Unified Equation of Fundamental Forces’ Coupling Values, and the Existence of Subsequent, Fifth and Other, Forces

Kaveh Mozafari

Department of Mathematical Modeling at ExcellenSation, Toronto, Canada.

DOI: 10.4236/jamp.2022.108168                

Abstract

This paper provides an equation to entangle all known fundamental forces by employing their coupling constants, i.e., strong (α_s), electromagnetic (α), weak (α_w), and gravitational (α_g) interaction coupling values. The constant coupling formulation is further indicative of many other fundamental forces with significantly weaker coupling values. As an example, the fifth fundamental force, Kashi’s Force, is found to have a coupling constant of 10^-1446, which is significantly smaller than the smallest known fundamental force, gravitational force, with an approximate coupling constant value of 10^-38. Additionally, the paper finds the sum of all fundamental forces based on the equation proposed is equal to 0.118065, which is within the range of effective world value of the strong coupling constant α_s(M²_z).

Keywords

Fundamental Forces, Coupling Value, Kashi’s Force, Unified Equation

1. Introduction

As of now, physics has discovered four known fundamental forces: strong nuclear force, electromagnetism, weak nuclear force, and gravitational interaction. For the author, the phrase “fundamental” could imply a commonality beyond what has been discussed in terms of physical property and origin. That is, a mathematical equation could untangle these forces. Such relation has been obtained arbitrarily through careful observation of the known coupling values. Each of the mentioned fundamental forces has particular properties and ranges. The current paper does not discuss the physical properties of such interaction; instead, it focuses on a few mathematical findings associated with them. Such findings intend to present a unique formulation to bound the forces to a single nature. It appears a mathematical equation can entangle the magnitudes of the four fundamental forces.

The mere existence of such a relationship can have significant implications for our understanding of the universe as a whole. One of the primary outgrowths of such a relationship can shed light on the actuality of other fundamental forces. The current paper insinuates one such formula, in which a single formula presents the relative magnitudes of all the forces. The submitted article focuses on mathematical findings rather than the physical implication of such a formulary. It is imaginable that provided the obtained results are acceptable, many compelling determinations could be made to advance our knowledge of our physical surroundings further. Discussion of such conclusions is beyond the scope of the presented paper, yet the author provides some hints for interested readers.

2. Math and Equations

In order to compare the magnitude of the known fundamental forces and ultimately formulating the relationship between them, the coupling constants are applied. The dimensionless electromagnetic coupling strength, $\alpha$ , is employed and presented as follows [1] .

$\alpha \equiv \frac{2\pi k{e}^2}{E_{photon}{\lambda }_{photon}}$

Provided the energy of a photon with 1 nm wavelength has an experimental value of 1240 eV, the alpha can be calculated as shown.

$\alpha \equiv \frac{2\pi \times 1.44\text{eV}\cdot \text{nm}}{1240\text{eV}\times 1\text{nm}}=\frac{1}{137}$

The strength of the dimensionless electromagnetic coupling can increase [2] in specific conditions that are in the focus of the presented paper; however, it would be utilized in proceeding approximation and formulation.

On the same note, the dimensionless coupling value of the strong interaction, ${\alpha }_s$ , is outlined bellow [3] .

${\alpha }_s\approx 1$

Similarly, the value of weak coupling, ${\alpha }_w$ , can be circumscribed as follows.

${\alpha }_w\approx {10}^{-6}$

Ultimately, the coupling strength for the gravitational force, ${\alpha }_g$ , can be determined by the following equation.

${\alpha }_g\equiv \frac{G{m}_p{m}_p}{1240\text{eV}\cdot \text{nm}}\approx {10}^{-38}$

Table 1 summarizes the obtained value, along with their approximation used in the current paper [1] .

Table 1. Coupling strength of the known fundamental forces.

For a more simplified formulation, the fundamental forces are ranked in an ascending order based on their strength, with the strong force being ${\alpha }_1$ . Table 2 depicts the approximate relative coupling strength values for the known fundamental forces.

Provided the approximated relative coupling values, Equation (1) can be appropriated to connect the values.

${\alpha }_{n+1}\approx {10}^{-\left(x_n^2+2\right)}$ (1)

where $x_n$ is the approximate power of the n^th coupling value, provided $x_1=0$ (Figure 1).

$x_n=-\left(x_{n-1}^2+2\right)$ (2)

That is, the following calculations can be interpreted for the coupling values. Note the n can only take integer values, and for its similarities with quanta concept, it can be called “Quanta Fo”.

The strong force coupling value can be retrieved using the following equation.

${\alpha }_1\approx {10}^0$

As $x_1=0$ , ${\alpha }_2$ , the electromagnetic force coupling value, can be obtained utilizing Equation (1).

${\alpha }_2\approx {10}^{-\left(0^2+2\right)}={10}^{-2}$

Similarly, since $x_2=-2$ , ${\alpha }_3$ , the weak force coupling value, can be calculated by utilizing Equation (1).

${\alpha }_3\approx {10}^{-\left({\left(-2\right)}^2+2\right)}={10}^{-6}$

Finally, ${\alpha }_4$ , the gravitational force coupling value, can be obtained substituting $x_3=-6$ in Equation (1).

${\alpha }_4\approx {10}^{-\left({\left(-6\right)}^2+2\right)}={10}^{-38}$

The next reasonable move would be to find the sum of coupling values where x is not equal to zero even though the x equals zero point integral would not affect the overall result. Accordingly, the integral of Equation (1) is assessed and presented in the following.

${\displaystyle {\int }_{-\infty }^{\infty }}{10}^{-\left(x^2+2\right)}\text{d}x=\underset{t\to \infty }{lim}{\left(\frac{1}{200}\sqrt{\frac{\pi }{\log 10}}\text{erf}\left(t\sqrt{\log 10}\right)\right)|}_{-t}^t$

Table 2. Approximate relative coupling strength of the known fundamental forces.

Figure 1. The figure coupling value versus as a function of x, where x is not equal to zero. The figure was prepared by utilizing the Matlab 2022 application.

where erf is defined as the error function.

${\displaystyle {\int }_{-\infty }^{\infty }}{10}^{-\left(x^2+2\right)}\text{d}x\approx \text{0}\text{.0117}$ (3)

The value obtained in Equation (3) is very close to the experimental value of the effective coupling constant of strong force 0.1 ${\alpha }_s\left(M_z^2\right)$ . This fact is depicted in Table 3 [4] - [16] .

That is, the sum of all possible forces found in Equation (3) is within the range of most accurate experimental effective strong interaction coupling constant of 0.1 ${\alpha }_s\left(M_z^2\right)$ [16] [17] .

It is also possible to find $x_0$ by extracting the $x_n$ from Equation (2) as presented in the following line.

$x_n=-\left(x_{n-1}^2+2\right)$

Since $x_1$ is equal to 0, then the following can be concluded (Figure 2).

Table 3. Selected world average values for ${\alpha }_s\left(M_z^2\right)$ since 1989 [10] , with current value of 0.1185 ± 0.0006 [16] .

Figure 2. Selected world average values for ${\alpha }_s\left(M_z^2\right)$ as a function of time, with current value of 0.1185 ± 0.0006. The figure was prepared by utilizing the Microsoft Excel application.

$x_1=-\left(x_0^2+2\right)$

$0=-\left(x_0^2+2\right)$

$\sqrt{-2}=x_0$

$\therefore x_0=\pm i\sqrt{2}$

Therefore, the quanta fo that generates the strong coupling interaction value is $\pm i\sqrt{2}$ .

3. Discussion

The methods of ascertaining values of coupling constants are subject to a lot of complexion; hence, variation [7] [19] [20] . Moreover, these values are restricted to specific conditions and physical properties. Therefore, a coherent conclusion is not the most logical approach, yet ignoring the findings is illogical.

The concept of formulation of coupling constants represented in Equation (1) indicates the possibility of other fundamental forces that are far weaker in nature than the proceeding known fundamental forces. As an example, provided the quanta fo, n, value is equal to 5, the coupling constant of the fifth force, Kashi’s Force, can be achieved by utilizing Equation (1), which is an entirely different concept from other proposed fifth forces [21] .

${\alpha }_5\approx {10}^{-\left(x_4^2+2\right)}={10}^{-\left(-{38}^2+2\right)}={10}^{-1446}$

Even though the proposed fifth force and subsequent forces have significantly smaller coupling constant; however, their value can be measurable and even significant on massive scales. The uncovered fundamental forces can be the answers to some of the demanding concepts in physics, such as dark matter and dark energy.

The other point worth noting, which requires more extensive research, is the origin of all fundamental forces. As all the presently recognized fundamental forces have been tied to a single equation (Equation (1)) and the sum of their constant coupling value is within the most precise estimations of 0.1 ${\alpha }_s\left(M_z^2\right)$ it can be inferred the other fundamental forces could be originated from the strong force. Such a claim asks for more extensive research and discussion to prove noble.

Another possible exciting finding that requires more detailed attention is the fact that Equation (1) resembles a normal curve. The normal nature of the curve has considerable implications, which are beyond the scope of the provided paper; however, it seems to be an intriguing topic for further research.

Moreover, the fact that Figure 1 resembles both positive and negative forces can imply the possibility of counter-forces, Ka Forces, for all existing fundamental forces. Such a finding requires more attention and could lead to more solutions and questions in the physical universe. It appears a mathematical equation can entangle the magnitudes of the four fundamental forces.

The mere existence of such a relationship can have significant implications for our understanding of the universe. One of the primary outgrowths of such a relationship can shed light on the actuality of other fundamental forces. The current paper insinuates one such formula: a single formula presents the relative magnitudes of all the forces. The submitted article focuses on mathematical findings rather than the physical implication of such a formulary. It is imaginable that provided the obtained results are acceptable, many compelling determinations could be made to advance our knowledge of our physical surroundings further. Discussion of such conclusions is beyond the scope of the presented paper, yet the author provides some hints for interested readers.

4. Conclusion

The purpose of the presented article was to seek a possible mathematical formulation between the known fundamental forces coupling values. Such formulation was achieved by approximating the already known values. The delivered equation explained the coupling values for the know fundamental forces. Additionally, the extrapolation of the formula enhances one to embrace possible new fundamental forces. The author decided to dobbed the fifth force as Kashi’s force, in memory of the great mathematician Jamshid Kashani. The Kashi’s force has an extremely small coupling value compared to the strong force, ${\alpha }_5\approx {10}^{-1446}$ , yet on a large scale, it can have a significant effect on the physical property of the university. Such correlation is beyond the range of the offered paper, yet the curious reader may find some avenue between Kashi’s force and large-scale phenomena, such as galactic motion in superclusters, within our know universe. Another interesting correlation could be discovered by overviewing the additional suggested fundamental forces and the proposed dark force in astrophysics.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References

[1]	Rohlf, J.W. (1994) “Modern Physics from aalpha to z0,” Modern Physics from aalpha to Z0. Wiley-VCH, Weinheim, 664.
[2]	Fritzsch, H. (2002) Fundamental Constants at High Energy. Fortschritte der Physik, 50, 518-524. https://doi.org/10.1002/1521-3978(200205)50:5/7%3C518::AID-PROP518%3E3.0.CO;2-F
[3]	Peskin, M.E. (2018) An Introduction to Quantum Field Theory. CRC Press, Boca Raton. https://doi.org/10.1201/9780429503559
[4]	Dissertori, G. (2016) The Determination of the Strong Coupling Constant. In: Maiani, L. and Rolandi, L., Eds., The Standard Theory of Particle Physics: Essays to Celebrate CERN’s 60th Anniversary, World Scientific, Singapore, 113-128. https://doi.org/10.1142/9789814733519_0006
[5]	Alekhin, S., Blümlein, J. and Moch, S. (2012) Parton Distribution Functions and Benchmark Cross Sections at Next-to-Next-to-Leading Order. Physical Review D, 86, Article ID: 054009. https://doi.org/10.1103/PhysRevD.86.054009
[6]	Martin, A., Stirling, W.J., Thorne, R.S. and Watt, G. (2009) Uncertainties on αS in Global PDF Analyses and Implications for Predicted Hadronic cross Sections. The European Physical Journal C, 64, 653-680. https://doi.org/10.1140/epjc/s10052-009-1164-2
[7]	Ball, R.D., Bertone, V., Del Debbio, L., Forte, S., Guffanti, A., Latorre, J.I., Lionetti, S., Rojo, J., Ubiali, M., Collaboration, N., et al. (2012) Precision NNLO Determination of αS (MZ) Using an Unbiased Global Parton Set. Physics Letters B, 707, 66-71. https://doi.org/10.1016/j.physletb.2011.11.053
[8]	Olive, K.A., Agashe, K., Amsler, C., Antonelli, M., Arguin, J.-F., Asner, D.M., Baer, H., Band, H.R., Barnett, R., Basaglia, T., et al. (2014) Review of Particle Physics. Chinese Physics C, 38, Article ID: 090001. https://doi.org/10.1088/1674-1137/38/9/090001
[9]	Flächer, H., Goebel, M., Haller, J., Höcker, A., Mönig, K., Stelzer, J., et al. (2009) Revisiting the Global Electroweak Fit of the Standard Model and beyond with Gfitter. The European Physical Journal C, 60, 543-583. https://doi.org/10.1140/epjc/s10052-009-0966-6
[10]	Altarelli, G. (1989) Experimental Tests of Perturbative QCD. Annual Review of Nuclear and Particle Science, 39, 357-406. https://doi.org/10.1146/annurev.ns.39.120189.002041
[11]	Trippe, T.G., Groom, D.E., Hayes, K.G., Manley, D., Höhler, G., Cutkosky, R., Wohl, C.G., Crawford, R.L., Gieselmann, K., Roos, M., et al. (1992) Review of Particle Properties, 1992-1993. Physical Review D, 46, 5210. https://doi.org/10.1103/PhysRevD.46.5210
[12]	Montanet, L., Gieselmann, K., Barnett, R., Groom, D., Trippe, T., Wohl, C., Armstrong, B., Wagman, G., Murayama, H., Stone, J., et al. (1994) Review of Particle Properties. Physical Review D, 50, 1173-1814. https://doi.org/10.1103/PhysRevD.50.1173
[13]	Schmelling, M. (1997) Status of the Strong Coupling Constant. arXiv preprint hep-ex/9701002.
[14]	Bethke, S. (2000) Determination of the QCD Coupling as. Journal of Physics G: Nuclear and Particle Physics, 26, Article No. R27. https://doi.org/10.1088/0954-3899/26/7/201
[15]	Bethke, S. (2007) Experimental Tests of Asymptotic Freedom. Progress in Particle and Nuclear Physics, 58, 351-386. https://doi.org/10.1016/j.ppnp.2006.06.001
[16]	Bethke, S., Dissertori, G. and Salam, G. (2014) Quantum Chromodynamics: Olive et al.(pdg). Chinese Physics C, 38, Article ID: 090001.
[17]	Montillet, J.-P. (2017) Sobolev Spaces, Schwartz Spaces, and a Definition of the Electromagnetic and Gravitational Coupling. Journal of Modern Physics, 8, 1700-1722. https://doi.org/10.4236/jmp.2017.810100
[18]	Glasman, C. (2008) Precision Measurements of αs at Hera. Journal of Physics: Conference Series, 110, Article ID: 022013. https://doi.org/10.1088/1742-6596/110/2/022013
[19]	Beane, S.R. and Savage, M.J. (2003) Variation of Fundamental Couplings and Nuclear Forces. Nuclear Physics A, 713, 148-164. https://doi.org/10.1016/S0375-9474(02)01268-X
[20]	Dmitriev, V. and Flambaum, V. (2003) Limits on Cosmological Variation of Quark Masses and Strong Interaction. Physical Review D, 67, Article ID: 063513. https://doi.org/10.1103/PhysRevD.67.063513
[21]	Kodukula, S.P. (2019) Values of Siva’s Constant “K” for All Fundamental Forces—A Review on Spin, Threshold Time and Quantum Entanglement. Journal of Modern Physics, 10, 466-476. https://doi.org/10.4236/jmp.2019.104032