Why is eta meson decay into 3 neutral pions forbidden?

[lLehner95]

Homework Statement

Explain why eta meson can't decay into 3 neutral pions. Use isospin and pions statistics.

Relevant Equations

`$|\eta >=|I=0,I_{z}=0>$ and $|\pi ^{0} >=|1,0>$

I started combining 2 pions:
$|\pi ^{0},\pi ^{0} >=\sqrt{\frac{2}{3}}|2,0>-\sqrt{\frac{1}{3}}|0,0>$
What should i do now? Should i continue combining the third pion or can i already say that it's forbidden? If yes, why? Is it because the state antisymmetric, impossible for two bosons?

 

[MichaelJ12]

Homework Statement: Explain why eta meson can't decay into 3 neutral pions. Use isospin and pions statistics.
Homework Equations: `$|\eta >=|I=0,I_{z}=0>$ and $|\pi ^{0} >=|1,0>$

I started combining 2 pions:
$|\pi ^{0},\pi ^{0} >=\sqrt{\frac{2}{3}}|2,0>-\sqrt{\frac{1}{3}}|0,0>$
What should i do now? Should i continue combining the third pion or can i already say that it's forbidden? If yes, why? Is it because the state antisymmetric, impossible for two bosons?

One way to tell it is forbidden is using G-parity conservation.

The eta meson has G-parity 1, so it can only decay to something whose total G-parity is 1. Each pion has G-parity of -1. Since the G-parity of a system of two or more particles is just the product of the G-parity of each of these particles (it is multiplicative), we conclude that the eta meson cannot decay to an odd number of pions.

You can read more about G-parity here https://en.wikipedia.org/wiki/G-parity.

 

[lLehner95]

Thank you! However, we haven't studied G-parity yet, and our teacher wants us to solve the problem using isospin and boson statistics.