When the Higgs field finds its minimum at a certain vacuum expectation value, the field can still oscillate around that minimum.
One direction x of the oscillation has a potential term of a form
V(x) = k x^2,
where x is the displacement from the minimum.
Other directions y of oscillation have the respective
V(y) = 0.
Each such direction y corresponds to the massless Klein-Gordon equation, and a massless scalar particle, a Goldstone boson.
A Brout-Englert-Higgs trick is to do away with the motion in the directions y by "rotating" the local frame so that all the displacements y stay zero. This requires adjusting the gauge fields in such a way that they "simulate" the effect that a Goldstone field would have had on the value of the lagrangian. The trick is described as "eating" the Goldstone bosons.
If a gauge field is only coupled through its derivatives to other fields, then one has a large freedom to transform the gauge field, and the physics stays the same. An example is the electromagnetic field, where one has a very large freedom to modify the 4-vector potential (= the gauge field). However, when the gauge field is coupled to, e.g., the Higgs field, then transformations are very much restricted.
Some people seem to think that one can carelessly transform the gauge fields and do away with the Goldstone bosons, and still keep the physics the same. That is a mistake.
http://philsci-archive.pitt.edu/10962/1/Sebastien_Rivat_-_Spontaneous_symmetry_breaking_-_2.pdf
Sebastien Rivat has observed that there is a problem in eating the Goldstone bosons.
As we wrote in our blog on August 24, 2019, the gauge symmetry of the electric potential does not work if an electron is present: the inertial mass of the electron depends on its potential difference relative to far-away space. The interaction with the electron spoils the gauge symmetry. A similar thing happens in the above analysis with the Higgs field: the interaction with the Higgs field spoils the gauge symmetry in the gauge fields.
It is not clear to us yet, what implications the various problems in gauge symmetry have for the standard model.
https://physicstoday.scitation.org/doi/full/10.1063/PT.3.2196
Eating up the Goldstone bosons was the revolution of 1964 which earned a Nobel prize for Englert and Higgs.
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.13.508
Above is a link to a freely readable copy of the original Higgs paper published in October 1964. The paper is just two pages.
Peter Higgs first defines two real scalar fields which interact with a vector field A.
In equation (3) he defines a vector field B which is based on A and the Goldstone field
Δφ_1.
He claims that B in equation (4) describes a vector field whose quanta have a non-zero mass.
But a disturbance in the Goldstone field moves at the speed of light. How can we model it with a new field B where the quantum moves slower than light?
Also, the vector field A is usually understood as an external field. For example, in the Dirac equation with the minimal coupling, A is an external field. Does it make sense to define a field B which is a mix of the particle fields (the scalar fields) and the vector field?
The Aharonov-Bohm effect
https://en.wikipedia.org/wiki/Aharonov–Bohm_effect
The Aharonov-Bohm effect shows that the absolute value of the electromagnetic vector potential A has observable effects. It is not just the electric field E and the magnetic field B which affect the observable behavior of electrons.
One cannot carelessly transform the electromagnetic vector potential A and still keep the observable physical behavior of the system same. The coupling between the electron field and the electromagnetic field spoiled the gauge symmetry.
Similarly, the coupling of the Higgs field to the gauge field probably spoils the gauge symmetry. The trick by Higgs and others of eating the Goldstone bosons by moving the contribution of the Goldstone field to the gauge field probably does not work.
Why have we not observed the massless Goldstone bosons? Would they show up in the LHC accelerator?
Or is the Z and W boson mass creation mechanism very different from what the standard model claims?
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