Symplectic Perturbation Theory in Massive Ambitwistor Space: A Zig-Zag Theory of Massive Spinning Particles

We develop a theory of massive spinning particles interacting with background fields in four spacetime dimensions in which holomorphy and chirality play a central role. Applying a perturbation theory of symplectic forms to the massive twistor space as a K\"ahler manifold, we find that the spin precession behavior of a massive spinning particle is directly determined from the manner in which self-dual and anti-self-dual field strengths permeate into "complex spacetime." Especially, the particle shows the minimally coupled precession behavior if self-dual field strength continues holomorphically into the complex: the Newman-Janis shift. In general, computing the momentum impulse shows that the parameters that control generic non-holomorphic continuations are directly related to the coupling constants in the massive-massive-massless spinning on-shell amplitude of Arkani-Hamed, Huang, and Huang, and thus they are interpreted as the single-curvature Wilson coefficients given by Levi and Steinhoff, redefined on complex worldlines. Finally, exact expressions for Kerr and $\sqrt{\text{Kerr}}$ actions are bootstrapped in monochromatic self-dual plane-wave backgrounds from symplectivity and a matching between classical scattering and the on-shell amplitude, from which we obtain all-order exact impulses of classical observables.