Electroweak Unification and Electroweak scale 
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*We said that the EM and weak interactions are unified as Electroweak Unification. However, for usual Grand Unification, it requires that the gauge group G is simple Lie group -- which means the G has no nontrivial normal subgroup other than itself.

However, neither $U(1) \times SU(2)$ nor $U(2)$ are simple Lie groups. So why do we call Electroweak Unification? They are still two different forces with two different couplings in Weinberg model?


*There are many energy scales related to Electroweak scale: https://en.wikipedia.org/wiki/Electroweak_scale


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*Higgs mass 125GeV.


*the temperature of electroweak symmetry breaking - 159.5±1.5 GeV


*Fermi scale:  vacuum expectation value $v = (G_F \sqrt{2})^{-1/2}- $ 246 GeV
What are these 3 scales so close to each other around 200 GeV? Are they related in some way or they are totally independent in the parameters of models? e.g. in Electroweak Unification, GUT or other models? For example, the Higgs potential has two parameters, the quadratic and quartic terms, so they seem not need to be so close in 200 GeV scales?
 A: *

*I don't know where that "we" came from, but the SM "unification" is partial, more of a Weinberg-angle tilt: It involves EM and WI inextricably linked and mixed. The different masses of W and the Z mirror the two couplings. For vanishing Weinberg angle, there would be no mixing and e=g'.  The question then is just a matter of terminology, loose as is often in physics.


*The SSB scale is the standard textbook Fermi scale, v = 246 GeV. The mass of the Higgs, 125 GeV is conceptually different, and, "in principle" could be very different; as different as the huge range of guesses/"predictions" covered, before the discovery of the Higgs, not to mention Higgsless models. Its value being so close to the Fermi scale is still a "mystery" to the hyper-ambitious.



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*The thermal QFT crossover scale (between the symmetric and SSB phases), 159.5±1.5 GeV,  utilizes both of the above scales into effectively modeling the thermal theory and checks by lattice simulation that it is in the same order of magnitude, almost certainly not coincidental. D’Onofrio, Rummukainen,   & Tranberg (2014). Phys Review Lett 113 (14), 141602. It is the temperature/energy scale below which the universe slips into the SSB phase, ultimately with the above v.

