  We study the $S$-matrix of Goldstones in the renormalizable theory of a
$U(1)$ complex scalar at finite charge, i.e. in a state that breaks Lorentz
invariance. The theory is weakly coupled so that this $S$-matrix exists at all
energies. Unlike the Lorentz invariant case, the resulting $S$-matrix is not
analytic in the exchanged (complexified) four-momentum. The non-analyticities
stem from the LSZ reduction formula, as a consequence of the energy-dependent
mixing between the radial and Goldstone modes.
