Higgs Confusion (The Higgs Era )

[csmcmillion]

Higgs Confusion (The Higgs "Era")

I thought I understood the basic concepts behind the Higgs mechanism (spontaneous symmetry breaking and such). But I've read a couple of sources that talk about the "Higgs Era" - the period shortly after the BB when the U was still hot/dense enough for the Higgs Bosons (HB) to still be around. Obviously, the energies required to reproduce that era and the massive HB explain the need for the LHC.

So here's my disconnect: If the HB is the quantum of the Higgs Field, which is the field that lends mass to other particles, shouldn't HBs still be everywhere? How can the Higgs Field still permeate all of space without its quanta also permeating all of space?

Simple answers appreciated!

Thanks.

 

[kurros]

The Higgs field is a very weird sort of field: when there are no quanta flying around, it still has a value, or a non-zero vacuum expectation value. I suppose other fields strictly do have this too but we can subtract that annoying stuff away through renormalisation tricks, while the Higgs non-zero value is already there at the classical level. Excitations above this non-zero field are then quantised and give you physical Higgs bosons.

 

[AdrianTheRock]

In the Glashow-Weinberg-Salam formulation, the Higgs field is a complex doublet prior to symmetry breaking, thus has four degrees of freedom. We then introduce an energy potential of the form (\Phi² - λ²)² for the Higgs field, where \Phi = (\phi₁, \phi₂, \phi₃, \phi₄), \phi_i real, represents the four components of the field. This has a non-zero energy of λ⁴ when \phi₁ = \phi₂ = \phi₃ = \phi₄ = 0 and a minimum energy state when |\Phi| = λ.

We then postulate that the fields (\phi₁ + i\phi₂) and (\phi₃ + i\phi₄) form a doublet under a local SU(2) gauge symmetry for which we introduce vector gauge fields (W₀, W₁, W₂) plus a local U(1) gauge symmetry for which we introduce a further vector gauge field B. The other three

We note that, at this stage, the set of \Phi states for which the energy is minimum form a 4-sphere. We then postulate that the symmetries are broken in such a way that \phi₂ = \phi₃ = \phi₄ = 0, leaving only \phi₁ free. Noting that the ground state is now simply \phi₁ = λ, we can write \phi₁ = λ + h, where h is the difference between the actual \phi₁ value at any given point in spacetime and its vacuum expectation value λ.

h is the Higgs boson we are looking for at the LHC.

When we do the maths, the presence of the \phi₂, \phi₃ and \phi₄ dimensions in the equation not only results in an intermixing of the constant λ with the components of the W_i and B fields, but intermixes in a way that generates mass terms for three of these four vector fields. These three \phi_i become Goldstone bosons that are 'eaten' by the gauge fields.

With some further algebraic maniuplations, we end up with

W⁺ = (1/√2) (W₁ - iW₂)
W^- = (1/√2) (W₁ + iW₂)
Z⁰ = W₃ cos \theta_W - B sin \theta_W
A = W₃ sin \theta_W + B cos \theta_W​

where \theta_W is defined so as to eliminate any coupling between the A field and the Goldstone bosons.

Thus the three weak gauge bosons that we observe in experiments gain rest masses while the fourth gauge boson remains massless and can thus be matched to the electromagnetic field.

So, in terms of the original \phi_i fields, the ground state everywhere is now \phi₁ = λ, but this corresponds to nothing directly observable because it is the ground state. The effects of the other \phi_i are observed as the masses of the weak bosons.

In the full theory, we also introduce interactions between the \Phi fields and the fundamental fermions (leptons and quarks). To do this, however, we have to introduce specific coupling constants between \Phi and each individual fermion field. This is in contrast to the weak boson mass terms, whose values come out of the maths as fixed formulae based only on λ, \theta_W and the coupling strength of the forces, for which the electromagnetic charge strength e can be taken as the parameter. (\theta_W itself turns out to be based on the ratio between the coupling strengths of the W_i and B fields). So the fermion mass generation is a bit of a 'bolt-on', and there are 'fermiphobic' versions of the theory in which this part is omitted, and the fermions are thus assumed to get their masses by some other, unknown means.

The fact the fermions come in three generations/families, each with the same sets of charges but differing only in the rest masses of the respective particles, strongly suggests we haven't yet found the complete theory for this area. It is as if, for each flavour of fermion, the mechanism that gives it its rest mass works like an operator that has three distinct eigenstates, hence giving the three generations of particles we observe.

 

[csmcmillion]

In the Glashow-Weinberg-Salam formulation, the Higgs field is a complex doublet prior to symmetry breaking, thus has four degrees of freedom. We then introduce an energy potential of the form (\Phi² - λ²)² for the Higgs field, where \Phi = (\phi₁, \phi₂, \phi₃, \phi₄), \phi_i real, represents the four components of the field. This has a non-zero energy of λ⁴ when \phi₁ = \phi₂ = \phi₃ = \phi₄ = 0 and a minimum energy state when |\Phi| = λ.

:-O

Simple answers appreciated!

 

[kurros]

Ok try this: Consider the electron field. This is a different thing to electrons themselves; electrons are quantised excitations of the electron field. The electron field fills all of space, but that doesn't mean that electrons fill all of space. It is a bit like ripples in a pond. The pond water fills the pond but there don't have to be any ripples in it unless you make some.

 

[csmcmillion]

Ok try this: Consider the electron field. This is a different thing to electrons themselves; electrons are quantised excitations of the electron field. The electron field fills all of space, but that doesn't mean that electrons fill all of space.

Now that computes!

Thanks