Physics Letters B 783 (2018) 193–199
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Solitons in a cavity for the Einstein-SU(2) Non-linear Sigma Model
and Skyrme model
Alex Giacomini
a
, Marcela Lagos
b,∗
, Julio Oliva
b
, Aldo Vera
b
a
Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia, Chile
b
Departamento de Física, Universidad de Concepción, Casilla, 160-C, Concepción, Chile
a r t i c l e i n f o a b s t r a c t
Article history:
Received
29 May 2018
Received
in revised form 14 June 2018
Accepted
15 June 2018
Available
online 21 June 2018
Editor:
M. Cveti
ˇ
c
In this work, taking advantage of the Generalized Hedgehog Ansatz, we construct new self-gravitating
solitons in a cavity with mirror-like boundary conditions for the SU(2) Non-linear Sigma Model and
Skyrme model. For spherically symmetric spacetimes, we are able to reduce the system to three
independent equations that are numerically integrated. There are two branches of well-behaved solutions.
The first branch is defined for arbitrary values of the Skyrme coupling and therefore also leads to a
gravitating soliton in the Non-linear Sigma Model, while the second branch exists only for non-vanishing
Skyrme coupling. The solutions are static and in the first branch are characterized by two integration
constants that correspond to the frequency of the phase of the Skyrme field and the value of the Skyrme
profile at the origin, while in the second branch the latter is the unique parameter characterizing the
solutions. These parameters determine the size of the cavity, the redshift at the boundary of the cavity,
the energy of the scalar field and the charge associated to a U (1) global symmetry. We also show that
within this ansatz, assuming analyticity of the matter fields, there are no spherically symmetric black
hole solutions.
© 2018 The Author(s). 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
Non-linear Sigma models appear in many contexts, as for ex-
ample
to describe the dynamics of Goldstone bosons [1], in con-
densed
matter systems [2], in supergravity [3], as well as being
the building blocks of classical string theory. In the case of light
mesons, it can be shown that the low energy dynamics can be
correctly described by a Non-linear Sigma Model for SU(2). In
such low energy processes, the mesons can be seen as Goldstone
bosons. In flat spacetime, the inclusion of the Skyrme term allows
to construct static regular solitons with finite energy, which de-
scribe
baryons [4]. In the latter scenario the ansatz for the SU(2)
group element is given by U
sol
=exp(iF(r)
τ ·
ˆ
x), with
τ the SU(2)
generators. A more general ansatz is defined by the Generalized
Hedgehog Ansatz, which includes U
sol
as a particular case, and is
defined by
U
±1
= Y
0
1 ± Y
i
t
i
, (1)
*
Corresponding author.
E-mail
addresses: alexgiacomini@uach.cl (A. Giacomini), marcelagos@udec.cl
(M. Lagos),
juoliva@udec.cl (J. Oliva), aldovera@udec.cl (A. Vera).
where 1 is the 2 ×2identity matrix, t
i
=−iσ
i
the SU(2) genera-
tors,
σ
i
being the Pauli matrices and
Y
0
=cosα(x
μ
), Y
i
=
n
i
sin α(x
μ
), (Y
0
)
2
+Y
i
Y
i
=1 , (2)
with a generalized radial unit vector
n
1
=cos(x
μ
) sin F (x
μ
),
n
2
=sin(x
μ
) sin F (x
μ
),
n
3
=cos F (x
μ
). (3)
Here α, and F are arbitrary functions of the space–time co-
ordinates.
This ansatz was originally introduced in the context
of the Gribov problem in regions with non-trivial topology [5],
and has been shown to provide a very fruitful arena to construct
new solutions of the theory. In reference [6], the compatibility
of this ansatz on the Einstein–Skyrme theory was thoroughly ex-
plored
considering a space–time which is a warped product of a
two-dimensional space–time with an Euclidean constant curvature
manifold. Also, within this ansatz, a novel non-linear superposi-
tion
law was found in [7]for the Skyrme theory, which was latter
extended to the curved geometry of AdS
2
× S
2
in reference [8].
Even more, the ansatz allows for exact solitons with a kink pro-
file
[9]. Asymptotically AdS wormholes and bouncing cosmologies
with self-gravitating Skyrmions were constructed in [10]as well as
https://doi.org/10.1016/j.physletb.2018.06.036
0370-2693/
© 2018 The Author(s). 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
.