$$\label{eq:LOlag}
\mathcal{L}_2 = \frac{F^2}{4}{\ensuremath{\left\langle u_\mu u^\mu+\chi_+ \right\rangle}}$$ where ${\ensuremath{\left\langle \dots \right\rangle}}$ denotes the $n_f$-dimensional flavour trace. The notation we will use for [$n_f=2$]{} is $F$ and $B$, and for [$n_f=3$]{} $F_0$ and $B_0$ for the constants in (\[eq:defs\]) and (\[eq:LOlag\]). The Lagrangians at next-to-leading order, $p^4$, were constructed in [@Gasser:1983yg; @Gasser:1984gg] for [$n_f=2$]{} and [$n_f=3$]{}. The [$n_f=2$]{} Lagrangian is $$\begin{aligned}
\label{eq:NLOlag2}
\mathcal{L}_4^{{\ensuremath{n_f=2}}}&=&
\frac{l_1}{4}{\ensuremath{\left\langle u_\mu u^\mu \right\rangle}}{\ensuremath{\left\langle u_\nu u^\nu \right\rangle}}
+\frac{l_2}{4}{\ensuremath{\left\langle u_\mu u_\nu \right\rangle}}{\ensuremath{\left\langle u^\mu u^\nu \right\rangle}}
+\frac{l_3}{16}{\ensuremath{\left\langle \chi_+ \right\rangle}}^2 +\frac{i l_4}{4}{\ensuremath{\left\langle u_\mu \chi_-^\mu \right\rangle}}
\nonumber\\&&
+ \frac{l_5}{4}{\ensuremath{\left\langle f_+^2-f_-^2 \right\rangle}}
+\frac{i l_6}{2}{\ensuremath{\left\langle f_{+\mu\nu}u^\mu u^\nu \right\rangle}}
- \frac{l_7}{16}{\ensuremath{\left\langle \chi_- \right\rangle}}^2
\nonumber\\&&
+ ~~{\rm 3 ~contact~terms} ~.\end{aligned}$$

The [$n_f=3$]{} Lagrangian is $$\begin{aligned}
\label{eq:NLOlag3}
\mathcal{L}_4^{{\ensuremath{n_f=3}}}&=&
 L_1{\ensuremath{\left\langle u_\mu u^\mu \right\rangle}}{\ensuremath{\left\langle u_\nu u^\nu \right\rangle}}
+L_2{\ensuremath{\left\langle u_\mu u_\nu \right\rangle}}{\ensuremath{\left\langle u^\mu u^\nu \right\rangle}}
+L_3{\ensuremath{\left\langle u_\mu u^\mu u_\nu u^\nu \right\rangle}}
\nonumber\\&&
+L_4{\ensuremath{\left\langle u_\mu u^\mu \right\rangle}}{\ensuremath{\left\langle \chi_+ \right\rangle}}
+L_5{\ensuremath{\left\langle u_\mu u^\mu\chi_+ \right\rangle}}
+L_6{\ensuremath{\left\langle \chi_+ \right\rangle}}^2
+L_7{\ensuremath{\left\langle \chi_- \right\rangle}}^2
\nonumber\\&&
+\frac{L_8}{2}{\ensuremath{\left\langle \chi_+^2+\chi_-^2 \right\rangle}}
-iL_9{\ensuremath{\left\langle f_{+\mu\nu}u^\mu u^\nu \right\rangle}}
+\frac{L_{10}}{4}{\ensuremath{\left\langle f_+^2-f_-^2 \right\rangle}}
\nonumber\\&&
+ ~~{\rm 2 ~contact~terms} ~.
$$

The Lagrangian for the general $n_f$-flavour case can be found in [@Bijnens:1999sh]. Terms that vanish due to the equations of motion have been dropped. This is discussed in detail in [@Bijnens:1999sh].

The Lagrangians at 