  In this work, we study the analytic properties of S-matrix for unstable
particles, which is defined as the residues on the unphysical sheets where
unstable poles reside. We demonstrate that anomalous thresholds associated with
UV physics are unavoidable for unstable particles. This is in contrast to
stable particles, where the anomalous thresholds are due to IR physics, set by
the scale of the external kinematics. As a result, any dispersive
representation for the amplitude will involve contributions from these
thresholds that are not computable from the IR theory, and thus invalidate the
general positivity bound. Indeed using toy models, we explicitly demonstrate
that the four-derivative couplings for unstable particles can become negative,
violating positivity bounds even for non-gravitational theories. Along the way,
we show that contributions from anomalous thresholds in a given channel can be
captured by the double discontinuity of that channel.
