  The coupling of spin 0 and spin 1 external fields to Dirac fermions defines a
theory which displays gauge chiral symmetry. Quantum mechanically, functional
integration of the fermions yields the determinant of the Dirac operator, known
as the chiral determinant. Its modulus is chiral invariant but not so its
phase, which carries the chiral anomaly through the Wess-Zumino-Witten term.
Here we find the remarkable result that, upon removal from the chiral
determinant of this known anomalous part, the remaining chiral invariant factor
is just the square root of the determinant of a local covariant operator of the
Klein-Gordon type. This procedure bypasses the integrability obstruction
allowing to write down a functional that correctly reproduces both the modulus
and the phase of the chiral determinant. The technique is illustrated by
computing the effective action in two dimensions at leading order in the
derivative expansion. The results previously obtained by indirect methods are
indeed reproduced.
