spontaneous breakdown of the $SU(n_f)_L\times SU(n_f)_R$ chiral symmetry of QCD with $n_f$ massless flavours to the vector subgroup $SU(n_f)_V$. These are parametrized by a special $n_f\times n_f$ unitary matrix $U$. The transformations under a chiral symmetry transformation $g_L\times g_R\in SU(n_f)_L\times SU(n_f)_R$ are $$\begin{aligned}
U &\to& g_R U g_L^\dagger\,,{\qquad\qquad\qquad s+ i p\to g_R (s + i p)
g_L^\dagger}\,, \nonumber\\
l_\mu&\to& g_L l_\mu g_L^\dagger -i\partial_\mu g_L g_L^\dagger\,,\quad
r_\mu\to g_R r_\mu g_R^\dagger -i\partial_\mu g_R g_R^\dagger\,.\end{aligned}$$ In addition we define $u$ with $u^2=U$ and $h(u,g_L,g_R)$ transforming as $$u\to g_R u h^\dagger = h u g_L^\dagger\,.$$

The easiest way to construct Lagrangians is to use objects $X$ that transform under chiral symmetry as $X\to h X h^\dagger$. For the present paper these are $u_\mu,f_\pm^{\mu\nu}$, $\chi_\pm$ and $\chi_-^\mu$ defined by $$\begin{aligned}
\label{eq:defs}
u_\mu &=& i\left[u^\dagger\left(\partial_\mu-ir_\mu\right)u
                -u\left(\partial_\mu-il_\mu\right)u^\dagger\right]\,,
\qquad \chi = 2B\left(s+ip\right)\,,
\nonumber\\
\chi_\pm &=& u^\dagger \chi u^\dagger\pm u\chi^\dagger u\,,
\qquad\qquad\qquad
\chi_-^\mu = u^\dagger D^\mu \chi u^\dagger - u D^\mu \chi^\dagger u\,,
\nonumber\\
f_\pm^{\mu\nu} &=& u F^{\mu\nu}_L u^\dagger \pm u^\dagger
F_R^{\mu\nu}u\,, \qquad\qquad ~~
D_\mu \chi =\partial_\mu \chi-i r_\mu \chi+i\chi l_\mu\,,
\nonumber\\
F_L^{\mu\nu}&=&\partial^\mu l^\nu-\partial^\nu l^\mu -i\left[l^\mu,l^\nu\right]\,,
\qquad
F_R^{\mu\nu}=\partial^\mu r^\nu-\partial^\nu r^\mu -i\left[r^\mu,r^\nu\right]\,.
$$ The Lagrangian at lowest order, $p^2$, is known since long ago and is in the present notation 