Soliton Solutions of Relativistic Hartree's Equations

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 $\mu_0=0$.