The $ζ$-function answer to parity violation in three dimensional gauge theories

We study parity violation in $2+1$-dimensional gauge theories coupled to massive fermions. Using the $ζ$-function regularization approach we evaluate the ground state fermion current in an arbitrary gauge field background, showing that it gets two different contributions which violate parity invariance and induce a Chern-Simons term in the gauge-field effective action. One is related to the well-known {\em classical} parity breaking produced by a fermion mass term in 3 dimensions; the other one, already present for massless fermions, is related to peculiarities of gauge invariant regularization in odd-dimensional spaces.

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