Positivity Bounds for Scalar Theories
  Assuming the existence of a local, analytic, unitary UV completion in a
Poincar\'{e} invariant scalar field theory with a mass gap, we derive an
infinite number of positivity requirements using the known properties of the
amplitude at and away from the forward scattering limit. These take the form of
bounds on combinations of the pole subtracted scattering amplitude and its
derivatives. In turn, these positivity requirements act as constraints on the
operator coefficients in the low energy effective theory. For certain theories
these constraints can be used to place an upper bound on the mass of the next
lightest state that must lie beyond the low energy effective theory if such a
UV completion is to ever exist.
