\phi_a$ as having vanishing vacuum expectation values in the asymmetric phase).

The expression for the one-quark-loop action $W_{\rm ql}$ has been obtained by using a modified inverse mass expansion of the heat kernel associated to the given Laplacian [@Osipov:2001]. The procedure takes into account the differences $\Delta_{us}=M_u^2-M_s^2$ in the nonstrange and strange constituent quark masses in a chiral invariant way at each order of the expansion, $b_i$ being the generalized Seeley–DeWitt coefficients of the new series. This modification distinguishes our calculation from the one made in [@Ebert:1986]. In fact we consider the series up to and including the order $b_2$ that corresponds to the first nontrivial step in the expansion of the induced effective hadron Lagrangian at long distances. At this stage meson fields obtain their kinetic terms, but are still considered to be elementary objects. The information about their quark-antiquark origin enters only through the coefficients such as the average $$\label{I-i}
   I_i=\frac{1}{3}\left[J_i(M_u^2)+J_i(M_d^2)+J_i(M_s^2)\right]$$ over the 1-loop euclidean momentum integrals $J_i$ with $i+1$ vertices ($i=0,1,\ldots$) $$\label{Ji}
    J_i(M^2)=16\pi^2\Gamma (i+1)\!\int              \frac{{\mathrm{d}}^4p_E}{(2\pi)^4}\,\hat\rho_\Lambda 
             \frac{1}{(p_E^2+M^2)^{i+1}}.$$ For the explicit evaluation of $J_i(M^2)$ we use the Pauli–Villars regularization method with two subtractions in the integrand. The procedure is fully defined by the insertion of the particular operator $$\label{reg-2}
    