Some properties of the finite temperature chiral phase transition

We study the phase transition of the (3 + 1)-dimensional Yukawa model at finite temperature. We calculate the critical exponents in the $\frac{1}{N}$ expansion and clarify certain subtleties involved in such a calculation. In the leading order we do not find the presence of any of the metastable states which were claimed in the literature. To this order, the exponents are the mean field, but corrections shift them to the usual nontrivial values. Dimensional reduction of this model is studied with special attention paid to the discrete symmetries of the Lagrangian before and after reduction. In the reduced $d=3\mathrm{}$ theory there are two possible types of mass terms, one of which is allowed by chiral symmetry. It is the different discrete symmetries of these two mass terms which force the finite temperature 3 + 1 Yukawa Lagrangian to reduce to the usual scalar universality class (characterized by a conformally invariant $\sigma$ model) rather than the chiral universality class (characterized by $d=3\mathrm{}$ conformally invariant NJL-type model).