Let use denote a 4-potential with A.
We get an equivalent physical system if we add the gradient of any sufficiently smooth function g to A.
Adding a gradient is called a gauge transformation because it retains the physics of the system as is.
A real-valued Higgs field which gives the mass for a gauge boson
Let us then introduce a real-valued Higgs field φ whose vacuum expectation value is different from zero. To achieve such a vacuum expectation value, we assume that there is a "potential" on φ of the form
V(φ) = k (φ - v)²,
where k is a constant and v is the vacuum expectation value.
We once again use the framework of Kien Nguyen (2009).
Let us write
φ = v + h.
We couple the Higgs field φ to the 4-potential A of a gauge boson field.
In the new lagrangian, there is a term
e² v² A².
This term gives the mass to the gauge boson field A. We ignore the terms of type
e² v h A²
and
e² h² A².
The part of the lagrangian for h is of the the same form as for a massive Klein-Gordon field:
The mass m is ~ sqrt(k).
An excitation of the field h is the Higgs boson.
What did we accomplish with the real-valued Higgs field?
We were able to give a mass to the gauge boson of the field A.
The real-valued Higgs field is a very well-behaving massive Klein-Gordon field. Its equation is linear.
The potential of the Higgs field is the familiar quadratic (harmonic) potential, not the complicated Mexican hat potential of the ordinary Higgs field.
There are no defects in the real-valued Higgs field.
There is no self-interaction in the real-valued Higgs field. The self-interaction in the ordinary Higgs field comes from the quartic potential φ⁴.
There is no symmetry breaking process.
The real-valued Higgs field seems to be much better than the ordinary complex-valued Higgs field. What is the problem with the real-valued field?
The problem is that when we do a gauge transformation of the gauge boson field A, then the value of the lagrangian changes. The physics is not conserved.
This is the reason why Peter Higgs among others devised a complex-valued Higgs field which can be transformed along a gauge transformation of A. The goal was to keep the value of the lagrangian the same. In a gauge transformation, the complex value of the Higgs field is rotated just in the right way to conserve the value of the lagrangian.
Let us analyze the whole process by which Peter Higgs gave a mass to the gauge boson.
The introduction of a scalar Higgs field whose vacuum expectation value is an almost constant v ≠ 0, is somewhat ad hoc, but it is required to give A a mass.
What is the value in devising a complex Higgs field, its potential function, its transformation (rotation), and writing a new lagrangian for those? If we can find simple formulae for the complex field, fine. But do we learn anything from the fact that we were able to find such formulae?
It is not clear to us if finding the details for the complex-valued Higgs field system teaches us anything. As if the the whole task would be unnecessary.
Defining the real-valued Higgs field gauge transformation from the Coulomb gauge
Suppose that we are able to do the physics that we want in the Coulomb gauge and using the real-valued Higgs field.
Can we define the gauge transformation of our real-valued field in a trivial way? We do not try to calculate any complicated formulae for the transformed fields nor find a special lagrangian which works with these fields. We simply define that the gauge transformed system must describe the exact same physical behavior as the Coulomb gauge system does with the real-valued Higgs field.
Let the lagrangian of the system in the Coulomb gauge be
L₀(φ, A).
Let the gauge transformation for A be f. We let the gauge transformation for φ be the identical mapping.
We define the lagrangian L for an arbitrary gauge by the following formula
L(φ, A) = L₀(φ, f(A)),
where f is the mapping which transforms A to the Coulomb gauge.
Now we have a lagrangian L which trivially is invariant under a gauge transformation. The precise mathematical formula of this lagrangian may look ugly, but is there anything fundamentally wrong with it?
Is there a rule that a lagrangian should have a simple arithmetic formula in all gauges, not just in the Coulomb gauge?
Trade-off of a simple arithmetic formula versus nice properties of a field
Let us again analyze what Peter Higgs and others did at the beginning of the 1960's.
Superconductivity and condensed matter physics inspired the idea of using a charged Higgs field φ to give a mass to gauge bosons in a field A.
The Higgs field must have an almost constant vacuum expectation value v ≠ 0 to do the trick. A simple way to accomplish that is to introduce a potential V whose minimum is at v.
The remaining problem is to make the Higgs field to respect the gauge invariance of A. Higgs and others found out that a complex-valued φ works if its gauge transformation is rotations around the origin of the complex plane, and the potential is the Mexican hat.
Higgs assumed that "symmetry has been broken" and the field has collapsed to be almost precisely equal to a fixed value v.
We can then study the behavior of the field close to this value v. We are able to eliminate the angle parameter from the complex value of φ and restrict the value of φ to vary over the real axis, in the vicinity of the real value v. We define φ = v + h, where all are real.
This endeavor allowed Peter Higgs and others to define the real-valued Higgs field which we suggested in an earlier section of this blog post!
That is, Peter Higgs and others built a machine which, when assuming symmetry breaking and analyzing the system close to v, behaves like the simple model we introduced above.
Certainly one can build other machines which, when restricted close to v, behave like the simple real-valued Higgs model. Why is the standard Higgs model good? Its formula is very simple. That is its main virtue.
However, the standard Higgs model has three ugly features:
1. It assumes symmetry breaking to a random minimum value v in the valley of the Mexican hat. That kind of behavior has never been observed in other fundamental fields.
2. The Mexican hat potential may allow defects to form. We have not observed defects in other fundamental fields. If defects exist, they are not matter which consists of particles. That would be very strange.
3. The quartic φ⁴ potential means that there is self-interaction in the Higgs field. The probability distribution of the momentum of a single particle may spread as time progresses, even though the particle does not interact with anything! That may contradict quantum mechanics.
Our own real-valued Higgs field avoids all those three ugly features. The drawback of our model is that its arithmetic formula is ugly. The lagrangian L is defined through the Coulomb gauge. The arithmetic formula may be very complicated in another gauge.
Should we favor a simple formula for the Higgs lagrangian, or beautiful properties of the field?
In this blog we think that beautiful properties of the field are more important.
Symmetry breaking and unification in the real-valued Higgs model
Symmetry breaking and unification of the elecromagnetic and weak fields at high energies is something which may be deduced from the Mexican hat potential.
In the real-valued Higgs model, with the quadratic potential, at high temperatures the mass of bosons W and Z will behave erratically. Is there some kind of a unification?
If W and Z particles are very dense, the hamiltonian will attain its minimum value at φ = 0. That would be a unification.
If the density is lowered, then φ slides to its value v. Should we call this symmetry breaking? In some sense, it is.
Conclusions
We showed that a real-valued Higgs field has beautiful properties - but we then have to define the lagrangian through the Coulomb gauge. The arithmetic formula of the lagrangian in other gauges may be terribly complicated.
We believe that beautiful properties of the field are more important than what the arithmetic formula looks like.
The vacuum in the lowest energy state contains a Higgs field whose value is v everywhere. This is ugly compared to other fields whose value is 0 everywhere. Is there a way to remove this ugliness? In this blog we have claimed that the vacuum is truly empty. It contains nothing. The fact that the Higgs field has the value v does not look nice.
In our blog our view is that matter is excitations of fields, and the excitations can be quantized, giving us "particles". The real-valued Higgs model conforms to this. There are no defects.
The notion of a scalar field with a spin zero particle is ugly since other fundamental fields that we know are not scalar fields. Could it be that a scalar field is a vector field in a disguise? In earlier blog posts we noted that the simple φ² probability or energy density is not Lorentz covariant for a massive Klein-Gordon field.
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