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Philip Mannheim
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Anomalous Dimensions and the Renormalizability of the Four-Fermion Interaction
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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.
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