Linear sigma model at finite density in the $1/N$ expansion to next-to-leading order

We study relativistic Bose-Einstein condensation at finite density and temperature using the linear sigma model in the one-particle-irreducible $1/N$ expansion. We derive the effective potential to next-to-leading (NLO) order and show that it can be renormalized in a temperature-independent manner. As a particular application, we study the thermodynamics of the pion gas in the chiral limit as well as with explicit symmetry breaking. At nonzero temperature we solve the NLO gap equation and show that the results describe the chiral-symmetry-restoring second-order phase transition in agreement with general universality arguments. However, due to nontrivial regularization issues, we are not able to extend the NLO analysis to nonzero chemical potential.

Figure 1

Phase diagram for pion condensation at the leading order. The upper curve is the chiral limit and the lower curve corresponds to the physical point.

Figure 2

Pion condensate in the chiral limit as a function of temperature and chemical potential.

Figure 3

Pion condensate at the physical point as a function of temperature and chemical potential.

Figure 4

Solution of the NLO gap equation in the chiral limit. Upper panel: the chiral condensate; lower panel: the mass parameter $M$. For reference, the LO values are shown by the dashed lines.

Figure 5

Solution of the NLO gap equation at the physical point (solid lines). Upper panel: the chiral condensate; lower panel: the mass parameter $M$. For reference, the LO values are shown by the dashed lines. We also show the NLO chiral condensate calculated using Eqs. (28) (dash-dotted) and (30) (dotted), respectively.

Figure 6

Critical temperature for chiral symmetry breaking as a function of the coupling. The dashed line denotes the LO critical temperature while the dotted one indicates the weak-coupling NLO limit (27).