Gravity might play a bigger role in the formation of elementary particles than scientists used to believe. A team of physicists from RUDN University obtained some solutions of semi-classical models that describe particle-like waves.

Scientists believe that gravity plays a significant role in the formation of elementary particles. Certain data confirm this assumption.
But due to their small size, the gravitational interaction between elementary particles (electrons, protons, and neutrons) is weak compared to Coulomb forces.
For instance, negatively charged electrons move around the atomic nucleus that contains positively charged protons. Hence, the proportion of Newtonian attraction to Coulomb repulsion (or ) is negligible. Be that as it may, on the Planck scale, for example, at distances around 1.6?10?35 m, these forces become equivalent.
A team of physicists from RUDN University has found solutions of existing models that correspond to particles in Planck’s range. They obtained some solutions of semi-classical models that describe particle-like waves. What’s more, they also calculated the ratio between the gravitational interaction of particles and the interaction of their charges.
Vladimir Kassandrov, Ph.D., and an Assistant Professor of the Institute of Gravitation and Cosmology, RUDN University, said, “ is considered a’ magical’ dimensionless number, and we are unaware of any serious attempts to theoretically obtain such a small value of 10-40. We presented a simple model that allowed for naturally obtaining this particular value.”
In particular, scientists used semi-classical models based on electromagnetic field equations. Such models have multiple solutions for particles and solitons. Gravity is not considered in such equations and replaced with a nonlinear correction that is chosen almost arbitrarily. Here is the main issue lies.
Although, it can be solved by adding the equations of three fundamental fields to the system. Then, following the requirements of gauge invariance (that prevent physical values from changing simultaneously with the fields’ transformation), the form of nonlinearity becomes strictly defined.
Using the same approach, scientists found solutions that matched the characteristics of typical elementary particles. The existence of such solutions would confirm the fundamental role of gravity in the formation of particles.
Scientists could not find solutions in which the charge and mass matched elementary particles at <0.9, and the very possibility of their existence remains questionable. But, they successfully obtained solutions to the model for ~1. They describe charged semi-quantum objects in the Planck range.
The physicists are still unsure what these solutions correspond to. Hypothetical particles with these parameters are called maximons or planckeons.
This is the first time scientists managed to obtain a discreet energy spectrum for objects with tending to infinity.
Vladimir Kassandrov from RUDN University said, “Although our attempt to calculate probability characteristics at <0.9 was not successful, the model still could have such particle-like solutions. In the future, we would like to shed light on this problem that is intriguing for physicists by extremely complex from the point of view of mathematics. We want to find out if solutions for elementary particles exist in the three-field model.”
Journal Reference:

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