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

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 $Q^{2}$ (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 $\lnμ^{2}$ 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 $β(e)$. 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 $β(e)$.

💡 Research Summary

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The paper investigates the renormalization of the electric charge in massless quantum electrodynamics (QED) when it is coupled to a scale‑invariant quadratic theory of gravity. Two notions of running coupling are compared: the conventional μ‑running obtained from the ultraviolet (UV) pole in minimal‑subtraction (MS) renormalization, and a “physical” running extracted from the logarithmic dependence of amplitudes on a hard external momentum scale Q² (such as p² or a Mandelstam invariant) after infrared (IR) effects have been removed.

First, the authors set up the model. The action contains the usual Maxwell term, a Dirac fermion, and quadratic curvature terms R² and R_{μν}R^{μν} with dimensionless couplings f₀ and f₂. Expanding the metric around flat space, they derive the free propagators for the fermion, photon, and graviton. The graviton propagator behaves as 1/p⁴, reflecting the improved UV behavior of the quadratic theory. Gauge‑fixing terms for both the photon (parameter ξ_a) and the graviton (parameter ξ_g) are introduced, together with the corresponding ghost sector (which does not contribute at the order considered).

The core of the analysis is the one‑loop photon vacuum polarization Π^{μν}(p). The authors first compute the pure‑QED contribution (diagram 1.1) using dimensional regularization and an infrared mass regulator m (replacing 1/k² → 1/(k²−m²)). The result can be written as
Π^{μν}(p)= (p² η^{μν}−p^{μ}p^{ν}) Π(p) ,
with Π(p) containing a UV pole 1/ε and a logarithm ln(−p²/μ²). After adding the counterterm diagram, the MS prescription yields the familiar β‑function
β_μ(e)=μ d e/dμ = e³/(12π²).

Next, the same quantity is renormalized by imposing a physical condition at a Euclidean momentum p²=−M² (the “p‑running” scheme). This introduces a logarithm ln(−p²/M²) in the finite part, and differentiating with respect to M reproduces exactly the same β‑function. Thus, in pure QED the two definitions are equivalent.

The novel part of the work is the inclusion of the quadratic‑gravity corrections. The relevant one‑loop diagrams (Figs. 1.2 and 1.3) involve a graviton exchanged between two photon legs, with vertices derived from the expansion of the Maxwell term in the curved background. The amplitude can be expressed in terms of Passarino‑Veltman scalar integrals B₀(0,m²,m²) and B₀(p²,m²,m²). Crucially, the combination B₀(0,m²,m²)−B₀(p²,m²,m²) is UV‑finite: the ln μ² pieces cancel completely. What remains are soft logarithms of the form ln(−p²/m²) (and constants) that depend on the IR regulator m and on the graviton gauge parameter ξ_g. These terms are non‑local, gauge‑dependent, and encode soft/collinear physics; they must not be interpreted as contributions to the UV running of the electric charge.

Because the UV logarithms cancel, the quadratic‑gravity sector does not modify the coefficient of the ln μ² term in the photon self‑energy. Consequently, the one‑loop β‑function of the electric charge remains unchanged: β(e)=e³/(12π²). The “physical” running extracted after removing the soft logs coincides with the μ‑running obtained from MS.

In the discussion, the authors present a general decomposition for any one‑loop correction to a two‑point function or scattering amplitude:
A(Q²,Ω)=A_UV(Ω) ln(Q²/μ²)+A_soft(Ω) ln(Q²/m²)+finite.
Here A_UV is gauge‑ and process‑independent and determines the renormalization of the coupling, while A_soft encodes IR dynamics, may be angular dependent, and can be gauge‑dependent. Only A_UV should be used to define the physical running.

The paper concludes that, once UV and IR logarithms are cleanly separated, the two notions of running are equivalent even in the presence of higher‑derivative gravity. The apparent discrepancy reported in earlier works stems from misidentifying soft, gauge‑dependent logs as UV contributions. This clarification provides a robust framework for extracting gauge‑invariant running couplings in theories where gravity or other higher‑derivative interactions are present.