Chiral phase transitions in d = 3 and renormalizability of four-Fermi interactions

We consider the Z₂ chiral phase transition in 2 + 1 dimensions. It is shown that the infrared fixed point defining this universality class is the critical four-Fermi interaction theory. We calculate, using 1/N expansion, the UV dimension of various local operators. It is found that the fermionic mass and four-Fermi interaction operators are relevant with UV dimensions 1 + O(1/N) and 2 + O(1/N), respectively. This is the reason behind the non-perturbative renormalizability of four-Fermi interaction theories. The six-Fermi operator has dimension 3 + 8/π²N + O(1/N²) and is therefore irrelevant. The generalization to universality classes corresponding to breakdown of larger chiral groups is discussed.