On the physical running of the electric charge in a dimensionless theory of gravity

TL;DR

This paper analyzes the renormalization of the electric charge in massless QED coupled to quadratic gravity, clarifying the distinction between UV and IR contributions to the running of the coupling.

Contribution

It demonstrates that quadratic gravity does not affect the one-loop beta function of QED and clarifies how to properly define gauge-independent running in gravitational contexts.

Findings

UV and IR logarithms are clearly separated in the photon vacuum polarization.

Quadratic gravity does not modify the one-loop UV coefficient of the beta function.

Soft IR logs should not be interpreted as contributions to the UV running of the charge.

Abstract

We revisit the renormalization of the gauge coupling in massless QED coupled to a scaleless quadratic theory of gravity. We compare two alternative prescriptions for the running of the electric charge: (i) the conventional $\mu$-running in minimal subtraction, and (ii) a ''physical'' running extracted from the logarithmic dependence of amplitudes on a hard scale $Q^{2}$ (e.g., $p^{2}$ or a Mandelstam invariant) after removing IR effects. At one loop, using dimensional regularization with an IR mass regulator $m$, we compute the photon vacuum polarization. We find a clean separation between UV and soft logarithms: the former is gauge and process independent and fixes the beta function, whereas the latter encodes nonlocal, IR-dominated contributions that may depend on gauge parameters and must not be interpreted as UV running. In the quadratic-gravity sector, the photon self-energy is UV finite--the $\ln\mu^{2}$ pieces cancel--leaving only $\ln(Q^{2}/m^{2})$ soft logs. Consequently, quadratic gravity does not modify the one-loop UV coefficient and thus does not alter $\beta(e)$. Therefore, the physical running coincides with the $\mu$-running in QED at one loop. Our analysis clarifies how to extract a gauge and process independent running in the presence of gravitational interactions and why soft logs from quadratic gravity should not contribute to $\beta(e)$.