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In every textbook I've read the only argument why gauge bosons can't have mass is "because a mass term would break gauge invariance". Even fermions can't have mass because it would break the $SU(2)_L$ electroweak invariance. But now I wonder why we'd like to have a gauge invariant Lagrangian in the first place. Why is this necesary? Gauge theory is just a great way of finding interaction terms but we don't need to actually have those symmetries.

What would be the problem if a fermion had its own mass term? Or a gluon? Or any other particle? I've read the following questions but I haven't found them useful to answer my question: Why can't gauge bosons have mass?, What goes wrong if we add a mass term for gauge bosons without the Higgs mechanism?

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Sometimes, you have no choice in the matter of whether or not your theory is a gauge theory. If you have massless vector bosons, you have a gauge symmetry. You cannot have a massless vector boson without a gauge symmetry because you need the redundancy of the gauge choice to eliminate one of the three degrees of freedom that a vector boson naively has, since states created by massless vectors can only have two degrees of freedom. The reason is basically that the representation theory of the Poincaré group does not allow massless particles of spin 1 to have an "$S_z=0$" state, see this answer of mine. So for QCD, where the gluon has experimental limits on its mass that make it reasonable to assume it really is massless, we must have a gauge symmetry, and so the Lagrangian needs to be gauge invariant.

It's a little bit different for the weak force, but once you know the Higgs mechanism, you realize that the pattern of electric and weak charges fits perfectly into the Standard Model of a spontaneously broken electroweak gauge symmetry. Since this description matches experiment well, we use a gauge theory also for this.

In the end, the answer to "why do we want gauge invariant Lagrangians/gauge theory" is simply because their predictions match experiment. Within quantum field theory, there is no a priori reason why physics should incorporate any specific gauge group.

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