# On a Crucial Role of Gravity in the Formation of Elementary Particles

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## Abstract

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## 1. Introduction: Gravity in Particle Physics

- It is a mini-boson star if the only non-kinetic term present in the scalar field Lagrangian is the mass term;
- It is a boson star if additional nonlinearity is inserted into the Klein–Gordon equation;
- (Mini)-soliton stars correspond to solutions which have a flat limit, i.e., to the soliton-like solutions that exist in the Minkowski space-time.

## 2. Rosen’s Model and Its Counterpart on a Curved Manifold

## 3. General Characteristics of the Three-Field Model

**we cannot set it to zero, since regular solutions disappear in this case**.

## 4. Stationary Spherically Symmetric Ansatz

## 5. Regular Solutions: Procedure of Numerical Integration

## 6. Regular Solution: Characteristics and Dependence on Parameters

- The solutions exist only in a narrow range of the frequency parameter $\omega \sim 1$ near the boundary value $\omega =1$. For $\omega =1$, the asymptotic of the scalar field differs from the Yukawa’s form and corresponds 5 to $\phi \sim exp(-\sqrt{8qr\phantom{\rule{3.33333pt}{0ex}}})$;
- All three characteristics $q,M,\omega $ have a minimum of almost equal values of the free parameter B;
- There are some additional local minima for the values of electric charge and energy (in other range of the parameter B, not represented in the figures).

## 7. Electrically Neutral Solutions

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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1. | |

2. | To avoid radicals in the equations we define metric functions via squares. Below we shall see that for any r and any solution both of these are positive. |

3. | |

4. | The solution with $\gamma =1$ for a particular value of $\omega $ had been previously obtained in [28]. |

5. | In the flat case such “extreme” solutions have been discovered in [23]. |

**Figure 1.**Typical form of the $\phi \left(r\right)$ distribution exponentially decreasing at large r.

**Figure 2.**Typical form of the (shifted) electrostatic potential $\chi \left(r\right)$, $\chi (\infty )=\omega $.

**Figure 3.**Typical form of a (horizon-free) metric function $\sqrt{{g}_{tt}}=f\left(r\right)$; note that $f\left(0\right)\ne 0,f(\infty )=1$.

**Figure 4.**Typical form of a (horizon-free) metric function $\frac{-1}{\sqrt{{g}_{rr}}}=h\left(r\right)$; note that $h\left(0\right)=h(\infty )=1$.

**Figure 8.**Proper energy (mass) M against the free parameter $A=f\left(0\right)$ for the neutral solutions.

**Figure 9.**Frequency $\omega $ against the free parameter $A=f\left(0\right)$ for the neutral solutions.

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Alharthy, A.; Kassandrov, V.V.
On a Crucial Role of Gravity in the Formation of Elementary Particles. *Universe* **2020**, *6*, 193.
https://doi.org/10.3390/universe6110193

**AMA Style**

Alharthy A, Kassandrov VV.
On a Crucial Role of Gravity in the Formation of Elementary Particles. *Universe*. 2020; 6(11):193.
https://doi.org/10.3390/universe6110193

**Chicago/Turabian Style**

Alharthy, Ahmed, and Vladimir V. Kassandrov.
2020. "On a Crucial Role of Gravity in the Formation of Elementary Particles" *Universe* 6, no. 11: 193.
https://doi.org/10.3390/universe6110193