Doubt on Ademollo-Gatto theorem proof

[Einj]

I'm studying the semileptonic decay of the kaon and I'm currently reading about the Ademollo-Gatto theorem. The question is probably silly. However, here it is.
The proof starts by just considering that $[V_-,V_+]=V_3$, where V is the V-spin. Now, we consider the matrix element $\langle K^0|V_3|K^0\rangle=1$.

I understand that the $|K^0\rangle$ state is an eigenstate of V₃ (is the same thing that happens for the usual isospin), but why exactly 1? Consider for example the third component of isospin. As K⁰ has I₃=1/2 I'd expect that matrix element to be 1/2. Why is 1?

Thanks

 

[Bill_K]

Actually isn't the V-spin of the K⁰ equal to 1? It belongs to a V-spin triplet along with K⁰ bar and some linear combination of π⁰ and η⁰.

 

[Einj]

I'm not really sure but I think that that is the U-spin. The V-spin should be the operator that allows the transition from K⁰ to π^- and so the K⁰ should belong to the doublet (K⁰,π^-), while the V-spin triplet should be composed by K⁺, K^- and π⁰/η₈. But, again, I'm not really sure.

 

[Bill_K]

Whoops, you are correct!