A freely readable copy of the Peter Higgs 1964 paper (just 2 pages) is available at the link.
The paper introduces two scalar fields and a gauge field A.
A somewhat analogous system would have the electromagnetic 4-potential A as the gauge field, and an electrically charged boson particle field as the scalar fields. An electron is a spinor field in the Dirac equation.
We know that if we have a free electromagnetic 4-potential A (of electromagnetic waves), then we can add the gradient of an arbitraty differentiable function f to A, and nothing changes in the physics. It is called a gauge symmetry.
Our August 24, 2019 post showed that there is no gauge symmetry if an electric charge is present: the inertial mass of the charge varies if its electric potential changes relative to far-away space. The electric potential of far-away space defines a preferred frame.
The perturbation in the Higgs paper
The fields are classical at this point. Higgs introduces a Mexican hat potential which causes the scalar fields to assume a constant non-zero value at a minimum energy configuration. We have defined the coordinates of the scalar fields such that the constant value of the two scalar fields is
(0, v).
That is, the minimum energy happens when the fields have that constant value everywhere in the Minkowski space.
(0, v).
That is, the minimum energy happens when the fields have that constant value everywhere in the Minkowski space.
Higgs studies a small perturbation of the field values
(Δφ_1, Δφ_2)
around the minimal energy value (0, v).
We have not seen a proof anywhere that the perturbation series converges. Thus, we do not know if the quadratic approximation by Higgs approximates the original system of equations.
The equations are rather complicated. It might be hard to prove the convergence.
One may ask if it matters at all if the perturbation series converges. The perturbation equations by Higgs look nice and we do not really need to care about the original equations.
(Δφ_1, Δφ_2)
around the minimal energy value (0, v).
We have not seen a proof anywhere that the perturbation series converges. Thus, we do not know if the quadratic approximation by Higgs approximates the original system of equations.
The equations are rather complicated. It might be hard to prove the convergence.
One may ask if it matters at all if the perturbation series converges. The perturbation equations by Higgs look nice and we do not really need to care about the original equations.
Introducing the "Unitary gauge": why does the Goldstone boson disappear?
Higgs defines a new 4-vector potential B by subtracting the gradient of Δφ_1 from A and writes equivalent equations using B. We say that he moved to the "unitary gauge" for the 4-vector potential.
He writes that the new equations describe a massive scalar boson and a massive gauge boson.
The Goldstone boson, which travels at the speed of light, disappeared. Where did it go?
The Goldstone boson is not directly visible in the new equations. But it still exists in the system. A mathematical manipulation of the equations does not change the physics. There still exist waves that carry energy at the speed of light.
What are the "real" physical fields described by the equations? We cannot answer that question by defining a new variable and removing an old variable.
Does the Higgs change of a variable prove that there is no speed-of-light energy transfer in the system? Or does it just hide the obvious speed-of-light waves behind complex equations?
A way to study this is to keep using the old field A and not move to B. The physics have to be the same. Is there some mechanism in the equations which couples the light-speed waves tightly to slower waves and prevents energy from being transferred at the speed of light?
Light within a medium moves slower than the light speed in the vacuum. If an electron is shot at a high speed into the medium, it will move faster than the local speed of light, and will emit Cherenkov radiation. Could something similar happen in the Higgs model? A Goldstone wave would lose energy to the gauge field and soon disappear? Since a Goldstone boson can have a low energy, it cannot produce massive gauge bosons. In the Higgs paper there are no massless gauge bosons, but in the Standard Model there is the photon.
The Goldstone bosons of the Higgs field may quickly decay to photons, and would not be noticed in the LHC instruments.
The Goldstone bosons of the Higgs field may quickly decay to photons, and would not be noticed in the LHC instruments.
The analogue of light within a medium
Light moves slower in a medium, for example, inside glass than in vacuum.
There are charges in glass which can oscillate in synchrony with the light waves. The massless photon field is coupled to a massive "field" of the charges. The photon "acquires a mass" through the interaction, and moves thus slower than the light speed in vacuum.
This effect is clearly analogous to the model in the Higgs 1964 paper.
When a wave of light enters glass, the wave quicly loses some energy to the oscillation of the charges. The end result is an oscillation wave moving slower than light in vacuum.
We may imagine that a photon has been converted to a quantum of an oscillation wave. A photon became a phonon.
Does a photon "exist" within glass? If we have no way of creating a photon inside the glass without at the same time creating the corresponding oscillation wave, then we might say that a photon cannot exist. It has been "eaten" by the oscillation wave.
We may imagine that charges in the glass are attached to the solid matter with springs. There are rubber bands between adjacent charges.
The electromagnetic field has the role of the gauge field.
The lagrangian in the Higgs paper
It is easiest to analyze the model in the Higgs paper by looking at the lagrangian.
We first analyze the fields as classical.
It is best to keep the original variables. Let
φ_1 = 0
φ_2 = v
approximately, where v is constant and non-zero. We will look at small deviations of the φ_i fields from those values.
Let us first forget A. Then a disturbance in φ_1 moves at the speed of light.
A disturbance in φ_2 moves much slower because a deviation from v stores a lot of energy in the lagrangian formula. It corresponds to a massive particle.
Let us then couple A to the system. We keep φ_2 a constant v. If we let φ_1 = 0, we see that a deviation of A from 0, A^2 stores energy in the lagrangian. That means that A is massive and a disturbance in A moves slowly.
We have now coupled A. What happens if we first set A = 0 in all space, and then disturb φ_1 at a point in space?
The lagrangian shows an interaction between φ_1 and A. Obviously, energy will start leaking from φ_1 to A immediately. The disturbance in φ_1 keeps moving at the speed of light, and it keeps leaking energy to A for its whole trip. The energy stored in A moves much slower.
Let us then use quantum mechanics. A disturbance in φ_1 corresponds to a massless boson. The energy E of a quantum may be very small.
The mass of a quantum of A may be larger than E. Then the quantum of φ_1 cannot decay into a quantum of A.
The reasoning above suggests that a massless Goldstone boson cannot be eaten by a massive gauge boson.
Can we excite φ_1 without exciting the field A at the same time? In the electroweak interaction, the gauge field A interacts with many different particles. We can probably excite A separately from the Goldstone boson field. Then the Goldstone boson field cannot be merged to the gauge field A.
The coupling "constant" (not really a constant) between φ_1 and A in the lagrangian is
2e φ_2 = 2ev.
If we set the constant e very small, then we clearly have two separate fields, φ_1 and A which interact very little. How fast does energy leak to A?
Decoupling the Goldstone bosons?
If the only equations of our physical system are the ones given in the Higgs paper, then moving to the new variable B can bee seen as decoupling the free degree of freedom in A and the Goldstone field.
Once one has solved B, one is free to choose any A and the Goldstone field φ_1, as long as
B = A - 1 / (ve) * gradient(φ_1).
This requires that all the communication to these fields go through the "interface" given by B. Then we may say that we decoupled a degree of freedom.
However, in more complicated lagrangians, one can generally communicate with A in many ways. Then we cannot do a decoupling in the way given.
One may argue that the field φ_1 does not exist at all if we use the new equations with B. Then the claim that there was a symmetric potential for φ_i and the symmetry was broken is vacuous. The system where the symmetry broke does not exist.
If we claim that φ_1 really existed, and exists right now, then we probably can excite φ_1 separately from A, and the Goldstone boson exists.
What fields "exist" if some fields are decoupled?
One may argue that the field φ_1 does not exist at all if we use the new equations with B. Then the claim that there was a symmetric potential for φ_i and the symmetry was broken is vacuous. The system where the symmetry broke does not exist.
If we claim that φ_1 really existed, and exists right now, then we probably can excite φ_1 separately from A, and the Goldstone boson exists.
Conclusion
If the scalar fields existed, and do exist now, and the fields fell into an energy minimum of the Mexican hat, then the Goldstone boson does exist, contrary to the claim in the Higgs 1964 paper. The gauge field cannot eat the Goldstone boson.
Maybe it is hard to excite the Goldstone field, and that is the reason why the boson has not been found.
It might also be that φ_1 does not exist at all. There was no symmetry breaking nor a Mexican hat potential. There is just the field φ_2 and its potential curve which has the minimum at a non-zero value v.
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