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Physical running of the electric charge in a dimensionless theory of gravity

M. Gomes1,*, A. C. Lehum2,†, and A. J. da Silva1,‡

  • *Contact author: mgomes@fma.if.usp.br
  • †Contact author: lehum@ufpa.br
  • ‡Contact author: ajsilva@fma.if.usp.br
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Phys. Rev. D 113, 065026 – Published 26 March, 2026

DOI: https://doi.org/10.1103/gc68-584n

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 μ running in minimal subtraction, and (ii) a physical running extracted from the logarithmic dependence of amplitudes on a hard scale Q2 (e.g., p2 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 lnμ2 pieces cancel—leaving only ln(Q2/m2) soft (IR) logs, which do not contribute to the running of the coupling once IR effects are consistently removed. Consequently, quadratic gravity does not modify the one-loop UV coefficient and thus does not alter β(e). Therefore, the physical running coincides with the μ 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 β(e).

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Phys. Rev. D 113, 065026– Published 26 March, 2026
  • Received 21 December 2025
  • Accepted 2 March 2026
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