In the standard model, Higgs particles have negative mass squared term in the SM lagrangian and hence, the Higgs is tachyonic in nature. This isn't a problem since -at low energy (temperature)- the Higgs particle will stay close to the local minima of the the Mexican hat potential and by doing so, it breaks $SU(2)$ gauge and acquires a non-negative real mass. But if we consider the high energy situation (assuming that the energy regime is higher than the unstable maximum of the higgs potential), the gauge symmetry should be restored and - as far as I understand - the higgs should have an imaginary mass.
Can Higgs particles move faster than light if the temperature/energy is high enough?
