# Same $U(1)$ charge for the $SU(2)$ doublet

Consider the electroweak gaugesymmetry $$SU(2)_L\times U(1)_Y$$. The entries of $$SU(2)_L$$ doublet will have same $$U(1)$$-charge. How can this be shown mathematically?

• If they didnt', the U(1) transformation would change the SU(2) state, which would make the system not invariant under the whole gauge group. Feb 7, 2014 at 15:18

1. If a theory is declared to have a symmetry group $$G$$, it means more abstractly that the group $$G$$ acts on the constituents (fields etc.) according to some rules and the theory (Lagrangian etc) stays invariant under such transformations.

2. Often the constituents (fields etc.) form a (linear) representations $$V$$ of the group $$G$$. If the representation is (completely) reducible we can decompose it in irreps. The fundamental objects (fields etc) [that we consider] are for this reason often chosen to transform as irreps of the theory.

3. Now an irrep $$V$$ of a product group $$G=G_1\times G_2$$ is of the form of a tensor products $$V\cong V_1 \otimes V_2$$ of irrep $$V_1$$ and $$V_2$$ for the groups $$G_1$$ and $$G_2$$, respectively.

4. The irreps of the Abelian group $$U(1)$$ are all $$1$$-dimensional and labelled by an integer $$n\in \mathbb{Z}$$ called the charge.

5. So to return to OP's question, in the electroweak theory with group $$G=SU(2)\times U(1)$$, the field transform by definition as an irrep $$V\cong V_1 \otimes V_2$$ of $$SU(2)\times U(1)$$. In particular, the irrep $$V$$ carries a $$U(1)$$ charge, which (modulo various normalization conventions) is the weak hypercharge. To summarize: The main point is that the weak hypercharge is fixed by definition/construction.

6. Perhaps the following comment is helpful: If we are given a tensor product $$V=V_1\otimes V_2$$, where we assume that

• (i) $$V$$ is a (completely) reducible representation of $$SU(2)\times U(1)$$,

• (ii) $$V_1$$ is an irrep of $$SU(2)$$, and

• (iii) $$V_2$$ is a $$1$$-dimensional representation of $$U(1)$$,

then it follows that $$V_2$$ (and $$V$$) must be irreps as well. And hence $$V_1$$ carries a fixed weak hypercharge, cf. OP's title question.