= Electromagnetic coupling of a weak multiplet
{title2=$\mathcal L_{\mathrm{em}}=-eA_\mu\bar\psi\gamma^\mu(T_3+Y)\psi$}
With the plus-sign convention $D_\mu=\partial_\mu+igT_3W_\mu^3+ig'YB_\mu$, let $\tan\theta_W=g'/g$ and $A_\mu=\sin\theta_WW_\mu^3+\cos\theta_WB_\mu$. The coefficient of $A_\mu$ is $g\sin\theta_WT_3+g'\cos\theta_WY=e(T_3+Y)$, where $e=g\sin\theta_W=g'\cos\theta_W$. 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>.
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