\hat\rho_\Lambda =1-\left(1-\Lambda^2\frac{\partial}{\partial M^2}
    \right) \exp\left(\Lambda^2\frac{\partial}{\partial M^2}\right).$$ Here the covariant cut-off $\Lambda$ is a free dimensionful parameter which characterizes the scale of the chiral symmetry breaking in the effective model considered. To the order of the heat kernel series truncated, only the integrals $J_0,J_1$ are needed. These are quadratic and logarithmic divergent respectively with $\Lambda\to\infty$, all other $J_i$ are finite. Note that the recurrence relation $$\label{rec}
    J_{i+1}(M^2)=-\frac{\partial}{\partial M^2} J_i(M^2)$$ is fulfilled. If $J_i$ is known for one value of $i$, then the function may be computed for other values of $i$ by successive applications of the relation.

In ${\cal L}_{{{\mathrm{st}}}}$ the $h_a$ are determined via the stationary phase conditions. These conditions and the pattern of explicit symmetry breaking show that in general $h_a$ can have only three non-zero components at most with indices $a=0,3,8$, i.e. $h_a\lambda_a=\mbox{diag} (h_u,h_d,h_s)$, which can be found from a system of three independent equations $$\label{SPA}
   \left\{ \begin{array}{l}
\vspace{0.2cm}   
   Gh_u + \Delta_u +\displaystyle\frac{\kappa}{16}\ h_dh_s
   +\displaystyle\frac{g_1}{4}\ h_u(h_u^2+h_d^2+h_s^2)
   +\displaystyle\frac{g_2}{2}\ h_u^3=0, \\
\vspace{0.2cm}   
   Gh_d + \Delta_d +\displaystyle\frac{\kappa}{16}\ h_uh_s
   +\displaystyle\frac{g_1}{4}\ h_d(h_u^2+h_d^2+h_s^2)
   +\displaystyle\frac{g_2}{2}\ h_d^3=0, \\
\vspace{0.2cm}   
   Gh_s + \Delta_s +\displaystyle\frac{\kappa}{16}\ h_uh_d
   +\displaystyle\frac{g_1}{4}\ h_s(h_u^2+h_d^2+h_s^2)
   +\displaystyle\frac{g_2}{2}\ h_s^3=0. 
   \end{array} \right.$$ Here $\Delta_f=M_f-m_f$, $f=u,d,s$. The matrix valued constants of higher order, like for instance $h_{ab}^{(1,2)}$ in ${\cal 