Heavy-Particle Effective Theory: from On-Shell Formalism to Gravitational Compton Amplitudes

We formulate on-shell heavy-particle effective theory (HPET) amplitudes by projecting the spinor-helicity variables onto the $SU(2)$ little group of a timelike reference velocity $v^μ$. Retaining both large and small components of the velocity-projected spinors gives an intermediate velocity-projected amplitudes in which reparameterization invariance can be imposed before expanding in the inverse heavy-particle mass. Integrating out the small component then yields HPET amplitudes organized by heavy-particle power counting, with a direct correspondence to local operators at fixed order. We determine the three-point velocity-projected amplitudes for two equal-mass particles of arbitrary spin $S$ coupled to a soft massless boson of helicity $h$. Four-point gravitational Compton amplitudes can be constructively obtained using velocity-projected three-point amplitudes, and expanded into HPET amplitudes which is free of spurious poles up to sub-leading order. As a gravitational application, we show that the projected gravitational minimal coupling reproduces the Kerr spin-induced multipoles up to order $2S$, and find the classical limits of the tree-level elastic gravitational scattering and gravitational Compton amplitudes agreed with the literature up to contact terms undetermined by factorization.