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Article

The Models of Primary Particles

This version is not peer-reviewed.

Lan Fu  *

Submitted:

19 January 2024

Posted:

19 January 2024

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Abstract
If we assume that: a. The four fundamental forces of nature are independent waves without rest masses, and their speeds are constant in a vacuum, just like light. b. Light or electromagnetic waves and gravity are comparable in structures. Weak and strong interactions are similar in structures. c. Light and weak interaction have the same speed cL with spin number +1 or –1. d. Gravity and strong interaction have the same speed cG without spin. e. The primary particles, namely electrons, electron neutrinos, and dark neutrinos in this paper, are made by the above four waves. We can find and describe some fundamental characteristics of the primary particles (e.g., their sizes, energies, and interactions) and introduce new attractive results from them (e.g., the source of the Pauli exclusion principle, the solution to the Einstein-Podolsky-Rosen paradox, and cG slightly faster than cL).
Keywords: 
models of particles; electrons; electron neutrinos; dark matter; sizes and energies of particles; interactions between two particles; Pauli exclusion principle; the solution to the Einstein-Podolsky-Rosen paradox; Gravity speed
Subject: 
Physical Sciences  -   Theoretical Physics

1. Introduction

We know intimately the term "atom" which comes from ancient Greek and means "uncuttable" or translated as "indivisible." In the early 19th century, the scientist John Dalton introduced the modern definition of an atom to characterize chemical elements. It was discovered that Dalton's atoms are not actually indivisible about a century later. An atom consists of three basic types of subatomic particles: electrons, protons, and neutrons, which occupy the tiny space in an atom. Protons and neutrons form the nucleus that contains most of an atom's mass. Electrons are the lightest charged particles in nature and revolve around the nucleus of an atom. An electron is seemingly indivisible yet. Until today, we have not split an electron into two or more smaller particles. We only make the positive and negative electron annihilation. A free neutron is unstable, decaying into a proton, electron, and neutrino. However, a free proton is stable, and is composed of two up quarks and one down quark in the modern Standard Model. Furthermore, whether a quark can be cut into smaller parts or whether the matter is infinitely divisible.
This paper tries to answer the above questions from a different perspective. What will happen when light or electromagnetic, gravity waves and other waves make up the primary particles that constitute the fundamental elements of matter? The assumptions are derived then:
  • Light or electromagnetic waves, weak interaction, gravity, and strong interaction are independent waves without rest masses. But their structures are different.
  • Light and gravity can be described by the wave equation with the field strength Ê and speed c.
2 Ê = 1 c 2 2 Ê t 2
c
Weak and strong interaction can be described by the 4-dimensional Laplace equation with field strength Ê' and speed c'.
2 Ê ' = 1 c ' 2 2 Ê ' t 2
d
According to the unified electro-weak theory, light and weak interaction have the same speed cL with spin number +1 or –1.
e
Gravity and strong interaction have the same speed cG without spin, and cG is constant in a vacuum.
f
The primary particles, which are electrons, electron neutrinos, and dark neutrinos in this paper, are made by the above four types of waves.

2. The Formation of Primary Particles

To suppose that the birth of primary particles may divide into the following two parts:
  • The light and weak interaction couple together (hereafter referred to as the E-W couple) when they have the same spin number and the second-order partial derivatives of their fields with respect to time 2 Ê t 2 are equal. The gravity and strong interaction couple together too (hereafter referred to as the G-S couple) when the second-order partial derivatives of their fields with respect to time 2 Ê ' t 2 are equal. So we have
    c L 2 2 ( Ê e + Ê w ) = 0 ,
    and
    c G 2 2 ( Ê G + Ê S ) = 0 .
    Where Êe, Êw, ÊG, and ÊS are electric, weak interaction, gravitational, and strong interaction fields.
  • The E-W couple has no spin. The original spins of the coupled waves convert the electric or weak charge property.
  • It makes a primary particle when the two coupled waves attract each other and shrink to a tiny sphere. One E-W couple and one G-S couple produce an electron or a positron whose charge property depends on the original spin of the E-W couple. The dark neutrinos are composed of two G-S couples. Two E-W couples with different original spin compress themselves into an electron neutrino. But they cannot attract each other with the same original spin.
Further, we assume that each field around the primary particle is time-independent and spherically symmetrical. Thus, in the spherical coordinate system, we have the uniform Laplace equation for the above equations (3), (4)
c 2 1 r 2 r r 2 Ê r + 1 r 2 sin θ θ sin θ Ê θ + 1 r 2 sin 2 θ 2 Ê φ 2 = 0 ,
whose general solution is
Ê = 1 c 2 j = 0 + k = 0 j A j r j + B j r j + 1 P j k ( cos θ ) e i k φ 1 c 2 j = 0 + k = 0 j A j r j + B j r j + 1 P j k ( cos θ ) e i k φ ,
where Aj and Bj are constants, P j k ( cos θ ) are associated Legendre polynomials, j and k are integers, j = 0, 1, 2, 3, …, kj, and j is called the degree of associated Legendre polynomials.

3. The Fields and Binding-Energy

We can now derive the electric, gravitational, weak interaction, and strong interaction fields Ê based on equation (6) and existing physical laws and data.
It is reasonable that equation (6) can be transformed into a pair of conjugate solutions.
Ê = 1 c 2 A j r j + B j r l + 1 k = 0 j P j k ( cos θ ) e i k φ Ê * = 1 c 2 A j r j + B j r j + 1 k = 0 j P j k ( cos θ ) e i k φ
Clearly, there is
[ E ^ a + E ^ b ] * = E ^ a * + E ^ b *
where the subscripts a and b denote electric, gravitational, weak interaction, or strong interaction.
And we let
( E ^ * ) * = E ^
The field Ê may be split into the macroscopic item 1 c 2 A j r j + B j r j + 1 and the quantum factors k = 0 j P j k ( cos θ ) e i k φ , k = 0 j P j k ( cos θ ) e i k φ . The degree of associated Legendre polynomials j rules properties of the field Ê because it is not only an exponent of r in the macroscopic item but also impacts forms of the quantum factors.
Compared equation (7) to Gauss's law of electrostatics and Newton's law of gravity, the electric field Êe is
E ^ e = ± 1 c L 2 B 1 r 2 k = 0 1 P 1 k ( cos θ ) e i k φ = ± q ^ c L 2 r 2 ( cos θ + sin θ e i φ ) E ^ e * = ± 1 c L 2 B 1 r 2 k = 0 1 P 1 k ( cos θ ) e i k φ = ± q ^ c L 2 r 2 ( cos θ + sin θ e i φ ) ,
and the gravitational field ÊG is
E ^ G = 1 c G 2 B 1 r 2 k = 0 1 P 1 k ( cos θ ) e i k φ = m ^ c G 2 r 2 ( cos θ + sin θ e i φ ) E ^ G * = 1 c G 2 B 1 r 2 k = 0 1 P 1 k ( cos θ ) e i k φ = m ^ c G 2 r 2 ( cos θ + sin θ e i φ ) ,
where '–' means attractive interaction, and '+' means repulsive interaction, as usual in this paper, q ^  is the mathematical electric charge, and m ^  is the mathematical mass or the gravitational charge.
Weak interaction has an intensity of a similar magnitude to the electromagnetic force at very short distances (around 10-18 meters), but this starts to decrease exponentially with increasing distance. Its effective range is about 10-17 to 10-16 meters[1, 2, 3]. All of the above can help us to determine weak interaction field Êw from equation (7). Êw is so supposed to equal to
E ^ w = ± 1 c L 2 B 1 r 2 k = 0 1 P 1 k ( cos θ ) e i k φ = ± w ^ c L 2 r 2 ( cos θ + sin θ e i φ )    r R c w ± 1 c L 2 B m r m + 1 k = 0 m P m k ( cos θ ) e i k φ = ± w ^ R c w m 1 c L 2 r m + 1 k = 0 m P m k ( cos θ ) e i k φ    r > R c w E ^ w * = ± 1 c L 2 B 1 r 2 k = 0 1 P 1 k ( cos θ ) e i k φ = ± w ^ c L 2 r 2 ( cos θ + sin θ e i φ )    r R c w ± 1 c L 2 B m r m + 1 k = 0 m P m k ( cos θ ) e i k φ = ± w ^ R c w m 1 c L 2 r m + 1 k = 0 m P m k ( cos θ ) e i k φ    r > R c w
where  w ^  is the mathematical weak charge, Rcw is the critical radius of weak interaction, and m is an integer greater than 1.
The strong force is a short-range interaction (around 10-15 meters) similar to the weak force. But its range is more complex than the weak force. At distances comparable to the diameter of a proton, it is approximately 100 times as strong as the electromagnetic force. At smaller distances, however, it becomes weaker. In particle physics, this effect is known as asymptotic freedom[4, 5, 6]. Moreover, it is supposed that the fields of the four fundamental forces have a unified form in a very tiny range. Hence equation (6) can be translated into the strong interaction field ÊS
E ^ S = B 1 c G 2 r 2 k = 0 1 P 1 k ( cos θ ) e i k φ = s ^ c G 2 r 2 ( cos θ + sin θ e i φ )    r R c S 1 A 1 r c G 2 k = 0 1 P 1 k ( cos θ ) e i k φ = s ^ r c G 2 R c S 1 3 ( cos θ + sin θ e i φ )    R c S 1 < r R c S 2 1 c G 2 B n r n + 1 k = 0 n P n k ( cos θ ) e i k φ = s ^ R c S 2 n + 2 c G 2 R c S 1 3 r n + 1 k = 0 n P n k ( cos θ ) e i k φ    r > R c S 2 E ^ S * = B 1 c G 2 r 2 k = 0 1 P 1 k ( cos θ ) e i k φ = s ^ c G 2 r 2 ( cos θ + sin θ e i φ )    r R c S 1 A 1 r c G 2 k = 0 1 P 1 k ( cos θ ) e i k φ = s ^ r c G 2 R c S 1 3 ( cos θ + sin θ e i φ )    R c S 1 < r R c S 2 1 c G 2 B n r n + 1 k = 0 n P n k ( cos θ ) e i k φ = s ^ R c S 2 n + 2 c G 2 R c S 1 3 r n + 1 k = 0 n P n k ( cos θ ) e i k φ    r > R c S 2
where s ^ is the mathematical strong charge, RcS1 and RcS2 are the 1st and 2nd critical radius of the strong interaction, and n is an integer greater than 1.
Further, it is assumed that RcS1 << Rcw < RcS2 and mn.
Now we turn to determine the energy. The energy density of wave equation (1) is given by E ^ 2 , and the equations (1) and (2) are Lorentz invariance. So we hope the equation of energy is Lorentz invariance, too, and let the binding-energy Ea-b of the four fields be
E a b 1 t 2 t 1 V c t 1 c t 2 E ^ a E ^ b * d x d y d z d c t
Note that E ^ a and E ^ b * are independent of time, and there are
0 2 π e i m φ e i n φ d φ = 2 π δ m n
and
0 π P j m ( cos θ ) P k m ( cos θ ) s i n θ d θ = ( j + m ) ! ( j m ) ! 2 2 j + 1 δ j k = N j m δ j k .
When we use light as a measurement medium to determine the energy as usual, we can translate equation (14) into the spherical coordinate system with the optical medium, which is
E a b = 1 ( t 2 t 1 ) 2 π k = 0 j N j k R 0 π 0 2 π c L t 1 c L t 2 E ^ a E ^ b * r 2 sin θ d r d θ d φ d c L t = c L 2 2 π k = 0 j N j k R 0 π 0 2 π E ^ a E ^ b * r 2 sin θ d r d θ d φ
where R is the radius at which two fields begin to interact with each other.
The general energy expression can be calculated when equation (7) is substituted into equation (17).
E a b = c L 2 2 π k = 0 j N j k R 0 π 0 2 π E ^ a E ^ b * r 2 sin θ d r d θ d φ = c L 2 2 π k = 0 j N j k R 0 π 0 2 π 1 c a 2 A a j r j + B a j r j + 1 k = 0 j P j k ( cos θ ) e i k φ × 1 c b 2 A b j r j + B b j r j + 1 k = 0 j P j k ( cos θ ) e i k φ r 2 sin θ d r d θ d φ = c L 2 2 π k = 0 j N j k R 0 π 0 2 π 1 c b 2 A b j r j + B b j r j + 1 k = 0 j P j k ( cos θ ) e i k φ × 1 c a 2 A a j r j + B a j r j + 1 k = 0 j P j k ( cos θ ) e i k φ r 2 sin θ d r d θ d φ = c L 2 2 π k = 0 j N j k R 0 π 0 2 π E ^ b E ^ a * r 2 sin θ d r d θ d φ = E b a = c L 2 c a 2 c b 2 R A a j r j + B a j r j + 1 A b j r j + B b j r j + 1 r 2 d r
Based on equation (18) and associated with equations (10) to (13), we first compute the self-binding-energy of the four fields. The self-binding-energy of the four fields Ee-e, EG-G, Ew-w, and ES-S are
E e e = c L 2 r q ^ 2 c L 4 r 2 d r = q ^ 2 c L 2 r ,
E G G = c L 2 r m ^ 2 c G 4 r 2 d r = c L 2 m ^ 2 c G 4 r ,
E w w = c L 2 r R c w w ^ 2 c L 4 r 2 d r + c L 2 R c w w ^ 2 R c w 2 m 2 c L 4 r 2 m d r = w ^ 2 c L 2 1 r 2 m 2 ( 2 m 1 ) R c w    r R c w c L 2 r w ^ 2 R c w 2 m 2 c L 4 r 2 m d r = w ^ 2 R c w 2 m 2 c L 2 ( 2 m 1 ) r 2 m 1    r > R c w ,
and
E S S = c L 2 r R c S 1 s ^ 2 c G 4 r 2 d r + c L 2 R c S 1 R c S 2 s ^ 2 r 4 c G 4 R c S 1 6 d r + c L 2 R c S 2 s ^ 2 R c S 2 2 n + 4 c G 4 R c S 1 6 r 2 n d r = c L 2 s ^ 2 c G 4 1 r 6 5 R c S 1 + 2 ( n + 2 ) R c S 2 5 5 ( 2 n 1 ) R c S 1 6    r R c S 1 c L 2 r R c S 2 s ^ 2 r 4 c G 4 R c S 1 6 d r + c L 2 R c S 2 s ^ 2 R c S 2 2 n + 4 c G 4 R c S 1 6 r 2 n d r = c L 2 s ^ 2 c G 4 1 5 R c S 1 6 R c S 2 5 r 5 + R c S 2 5 ( 2 n 1 ) R c S 1 6    R c S 1 < r R c S 2 c L 2 r s ^ 2 R c S 2 2 n + 4 c G 4 R c S 1 6 r 2 n d r = c L 2 s ^ 2 c G 4 × R c S 2 2 n + 4 ( 2 n 1 ) R c S 1 6 r 2 n 1    r > R c S 2 .
The binding-energy of the four fields, such as Ee-G, Ee-w, Ee-S, etc. are
E e G = c L 2 r q ^ c L 2 r 2 m ^ c G 2 r 2 r 2 d r = q ^ m ^ c G 2 r ,
E e w = c L 2 r R c w q ^ c L 2 r 2 w ^ c L 4 r 2 r 2 d r + 0 = q ^ w ^ c L 2 1 r 1 R c w    r R c w 0    r > R c w ,
E e S = c L 2 r R c S 1 q ^ c L 2 r 2 s ^ c G 2 r 2 r 2 d r + c L 2 R c S 1 R c S 2 q ^ c L 2 r 2 s ^ r c G 2 R c S 1 3 r 2 d r + 0 = q ^ s ^ c G 2 1 r 3 2 R c S 1 + R c S 2 2 2 R c S 1 3    r R c S 1 c L 2 r R c S 2 q ^ c L 2 r 2 s ^ r c G 2 R c S 1 3 r 2 d r + 0 = q ^ s ^ 2 c G 2 R c S 1 3 R c S 2 2 r 2    R c S 1 < r R c S 2 0    r > R c S 2 ,
E G w = c L 2 r R c w m ^ c G 2 r 2 w ^ c L 2 r 2 r 2 d r + 0 = m ^ w ^ c G 2 1 r 1 R c w    r R c w 0    r > R c w ,
E G S = c L 2 r R c S 1 m ^ c G 2 r 2 s ^ c G 2 r 2 r 2 d r + c L 2 R c S 1 R c S 2 m ^ c G 2 r 2 s ^ r c G 2 R c S 1 3 r 2 d r + 0 = c L 2 m ^ s ^ c G 4 1 r 3 2 R c S 1 + R c S 2 2 2 R c S 1 3    r R c S 1 c L 2 r R c S 2 m ^ c G 2 r 2 s ^ r c G 2 R c S 1 3 r 2 d r + 0 = c L 2 m ^ s ^ 2 c G 4 R c S 1 3 R c S 2 2 r 2    R c S 1 < r R c S 2 0    r > R c S 2 ,
and
E w S = c L 2 r R c S 1 w ^ c L 2 r 2 s ^ c G 2 r 2 r 2 d r + c L 2 R c S 1 R c w w ^ c L 2 r 2 s ^ r c G 2 R c S 1 3 r 2 d r + 0 = w ^ s ^ c G 2 1 r 3 2 R c S 1 + R c w 2 2 R c S 1 3    r R c S 1 c L 2 r R c w w ^ c L 2 r 2 s ^ r c G 2 R c S 1 3 r 2 d r + 0 = w ^ s ^ 2 c G 2 R c S 1 3 R c w 2 r 2    R c S 1 < r R c w 0    r > R c w .
For the convenience of subsequent calculations, we require the results of our defined energy to be consistent with those of the conventional physical method. Compared equations (19) and (20) to electric and gravitational potential energy formulas, it can easily find that the relations of mathematical electric charge ê and mathematical mass m ^ to electric charge q and mass m are
e ^ = c L k q
and
m ^ = c G 2 G m c L
where k is the Coulomb constant, and G is the gravitational constant.

4. The Structures of Primary Particles

A primary particle looks like a tiny spheroidal balloon with two envelopes (Figure 1). Each of them is made by an E-W couple or a G-S couple. The envelopes can characterize as:
  • The whole biding-energy of the coupled waves concentrates on the envelopes.
  • The macroscopic items of combined field strengths of the two coupled waves are equal on the envelopes. Outside the envelopes, the coupled waves become two independent static fields. But there are no fields inside the envelopes.
  • The size of the envelopes, which means the size of a primary particle, too, depends on the critical radius of weak or strong interaction.
  • The two envelopes have the same inherent frequency νin, although this is not mathematically required.
  • The degree of associated Legendre polynomials j is the same on the two envelopes.
  • Behaviors of the two envelopes obey the Self-Conjugate Mechanism, which requires that one occupies the surface of k = 0 j P j k ( cos θ ) e i k φ and the other must take up k = 0 j P j k ( cos θ ) e i k φ , or they are conjugate to each other.
Hence, the biding-energy of a primary particle Epri can be generally described as
E p r i = c L 2 2 π k = 0 j N j k R 0 π 0 2 π [ E ^ a + E ^ b ] [ E ^ c + E ^ d ] * r 2 sin θ d r d θ d φ = c L 2 2 π k = 0 j N j k R 0 π 0 2 π E ^ a E ^ c * + E ^ a E ^ d * + E ^ b E ^ c * + E ^ b E ^ d * r 2 sin θ d r d θ d φ = E a c + E a d + E b c + E b d ,
based on equation (17) Epri is clearly equivalent to the rest mass.
Evidently, the total energy of a primary particle comprises the biding-energy or the rest mass and the energy in static fields.

4.1. An Electron Neutrino

An electron neutrino is composed of two E-W couples with different original spin. In order to explore its structure, these assumptions should be adopted:
  • Its radius re_ν is equal to the critical radius of weak interaction Rcw.
  • The charges in equations (10) and (12) are equal and minimal for an electron neutrino, which means ê e _ ν = w ^ e _ ν when ê e _ ν and w ^ e _ ν are the mathematical electric charge and the mathematical weak charge of an electron neutrino.
Integrating equations (10), (12), and the above characters, we have two field equations on envelopes of an electron neutrino ÊE, e_ν
E ^ E , e _ ν E ^ e + E ^ w r = R c w = q ^ e _ ν + w ^ e _ ν c L 2 R c w 2 ( cos θ + sin θ e i φ ) E ^ e + E ^ w r = R c w * = q ^ e _ ν + w ^ e _ ν c L 2 R c w 2 ( cos θ + sin θ e i φ )
where '–' only indicates that two E-W couples are attracted to each other on the envelopes.
Based on equation (31) and associated equations (19), (21), and (24), we can easily compute the biding-energy of an electron neutrino Ee_ν.
E e _ ν = E e e + 2 E e w + E w w r = R c w = q ^ e _ ν 2 c L 2 r + 2 q ^ e _ ν w ^ e _ ν c L 2 1 r 1 R c w + w ^ e _ ν 2 c L 2 r 1 r 2 m 2 ( 2 m 1 ) R c w r = R c w = 1 c L 2 R c w q ^ e _ ν 2 + w ^ e _ ν 2 2 m 1 = 2 m ( 2 m 1 ) R c w q ^ e _ ν 2 c L 2 = 2 m ( 2 m 1 ) R c w w ^ e _ ν 2 c L 2
Combining the envelopes' characters b. with equations (10), (12), and (32), the fields around an electron neutrino Êe_ν can be directly written as
E ^ e _ ν r > R c w E ^ e e _ ν + E ^ w e _ ν = ± q ^ e _ ν c L 2 r 2 ( cos θ + sin θ e i φ ) ± w ^ e _ ν R c w m 1 c L 2 r m + 1 k = 0 m P m k ( cos θ ) e i k φ E ^ e e _ ν + E ^ w e _ ν * = q ^ e _ ν c L 2 r 2 ( cos θ + sin θ e i φ ) w ^ e _ ν R c w m 1 c L 2 r m + 1 k = 0 m P m k ( cos θ ) e i k φ r > R c w .

4.2. Dark Neutrinos

Two G-S couples make a dark neutrino. However, the strong interaction field has two critical radii, so there are two types of dark neutrinos, and they are named Dark I and Dark II. Similar to Section 4.1, it is assumed that:
  • The sizes of Dark I and II are equal to the 1st and 2nd critical radius of strong interaction.
  • Dark I and II have the same mathematical mass and mathematical strong charge.
  • The mathematical mass m ^ D _ ν and the mathematical strong charge s ^ D _ ν are equal, i.e., m ^ D _ ν = s ^ D _ ν . s ^ D _ ν is minimal.
Replicating the process of the previous section, we have the fields of a Dark I on the envelopes ÊE, D_νI
E ^ E , D _ ν I E ^ G + E ^ S r = R c S 1 = m ^ D _ ν + s ^ D _ ν c G 2 R c S 1 2 ( cos θ + sin θ e i φ ) E ^ G + E ^ S r = R c S 1 * = m ^ D _ ν + s ^ D _ ν c G 2 R c S 1 2 ( cos θ + sin θ e i φ ) ,
where '–' only means that two G-S couples are attracted to each other on the envelopes.
Based on equation (31) and associated equations (20), (22), and (27), we can compute the biding-energy of a Dark I ED_νI
E D _ ν I = E G G + 2 E G S + E S S r = R c S 1 = c L 2 c G 4 m ^ D _ ν 2 r + m ^ D _ ν s ^ D _ ν 1 r 3 2 R c S 1 + R c S 2 2 2 R c S 1 3 + s ^ D _ ν 2 1 r 6 5 R c S 1 + 2 ( n + 2 ) R c S 2 5 5 ( 2 n 1 ) R c S 1 6 r = R c S 1 2 ( n + 2 ) R c S 2 5 5 ( 2 n 1 ) R c S 1 6 c L 2 s ^ D _ ν 2 c G 4 = 2 ( n + 2 ) R c S 2 5 5 ( 2 n 1 ) R c S 1 6 c L 2 m ^ D _ ν 2 c G 4
To get the fields of a Dark II on the envelopes ÊE, D_νII and the biding-energy of a Dark II ED_νII, we imitate the last process and have
E ^ E , D _ ν I I E ^ G + E ^ S r = R c S 2 = 1 c G 2 m ^ D _ ν R c S 2 2 + R c S 2 R c S 1 3 s ^ D _ ν ( cos θ + sin θ e i φ ) E ^ G + E ^ S r = R c S 2 * = 1 c G 2 m ^ D _ ν R c S 2 2 + R c S 2 R c S 1 3 s ^ D _ ν ( cos θ + sin θ e i φ ) ,
and
E D _ ν I I = E G G + 2 E G S + E S S r = R c S 2 = c L 2 c G 4 m ^ D _ ν 2 r + s ^ D _ ν 2 5 R c S 1 6 R c S 2 5 r 5 + R c S 2 5 s ^ D _ ν 2 ( 2 n 1 ) R c S 1 6 r = R c S 2 R c S 2 5 ( 2 n 1 ) R c S 1 6 c L 2 s ^ D _ ν 2 c G 4 = R c S 2 5 ( 2 n 1 ) R c S 1 6 c L 2 m ^ D _ ν 2 c G 4 5 2 ( n + 2 ) E D _ ν I .
Following the computations of the particle external field in the previous section, we can obtain the fields around a Dark I ÊD_νI
E ^ D _ ν I r > R c S 1 E ^ G D _ ν I + E ^ S D _ ν I = E ^ G D _ ν I s ^ D _ ν r c G 2 R c S 1 3 ( cos θ + sin θ e i φ )    R c S 1 < r R c S 2 E ^ G D _ ν I s ^ D _ ν R c S 2 n + 2 c G 2 R c S 1 3 r n + 1 k = 0 n P n k ( cos θ ) e i k φ    r > R c S 2 where E ^ G D _ ν I = m ^ D _ ν c G 2 r 2 ( cos θ + sin θ e i φ ) E ^ G D _ ν I + E ^ S D _ ν I * = E ^ G D _ ν I * s ^ D _ ν r c G 2 R c S 1 3 ( cos θ + sin θ e i φ )    R c S 1 < r R c S 2 E ^ G D _ ν I * s ^ D _ ν R c S 2 n + 2 c G 2 R c S 1 3 r n + 1 k = 0 n P n k ( cos θ ) e i k φ    r > R c S 2 where E ^ G D _ ν I * = m ^ D _ ν c G 2 R c S 1 2 ( cos θ + sin θ e i φ )
and the fields around a Dark II ÊD_νII
E ^ D _ ν I I r > R c S 2 E ^ G D _ ν I I + E ^ S D _ ν I I = m ^ D _ ν c G 2 r 2 ( cos θ + sin θ e i φ ) s ^ D _ ν R c S 2 n + 2 c G 2 R c S 1 3 r n + 1 k = 0 n P n k ( cos θ ) e i k φ E ^ G D _ ν I I + E ^ S D _ ν I I * = m ^ D _ ν c G 2 R c S 1 2 ( cos θ + sin θ e i φ ) s ^ D _ ν R c S 2 n + 2 c G 2 R c S 1 3 r n + 1 k = 0 n P n k ( cos θ ) e i k φ r > R c S 2 .
Comparing equations (36) with (38) reveals that, as far as measurements, namely energy, are concerned, there is little difference between Dark I and Dark II. However, their volumes are significant differences in the microscopic domain. There should only be Dark IIs in most cases following the principle of energy minimization.

4.3. An Electron or A Positron

Electrons and positrons have the same structure. We will not distinguish significantly between electrons and positrons during the subsequent descriptions and computations. One E-W couple and one G-S couple attract each other to form an electron or a positron, so its structure is the most complex in primary particles. Following the assumptions about dark neutrinos, it is supposed that:
  • The radius of an electron re equals the critical radius of weak interaction Rcw, although there are three critical radii for weak and strong interactions.
  • The mathematical electric charge q ^ e and the mathematical weak charge w ^ e are equal, i.e., q ^ e = w ^ e .
  • The mathematical strong charge s ^ e are minimal, which means s ^ e = s ^ D _ ν .
Referring to the way we did in the previous sections, we can obtain the field of an electron on the envelopes ÊE, e and the biding-energy of an electron Ee.
E ^ E , e E ^ e + E ^ w r = R c w = q ^ e + w ^ e c L 2 R c w 2 ( cos θ + sin θ e i φ ) E ^ G + E ^ S r = R c w * = 1 c G 2 m ^ e R c w 2 + s ^ e R c w R c S 1 3 ( cos θ + sin θ e i φ ) q ^ e + w ^ e c L 2 R c w 2 = 1 c G 2 m ^ e R c w 2 + s ^ e R c w R c S 1 3 ,
and
E e = E e G + E e S + E G w + E w S r = R c w = q ^ e m ^ e c G 2 r + q ^ e s ^ e 2 c G 2 R c S 1 3 R c S 2 2 r 2 r = R c w = q ^ e m ^ e c G 2 R c w + q ^ e s ^ e 2 c G 2 R c S 1 3 R c S 2 2 R c w 2 .
According to the previous assumptions RcS1 << Rcw < RcS2, q ^ e = w ^ e , and s ^ e = s ^ D _ ν , we have
2 q ^ e c L 2 R c w 2 = 2 w ^ e c L 2 R c w 2 s ^ D _ ν R c w c G 2 R c S 1 3
and
E e q ^ e s ^ D _ ν 2 c G 2 R c S 1 3 R c S 2 2 R c w 2 = q e 2 c L 2 R c w R c S 2 2 R c w 2 1 .
Now we directly give the result for fields around an electron Êe.
E ^ e r > R c w E ^ e e + E ^ w e = ± q ^ e c L 2 r 2 ( cos θ + sin θ e i ϕ ) ± w ^ e c L 2 r 2 ( cos θ + sin θ e i φ )    r > R c w E ^ G e + E ^ S e * = m ^ e c G 2 r 2 ( cos θ + sin θ e i ϕ ) s ^ e r c G 2 R c S 1 3 ( cos θ + sin θ e i φ )    R c w < r R c S 2 m ^ e c G 2 r 2 ( cos θ + sin θ e i ϕ ) s ^ D _ ν R c S 2 n + 2 c G 2 R c S 1 3 r n + 1 k = 0 n P n k ( cos θ ) e i k φ    r > R c S 2
Next, the examination of equations (41) and (42) reveals that the 2nd critical radius of strong interaction RcS2 should be the geometric characterization parameter of a G-S envelope rather than the 1st critical radius of strong interaction RcS1. It is further assumed that Rcw and RcS2 are proportional to the wavelengths of the E-W and the G-S couple, respectively, i.e., R c w = ξ λ E W , R c S 2 = ξ λ G S . Thus, we can directly write with the envelopes characters d.
R c S 2 R c w = λ G S λ E W = c G c L ,
which shows that the gravity speed cG is faster than the light speed cL when the above equation compares with equation (42).

5. The Interactions Between Two Primary Particles

Imagine that two static primary particles are initially rested on each other and then separated by a repulsive force for a distance of l. Particles I and II have the potential and kinetic energy in the separated state (Figure 2). At the initial state, particles I and II are equivalent in that they share a common center of the sphere (Figure 2) because there are no fields in envelopes of primary particles.
Based on the law of conservation of energy, the initial energy of particles I and II is equal to the sum of the potential energy and kinetic energy (including magnetic energy for electrons) after their separation. Therefore, referring to equation (17), the potential energy of the two primary particles E P 1 ^ P 2 can be defined as
E P 1 ^ P 2 = 1 ( t 2 t 1 ) 2 π k = 0 j N j k × R c 0 π 0 2 π c L t 1 c L t 2 E ^ P 1 E ^ P 2 * r 2 sin θ d r d θ d φ d c L t R c l 0 π 0 2 π c L t 1 c L t 2 E ^ P 1 E ^ P 2 * r 2 sin θ d r d θ d φ d c L t = 1 ( t 2 t 1 ) 2 π k = 0 j N j k l 0 π 0 2 π c L t 1 c L t 2 E ^ P 1 E ^ P 2 * r 2 sin θ d r d θ d φ d c L t
where E ^ P 1 and E ^ P 2 are the external fields of primary particles I and II, l is the distance between the two particles (Figure 2), Rc is the larger of the two particles’ radii, and R c l 0 π 0 2 π c L t 1 c L t 2 E ^ P 1 E ^ P 2 * r 2 sin θ d r d θ d φ d c L t converts to kinetic energy (including magnetic energy for electrons).
Equation (47) must still hold certainly when two primary particles move in the opposite mode of Figure 2, i.e., two rested on each other particles are attracted at the initial distance l and approach each other until they come together. It is therefore assumed that the Self-Conjugate Mechanism remains between two interacting primary particles. According to existing physics knowledge, the Self-Conjugate Mechanism makes two sets of fields around one particle conjugate to two sets of fields around another depending on the rotation of two particles. In other words, two particles have achieved the Self-Conjugate after they rotate one cycle with angular velocity ω. There are Self-Conjugate lines in pairs that are further presumed at I :   ( r c ,   θ ,   ω t c ) for particle I and I I :   ( r c ,   θ ,   ω t c + π ) for particle II in their respective spherical coordinate systems, i.e., these lines are in the opposite position while rI = rII = r, θI = θII = θ (Refer to Figure 2). So when we take the rI-coordinate system of the particle I as a reference (Refer to Figure 2), in equation (47), the item R c 0 π E ^ P 1 ( r I ) E ^ P 2 * ( r I I ) r I 2 sin θ I d r I d θ I = R c 0 π E ^ P 1 ( r ) E ^ P 2 * ( r ) r 2 sin θ d r d θ , and the item 0 2 π e i m φ I e i n φ I d φ I
0 2 π e i m φ I e i n φ I d φ I = 2 η π 2 η π + 2 π e i m ( ω t c ) e i n ( ω t c + π ) d ( ω t c ) = 2 π δ m n      Same   rotation   direction   of   ω 2 η π 2 η π + 2 π e i m ( ω t c ) e i n ( ω t c + π ) d ( ω t c ) = 0      Opposite   rotation   direction   of   ω ,
where tc is the time of conjugation of two primary particles, and η = 0, 1, 2, ……, η. Hence, equation (47) still holds, just with a minus sign difference. Here, the minus sign only indicates that energy is gathering in this process, contrary to equation (47). Later, we continue to use equation (47) without distinguishing whether the energy is spreading or gathering in the motion of two primary particles.
Thus we can follow the results from the previous chapters when there is the potential energy between two primary particles, and the distance of l remains constant. There are only two conjugate forms between the two particles because each of the two particles comprises two coupled waves. Hence, equation (47) can be translated into
E P 1 ^ P 2 = c L 2 2 π k = 0 j N j k l 0 π 0 2 π E ^ P 1 E ^ P 2 * r 2 sin θ d r d θ d φ = c L 2 2 π k = 0 j N j k × l 0 π 0 2 π ± [ E ^ a P 1 + E ^ b P 1 ] [ E ^ c P 2 + E ^ d P 2 ] * ± [ E ^ w P 1 + E ^ x P 1 ] [ E ^ y P 2 + E ^ z P 2 ] * r 2 sin θ d r d θ d φ = E a P 1 c P 2 + E a P 1 d P 2 + E b P 1 c P 2 + E b P 1 d P 2 r = l + R c + E w P 1 y P 2 + E w P 1 z P 2 + E x P 1 y P 2 + E x P 1 z P 2 r = l + R c
where the subscripts a to d, and w to z denote electric, gravitational, weak interaction, or strong interaction.
According to Newtonian mechanics, the work done is the same as the potential energy when primary particles I and II move relative to each other. Reversing the Newtonian mechanics definition of work, i.e., F = W = E P 1 ^ P 2 , the force between two primary particles F P 1 ^ P 2 is therefore
F P 1 ^ P 2 = ± d E P 1 ^ P 2 d l .
Two Self-Conjugate primary particles have no initial phase difference in zenith and azimuthal angle under the requirements of the assumption of Self-Conjugation lines. From equation (48), they have potential energy or force when they rotate in the same direction, while they have zero potential energy or rest when they do in opposite directions. Therefore, the force mode has two forms, all right-handed or up rotation and all left-handed or down rotation, which shows each primary particle has spin values of ± 1 2 .

5.1. Two Particles of the Same Type

Start by computing the interaction between two electron neutrinos. Combining equations (19), (21), and (49), we have the potential energy Ee_ν^e_ν
E e _ ν ^ e _ ν = 2 E e e _ ν e e _ ν + 2 E e e _ ν w e _ ν + E w e _ ν w e _ ν r = l + R c w = 2 q ^ e _ ν 2 c L 2 ( l + R c w ) + R c w 2 m 2 w ^ e _ ν 2 c L 2 ( 2 m 1 ) ( l + R c w ) 2 m 1 2 q ^ e _ ν 2 c L 2 ( l + R c w ) = 2 w ^ e _ ν 2 c L 2 ( l + R c w )   when   l R c w .
Since two equations (34) have attractive and repulsive states under the Self-Conjugate Mechanism, from equation (50), the force of two electron neutrinos Fe_ν^e_ν is
F e _ ν ^ e _ ν = ± 2 q ^ e _ ν 2 c L 2 ( l + R c w ) 2 + R c w 2 m 2 w ^ e _ ν 2 c L 2 ( l + R c w ) 2 m 2 ± 2 q ^ e _ ν 2 c L 2 ( l + R c w ) 2 = ± 2 w ^ e _ ν 2 c L 2 ( l + R c w ) 2   when   l R c w ,
where the sign '–' or '+' depends on the Self-Conjugate forms between two electron neutrinos, and the "–" or "+" should be random.
Association equation (49) with equations (20), (22), (27), and (39), we can compute the potential energy between two Dark Is ED_νI^D_νI
E D _ ν I ^ D _ ν I = 2 E G D _ ν I G D _ ν I + 2 E G D _ ν I S D _ ν I + E S D _ ν I S D _ ν I r = l + R c S 1 = 2 c L 2 c G 4 m ^ D _ ν 2 l + R c S 1 + m ^ D _ ν s ^ D _ ν R c S 1 3 R c S 2 2 ( l + R c S 1 ) 2 + s ^ D _ ν 2 R c S 1 6 R c S 2 5 ( l + R c S 1 ) 5 5 + R c S 2 5 2 n 1 2 c L 2 s ^ D _ ν 2 c G 4 R c S 1 6 R c S 2 5 ( l + R c S 1 ) 5 5 + R c S 2 5 2 n 1 = 2 c L 2 m ^ D _ ν 2 c G 4 R c S 1 6 R c S 2 5 ( l + R c S 1 ) 5 5 + R c S 2 5 2 n 1    0 l R c S 2 R c S 1 2 c L 2 c G 4 m ^ D _ ν 2 l + R c S 1 + R c S 2 2 n + 4 ( 2 n 1 ) R c S 1 6 s ^ D _ ν 2 ( l + R c S 1 ) 2 n 1 = 2 c L 2 s ^ D _ ν 2 c G 4 ( l + R c S 1 ) 1 + R c S 2 2 n + 4 ( 2 n 1 ) R c S 1 6 ( l + R c S 1 ) 2 n 2 = 2 c L 2 m ^ D _ ν 2 c G 4 ( l + R c S 1 ) 1 + R c S 2 2 n + 4 ( 2 n 1 ) R c S 1 6 ( l + R c S 1 ) 2 n 2    l > R c S 2 R c S 1 ,
and the force between two Dark Is FD_νI^D_νI
F D _ ν I ^ D _ ν I = 2 c L 2 c G 4 m ^ D _ ν 2 ( l + R c S 1 ) 2 + 2 m ^ D _ ν s ^ D _ ν R c S 1 3 ( l + R c S 1 ) + s ^ D _ ν 2 R c S 1 6 ( l + R c S 1 ) 4 2 c L 2 s ^ D _ ν 2 c G 4 R c S 1 6 ( l + R c S 1 ) 4 = 2 c L 2 m ^ D _ ν 2 c G 4 R c S 1 6 ( l + R c S 1 ) 4    0 l R c S 2 R c S 1 2 c L 2 c G 4 m ^ D _ ν 2 ( l + R c S 1 ) 2 + R c S 2 2 n + 4 R c S 1 6 s ^ D _ ν 2 ( l + R c S 1 ) 2 n = 2 c L 2 s ^ D _ ν 2 c G 4 ( l + R c S 1 ) 2 1 + R c S 2 2 n + 4 R c S 1 6 ( l + R c S 1 ) 2 n 2 = 2 c L 2 m ^ D _ ν 2 c G 4 ( l + R c S 1 ) 2 1 + R c S 2 2 n + 4 R c S 1 6 ( l + R c S 1 ) 2 n 2    l > R c S 2 R c S 1 .
Duplicating the last process yields the potential energy between two Dark IIs ED_νII^D_νII
E D _ ν I I ^ D _ ν I I = 2 E G D _ ν I I G D _ ν I I + 2 E G D _ ν I I S D _ ν I I + E S D _ ν I I S D _ ν I I r = l + R c S 2 = 2 c L 2 c G 4 m ^ D _ ν 2 l + R c S 2 + R c S 2 2 n + 4 ( 2 n 1 ) R c S 1 6 s ^ D _ ν 2 ( l + R c S 2 ) 2 n 1 = 2 c L 2 s ^ D _ ν 2 c G 4 ( l + R c S 2 ) 1 + R c S 2 2 n + 4 ( 2 n 1 ) R c S 1 6 ( l + R c S 2 ) 2 n 2 = 2 c L 2 m ^ D _ ν 2 c G 4 ( l + R c S 2 ) 1 + R c S 2 2 n + 4 ( 2 n 1 ) R c S 1 6 ( l + R c S 2 ) 2 n 2 ,
and the force between two Dark IIs FD_νII^D_νII
F D _ ν I I ^ D _ ν I I = 2 c L 2 c G 4 m ^ D _ ν 2 ( l + R c S 2 ) 2 + R c S 2 2 n + 4 R c S 1 6 s ^ D _ ν 2 ( l + R c S 2 ) 2 n = 2 c L 2 s ^ D _ ν 2 c G 4 ( l + R c S 2 ) 2 1 + R c S 2 2 n + 4 R c S 1 6 ( l + R c S 2 ) 2 n 2 = 2 c L 2 m ^ D _ ν 2 c G 4 ( l + R c S 2 ) 2 1 + R c S 2 2 n + 4 R c S 1 6 ( l + R c S 2 ) 2 n 2 .
The potential energy between two electrons Ee^e has two forms because an electron is composed of one E-W couple and one G-S couple. Same as electron neutrinos, the two forms should be random and rely on the Self-Conjugate forms between two electrons. Combining equations (19) to (28) and (49), we can compute the two forms of the potential energy. One is Ee^eI when the two E-W couples are conjugate, and the two G-S couples are conjugate.
E e ^ e I = E e e e e + 2 E e e w e + E w e w e + E G e G e + 2 E G e S e + E S e S e r = l + R c w = q ^ e 2 c L 2 ( l + R c w ) + R c w 2 m 2 w ^ e 2 c L 2 ( 2 m 1 ) ( l + R c w ) 2 m 1 + c L 2 m ^ e 2 c G 4 ( l + R c w ) + c L 2 m ^ e s ^ e c G 4 R c S 1 3 R c S 2 2 ( l + R c w ) 2 + c L 2 s ^ e 2 c G 4 1 5 R c S 1 6 R c S 2 5 ( l + R c w ) 5 + R c S 2 5 ( 2 n 1 ) R c S 1 6    0 l R c S 2 R c w q ^ e 2 c L 2 ( l + R c w ) + R c w 2 m 2 w ^ e 2 c L 2 ( 2 m 1 ) ( l + R c w ) 2 m 1 + c L 2 m ^ e 2 c G 4 ( l + R c w ) + c L 2 R c S 2 2 n + 4 c G 4 ( 2 n 1 ) R c S 1 6 s ^ e 2 ( l + R c w ) 2 n 1    l > R c S 2 R c w q ^ e 2 c L 2 ( l + R c w ) + c L 2 m ^ e 2 c G 4 ( l + R c w )    when   l   is   large
Another is Ee^eII when the two E-W couples are conjugate to the two G-S couples.
E e ^ e I I = 2 E e e G e + E e e S e + E w e G e + E w e S e r = l + R c w = 2 q ^ e m ^ e c G 2 ( l + R c w ) + q ^ e s ^ e c G 2 R c S 1 3 R c S 2 2 ( l + R c w ) 2    0 l R c S 2 R c w 2 q ^ e m ^ e c G 2 ( l + R c w )    l > R c S 2 R c w
Derivation of the last two equations can yield the forces between two electrons in both forms that are
F e ^ e I = ± q ^ e 2 c L 2 ( l + R c w ) 2 ± R c w 2 m 2 w ^ e 2 c L 2 ( l + R c w ) 2 m c L 2 m ^ e 2 c G 4 ( l + R c w ) 2 2 c L 2 m ^ e s ^ e c G 4 R c S 1 3 ( l + R c w ) c L 2 s ^ e 2 c G 4 R c S 1 6 ( l + R c w ) 4    0 l R c S 2 R c w ± q ^ e 2 c L 2 ( l + R c w ) 2 ± R c w 2 m 2 w ^ e 2 c L 2 ( l + R c w ) 2 m c L 2 m ^ e 2 c G 4 ( l + R c w ) 2 c L 2 R c S 2 2 n + 4 c G 4 R c S 1 6 s ^ e 2 ( l + R c w ) 2 n    l > R c S 2 R c w ± q ^ e 2 c L 2 ( l + R c w ) 2 c L 2 m ^ e 2 c G 4 ( l + R c w ) 2    when   l   is   large
and
F e ^ e I I = 2 q ^ e m ^ e c G 2 ( l + R c w ) 2 2 q ^ e s ^ e c G 2 R c S 1 3 ( l + R c w )    0 l R c S 2 R c w 2 q ^ e m ^ e c G 2 ( l + R c w ) 2    l > R c S 2 R c w ,
plus s ^ e = s ^ D _ ν in equations (57) to (60).
Comparing equations (57) and (58), (59) and (60) shows that the potential energy and force between two electrons are very different in the two Self-Conjugate forms, with the smaller one close to zero.

5.2. Two Particles of the Different Type

Similar to the last section, this section still starts by computing the interaction of an electron neutrino with another primary particle. Combining equations (23) to (28), (49), and (50), the potential energy between an electron neutrino and a Dark I Ee_ν^D_νI is
E e _ ν ^ D _ ν I = 2 E e e _ ν G D _ ν I + E e e _ ν S D _ ν I + E w e _ ν G D _ ν I + E w e _ ν S D _ ν I r = l + R c w = 2 q ^ e _ ν m ^ D _ ν c G 2 ( l + R c w ) + q ^ e _ ν s ^ D _ ν c G 2 R c S 1 3 R c S 2 2 ( l + R c w ) 2    0 l R c S 2 R c w 2 q ^ e _ ν m ^ D _ ν c G 2 ( l + R c w )    l > R c S 2 R c w ,
and the force between an electron neutrino and a Dark I Fe_ν^D_νI is
E e _ ν ^ D _ ν I = 2 q ^ e _ ν m ^ D _ ν c G 2 ( l + R c w ) 2 2 q ^ e _ ν s ^ D _ ν c G 2 R c S 1 3 ( l + R c w )    0 l R c S 2 R c w 2 q ^ e _ ν m ^ D _ ν c G 2 ( l + R c w ) 2    l > R c S 2 R c w ,
plus m ^ D _ ν = s ^ D _ ν in above two equations.
Repeating the previous processes gives the potential energy between an electron neutrino and a Dark II Ee_ν^D_νII
E e _ ν ^ D _ ν I I = 2 E e e _ ν G D _ ν I I + E e e _ ν S D _ ν I I + E w e _ ν G D _ ν I I + E w e _ ν S D _ ν I I r = l + R c S 2 = 2 q ^ e _ ν m ^ D _ ν c G 2 ( l + R c S 2 ) = 2 q ^ e _ ν s ^ D _ ν c G 2 ( l + R c S 2 ) ,
the force between an electron neutrino and a Dark I Fe_ν^D_νII
F e _ ν ^ D _ ν I I = 2 q ^ e _ ν m ^ D _ ν c G 2 ( l + R c S 2 ) 2 = 2 q ^ e _ ν s ^ D _ ν c G 2 ( l + R c S 2 ) 2 ,
the potential energy between an electron neutrino and an electron Ee_ν^e
E e _ ν ^ e = E e e _ ν e e + E e e _ ν w e + E w e _ ν e e + E w e _ ν w e r = l + R c w + E e e _ ν G e + E e e _ ν S e + E w e _ ν G e + E w e _ ν S e r = l + R c w = q ^ e _ ν q ^ e c L 2 ( l + R c w ) + R c w 2 m 2 w ^ e _ ν w ^ e c L 2 ( 2 m 1 ) ( l + R c w ) 2 m 1 + q ^ e _ ν m ^ e c G 2 ( l + R c w ) + q ^ e _ ν s ^ e 2 c G 2 R c S 1 3 R c S 2 2 ( l + R c w ) 2    0 l R c S 2 R c w q ^ e _ ν q ^ e c L 2 ( l + R c w ) + R c w 2 m 2 w ^ e _ ν w ^ e c L 2 ( 2 m 1 ) ( l + R c w ) 2 m 1 + q ^ e _ ν m ^ e c G 2 ( l + R c w )    l > R c S 2 R c w q ^ e _ ν q ^ e c L 2 ( l + R c w ) 1 + R c w 2 m 2 ( 2 m 1 ) ( l + R c w ) 2 m 2 + q ^ e _ ν s ^ D _ ν 2 c G 2 R c S 1 3 R c S 2 2 ( l + R c w ) 2    0 l R c S 2 R c w q ^ e _ ν q ^ e c L 2 ( l + R c w ) 1 + R c w 2 m 2 ( 2 m 1 ) ( l + R c w ) 2 m 2    l > R c S 2 R c w q ^ e _ ν q ^ e c L 2 ( l + R c w ) = w ^ e _ ν w ^ e c L 2 ( l + R c w )   when   l R c w ,
and the force between an electron neutrino and an electro Fe_ν^e
F e _ ν ^ e = ± q ^ e _ ν q ^ e c L 2 ( l + R c w ) 2 ± R c w 2 m 2 w ^ e _ ν w ^ e c L 2 ( l + R c w ) 2 m q ^ e _ ν m ^ e c G 2 ( l + R c w ) 2 q ^ e _ ν s ^ e c G 2 R c S 1 3 ( l + R c w )    0 l R c S 2 R c w ± q ^ e _ ν q ^ e c L 2 ( l + R c w ) 2 ± R c w 2 m 2 w ^ e _ ν w ^ e c L 2 ( l + R c w ) 2 m q ^ e _ ν m ^ e c G 2 ( l + R c w ) 2    l > R c S 2 R c w q ^ e _ ν q ^ e c L 2 ( l + R c w ) 2 1 + R c w 2 m 2 ( l + R c w ) 2 m 2 + q ^ e _ ν s ^ D _ ν c G 2 R c S 1 3 ( l + R c w )    0 l R c S 2