Scattering amplitudes in dimensionless quadratic gravity coupled to QED

This paper presents compact analytic expressions for tree-level 2 → 2 scattering amplitudes in agravity coupled to QED, systematically including photon-graviton interference contributions and demonstrating characteristic ultra-Planckian scaling and infrared enhancements.

The Gist

The High-Energy Dance of Particles

A simple guide to "Scattering amplitudes in dimensionless quadratic gravity coupled to QED"

Imagine the universe is a giant dance floor. Usually, the dancers are divided into two groups: the Gravity Dancers (who care about mass and space) and the Quantum Dancers (who care about tiny particles like electrons and light). The problem is, they speak different languages. When they try to dance together at super-high speeds, the music usually breaks down, and the math explodes.

This paper is about teaching these two groups how to dance together without tripping over each other, specifically at energies so high they are "ultra-Planckian" (far beyond what our current particle colliders can reach).

Here is the breakdown of what the scientists did and what they found.

1. The New Dance Rules (Agravity)

Standard gravity (Einstein’s General Relativity) is great for planets and stars, but it gets messy at the quantum level. To fix this, the authors use a modified version of gravity called "Agravity."

• The Analogy: Imagine standard gravity is like a trampoline. If you put a bowling ball on it, it curves. But if you put a tiny marble on it at super-speed, the math describing the bounce gets crazy.
• The Fix: Agravity adds "extra springs" to the trampoline (mathematically, these are squared curvature terms). This makes the trampoline stable even when the marble is moving at impossible speeds. It also makes the rules "scale-free," meaning the dance looks the same whether you zoom in or out, until you hit the very smallest limits.

2. The Players

The paper looks at specific collisions between four types of "dancers":

1. Photons: Particles of light.
2. Electrons: Charged matter particles.
3. Scalars: Hypothetical charged particles (like a cousin to the Higgs boson).
4. Gravitons: The invisible particles that carry the force of gravity.

3. The Collision (Scattering)

The scientists calculated what happens when these particles crash into each other. In physics, this is called "scattering."

• The Old Way: Usually, we calculate how electrons bounce off each other using just electricity (photons).
• The New Way: In this paper, they calculated how electrons bounce off each other using both electricity and gravity at the same time.
• The Interference: Imagine throwing two stones into a pond. The ripples meet. Sometimes they add up to make a big wave; sometimes they cancel out. The paper calculates exactly how the "gravity ripples" and "light ripples" mix together.

4. Key Discoveries

After doing the complex math, the authors found three main things:

A. The "Grazing" Effect
When particles collide at these super-high energies, they don't usually bounce straight back. They tend to "graze" past each other.

• The Analogy: Think of two cars driving past each other on a highway. They don't crash head-on; they just pass close by.
• The Finding: The new gravity rules make this "grazing" even more likely. The particles prefer to scatter at very small angles (forward or backward). The math shows a huge spike in probability when the angle is tiny.

B. The Universal Volume Knob
No matter which particles are dancing (electrons, photons, or scalars), the "loudness" of the interaction follows a specific rule as energy goes up.

• The Finding: The probability of a specific type of collision drops in a predictable way as the energy increases. It’s like a universal volume knob that turns down the interaction strength in a consistent pattern, regardless of the particle type.

C. The Invisible Handshake
One of the most important checks in physics is ensuring the math doesn't depend on arbitrary choices made by the scientist (called "gauge-fixing").

• The Finding: The authors proved that their results are solid. No matter how they set up their mathematical "camera angles," the final answer for the particle collisions remained the same. This means the theory is consistent.

5. Why Does This Matter?

You might ask, "We can't build a machine to reach these energies, so why bother?"

• The Blueprint: This paper provides a "blueprint" for how the universe might work at its very beginning (like the Big Bang).
• The Stress Test: It helps physicists test if their theories of gravity are actually mathematically possible. If the math breaks, the theory is wrong.
• The Bridge: It builds a bridge between the world of light (QED) and the world of gravity, showing us how they interfere with each other when things get really hot and fast.

Summary

In short, this paper is a detailed instruction manual for how particles behave when they collide at speeds where gravity and quantum mechanics fight for control. They found that with this specific type of "super-gravity," particles prefer to skim past each other, and the math holds together nicely without breaking. It’s a step toward understanding the ultimate rules of the universe.

Technical Summary

Here is a detailed technical summary of the paper "Scattering amplitudes in dimensionless quadratic gravity coupled to QED" by I. F. Cunha and A. C. Lehum.

1. Problem Statement

The paper addresses the behavior of quantum gravity at ultra-Planckian energies ($E \gg M_P$) when coupled to standard matter fields. Standard Einstein-Hilbert General Relativity is non-renormalizable in a perturbative expansion. Quadratic quantum gravity, which includes curvature-squared terms ($R^2, R_{\mu\nu}^2$), offers a power-counting renormalizable alternative. Specifically, the authors focus on "agravity," a scale-free realization of quadratic gravity where the classical action contains no Einstein-Hilbert term and no explicit mass parameters.

While previous studies have analyzed purely graviton-mediated processes, there is a lack of systematic understanding regarding how higher-derivative gravity interferes with standard gauge interactions (specifically QED) in the presence of charged matter (fermions and scalars). The core problem is to determine the tree-level scattering amplitudes in this coupled system, explicitly accounting for photon-graviton interference, and to analyze the resulting ultraviolet (UV) scaling and infrared (IR) singularities.

2. Methodology

The authors employ a perturbative field-theoretic approach within the framework of agravity coupled to an Abelian gauge theory (QED).

• Lagrangian Formulation: The classical action includes quadratic curvature invariants (governed by dimensionless couplings $f_0$ and $f_2$), the Maxwell term, Dirac fermions, complex scalars, and a non-minimal scalar-curvature coupling ($\xi$). The theory is expanded around flat spacetime ($g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$).
• Propagators and Gauge Fixing:
  • The graviton propagator in agravity behaves as $1/p^4$at large momentum (due to the$1/p^4$nature of the quadratic action), distinct from the$1/p^2$ behavior in standard GR.
  • Gauge fixing is implemented using a de Donder-type term for gravity (parameter $\zeta_g$) and a covariant gauge for the photon (parameter $\xi$).
• Scattering Calculation: The authors compute tree-level $2 \to 2$ scattering amplitudes for a broad set of processes involving charged fermions, scalars, and photons.
• Observables: They calculate the unpolarized squared matrix elements ($|M|^2$), summed over final spins/helicities and averaged over initial states.
• Consistency Checks: The calculations explicitly verify that the final physical results are independent of the gravitational gauge-fixing parameter $\zeta_g$, ensuring diffeomorphism gauge invariance at the tree level.

3. Key Contributions

• Systematic Amplitude Catalog: The paper provides a unified set of compact analytic expressions for the unpolarized squared matrix elements across eight distinct scattering channels (e.g., Møller scattering, Compton scattering, annihilation).
• Interference Terms: Unlike analyses restricted to pure graviton exchange, this work systematically retains and calculates the photon-graviton interference contributions. This is crucial for understanding how higher-derivative gravity modifies familiar QED processes.
• Spin-0 vs. Spin-2 Dynamics: The authors clarify the role of the spin-0 component of the quadratic gravity propagator. They demonstrate that the spin-0 mode contributes to scalar scattering (via the trace of the energy-momentum tensor) but vanishes for on-shell massless fermions and photons, where the trace of the energy-momentum tensor is zero.
• Gauge Invariance Verification: Explicit proof is provided that the physical squared amplitudes do not depend on the arbitrary gravitational gauge parameter $\zeta_g$.

4. Key Results

The paper derives specific formulas for processes such as $e^-e^- \to e^-e^-$, $\gamma\gamma \to \gamma\gamma$, $e^-e^+ \to \gamma\gamma$, and scalar-fermion scattering. The results highlight several universal features:

• Angular Enhancement: All amplitudes display characteristic forward and backward enhancements associated with small momentum transfer. Due to the $1/p^4$graviton propagator, these enhancements are amplified compared to standard gravity. For example, in the center-of-momentum (CoM) frame, the squared amplitudes often scale with factors like$\csc^8 \theta$(e.g., Eq. 16 and 19), indicating a strong singularity as$\theta \to 0$or$\pi$.
• UV Scaling: In the ultra-Planckian regime, the theory is classically scale-free. Consequently, the squared matrix elements $|M|^2_{\text{CoM}}$ depend only on the scattering angle $\theta$ and dimensionless couplings, with no residual dependence on the total energy $s$.
• Differential Cross Section: When forming the differential cross section, the universal scaling is recovered:
  $$\frac{d\sigma}{d\Omega} \propto \frac{1}{s}$$
  This reflects the dimensionless nature of the gravitational couplings in agravity.
• Interference Structure: The squared amplitudes typically take the form of a square of a sum, e.g., $(\text{QED term} + \text{Gravity term})^2$. This reveals how the spin-2 agravity correction ($f_2$) interferes with the electromagnetic contribution ($e$).
• IR Singularities: The massless exchange leads to strong IR/collinear amplification in the forward direction. The authors note that integrated observables require standard experimental angular cuts (or $|t|$ cuts) to regulate these singularities, similar to standard gauge theories but with stronger sensitivity.

5. Significance

• Phenomenological Building Blocks: The analytic expressions provided serve as essential inputs for phenomenological studies of quantum gravity effects at high energies, particularly in scenarios where the Planck scale might be effectively lowered or where scale-invariant gravity is relevant.
• UV Consistency: The results support the consistency of agravity as a UV-complete theory when coupled to matter, showing that unitarity and gauge invariance are maintained at the tree level despite the higher-derivative nature of the propagator.
• Conceptual Insight: The work clarifies the interplay between higher-derivative gravitational dynamics and standard gauge interactions. It demonstrates that while gravity reshapes familiar QED amplitudes (particularly in the collinear regime), the fundamental scaling laws of a dimensionless theory remain robust.
• Future Directions: The paper lays the groundwork for future studies involving renormalization group (RG) running of the couplings, loop corrections, and extensions to non-Abelian gauge theories (QCD).

In summary, this paper establishes a rigorous analytic foundation for understanding how scale-free quadratic gravity modifies standard model scattering processes at ultra-high energies, emphasizing the critical role of interference terms and the specific kinematic signatures of the $1/p^4$ graviton propagator.