# Invariant Lagrangian under $SU(2)\times U(1)$

I am studying QFT and I was reading the chapter 10 about Spinors at the Schwartz's Quantum field theory and the standard model. After that chapter I got the task of constructing a lagrangian that is invariant under $$SU(2)\times U(1).$$

It was given a scalar field H ~ (2, 1/2), and fermion fields $q_l = (2, 1/6)$, $u_r = (1, 2/3)$ and $d_r = (1, -1/3)$, where the firs two are doublet of $SU(2)$ and the last two are singlet of $U(1)$. The second number inside the brackets are hypercharges.

I am not sure of what to do. I mean, I know that a lagrangian invariant under $SU(2)\times U(1)$ must be invariant under $exp (i\alpha(x)/2)U(x)\phi$ where $\phi$ is the field. But I don't know what to do with the hypercharges and how to construct this lagrangian using just symmetry arguments. I mean this is the electroweak lagrangian, right? When Schwartz construct invariant lagrangians in the mentioned chapter of his book he starts with the symmetry group under which he wants his lagrangian to be invariant and he tackle term after term to find which ones are lorentz invariant. I did it this way as exercises for the complex scalar field, then for the $U(1)$ group, and for $SU(2)$ doublet alone. But now I am not sure of what to do with all of those field terms because I am not sure of exactly what a fermion field is and how to put all of those fields inside the lagrangian.

Does anyone have any hint/help?

• So what are your concrete questions? – Name YYY Oct 13 '17 at 20:41
• @NameYYY21 my question are related to the hypercharge, see I have few experience with QFT and I am not sure on how to proceed to write down the lagrangian which is invariant under the symmetry group with the fermion fields given. I saw one example on the subject, namely the one Schwartz did in his book, and I was able to proceed on a few basic exercises, but this one is troubling me out. – Dimitri Oct 14 '17 at 18:42
• @NameYYY sorry, forgot to tag you properly in the previous comment – Dimitri Oct 14 '17 at 21:15
• You also don't understand how to construct Lorentz invariant fermion lagrangian, or the only problem is constructing the $SU(2)\times U(1)$ lagrangian from the fermion fields only? – Name YYY Oct 15 '17 at 7:56