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High Energy Scattering in Perturbative Quantum Gravity at Next to Leading Power

Ratindranath Akhoury, Ryo Saotome, George Sterman

TL;DR

The paper addresses high-energy, small-angle gravitational scattering of an ultra-relativistic light scalar φ off a heavy scalar σ, developing a diagrammatic expansion around the eikonal limit and computing the first power correction in the small parameter Δ/Eφ. It demonstrates that the leading eikonal phase χ0 exponentiates in impact-parameter space and that the first nonleading power corrections arise from both light-propagator expansions and graviton-tree insertions, with explicit results in arbitrary dimensions and a detailed four-dimensional analysis. In 4D, the next-to-eikonal corrections from propagator expansions vanish (χ2^a = 0) while graviton-tree contributions (χ2^b) remain finite and shift the saddle point by an amount of order the Schwarzschild radius, implying a nontrivial but controlled deviation from the leading eikonal behavior. The findings reinforce the infrared-dominated, long-distance nature of perturbative quantum gravity at small angles and lay groundwork for further power-corrections in diagrammatic or related approaches, with potential relevance to scattering off black holes.

Abstract

We consider the relativistic scattering of unequal-mass scalar particles through graviton exchange in the small-angle high-energy regime. We show the self-consistency of expansion around the eikonal limit and compute the scattering amplitude up to the next-to-leading power correction of the light particle energy, including gravitational effects of the same order. The first power correction is suppressed by a single power of the ratio of momentum transfer to the energy of the light particle in the rest frame of the heavy particle, independent of the heavy particle mass. We find that only gravitational corrections contribute to the exponentiated phase in impact parameter space in four dimensions. For large enough heavy-particle mass, the saddle point for the impact parameter is modified compared to the leading order by a multiple of the Schwarzschild radius determined by the mass of the heavy particle, independent of the energy of the light particle.

High Energy Scattering in Perturbative Quantum Gravity at Next to Leading Power

TL;DR

The paper addresses high-energy, small-angle gravitational scattering of an ultra-relativistic light scalar φ off a heavy scalar σ, developing a diagrammatic expansion around the eikonal limit and computing the first power correction in the small parameter Δ/Eφ. It demonstrates that the leading eikonal phase χ0 exponentiates in impact-parameter space and that the first nonleading power corrections arise from both light-propagator expansions and graviton-tree insertions, with explicit results in arbitrary dimensions and a detailed four-dimensional analysis. In 4D, the next-to-eikonal corrections from propagator expansions vanish (χ2^a = 0) while graviton-tree contributions (χ2^b) remain finite and shift the saddle point by an amount of order the Schwarzschild radius, implying a nontrivial but controlled deviation from the leading eikonal behavior. The findings reinforce the infrared-dominated, long-distance nature of perturbative quantum gravity at small angles and lay groundwork for further power-corrections in diagrammatic or related approaches, with potential relevance to scattering off black holes.

Abstract

We consider the relativistic scattering of unequal-mass scalar particles through graviton exchange in the small-angle high-energy regime. We show the self-consistency of expansion around the eikonal limit and compute the scattering amplitude up to the next-to-leading power correction of the light particle energy, including gravitational effects of the same order. The first power correction is suppressed by a single power of the ratio of momentum transfer to the energy of the light particle in the rest frame of the heavy particle, independent of the heavy particle mass. We find that only gravitational corrections contribute to the exponentiated phase in impact parameter space in four dimensions. For large enough heavy-particle mass, the saddle point for the impact parameter is modified compared to the leading order by a multiple of the Schwarzschild radius determined by the mass of the heavy particle, independent of the energy of the light particle.

Paper Structure

This paper contains 13 sections, 77 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Relativistic Scattering
  3. Eikonal Phase
  4. One Graviton Exchange
  5. Two Graviton Exchange
  6. $n$-Graviton Exchange and Expansion Around the Eikonal Limit
  7. Next-to-Eikonal Power at One Loop
  8. Graviton trees at one loop
  9. The first power correction to multi-graviton exchange
  10. $n$-Graviton Ladders
  11. Combining a graviton tree with ladders
  12. The full next-to-leading power
  13. Conclusions

Figures (6)

  • Figure 1: A scattering process with a single graviton exchanged. The heavy scalar is the solid line and the light scalar is the dotted line.
  • Figure 2: A two-graviton ladder diagram. The heavy scalar is the solid line and the light scalar is the dotted line.
  • Figure 3: A two-graviton crossed ladder diagram. The heavy scalar is the solid line and the light scalar is the dotted line.
  • Figure 4: A diagram with a seagull interaction. The heavy scalar is the solid line and the light scalar is the dotted line.
  • Figure 5: A diagram with a triangle interaction. The heavy scalar is the solid line and the light scalar is the dotted line.
  • ...and 1 more figures