Understand Eta Particle Decay into 3 Pions

[eendavid]

1. The statement, all variables and given/known data
I am studying the decay of the $$\eta$$-particle. Povh et Al, 'Particles and nuclei' say that a decay into 3 pions is not possible via the strong interaction. "For reasons of symmetry 3 pions (isospin equals 1) can not couple to zero isospin." This explains the long lifetime of the eta-particle. I do not understand this.

Homework Equations

Denote by $$D^{(j)}$$ the non-reducible representations of a rotation in isospin space, then $$D^{(1)}\otimes D^{(1)}=D^{(2)}\oplusD^{(1)}\oplusD^{(0)}$$


3. The problem
As
$$D^{(1)}\otimes D^{(1)}\otimes D^{(1)}= ...\oplus D^{(1)}\otimes D^{(1)}\oplus...$$, their are $$D^{(0)}$$ components in the 3 pion system. Am I making a misinterpretation somewhere?

 

[eendavid]

For the purpose of archiving, here's the answer: the decay violates conservation of G-parity, which is conserved in the strong interaction.