Dark Energy Paradigm for Vacuum Energy

Dark Energy Modeled with $\Lambda CDM$

Table 1. Planck’s dimensional analysis

Table 2. Equations of three generations to predict $\Lambda$ and lepton masses

Prediction of Lepton Masses as Evidence

Energy conservation and force equilibrium

The conservation of energy states that the total energy of an isolated system remains constant. In the dark energy paradigm, the equal amount of dark energy is needed for a particle to be created in the vacuum. Dark energy can have only a certain energy value in force equilibrium which corresponds to a particle’s mass. The process for predicting a mass consists of three steps. First, the velocity v and the Lorentz factor $\gamma$ are obtained and inserted into the matter wave equation $\gamma mvl=\hslash$. Second, the equation is rearranged so that mass m and wavelength l can be derived from v and $\gamma$. Third, the dark energy density $\rho =m/l^3$ is evaluated in force equilibrium to predict the particle’s mass m. Sometimes mass m can be directly obtained and the steps can be further simplified.

Table 3. Predicted vs. observed lepton masses

Table 3. Predicted vs. observed lepton masses

mass	particle	predicted	observed	difference
$m_{{\nu }_e}$	${\nu }_e$	$6.0\times {10}^{-7}$	$\le 8.0\times {10}^{-7}$	◯
$m_{{\nu }_{\mu }}$	${\nu }_{\mu }$	$0.17$	$\le 0.17$	◯
$m_{{\nu }_{\tau }}$	${\nu }_{\tau }$	$17.5$	$\le 18.2$	◯
$m_e$	e	$0.511$	$0.511$	$0.000$
$m_{\mu }$	$\mu$	$105.105$	$105.658$	$-0.005$
$m_{\tau }$	$\tau$	$1766.622$	$1776.860$	$-0.006$

If any one of the masses is known (such as $m_e$), all the rest of the masses can be calculated. Observed data of $e,\mu$, and $\tau$ are from the particle data group [41]. Observed data of ${\nu }_e,{\nu }_{\mu }$ and ${\nu }_{\tau }$ are from the KATRIN Collaboration [42], Assamagan et al. [43] and ALEPH Collaboration [44], respectively. ‘◯’ means the prediction within the observed range. ‘Difference’ means the normalized difference between predicted and observed data. Mass units are in $MeV$.

Conclusion

Methods

Particle and antiparticle interaction

Let’s assume that each fermion particle has a fractional charge $e/n$ which generates the Coulomb force between a particle and its antiparticle. Here, all fermions share the same matter wave equation

$$pl=\gamma mvl=\hslash ,$$

(52)

where $p=\gamma mv$ is momentum, $\gamma$ is the Lorentz factor, m is mass, v is velocity, and l is wavelength (divided by $2\pi )$. Assuming a particle with $e/n_1$ charge and its antiparticle with $e/n_2$ charge form a circular orbit, the centripetal force is equal to the Coulomb force between $e/n_1$ and $e/n_2$ fractional charges (See Table 4)

$$\frac{m{v}^2}{l}=\frac{k(e/n_1)(e/n_2)}{l^2},$$

(53)

where k is the Coulomb constant and $v\ll c(\gamma =1)$. After rearranging the terms, v is

$$v=\frac{k(e/n_1)(e/n_2)}{\hslash }=\alpha c/\left(n_1{n}_2\right),$$

(54)

where $\alpha$ is the fine structure constant and $k{e}^2=\hslash \alpha c$. Here, fractional charge parameters for leptons are $n_1=n_2=1$, where v is

$$v=\alpha c,$$

(55)

which is the velocity of the 1^st generation of leptons. Thus, $v_1$ at the creation of a lepton particle has the same value as the velocity of an electron in the Bohr model. It is important to note that neutrino charge is unique in that it only interacts with neutrino charge, while electron and quark charges interact with each other.

Now the fractional charge parameters for down quarks are $n_1=n_2=3$, where v is

$$v=\alpha c/9,$$

(56)

and fractional charge parameters for up quarks are $n_1=3/2$ and $n_2=3$, where v is

$$v=2\alpha c/9.$$

(57)

Thus, fractional charges in force equilibrium play an important role in determining velocity v at the creation of a particle and antiparticle pair.

Table 4. Fractional charge parameters for fermions

Table 4. Fractional charge parameters for fermions

	neutrino	electron	down quark	up quark
$n_1$ (particle)	1	1	3	$3/2$
$n_2$ (antiparticle)	1	1	3	3
$n_1{n}_2$	1	1	9	$9/2$

Neutrino charge is unique in that it only interacts with neutrino charge.

The 3^rd generation lepton masses

For the 3^rd generation, the matter wave equation is

$${\gamma }_3{m}_3{v}_3{l}_3=\hslash ,$$

(63)

where the Lorentz factor ${\gamma }_3$ is

$${\gamma }_3=\frac{1}{\sqrt{1-{(v_3/c)}^2}}.$$

(64)

If $m_1$ gets excited where $v_3\approx c$, $m_3$ is (See Equation (41))

$$m_3=\alpha {\gamma }_3^3{m}_1,$$

(65)

and replacing $m_1$ by $m_2$ yields (See Equation (40))

$$m_3={({\gamma }_3/{\gamma }_2)}^3{m}_2,$$

(66)

where $m_3$ is a function of $m_2$.

While the 2^nd generation is generated by the 2-dimensional interaction

$${\rho }_r={\rho }_{\theta },$$

(67)

the 3^rd generation is caused by the 3-dimensional interaction (See Figure 2 and Table 5)

$${\rho }_r=|{\overrightarrow{\rho }}_{\theta }+{\overrightarrow{\rho }}_{\varphi }|,$$

(68)

where vector $\overrightarrow{\rho }$ corresponds to force vector $\overrightarrow{F}$ (See Table 1)

$$\overrightarrow{F_{\theta }}=G\frac{\hslash }{c}\overrightarrow{{\rho }_{\theta }},\overrightarrow{F_{\varphi }}=G\frac{\hslash }{c}\overrightarrow{{\rho }_{\varphi }},$$

(69)

where ${\rho }_{\theta }$ is dark energy density in a particle’s moving direction which is under ${\gamma }_2$ length contraction. Thus, ${\rho }_{\theta }$ changes to (See Equation (46))

$${\rho }_{\theta }=\frac{{\gamma }_2{m}_o}{l_o(l_o/{\gamma }_2^3)(l_o/{\gamma }_2)}={\gamma }_2^5{\rho }_o,$$

(70)

where $l_{\varphi }=l_o/{\gamma }_2$ is for the additional length contraction in the $\varphi$ direction. Assuming ${\rho }_{\varphi }={\rho }_{\theta }$ in force equilibrium, ${\rho }_{\varphi }$ is

$${\rho }_{\varphi }={\gamma }_2^5{\rho }_o,$$

(71)

and the magnitude of the vector sum is (See Equation 68)

$${\rho }_r=|{\overrightarrow{\rho }}_{\theta }+{\overrightarrow{\rho }}_{\varphi }|=\sqrt{2}{\gamma }_2^5{\rho }_o,$$

(72)

where $\sqrt{2}$ is due to the orthogonal vector sum in $\theta$ and $\varphi$ directions (See Figure 2). Here, the matter wave equation is

$$m_{o}c{l}_o=\left({\gamma }_3{m}_o\right)c(l_o/{\gamma }_3)=\hslash ,$$

(73)

where $l_o/{\gamma }_3$ represents the length contraction, and ${\rho }_r$ in terms of ${\gamma }_3$ is

$${\rho }_r=\frac{{\gamma }_3{m}_o}{{(l_o/{\gamma }_3)}^3}={\gamma }_3^4{\rho }_o.$$

(74)

Thus, the relation between ${\gamma }_3$ and ${\gamma }_2$ is

$${\rho }_r={\gamma }_3^4{\rho }_o=\sqrt{2}{\gamma }_2^5{\rho }_o.$$

(75)

After rearranging the terms, ${\gamma }_3$ is

$${\gamma }_3=2^{\frac{1}{8}}{\gamma }_2^{\frac{5}{4}}=1590.1,$$

(76)

where ${\gamma }_2=339.7$ (See Equation (43)), and $m_3$ for tau neutrino is (See Equation (66))

$$m_{{\nu }_{\tau }}=m_3={({\gamma }_3/{\gamma }_2)}^3{m}_2=17.5MeV,$$

(77)

where $m_3$ is within the observed range of a tau neutrino (See Table 3).

On the other hand, ${\gamma }_3$ for tau is

$${\gamma }_3=2^{\frac{1}{8}}{\gamma }_2^{\frac{5}{4}}=77.9,$$

(78)

where ${\gamma }_2=30.4$ (See Equation (50)), and $m_3$ for tau is (See Equation (51) and Equation (66))

$$m_{\tau }=m_3={({\gamma }_3/{\gamma }_2)}^3{m}_2=1766.622MeV,$$

(79)

which agrees with the observed data within 1% (See Table 3).

Table 5. Matter wave & energy density equations for three generations

Table 5. Matter wave & energy density equations for three generations

generation	mass	neutrino	electron	matter wave	energy density
$1$	$m_1$	$m_{{\nu }_e}$	$m_e$	${\gamma }_1{m}_1{v}_1{l}_1$=ℏ	${\rho }_r$
$2$	$m_2$	$m_{{\nu }_{\mu }}$	$m_{\mu }$	${\gamma }_2{m}_2{v}_2{l}_2$=ℏ	${\rho }_r$=${\rho }_{\theta }$
$3$	$m_3$	$m_{{\nu }_{\tau }}$	$m_{\tau }$	${\gamma }_3{m}_3{v}_3{l}_3$=ℏ	${\rho }_r$=$|\overrightarrow{{\rho }_{\theta }}$+$\overrightarrow{{\rho }_{\varphi }}|$

$\gamma$ is the Lorentz factor, m is mass, v is velocity, and l is wavelength (divided by $2\pi$). ${\rho }_r$, ${\rho }_{\theta }$ and ${\rho }_{\varphi }$ are dark energy densities in r, $\theta$ and $\varphi$ directions, respectively (See Figure 2). ${\overrightarrow{\rho }}_{\theta }$ and ${\overrightarrow{\rho }}_{\varphi }$ correspond to repulsive force vectors (See Table 1).

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