Let me anwser a closely related quenstion:
*Consider a U(1) gauge theory with massless gauge bosons, can any small perturbations give the gauge boson an mass.*

Amazingly, the anser is NO. The masslessness of the gauge boson is topologically robust.
No small perturbations can give the gauge boson an mass.
For detail, see my article.

Let me make the statement more precise. Here we consider
a compact U(1) gauge theory with a finite UV cutoff (such as a lattice gauge theory),
that contains gapless gauge bosons at low energies.
Then no small perturbations to this compact U(1) gauge theory with a finite UV cutoff
can give the gauge boson an mass, even for the small perturbations that break the gauge invariance.

So the masslessness of gauge boson is a stable universal property of a quantum phase.
Only a phase transition can make the gauge boson massive.

This post imported from StackExchange Physics at 2014-04-05 17:34 (UCT), posted by SE-user Xiao-Gang Wen