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?