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

We consider the model of minimally interacting electromagnetic, gravitational and massive scalar fields free of any additional nonlinearities. In the dimensionless form, the Lagranginan contains only one parameter $\gamma ={(m\sqrt{G}/e)}^2$ which corresponds to the ratio of gravitational and electromagnetic interactions and, for a typical elementary particle, is about ${10}^{-40}$ in value. However, regular (soliton-like) solutions can exist only for $\gamma \ne 0$, so that gravity would be necessary to form the structure of an (extended) elementary particle. Unfortunately (in the stationary spherically symmetrical case), the numerical procedure breaks in the range $\gamma \le 0.9$ so that whether the particle-like solutions actually exist in the model remains unclear. Nonetheless, for $\gamma \sim 1$ we obtain, making use of the minimal energy requirement, a discrete set of (horizon-free) electrically charged regular solutions of the Planck’s range mass and dimensions (“maximons”, “planckeons”, etc.). In the limit $\gamma \to \infty$, the model reduces to the well-known coupled system of the Einstein and Klein–Gordon equations. We obtain—to our knowledge—for the first time, the discrete spectrum of neutral soliton-like solutions (“mini-boson stars”, “soliton stars”, etc.) View Full-Text