# Isospin of pions

Suppose that I have two pions with zero relative angular momentum. I want to find possible total isospin values. What I'm thinking is that their state should be symmetric since they are bosons. Each pion is in isospin $1$ representation of $SU(2)$ so I take tensor product of spin $1$ reps. I can decompose $$1 \otimes 1 = 2 \oplus1\oplus0.$$ Since representation with total isospin $T=2$ is symmetric I can get all the values from $-2$ to $2$. Am I correct? Are there any other restrictions?

• It seems that You're right. – Name YYY Dec 1 '16 at 15:53
• $1\times 1 =1 \ne 3 =2 +1+0\;$ so your decomposition is incorrect. Does there exist 0-dimensional linear spaces ??? – Frobenius Jul 30 '18 at 4:41

I think you must see it by analogy in the frame of angular momentum : $$(2j_{\alpha}\boldsymbol{+}1)\boldsymbol{\otimes} (2j_{\beta}\boldsymbol{+}1)=\bigoplus_{j_{\rho}\boldsymbol{=}\boldsymbol{\vert} j_{\alpha}\boldsymbol{-}j_{\beta}\boldsymbol{\vert}}^{j_{\rho}\boldsymbol{=} j_{\alpha}\boldsymbol{+}j_{\beta}}(2j_{\rho}\boldsymbol{+}1) \tag{01}\label{eq01}$$ For $\;j_{\alpha}\boldsymbol{=}1\boldsymbol{=}j_{\beta}\;$
$$\boldsymbol{3}\boldsymbol{\otimes}\boldsymbol{3}=\boldsymbol{1}\boldsymbol{\oplus}\boldsymbol{3}\boldsymbol{\oplus}\boldsymbol{5} \tag{02}\label{eq02}$$