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

This paper demonstrates that in massless QED coupled to scaleless quadratic gravity, the physical running of the electric charge coincides with the conventional $\mu$-running at one loop because the gravitational sector contributes only infrared-dominated soft logarithms that do not alter the gauge-independent UV beta function.

The Gist

Imagine the universe as a giant, complex machine. Physicists have two main blueprints for how this machine works: one for the tiny world of particles (like electrons and photons) and one for the massive world of gravity.

For a long time, scientists have struggled to combine these two blueprints. The standard theory of gravity (Einstein's) breaks down when you try to use it on the tiniest scales, while the theory of particles (Quantum Electrodynamics, or QED) works perfectly on its own but ignores gravity.

This paper tackles a specific question: If we add a "fix" to gravity to make it work better at tiny scales, does it change how electric charge behaves?

Here is the breakdown using simple analogies:

1. The "Running" Charge

In physics, the strength of an electric charge isn't actually a fixed number like a rock. It's more like a volume knob on a stereo. Depending on how close you are to the source or how much energy you use, the "volume" (the strength) seems to change. This is called "running."

Scientists have two ways to measure this volume knob:

• Method A (The "µ-running"): This is like checking the volume knob in a soundproof, controlled lab. You look strictly at the mathematical "glitches" (divergences) that happen at the highest energies and ignore the messy background noise. This is the standard, accepted way to calculate the charge.
• Method B (The "Physical running"): This is like listening to the music in a noisy room. You measure how the volume changes based on the actual sound waves hitting your ear (the momentum of the particles).

2. The New Gravity Theory

The authors are testing a specific version of gravity called "Quadratic Gravity."

• The Problem: Standard gravity is like a car with a flat tire; it gets stuck when you try to drive it at super-high speeds (the Planck scale).
• The Fix: Quadratic gravity adds extra "suspension" (mathematical terms) to the car. This makes the ride smoother at high speeds, theoretically fixing the flat tire. However, this new suspension is tricky and might introduce "ghosts" (unphysical particles) or weird behaviors.

3. The Experiment: Mixing QED and Gravity

The authors asked: If we drive our electric car (QED) on this new, fancy Quadratic Gravity road, does the volume knob (electric charge) turn differently?

They ran the numbers (calculated the "one-loop" corrections) using both Method A and Method B to see if the results matched.

4. The Big Discovery: Separating the Signal from the Noise

Here is the clever part of their finding. When they added gravity to the mix, they saw a lot of new "noise" in the data.

• The Noise (IR Effects): Think of this as the wind, rain, and road rumble. In physics, these are "soft" effects related to low energy and long distances. They depend on how you set up your experiment (gauge parameters) and are messy.
• The Signal (UV Effects): This is the actual engine performance. It represents the high-energy, fundamental changes to the theory.

The authors found that the "noise" from Quadratic Gravity was very loud. It looked like the volume knob was changing wildly. But, when they carefully filtered out the wind and rain (the soft, infrared effects), they realized the engine itself hadn't changed at all.

5. The Conclusion

• The Verdict: Quadratic gravity does not change the fundamental way electric charge runs. The "volume knob" stays exactly the same as it was in the old theory without gravity.
• The Warning: Previous studies that claimed the charge did change were likely fooled by the "noise." They mistook the wind and rain (soft, low-energy effects) for a change in the engine (high-energy physics).
• The Takeaway: The two methods of measuring the charge (Method A and Method B) actually agree with each other, provided you are careful to separate the high-energy signal from the low-energy noise.

In short: Adding this specific type of "super-suspension" to gravity makes the ride smoother at high speeds, but it doesn't change the engine's fuel efficiency. The electric charge behaves exactly as physicists expected, and any claims that it changes were just a misunderstanding of the background noise.

Technical Summary

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

Problem Statement
The paper addresses a recent controversy regarding the definition of "physical running" for coupling constants in quantum gravity. While the standard Minimal Subtraction (MS) scheme defines the beta function via the renormalization scale $\mu$ (µ-running), alternative approaches propose extracting the running from the logarithmic dependence of scattering amplitudes on external momenta ($p^2$), referred to as "physical" or $p$-running. Recent literature (Refs. [28, 32]) suggested that in quadratic gravity, these two prescriptions yield different beta functions, arguing that the physical running should be determined by the $\ln(-p^2)$ dependence. However, previous analyses were conducted in pure gravity and specific gauges, leaving open questions about gauge dependence and the separation of Ultraviolet (UV) and Infrared (IR) effects. This work investigates whether quadratic gravity modifies the one-loop beta function of the electric charge in massless QED and clarifies the equivalence (or lack thereof) between µ-running and physical running when gravitational interactions are present.

Methodology
The authors analyze massless Quantum Electrodynamics (QED) coupled to a scaleless quadratic theory of gravity. The model includes curvature-squared terms ($R^2$ and $R_{\mu\nu}R^{\mu\nu}$) which render the theory perturbatively renormalizable, alongside standard Maxwell and Dirac fields.

1. Formalism: The action is expanded around a flat Minkowski background ($g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$) to derive the effective Lagrangian for small fluctuations. Propagators for the fermion, photon, and graviton are established, including a general gravitational gauge parameter $\xi_g$.
2. Calculation: The authors compute the one-loop photon vacuum polarization (self-energy) $\Pi_{\mu\nu}(p)$. This involves evaluating Feynman diagrams corresponding to standard QED corrections and those involving graviton exchange (Figs. 1.2 and 1.3).
3. Regularization: Dimensional regularization ($D = 4 - 2\epsilon$) is used for UV divergences, while a small IR mass regulator $m$ is introduced to handle IR divergences (modifying propagators $1/k^2 \to 1/(k^2 - m^2)$).
4. Prescriptions: Two methods for extracting the beta function are compared:
   • µ-running: Derived from the UV-divergent counterterm in the MS scheme.
   • Physical running: Derived from the logarithmic dependence on the external momentum scale $Q^2$ (e.g., $p^2$) after removing IR effects, following the prescription of Peskin and Schroeder.

Key Contributions and Results
The central result is a clean separation between UV and IR logarithmic contributions in the photon self-energy:

1. QED Sector: In pure QED, the vacuum polarization contains a UV logarithm $\ln(-p^2/\mu^2)$ and soft IR logarithms. The authors confirm that both the µ-running and the physical running (defined via the renormalization scale $M$) yield the identical one-loop beta function $\beta(e) = e^3/12\pi^2$.
2. Quadratic Gravity Sector: The gravitational corrections to the photon self-energy are found to be UV finite. Specifically, the combination of scalar integrals arising from graviton loops, $B_0(0, m^2, m^2) - B_0(p^2, m^2, m^2)$, results in the cancellation of all $\ln(\mu^2)$ terms.
3. Nature of Gravitational Logs: While the gravitational contribution contains terms proportional to $\ln(-p^2/m^2)$, these are identified as soft IR logarithms, not UV running.
   • These terms depend on the IR regulator $m$ and the gravitational gauge parameter $\xi_g$.
   • They encode nonlocal, IR-dominated physics (soft/collinear dynamics) rather than the renormalization group evolution of the coupling.
4. Beta Function Equivalence: Because the gravitational sector contributes no UV-divergent $\ln(\mu^2)$ terms, the coefficient of the UV logarithm (which determines the beta function) remains unchanged from the pure QED case. Consequently, the "physical" running, once IR effects are properly separated and discarded, coincides exactly with the conventional µ-running. The quadratic gravity sector does not modify the one-loop beta function of the electric charge.

Significance and Claims
The paper claims to resolve the apparent discrepancy between µ-running and physical running in the context of quadratic gravity. The authors argue that the differences reported in previous works (Refs. [28, 32]) likely stem from the premature omission of UV-divergent integrals and the subsequent misidentification of the logarithmic dependence of scalar integrals (like $B_0(p^2, m^2, m^2)$) as physical running.

The authors emphasize that the mere presence of a $\ln(-p^2)$ term does not signal UV running; one must distinguish between:

• UV logs: $\ln(Q^2/\mu^2)$, which are gauge-independent, process-independent, and fix the beta function.
• Soft IR logs: $\ln(Q^2/m^2)$, which are gauge-dependent, process-dependent, and must not be interpreted as contributions to the renormalization group evolution.

By demonstrating that the gravitational corrections to the photon self-energy are UV finite and that the remaining momentum logarithms are purely IR in origin, the paper establishes that the two prescriptions for the running of the electric charge are equivalent at one loop. The analysis provides a rigorous framework for extracting gauge- and process-independent running couplings in the presence of gravitational interactions, cautioning against interpreting soft IR logarithms as evidence of modified UV running.