Effective scalaron--photon interaction in $f(R)$ gravity

Abstract

We revisit the effective coupling of the scalaron to gauge fields in $f(R)$ gravity minimally coupled to the Standard Model, focusing on the scalaron decay into two photons. Treating the scalaron as an intrinsic component of the Jordan-frame metric, we analyse quantum effects arising from its coupling to the energy--momentum tensor. In this framework, the trace anomaly contributes to the scalaron--gauge boson interaction. Using Fujikawa's approach, we obtain the trace-anomaly contribution, associated with the transformation of matter fields between the Jordan and Einstein frames, first in QED and then in the full Standard Model. The resulting effective scalaron--photon interaction agrees with direct perturbative calculations that include the trace anomaly and differs from the result obtained when only the classical expression for the trace of the energy--momentum tensor is taken into account in the scalaron interaction. In the limit where the scalaron mass is much smaller than the masses of particles circulating in the loop, the diagrammatic contribution from the classical expression of the trace of the energy--momentum tensor cancels the anomaly-induced term, leading to a vanishing effective coupling and a strong suppression of the scalaron decay rate into photons. These results clarify the origin of discrepancies in the literature concerning the effective scalaron coupling to massless gauge fields. They stem from inequivalent prescriptions for incorporating quantum loop effects in $f (R)$ gravity, leading to different scalaron--gauge field interactions with direct implications for scalaron dark-matter phenomenology.