Removing Higgs field defects: set the potential infinite at the origin of the complex plane

UPDATE June 30, 2022: the Mexican hat potential may contradict quantum mechanics. Let us create a very high energy Higgs particle. If the Higgs potential is not the harmonic potential

      k (r - v)²,

where v is the minimum energy value and k is a constant, then the wave of the particle may be disturbed by the strange form of the potential function V(x). The wave would be scattered from itself. Is it possible in quantum mechanics that a particle scatters itself?

A possible solution: the wave which makes the field slip over the center of the Mexican hat necessarily contains a lot of energy, and very many Higgs particles. We can treat the wave as a classical wave. The next question is if the classical wave conserves energy and momentum?

If the action is conserved in translations of position and time, then momentum and energy are conserved. This is true for the Higgs lagrangian. Thus, the classical wave does conserve energy and momentum in newtonian mechanics.

However, is the Higgs system Lorenz covariant? Suppose that we have on observer X moving at a very high speed. He has displaced the Higgs field in a volume S to a value z. He used the energy

       E = (S, z).

A static observer thinks that the volume is only 1/2 S, because of length contraction. He thinks that the energy is less. Does this break Lorentz covariance?

The concept of a scalar field is suspicious from the point of view of Lorentz covariance. The electromagnetic field has to be Lorentz transformed. Can a scalar field be Lorentz covariant?

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In our previous blog post we suggested that one can eliminate sources in the Higgs field by using a Mexican hat potential which is infinite at the origin of the complex plane.

https://en.wikipedia.org/wiki/Higgs_mechanism

If the Higgs field originally is without sources, then it (probably) cannot obtain sources even at a high temperature because the value of the field cannot "slip" over the center of the hat. Then no defects or crystal boundaries can appear in the field.

The value of the field is then given by

       φ(x) = r exp(i α),

where the radius r > 0 is real and the angle α is real.

One may identify the space (r, α) with the Archimedes' screw where the origin r = 0 has been removed.

But is it sensible to remove the origin from the space of the values of the Higgs field?

https://www.theorie.physik.uni-muenchen.de/lsfrey/teaching/archiv/sose_09/rng/higgs_mechanism.pdf

Let us look at the article by Kien Nguyen (2009):

The coupling between the Higgs field φ and the gauge boson 4-potential field A looks like the minimal coupling of electrodynamics.

The gauge transformation

The value of the lagrangian without the Higgs field is conserved by adding the gradient of an arbitrary function η(x) to the 4-potential A. This is just like in electrodynamics. The addition changes the gauge.

When changing the gauge of A, we have to transform the Higgs field, too, to conserve the value of the lagrangian. The full gauge transformation is the lowest pair of equations above.

The gauge transformation of A and φ conserves the value of the lagrangian: the physics of the system is not affected by the gauge transformation.

Since φ(x) is transformed by multiplying it by

       exp(i e η(x))

we do not need to include the origin 0 among the possible values of the Higgs field.

Removing the origin from the possible values of the Higgs field

Is it an ugly modification to remove the origin 0 from the possible values of the Higgs field?

The motivation for the introduction of the Higgs field is to create masses for the W and Z bosons.

The Higgs field has to have an almost fixed, non-zero vacuum expectation value to serve this function.

But it is not required that the value of the Higgs field must be able to "slip" over the center of the Mexican hat. We can claim that the "slip" feature is superfluous and makes the model more ugly than a model where slipping is prohibited.

In the gauge transformation, we have to transform φ(x) by rotating its value around the origin of the complex plane. Slipping is something very different. Why should slipping be allowed?

Higgs field defects as cosmological objects

https://www.researchgate.net/publication/262953976_Tom_Kibble_and_the_early_universe_as_the_ultimate_high_energy_experiment

Neil Turok (2014) writes about defects in cosmology.

No defects have been observed. Do they exist at all?

If defects would exist, they would be a form of matter which is radically different from the particle matter which we are familiar with.

Ockham's razor suggests that defects do not exist. If defects would exist, why we have seen no such matter so far?

Wormholes in general relativity can be regarded as "defects". They differ in the topology from flat Minkowski space. In this blog we have argued that Minkowski is the true structure of spacetime and wormholes cannot exist. In the case of the Higgs field, defects would exist in the combination of Minkowski space plus the complex number space of the Higgs field values.

We do not like defects in general relativity. It is logical that we do not like defects in the Higgs field either.

Defects in crystals and superfluids

Defects do exist is crystal structures of solid matter, as well as in the vortex structure of superfluids. Solid matter is a complex structure which has many parts, atoms. We do not know the structure of a superfluid. The structure might be complex.

Defects in the case of crystals are emergent objects which "live" above a complex substructure.

If there were defects in the Higgs field, they would live in a very simple and fundamental field of nature. Is it a plausible assumption that fundamental fields can contain defects?

The LHC particle accelerator and defects in the Higgs field

We need to check if the conventional Mexican hat potential would allow defects to be created at the 7 TeV energy of the LHC.

In LHC, individual particles can obtain the 246 GeV energy of the center of the Mexican hat. Is that enough to create a defect.

The LHC has not observed any defects.

The Higgs field is "almost" real-valued

The value of the radius r in the value of the Higgs field

       φ(x) = r exp(i α)

must be almost constant. Otherwise, the mass of W and Z bosons would vary.

Why r cannot be exactly constant? We do not know. The Higgs boson would not exist if r would have a strictly constant value. The Higgs boson consists of the oscillation of r around the minimum potential.

Most of the variability of the value of the Higgs field is in the angle α. In this sense, the value of the Higgs field is almost a real number.

Conclusions

We have argued that defects in the Higgs field would be an ugly feature.

We have to check if anyone has calculated the minimum energy of a defect in the Higgs field, and if the LHC would be able to create defects.

There is no "symmetry breaking" in the Higgs field, after all; galaxy clusters

UPDATE June 27, 2022: Suppose that we define the Higgs field as a "helix":

       φ(x) = (r, α),

where r >= 0 is real and the angle α is any real number.

The "value" of φ in an equation is the complex number defined by r and α, but after one round around the origin of the complex plane we do not return to the same point. It is like an Archimedes' screw.

We require the Higgs field to be continuous in spacetime. Could it be that then there will be no "crystal boundaries" nor defects?

But in this case there is a singularity at φ(0, 0).

Another attempted solution: require that the vector field defined by φ never contains a source. We can ensure that it stays that way by putting the Higgs potential infinite for φ(x) = 0. Is there any reason why the Higgs potential should be finite there?

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The classical complex Higgs field is assumed to fill the entire spacetime, and its vacuum expectation value in the minimum energy state is a constant complex number

       v exp(i α),

where v is the radius of the groove, or the circular valley, of the Mexican hat potential, and α is an angle from the real axis.

https://en.wikipedia.org/wiki/Higgs_mechanism

The field rolls to some angle α in the groove of the Mexican hat potential.

We in this blog are somewhat worried of the following possibilities:

A. Even in the minimum energy state, spacetime seems to contain information: the angle α.

B. If the field rolls down from an arbitratry state in a large spatial volume, then the angle α will differ from place to place. There will probably be something like boundaries of "crystals" or defects in the state of the field. Could there even be singularities?

https://www.theorie.physik.uni-muenchen.de/lsfrey/teaching/archiv/sose_09/rng/higgs_mechanism.pdf

Kien Nguyen (2009) has written a simple introduction to symmetry breaking.

Problem A: the information in "empty space"

We may define empty space, or the vacuum, as the lowest energy state of the classical fields.

For most fields, that means that they are identically zero everywhere.

However, the Higgs field is strange: it is not zero, and also seems to contain information, the angle α.

This would be ugly, but it looks like that we cannot in any way find out the value of α. We can only see that α is the same throughout space.

If α is not the same constant everywhere, then the electroweak 4-potential A is not zero everywhere, and we should see some kind of "matter" in space.

Only differences in α are observable.

In the paper of Kien Nguyen the angle α is eliminated altogether, and the 4-potential A will do its job.

Thus, do we have a broken symmetry or not?

Not in empty space. But if the space is not empty? Then the existence of matter breaks the symmetry.

https://arxiv.org/abs/1102.0468

Ward Struyve (2011) argues that there is no symmetry breaking.

He writes that Peter Higgs himself in his 1966 paper presented an interpretation where the symmetry is not broken.

Problem B: are singularities possible in the Higgs field? No

For low energies, the Higgs field is approximately determined by the angle α at each location.

Since α can vary from place to place, it may happen that we will have something which is analogous to crystal boundaries or defects.

We are saved from singularities, because the potential of the Higgs field at zero, or at the center of the Mexican hat, is finite.

There could still exist crystal boundaries. We need to check the work of Tom Kibble. He has written about defects.

Conclusions

Empty space is still "empty", even with the Higgs field. There is no genuine symmetry breaking where the Higgs field in empty space would determine a preferred direction in the complex plane.

The Higgs field is "scalar": it does not determine any direction in spacetime.

We do not like singularities. Fortunately, singularities cannot form.

We do not like that the Higgs field in empty space has a non-zero value nor that it has a charge which fills all the space. The Higgs field is different from all the other fields in that respect. But we have not found a way to build a model where the value of the Higgs field is zero: we cannot find a way to give a mass to the W and Z bosons if the field is zero.

Have astronomers found anything which might be crystal boundaries in the Higgs field? The large-scale structure of galaxy clusters looks like "filaments". Could they come from defects in the Higgs field?