  In effective quantum field theory, a spin-1 vector boson can have a
technically natural small mass that does not originate from the Higgs
mechanism. For such theories, which may be written in St\"uckelberg form, there
is no point in field space at which the mass is exactly zero. I argue that
quantum gravity differs from, and constrains, effective field theory:
arbitrarily small St\"uckelberg masses are forbidden. In particular, the limit
in which the mass goes to zero lies at infinite distance in field space, and
this distance is correlated with a tower of modes becoming light according to
the Swampland Distance Conjecture. Application of Tower or Sublattice variants
of the Weak Gravity Conjecture makes this statement more precise: for a spin-1
vector boson with coupling constant $e$ and St\"uckelberg mass $m$, local
quantum field theory breaks down at energies at or below $\Lambda_{\rm UV} =
\min((m M_{\rm Pl}/e)^{1/2}, e^{1/3} M_{\rm Pl})$. Combined with
phenomenological constraints, this argument implies that the Standard Model
photon must be exactly massless. It also implies that much of the parameter
space for light dark photons, which are the target of many experimental
searches, is compatible only with Higgs and not St\"uckelberg mass terms. This
significantly affects the experimental limits and cosmological histories of
such theories. I explain various caveats and weak points of the arguments,
including loopholes that could be targets for model-building.
