Physics Letters B 777 (2018) 340–345
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
A two-dimensional soliton system of vortex and Q-ball
A.Yu. Loginov
Tomsk Polytechnic University, 634050 Tomsk, Russia
a r t i c l e i n f o a b s t r a c t
Article history:
Received
7 November 2017
Received
in revised form 11 December 2017
Accepted
19 December 2017
Available
online 27 December 2017
Editor:
N. Lambert
Keywords:
Vortex
Flux
quantization
Q-ball
Noether
charge
The (2 + 1)-dimensional gauge model describing two complex scalar fields that interact through a
common Abelian gauge field is considered. It is shown that the model has a soliton solution that
describes a system consisting of a vortex and a Q-ball. This two-dimensional system is electrically
neutral, nevertheless it possesses a nonzero electric field. Moreover, the soliton system has a quantized
magnetic flux and a nonzero angular momentum. Properties of this vortex-Q-ball system are investigated
by analytical and numerical methods. It is found that the system combines properties of topological and
nontopological solitons.
© 2017 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Topological solitons of (2 + 1)-dimensional field models play
an important role in field theory, physics of condensed state, cos-
mology,
and hydrodynamics. First of all, it is necessary to men-
tion
vortices of the effective theory of superconductivity [1] and
vortices of the (2 + 1)-dimensional Abelian Higgs model [2]. An-
other
important example is given by the soliton solution of the
(2 + 1)-dimensional nonlinear O (3) σ model [3] that effectively
describes the behavior of a ferromagnet in the critical region.
Two-dimensional
soliton solutions of Abelian Maxwell gauge
models are necessarily electrically neutral. This is because the
(2 + 1)-dimensional Maxwell electrodynamics does not admit the
existence of electrically charged spatially localized solutions with
finite energy [4], in contrast to the (3 +1)-dimensional case. How-
ever,
the electrical neutrality does not forbid the existence of two-
dimensional
solitons possessing an electric field.
In
this Letter we consider a two-dimensional soliton system
consisting of an Abelian vortex and a Q-ball. The vortex and the
Q-ball interact through a common Abelian gauge field. This electri-
cally
neutral soliton system possesses a radial electric field, carries
a quantized magnetic flux, and has a nonzero angular momentum.
The soliton system combines the properties of vortex and Q-ball.
The interaction between the vortex and the Q-ball by means of a
common gauge field leads to a significant change of their shapes.
E-mail address: aloginov@tpu.ru.
2. Lagrangian and field equations of the model
The (2 +1)-dimensional model we are interested in is described
by the Lagrangian density
L =−
1
4
F
μν
F
μν
+
D
μ
φ
∗
D
μ
φ − V
(
|
φ
|
)
+
D
μ
χ
∗
D
μ
χ − U
(
|
χ
|
)
,
(1)
where φ and χ are complex scalar fields that are minimally cou-
pled
to the Abelian gauge field A
μ
through covariant derivatives:
D
μ
φ = ∂
μ
φ − ieA
μ
φ, D
μ
χ = ∂
μ
χ − iqA
μ
χ. (2)
The self-interaction potentials V
(
|
φ
|
)
and U
(
|
χ
|
)
are
V
(
|
φ
|
)
=
λ
2
φ
∗
φ − v
2
2
,
U
(
|
χ
|
)
=
m
2
χ
∗
χ − g
χ
∗
χ
2
+h
χ
∗
χ
3
, (3)
where λ, g, and h are the positive self-interaction constants, m is
the mass of the scalar χ -particle, and v is the vacuum average of
the complex scalar field φ. We suppose that the potential U
(
|
χ
|
)
has the global minimum at χ = 0 and a local one at some
|
χ
|
=
0;
hence we have the following condition for the parameters m, g,
and h:
g
2
4m
2
< h <
g
2
3m
2
. (4)
https://doi.org/10.1016/j.physletb.2017.12.054
0370-2693/
© 2017 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.