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arxiv: 2512.11742 · v1 · submitted 2025-12-12 · ✦ hep-th · hep-ph

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

M. Gomes , A. C. Lehum , A. J. da Silva This is my paper

Pith reviewed 2026-05-16 22:59 UTC · model grok-4.3

classification ✦ hep-th hep-ph
keywords QEDquadratic gravitybeta functionrunning couplingvacuum polarizationdimensional regularizationinfrared regulatorultraviolet divergences
0
0 comments X p. Extension

The pith

Quadratic gravity leaves the one-loop beta function of the electric charge unchanged in massless QED.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that in massless QED coupled to a scaleless quadratic gravity theory, the gravitational sector produces no ultraviolet divergences in the photon vacuum polarization at one loop. This means the conventional mu-running of the gauge coupling in minimal subtraction coincides exactly with a physical running extracted from the hard momentum scale dependence after infrared effects are subtracted. The key step is showing that any logarithmic terms generated by gravity depend on the infrared regulator mass and on gauge parameters, so they cannot be read as ultraviolet running. A reader would care because this distinction prevents misinterpreting infrared logarithms as modifications to the beta function when gravity is included. The result clarifies how to define a gauge- and process-independent running coupling in the presence of gravitational interactions at this perturbative order.

Core claim

Using dimensional regularization with an infrared mass regulator, the photon self-energy separates cleanly into ultraviolet poles that are gauge-independent and process-independent, fixing the same beta function for the electric charge as in pure QED, and soft logarithms of the form ln(Q²/m²) that arise from the quadratic-gravity vertices. These soft logarithms cancel in the ultraviolet after all diagrams are summed and must be discarded for the purpose of extracting ultraviolet running. Consequently, quadratic gravity contributes zero to the one-loop ultraviolet coefficient and the physical running extracted from a hard scale Q² matches the mu-running.

What carries the argument

The separation of ultraviolet poles from soft logarithms in the one-loop photon vacuum polarization, where the former determine beta(e) and the latter are identified as infrared by their dependence on the regulator mass m and gauge parameters.

If this is right

  • The one-loop ultraviolet coefficient in the beta function for e receives no correction from quadratic gravity.
  • Physical running extracted from hard-scale dependence in amplitudes coincides with minimal-subtraction running at this order.
  • Any apparent running induced by quadratic gravity is confined to infrared logarithms that depend on the regulator and gauge choice.
  • Gauge and process independence of the ultraviolet running survives the inclusion of quadratic gravity at one loop.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same separation technique could be applied to other matter couplings to isolate whether higher-order gravitational corrections remain ultraviolet-finite.
  • If soft logarithms are reclassified as ultraviolet in a different regularization scheme, the physical running would appear modified, but the paper's regulator choice and cancellation pattern argue against that reclassification.
  • The result suggests that dimensionless quadratic gravity may preserve standard-model-like running for gauge couplings through one loop, with any deviations pushed to higher orders or to infrared observables.

Load-bearing premise

That logarithms depending on the infrared regulator mass and gauge parameters after ultraviolet poles cancel represent purely soft effects and cannot be reinterpreted as ultraviolet running.

What would settle it

An explicit two-loop calculation of the photon self-energy that produces additional ultraviolet poles proportional to the quadratic-gravity couplings would falsify the claim that gravity does not modify beta(e) beyond one loop.

Figures

Figures reproduced from arXiv: 2512.11742 by A. C. Lehum, A. J. da Silva, M. Gomes.

Figure 1
Figure 1. Figure 1: Photon field self-energy. Wavy, continuos and wiggly lines represent the photon, fermionic and [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
read the original 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 $\beta(e)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper computes the one-loop photon vacuum polarization in massless QED coupled to quadratic gravity using dimensional regularization supplemented by an IR mass regulator m. It reports a clean separation in which the UV logarithms (ln μ²) from the gravitational sector cancel exactly, leaving only soft logarithms of the form ln(Q²/m²) that depend on the IR cutoff and gauge parameters. Consequently the UV coefficient fixing β(e) is identical to the pure-QED value, so that the physical running extracted from hard-scale dependence after IR subtraction coincides with the conventional μ-running at this order.

Significance. If the separation of UV and soft logarithms is robust, the result clarifies how to define a gauge- and process-independent physical running in the presence of higher-derivative gravity. It shows that quadratic gravity introduces no additional one-loop UV correction to the QED beta function, thereby removing a potential source of confusion between infrared logarithms generated by the gravitational sector and genuine ultraviolet running. This distinction is useful for future work on renormalization in gravity-matter systems.

minor comments (2)
  1. The precise operational definition of the 'physical running' (how the hard scale Q is inserted into the amplitude and how IR subtraction is performed) would benefit from an explicit formula or short subsection, even if the underlying computation is standard.
  2. A short table or equation listing the separate contributions (QED, graviton, ghost, etc.) to the coefficient of ln μ² before cancellation would make the UV finiteness of the gravitational sector immediately verifiable.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful and accurate summary of our manuscript, as well as for the positive recommendation of minor revision. The referee's description precisely reflects our one-loop calculation of the photon vacuum polarization and the resulting separation between UV and soft logarithms.

read point-by-point responses

  1. Referee: The paper computes the one-loop photon vacuum polarization in massless QED coupled to quadratic gravity using dimensional regularization supplemented by an IR mass regulator m. It reports a clean separation in which the UV logarithms (ln μ²) from the gravitational sector cancel exactly, leaving only soft logarithms of the form ln(Q²/m²) that depend on the IR cutoff and gauge parameters. Consequently the UV coefficient fixing β(e) is identical to the pure-QED value, so that the physical running extracted from hard-scale dependence after IR subtraction coincides with the conventional μ-running at this order.

    Authors: We agree with this summary. Our explicit computation in dimensional regularization with the IR regulator m demonstrates that all UV logarithms arising from the quadratic-gravity diagrams cancel exactly, independent of gauge choice. The surviving logarithms are purely soft, of the form ln(Q²/m²), and carry the expected dependence on the IR cutoff and gauge parameters. This leaves the UV coefficient in the beta function unchanged from its pure-QED value, so that the physical running defined via hard-scale dependence after IR subtraction coincides with the conventional μ-running at one loop. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper performs an explicit one-loop computation of the photon vacuum polarization in dimensional regularization with an IR mass regulator m. The UV poles (which fix β(e)) arise solely from the standard QED diagrams; quadratic-gravity diagrams are shown to cancel their ln μ² terms and leave only soft ln(Q²/m²) contributions that depend on the IR cutoff and gauge parameters. This separation is the conventional QFT procedure for isolating the beta-function coefficient and does not reduce to any fitted parameter, self-definition, or load-bearing self-citation. The result is therefore self-contained and independent of the target claim.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard one-loop perturbation theory in QED plus quadratic gravity, dimensional regularization, and the physical interpretation that only UV logarithms contribute to the beta function. No new free parameters, axioms beyond conventional QFT, or invented entities are introduced.

axioms (2)
  • domain assumption Dimensional regularization with an auxiliary IR mass regulator m separates UV and soft logarithms unambiguously
    Invoked in the computation of the photon vacuum polarization to isolate the UV piece that fixes β(e)
  • domain assumption Only the gauge- and process-independent UV logarithms define the physical running of the charge
    Stated explicitly when discarding the soft ln(Q²/m²) terms from quadratic gravity

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