  Ambitwistor-strings are chiral strings whose targets are spaces of complex
massless particles, and whose correlation functions directly lead to simple,
compact formulae for scattering amplitudes and loop integrands for massless
gauge and gravity theories. This article extends this framework to worldsheet
models for massive particles in 4d, obtained via a symmetry reduction of a
higher dimensional massless model. The target space of the resulting models
turns out to be the phase space of 4d massive particles in a twistorial
representation, and so the worldsheet theory agrees with the two-twistor string
previously introduced by the authors. However, the paper has been written so as
to be largely self-contained. We discuss two interesting classes of massive
theories in detail. For gauge theories, the reduction procedure is explicitly
adapted to supersymmetric gauge theories on the Coulomb branch. For
supergravity theories, the reduction is adapted to give theories obtained via
Cremmer, Scherk and Schwartz (CSS) reduction, with broken supersymmetry and
massive multiplets. The reduction procedure gives explicit and systematic rules
to obtain amplitudes for all these theories and their amplitudes from two
compact master formulae that have their origins in 6d based on the polarized
scattering equations; in the CSS case the formulae are new, and in both cases
their derivation is systematic. The freedom to include mass allows the
definition of a loop insertion operator, thereby extending the formulae to
1-loop. Unlike the massless 4d twistorial models, these all display a perfect
double copy structure, here incorporating massive particles in the relationship
between gauge theory and CSS supergravity amplitudes.
