Abstract

 

There are currently three important connected major unanswered questions in physics and astrophysics.

(1)  How can the theories of symmetry and the Higgs field be used to calculate the masses of the fundamental particles?

(2)  How can dark matter be explained and described?

(3)  How can dark energy be explained and described?

A self-consistent theory (called Model 1 in this paper) has been developed that appears to answer questions 2 and 3 quantitatively. In order to derive and justify Model 1, however, it became necessary to calculate the mass-energy of the proton and other fundamental particles that make it up, which answers question 1. This procedure then gave a path for calculating the mass of the Super Particle, which is the primary particle of Model 1. The super particle was found to be a Grand Unified Particle, which unifies the electromagnetic, weak and the strong forces. In investigating the properties of this super particle, it became obvious that it had the properties of dark matter, and when it breaks down, it generates dark energy. This paper describes the means of calculating mass-energy for any particle and the properties of the shielded super particle. In a sequence of related papers (Wald, Model 1-B; Wald, Model 1-C; Wald, Model 1-D; Wald, Model 1-E; and Wald, Model 1-F), Model 1 is detailed and expanded.

 

 

 

I. THE PROBLEM

Schumm points out that mass appears to be the charge for the gravitational field (Schumm, 10). Higgs found out that elementary particle mass is connected to a field that permeates space (Kane, 97). The mass comes from the interaction of the Higgs field and the mass charge of the particle, so it should be calculable using quantum mechanical techniques. The ability to calculate the mass of particles allows us to calculate the mass and characteristics of the grand unified particle or super particle. The super particle is the basic particle of Model 1, and it appears to explain dark matter and dark energy. The unique features of Model 1 are:

• There are two spaces in the universe, low-energy particle space and high-energy quantum vacuum space. A potential barrier 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 particle space to vacuum space through black holes where they are converted into super particles (energy ~10¹⁷GeV).  They are then wrapped with a potential energy barrier shield (~10¹⁹GeV) to stabilize them. The barrier shield forms the boundary of high-energy vacuum space. These stabilized, shielded super particles are then able to escape from the black hole into particle space. The shielded super particles have a low interaction cross-section with ordinary particles except through gravity, and so are observed as dark matter.

• Dark matter particles interact with each other and form a slowly building bubble centered on a galaxy that stabilizes its outer edges. Corridors of dark matter are also generated which form a cosmic web between the galaxies. These corridors guide the development of new galaxies.

• The super particles can tunnel through the barrier into particle space. Upon reaching particle space, the super particles become unstable and break down into particles (cosmic ray protons) with ultra high kinetic energy (UHECR’s). In doing so, they give up potential energy from their barrier shields into particle space which becomes the dark energy that we observe as the cause of our accelerating, expanding universe.

 

In this paper, we will show that the mass of particles can be calculated by adding a Higgs state to the Standard Model particle states. Further, we will extend the theory that allows us to determine the standard model masses to deduce the existence of the shielded super particle of Model 1. The super particle will also be shown to be a Grand Unification Particle. The shield for this particle will further be shown to be a Complete Unification Field unifying electromagnetic, weak, strong and gravitational forces. Finally, the super particle will be shown to have the characteristics of dark matter, and yield dark energy and cosmic rays when it tunnels through the shield and breaks down.

 

II. THE SOLUTION

This paper will proceed by:

1. Developing the theory that allows us to calculate the mass of particles.
2. Proving the accuracy of the theory by calculating the masses of the particles, and comparing them with experiment.
3. Calculating the mass of the Grand Unified Particle, showing that it fits the characteristics of the Super Particle, and then calculating the mass of the barrier shell needed to stabilize the super particle.
4. Showing how a black hole in the center of a galaxy can form shielded super particles, and then showing how it can escape the black hole as dark matter.
5. Showing how escaping dark matter can stabilize the edge of a galaxy.
6. Showing how super particles can penetrate the barrier shell by quantum tunneling, and then break down to form ultra high-energy protons and potential energy (dark energy). 

1. Particle Mass Theory 

Kane (Kane, 261) describes mass as due to three sources:

(1). Higgs free mass of the particles

(2). Confinement mass-i.e. the mass-energy that holds the particles together.

(3). Interaction mass.

Free mass is given to quarks and leptons through interaction of symmetry with the Higgs field. As will be shown below, confined mass is given to confined particles by interaction of quarks confined into protons with the Higgs field. The interaction mass is small and calculable and will not be pursued further here because it is treated elsewhere (Kane, 262). Here we will work out the theory for, and the values of, each component.

 

(1). Higgs Free Mass of the Particles

Here we give a description of the quantization of the fermion and Higgs particles. For comparison, let us begin with a known physical phenomenon-electrons in quantum shells around a proton nucleus-the atom. According to classical mechanics, the potential energy levels of this system can be obtained by noting that the electric field around the nucleus is:

            ε = 10^-7 c²q_e/r² (nt/coul)             

Then the force on the electron in orbit is:

            f = 10^-7c²q_e²/r² = εq_e (nt)                    

And the potential energy of the electron in its shell is:

            V = -(10^-7 c²q_e/r²) q_er = εq_er (J)

So if the electron orbit moves outward a distance d, the amount of potential energy given up is:

            ΔV = -(10^-7c²q_{e /} r²) q_ed = εq_ed (J)     

But the probability of presence of the electron in the shell n is low due to destructive interference of the electron with itself unless the radius is such that the electron orbit distance traveled around its shell is exactly n electron wavelengths long (n = 1,2,3…). Then the electron interferes constructively, and the potential energy of the electron as it moves around the orbit in its shell is 

            V_n = -(1)hcR_∞/n²  (J)

Where the field (1)hcR_∞  replaces 10^-7c²q_e / r² (J) and becomes ε. Note the quantity (1) is the quantum symmetry term for symmetry U(1), which applies to the electromagnetic force. The quantum number n appears in the denominator to replace r as the measure of how many wavelengths are needed to span an orbit. Note also that the charge q_e is incorporated into the term R_∞.       

And if the electron moves from orbit 1 to orbit 2 and loses potential energy, a photon of wavelength λ_l and energy hc/ λ_l is emitted, according to the equation 

            1/ λ_l  = R_∞(1/ n₁² – 1/ n₂²)

            Where:

            n = principal quantum number

            R_∞ = Rydburg number (1/meters)

            h =Planck’s constant (J sec)

            c = speed of light (m/sec)

            q_e = electromagnetic charge

Now we will use the same procedure to develop the mass of a particle from the Higgs field and compare it with the quantum electron case. Kane starts with the Lagrangian (see Kane, 98) of a Higgs field of a particle passing through an ambient Higgs field caused by many particles in a vacuum.

            T – V = ½∂_μφ ∂ ^μφ – (½μ²φ ²+ ¼ λφ ⁴)

Note that the potential energy V is related to the Higgs field as follows

            V = ½μ²φ ²+ ¼ λφ  

            Where:

            λ = self interaction coefficient

            f  = Higgs field (a complex quantity), so f ² = f^† f  

The first term appears to be the interaction of the particle Higgs field with the ambient Higgs field. The second term appears to be the interaction of the particle field with itself and the ambient Higgs field. To find the excitation energies, and thus the masses, we must find the minimum of the potential and expand around the minimum to get excitations, which are the particles. In field theory, it is conventional to call the minimum the ground or vacuum state, and the perturbation terms are excitations. The form of the Lagrangian determines the mass of the particles. If these operations are performed, the result is:

            m _η²  = -2 μ² = mass               

Note m _η is a complex quantity, so to get particle mass m_p (real), we must use m _η² to get the real quantity. Then, V can be written:          

            V = – m _η²φ ²+ ¼ λφ ⁴= -(m _η²– ¼ λ_tφ ²)φ ²     (1)

And we note that particle mass energy is quantized as m_pη, then if the quantum state of the Higgs potential energy changes, a massive particle is formed as follows:

            ΔV_η = m_pη = m_η²φ ² (1- φ ² ¼ ( λ_t /m _η²)) = particle mass energy        (2)

Now we note that as with the electron, each charged particle is moving in a reentrant pattern determined by its symmetry, so the mass energy is low due to destructive interference unless the particle orbit distance traveled is exactly n particle wavelengths long (n = 1,2,3…). Then the particle interferes constructively, and the potential energy of a particle as it moves around its orbit in its symmetry pattern is the product of factors (see Kane, 90) as shown in the equation:

            V_n= m _η²φ ² (1- φ ² ¼ ( λ_t /m _η²))  = (symmetry factor)(charge factor)(field factor)

The value of each factor is as follows.

The Symmetry factor specifies which symmetry is active in a particle, and because of constructive interference, each term becomes:

            (Electromagnetic factor U(1)) = (1)

            (Isospin factor SU(2)) = (2)

            (Color factor SU(3)) = (3)

            (Higgs factor SU(4)) = (4)

So the (symmetry factor) = (S) = (1) or (2×1) or (3x2x1) or (4x3x2x1) = (n!) for each possible symmetry number (n).  Note that only the quantum numbers for the symmetries active in the particle appear in the factor. Note also that a particle is formed for each set of active symmetries, and the families of particles stem from the sequence of sets.

The Charge factor specifies which charge is active in a particle:

            (Electromagnetic charge) = q_e

            (Isospin charge) = q_i

                (Color charge) = q_c

                (Mass charge) = m_p = the charge that generates mass-energy (ref 2, Model 1B).    

            So the (charge factor) = (q_e q_i q_c m_p) = q_t

            where only the charges active in the particle appear in the factor 

The field factor is more complicated. Two Higgs fields are important in mass generation, the ambient and the self-interaction fields. 

The first Higgs field is from the distributed mass in the universe, which is uniform on a large scale. We call this the ambient field. It can be described by the equation:

         φ_h1² n_p = K_hohc(S)(Σ q_t)n_p

                       = K_hfhc(S)(Σ q_e q_i q_c)n_f(for fermions)

                       = K_hbhc(S)(Σ m_p)n_b(for bosons)

In writing this equation, it is noted that the density of particles is small, so on average one particle is not close to another particle. It is also noted that enough total particles exist so that the total summed field density from all particles K_ho Σ q_{t η}²/r² is above the threshold needed to generate a particle. Then the r dependence is averaged out, the field is quantized, and f_h1² becomes K_hohc(Σ q_t). It is finally noted that the particles have a closed orbit in  the ambient field, and the probability of orbit occupation of a Fermion or Boson is low due to destructive interference of the particle with itself unless the distance traveled by the particle around its closed loop is exactly n_p wavelengths long (n_p = 1,2,3…). It will be seen, then, that the quantum number n_p is in the numerator and linear rather than the denominator and squared because the summed Higgs field strength is uniform in space rather than radially dependent as it was for the electron in an atom.

The second Higgs field is due to self-interaction, and so is controlled by the quantum number of the Higgs particles m _ηthat are close. The field is determined by the quantum number of the Higgs particle that is orbiting near the charge of the particle. This field can then be described for the Higgs quantum number m by the equation:

            φ_h2² m =  mrK_ho ¼ ( λ_t /m _ηh²) m _ηh²/r².= m K_ho ¼ ( λ_t )/r

Recalling that the field is small unless r is such that the distance traveled in a pattern orbit is an integral multiple of the particle wavelength, we find for a particle influenced by an orbiting Higgs particle with quantum number m:

            φ_h2² m = K_ho Σ λ_t (m/4),

If we insert the two Higgs terms into equation (2), we get:

            φ_h² =  φ_h1²(1- φ_h2² )=  n_p² (1- λ_t (m/4))             (3)

But as Kane noted (Kane, 105) as part of the Higgs mechanism, that the Higgs field must be assigned an SU(2) doublet m’(up or down) with the following characteristics:

            m’ = either of the two quantities:

            down- (m) = variable internal field term,

            (m) = 0,1,2,3,4,6,10

Note that λ_t remains constant at 1, and

            (1-λ_t(4-m)/4) = (1-4/4), (1-3/4), (1-2/4), (1-1/4), (1-0/4), (1-0/4)

            up- (m+(m-1)) = constant ambient field term,

            (m) = 1,2,3,4,6,10 (the m=0 case does not exist)

            (m+(m-1)) = 1,3,5,7,…

Note that λ_t has the values λ_t = 2/1, 2/2, 2/3, 2/4,… and

            (1-λ_t(m/4))  = (1-2/4), (1-2/4), (1-2/4), (1-2/4),…

Then comparison with the field equation (2) allows us to replace it with the following equation to get the quantized field factor:

            (Field factor) = n_p^2m’(1- λ_t(m/4)),              (4)                                  

Then by putting all factors together, we get for the total potential energy:

            V_n= (symmetry factor)(charge factor)(field factor)

                 = (S)K_hohc(Σ q_t)n_p^2m’ (1- λ_tm/4)          (5)

                 = mass energy of the primary particles       

 

            Where:

            (S)= symmetry = (1) or (2×1) or (3x2x1) or (4x3x2x1)

            Σ q_t = sum of charges

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

            φ_h1² = K_hohc(S)(Σ q_t).

            λ_t = self interaction factor

            m’ = doublet shown above

    (2). Confinement mass

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 for primary particles 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))        (6)

       (3). Modifications for Multiple Particles Acting Together

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

(1). The primary case is: 

            (Field factor) = n_p^2m’(1- λ_t(m/4)),             

(2). 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

(3). 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)

(4). 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)

   (4). Mass-energy Units

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

(1)  Fermions

If K_hfhc(S)(Σ q_e q_i q_c) = 1, the energy for each Fermion is in units of MeV

(2) Bosons

If K_hbhc(S)(Σ m_p)  = 1, the energy for each Boson is in units of GeV 

 

  2.    Primary Particle Mass Calculation

 

In order to prove the accuracy of this mass calculation procedure, we will now calculate the mass of all the principle fermions and bosons.

 

Table I. Leptons 

(n,m) values

 (1,4)                (1,(1+0))         (2,4)                  (2,(2+1))          (2,4)                 (2,(3+2))                                                                                                                    

Experimental lepton mass (Kane,8)

ν_e(<2.2eV)      e(0.511MeV)  ν_μ(<0.17MeV) μ(105.7MeV)  ν_τ(<15.5MeV) τ(1,777MeV)

Lepton masses from Higgs charges and fields

(1-4/4)             (1-2/4)            (1-4/4)             (1-2/4×1/3)     (1-4/4)              (1-2/4×1/3)

                          x(1×1)                                        x(1×2)                                          x(1×2)

                          x1^2×1                                          x2 ²⁽²⁺¹⁾                                     x2 ²⁽³⁺²⁾

(0eV)               (0.50MeV)      (0eV)               (106.2MeV)    (0eV)                (1,707MeV)

Notes

(a). We see that the Higgs field forms one family with SU(1), and produces an electron with mass close to the observed mass.

(b). Note in case (2,(3+2)), that the Higgs quantum number (3) can exceed the symmetry number (2).

(c). The neutrino does not interact with the Higgs field, so the m value remains 4 for all neutrinos, and the primary mass-energy of the neutrino remains 0 for all neutrinos. However, the neutrino does exist, so it is shown.

(d). The Higgs field forms two families with SU(2), and gives two new particles, μ and τ with mass close to the observed mass. Again, the corresponding neutrinos do not interact with the Higgs field, so the primary mass energy is zero.

 

Table II Fermions

(n,m) values

 (3,3)                (3,(1+0))          (3,2)                 (3,(2+1))         (3,1)                 (3,(3+2))                                                                                                                    

Experimental quark mass

d(2.3MeV)      u(4.8MeV)       s(95MeV)       c(1.275GeV)   b(4.18GeV)     t(173.2GeV)

Quark masses from Higgs charges and fields

(1-3/4)             (1-2/4)     (1-2/4(2/3)5/4)  (1-2/4(1/3)      (1-0(1/4?))      (1-2/4)

x(1×1)               x(1×1)               x(1×2)              x(1×2)              x(1x2x3)          x(1x2x3)

x3^2×1                x3^2×1                x3^2×2               x3^2x(2+1)         x3^2×3                x3^2x(3+2)          

d(2.25MeV)    u(4.5MeV)      s(94.5MeV)    c(1.210GeV)   b(4.37GeV)     t(177GeV)                  

Notes

(a). The first (n,m) value is (3,4), but that yields an energy of zero and the particle does not exist, so it is not shown.

(b). For the (3,2) case, three bosons and two fermions are acting together at a time, so the term becomes:

(1-2/4(2/3(2/4+3/4)) = (1– 2/4(2/3)5/4) = (1-5/12)

(c). For the (3,(2+1)) case, three bosons are acting together, one at a time, so the term becomes:

            (1-2/4(1/3))

(d). The (3,3) case should be (1-1/4), not (1-0). The (1-0) case agrees with the experimental data, but not with the quantum formula (5), which is surprising since it is the only disagreement. This case is significant, so more will be said about it below.

 

Table III Higgs Bosons Including Possible Grand Unification SU(6) and Complete Unification SU(10) Bosons

(n,m) values

(4,0)                (4,1)                 (4,2)                 (4,3)                     (4,6)                (4,10)                                                                                                     

Experimental and other possible Higgs masses

                         H^o(125GeV)                                                                                    (~10¹⁹G)

Masses from Higgs charges and fields

(1-4/4)             (1-3/4)            (1-2/4)             (1-1/4)                 (1-0/4)            (1-0/4)

x(1)                   x(1×2)              x(1x2x3)           x(1x2x3x4)        x(1×2…x6)       x(1×2…x10)

x4^2×1                      x4^2×2                   x4^2×3                      x4^2×4                       x4^2×6                x4^2×10

 (0GeV)            (128G)            (12,300G)        (1.18×10⁶G)      (1.2×10¹⁰G)      (4.00×10¹⁸G)

Notes

(a). There is no (m, m+(m-1)) column for the Higgs because the Higgs interacts only with the ambient field.

(b). The second Higgs family generates a Higgs particle with a mass-energy of 128G versus measured 125G. Also, there is a known interaction between the SU*(4) Higgs field and SU(2) particles in the weak force.

(c). One other possible Higgs particle shows on the table-(4,4), but it has a 0GeV energy, so it may exist. No lower energy Higgs particle has been found, but a systematic search has not been made.

(d). The Higgs field (4,3) appears to be the one that reacts with (3,3) to make a quark.

(e). The next Higgs field (4,2) appears to be the one that interacts with (3,1) to make a heavier quark.

(f). The (4,10) Higgs appears to match the expected value for the barrier shell. 

(g). Note again in the (4,6) and (4,10) cases that the Higgs quantum numbers (6 and 10) can exceed the symmetry number (4).

 

Table IV Confinement Mass for the Proton and of the Grand Unification Proton

The confinement mass component for the proton can be described as the energy component due to the exchange of the gluons between the quarks to maintain the confinement of the quarks. The quark combinations to make nuclei are as follows:

• u, u, d. This is the common proton in particle space
• c, c, s. This is probably a proton in particle space, but it is low probability event.
• t, t, b. This may be a grand unification proton or super proton in vacuum space

A different combination of u and d gives the neutron, but the neutron is not important for the purposes of this paper, so it will not be pursued here.   

           

Table IV Confinement Mass for the Proton and of the Grand Unification Proton

(n,m) values

            (3,(1+0))                                 (3,(2+1))                                  (3,(6+5))                                                                                                                    

Experimental proton mass

            uud(938MeV)                         ccs(?GeV)                               ttb(~10¹⁷GeV)

Proton masses from Higgs charges and fields

d(2.25MeV)    (1-5/8)               c(1.21GeV)     (1-5/8)               t(177GeV)       (1-5/8)

u(4.5MeV)      x(1x2x3)             s(107MeV)     x(1x2x3)            b(4.37GeV)     x(1x2x3x6)

uud=45.6M     x3^2x(1+0)           ccs=157G        x3^2x(2+1)           ttb=1.37 10⁵    x3^2x(6+5)          

            uud(923MeV)                         ccs(257GeV)                           ttb(5.8 10¹⁶G)

Notes

(a). The mass energy for a proton matches the experimental data, and so there is reason to believe the formula is correct.

(b). The mass energy for ttb super proton matches the value for Grand Unified Force (Kane, 281) obtained by other means. This is strong evidence for the existence of a super proton.

(c). The self-interaction factor for the proton (uud) is (1-5/8), and fits the data well. Note that this factor results from the fact that the self-interaction term comes from interaction with two Higgs bosons rather than one as for a particle, and so is:

            (1-λ_tf ^2p/m _η²)= (1-2/4(2/4+3/4)) = (1-2/4(5/4)) = (1–5/8)

 3.    Shielded Super Particle Mass Calculation 

Kane notes that three forces appear to achieve the same value at ~10¹⁷ GeV. It also appears that four forces (including gravity) may unify at a somewhat higher value (10¹⁹ GeV-see Kane, 281), and this fact fuels the speculation that the forces unify at high energy. We note the following combinations:

• U(1)xSU(2)xSU(3) = SU(6) = The electromagnetic, weak and strong forces are all combined (Grand Unification)
• U(1)xSU(2)xSU(3)xSU(4) = SU(10) = All forces are combined. (Complete Unification)

In contrast, we note the following:

• SU(5) = SU(2)xSU(3) = unification of only weak and strong forces.
• SU(7), SU(8) and SU(9) have the same problem. They can unify only some of the forces.

The symmetry term in section A above, demands that we must account for the symmetry of all of the lower symmetry forces in order to calculate the mass of a particle, however. Thus all higher combinations are forbidden except SU(6) and SU(10). As a result, the proton decay lifetime experiment, with lifetime calculations based on SU(5), is expected to fail, as it appears to be doing (Kane, 289). But the super proton SU(6) and the barrier shell SU(10) are expected to be formed along with the standard model particles if the energy conditions are correct.

For the quark (3,1) case above, the (1-1/4) value for internal states was expected, which would give a mass value of 3.28GeV, but this value is a poor fit to data. However, the (1-0) value, which was used, gave the best fit to data. The (3,(3+2)) case would give the same value for either case. But the (1-0) value goes with the (4,6) case, which is the Grand Unification value. Note that the (3,2) value is too high as well, and it is also a down quark. It appears that Grand Unification case dominates the particle formation process. It appears that we have accidentally stumbled on the components of a Grand Unification proton in cases (3,3) and (3,(3+2)). When we combine these quarks into a super proton, as shown above, we get a mass energy value of ttb-5.8×10¹⁶ GeV, a value very close to the expected ~10¹⁷ GeV. Also, the τ particle (1,777MeV), case (2,(3+2)) appears to be the electron equivalent for this super particle, and will be called the super electron.

The super proton is expected to be unstable in low energy particle space, and break down into an ultra high kinetic energy proton and give up potential energy. In order to maintain a high potential energy environment where it can recombine into a super particle and so maintain its lifetime, a potential energy barrier shell is required. It must have a potential energy greater than the super proton energy (5.8×10¹⁶ GeV) to contain it, but less than the Planck energy (1.22×10¹⁹ GeV), say ~10¹⁹ GeV. The (4,10) Higgs field appears to fit at potential energy 4.00×10¹⁸ GeV. Note also that as a Higgs field, it acts on mass, so electromagnetic force and photons penetrate the barrier, but mass does not, except under special circumstances.

The barrier shell can be approximated by a thin potential energy shell of thickness “a” that obeys the following equations (see Wald, Model 1-D Appendix H for more details).

            If E> V₀

            T = 1/(1+V₀²sin²(k₁a)/4E(E-V₀)               (7)

                = transmission through the barrier shell

            Where:

            k₁= (8p²m(E-V₀)/h²)) ^1/2

            E = particle kinetic energy

            V₀ = shell potential energy    

 

            If E< V₀,

            T = 1/(1+V_o² sinh²(k₁a)/4E(V₀-E),

            Where:

            k₁= (8p²m(V₀-E)/h²)) ^1/2

We see, then that massive particles with kinetic energy less than the potential energy (4.00×10¹⁸GeV) are mostly reflected, but some can tunnel through.

 

4    Formation of Shielded Super Particles

 To determine how the super particles and barrier shells are formed and operate, we must determine the equations for the processes that are valid in black holes. The expansion and contraction of space in high- energy zones are controlled by the following field density (ρ_f) and pressure (p) equations from general relativity (see Peebles, 396):

             ρ_f = φ’ ²/2+ V = field density

             p = φ’ ²/2 – V = pressure                   (8)

            Where:

            V = a potential energy density

            φ= a new real scalar field = Higgs field

            φ’=the time rate of change of the field

            φ’ 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 average mass-energy density (r_m), pressure (p) and the cosmological constant (Λ) (see Peebles, 75):

            ä/a = -4/3πG (ρ_m+ 3p) + Λ                 (9)

                    = acceleration of the cosmological expansion parameter         

            Where:

            m = particle mass

            ρ_m = Σm/vol

            vol = volume of space containing the particles

Note from the field pressure equation (8), that if the potential energy density exceeds the field kinetic energy, the pressure is negative. The field kinetic energy term is slowly varying, however, because it depends primarily on the total mass in the general vicinity (see Wald, Model 1-B), so the potential energy controls the pressure. Then, if the potential energy increases enough, the negative pressure term in equation (9) can become large enough to exceed the mass density term in equation (9), and the acceleration of the cosmological expansion parameter (ä/a) turns positive, and space will eventually expand. If the potential energy V is small compared to the field kinetic energy term, however, the field pressure term is positive, and if Λ is small, the acceleration ä/a becomes negative, and space will eventually contract. Even if the (ä/a) term is negative, however, space will continue to expand for a while, thus maintaining its prior state, but eventually, expansion velocity will reduce below zero and space will contract. Thus, if ä/a is positive, space will expand faster and faster as time goes on. On the other hand, if ä/a is negative, space will expand slower and slower until the expansion velocity reverses sign, and then space will contract.

Note from the field pressure equation, that if the potential energy exceeds the field kinetic energy, the field pressure is negative. Also if the field pressure is large enough, it will exceed the mass density term, and the acceleration of the cosmological expansion parameter ä/a will turn positive, and space will expand. If the potential energy V is small compared to the field kinetic energy term, field pressure is positive, the acceleration ä/a is negative, and space contracts.

Further, it has been suggested that mass is the conserved charge for the gravitational (Higgs) field by Schumm (Schumm, 10) and others. If this suggestion is true, we would expect the field φ to obey the following equation as it emanates from a massive particle:

            φ = K_ho m /r²              (10)           

The field described above is a real, scalar field, and is associated with spatial curvature as is mass-energy, and has the general characteristics of the real part of the Higgs field. So we will assume that the real part of the Higgs field accounts for the expansion and contraction of space.

Higgs found that potential energy is connected (see equation (1) above) to the Higgs field and mass. Also, the Higgs field is related to mass through equation 10. Now as the radius from the black hole center decreases, ρ_m increases, and so the field density (φ/vol) increases as well. From equation (1), we see that V increases faster than ρ_m, and so a contraction zone gradually turns into an expansion zone as a particle descends toward the center of the black hole. Therefore at some radius from the black hole center, ä/a will turn positive, form a barrier and reflect most of the particles. So a particle will descend through the contraction zone toward the black hole center, and then be reflected back by the expansion zone inside it, and then return again in the contraction zone under the influence of gravity. This cycling back and forth in two zones characterizes the high-energy diffusion zone where super particles are formed. Note that the volume of the diffusion zone is directly dependent on the mass of the black hole because of the r dependence of ρ_m.

As the particle kinetic energy, and the spatial potential energy approach the unification energy (10¹⁷ GeV), the particles can convert to super particles. At the same time, potential energy barrier shells are being formed in the higher energy of the expansion zone. The super particles increase their kinetic energy during recycling to nearly 10¹⁹ GeV, where they can penetrate the shell and become stabilized as recycling shielded super particles. These shielded super particles are dark matter.

Once the shielded super particles are formed, they can then penetrate the black hole event horizon in the following way. The speed of light is determined by the energy in the high-energy granules of Planck space (Wald, Model 1-D Appendix A for more details). When the kinetic energy of a particle nears the Planck energy (1.22×10¹⁹ GeV), the speed of light increases above 2.99×10¹⁰ cm/sec in order for the Planck length to remain constant under relativistic foreshortening. Thus it is above the escape velocity (2.99 x 10¹⁰ cm/sec) of the event horizon formed at lower energies, so the shielded super particle escapes. In a general relativistic way of looking at this problem, a particle near Planck energy exceeds the ability of the gravitational well generated by black hole particles to contain it.

We note that low energy protons are shielded from interacting with super particles by the barrier shell. This interaction cross section is similar to the interaction of a proton with a neutron, and it is calculable. The proton-shell interaction cross section has been calculated to be ~10^-45 cm² (Wald, Model 1-C Appendix2), and so is small enough to warrant the name “Dark Matter”.

 

 5.   Stabilization of Galaxies with Dark Matter

Beyond the event horizon, the density of shielded super particles is low enough that the mean free path is greater than the distance to the edge of the galaxy, and the super particles fly free to intergalactic space creating a dark matter density that reduces as ~1/r² from the black hole at the galactic center. Now Peebles (Peebles, 47) shows that the matter density ρ_t(r) in a galaxy at a radius r is:

ρ_t(r) = ν_c² / 4πGr²

Where

ν_c = circular rotation velocity as a function of radius.

G = gravitational constant.

The astronomical data for spiral galaxies show that ν_c starts lowtoward the center of the galaxy and rises to a maximum and then flattens out to a constant value. Newtonian gravity requires that ν_c reduce beyond the maximum, or the matter beyond the maximum would escape from the galaxy’s gravity. Since ν_c is constant beyond the maximum, the density must increase and vary as ~1/r² out there. If the density of the dark matter that is decreasing at a slower rate (~1/r²) becomes equal to the density of the visible matter (~ν_c² / 4πGr² ) at the maximum ν_c , then the density of the dark plus the visible matter would vary as ν_c ~1/r². This, of course is what we observe.

In addition to this relation, it has been observed that there is a relation between the mass of the central black hole of a galaxy and velocity dispersion of the stars of the bulge in those galaxies. This relation is called the M-sigma relation (Ferraese, 539). Note in section D that the volume of the diffusion zone where shielded super particles are formed, and thus the number of shielded super particles formed, is directly dependent on the black hole mass (the number increases as the mass increases). Thus when the shielded super particle mass equals the visible matter mass, matter with velocity greater than the escape velocity of visible plus dark matter will escape, causing the dependence of the black hole mass on the velocity dispersion 

Note finally that the dark matter moves away from the black holes in each galaxy preferentially in the direction of other black holes, thus forming a net like structure with galaxies as nodes.(the cosmic web). So intergalactic gas and dust tend to form galaxies around this structure in groups, strings and walls 

These observed data imply that shielded super particles satisfy three of the primary requirements of dark matter. More details on this dark matter halo are given in Wald, Model 1-C.

 

6.     Tunneling Super Particles, Cosmic Rays and Dark Energy 

It has been noted that a super particle is unstable outside of its barrier shell, so when it tunnels through the barrier shell, it will become a proton with ultra high energy (up to ~10¹⁹ GeV) and a significant amount of potential energy that came from the barrier shell and is added to particle space. Thus Model 1 predicts the existence of ultra high-energy cosmic rays (UHECR’s), and a buildup of potential energy in particle space. 

Such cosmic ray events have been observed with energy up to ~ 10²² EV, and there is evidence that the energy goes higher. These events should not exist, since they are beyond the GZK cutoff, and no known local omni directional cosmic ray sources with this energy exist except for tunneling super particles. The GZK cutoff is formed when high-energy protons above a certain energy interact with microwaves to form lower energy particles. Since microwaves are everywhere (the GZK energy), cosmic rays should be limited to that energy. Yet such cosmic rays are observed. They occur anywhere there is dark matter-i.e. anywhere within a galaxy, and at lesser frequency, beyond (see Wald, Model 1-E for details). 

In addition to ultra high-energy protons, the tunneling super particles yield potential energy from the barrier shells upon breaking down. It has been found (Wald, Model 1-E Appendix 1) that a barrier shell thickness of 10^-31 cm will yield enough tunneling super particles over ~10¹⁰ years to account for the dark energy (potential energy density) we observe in our universe (~10^-4 GeV/cc).

 

III. SUMMARY/CONCLUSIONS

In this paper, a procedure has been developed that allows us to calculate the mass of the particles of the universe with considerable accuracy from the Higgs theory and the theory of Symmetry.

Following Model 1, the mass of a super particle was calculated, and found to match the expected Grand Unification energy (~10¹⁷GeV) (Kane, 281), and consists of a quantum combination of the t and b quarks. We found finally, that the barrier shell of the shielded super particle fits the characteristics of a Higgs field that satisfies the SU(10) symmetry (Total Unification of the electromagnetic, weak, strong, and gravitational forces). 

This shielded super particle was found to have the low interaction cross section of dark matter, stabilize the outer edges of galaxies, explain the existence of ultra high-energy cosmic rays (UHECR’s), and explain the dark energy that is expanding the universe.

 

IV. REFERENCES

 

1. L. H. Wald, Model 1-B “The Origin of the Higgs Field for Model 1” www.Aquater2050.com/2017/01/
2. L. H. Wald, Model 1-C “Shaping the Dark Matter Cloud” www.Aquater2050.com/2017/02/
3. L. H. Wald, Model 1-D “The Recycling Universe” www.Aquater2050.com/2017/02/
4. L. H. Wald, Model 1-E “The Super Particle as a Cosmic Ray” www.Aquater2050.com/2017/03/
5. L. H. Wald, Model 1-F “How to Prove a Theory’s Correctness” www.Aquater2050.com/2017/03/
6. B. A. Schumm, Deep Down Things, Baltimore, Johns Hopkins University Press, 2004.
7. G. Kane, Modern Elementary Particle Physics, Ann Arbor, Michigan, Perseus Publishing, 1993.
8. P. J. E. Peebles, Principles of Physical Cosmology, Princeton, New Jersey, Princeton University Press.
9. Ferraese, L. and Merritt, D. “A Fundamental Relation Between Supermassive Black Holes and their Host Galaxies” The Astrophysical Journal The American Astronomical Society. 539 (1) (2000-08-10)