Continuous-Spin Particles, On Shell

We study on-shell scattering amplitudes for continuous-spin particles (CSPs). Poincar\'e invariance, little-group $ISO(2)$ covariance, analyticity, and on-shell factorisation (unitarity) impose stringent conditions on these amplitudes. We solve them by realizing a non-trivial representation for all little-group generators on the space of functions of bi-spinors. The three-point amplitudes are uniquely determined by matching their high-energy limit to that of definite-helicity (ordinary) massless particles. Four-point amplitudes are then bootstrapped using consistency conditions, allowing us to analyze the theory in a very transparent way, without relying on any off-shell Lagrangian formulation. We present several examples that highlight the main features of the resulting scattering amplitudes. We discuss CSP's amplitudes as a new infrared deformation of ordinary massless amplitudes, which is controlled by the scale of the Pauli-Lubanski vector squared, as opposed to the familiar mass deformation. Finally, we explore under which conditions it is possible to relax some assumptions, such as strict on-shell factorisation, analyticity, or others. In particular, we also investigate how continuous-spin particles may couple to gravity and electromagnetism, in a loose version of $S$-matrix principles.