When Yang-Mills field theory was introduced, a problem is that the gauge invariance can not allow mass for the gauge field. Later people invented spontaneous symmetry breaking and Higgs mechanism to give the gauge field mass. The Higgs particle is almost confirmed at LHC.
My question is, since there is symmetry non-conservation (P/CP) in nature, why not simply put a mass on Yang-Mill's theory directly, got a non-abelian Proca action, say gauge symmetry breaking (although gauge may not be a symmetry actually Gauge symmetry is not a symmetry?; actually this point is more subtle, one can also take a Stueckelberg action then fixing the gauge, it leads to the same Lagrangian)? Is there any theoretical reason for not adding mass by hand on Yang-Mills theory? Or just because Higgs particle was found, it works, that's it.
My friend has a guess, that gauge invariance implies BRST symmetry, which restricts the possible form of Lagrangian. If one did a renormalization flow transformation to lower energy scale without BRST symmetry, there will be other coupling in the effective Lagrangian at lower energy scale. I am not sure about this reasoning, because BRST symmetry can restrict the possible counter terms, can it also restrict the possible terms in the effective Lagrangian?
Best Answer
In a quantum theory, gauge symmetry is an inevitable consequence of Poincare invariance and long range interactions at the classical level (the weak and strong interactions aren't long range because of quantum effects, such as confinement and the Higgs mechanism). If one "breaks" a gauge symmetry (what it doesn't have much since since gauge symmetries are mathematical ambiguities rather than physical symmetries), the one has to give up either:
Note that breaking a gauge symmetry is different from formulating a theory without gauge invariance. For example, classical electrodynamics in terms of the electric and magnetic field doesn't have a gauge symmetry, but it doesn't break it.