From renormalized stress-energy tensor to the effective action for Dirac fields in curved spacetime

We present a detailed analysis of the renormalization of Dirac fermions coupled to curved spacetime geometry with torsion and scalar fields, employing the point-splitting method and heat kernel techniques. Building upon the classical Dirac action, we compute one-loop quantum corrections to the effective action, focusing on ultraviolet divergences and their cancellation via counterterms. Our approach incorporates non-minimal curvature couplings, Yukawa interactions with scalar fields, and torsion-induced axial vector couplings. By deriving explicit expressions for the Seeley–DeWitt coefficients and performing spinor trace computations, we determine the beta functions governing the renormalization group flow of gravitational and matter couplings. These results provide crucial insights for quantum field theories in geometrically rich backgrounds, with potential applications in early-universe cosmology, quantum gravity, and supergravity models.