An analytic study of the tricritical line in the U(Nf)xU(Nf) sigma model
  The tricritical line of the first-order chiral phase transition is investigated in the U(Nf)xU(Nf) sigma model by means of the ring improved one-loop finite temperature effective potential. To locate the tricritical line in the space of the coupling constants, we expand the effective potential up to third order in the high temperature expansion, and up to sixth order in the order parameter expansion. In this approximation, the tricritical line can be evaluated to the lowest order of the coupling constants and it follows an analytic relation between the tree-level masses of the scalar bosons. The validity of the high temperature expansion, the order parameter expansion, and the ring improved perturbation is critically examined. This result does not alter if one includes the massless fermions with the small Yukawa couplings. 
Junichiro Yasuda