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If we assume the GUT unification energy is some value E, (say $10^{16}GeV$). Can we work out the running of the particle masses for example the electon, muon and tau at various energies? And will they all have the same mass at the Unification energy?

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  • $\begingroup$ I think their masses will all go to zero as the Higgs field loses its expectation value. $\endgroup$ – G. Smith May 25 at 3:16
  • $\begingroup$ There is no "the GUT unification energy" - there is a lot of different theories for how a GUT is supposed to work, and it's not clear why you think that the behaviour of the particle masses would be independent of the details of the theory. $\endgroup$ – ACuriousMind May 25 at 9:39
  • $\begingroup$ @ACuriousMind Well it might just be a formula that depends only on the rest mass and the Unification Energy. Such as $M_e exp(-E/E_U)$ as a made up example. Lots of times you get simple formulae independent of the details. For example, I wondered if the formula was like an exponential, or like a power law, or like a polynomial law. $\endgroup$ – zooby May 25 at 14:10
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These particles acquire their masses via the yukawa coupling to the Higgs field. Above the electroweak phase transition, the Higgs vev is zero and so the mass is zero (ignoring any effective "thermal mass" that may arise in a finite-temperature environment). However, the yukawa still has a meaning, as the amplitude for the particle to emit a Higgs boson, and its running can be calculated.

How masses/yukawas run up to the GUT scale, depends on the theory! If there is nothing but standard model, they will run one way; if there are intermediate scales at which new physics enters, they will run differently.

There is definitely no reason for the electron, muon, tau yukawas to be the same at the GUT scale, and in fact I am not even sure that it's possible. In the standard model, it's the gauge couplings which (almost) unify at the GUT scale, but the yukawas are a separate set of quantities and go their own ways.

What does occur in some GUT models, is a unification of the masses of the heaviest particles in each family - look for "bottom-tau unification" and "top-bottom-tau unification". There are also models in which some of the 3x3 yukawa mass matrices are "democratic" - every element equal to one. But to get the yukawas for the mass eigenstates, you have to diagonalize those matrices, and then the diagonal elements are no longer equal - instead you get something of the form diag(m,0,0), reflecting the pattern whereby the third generation is much heavier than the first two.

This 2007 paper calculates the running of lepton and quark masses up to high scales, for standard model and MSSM. But as I said, for a different theory, the beta functions which describe the running will be different, and so the high-energy behavior will be different.

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