Quantum electrodynamics mediated by a photon with continuous spin

In recent work with Zhou, we studied classical interactions of matter with a "continuous spin" field, whose excitations have zero mass but nonzero spin Casimir (spin scale) $\rho$. We reported evidence that such theories are a calculable infrared deformation of familiar massless theories including electromagnetism. Building on the aforementioned work, this paper presents rules for computing scattering amplitudes involving charged scalar matter and continuous spin photons with nonzero spin scale. As a concrete example, we compute the pair annihilation and Compton scattering amplitudes in this theory and comment on their physical properties, including unitarity and scaling behavior at small and large $\rho$. Like the classical physics and soft factors studied in prior work, these amplitudes exhibit "helicity correspondence," recovering familiar amplitudes at energy scales $\gg \rho$, but with calculable deviations that could be experimentally tested. These results bolster the conjecture that continuous spin particles with a very small $\rho$ could mediate long-range forces in nature (e.g., the photon could be a continuous spin particle), and extend the range of physical effects that could be used to test this hypothesis.