S-matrix positivity without Lorentz invariance

Positivity bounds on the S-matrix provide powerful constraints on effective field theories in flat space. Extending these results to cosmological backgrounds requires an understanding of the analytic structure of amplitudes in theories with broken Lorentz invariance. A paradigmatic example is provided by relativistic superfluids. We investigate the S-matrix in a UV complete model of relativistic superfluid: a complex scalar with quartic interactions at finite density. We stop at tree level since there are no absolutely stable single-particle states, although the phonons lifetime can be made arbitrarily long at low density or weak coupling. For 2 → 2 processes in center-of-mass configurations, we confirm the absence of crossing symmetry in its simplest implementation. Due to a non-analyticity in the phonon dispersion relation, a new branch cut appears in the amplitude, at tree level, for unphysical kinematics. The branch point defines a new scale in the problem, which is inversely proportional to the chemical potential. An interesting consequence is a non-decoupling effect in the high energy limit. In the model we study, even derivatives of the forward amplitude are positive, while odd derivatives are negative. This pattern can be understood in a general way in the limit of low density, or weak Lorentz breaking.