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I have a very very simple and naive question, which I hope I shouldn't be ashamed to ask, but I would like to clarify an issue in my mind.

We hear everywhere that the value of the ew vacuum is v ~ 246GeV, fixed by the Fermi Constant G_F, v = (\sqrt(2)*G_F)^(-1/2).

Now this is also the vacuum of the Higgs potential in the standard model, which is determined by the two parameters, say mu^2 and \lambda, in the Higgs potential, schematically

V(\phi) = mu^2 phi^2 + \lambda \phi^4.

The mechanism does not have anything to do with G_F at this point, it exists completely independently of the rest of the Lagrangian which can contain any other interactions.

Therefore the question is, at which step of the implementation of the mechanism in the EW SU(2)xU(1) and how exactly do we get to say that this v is actually connected to G_F, and in particular it is exactly equal to (\sqrt(2)*G_F)^(-1/2)? Does one necessarily need to compute, for example, the process

\mu -> \nu + \bar{\nu} + e^+

in the SM in terms of the parameter v, then take the limit M_W->\infty, get the corresponding "effective" 4-point interaction and then compare it to what Fermi had called G_F, or can one somehow make this statement"a priori" from more general considerations?

Also, once one agrees on this, what is the most precise method to measure G_F?

Thanks guys and sorry if this sounds a bit silly...

Sleuth

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# Origin of the value of the vacuum v ~ 246 GeV

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