^{3}

We numerically construct stationary and spherically symmetric nontopological soliton solutions in a system composed of a complex scalar field, a U(1) gauge field and a complex Higgs scalar field that causes spontaneous symmetry breaking. It is shown that the charge of the soliton is screened by counter charge everywhere.

Nontopological solitons, which are energy-minimum solutions under the condition of fixed conserved U(1) charge in classical field theories, appear in various theories: a coupled system of a complex scalar field and a real scalar field [^{1}

Gauge theory with spontaneous symmetry breaking is the most fundamental framework in modern physics. We present, in this article, nontopological soliton solutions in a system composed of a complex scalar field coupled to a U(1) gauge field, and a complex Higgs scalar field that causes spontaneous symmetry breaking. This is a generalization of Friedberg–Lee–Sirlin’s model [

We show that the charges of nontopological solitons are perfectly screened [

We consider the system described by the action

The action (

The energy of the system is given by

Equivalently, the fields should take the form

By varying Eq. (

We assume that the fields are stationary and spherically symmetric in the form

Substituting Eq. (

In Eqs. (

The set of Eqs. (

If we regard the coordinate

Using the ansatz (

At the origin, we impose the regularity conditions for the spherically symmetric fields as

To obtain numerical solutions to the coupled ordinary differential equations (

In

Numerical solutions

We show the charge densities

The charge densities,

The total charge of the scalar field

The total charge of

First, we determine

If we require the solutions to be localized in a finite region, the parameter

Next, we determine

Trajectory of the numerical solution for

Here,

From the numerical calculations, we see that this occurs for

Therefore, the allowed range of

Then, the nontopological soliton solution exists for the model parameters satisfying

The nontopological soliton obtained in this paper can be regarded as a condensate of particles of the scalar field

In

The ratio

The nontopological soliton in the range (^{2}

The ratio

In this article, we have shown that nontopological solitons exist stably in a system consisting of a complex scalar field coupled to a U(1) gauge field and a complex Higgs scalar field that causes spontaneous symmetry breaking. The characteristic property of the solitons in this system is the perfect charge screening [

Owing to the perfect charge screening, infinitely heavy solitons are allowed in this system. Then, two solitons would merge by collision and form a larger soliton [

The system considered here should be embedded in more realistic field theories. Generalization of the model is an interesting issue. Furthermore, the stability of the solutions should be clarified from various points of view [

We are grateful to K.-i. Nakao, H. Itoyama, Y. Yasui, N. Maru, N. Sakai, and M. Minamitsuji for valuable discussions. H.I. was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 16K05358.

Open Access funding: SCOAP

^{1}Potentials inspired by supersymmetric theories also allowed nontopological soliton solutions [

^{2}Large gauged