Abstract

Of the major unanswered questions in physics and astrophysics, three are arguably the most important.

How can the theories of symmetry and the Higgs field be used to identify the origin of the Higgs field and calculate the masses of the fundamental particles?
How can dark matter be explained and described?
How can dark energy be explained and described?

A self-consistent theory called Model 1 has been developed that answers these questions quantitatively. The unique features of this model are:

There are two spaces in the universe, particle space and quantum vacuum space. A potential barrier composed of Higgs field separates them. One space contains visible matter, and the other space contains dark matter.

There is a cycling of mass-energy between these spaces through the black holes that connect them. Particles pass from visible matter space through black holes where they are converted into super particles (energy ~10¹⁷GeV) that operate with unified force behind the potential barrier (potential energy ~10¹⁹GeV), and become shielded super particles or dark matter.

Considerable work has been done to show how Model 1 answers the important questions named above and is an accurate description of nature. While working on Model 1, it was noted that the Higgs field appears the same as the real, scalar field of general relativity that controls space expansion (Peebles, 396). These facts introduce an important issue, namely, the connection between the Higgs field, the inertial effects of mass, the expansion and contraction of space, general relativity, and mass formation. This connection also answers the three questions posed above.

The Problem

Many astrophysicists consider three questions to be the most important in astrophysics.

How can the theories of symmetry and the Higgs field be used to identify the origin of the Higgs field and calculate the masses of the fundamental particles?
How can dark matter be explained and described?
How can dark energy be explained and described?

A self-consistent theory called Model 1 has been developed that answers these questions quantitatively. The unique features of this model are:

There are two spaces in the universe, particle space and quantum vacuum space. A potential barrier composed of Higgs field separates them. One space contains visible matter, and the other space contains dark matter.

There is a cycling of mass-energy between these spaces through the black holes that connect them. Particles pass from visible matter space through black holes where they gain energy to the force unification level and are converted into super particles (energy ~10¹⁷GeV) that operate with unified force behind the potential barrier (potential energy ~10¹⁹GeV), and become shielded super particles of dark matter.

Dark matter particles interact with each other and form a slowly building and moving bubble centered on a galaxy. Corridors of dark matter form a cosmic web between the galaxies, which guide the development of new galaxies and connect the bubbles of dark matter to each other.

While in particle space, the super particles tunnel through the barrier shield into particle space. The super particles without their shield are unstable and break down into cosmic rays (protons) with ultra high kinetic energy (UHECR’s). In doing so, they give up potential energy into particle space. The potential energy gradually builds up to become the dark energy that we observe as the cause of our accelerating, expanding universe. The ultra high energy protons are observed as cosmic rays with energy between the energy of force unification (~10¹⁷GeV) and the barrier energy (~10¹⁹GeV), which is beyond the GZK cutoff.

Higgs found out that elementary particle mass-energy is connected to a field that permeates space (Kane, 97). Schumm points out that mass-energy appears to be the charge for the gravitational attraction (Schumm, 10). Shumm also notes that mass can be modeled by a screening effect with the Higgs field (see Shumm, 293-299). Most important, Einstein described a background independent way to connect mass to the curvature of space-time in the field equation (Misner, 431 and 41). In all of this, it seems that there is not an intrinsic mass in the fundamental particles of the nucleus. The mass we observe appears to come from the interaction of the Higgs field and the mass charge of the particle, and this mass-energy can be calculated. Further, it was noted that the Higgs field appears to be the same as the real, scalar field of general relativity that controls space expansion (Peebles, 396). These facts introduce an important issue, namely, the connection between the Higgs field, the inertial effects of mass, the expansion and contraction of space, general relativity, and mass formation. This issue will be pursued here.

The Solution

There are five effects connected with mass that are usually described separately.

The origin of the Higgs field
The inertial effects of mass
The expansion and contraction of space
The gravitational effects of mass
The quantum connection between mass and the Higgs field

These effects are also described with different theories, but they are closely related, and seem to require a unified theory. Such a unified theory will be described here.

Origin of the Higgs Field

We expect a field to have a source-its charge. For example, the electric field has a source-the conserved electric charge Q. This charge results from the invariance of the phase of electromagnetic radiation throughout space (Noether’s theorem). The result is the equation for the electric field (ε):

ε = K_eQ/r²

Now the idea that mass is a charge has been suggested by Schumm (Schumm, 10) and others. If this suggestion is true, it will result in the following equation for a field emanating from mass, which we will call φ_h.

φ_h = K_ho m /r²

In Appendix A, the generation of particle mass m_p by the Higgs field is described in some detail, so we see a connection between them. Let us assume that f_h is the Higgs field, and see the consequences of this assumption.

We expect the Higgs field to be uniform in space, because the mass we measure, which is generated by the Higgs field, is uniform in space, but the above equation indicates that the Higgs field might be lumpy, i.e. concentrated around mass particles. To explain why this conflict does not occur, we must look at three cases.

The Higgs field of particle space. Here, in the particle space in which we live, the Higgs field comes mostly from the total distribution of mass in the universe. Mass distribution is uniform but low in value on a large scale, even though it is not on a small scale. Thus the Higgs field is smooth, but low in value on a large scale and dominated by the average mass density in the universe. Now mass formation is a quantum effect (equation 4, below), and so dominated by thresholds. If the Higgs field is above the threshold needed to form the standard model particles, particles are formed, and the mass achieved does not change until it exceeds the next quantum threshold, where it makes a quantum leap to a new value. A local change in Higgs field around a massive object would not be large enough or long lasting enough to exceed the next quantum threshold. Thus the mass-energy of a particle would remain constant as long as the average value of the Higgs field is above the threshold for particle space, but below the threshold for a higher energy (vacuum) space.
The Higgs field in vacuum space. Here, in the vacuum bubble that surrounds the super particle, the Higgs field comes mostly from the mass of the bubble shell, which surrounds the super particle, and thus provides a smooth, and very high-energy field density. A significant amount of field also comes from the kinetic energy of the super particle moving within the bubble shell, but it again does not exceed the next quantum threshold. Thus the mass-energy would remain constant as long as the average value of the Higgs field is above the threshold for vacuum space, but below the threshold for a higher energy (Planck) space.
The Higgs field in a black hole. Here, the Higgs field is dominated by the average energy of the many particles in the black hole. However, the field is stratified in layers. The closer the layer is to the center of the black hole, the higher the mass-energy density, and so the higher the Higgs field. There exists a layer with enough energy to allow a super particles to form, and the mass-energy would remain constant for this layer.

From Noether’s theorem, we expect that if there is an invariance in physical laws in different frames of reference, there must be a conserved quantity or charge connected with it. We have assumed that mass is the charge for the generation of the Higgs field, we have to ask, then, what is the invariance that causes the mass charge. As examples, let us start with the case of dynamics without acceleration. Here, we have observed that dynamics without acceleration follows the same physical laws if we move the frame a distance L in space, and this invariance generates a conserved quantity–momentum. Also, if we rotate the frame an angle θ, there is no change in physical laws, and this generates a conserved quantity—angular momentum. Further, if we move the time forward or backward, the laws do not change. This generates a conserved quantity—energy. Now, let us continue with the case of dynamics with acceleration. Einstein showed that there is no special frame of reference even for masses being accelerated by gravity. Thus we are justified by Noether’s theorem in taking mass as a conserved quantity for this case, and we find by experiment that it is indeed conserved.

We see that we have found that mass satisfies the requirements of a conserved charge that it is connected to the Higgs field, and that it can be the source of the Higgs field. Since mass has all the characteristics of the charge and generates a Higgs-like field, we will assume it is the Higgs charge until we find reason to doubt this assumption.

The Inertial Effects of Mass Through the Higgs Field

The inertial effects of mass such as resistance to acceleration can be explained by a screening effect from the Higgs field (see Shumm, 293-299) that explains inertial mass formation. To explain this effect, we compare it with the screening effect of electrons on photons. If a photon passes through a medium filled with free electrons, its oscillating electric field oscillates the electron charges, and they generate opposing photons that tend to interfere with the oscillation of the original photon field. Thus the electric field in the medium will oppose the constant oscillation of the original photon field, and reduce it. This tendency is called screening, and it gives a finite range to a photon in conducting media. We can explain this effect by saying that the screening electric field generates an “effective mass” for the photon even though the photon has zero mass and infinite range. This screening range is similar to the range of a field with massive exchange particles, shown in the equation (see Kane, 29).

f = f_o /4πr exp(-mr) (natural units)

Note that the field is significant in size only out to a range of r ~ 1/m (natural units), where m is the “effective mass”.

In a similar manner, we can explain the mass of a quark with a scalar (Higgs) field that “drags” on the quark as it accelerates. If an accelerating quark passes through a medium (vacuum) filled with massive particles that create the field (visible and dark matter), the accelerating quark generates an accelerating Higgs field that interferes with the ambient Higgs field in the vacuum, and causes it to push back to reduce the relative acceleration toward zero. Thus the ambient Higgs field will oppose the acceleration, and slow it down toward a constant velocity. We can then explain this effect by saying that the screening Higgs field generates an “effective mass” for the accelerating quark as shown above. There does not appear to be any free inertial mass to a quark other than the “effective” mass generated by the Higgs-like field, so we will again assume that the Higgs field accounts for the inertial effects of mass until we find reason to doubt this assumption.

The Expansion and Contraction of Space

In order to deal with the expansion and contraction of space, Peebles postulates the existence of a new real scalar field, and develops the equations for the field density r_f, and the pressure p in space from general relativity.The expansion and contraction of space in this zone is controlled by the following field density (r_f) and pressure (p) equations from general relativity (see Peebles, 396):

r_f = f’ ²/2+ V = field density (1)

p = f’ ²/2 – V = pressure (2)

Where:

V = a potential energy density

f = a new real scalar field

f’= the time rate of change of the field

f’ ²/2 = a field kinetic energy term

Also, from the field equation of general relativity, Peebles develops the cosmological equation for the time evolution of the expansion parameter (a(t)) due to mass-energy density (r_m) and pressure (p) (see Peebles, 75):

ä/a = -4/3πG (r_m+ 3p) (3)

= acceleration of the cosmological expansion parameter

Note from the field pressure equation (2), that if the potential energy exceeds the field kinetic energy, the field pressure is negative. Also, if the negative field pressure is large enough, it can exceed the mass density. Then, if the negative field pressure term is large enough to exceed the mass density term, the acceleration of the cosmological expansion parameter (ä/a) turns positive, and space expands. If the potential energy V is small compared to the field kinetic energy term, however, the field pressure term is positive, the acceleration ä/a is negative, and space contracts.

It is important to understand that the field described here as a real, scalar field, is associated with spatial curvature as is mass-energy, and has the general characteristics of the real part of the Higgs field. So the real part of the Higgs field accounts for the expansion and contraction of space, and mass-energy also accounts for the expansion and contraction of space. we have reason to ascribe this effect to the Higgs field.

Gravitational Attraction from Mass-energy

Einstein described a background independent way to show how the curvature of space-time can be related to the local distribution of mass-energy in the field equation (Misner, 42). He then used the field equation to show that gravitation is a local phenomenon of massive particles in free fall in curved space-time, and does not require a force of attraction acting at a distance as indicated by Newton. Only the local distribution of mass-energy is important. The mass-energy curves space-time and gives an apparent attraction between massive objects that we call gravitational attraction.

Always remember, however, that mass is not the only thing that locally controls space-time curvature. There is a real, scalar field embedded in space-time that controls the shape of space as well (see Peebles, 396), and this field appears to be the real part of the Higgs field.

The Quantum Expression for Particle Mass from the Higgs field

The total potential energy term for the primary particles is (see Wald, Model 1A):

V_n= m _η²φ ² = (symmetry factor)(charge factor)(field factor)

= n! q_t hK_o n^2m’ (1- λ_tm/4)

= mass energy of the primary particles

Where:

q_t= total charge

hK_o= Higgs mass-energy constant (analogous to R_∞(q_e) for the electron case)

λ_t= self interaction factor

m = Higgs field number

Now we have the power to calculate the mass of the primary particles, but we still must establish how to calculate the confinement mass of composite particles such as the proton and the neutron. For composite particles, a procedure similar to the above one yields:

V_nc= (Particle 1 factor)(Particle 2 factor)(Particle 3 factor)

(Symmetry factor)(Charge factor)(Field factor)

= (V_n1)(V_n2)(V_n3)(n! q_t hK_o n^2m’ (1- λ_tm/4))

We note here that if many particles become active together, we get modifications in equation (5). There are three cases.

(1) If three bosons are acting together on fermions (as in two cases of the Higgs mechanism with SU(2) symmetry), the self interaction term becomes:

(1-λ_tm/4) = (1-λ_t(1/3×2/4)) if one particle of three is acting on m at a time

= (1-λ_t(2/3×2/4)) if two particles of three are acting on m at a time

(2) If three bosons and two fermions are acting together at a time (as in the second case of the Higgs mechanism with SU(2) symmetry), then:

(1-λ_tm/4) = (1-λ_t2/3(2/4(2/4+3/4)) = (1–λ_t 2/3×5/8)

(3) If two fermions are acting together (as in proton formation from quarks), the self interaction term becomes:

(1-λ_tm/4) = (1-λ_t(2/4(2/4+3/4)) = (1-λ_t5/8)

Finally, we must specify the units of the mass-energy for the particles. There are two cases.

(1) If hK_o= 1, and λ_t= 1, the energy for each Fermion is given in MeV

(2) If hK_o= 1, and λ_t= 1, the energy for each Boson is given in GeV

Summary

We see that a complex Higgs field originating from the mass-energy of the particles in the universe and imbedded in space can account for all the effects of mass. We notice that:

Mass generates the Higgs field.
The Higgs field generates the inertial effects of mass
The Higgs field works with the mass generated curvature of space to generate the expansion and contraction of space.
The mass generated curvature of space works with the massive particles to generate the gravitational force.
The complex Higgs field works with particle charge to generate the mass-energy of the basic particles.

In a simpler form, we may say that gravitational charge (mass-energy) generates the Higgs field, the Higgs field generates mass-energy, the mass energy generates the curvature of space and thus the gravitational force. The Inertia and expansion-contraction are side effects of the process.

Conclusion

We see that the real, scalar field is compatible with the Higgs field, but the real part of the field contains the dynamics, and the imaginary part is required to generate mass.

References

B. A. Schumm, Deep Down Things, Baltimore, Johns Hopkins University Press, 2004.
G. Kane, Modern Elementary Particle Physics, Ann Arbor, Michigan, Perseus Publishing, 19933.
L. H. Wald, “AP4.7 DARK MATTER AND ENERGY-FUNDAMENTAL PROBLEMS IN ASTROPHYSICS” www.Aquater2050.com/2016/10/
Misner, Thorne and Wheeler, Gravitation, New York, Freeman and Co., 1973.
L. H. Wald, “AP4.7D HOW TO PROVE A THEORY’S CORRECTNESS” www.Aquater2050.com/2015/12/
L. H. Wald, “AP4.7I THE SUPER PARTICLE AS A COSMIC RAY” www.Aquater2050.com/2015/11/
L. H. Wald, “AP4.7L EXTRACTING SUPER PARTICLES FROM THE BARRIER SHELL” www.Aquater2050.com/2016/05/
L. H. Wald, “Model 1A Mass and Function of the Standard Model Particles”, www.Aquater2050.com/2017/01/
L. H. Wald, “AP4.7G ORIGIN OF THE NEW SCALAR FIELD” www.Aquater2050.com/2015/12/