  We constrain effective field theories by going beyond the familiar positivity
bounds that follow from unitarity, analyticity, and crossing symmetry of the
scattering amplitudes. As interesting examples, we discuss the implications of
the bounds for the Galileon and ghost-free massive gravity. The combination of
our theoretical bounds with the experimental constraints on the graviton mass
implies that the latter is either ruled out or unable to describe gravitational
phenomena, let alone to consistently implement the Vainshtein mechanism, down
to the relevant scales of fifth-force experiments, where general relativity has
been successfully tested. We also show that the Galileon theory must contain
symmetry-breaking terms that are at most one-loop suppressed compared to the
symmetry-preserving ones. We comment as well on other interesting applications
of our bounds.
