arxiv: 2603.05476 · v2 · submitted 2026-03-05 · ✦ hep-th

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Scattering amplitudes in dimensionless quadratic gravity coupled to QED

I. F. Cunha , A. C. Lehum

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Pith reviewed 2026-05-15 14:49 UTC · model grok-4.3

classification ✦ hep-th

keywords scattering amplitudesquadratic gravityQEDultra-Planckian scatteringdifferential cross sectionshigher-derivative gravitytree-level processes

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The pith

In quadratic gravity coupled to QED, all analyzed ultra-Planckian scattering channels exhibit differential cross sections scaling as 1/s.

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

The paper computes analytic expressions for tree-level scattering amplitudes in a scale-free quadratic gravity model coupled to quantum electrodynamics. It includes processes with charged fermions and scalars interacting via photons and higher-derivative gravitons, keeping photon-graviton interference terms. The results show that the differential cross sections for all channels follow dσ/dΩ ∝ 1/s at high energies. This matters because it gives a concrete way higher-derivative gravity alters familiar QED scattering when energies approach the Planck scale.

Core claim

In agravity, the dimensionless quadratic gravity theory, coupled to QED, the unpolarized squared matrix elements for 2→2 scatterings of photons, fermions, and scalars are computed at tree level. The amplitudes feature the characteristic 1/p^4 graviton propagator leading to forward enhancements, and after integration, the differential cross sections exhibit the universal ultra-Planckian scaling dσ/dΩ ∝ 1/s independent of the specific channel and gravitational gauge choice.

What carries the argument

The higher-derivative graviton propagator proportional to 1/p^4 arising from the quadratic curvature terms in the agravity action, which ensures dimensionless couplings and dictates the high-energy scaling when interfering with photon exchange.

If this is right

Analytic expressions are provided for photon-photon, fermion-fermion, fermion-photon, scalar-fermion, scalar-photon, scalar-scalar, and annihilation channels.
Results are independent of the gravitational gauge-fixing parameter.
The scaling holds with the inclusion of photon-graviton interference contributions.
Forward and backward peaks are amplified by the graviton propagator.

Where Pith is reading between the lines

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

This universal scaling could provide a basis for defining infrared behaviors in agravity without additional assumptions.
It suggests that similar high-energy behaviors might appear in other renormalizable gravity-matter models.
Extensions to include loop corrections could test if the tree-level scaling persists.

Load-bearing premise

The assumption that tree-level calculations with the higher-derivative graviton propagator suffice to describe the dynamics at ultra-Planckian energies without higher-order or non-perturbative effects.

What would settle it

A calculation of one-loop corrections that changes the 1/s scaling or an experimental observation of high-energy scattering cross sections deviating from this behavior would falsify the claim.

Figures

Figures reproduced from arXiv: 2603.05476 by A. C. Lehum, I. F. Cunha.

**Figure 2.** Figure 2: Tree-level graviton-exchange diagrams for [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

**Figure 3.** Figure 3: Tree-level diagrams for e −e + → γγ, including the standard QED contributions and the additional graviton-exchange diagram. γ e γ e e 1 γ e γ e h 2 γ e γ e e 3 Scattering [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

**Figure 4.** Figure 4: Tree-level diagrams for Compton scattering [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗

**Figure 5.** Figure 5: Tree-level diagrams contributing to ϕ e− → ϕ e− (charged scalar–electron scattering), including photon exchange and graviton exchange. Straight, wavy, and curly lines denote, respectively, electrons, photons, and gravitons, while dashed lines denote the charged scalar field [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗

**Figure 6.** Figure 6: Tree-level diagrams for the annihilation channel [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗

**Figure 7.** Figure 7: Tree-level Feynman diagrams for γ ϕ → γ ϕ (scalar Compton scattering), showing the photon-exchange contribution and the graviton-exchange contribution. ϕ ϕ ϕ ϕ 1 ϕ ϕ ϕ ϕ h 2 ϕ ϕ ϕ ϕ γ 3 ϕ ϕ ϕ h ϕ 4 ϕ ϕ ϕ γ ϕ 5 Scattering [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗

**Figure 8.** Figure 8: Tree-level diagrams contributing to ϕ ϕ → ϕ ϕ (elastic charged scalar scattering), including the quartic scalar interaction, photon exchange, and graviton exchange [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗

read the original abstract

We study ultra-Planckian $2\to2$ scattering in an Abelian gauge theory coupled to agravity, the scale-free and renormalizable realization of quadratic quantum gravity. Focusing on charged fermions and scalars interacting with the photon and the higher-derivative graviton, we present compact analytic expressions for the unpolarized squared matrix elements for a broad set of tree-level processes, including photon--photon, fermion--fermion, fermion--photon, scalar--fermion, scalar--photon, scalar--scalar, and annihilation channels. In contrast to purely graviton-mediated analyses, we retain systematically the photon--graviton interference contributions and verify explicitly the independence of the results on the gravitational gauge-fixing parameter. The amplitudes display characteristic forward/backward enhancements associated with small momentum transfer, amplified by the $1/p^{4}$ graviton propagator, while their high-energy scaling reflects the underlying dimensionless gravitational couplings. Moreover, for all channels analyzed the corresponding differential cross sections exhibit the universal ultra-Planckian scaling $d\sigma/d\Omega \propto 1/s$, where $s$ is the Mandelstam invariant (the squared center-of-momentum energy). Our results furnish a unified amplitude-level description of how higher-derivative gravity reshapes familiar QED scattering at ultra-Planckian energies and provide analytic building blocks for further studies of IR definitions and UV consistency in agravity with matter.

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.

Desk Editor's Note private letter to a colleague

This paper supplies explicit tree-level amplitudes for gravity-QED scattering with interference terms and confirms the universal 1/s high-energy scaling of the cross sections.

read the letter

This paper works out the tree-level scattering amplitudes for various 2-to-2 processes in a scale-free quadratic gravity model coupled to QED. The key result is a set of compact analytic expressions for the unpolarized |M|^2 that include both graviton and photon exchanges plus their interference, and they all lead to differential cross sections that fall as 1/s in the ultra-Planckian regime. What the authors do well is extend the pure-gravity calculations by systematically adding charged matter and keeping the interference terms. They also verify gauge independence for the gravitational part, which is a straightforward but necessary check. The scaling argument follows directly from the dimensionless couplings and the 1/p^4 propagator, so the 1/s behavior is not surprising once you see the structure. The main limitation is that this remains a tree-level analysis. At energies far above the Planck scale, one could question whether the effective description holds without resummation or other corrections, but the paper presents it as a direct computation from the Lagrangian without claiming broader validity. No numerical cross-checks or comparisons to other methods are mentioned, though the analytic forms are the main deliverable. This kind of work is useful for theorists who need explicit amplitudes in quadratic gravity with matter, perhaps for studying unitarity or black hole scattering in these models. A reader interested in quantum gravity phenomenology or renormalizable gravity extensions would get value from the formulas. It is solid enough on its own terms to warrant peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript computes tree-level 2→2 scattering amplitudes in dimensionless quadratic gravity (agravity) coupled to QED for channels involving charged fermions, scalars, and photons. It derives compact analytic expressions for the unpolarized |M|^2, retains photon-graviton interference, verifies gravitational gauge-parameter independence, and reports that all differential cross sections exhibit the universal ultra-Planckian scaling dσ/dΩ ∝ 1/s.

Significance. If the analytic results hold, the work supplies explicit, reusable building blocks for high-energy scattering in a renormalizable gravity-matter model. The universal 1/s scaling follows directly from the dimensionless couplings and 1/p^4 graviton propagator, distinguishing the model from Einstein gravity and offering concrete input for studies of IR definitions and UV consistency.

minor comments (2)

The abstract states that compact analytic expressions are presented, but the manuscript would benefit from a summary table listing each process, its |M|^2 form, and the leading high-s term to improve readability and allow quick cross-checks.
A short paragraph clarifying the kinematic regime (e.g., fixed-angle ultra-Planckian limit) and the neglect of higher-loop or non-perturbative effects would strengthen the discussion of the tree-level results without altering the central scaling claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript and the recommendation for minor revision. The assessment correctly identifies the key results: compact analytic expressions for tree-level amplitudes in agravity-coupled QED, retention of photon-graviton interference, gauge-parameter independence, and the universal ultra-Planckian scaling dσ/dΩ ∝ 1/s. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

Direct Feynman-rule computation from Lagrangian; no circularity

full rationale

The paper derives analytic |M|^2 expressions for tree-level 2→2 processes directly from the agravity+QED Lagrangian using standard Feynman rules. The claimed universal dσ/dΩ ∝ 1/s scaling at high s follows immediately from the dimensionless gravitational couplings (propagator ~1/p^4, vertices ~s) making |M| s-independent and the flux factor supplying the extra 1/s; this is ordinary dimensional analysis, not a fit or self-definition. No self-citations are load-bearing for the central result, no parameters are fitted to data and renamed as predictions, and the gauge-independence check is an internal consistency verification rather than a reduction to prior work. The derivation chain is self-contained against the model Lagrangian.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard QFT perturbation theory and the definition of agravity as scale-free renormalizable quadratic gravity; no free parameters or new entities are introduced in the abstract description.

axioms (2)

standard math Standard quantum field theory Feynman rules and tree-level perturbation theory apply to the coupled agravity-QED Lagrangian
Invoked for computing all listed 2-to-2 amplitudes
domain assumption The agravity model is a consistent, renormalizable realization of quadratic quantum gravity at ultra-Planckian energies
Required for the validity of the higher-derivative graviton propagator and the claimed scaling

pith-pipeline@v0.9.0 · 5547 in / 1254 out tokens · 46181 ms · 2026-05-15T14:49:10.964125+00:00 · methodology

discussion (0)

Reference graph

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