# Rho meson decay to three neutral pions

In my assignment, I am asked to show whether $\rho^0\to3\pi^0$ is allowed. From the particle data group, I cannot find the decay mode; hence, I am guessing it is not allowed. Based on the knowledge that $\rho^0\to2\pi^0$ is not allowed, I focus on parity. However, the action contains an angular momentum of 3 bodies. Then I get stuck. Can someone tell me whether this action is allowed? Or at least give me some hint. (I have not learned weak interaction and strong interaction yet.)

• Well, the decay certainly violates G-parity, but so does the suppressed but allowed decay $\pi^0\pi^-\pi^+$, since isospin is an imperfect symmetry, hence also G. Parity violation would make the decay weak, so enormously suppressed. Note your 3-body wf must be completely symmetric, but, can it be? – Cosmas Zachos Feb 7 '18 at 2:28
• @CosmasZachos Yes, my question now becomes for many bodies (or 3 bodies) problem, will the orbital angular momentum still have the same spherical harmonic function? If so, then the $Y_1^m$ will change the sign under parity; hence, 3 $\pi^0$ cannot be in the same orbit because they are indistinguishable. – Hamio Jiang Feb 7 '18 at 2:38

But G-parity relies on isospin, which is broken somewhat in the strong decay, so there is also the 3π mode at the $10^{-4}$ level, where all 3π s are different, instead of your 3π0 which has to be fully symmetric.
However, this is impossible; ignoring normalizations, $$(12+21)3-3(12+21)= 123+213-312-321= (123-321)+(-312+213),$$