If The Higgs boson and field ceased to exist (starting now), then would particles zip around at the speed of light, or would they gain mass from $E=mc^2$?
Or would something else happen?
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If the Higgs Boson somehow stopped existing, leptons like the electron, and the gauge bosons of electroweak theory would lose their mass. As $m = 0$ for these particles they would have no energy from $E = mc^2$. In addition, electroweak symmetry would be restored, so there would no longer be a separate electromagnetic force and weak force.
Baryonic matter like protons and neutrons would be slightly different --- the individual quarks would lose their mass but protons would still be massive due to the strong nuclear force, as almost 99% of the mass of a proton comes from the its QCD binding energy.
This scenario is not so realistic, because as far as we are aware a field can't simply stop existing. However, a similar scenario of vacuum decay, where the Higgs expectation value spontaneously changes across a bubble expanding at the speed of light, is a topic of current research.