Soliton Solutions of Relativistic Hartree's Equations

TL;DR

This paper derives exact soliton solutions for relativistic Hartree's equations modeling a composite particle with scalar fields, analyzing stability and mass dependence on parameters, including a closed-form mass formula for massless exchange particles.

Contribution

It provides the first exact ground state solutions and stability analysis for relativistic Hartree's equations with self-interaction, including a closed-form mass formula for massless exchange particles.

Findings

Exact ground state solution derived using integral transform method.

Proved local stability of the solutions.

Mass formula for massless exchange particle obtained in closed form.

Abstract

We study a model based on $N$ scalar complex fields coupled to a scalar real field, where all fields are treated classically as c-numbers. The model describes a composite particle made up of $N$ constituents with bare mass $m_0$ interacting both with each other and with themselves via the exchange of a particle of mass $\mu_0$. The stationary states of the composite particle are described by relativistic Hartree's equations. Since the self-interaction is included, the case of an elementary particle is a nontrivial special case of this model. Using an integral transform method we derive the exact ground state solution and prove its local stability. The mass of the composite particle is calculated as the total energy in the rest frame. For the case of a massless exchange particle the mass formula is given in closed form. The mass, as a function of the coupling constant, possesses a well pronounced minimum for each value of $\mu_0/m_0$, while the absolute minimum occurs at $\mu_0=0$.