{{\cal Q}}^{\rm em}_L &=& 
u (D_\mu Q_L^{\rm em}) u^{\dagger}, {\nonumber}\\
{\widehat}\nabla_\mu {{\cal Q}}^{\rm em}_R &=& 
u^{\dagger}(D_\mu Q^{\rm em}_R) u ,\end{aligned}$$ with $$\begin{aligned}
\label{eq:covder2}
D_\mu Q^{\rm em}_L &=& \partial_\mu Q^{\rm em}_L 
- i[l_\mu,Q^{\rm em}_L], {\nonumber}\\
D_\mu Q^{\rm em}_R &=& \partial_\mu Q^{\rm em}_R - i[r_\mu,Q^{\rm em}_R]~.\end{aligned}$$ In the presence of dynamical photons and leptons, the external fields $l_\mu,r_\mu$ are modified as $$\begin{aligned}
\label{eq:lrmu}
l_\mu & \longrightarrow & v_\mu - a_\mu - e Q_L^{\rm em} A_\mu + \sum_{\ell=e,\mu}
(\bar \ell \gamma_\mu \nu_{\ell L} Q_L^{\rm w} + {\overline}{\nu_{\ell L}} 
\gamma_\mu \ell Q_L^{{\rm w}{\dagger}}), {\nonumber}\\
r_\mu & \longrightarrow & v_\mu + a_\mu - e Q_R^{\rm em} A_\mu\end{aligned}$$ where $A_\mu$ is the photon field and the weak charge matrix is defined as $$\begin{aligned}
\label{eq:Qw}
Q_L^{\rm w} = - 2 \sqrt{2}\; G_F \left( \begin{array}{ccc}
0 & V_{ud} & V_{us} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right) ~, &\qquad
{{\cal Q}}_L^{\rm w} = u Q_L^{\rm w}  u^{\dagger}~.\end{aligned}$$ $G_F$ is the Fermi coupling constant and $V_{ud}$, $V_{us}$ are Kobayashi-Maskawa matrix elements.

For radiative corrections in semileptonic processes, one needs in addition the leptonic Lagrangian [@Knecht:1999ag] $$\begin{aligned}
\label{eq:Llept}
{\mathcal{L}}_{\rm lept} &=& 
e^2 \sum_{\ell} \left \{ {F_0}^2 \left[  
X_1 {\overline}{\ell} \gamma_\mu \nu_{\ell L} 
\langle 