Electromagnetic coupling of a weak multiplet (${\mathcal{L}}_{\mathrm{em}}=-e{A}_{\mu }\bar{\psi }{\gamma }^{\mu }(T_3+Y)\psi$)

With the plus-sign convention $D_{\mu }={\partial }_{\mu }+ig{T}_3{W}_{\mu }^3+i{g}^{\prime }Y{B}_{\mu }$, let $\tan {\theta }_W=g^{\prime }/g$ and $A_{\mu }=\sin {\theta }_W{W}_{\mu }^3+\cos {\theta }_W{B}_{\mu }$. The coefficient of $A_{\mu }$ is $g\sin {\theta }_W{T}_3+g^{\prime }\cos {\theta }_{W}Y=e(T_3+Y)$, where $e=g\sin {\theta }_W=g^{\prime }\cos {\theta }_W$. The extra minus sign in the interaction Lagrangian comes from the $i$ in $\bar{\psi }i{\gamma }^{\mu }{D}_{\mu }\psi$. It reproduces the electric charges of every component of an Standard Model representation.