It is well known that Einstein worked on the theory of unified fields (UFT) from the 1930s until 1955, during which he proposed several hypothetical field equations to describe the basic mathematical rules governing two fundamental interactions, though without success. Prior to the 1960s, only gravity and electromagnetic forces were recognized in the scientific community. There was no clear understanding of the third and fourth fundamental interactions in the universe at that time. In 1967, S. Weinberg published the theory of electroweak interactions [1] , which introduced the concept of unifying forces through the Lie group by SU(2) × U(1), although he did not extend this unification to the nuclear force. Around the same time, the Yang-Mills theory provided a framework for understanding the representation of the Lie group by SU(3), which governs SU(3) as the rules for Quantum Chromodynamics (QCD) scales(Λ_QCD) [2] . The strong force, also known as the nuclear force, adheres to these rules, and QCD dictates that quarks must obey color confinement and color symmetry. This led scientists to establish a set of rules to construct models or physical frameworks to understand the behavior of elementary particles, a model now known as the Standard Model (SM). However, as advancements in high-energy physics continued, physicists increasingly encountered phenomena that could not be explained within the Standard Model, such as hypothetical supersymmetric (SUSY) particles, often referred to as “ghost particles.” Although these particles have not been experimentally verified, they have posed significant challenges in scientific data analysis. It is widely believed that the development of a theory unifying the three cosmic interactions is essential, as it is seen as the most promising candidate among all Grand Unified Theories (GUTs), regardless of whether it includes color symmetry. In this context, data and results from CERN (2019) suggest the need for a theory that can provide a robust explanation for running coupling constants, which were precisely measured at values of 0.02 and 0.04 at high energy scales [3] . This raises critical questions: What is the significance of these findings? How do they relate to Einstein’s UFT? These are sensitive and important issues (and if Dr. Pauli were alive today, he would likely pose these questions sharply). It is important to note that the discussion above is based on the behavior of fine structure constants, which also vary at high energy scales [4] - [8] . This paper is primarily based on the theory of electroweak interactions as proposed by S. Weinberg in 1967. For a long time, the widely known four fundamental cosmic interactions have been confined to the categories defined by this theory and the existing Grand Unified Theories (GUTs). Despite extensive research, there has been little progress in unifying these fields. Some GUT-related theories even predict proton decay, but no experimental evidence has yet confirmed this phenomenon. In this work, we aim to reconstruct electroweak interactions and calculate the running points of the fine structure constant, addressing a longstanding question in physics. Our study also responds to the famous physicist Richard Feynman’s remarks on the fine structure constant, often referred to as “God’s number.” Furthermore, we introduce a new interpretation of the Yukawa potential turning phase, considering it as a spectral mass gap in interactions (electro-strong or strong nuclear force), which has long been hidden but can be explained within the framework of electroweak interactions.

For a significant period, the equations of Yang-Mills have remained unsolved at energy scales relevant to describing atomic nuclei. We demonstrate that the half-wavelength of glueballs can be denoted as $\lambda /2=r_{Yuka}/2$, which constitutes plane waves of glueballs interacting with particles, and this finding is in complete agreement with experimental data [9] [10] . The motivations of studying unique connection of color charges (e.g., the SU(5) model by SUSY, which is Beyond the Standard Model (BSM) and is behind widely-studied models) [11] . Therefore, we determine that the quark freedom in QCD scales is six, by using the method described in this paper and the well-known process of strong fine-tuning [12] . A key highlight of our work is the demonstration that the running points of coupling constants at high energy scales are actually connected to the vacuum light speed C, as derived in this paper. We conclude that, through the method of approximation, our findings are eventually linked to the work on Two- and Four-Point Functions by Sander Mooij and Mikhail Shaposhnikov [13] . Finally, this paper provides an answer to the outstanding problem concerning the value of the cutoff Λ.

This work is grounded in the principles of S. Weinberg’s theory of electroweak interactions (1967) and explores the transition from the Standard Model (SM) to Beyond Standard Model (BSM) physics at different energy scales. The method employed in this paper relies entirely on hand calculations, without the use of computers or analytical instruments. If one wonders about the exact relationship between strong and electromagnetic forces or how Yang-Mills theory explains the physics of nuclei and their constituents, this paper provides a simple yet profound answer, building consecutively on the work of S. Weinberg. By utilizing helicity, we can explore the spin-spin couplings of massless particles, that is $\hat{L}=\hat{r}\times \hat{p}=const$, $\hat{L}=\hat{r}\times \underset{m\to \infty }{\lim }{\displaystyle \underset{i=1}{\overset{\infty }{\sum }}{m}_i{\stackrel{\to }{v}}_j}=const$, in the early universe. Given that the cosmic mass was infinite during this period, we arrive at the second formula. It requires $\hat{r}\equiv \hat{x}+\hat{y}+\hat{z}=0$, where $S^{\prime }=0$ is denoted as the scalar boson, while the angular momentum is projected onto the z-axis, implying the inflaton spin with $S^{\prime }=0$.

Massless particles, such as Sgluons with eight types of color charges and massless Binos¹, which are colorless, play a crucial role. The combination of Sgluons and Binos clearly violates color symmetry. However, the colorlessness of Binos serves as a “center” for the color symmetry of Sgluons. The combination relationships of colors are demonstrated with a deduction of colorlessness:

$\begin{array}{l}{g}_{i,\tilde{B}}+g_{i,\tilde{g}}\stackrel{\left(L.E.\right)}{{\displaystyle =}}{g}_{i,\tilde{g}}\ne 0,\\ \tilde{g}\stackrel{{\Lambda }_{GUT}}{\leftrightarrow }\tilde{B},\forall r=r_{Yuka}=\infty ,{\tilde{m}}_{\tilde{g}}={\tilde{m}}_{\tilde{B}}=0,\\ g_{i,\tilde{g}}\stackrel{\left(loop\right)}{{\displaystyle =}}{g}_{i,\tilde{B}}=0\end{array}$(1)

where widely-known $\left[g_i,g_j\right]=i{f}^{ijk}{g}_k,i=1,2,3,\cdots ,8$.

In Quantum Chromodynamics (QCD), the concept of colorlessness is com-monly represented by white, which is the color symmetry before mixing with higgsinos. This will be discussed further in the next section, where we refer to their common eigenstates. Note that $r_{Yuka}$ as the “center” at high energy scales (in the context of Yang-Mills theory where particles are massless) is not arbitrary; it has a specific value that will be derived in later sections. This $r_{Yuka}$ is positioned at the interacting top-point with ${\cos {\theta }_w|}_{{\theta }_w=\pi /4}$ in Feynman diagrams (see Figure 1 ). The corresponding values $T_a={\lambda }_a/\sqrt{2}$ can be represented in Gell-Mann matrices. Similar to how neutralinos have a mixed state that produces four common eigenstates, this paper presents a compelling assumption. The symmetrical group of color confinement is represented by SU(3)_C, while SU(3)_Diag denotes asymptotic freedoms without color attributes (R, G, B).

A) Tensors and the rotation matrix (with chirality versus helicity chart)

Given the cascade decays, assuming that

$\begin{array}{l}\tilde{g}\stackrel{A^0}{\to }\tilde{B},\forall t>t_0\\ \tilde{B}\stackrel{A^0}{\to }\tilde{g},\forall t<t_0,CPT\\ {\tilde{m}}_{\tilde{g}}={\tilde{m}}_{\tilde{B}}=0\\ \tilde{g}\stackrel{{\Lambda }_{GUT}}{\leftrightarrow }\tilde{B},\forall r=r_{Yuka}=\infty \end{array}\}$(2)

where high energy scales is denoted by ${\Lambda }_{GUT}$. $A^0$ indicated as GUT-scale QCD axions. $\tilde{g}$ indicated as Sgluons. And $\tilde{B}$ indicated as Binos.

The term ${\tilde{m}}_{\tilde{g}}={\tilde{m}}_{\tilde{B}}=0$ refers to one of the common eigenstates of their masses.

The second reaction corresponds to the evolution of reversal time (t < t₀).

For the massless gaugino, denoted by ${\tilde{m}}_{\tilde{g}}={\tilde{m}}_{\tilde{B}}=0$, the relationships (rotation matrix) are shown below for an arbitrary matrix in physical mathematics. Specifically,

$M=\exp \left(A\right)={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{A^k}{k!}}$(3)

where A represents the EM-tensor and is a Skew-symmetric matrix, shown as below. In U(1)_EM group, let A be a diagonal matrix with $a_{ij}=a_{ji}=0$:

$A_{\alpha \beta }\equiv \left[\begin{array}{cccc}0& \frac{E_x}{c}& \frac{E_y}{c}& \frac{E_z}{c}\\ -\frac{E_x}{c}& 0& -B_z& B_y\\ -\frac{E_y}{c}& B_z& 0& -B_x\\ -\frac{E_z}{c}& -B_y& B_x& 0\end{array}\right]$(4)

In this context, all components of the electric field (E-field) are forced to be zero. Due to the chaotic period during the early universe, this leads to the presence of a magnetic monopole, denoted by $\ln \hslash$, which ensures that all components of the magnetic field (B-field) are non-zero and must exist (indicating a non-zero matrix of $m\times n=4\times 4$). Particles during the inflation period possess supersymmetry (SUSY), which implies that they have magnetic monopoles associated with $\ln \hslash$. These monopoles can be incorporated into the electromagnetic tensor to ensure it is non-zero, thereby enabling their inclusion in subsequent calculations involving $M=\exp \left(A\right)$.

M represents the strong-force tensor within the SU(3) groups. Using this approach, we can also apply the electroweak tensor ${\theta }_w$ as shown below. The spontaneous breaking of SUSY leads to the combination of gauginos into two different bosons. Specifically:

$M\equiv \left(\begin{array}{c}g\\ \gamma \end{array}\right)=\left(\begin{array}{cc}\cos {\theta }_w& \sin {\theta }_w\\ -\sin {\theta }_w& \cos {\theta }_w\end{array}\right)\left(\begin{array}{c}\tilde{g}\\ \tilde{B}\end{array}\right)$(5)

And

$M=\exp \left(A\right)={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{A^k}{k!}}$(6)

Because of the zero charges associated with $\left(q_L=q_R=0\right)$ in Weinberg’s plane, and based on the pattern of weak isospin with ${\theta }_w=0$, $\cos {\theta }_w=1$, this can be indicated in progress as

$Const\cdot \alpha =\frac{\pi e_L{e}_R}{\left|{\stackrel{\to }{Y}}^{\prime }\right|}\stackrel{\left(Singul.\right)}{\equiv }\frac{e_L{e}_R}{\hslash c}$(7)

where $Const$ as slope, and ${\alpha }^{-1}$ as x’-axis on W-plane. It yields

$\alpha ={\frac{{\left|r\right|}^2\left(e_L{e}_R\right)}{Const\cdot \hslash c}|}_{e_L=e_R}\propto \frac{e^2}{4\pi \hslash c}$(8)

The constant selected depends precisely on the Weinberg angle $\theta =2\pi$, which determines $Const=4\pi {\left|r\right|}^2$ as the surface area of the singular point, represented as a tight loop in high dimensions. In this following, $1/{\epsilon }_0$ is chosen to ensure no units in the equation for $\alpha$ in the case of Equation (8). Consequently, this yields $\alpha =\frac{e^2}{4\pi {\epsilon }_0\hslash c}$.

Top (CP Conservation): $r_{Yuka}=finite$ (localization) is located at the interacting top-point and serves as a closure at low energy scales.

Bottom (CP Violation): $r_{Yuka}=\infty$ (globalization) forms a loop and diverges at high energy scales (e.g., in the early universe with $\left|v_i\right|=\left|v_j\right|=c,i\ne j$, where massless particles exist). To maintain translational invariance in quantum mechanics (ensuring energy conservation), additional terms are included in the diagram (e.g., antiparticles) with reversal in time ( $t<t_0$). This diagram does not violate CPT symmetry; CP remains conserved while $t<t_0$, leading to the conservation of U(1). Consequently, the photon’s R-parity is +1, while the Bino’s R-parity is −1, and similar considerations apply to gluons and sgluons. The loop contributes its energy to mix RGB colors to zero, thereby restoring color symmetry and ultimately returning to CP conservation.

Therefore the matrix is simplified, producing the eigenstate as a result of mixing states:

$M\equiv \left(\begin{array}{c}g\\ \gamma \end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\underset{eigenstate}{\underset{︸}{\left(\begin{array}{c}\tilde{g}\\ \tilde{B}\end{array}\right)}}$ (9)

In the case of zero colors, which leads to a colorless state in Equation (9) at high energy scales, this further produces the common eigenstate (e.g., mass eigenstate) of the system. Consequently, their masses are equivalent to being massless:

$m_{\tilde{g}}=0=\frac{m_{\tilde{B}}}{\cos {\theta }_w}$ (10)

Given the high-energy state during the inflation era, A and M could be interchanged arbitrarily within an infinitesimally short time, as shown in Sect. III.A. However, the prohibition rules must still be adhered to.

By allowing $M\stackrel{\left({\Lambda }_{GUT}\right)}{\equiv }A\cdot Const$ to be interchanged, a satisfactory approach for unifying the two fundamental cosmic forces can be initiated:

$\begin{array}{l}A=\exp \left(M\right)={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{M^k}{k!}},\\ A=\underset{\begin{array}{l}\equiv 0\\ \left(flat.Space\right)\end{array}}{\underset{︸}{\frac{M^0}{0!}}}+\underset{\begin{array}{l}=0\\ \left(Chiaral.Symmetry\right)\end{array}}{\underset{︸}{\frac{M^1}{1!}+\frac{M^2}{2!}+\frac{M^3}{3!}+\frac{M^4}{4!}}}+\underset{\begin{array}{l}\ne 0\\ \left(SUSY.Breakings\right)\end{array}}{\underset{︸}{\frac{M^5}{5!}+\frac{M^6}{6!}}}+\underset{\begin{array}{l}=0\\ \left(End.of.Inflation\right)\end{array}}{\underset{︸}{H.O.T\cdots }},\\ A\approx \frac{1}{144}M\end{array}$ (11)

where

$\begin{array}{l}\frac{M^1}{1!}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\left(\begin{array}{c}\tilde{g}\\ \tilde{B}\end{array}\right)=\left(\begin{array}{c}\tilde{g}\\ \tilde{B}\end{array}\right)\equiv M,\\ \begin{array}{l}\frac{M^5}{5!}=\frac{1}{5!}\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\left(\begin{array}{c}\tilde{g}\\ \tilde{B}\end{array}\right)=M/120\ne 0\\ \frac{M^6}{6!}=\frac{1}{6!}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\left(\begin{array}{c}\tilde{g}\\ \tilde{B}\end{array}\right)=-M/720\ne 0\end{array}\}invisible\end{array}$ (12)

The first term of the series corresponds to the flat space predicted by the Big Bang in the current Λ CDM model. The term following the second one will be discussed in the subsequent sections (refer to the three formulae below). Note that in Equation (12), the second and third formulae are concealed. At the instant of inflation, supersymmetry (SUSY) was spontaneously broken, forcing the eigenstates of mass, including $\left(\begin{array}{c}\tilde{g}\\ \tilde{B}\end{array}\right)\stackrel{\left(L.E.\right)}{{\displaystyle \ne }}0$ (gluons ( $\tilde{g}\to g$)) and graviphotons ( $\tilde{B}\to \text{LKPs}$with motional mass 10⁸ eV and a lifetime 10⁻⁸ s), to be produced at 10⁻⁶ s. The numerous LKPs were scattered into pairs of photons and gravitons at a specific temperature level, traveling through the later universe at $t>{10}^{-6}\text{s}$. This type of exchange was restricted during the SUSY epoch, where high-dimensional considerations were allowed, and magnetic monopoles existed. By the end of inflation, all magnetic monopoles had degenerated (through an unknown “absorption mechanism”) into SUSY particles, leading to the spontaneous breaking of SUSY at $t>{10}^{-6}\text{s}$. Despite this, the gauge conditions, including the hidden terms in Equation (12), remained intact.

Due to Equation (11) and (12), obviously all running points of fine structure constant is hidden in higher dimensions with $n=5$~ $n=6$ on expansions of matrix. Based on this, therefore assuming that $n\equiv 5.472$, hence $n!=5.472!\approx 137\cdot 2$. Such leads in

Due to Equations (11) and (12), it is evident that all running points of the fine structure constant are concealed within higher dimensions, particularly with $n=5$~ $n=6$ in the expansions of the matrix. Based on this assumption, if $n\equiv 5.472$, then $n!=5.472!\approx 137\cdot 2$. This, in turn, leads to

$\frac{M^{5.472}}{5.472!}=\frac{1}{5.472!}\left[\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)+{\left(\begin{array}{cc}\cos {\theta }_w& \sin {\theta }_w\\ -\sin {\theta }_w& \cos {\theta }_w\end{array}\right)}^{0.472}\right]\left(\begin{array}{c}\tilde{g}\\ \tilde{B}\end{array}\right)$ (13)

Thus, we obtain that the above expression is rotating on the plane of weak isospin, specifically considering the coterminal angles of the Weinberg angle:

$\frac{M^{5.472}}{5.472!}={\frac{1}{5.472!}\left[\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)+\left(\begin{array}{cc}\cos {\theta }_w& \sin {\theta }_w\\ -\sin {\theta }_w& \cos {\theta }_w\end{array}\right)\right]\left(\begin{array}{c}\tilde{g}\\ \tilde{B}\end{array}\right)|}_{{\theta }_w=\left(0.472\right)\cdot 2\pi }$ (14)

Namely²

$\frac{M^{5.472}}{5.472!}\approx \frac{1}{5.472!}\left[\underset{=1}{\underset{︸}{\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)}}+\underset{\equiv 1}{\underset{︸}{\left(\begin{array}{cc}\text{0}\text{.9985}& \text{0}\text{.0548}\\ -\text{0}\text{.0548}& \text{0}\text{.9985}\end{array}\right)}}\right]\left(\begin{array}{c}\tilde{g}\\ \tilde{B}\end{array}\right)$ (15)

Moreover,

$\frac{M^{5.472}}{5.472!}\approx \frac{2}{5.472!}\left(\begin{array}{c}\tilde{g}\\ \tilde{B}\end{array}\right)\stackrel{\left({\Lambda }_{GUT}\right)}{\approx }\frac{1}{\text{136}\text{.9038}}\left(\begin{array}{c}\tilde{g}\\ \tilde{B}\end{array}\right)\approx \frac{1}{\text{137}}\left(\begin{array}{c}\tilde{g}\\ \tilde{B}\end{array}\right)$ (16)

Thus we obtain

$A=\exp \left(M\right)=\frac{M^{5.472}}{5.472!}\stackrel{\left(L.E.\right)}{\approx }\frac{1}{\text{137}}M$ (17)

Equation (17) is indicated as an observation at low energy. Thus, we establish a complete relationship between electromagnetic forces and the strong force through these deductions. It is important to note that this is not a coincidence; the primary rotation of the Weinberg angle was specifically performed to obtain the factor of 2. This factor, in conjunction with the Λ CDM model, chiral symmetry (as seen in Equation (11)), and supersymmetry (SUSY), was derived to

accurately determine the factor in terms of $\frac{2}{5.472!}\approx \frac{1}{137}$. At this point, we can revisit the historical context with a clearer understanding. Additionally, the magnitude of $\alpha \left(t\right)\approx \frac{1}{\text{136}\text{.9038}}$ complies with the indicated data from the sets.

B) The Statement: The Ising model Phase Path r_Yuka

The famous Yukawa potential is expressed as:

$V\left(r\right)=-g^2\frac{{\text{e}}^{\kappa mr}}{r}$ (18)

where g is the coupling constant, $\kappa \equiv c/\hslash$ represents a specific term in the interaction, m is the mass of the interaction particles, and r denotes the particle-waves, such as meson waves radiated from a nucleus or the core of a particle or boson (e.g., a photon, meson, or gluon). It is evident that the quantum system exhibits duality, possessing two or more classical limits, as represented by Equation (18) and Equation (19). According to the Ising model:

$\langle \psi |\hat{F}|\psi \rangle =-\frac{\partial V\left(r\right)}{\partial r}=-g^2\frac{\kappa mr{\text{e}}^{\kappa mr}-{\text{e}}^{\kappa mr}}{r^2}$ (19)

The strong force must vanish at a unique critical phase (e.g., a phase path $r=r_{Yuka}$) in space, converting into the electromagnetic force governed by linear ordinary differential equations (ODEs). Based on this idea³, therefore

$\begin{array}{l}0=-g^2\frac{\kappa mr{\text{e}}^{\kappa mr}-{\text{e}}^{\kappa mr}}{r^2},\\ 0=g^{\prime }\frac{-\kappa mr{\text{e}}^{\kappa mr}+{\text{e}}^{\kappa mr}}{r^2}=g^{\prime }\frac{1}{r^2}\left[i\kappa mr\left(i{\text{e}}^{\kappa mr}\right)+{\text{e}}^{\kappa mr}\right],g^{\prime }\ne g\end{array}$ (20)

Note that $-\infty <r<\infty ,r\ne 0$. The precise ranges will be provided later. Referencing Equation (5), it actually approaches zero for massless particles during the inflationary epoch. However, towards the end of this epoch, at the instant of 10⁻⁶ s, LKPs appear and acquire mass 10⁸ eV due to SUSY breaking. Consequently, the second term $\tilde{g}\to g$ in the matrix of Equation (5) is forced to vanish, i.e.,

$0=g^{\prime }\frac{1}{r^2}\left(\underset{\begin{array}{l}\ne 0\\ \left(LKP,m={10}^8\text{eV}\right)\end{array}}{\underset{︸}{i\kappa mr\left(i{\text{e}}^{\kappa mr}\right)}}+\underset{\begin{array}{l}\equiv 0\\ \left(gluon\right)\end{array}}{\underset{︸}{{\text{e}}^{\kappa mr}}}\right)=g^{\prime }\frac{\kappa mr\left(i{\text{e}}^{\kappa mr}\right)}{r^2}=g^{\prime }\frac{\kappa mr\left(\sin \left(\kappa mr\right)\right)}{r^2}$ (21)

where we let the matrix element of ${\sin {\theta }_w|}_{{\theta }_w=0,2\pi }\equiv 0$ be defined. This allows the extracted $\kappa mr=2\pi$ to hold significant importance. Furthermore⁴,

$r=\frac{h}{mc}$ (22)

Considering that causality involves a “second-order constant perturbation” with factors of 99 (refer to Appendix A for the origin of 99),

$r^{\prime }={\frac{h}{\eta mc}|}_{\eta =99}$ (23)

Notably the starting point of $r^{\prime }$ is indicated from $r^{\prime }=0$ to $r^{\prime }=2r$, based on the principle of causality.

$r=\frac{h}{2\cdot 99mc}$ (24)

Substituting the cosmic physical constants by $h,m,c$, we obtain:

$r_{Yuka}=\frac{1.05\times {10}^{-34}\text{J}\cdot \sec }{\left(2\right)\left(99\right)\left(9.11\times {10}^{-31}\text{kg}\right)\cdot 3\times {10}^8\text{m}\cdot {\sec }^{-1}}$ (25)

Results with $r_{Yuka}\approx \text{1}\text{.9404 fm}$ were obtained for the phase path of interaction carriers (e.g., bosons such as hypothetical glueballs; see Appendix B). Additionally, by considering causality for electron matter-waves during electron collisions,

${\bar{r}}_{Yuka}\approx \frac{\left(\text{1}\text{.9404}+\text{1}\text{.9560}\right)\text{fm}}{2}\approx \text{1}\text{.950}\text{fm}$ (26)

In the case of electron-electron interactions (at the end of inflation⁵, $m=m_e$,

${10}^{-36}\text{s}\le t\le {10}^{-32}\text{s}$), the above expression ${\bar{r}}_{Yuka}$ aligns with Ting’s experimental results from 1967 (i.e., $\underset{\approx 1.954\text{fm}}{\underset{︸}{{\langle r\rangle }_{\min }}}-{\bar{r}}_{Yuka}=4\times {10}^{-18}\text{m}~{10}^{-18}\text{m}$).

Notably, $r_{Yuka}\approx \text{1}\text{.9404}\text{fm}$, as derived from Equation (25) to Equation (26), reveals that an estimated value for the spectrum mass gap has been missing from quantum Yang-Mills theory.

Remark.

See Equations (19) and (20), both of which are relevant to their corresponding matrices.

C) Representation by r_Yuka (a spectrum mass gap in r-basis)

This section presents a representation of $r_{Yuka}$ in nuclear physics as part of the conclusions. Deduction: Following the establishment of Coulomb’s law, the physical system of the universe evolved into a reactive equation at 10⁻⁶ s. During this period, graviphotons were produced in large quantities (with $\tau ={10}^{-8}\text{s}$ for LKPs), but they were unstable and quickly scattered to form electromagnetic fields (photons) and gravitational fields in free space, which then propagated throughout the universe after inflation. As the uniform temperature decreased, the Weinberg angle continued to rotate until the particles reached a stable state. The rotation of the matrix indicates the constant interchange of roles among massless particles within the matrix.

$\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\stackrel{\pm \frac{\pi }{2}}{\to }\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ (27)

Therefore

$\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\left(\begin{array}{c}g\\ {\gamma }_g\end{array}\right)\stackrel{\pm \frac{\pi }{2}}{\to }\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\left(\begin{array}{c}g\\ {\gamma }_g\end{array}\right)$ (28)

This reveals that gluons participate in chromodynamics (QCD) within $\left|r_{Yuka}\right|\le \text{1}\text{.9404}\text{fm}$, and graviphotons ( ${\gamma }_g$) are instantaneously scattered into pairs of photons and gravitons, both of which have zero mass after scattering.

$F\left(r\right)=-g^{\prime }{\frac{1}{r^2}\left[i\kappa mr\left(i{\text{e}}^{\kappa mr}\right)\right]|}_{{\text{e}}^{\kappa mr}\equiv 1}=-g^{\prime }i\frac{\kappa \left(im\right)r}{r^2},im\ne 0$ (29)

At this moment, virtual photons mediate the electromagnetic force, resulting in a non-zero imaginary mass. Without this, the discussions would lose their significance. Mathematically arranging this, we obtain:

$F\left(r\right)=g^{\prime }\frac{{\kappa mr|}_{r=r_{Yuka}}}{r^2}$ (30)

Note that r in the numerator holds significance as $r_{Yuka}$, as it is associated with the matrix from previous calculations. At $t\ge {10}^{-6}\text{s}$, it is evident that electron-electron interactions are governed by Equation (26) as shown below:

$\begin{array}{l}\kappa m{r}_{Yuka}=\frac{mc{r}_{Yuka}}{\hslash },\\ \kappa m{r}_{Yuka}=\frac{\left(9.11\times {10}^{-31}\text{kg}\right)\left(2.997\times {10}^8\text{m}\cdot {\sec }^{-1}\right)\left(1.94\times {10}^{-15}\text{m}\right)}{\text{1}\text{.05456}\times {10}^{-34}\text{J}\cdot \sec },\\ \kappa m{r}_{Yuka}\approx \text{50}\text{.2268}\times {10}^{-4}\frac{\text{kg}\cdot {\text{m}}^2}{\text{J}\cdot {\sec }^2}\end{array}$ (31)

The above is established in the context of high energy scales. (We will later see that this corresponds to the elementary charge of e.)

The strong force and electromagnetic forces undergo a linear transformation at point $r_{Yuka}$ (i.e., the unification of electrostrong interactions). Due to the nature of the opposing directions of their tensors, the calculations are performed within a repulsion field (where a negative sign for $-g^{\prime }$ is assigned to electrons). Therefore, the real calculation should include the hidden terms⁶.

$-\frac{{\left(\kappa m{r}_{Yuka}\right)}^6}{{10}^5}\stackrel{\left({\Lambda }_{GUT}\right)}{\equiv }-e\approx -\text{1}\text{.6055}\times {10}^{-19}\text{C}$ (32)

(LHS: at high-energy scales; RHS: at low-energy scales)

(Where $n=5,n=6$ originates from higher dimensions or general repulsion fields. The indication of 10⁵ could be derived from the coupling constant of $g^{\prime }$. The chaotic state of the early universe naturally normalized these units at the current low energy, a result confirmed by modern experiments (see Table 1 for proof). Therefore, constant verification is unnecessary.) Substituting the above expression into Equation (30), we derive Coulomb’s law:

$F\left(r\right)={\xi g^{\prime }\frac{-e}{r^2}|}_{t<{10}^{-6}\text{s}}={\frac{1}{4\pi {\epsilon }_0}\frac{e^2}{r^2}|}_{t={10}^{-6}\text{s}}$ (33)

Using cosmic extra dimensions to rigorously define

$g^{\prime }\equiv \frac{-e}{4\pi {\epsilon }_0\xi },\xi \equiv {10}^{\left[\beta \right]},\begin{array}{l}5\le \left[\beta \right]\le 11,repulsions\\ \beta =0,attractions\end{array}\}$ (34)

The denominator parameter $\xi$ represents the sum of diagonal elements after the matrix of the EM-tensor is diagonalized. The term $\beta$ originates from the upper index of the four-vector potential in the Euclidean space metric. In this context, the numerator only includes -e and cannot be expressed as e² mathematically; otherwise, e could not have been resolved in the previous discussions. In the specific case of physics based on the Yukawa potential, due to its mathematical nature, the parameter settings for the linear transformation to Coulomb’s law (e.g., massless photon) are flawed: the Yukawa potential V(r) has not been properly differentiated and substituting $m=0$ at that point is incorrect.

Based on the M-matrix conversion results presented in this paper, $m=0$ correctly refers to the massless gluon (not the massless photon) and corresponds to the definition of the EM-coupling constant g². However, this definition is incomplete in both physics and mathematics (particularly the latter), making it challenging to theoretically solve for the elementary charge, even though its value was experimentally determined by Millikan in the 1910s. According to our interpretation, by inversely solving for the charge and substituting it back into the correlation equations, Coulomb’s law is derived theoretically.

D) Reconstruction for photon energy of $E=n\hslash \omega$ by Einstein

The representation by $r_{Yuka}$ clearly indicates the point of SUSY breaking for massless particles (or spontaneous symmetry breaking (SSB) for gluons and photons at low energy scales). Consequently, $r_{Yuka}$ can be fully incorporated into the matrix of Equation (4). Following this, in the context of the neutral current, as conceptualized by S. Weinberg, Sheldon Lee Glashow, and Abdus Salam (since photons are radiated by electrons), a dynamic framework for it, such as a Lagrangian, is required. Therefore,

$L_N=T-V\left(r\right)$ (35)

Given that $r_{Yuka}$ corresponds to an electron, and $V\left(r\right)=0$ is associated with a free particle (such as a photon), after SUSY breaking or SSB,

$L_N=T={\frac{m{\left(\stackrel{\dot{}}{r}\right)}^2}{2}|}_{r=r_{Yuka}}$(36)

The input event occurs at high energy scales and is subsequently output at low energy for all physical units. This process is based on the eigenstate properties of the rotation matrix in Equation (9).

Remark.

a) Equation (27) is clearly a representation.

b) Based on the previous discussions, we conclude that the range of the strong interaction is confined within $r\le 1.9404fm$, consistent with inherent physical principles. The typical effective range of electro strong interactions is situated at $-1.9404fm\le r<\infty ,r\ne 0$.

c) Although the results of these equations were obtained with sufficiently small probabilities, they are still valid.

Regarding Equations (23) and (24), after rearranging Equation (23), we obtain:

$2r=\frac{h}{\eta mc}$ (37)

Using Compton wavelengths of $\lambda =h/mc$ for an electron with mass m, this leads to:

$2r=\lambda /\eta$ (38)

When a photon is instantaneously radiated at $r_{Yuka}$ by an electron, there is no need to consider the perturbation term with $\eta$. Thus, we can directly set $\eta \equiv 1$. Therefore,

$2r=\lambda$ (39)

Leads in

${\stackrel{\dot{}}{r}|}_{r=r_{Yuka}}=\stackrel{\dot{}}{\lambda }/2$ (40)

Moreover⁷,

$L_N=\frac{1}{8}{m{\left(\stackrel{\dot{}}{\lambda }\right)}^2|}_{r=r_{Yuka}}$ (41)

This complies with one of the eight components of the Lagrangian due to the factor of 1/8. At low energy scales, the Lagrangian is averaged in thermal equilibrium (i.e., following the statistical Boltzmann distribution) when a photon is radiated. Considering the interactions between a fermion (an electron) and a gauge boson (a photon), we have⁸:

$L_N=e{J}_{\mu }^{em}{A}_{\mu }+\frac{g}{\cos {\theta }_W}\left(J_{\mu }^3-{\sin }^2{\theta }_W{J}_{\mu }^{em}{Z}^{\mu }\right)$ (42)

With the weak isospin of the right chiral state $T=0$ for a stable electron (which does not participate in strong interactions due to its isospin $I=0$), this leads to $J_{\mu }^3=0$. Additionally, with the weak hypercharge of the right chiral state $Y_W=-2$ for a stable electron, this yields ${\theta }_W=-\pi$. Associated with the above, therefore, Equation (42) becomes:

$L_N=e{J}_{\mu }^{em}{A}_{\mu }$ (43)

where $J_{\mu }^{em}$ denotes the electromagnetic current and $A_{\mu }$ represents the electromagnetic tensor in four dimensions. Evidently, Equation (43) provides the Lagrangian, i.e., the photon energy at $r_{Yuka}$. Based on the above, and further referencing Equation (37) with $\eta \equiv 1$ at $r_{Yuka}$, therefore

$m=\frac{h}{2rc}$ (44)

Using $E=m{c}^2$ with $2r=\lambda$ (Equation (39)) to calculate it, we obtain

$\begin{array}{l}E=\frac{hc}{2r}=\hslash \left(2\pi \frac{c}{2r}\right),\\ E=\hslash \left(2\pi c/\lambda \right)=\hslash \omega ,n=1\end{array}$ (45)

At this moment, we can finally understand why Einstein concluded the expression of photon energy as $E=n\hslash \omega$ in 1905.

Remark.

In the deductions where $\stackrel{\dot{}}{\lambda }$ notably represents the meson wave function varying with time at $r_{Yuka}$, the expressions of the neutral current in this section align with $J_{\mu }=i\epsilon \bar{\psi }{\gamma }_{\mu }\tau \psi$.

E) Importance: Light-speed of C in GUT scales (Factorial 2! given by loops)

The elementary charge e has been theoretically obtained in previous sections. Moving forward, we now require the mathematical-physical expression of the speed of light, C, which is particularly intriguing. Referencing Equation (32),

$-\frac{{\left(\kappa m{r}_{Yuka}\right)}^6}{{10}^5}\stackrel{\left({\Lambda }_{GUT}\right)}{\equiv }-e\approx -1.6055\times {10}^{-19}\text{C}$ (46)

Moreover,

$\begin{array}{l}{c}^6=\frac{{10}^5\left(1.6055\times {10}^{-19}\text{C}\right){\hslash }^6}{{\left(m{r}_{Yuka}\right)}^6}=\frac{{10}^5\left(1.6055\times {10}^{-19}\text{C}\right){\left(1.05\times {10}^{-34}\text{J}\cdot \sec \right)}^6}{{\left(9.11\times {10}^{-31}\text{kg}\right)}^{\text{6}}{\left(1.94\times {10}^{-15}\text{m}\right)}^6},\\ c={\left(7.06\times \left({10}^{50}\right)\right)}^{\frac{1}{\text{6}}}\approx \text{1}\text{.3850}\times {\text{10}}^{\text{8}\text{.}\overline{\text{33}}}=\text{1}\text{.3850}\times \left(\underset{\approx \text{2}\text{.15443}}{\underset{︸}{{\text{10}}^{\text{0}\text{.}\overline{\text{33}}}}}\times {\text{10}}^{\text{8}}\right)\\ \approx \text{2}\text{.9839}\times {\text{10}}^{\text{8}}\left[\text{m}/\text{sec}\right]\end{array}$(47)

Similar to the measurement results in the history of physics (with an error ≤±1.31%), note that the factor of 1.38(5) precisely reveals the coefficients of the famous Boltzmann constant. Therefore, its illustration is as follows: With the fixed Boltzmann constant and the constant speed of light C, note that Equation (47) can be rewritten as:

$c={\left(7.06\times \left({10}^{50}\right)\right)}^{\frac{1}{\text{6}}}={\left(Const\cdot {\left({10}^{50}\right)}^{\frac{1}{2}}\right)}^{\frac{1}{\text{3}}}=Const$ (48)

Obviously,

${\left({10}^{50}\right)}^{\frac{1}{2}}=Const$ (49)

So that

${10}^{25}\text{eV}\stackrel{\left({\Lambda }_{GUT}\right)}{=}{10}^{16}\text{GeV}$ (50)

Recall that the Grand Unified Theory (GUT) requires that three coupling constants precisely converge at point ${\Lambda }_{GUT}\approx {10}^{16}\text{GeV}\gg {\Lambda }_{QCD}$ (e.g., the points associated with the speed of light C within six-dimensional energy scales, which we currently hypothesize may be greater than 10⁸ GeV in this “section”). It is evident that this convergence is not coincidental; it is directly related to the Supersymmetry (SUSY) discussed in this paper, much like the motivations behind widely-studied models (e.g., the SU (5) model by SUSY, which is Beyond the Standard Model (BSM)).

Remark.

All running points converge at C, precisely aligning with the strong coupling constant of graviphotons (LKPs).

S-Duality: Here directly given that an approximation as below:

$\begin{array}{l}\left|\theta \right|\equiv \frac{1}{\sqrt{1+{\sin }^2{x}_1+{\cos }^2{x}_2}}{\tan }^{-1}\frac{\sqrt{1+{\sin }^2{x}_3+{\cos }^2{x}_4}}{\sqrt{1+{\sin }^2{x}_1+{\cos }^2{x}_2}}=2\cdot {\displaystyle \underset{x=0}{\overset{x_4/2}{\int }}\frac{\text{d}x}{1+x^2}}\\ \stackrel{\left({\Lambda }_{QCD}\right)}{\approx }{\displaystyle \underset{x=0}{\overset{g_s=1.214}{\int }}\frac{\text{d}x}{2+{\sin }^2{x}_1+{\cos }^2{x}_2+{\sin }^2{x}_3+{\cos }^2{x}_4}}\end{array}$ (51)

Here, $\left|\theta \right|$ represents the small angles between two fitted arc lengths, while $x_i,i=1,2,3,4$ (within QCD scaled ranges) denotes the coupling constants, also known as the Weinberg angle for force-charged carriers. These values are essentially approximated using the least squares method. The same slope starts from and is bounded at [10¹¹, 10¹⁸] GeV, as assigned by the two fitted arc lengths (see the triangle in Figure 2 ). It is not difficult to observe that the slope is indicated as

$\begin{array}{l}\frac{\text{d}{x}_i}{\text{d}E}\stackrel{\left({\Lambda }_{GUT}\right)}{=}{\frac{1}{\frac{1}{2}m{v}^2+\frac{3}{8}m\frac{v^4}{c^2}+\cdots }\frac{\text{d}{x}_i}{\text{d}v}|}_{v=c}=\tan \left|\theta \right|,\\ \left|\theta \right|\stackrel{\left({\Lambda }_{GUT}\right)}{\approx }{\tan }^{-1}\frac{\text{d}{x}_i}{m{c}^2\text{d}v}\end{array}$ (52)

Evidently, ${\stackrel{\dot{}}{x}}_i$ varies with the energy flow of $m{c}^2\text{d}{\Phi }_{GUT}$ (in ${\Lambda }_{GUT}$ scales). At such large energy scales, this directly results in $v=c$ and $\text{d}{x}_i=0$, leading to $\left|\theta \right|\stackrel{\left({\Lambda }_{GUT}\right)}{\approx }{\tan }^{-1}\left(0/0\right)$, ${\theta }^{\prime }\stackrel{\left(L.H.\right)}{=}{\left({\tan }^{-1}\left(0/0\right)\right)}^{\prime }$ intelligently. Equation (50) suggests that $v=c$ is located at the point of convergence of the coupling constants, meaning it runs to this point when using derivatives, as shown:

$\begin{array}{l}{\theta }^{\prime }\stackrel{\left({\Lambda }_{GUT}\right)}{\approx }\frac{1}{2+{\sin }^2{x}_1+{\cos }^2{x}_2+{\sin }^2{x}_3+{\cos }^2{x}_4+\cdots }\\ ={{\left[{\tan }^{-1}\frac{\text{d}{x}_i}{m{c}^2\text{d}v}\right]}^{\prime }|}_{\begin{array}{l}{x}_i=const\\ v=c\end{array}},\theta \ge 0\end{array}$ (53)

Resulting in the following sequence:

$\begin{array}{l}{\theta }^{\prime }\stackrel{\left({\Lambda }_{GUT}\right)}{\approx }\frac{1}{2+{\sin }^2{x}_1+{\cos }^2{x}_2+{\sin }^2{x}_3+{\cos }^2{x}_4+\cdots }={{\left[{\tan }^{-1}\frac{0}{0}\right]}^{\prime }|}_{x_i=const},\theta \ge 0,\\ {\theta }^{\prime }\stackrel{\left({\Lambda }_{GUT}\right)}{\approx }{\displaystyle \underset{i=1}{\overset{\infty }{\sum }}\frac{1}{1+{\sin }^2{x}_i+{\cos }^2{x}_i}}={{\left[{\tan }^{-1}Const\right]}^{\prime }|}_{x_i=const},\\ \theta \stackrel{\left({\Lambda }_{GUT}\right)}{\approx }{\displaystyle \underset{\stackrel{\dot{}}{x}}{\overset{\stackrel{\dot{}}{x}=c}{\int }}{\displaystyle \underset{i=1}{\overset{\infty }{\sum }}\frac{\text{d}x}{1+{\sin }^2{x}_i+{\cos }^2{x}_i}}}={\tan }^{-1}Const=Const,\\ Const=\tan \theta =\frac{\text{d}{x}_i}{m{c}^2\text{d}v}\end{array}$(54)

i.e., the speed of light in a vacuum, expressed for convenience in terms of its square, is given by

$c^2=Const\cdot {\frac{\text{d}{x}_i}{m\text{d}v}|}_{\begin{array}{l}{x}_i=const\\ m=0\end{array}}=Const$ (55)

Another approach could involve working within QCD scales, as outlined below. (Note: In Equation (55), $m=0$ pertains to massless particles at GUT scales.)

Let the coupling constants be represented as angles within QCD scales:

$x_1\approx \text{1}\text{.214},x_2\approx \text{0}\text{.302822},x_3\approx \text{0}\text{.6295},x_4\approx \text{0}\text{.7180}$ (56)

Based on smooth and continuous arc-lengths in the RGE scale $\mu$ (measured in GeV), the following two linear combinations are presented:

$\begin{array}{l}{\theta }^{\prime }\stackrel{\left({\Lambda }_{QCD}\right)}{\approx }\frac{1}{2+{\sin }^2\left(\text{1}\text{.214}\right)+{\cos }^2\left(\text{0}\text{.302822}\right)+{\sin }^2\left(\text{0}\text{.6295}\right)+{\cos }^2\left(\text{0}\text{.7180}\right)}\\ =\text{0}\text{.249980}\approx 1/4,\\ {\theta }^{\prime }\stackrel{\left({\Lambda }_{QCD}\right)}{\approx }\frac{1}{2+{\sin }^2\left(\text{0}\text{.302822}\right)+{\cos }^2\left(\text{1}\text{.214}\right)+{\sin }^2\left(\text{0}\text{.7180}\right)+{\cos }^2\left(\text{0}\text{.6295}\right)}\\ =\text{0}\text{.250024}\approx 1/4\end{array}$ (57)

i.e., all numbers would convergence onto 1/4:

$\begin{array}{l}\frac{1}{2+{\sin }^2{x}_1+{\cos }^2{x}_2+{\sin }^2{x}_3+{\cos }^2{x}_4+\cdots }=1/4,\\ {\sin }^2{x}_1+{\cos }^2{x}_2+{\sin }^2{x}_3+{\cos }^2{x}_4+\cdots =2\equiv 2!\end{array}$ (58)

Refer to the note below⁹ for its corresponding QFT (i.e., the equivalence of Equation (58) is denoted as Tree Level(s): ${\Gamma }_{tree}^{\left(2\right)}=i\left(k^2+m^2\right)>0$). Specifically, all constants in Equation (54) are indicated as 1/3. Generalizing this into Equation (55), the calculated result of 1/3 becomes a hidden power component of the constant in both Equation (48) and Equation (55), respectively. By taking the logarithm and applying the concepts from Table I, the units of C are omitted at ${\Lambda }_{GUT}$ scales. Therefore,

${\log }_{10}c=\frac{1}{\text{3!}}\cdot {\log }_{10}\left(Const\cdot \left({10}^{50}\right)\right)\equiv Const$ (59)

The origins from Equation (48) are self-consistent, and therefore, it was found that

${\log }_{10}c=\frac{1}{3}\cdot {\log }_{10}Const\equiv Const$ (60)

Starting from Equation (55), we find that Equation (59) is physically equivalent to Equation (60) via instantons. Thus, by aligning the form of Equation (59) with Equation (60), we see that they both represent the same concept—extremely small sizes or distances, very close to Planck scales. Consequently, this alignment yields the precise fit points of the running coupling constants at ${\Lambda }_{GUT}$ scales. The results are, indeed, remarkable¹⁰:

$\begin{array}{l}{\frac{1}{3}\cdot {\log }_{10}\left(3\times {10}^8\right)\cdot \frac{1}{10\cdot Cons{t}^{\prime }}|}_{Cons{t}^{\prime }=7.06}\approx 0.04,~{10}^{11}\text{GeV},\\ {\frac{1}{3!}\cdot {\log }_{10}\left(3\times {10}^8\right)\cdot \frac{1}{10\cdot Cons{t}^{\prime }}|}_{Cons{t}^{\prime }=7.06}\approx 0.02,~{10}^{18}\text{GeV}\end{array}$ (61)

where the anti-logarithm: ${{\log }_{10}{}^{-1}\left(Const\cdot \left({10}^{50}\right)\right)\equiv \frac{1}{10\cdot Cons{t}^{\prime }}|}_{Cons{t}^{\prime }=7.06}$

suppresses four curves on p-103 of the PDF “Axion and ALP couplings—CERN Indico (2019) where ${\alpha }_6$ could be regarded as being the fourth curve because of continuity: $\underset{x_i\to x_j}{\lim }f\left(x_i\right)=f\left(x_j\right)\equiv {\alpha }_6$), and the term of $\left(10\cdot Cons{t}^{\prime }\right)$ is assigned from Equation (48). Note that all numbers with constants could be merged ensemble. Above is not pointed out what ways calculating value of light-speed C, it is pointed out its vale is a constant regardless scenarios of running coupling constants, and in points of view of $\theta \equiv x_i$, it is exactly the convergence point of three coupling constants while all running in scales of ${\Lambda }_{GUT}$.

So far as, note that the indicated arc-lengths of picture are done by CERN (2019), one could preliminary refer to Figure 2 where the hypotenuse of triangle fits four curves. This section is majority-contributed by the first author in this paper, and supervised by the second author. Where ${\alpha }_6$ could be considered the fourth curve due to its continuity: $\underset{x_i\to x_j}{\lim }f\left(x_i\right)=f\left(x_j\right)\equiv {\alpha }_6$, the term C is derived from Equation (48). It’s important to note that all numbers with constants can be merged. While the specific method for calculating the value of $\left(10\cdot Cons{t}^{\prime }\right)$ is not detailed here, it is established that C remains a constant, irrespective of the scenarios involving running coupling constants. From the perspective of $\theta \equiv x_i$, this constant precisely marks the convergence point of the three coupling constants as they run across ${\Lambda }_{GUT}$ scales.

It is worthy to notice that, the arc-lengths cited here which are produced by CERN (2019), and one can preliminarily refer to Figure 2 , where the hypotenuse of a triangle aligns with the four curves. This section was primarily contributed by the first author of this paper, under the supervision of the second author.

Remark

The derivation in the section aligns with the work of Kane, Gordon L. Specifically, the speed of light, denoted as C, is shown as the convergence point for all running points at high energy scales.

(interactions) into a single force has been achieved in this work. We have addressed and overcome the challenges associated with running points at high energy scales. The derived results for running points at high energy scales—including those of the fine structure constant $\alpha \left(t\right)$, the elementary charge e, the speed of light C, and Coulomb’s law—have all been demonstrated in previous sections. Various difficulties in high-energy physics, such as running points at varying energy scales, massless particles, and the sources of photon energy, have largely been resolved through this work, which represents a significant scientific contribution. Aside from the Higgs mechanism, the representations for dynamics presented in this paper align with those of electroweak interaction theory, incorporating the novel concept of $r_{Yuka}\approx \text{1}\text{.9404}\text{fm}\equiv k{\Delta }_0>0,k>0$ in glueballs.

We found that the antiscreening colors with a trivial UV fixed point ${\alpha }_S\left({\mu }^2\right)=0$, therefore $G=SU{\left(3\right)}_{Diag}\times SU{\left(2\right)}_L\times U{\left(1\right)}_{EM}\subset SU\left(5\right)$ with ∆₀ > 0, which can be completely derived by asymptotic freedoms, primarily introduced in this work (i.e., providing an estimated value for the spectrum mass gap that has been elusive in Yang-Mills Theory). Consequently, this work also contributes to Grand Unified Theories (GUTs) without predicting proton decay. Additionally, the idea of a loop contributing its energy to maintaining color symmetry among massless particles at high energy scales is explored. Furthermore, we concluded our Beyond Standard Model (BSM) work with the formulation of $G=SU{\left(3\right)}_{Diag}\times SU{\left(2\right)}_L\times U{\left(1\right)}_{EM}$ (i.e., Yang-Mills theory with a non-Abelian symmetry group), whose Lagrangian was derived for both Quantum Chromodynamics (QCD) and electromagnetic fields, in compliance with the representation of $L_{YM}$. We have provided verification for the six quark flavors: up (u), down (d), strange (s), charm (c), bottom (b), and top (t). Most notably, the running coupling constants of 0.02 and 0.04 at [10¹¹, 10¹⁸] GeV, as reported by CERN in 2019, correspond to the speed of light c in a vacuum. Thus, we can conclude that the three fundamental cosmic forces are fully unified. This work could serve as a foundation for advancing the research to complete Einstein’s unfinished Unified Field Theory (UFT) manuscript from 1955. Additionally, we have addressed the issue of strong CP-violation in Appendix D, where the significance of ${\theta }_{CP}\to 0^{\pm }$ is thoroughly explained.

After completing this work, we would like to express our gratitude to Mr. Ben Jieu for his valuable and insightful comments on this paper.