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

TL;DR

This paper explores the role of gravity in forming elementary particles by analyzing a model with electromagnetic, gravitational, and scalar fields, revealing that gravity is essential for stable particle-like solutions, and identifying discrete spectra of such solutions.

Contribution

It demonstrates that gravity is crucial for the existence of regular, soliton-like solutions in a simplified field model and uncovers a discrete spectrum of neutral solutions.

Findings

Regular solutions exist only for non-zero gravity parameter b3.

Identifies a discrete set of charged solutions with Planck-scale mass and size.

First to find a discrete spectrum of neutral soliton-like solutions.

Abstract

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 = Gm^2/e^2 which corresponds to the ratio of gravitational and electromagnetic interactions and, for a typical elementary particle, is about 10^-40. 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 \approx 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.) %