Solution
= Solution
Use the <weak hypercharge> normalization $Q=T_3+Y$. Write $U(x)\in SU(2)_L$ and the hypercharge transformation as $e^{i\beta(x)Y}$. A left-handed doublet transforms as $\psi_L\mapsto e^{i\beta Y_L}U\psi_L$, while a right-handed singlet transforms as $\psi_R\mapsto e^{i\beta Y_R}\psi_R$. The <Higgs doublet> transforms as $\phi\mapsto e^{i\beta/2}U\phi$.
The representations per <fermion generation> are:
|| Field
|| $SU(3)_c$
|| $SU(2)_L$
|| $Y$ in $Q=T_3+Y$
|| $\widehat Y=2Y$ in $Q=T_3+\widehat Y/2$
| $Q_L=(u_L,d_L)^T$
| $\mathbf3$
| $\mathbf2$
| $1/6$
| $1/3$
| $u_R$
| $\mathbf3$
| $\mathbf1$
| $2/3$
| $4/3$
| $d_R$
| $\mathbf3$
| $\mathbf1$
| $-1/3$
| $-2/3$
| $L_L=(\nu_L,e_L)^T$
| $\mathbf1$
| $\mathbf2$
| $-1/2$
| $-1$
| $e_R$
| $\mathbf1$
| $\mathbf1$
| $-1$
| $-2$
| $\phi=(\phi^+,\phi^0)^T$
| $\mathbf1$
| $\mathbf2$
| $1/2$
| $1$
There is no right-handed <neutrino> in the minimal <Standard Model>. If such a sterile singlet were added, its hypercharge would be zero. The two hypercharge columns give equivalent normalizations, not different physical assignments. This is the <electroweak representation and hypercharge table>.
<Yukawa interactions> use $\phi$ for down-type quarks and charged leptons, but $\widetilde\phi=i\sigma^2\phi^*$ for up-type quarks. The conjugate doublet transforms as $e^{-i\beta/2}U\widetilde\phi$, using the pseudoreality of the $SU(2)$ fundamental representation. For example, the hypercharge sums in the down, charged-lepton, and up terms are
$$
-\frac16+\frac12-\frac13=0,\qquad
+\frac12+\frac12-1=0,\qquad
-\frac16-\frac12+\frac23=0.
$$
The doublet indices contract between $\bar\psi_L$ and the scalar, and colour indices contract in the quark terms. This verifies <gauge invariance>.
The indices $i,j=1,2,3$ label <fermion generations>, specifying which left-handed family and right-handed family are coupled. They are not weak-isospin indices. Each fermion type has its own complex <Yukawa matrix>. After symmetry breaking these matrices determine masses; the mismatch of the two quark left-handed mass rotations produces the <CKM matrix>. \b[Left fields are weak doublets, right fields are weak singlets, and the scalar is a hypercharge-$1/2$ doublet in the stated convention.]