Q}}_R^{\rm
  em}\rangle ~.$$ The spurion fields $$\begin{aligned}
\label{eq:QLQR}
{{\cal Q}}_L^{\rm em} =  u Q_L^{\rm em} u^{\dagger}~, \hspace*{.5cm} & \hspace*{.5cm}
{{\cal Q}}_R^{\rm em} = u^{\dagger}Q_R^{\rm em} u  \end{aligned}$$ are expressed in terms of the quark charge matrix $$\begin{aligned}
\label{eq:Qem}
Q_L^{\rm em} = Q_R^{\rm em} &=& \left( \begin{array}{ccc} 2/3 & 0 & 0 \\ 0 & -1/3 & 0
    \\ 0 & 0 & -1/3 \end{array} \right) ~.\end{aligned}$$ The lowest-order electromagnetic LEC $Z$ can be determined either directly from the pion mass difference ($Z \simeq 0.8$) or, in principle more reliably, from a sum rule in the chiral limit ($Z \simeq 0.9$) [@Moussallam:1998za].

Neglecting leptonic terms for the moment, the next-to-leading order Lagrangian of $O(e^2 p^2)$ was constructed by Urech [@Urech:1994hd]: $$\begin{aligned}
\label{eq:Le2p2}
{\mathcal{L}}_{e^2p^2} &=&
e^2 {F_0}^2 \left\{ \frac{1}{2} K_1 \; \langle ({{\cal Q}}^{\rm em}_L)^2 +
({{\cal Q}}^{\rm em}_R)^2\rangle \; \langle u_\mu  
u^\mu\rangle   + K_2 \; \langle {{\cal Q}}^{\rm em}_L {{\cal Q}}^{\rm em}_R\rangle 
\; \langle u_\mu u^\mu \rangle \right. {\nonumber}\\
&& \mbox{} - K_3 \; [\langle {{\cal Q}}^{\rm em}_L u_\mu\rangle 
\; \langle {{\cal Q}}^{\rm em}_L u^\mu
\rangle + \langle {{\cal Q}}^{\rm em}_R u_\mu\rangle 
\; \langle {{\cal Q}}^{\rm em}_R u^\mu\rangle ] {\nonumber}\\
&& \mbox{} + K_4 \; \langle {{\cal Q}}^{\rm em}_L u_\mu\rangle 
\; 