Gravitational Positivity for Phenomenologists: Dark Gauge Boson in the Swampland

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/Stückelberg mechanism. Under certain assumptions on gravitational amplitudes at high energy, we obtain a lower bound m_V≳Λ_UV^2 /g M_Pl on the gauge boson mass m_V, where g is the coupling constant of the gauge field, M_Pl is the reduced Planck mass and Λ_UV is the ultraviolet cutoff of the effective field theory. This bound can strongly constrain new physics models involving a massive gauge boson.