Gravitational deflection of vector photons via effective field theory

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 the integration-by-parts reduction techniques to precisely examine the long-range contributions governed by terms in the amplitude that are nonanalytic 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.