# Doubt on Ademollo-Gatto theorem proof

1. Nov 4, 2012

### 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

2. Nov 4, 2012

### 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.

3. Nov 4, 2012

### 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-), while the V-spin triplet should be composed by K+, K- and π08. But, again, I'm not really sure.

4. Nov 4, 2012

### Bill_K

Whoops, you are correct!