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In a pure U(1) gauge theory yes, but once you start adding other gauge symmetries you are extremely restricted with what you can write down.. for instance the SM is a chiral gauge theory and the fermionic mass terms are no longer gauge invariant
It's the opposite of a problem! that's why gauge theories are said to be maximally predictive, the more constrained we are the better The higgs transforms under the chiral symmetry group (like left handed fermions).. this way we can write down gauge invariant interactions.. 1/2
Oh wow I didn’t realize this before
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