paper we give a quantitative account of local multi-fermion forces on the phase diagram in the $(T,\mu )$ plane, by comparing the results for two sets of parameters, in the small and large coupling regimes of the $8q$ strengths, corresponding to the two above mentioned alternative mechanisms of chiral symmetry breaking.

Throughout the paper we work for simplicity in the isospin limit $m_u=m_d
\ne m_s$, breaking explicitly the chiral $SU(3)_L\otimes SU(3)_R$ symmetry to the $SU(2)_I\otimes U(1)_Y$ (isospin-hypercharge) subgroup, and take the same baryonic chemical potential $\mu$ for all quark species. Generalizations to take into account the nonzero isospin chemical potential can be implemented as for instace in [@Klimenko:2006].

Effective Lagrangian
====================

The explicit form of the multi-quark Lagrangian considered is presented in [@Osipov:2005b; @Osipov:2006a] $$\label{efflag}
  {\cal L}_{\mathrm{eff}}=\bar{q}(i\gamma^\mu\partial_\mu - m)q
          +{\cal L}_{4q} + {\cal L}_{6q}
          +{\cal L}_{8q}^{(1)}+{\cal L}_{8q}^{(2)}.$$ Quark fields $q$ have color $(N_c=3)$ and flavor $(N_f=3)$ indices which are suppressed, $\mu =0,1,2,3$. Here $$\begin{aligned}
\label{L4q}
   {\cal L}_{4q} &\!\! =\!\! & \frac{G}{2}\left[(\bar{q}\lambda_aq)^2+
                      (\bar{q}i\gamma_5\lambda_aq)^2\right], \\
\label{Ldet}
   {\cal L}_{6q} &\!\! =\!\! & \kappa (\mbox{det}\ \bar{q}P_Lq
                      + \mbox{det}\ \bar{q}P_Rq), \\  
   {\cal L}_{8q}^{(1)}&\!\! = \!\! & 
   8g_1\left[ (\bar q_iP_Rq_m)(\bar q_mP_Lq_i) \right]^2, \\ 
   {\cal L}_{8q}^{(2)}&\!\! = \!\!& 
   16 g_2\left[ (\bar q_iP_Rq_m)(\bar q_mP_Lq_j) 
   (\bar q_jP_Rq_k)(\bar q_kP_Lq_i) \right]. 