Linear sigma model at finite temperature and chemical potentials

The study of QCD at low energies is relevant in explaining the world around us but is extremely difficult due to the mathematical structure of the theory. The linear sigma model is a well known and simple effective model for low-energy QCD. One couples the O(4) linear sigma model to quark fields in order to study the effects of the quarks and mesons on the chiral phase transition as functions of the temperature T and the quark chemical potential muq. As an effective model for QCD, one hopes to reproduce some aspects of the QCD phase diagram, namely, the line of first order transitions that has a critical end-point at a second order transition. One studies how this line varies with changing pion mass. One uses the self-consistent Cornwall-Jackiw-Tomboulis method in an extended Hartree approximation using a summation over all daisy diagrams. One studies the mesonic and quark properties, including mean field, fluctuations and effective masses and how they relate to the transition structure.;Pion condensation also is thought to play an important part in different nuclear matter effects. It is widely thought that the dense baryonic matter that exists inside the cores of compact stars or in heavy ion collisions is just dense quark matter with an isospin asymmetry. To study this, one uses the O(4) linear sigma model with an isospin chemical potential mu I associated with two of the pion fields. One generalizes the self-consistent method used for the previous case so that it is applicable here. One studies the phase structure, including the possibility of pion condensation. One also studies the mean fields, fluctuations and effective masses and how they relate to the transition structure.