  We consider the quantum effective action of Dirac fermions on four
dimensional flat Euclidean space coupled to external vector- and axial
Yang-Mills fields, i.e., the logarithm of the (regularized) determinant of a
Dirac operator on flat R^4 twisted by generalized Yang-Mills fields. According
to physics folklore, the logarithmic divergent part of this effective action in
the pure vector case is proportional to the Yang-Mills action. We present an
explicit computation proving this fact, generalized to the chiral case. We use
an efficient computation method for quantum effective actions which is based on
calculation rules for pseudo-differential operators and which yields an
expansion of the logarithm of Dirac operators in local and quasi-gauge
invariant polynomials of decreasing scaling dimension.
