2 IV. THE ROLE OF THE CURRENT MASS AND THE PHASE DIAGRAM ( µ  µ e ) FIG. 2: Phase diagram of neutral matter in ( µ  m π ) plane. FIG. 1: The constituent quark mass M   the pion condensate N   and meson masses as a function of µ at T = 0 in the neutral phase for a toy value of the current quark mass m = 10 keV. In the past years  a large number of the analysis about π c condensation have been performed in the chiral limit. In this section we investigate the role of the finite current quark mass in π c condensation. In order to do this  we set the cutoff Λ and the coupling G to the values spec- ified above and we treat m as a free parameter. As a consequence  the pion mass at µ = T = 0  m$_{π}$ in the fol- lowing  is a free parameter as well. In Fig. 2 we report the phase diagram in ( µ  m$_{π}$ ) plane in the neutral case. The solid line represents the border between the two regions where chiral symmetry is broken and restored. The bold dot is the critical endpoint of the first order transition. The shaded region indicates the region where π c conden- sation occurs. In the chiral limit ( m$_{π}$ = 0) our results are in good agreement with those obtained in Ref. [16]. Indeed  there exist two critical values of the quark chem- ical potential  µ$_{c}$$_{1}$ and µ$_{c}$$_{2}$   corresponding to the onset and vanishing of π c condensation  respectively. When the current quark mass increases  a shrinking of the shaded region occurs till the point µ$_{c}$$_{1}$ ≡ µ$_{c}$$_{2}$ for m c π ∼ 9 MeV  corresponding to a current quark mass of m ∼ 10 keV. Hence the gapless π c condensation is extremely fragile with respect to the symmetry breaking effect of the cur- rent quark mass. As a final investigation  in Fig. 3 we report the phase diagram of quark matter in the ( µ  µ$_{e}$ ) plane when the current quark mass is tuned to m = 5 . 5 MeV. At each value of ( µ  µ$_{e}$ ) we compute the chiral and pion condensates by minimization of the thermody- namical potential. The solid line represents the first or- der transition from the π c condensed phase to the chiral symmetry broken phase without the π c condensate. The bold dot is the critical endpoint for the first order transi- tion  after which the second order transition sets in. The dashed line indicates the first order transition between the two regions where chiral symmetry is broken and restored  respectively. The dot-dashed line is the neu- trality line µ neut e = µ$_{e}$ ( µ ) which is obtained by requiring the global electrical neutrality condition  ∂ Ω /∂µ$_{e}$ = 0. the threshold for the onset of the condensation is found to be µ$_{e}$ = m$_{π}$   i.e.  when the absolute value of the electric chemical potential equals the vacuum pion mass. We now consider neutral quark matter at µ = 0 and T = 0 and we want to study the relation between the threshold of π c condensation at finite density and the in- medium pion masses in the neutral ground state (for fur- ther recent studies at finite baryon and/or isospin chem- ical potential see also Refs. [13  14]). In Ref. [10]  it is shown that at the physical point m = 5 . 5 MeV  there is no room for π c condensation in the neutral phase (sim- ilar results obtained also in Ref. [15]). Even though the picture changes when the current mass is lowered  for our discussion it is enough to state that we consider a current quark mass of the order of 10 keV. In Fig. 1 we plot M and N in the neutral phase as a function of µ . In this fig- ure M$_{π}$ 0   M$_{π}$ ± denote the in-medium pion masses defined by the poles of the pion propagators in the rest frame  computed in the randomt phase approximation (RPA) to the Bethe-Salpeter (BS) equation. The positive and negative solutions of the BS equation in ω corresponds to the excitation gaps for π + and π −   which are ( M$_{π}$ + + µ$_{e}$ ) and ( M$_{π}$ − − µ$_{e}$ )  respectively. From Fig. 1 we notice that the transition to the pion condensed phase is of second order and it occurs at the point where M$_{π}$ − = µ$_{e}$ . For a more detailed mathematical discussion of the numerical results shown in Fig. 1 we refer to [9].