$0^{-+}$ and $0^{++}$ states [@Osipov:2006a] only depend on the couplings $G$ and $g_1$ through the linear combination $\xi=G+
g_1(h_u^2+h_d^2+h_s^2)/4$, except for the $00,\, 08$ and $88$ states inside the scalar nonet. However as soon as $T$ or $\mu$ are finite, the $h_f$ start to change due to their intrinsic $T,\mu$ dependence, acquired through the coupling to the quark loop integrals in the gap equations, eqs. (\[gap\]) below. The $T,\mu$ dependence of the combination $\xi$ above is steered by the strength $g_1$. This is the main reason why the $8q$-interactions may strongly affect the thermodynamic observables, without changing the spectra at $T=\mu=0$.

Thermodynamic potential
=======================

Case of vanishing temperarure and chemical potential
----------------------------------------------------

Before addressing the thermodynamical potential it is instructive to briefly discuss the effective potential of the model at $T=\mu=0$. Using standard techniques [@Osipov:2004a], we obtain from the gap-equations $$\label{gap}
            h_f + \frac{N_c}{2\pi^2} M_f J_0(M_f^2)=0$$ the effective potential $V(M_f)$ as a function of three independent variables $M_f=\{M_u,M_d,M_s\}$. If the parameters of the model are fixed in such a way that eqs. (\[SPA\]) have only one real solution, the effective potential is $$\begin{aligned}
\label{effpot1}
     V(M_f)&\!\! =\!\!&-\frac{1}{2}\int_0^{M_f}\!\!\sum_{f=u,d,s}h_f {\mathrm{d}}M_f
           -\frac{N_c}{4\pi^2}\int_0^{M_f}\!\!\sum_{f=u,d,s}M_fJ_0(M_f^2){\mathrm{d}}M_f
           \nonumber \\
           &\!\! = \!\!&\frac{1}{16}\left.\left(4Gh_f^2+\kappa h_uh_dh_s 
           +\frac{3g_1}{2}\left(h_f^2\right)^2+3g_2h_f^4\right)\right|^{M_f}_0 
           +\frac{N_c}{8\pi^2}\sum_{f=u,d,s}\! J_{-1}(M_f^2),\end{aligned}$$ where $h_f^2=h_u^2+h_d^2+h_s^2, \,\, 