Anomalous Dimensions and the Renormalizability of the Four-Fermion Interaction

Abstract

We show that when the dynamical dimension of the $\bar{\psi}\psi$ operator is reduced from three to two in a fermion electrodynamics with scaling, a $g(\bar{\psi}\psi)^2+g(\bar{\psi}i\gamma^5\psi)^2$ four-fermion interaction which is dressed by this electrodynamics becomes renormalizable. In the fermion-antifermion scattering amplitude every term in an expansion to arbitrary order in $g$ is found to diverge as just a single ultraviolet logarithm (i.e. no log squared or higher), and is thus made finite by a single subtraction. While not necessary for renormalizability per se, the reduction in the dimension of $\bar{\psi}\psi$ to two leads to dynamical chiral symmetry breaking in the infrared, with the needed subtraction then automatically being provided by the theory itself through the symmetry breaking mechanism, with there then being no need to introduce the subtraction by hand. Since the vector and axial vector currents are conserved, they do not acquire any anomalous dimension, with the four-fermion $(\bar{\psi}\gamma^{\mu}\psi)^2$ and $(\bar{\psi}\gamma^{\mu}\gamma^5\psi)^2$ interactions instead having to be controlled by the standard Higgs mechanism.