Conformal Anomalies and the Gravitational Effective Action: The $TJJ$ Correlator for a Dirac Fermion

TL;DR

The paper computes the one-loop TJJ correlator for a Dirac fermion to illuminate gauge contributions to the gravitational conformal anomaly and its nonlocal effective action. By enforcing vector current Ward identities and a trace condition, it decomposes the amplitude onto a finite 13-term basis and identifies an anomaly pole in the form factor F1, separating it from conformal-invariant pieces via a pole term S_pole and a local remainder. The work analyzes both off-shell and on-shell, massive and massless regimes, clarifying how anomaly poles can dominate ultraviolet behavior while decoupling in the infrared for massive cases, and how mass corrections modify the trace anomaly. This provides a concrete perturbative realization of the anomaly’s nonlocal structure, connects to the running of the gauge coupling, and furnishes a framework for extending to the full Standard Model coupled to gravity and potential cosmological applications.

Abstract

We compute in linearized gravity all the contributions to the gravitational effective action due to a virtual Dirac fermion, related to the conformal anomaly. This requires, in perturbation theory, the identification of the gauge-gauge-graviton vertex off mass shell, involving the correlator of the energy-momentum tensor and two vector currents ($TJJ$), which is responsible for the generation of the gauge contributions to the conformal anomaly in gravity. We also present the anomalous effective action in the inverse mass of the fermion as in the Euler-Heisenberg case.

Figures (1)

• Figure 1: The complete one-loop vertex (a) given by the sum of the 1PI contributions called $V^{\mu \nu \alpha \beta} (p,q)$ (b) and $W^{\mu \nu \alpha \beta} (p,q)$ (c).