Question: Why can't we add a mass term for the gauge bosons of a non-abelian gauge theory?
In an abelian gauge theory one can freely add a mass and, while this breaks gauge invariance, as long as the coupling current is conserved everything works fine (i.e., the scalar modes decouple and the theory is renormalisable).
In non-abelian gauge theories, it is often stated that the only way to introduce a mass term is through the Higgs mechanism. If we added a mass term without introducing the Higgs field, but the coupling current is still conserved, at what point would the theory break down? It seems to me that the scalar modes decouple as well, at least to tree level. I failed to push the calculation to one loop order, so maybe the theory breaks down here. Is this the most immediate source of problems, or is there any simpler observable which fails to be gauge invariant?
One would often hear that if we break gauge invariance the theory is no longer renormalisable. I may be too naïve but it seems to me that a (gauge-fixed) massive gauge boson has a $\mathcal O(p^{-2})$ propagator and therefore (as long as the current in the vertices is conserved) the theory is (power counting) renormalisable. Or is it?
To keep things focused, let us imagine that we wanted to give gluons mass, while keeping self-interactions and the coupling to matter (and ghosts) unchanged. Could this work without a Higgs?
There are many posts about that are asking similar things. For example,
Can mass develop without the Higgs mechanism? asks about the mass of fermions, not of the gauge bosons.
Massive Gauge Bosons without Higgs Effect same as above.
Is Higgs mechanism necessary in QCD? same as above.
What are the alternatives to the Higgs mechanism? is a list of modifications/generalisations to the Higgs mechanics, but doesn't discuss if it is avoidable at all.