  We consider the one loop effective action in four dimensional Euclidean space
for a background chiral field coupled to a spinor field. It proves possible to
find an exact expression for this action if the mass $m$ of the spinor
vanishes. If $m$ does not vanish, one can make a perturbative expansion in
powers of the axial field that contributes to the chiral field, while treating
the contribution of the vector field exactly when it is a constant. The
analogous problem in two dimensions is also discussed.
