isospin and hypercharge conservation, if $m_u = m_d \neq m_s$.

The model has been bosonized in the framework of functional integrals in the stationary phase approximation leading to the following effective mesonic Lagrangian ${\cal L}_{{\rm bos}}$ at $T=\mu=0$ $$\begin{aligned}
\label{bos}
    &&{\cal L}_{\mathrm{eff}}\to {\cal L}_{{\rm bos}}={\cal L}_{{{\mathrm{st}}}}+{\cal L}_{{\rm ql}}, 
    \qquad
    {\cal L}_{{{\mathrm{st}}}}=h_a\sigma_a+\frac{1}{2} h_{ab}^{(1)}
                  \sigma_a\sigma_b+\frac{1}{2} h_{ab}^{(2)}
                  \phi_a\phi_b +{\cal O}({\mbox{f\mbox{}ield}}^3),
                  \nonumber\\
    &&W_{{\rm ql}}(\sigma,\phi )=\frac{1}{2}\mbox{ln}|{{\mathrm{det}}}D^{\dagger}_E D_E|=-\!\int\!
                 \frac{{\mathrm{d}}^4 x_E}{32 \pi^2}
                 \sum_{i=0}^\infty I_{i-1}\mbox{tr}(b_i)=\int\!{\mathrm{d}}^4 x_E
                 {\cal L}_{{\rm ql}},
                 \nonumber\\
    &&b_0 =1,\hspace{0.2cm} b_1=-Y,
                 \hspace{0.2cm} b_2=\frac{Y^2}{2} 
                 +\frac{\Delta_{us}}{\sqrt{3}}\lambda_8 
                 Y, \hspace{0.2cm} \ldots , 
                 \nonumber\\
    &&Y=i\gamma_{\alpha}(\partial_{\alpha}\sigma 
                 +i\gamma_5\partial_{\alpha}\phi )
                 +\sigma^2+\{{\cal M},\sigma\}+\phi^2 
                 +i\gamma_5[\sigma+{\cal M},\phi ]\end{aligned}$$ written in terms of the scalar, $\sigma=\lambda_a\sigma_a$, and pseudoscalar, $\phi=\lambda_a\phi_a$, nonet valued quantum fields. The result of the stationary phase integration at leading order, ${\cal L}_{{\mathrm{st}}}$, is shown here as a series in growing powers of $\sigma$ and $\phi$. The result of the remaining Gaussian integration over the quark fields is given by $W_{\rm ql}$. Here the Laplacian in euclidean space-time ${D^{\dagger}_E D}_E={\cal M}^2-
\partial_\alpha^2+Y$ is associated with the euclidean Dirac operator $D_E=i
\gamma_\alpha\partial_\alpha -{\cal M}-\sigma -i\gamma_5\phi$ (the $\gamma_\alpha,
\,\alpha =1,2,3,4$ are antihermitian and obey $\{\gamma_\alpha,\gamma_\beta\}=-2
\delta_{\alpha\beta}$); ${\cal M}=\mbox{diag}(M_u,M_d,M_s)$ is the constituent quark mass matrix (to explore the properties of the spontaneously broken theory, we define quantum fields $\sigma_a, 