direct derivation of [ChPT]{} from the underlying assumptions is given by Leutwyler [@Leutwyler:1993iq].

The perturbation in [ChPT]{} is not an expansion in a small coupling constant but an expansion in momenta and quark masses. Its consistency was shown in detail in [@Weinberg:1978kz] and is often referred to as Weinberg or $p$ power counting.

A more extensive introduction to [ChPT]{} can be found in [@Scherer:2012xha]. There are many reviews of [ChPT]{}. Those focusing on the meson sector are two at the one-loop level [@Ecker:1994gg; @Pich:1995bw] and one at the two-loop level [@Bijnens:2006zp].

In terms of a quark field $\bar q =(\bar u~\bar d)$ ([$n_f=2$]{}) or $\bar q =(\bar u~\bar d~\bar s)$ ([$n_f=3$]{}) the fermionic part of the QCD Lagrangian can be written as $$\label{QCD}
\mathcal{L}_{QCD} = \bar q i\gamma^\mu 
\left( \partial_\mu-ig_s G_\mu -i(v_\mu+\gamma_5 a_\mu)\right)q
 -\bar q s q+i\bar q p\gamma_5 q\,.$$ The external fields or sources $v_\mu, a_\mu, s$ and $p$ are $n_f\times n_f$ matrices in flavour space. They were introduced in [@Gasser:1983yg; @Gasser:1984gg] to make chiral symmetry explicit throughout the calculation and to facilitate the connection between QCD and [ChPT]{}. For later use we define $l_\mu=v_\mu-a_\mu$ and $r_\mu = v_\mu+a_\mu$.

The degrees of freedom are the Goldstone bosons of the 