistics theorem. The answers I have seen say that $L=1$ for the pions gives an anti-symmetric wavefunction. Assuming this is true, I understand why this is not allowed under the spin-statistics theorem, but I don't understand why the wavefunction anti-symmetric; that is the main question.
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The spatial wavefunction is $Y_L^m(\theta,\phi)$. When exchanging the two particles, the spatial wavefunction becomes to $Y_L^m(\pi-\theta,\pi+\phi)$. Mathematically, we have $Y_L^m(\pi-\theta,\pi+\phi)=(-1)^L Y_L^m(\theta,\phi)$. If $L$ is odd, the spatial part is antisymmetric, otherwise symmetric.
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