  The gravitational positivity bound gives quantitative "swampland''
constraints on low-energy effective theories inside theories of quantum
gravity. We give a comprehensive discussion of this bound for those interested
in applications to phenomenological model building. We present a practical
recipe for deriving the bound, and discuss subtleties relevant for realistic
models. As an illustration, we study the positivity bound on the scattering of
the massive gauge bosons in the Higgs/St\"{u}ckelberg mechanism. Under certain
assumptions on gravitational amplitudes at high energy, we obtain a lower bound
$m_{V} \gtrsim \Lambda_\mathrm{UV}^2 /g M_\mathrm{Pl}$ on the gauge boson mass
$m_V$, where $g$ is the coupling constant of the gauge field, $M_\mathrm{Pl}$
is the reduced Planck mass and $\Lambda_\mathrm{UV}$ is the ultraviolet cutoff
of the effective field theory. This bound can strongly constrain new physics
models involving a massive gauge boson.
