2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/190655Explicit exact formulas are presented, up to fourth order in a strict chiral covariant derivative expansion, for the normal parity component of the Euclidean effective action of even-dimensional Dirac fermions. The bosonic background fields considered are scalar, pseudo-scalar, vector and axial vector. No assumptions are made on the internal symmetry group and, in particular, the scalar and pseudo-scalar fields need not be on the chiral circle.24 pages, REVTEX, no figures. Minor correctionsHigh Energy Physics - Theorytext