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Gravitational Deflection of Vector Photons via Effective Field Theory

Yihan Ma, Ding-fang Zeng

TL;DR

This work uses effective field theory in the weak-field limit to study the gravitational deflection of vector photons scattering off a heavy scalar proxy for a black hole. By applying Integration-by-Parts (IBP) reduction and LiteRed, the authors reduce the one-loop photon–scalar amplitude to four master integrals and extract the long-range, non-analytic contributions that govern deflection. They derive the deflection angle using both geometric optics and the eikonal approximation, obtaining a leading classical term $\theta \approx \frac{4 G_N m}{b}$ plus quantum corrections scaling as $\frac{G_N^2 m^2}{b^2}$ and $\frac{G_N^2 m \, \hbar}{b^3}$, with additional infrared-log structures. The paper also discusses discrepancies with prior literature, tracing them to choices in the three-graviton vertex rules (background-field vs perturbative EFT), and notes limitations to the weak-field, low-energy regime where higher-loop corrections become increasingly complex.

Abstract

Gravitational scattering of the electromagnetic field from a heavy scalar field provides a fundamental testbed for understanding the deflection of light by massive bodies. In many approaches based on effective field theory, the calculation of scattering amplitudes quickly becomes complicated due to the large number of Feynman integrals required, especially beyond leading order. In this work, we study this problem using effective field theory in the weak field approximation. We utilize Integration-By-Parts reduction techniques to precisely examine the long-range contributions governed by terms in the amplitude which are non-analytic in momentum transfer. Using geometric optics and the eikonal approximation, we derive expressions for the deflection angle and find the origin of differences relative to earlier works.

Gravitational Deflection of Vector Photons via Effective Field Theory

TL;DR

This work uses effective field theory in the weak-field limit to study the gravitational deflection of vector photons scattering off a heavy scalar proxy for a black hole. By applying Integration-by-Parts (IBP) reduction and LiteRed, the authors reduce the one-loop photon–scalar amplitude to four master integrals and extract the long-range, non-analytic contributions that govern deflection. They derive the deflection angle using both geometric optics and the eikonal approximation, obtaining a leading classical term plus quantum corrections scaling as and , with additional infrared-log structures. The paper also discusses discrepancies with prior literature, tracing them to choices in the three-graviton vertex rules (background-field vs perturbative EFT), and notes limitations to the weak-field, low-energy regime where higher-loop corrections become increasingly complex.

Abstract

Gravitational scattering of the electromagnetic field from a heavy scalar field provides a fundamental testbed for understanding the deflection of light by massive bodies. In many approaches based on effective field theory, the calculation of scattering amplitudes quickly becomes complicated due to the large number of Feynman integrals required, especially beyond leading order. In this work, we study this problem using effective field theory in the weak field approximation. We utilize Integration-By-Parts reduction techniques to precisely examine the long-range contributions governed by terms in the amplitude which are non-analytic in momentum transfer. Using geometric optics and the eikonal approximation, we derive expressions for the deflection angle and find the origin of differences relative to earlier works.

Paper Structure

This paper contains 11 sections, 60 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Calculation of scattering amplitudes
  3. Scattering amplitude at tree-level
  4. Scattering amplitude at one-loop level
  5. The deflection angle of photon trajectory
  6. Geometric optics approximation
  7. Eikonal approximation
  8. Discussion on the result
  9. Conclusion
  10. Feynman Rules
  11. Master Intgrals

Figures (4)

  • Figure 1: Diagram of $\gamma-\phi$ scattering in the center-of-mass frame.
  • Figure 2: Tree-level Feynman diagram of t-channel
  • Figure 3: The one-loop Feynman graphs of the scattering of a photon from a heavy scalar field which need to be calculated
  • Figure 4: The left panel shows the photon momenta, while the right panel gives the trajectory of photons without external potentials. The wavy line in the right panel is the photon trajectory and $b$ is the impact parameter relative to the central black hole.