  We investigate the analytic structure of scattering amplitudes in theories in
which Lorentz invariance is spontaneously broken. We do so by computing and
studying the S-matrix for a simple example: a superfluid described by a complex
scalar with quartic interactions. The computation is confined to tree-level,
for there are no absolutely stable single-particle states, though the lifetime
can be made long by lowering the chemical potential. For the $2 \to 2$
amplitude in center-of-mass configurations, not only is crossing symmetry
violated, there appears a {\it tree level} branch cut for unphysical
kinematics. Its appearance is a consequence of non-analyticity in the
dispersion relation. The branch point defines a new scale in the problem, which
scales inversely with the chemical potential. In this example, 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 a small chemical
potential, or weak Lorentz breaking.
