QCD phase diagram from chiral symmetry restoration: analytic approach at high and low temperature using the Linear Sigma Model with Quarks

We use the linear sigma model with quarks to study the QCD phase diagram from the point of view of chiral symmetry restoration. We compute the leading order effective potential for high and low temperatures and finite quark chemical potential, up to the contribution of the ring diagrams to account for the plasma screening effects. We fix the values of the model couplings using physical values for the input parameters such as  the vacuum pion and sigma masses, the critical temperature at vanishing quark chemical potential and the conjectured end point value of the baryon chemical potential of the transition line at vanishing temperature. We find that the critical end point (CEP) is located at low temperatures and high quark chemical potentials $(\mu^{\text{CEP}}>320\ {\mbox{MeV}},T^{\text{CEP}}<40\ {\mbox{MeV}})$.