Skyrmions, Skyrme stars and black holes with Skyrme hair in five spacetime dimension
HJE
Skyrmions, Skyrme stars and black holes with Skyrme hair in ve spacetime dimension
Yves Brihaye 0 2 4
Carlos Herdeiro 0 2
Eugen Radu 0 2
D.H. Tchrakian 0 1 2 3
Belgium 0 2
Maynooth 0 2
Ireland 0 2
0 10 Burlington Road, Dublin 4 , Ireland
1 Department of Computer Science, Maynooth University
2 Campus de Santiago , 3810183 Aveiro , Portugal
3 School of Theoretical Physics
4 PhysiqueMathematique, Universite de MonsHainaut
5 Dublin Institute for Advanced Studies
We consider a class of generalizations of the Skyrme model to ve spacetime dimensions (d = 5), which is de ned in terms of an O(5) sigma model. A special ansatz for the Skyrme eld allows angular momentum to be present and equations of motion with a radial dependence only. Using it, we obtain: 1) everywhere regular solutions describing localised energy lumps (Skyrmions); 2) Selfgravitating, asymptotically nonsingular solitonic solutions (Skyrme stars), upon minimally coupling the model to Einstein's gravity; 3) both static and spinning black holes with Skyrme hair, the latter with rotation in two orthogonal planes, with both angular momenta of equal magnitude. In the absence of gravity we present an analytic solution that satis es a BPStype bound and explore numerically some of the nonBPS solutions. In the presence of gravity, we contrast the solutions to this model with solutions to a complex scalar eld model, namely boson stars and black holes with synchronised hair. Remarkably, even though the two models present key di erences, and in particular the Skyrme model allows static hairy black holes, when introducing rotation, the synchronisation condition becomes mandatory, providing further evidence for its generality in obtaining rotating hairy black holes.
Black Holes; Classical Theories of Gravity
1 Introduction and motivation
2 The O(5) Skyrme model in d = 4 + 1 dimensions
2.1
A BPS bound and the topological charge
2.2 Coupling to gravity
2.3 A codimension1 Skyrme eld ansatz
3 Flat spacetime Skyrmions
3.1
3.2
A spherical BPS Skyrmion in the quartic model
The general model 3.2.1 3.2.2 3.2.3
The e ective action and densities
A virial identity and scaling
The solutions
4 Skyrme stars
4.1
Spherical stars
4.2 Spinning stars
5 BHs with Skyrme hair
5.1 Spherically BHs
5.2 Spinning BHs 5.1.1 5.1.2
The probe limit  Skyrmions on a Schwarzschild BH background
Including backreaction 5.2.1
The probe limit  Skyrmions on a MyersPerry BH background
5.2.2 Including backreaction
6 Conclusions
A Solutions with a quartic term
B O(D + 1) Skyrme models on a RDEuclidean space
1
Introduction and motivation
The Skyrme model has more than fty years of history, starting from Skyrme's original
construction and its basic solution, with unit baryon number [1, 2]. It provided the very rst
explicit example of solitons in a relativistic nonlinear eld theory in d = 3 + 1 spacetime
dimensions. Such solutions, dubbed Skyrmions, have found interesting applications, e.g.
as an e ective description of low energy QCD [3, 4] and on the issue of proton decay [5].
{ 1 {
In its simplest version, the Skyrme model is described by a set of four scalars f ag,
a = 1; ::; 4, satisfying the sigmamodel constraint a a = 1, with a target space1 S3
SU(
2
)
and a Lagrangian density possessing a global O(
4
) symmetry. In addition to a standard
(quadratic) kinetic term and a potential, the model contains an extra (quartic) term which
is fourth order in derivatives2 allowing it to circumvent Derrick's nogo theorem for nite
energy lumplike solutions in
eld theory [9]. These Skyrmions possess some topological
properties, being characterized by the homotopy class 3(SU(
2
)) = Z. Moreover, the energy
functional of the Skyrme model has a Bogomol'nyitype bound, in terms of the topological
charge B associated with the homotopy, which is identi ed as the baryon number.
The solutions of the Skyrme model have been studied intensively over the last decades.
can be found in the monograph [13].
Skyrmions persist after taking their gravitational backreaction into account, within
the Skyrme model minimally coupled to Einstein's gravity (even including a cosmological
constant, e.g. [14]). The properties of such, hereafter Skyrme stars, have been considered
in [15{20]. Moreover, and following a generic rule in gravitating solitons [21{23], these
starlike con gurations are compatible with the addition of a (small) horizon at their center.
In the Skyrme model case, this construction was carried out by Luckock and Moss [24],
and provided the rst physically relevant counterexample to the nohair conjecture [25].
The construction in [24] was performed in the probe limit, i.e. a Skyrme test
eld on
a Schwarzschild black hole (BH) background, but subsequent work included the
backreaction [26]. This results in BHs with (primary) Skyrme hair, some of the solutions being
stable against spherical linear perturbations (a review of these aspects can be found in [27];
see also [28{32] for more recent work).
Most of the known Skyrmemodel solutions are static, but spinning generalizations have
been constructed. Spinning
at spacetime Skyrmions were considered by Battye, Krusch
and Sutcli e [33]. In contrast to the (conceptually simpler) spinning Qball solutions [34,
35], for spinning Skyrmions the angular momentum J is a continuous parameter that can
be arbitrarily small, so that they can rotate slowly and rotating solutions are continuously
connected to static ones. The e ects of gravity on these spinning Skyrmions has been
studied by Ioannidou, Kleihaus and Kunz in [36], revealing, in particular, a number of
con gurations which do not have a at space limit. No rotating BHs with Skyrme hair,
however, have been constructed so far, presumably due to the complexity of the numerical
problem.
As we shall see, the complexity just mentioned can be considerably alleviated by
considering the generalisation of the Skyrme model to higher (odd) spacetime dimensions.
1Usually the SU(
2
) group element U is employed, which in terms of the real eld a is given by U = a a,
with
a = (i i; 1I), and U 1 = U y =
a ~a, with ~a = ( i i; 1I), where i are the standard Pauli matrices.
2Higher derivative terms, up to the allowed sextic kinetic term, can also be included as corrections [6, 7]
and are generally expected, see e.g. the recent work [8].
{ 2 {
Moreover, in recent years, the interest in eld theory solutions in d 6= 4 increased
signi cantly. A recurrent lesson has been that well known results in d = 4 physics do not
have a simple extension to higher dimensions. For example, the BH solutions in d > 5
models of gravity are less constrained, with a variety of allowed horizon topologies [37]. In
the Skyrme case, however, the only other dimensions considered so far in the literature is
d = 3 [38{41], even though (gravitating) Skyrmions should also have d > 5 generalizations.
In this paper we shall consider a higher dimensional Skyrme model with the main goal
of testing the existence of rotating BHs with Skyrme hair, but also to understand how the
spacetime dimension a ects standard results for Skyrme physics. The Skyrme system to be
addressed is a generalisation of the usual d = 4 Skyrme model, containing higher derivative
HJEP1(207)3
terms in addition to the standard ones. This can be done, in principle, in all (including
even) dimensions; for concreteness we restrict our attention to the d = 4 + 1 case, where
we have carried out a detailed numerical study. A technical advantage of this case is the
possibility to consider con gurations with two equal angular momenta, for which there is a
symmetry enhancement. As such, the problem results in a system of ordinary di erential
equations (ODEs) which are easier to study.
We shall also contrast the Skyrme model with the better known and conceptually
simpler model of a complex scalar eld, for which one also nds at spacetime eld theory
solutions (Qballs [34, 35, 42]), gravitating solitons (boson stars [43]) and hairy BHs (Kerr
BHs with scalar hair [44{46]). In particular for the latter there are no static hairy BHs. But
in agreement with the latter, rotating BHs with Skyrme hair also require a synchronisation
condition, as described below.
The paper is organized as follows. In section 2 we introduce the model together with
a codimension1 Skyrme eld ansatz. The nongravitating and gravitating solitons of the
model  Skyrmions and Skyrme stars  are discussed in sections 3 and 4. BHs with
Skyrme hair are presented in section 5. In sections 3{5 we study both static and rotating
solutions. We give our conclusions and remarks in section 6. Numerical solutions with
a quartic term in the Skyrme action are reported in appendix A. Appendix B contains a
discussion of the Skyrme model for a general number of spacetime dimensions.
Conventions and numerical method.
Throughout the paper, mid alphabet latin let
ters i; j; : : : label spacetime coordinates, running from 1 to 5 (with x
5 = t); early latin
letters, a; b; : : : label the internal indices of the scalar eld multiplet. As standard, we use
Einstein's summation convention, but to alleviate notation, no distinction is made between
covariant and contravariant internal indices.
The background of the theory is Minkowski spacetime, where the spatial R4 is written
in terms of bipolar spherical coordinates,
ds2 = dr2 + r2(d 2 + sin2 d'12 + cos2 d'22)
dt2 ;
(1.1)
where
2 [0; =2] is a polar angle interpolating between the two orthogonal 2planes and
('1; '2) 2 [0; 2 ] are azimuthal coordinates, one in each 2plane. r and t denote the radial
and time coordinate, 0 6 r < 1 and
1 < t < 1, respectively.
{ 3 {
For most of the solutions the numerical integration was carried out using a standard
shooting method. In this approach, we evaluate the initial conditions at r = 10 5 (or
r = rH +10 5) for global tolerance 10 14, adjusting for shooting parameters and integrating
towards r ! 1. The spinning gravitating solutions were found by using a professional
software package [47]. This solver employs a collocation method for boundaryvalue ODEs
and a damped Newton method of quasilinearization. A linearized problem is solved at
each iteration step, by using a spline collocation at Gaussian points. An adaptive mesh
selection procedure is also used, such that the equations are solved on a sequence of meshes
until the successful stopping criterion is reached.
2
The O(5) Skyrme model in d = 4 + 1 dimensions
a
The Skyrme model can be generalised to an arbitrary number of dimension  cf.
appendix B. Restricting to d = 4 + 1 spacetime dimensions, the Skyrme model is de ned
P
in terms of the O(5) sigma model real elds f ag, a = 1; : : : ; 5, satisfying the constraint
a a = 1. It proves useful to introduce the following notation for all the allowed kinetic
terms, quadratic, quartic, sextic and octic:
HJEP1(207)3
in terms of which the kinetic terms are written:
a
(
1
) = i
ab
(
2
) = ij
abc
(
3
) = ijk
abcd
(
4
) = ijkl
iajbkc ld + jakbcl id + kli j ;
abc d
abcd abcd
i1j1k1l1 i2j2k2l2 gi1i2 gj1j2 gk1k2 gl1l2 :
Observe that a;i =
agij , ab;ij =
j
kablgikgjl, etc., where gij is the metric tensor of the ve
dimensional background geometry.
We shall always include in the action the quadratic term F 2. Then, for most solutions
in this paper, as will be justi ed in section 3.2.2 by a virial identity, only the sextic term
F 6 is also needed, so we will eschew the octic terms, which do not bring any qualitative
features to the solutions. We shall likewise drop the quartic term F 4, though in appendix A
we verify that its inclusion does not change the general features. The sole exception to this
pattern is the BPS solution of section 2.1 which relies only on the quartic term.
The Lagrangian density of the model considered throughout is
LS =
2 F
where V is the Skyrme potential whose explicit form will be discussed later and i > 0
are coupling constants. Observe these are dimensionful constants: [ 0] = length 5, [ 1] =
{ 4 {
(2.1)
(2.2)
for rotating solutions.
equations
length 3, [ 2] = length 1 and [ 3] = length. Inclusion of the potential term is mandatory
From the Lagrangian (2.2) it follows that the scalars a satisfy the EulerLagrange
da
d a
2 1 ri ia + 8 2 i r
b;k j jakb;i + 9 3 bc;jkri iajbkc + 0
= 0 :
(2.3)
A BPS bound and the topological charge
The model (2.2) with 0 = 1 = 3 = 0, i.e. the action
IS =
possesses some special properties, provided the spacetime geometry is ultrastatic (gtt =
1) and the Skyrme ansatz has no time dependence. Then the resulting system lives
e ectively on a four dimensional space with Euclidean signature (which, however, can be
curved), being conformal invariant.
After de ning the twoform Hodge dual of iajb as
? iajb = p
g aba1b1c iji1j15 a1b1;i1j1 c
;
we can state the Bogomol'nyi inequality
which implies that the massenergy of the model is bounded from bellow
a1a2
i1i2
? a1a2 2
i1i2
> 0 ;
M >
2
4
B ;
where B is the topological charge
B =
Z
d x
4 p
g T ; with
while the topological current is
M =
exist,3 being solutions of the 1st order equations
As we shall see in the next section, selfdual solutions saturating the above bound
It is clear that the bound (2.7) holds as well for the general Lagrangian (2.2), in which
case it can never be saturated, since the contribution of the supplementary terms is always
positive.
3This contrasts with the Skyrmions on R3 where no BPS solitons exist, while they do on S3 [48].
iajb =
? iajb :
{ 5 {
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
where LS is the Lagrangian density (2.2) for the Skyrme sector and G is Newton's constant.
Variation of (2.12) w.r.t. the metric tensor leads to the Einstein equations where the energymomemtum tensor is
in terms of the contributions of the distinct terms in (2.2), which read
Rij
1
2 gij = 8 G Tij ;
Tij = 0Ti(j0) + 1Ti(j1) + 2Ti(j2) + 3Ti(j3) ;
Ti(j0) =
Ti(j1) = i j
a a
gij V ( a);
1
2 gij F 2;
Ti(j2) = iakb1 jakb2 gk1k2
Ti(j3) =
1
6
1
4 gij F 4;
ik1l1 jakbc2l2 gk1k2 gl1l2
abc
1
6 gij F
1 + i 2 = sin F (r) sin ei('1 !t) ;
3 + i 4 = sin F (r) cos ei('2 !t) ;
5 = cos F (r) :
As usual, for a given ansatz, the gravity equations (2.13) are solved together with the
matter equations (2.3), subject to some physical requirements (e.g.asymptotic atness and
niteness of the total mass).
2.3
A codimension1 Skyrme eld ansatz
The O(5) solutions in this work are constructed within a Skyrme elds ansatz in terms of
a single function F (r):
Here, w > 0 is an input parameter  the frequency of the elds. The corresponding
expression of the topological charge density is:
For the Skyrme potential we shall take the usual `pion mass'type
which is a natural generalization of that used in the d = 4 model.
A remarkable feature of the ansatz (2.16), rst suggested in [49], albeit for a complex
doublet rather than a Skyrme
eld, is that for any geometry in this work the angular
dependence is factorized in a consistent way, and the Skyrme equations (2.3) reduce to a
single ODE for the function F (r).
{ 6 {
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
Flat spacetime Skyrmions
We start by considering solutions in the probe limit, i.e. we solve the Skyrme equations
on a
xed spacetime background. Apart from being technically simpler, we shall
nd
that they possess already a number of basic properties of the corresponding gravitating
generalizations.
A spherical BPS Skyrmion in the quartic model
In the simplest spherically symmetric case and a at background, the model with only a
quartic term (2.4) allows a simple analytical solution, found with the ansatz (2.16) (with
! = 0). Then the rst order eqs. (2.6) reduce to4
Restricting to the plus sign,5 the solution of the above equation reads
with r0 > 0 an arbitrary parameter. This is an everywhere regular con guration, with
F (r) interpolating between
(at r = 0) and zero (at r = 1). Its energy density is
F 0
sin F
r
= 0 :
F (r) = 2 arctan
r0
r
;
(r) =
Ttt =
96 2r04 ;
while all other components of Tij vanish. The total mass of solution is
It would be interesting to investigate the existence of static nonselfdual solutions of the
simple model (2.4). They are expected to exist, in analogy with instantonantiinstanton
solutions to YangMills theory [51].
3.2
3.2.1
The general model
The e ective action and densities
The simple quartic model (2.4) is too restrictive. Indeed, the selfduality eqs. (2.11) cannot
be satis ed for gtt 6=
1 or ! 6= 0. The situation in this restrictive case is the same as what
occurs in the EinsteinYangMills system [52], where the solutions of the usual YangMills
(YM) model F (
2
)2 do not survive when considering their backreaction on the spacetime
geometry. This is because the scaling requirement is violated. In that case, this defect was
remedied by adding the higherorder YM term F (
4
)2 and regular gravitating YM solutions
were constructed for d = 6; 7 spacetime dimension [53]. Subsequently, the case d = 5 was
considered as well [54], where also BH solutions were constructed.
4It is interesting to note that the solution (3.2) is related to the radially symmetric BPST instanton [50]
described by the form factor w(r), via w(r) = cos F .
5The minus sign solution reads F (r) =
2 arctan(r0=r) and possesses similar properties.
{ 7 {
Despite the fact that this describes an e ective onedimensional system, the con gurations
with ! 6= 0 are not spherically symmetric6 and carry an angular momentum density, j,
j
T't 1
sin2
=
T't 2
cos2
= ! sin2 F
1 + 2 2 F 02 +
gTtt = 2 2
dr r3 ; J1 = J2 = J =
Z
d x
4 p
gT't 1;2 =
2
0
S =
where
For the Skyrme system, the F
4 density scales as L 4, while the usual gravity scales as
L 2. Thus, as in the YM case, it is necessary to add higherorder terms. In the general
case with all i 6= 0 and a Skyrme ansatz given by (2.16), one can show that the equation
for F (r) can also be derived from the e ective action
drLe ;
with
Le = r3
0(1
Following d = 5 BH physics conventions, we de ne also the reduced angular momentum of
a spinning con guration as
3.2.2
A virial identity and scaling
The form (3.5) of the reduced allows the derivation of an useful Derricktype virial relation.
Let us assume the existence of a globally regular solution F (r), with suitable boundary
6This contrasts with the complex scalar elds model in [49], which possesses spherically symmetric Qball
solutions supported by the harmonic time dependence of the elds.
j
27
J 2
8 M 3 :
{ 8 {
conditions at the origin and at in nity. Then each member of the 1parameter family
F ( r) assumes the same boundary values at r = 0 and r = 1 and the action
S(F ) must have a critical point at
= 1, i.e. [dS =d ] =1 = 0. This results in the
following virial identity satis ed by the nite energy solutions of the eld equations
sin4 F
r4
The positivity of all terms in the previous relation shows that the existence of d = 5
Skyrmions can be attributed to a balance between: the attractive interaction provided by
the usual kinetic term in the Skyrme action together with the potential (left hand side
terms); and a repulsive interaction provided by the sextic term, plus the centrifugal force
in the rotating case (right hand side terms). Also, one can see that, as anticipated above,
the contribution of the quadratic term occurs with an overall w2factor only, and is not
mandatory for the existence of solutions.
The Skyrme Lagrangian (2.2) contains four input parameters i. However, the constant
multiplying the quadratic term can be taken as an overall factor for the Skyrme action.
Also, the equation for F is invariant under the transformation r !
r, 0= 1 !
2= 1 !
2=( 2 1), 3= 1 !
3=( 4 1) together with w ! w= , which can be used to x
the value of one of the constants 0; 2 or 3
. Then following the d = 4 case, we de ne
a characteristic length L and mass MS of the Skyrmion system as given by the constants
multiplying the quadratic and potential terms, with
L
r
1
0
;
MS
2
1 ;
0
(3.12)
the numerical results being obtained in units set by L and MS. However, to avoid cluttering
the output with a complicated dependence of L; MS we shall ignore these factors in the
displayed numerical results.
The problem still contains two free constants which multiply the quartic and sextic
terms in the Skyrme action. Moreover, in the presence of gravity, one extra parameter
occurs. Therefore, the determination of the domain of existence of the solutions would be
a lengthy task. In this work, in order to simplify the picture, we have chosen to solve a
model without the quartic term and with a unit value for the parameter multiplying the
sextic term. Thus, all reported numerical results below are found for the following choice
of the coupling constants
0 = 1 = 3 = 1;
2 = 0 :
(3.13)
To provide evidence that this choice does not restrict the generality of our results, we
present, in appendix A, solutions including the presence of a quartic term in the action,
2 6= 0, which indeed does not appear to a ect the qualitative properties of the solutions.
{ 9 {
The function F (r) satis es a nonenlightening second order di erential equation which we
shall not display here (its ! = 0 limit can be read o
by setting N (r) =
(r) = 1 in (4.8)).
This equation does not seem to possess exact solutions and it is solved numerically with
the boundary conditions7 F (0) =
regularity requirements.
and F (
1
) = 0, which follow from
nite energy and
The asymptotics of F (r) can be systematically constructed in both regions, near the
origin and for large r. The corresponding expression for small r is
F (r) =
+ f1r + f3r3 + O(r5); with f3 =
f1
0 + 1(2f12 + !2) + 6f14( 2
12[ 1 + 6f12( 2 + f12 3)]
3!2)
;
in terms of a single undetermined parameter f1 < 0. One notice that the energy density
at r = 0 is nonzero, with
while the angular momentum density j, given by (3.7), vanishes as O(r2).
An approximate solution valid for large r can be written in terms of the modi ed Bessel
function of the second kind, K,
F (r)
c
r
"
K 2; r
r
0
1
!2
#
c
r
2
e r
q 01 !2
r3=2
+ : : : ;
(3.16)
(with c a constant), which shows the existence of an upper bound on the scalar eld
frequency, !
p
0= 1 = 1=L, the solutions becoming delocalized for larger values of !.
Therefore, similar to other examples of spinning scalar solitons (e.g. [34, 35]), the presence
of a potential term in the action, 0 6= 0, is a prerequisite for the existence of nite mass
solutions. Note, however, that we have found numerical evidence for the existence of static
solitons with V = 0, which decay as 1=r3 at in nity.
Solutions interpolating between (3.14) and (3.16) are easily constructed 
gure 1. In
our approach, the control \shooting" parameter is f1 which enters the near origin
expansion (3.14)  see gure 1, inset of right panel. For a given frequency !, a single nodeless
solution is found for a special value of f1. The pro les of the energy density and of the
angular momentum density of typical solutions are shown in
gure 1, including the static
one, which has ! = 0. In
gure 2 we display the total mass and angular momentum of
the spinning Skyrmions as a function of their frequency. One can see that both quantities
increase monotonically with !, with a smooth ! ! 1 limit which maximizes the values of
mass and angular momentum. Also, at least for the considered values of the coupling
constants, the Skyrmions are never fast spinning objects, with a reduced angular momentum
j always much smaller than one.
7Solutions with F (r) interpolating between k (with k > 1) at r = 0 and F (
1
) = 0 do also exist.
However, they are more massive and likely to be unstable.
4.5
2
ω=0.75
ω=0.25
1
1.2
1
f1.4
0
2
r
background are shown together with their energy density. Right panel: the distribution of the
angular momentum density for spinning Skyrmions. The inset shows the values of the parameter
f1 which enters the small r expression of the solutions.
)
r
(
F
ω=0.95
ω=0
60
T t
T30
0
0
ω=0
Skyrmions on a at spacetime background are shown as a function of their frequency !.
The above solutions possess gravitating generalizations, which are found by solving the
Skyrme equation (2.3) together with the Einstein equations (2.13). A suitable metric for
spherically symmetric con gurations reads
+r2(d 2 +sin2 d'12 +cos2 d'22) N (r) 2(r)dt2 ; where N (r)
1
the function m(r) being related to the local massenergy density up to some overall factor.
For static, spherically symmetric solutions, the scalar ansatz is still given by (2.16) with
m(r)
r2 ;
(4.1)
Ls =
r
3
This form of the system allows us to derive, following [55, 56] a generalization of the at
spacetime virial identity (3.11). Following the same reasoning as in section 3.2.2, we nd
that the nite energy solutions satisfy the integral relation
which clearly shows that nontrivial gravitating solutions with
nite mass cannot exists in
a model without the sextic term. Indeed, in that case the right hand side would vanish
and all terms in the integrand of the left hand side of (4.4) would either vanish or be
strictly positive, making the equality impossible for a nontrivial con guration. Observe
that turning on gravity adds an extra attractive term, in addition to those provided by the
quadratic and potential terms.
By using the same dimensionless radial coordinate and rescaling as in the
nongravitating case (together with m ! m=L2), one nds that the gravitating system possess
one extra dimensionless coupling constant
2 = 4 G 13=2= 10=2 :
Then the Einstein equations used in the numerics reduce to (recall that we set 0 =
1 = 1 and 2 = 0, 3 = 1)
The equations of the model can also be derived from the reduced Lagrangian:
and
where
(4.2)
(4.3)
HJEP1(207)3
(4.5)
(4.6)
(4.7)
(4.8)
m0 =
0 =
3
r
2
+ 3
together with an extra constrain equation. The function F (r) satis es the 2nd order
sin F
N 1 + 6 2 sinr22 F + 6 3 sinr44 F
1 +
1
r2 sin2 F P1
F(r)
where
m(r)
1
α/
panel: the total mass M , the control parameter f1 and the value of the metric function
origin are shown as a function of coupling constant, , for spherically symmetric Skyrme stars.
P1 =
3
2
sin(2F )
rF 0 rN 0 + N
3 +
;
r 0
P2 = cos(3F ) + cos F (2N r2F 02
P3 = cos(3F ) + cos F (8N r2F 02
1) + 2rF 0 sin F rN 0 + N
1 +
;
(4.9)
1) + 4rF 0 sin F rN 0 + N
1 +
:
r 0
r 0
These equations are solved by imposing the boundary conditions8
at the origin (r = 0), while at in nity, the solutions satisfy
N (0) = 1;
(0) = 0 > 0; F (0) =
;
N ! 1;
! 1; F ! 0 :
The properties of the spherically symmetric Skyrme stars can be summarized as follows.
For all studied cases, m(r), (r), and F (r) are monotonic functions of r,9 the pro le of a
typical solution being presented in
gure 3 (left panel). For small values of
there is a
fundamental branch of solutions that reduces to the at space Skyrmion as
increases, the mass parameter M decreases, as well as the value (0). The solutions exist
up to a maximal value
max of the parameter . At the same time, the absolute value of the
\shooting" parameter f1 increases with . We found that a secondary branch of solutions
emerges at
max, extending backwards in
branch, both (0) and f1 decrease as

gure 3 (right panel). Along this second
decreases, while the value of the physical mass
! 0. When
8An approximate form of the solutions compatible with these conditions can easily be constructed. For
example, the small r expansion contains two essential parameters, F (0) and
(0).
9Excited solutions with nodes in the pro le of F , which would therefore be a nonmonotonic function,
are likely to exist as well.
(4.10)
(4.11)
M= 2 strongly increases. Some understanding of the limiting behaviour can be obtained
by noticing that the
! 0 limit can be approached in two di erent ways, as G ! 0 ( at
space, rst branch) or as 1 ! 0 (second branch). Then we conjecture that the limiting
solution on the upper branch corresponds to a gravitating model without the F 2term in
the Skyrme Lagrangian.
d = 5 rotating spacetimes generically possess two independent angular momenta. Here,
however, we focus on con gurations with equalmagnitude angular momenta which are
compatible with the symmetries of the matter energymomentum tensor we have chosen.
A suitable metric ansatz reads (note the existence of a residual gauge freedom which will
HJEP1(207)3
ds2 =
+ g(r)d 2 + h(r) sin2 [d'1
W (r)dt]2 + h(r) cos2 [d'2
W (r)dt]2
(4.12)
h(r)] sin2 cos2 (d'1
d'2)2
b(r)dt2 :
For such solutions the isometry group is enhanced from Rt
U(
1
)2 to Rt
U(
2
), where Rt
denotes the time translation. This symmetry enhancement allows factorizing the angular
dependence and thus leads to ordinary di erential equations.
The angular momentum and energy densities are given by j = =
T't1
sin2
(!
=
W )
b(r)
T't2
cos2
The complete ansatz, (4.12) and (2.16), can be proven to be consistent, and, as a
result, the EinsteinSkyrme equations reduce to a set of ve ODEs (in the numerics, we x
the metric gauge by taking g(r) = r2).
We seek asymptotically at solutions, subject to the following boundary conditions as
r ! 1: f = b = g(r)=r2 = 1 and F = W = 0. The total (ADM) mass M and angular
momenta J1 = J2 = J , are read o from the asymptotic behaviour of the metric functions,
gtt =
1 +
8GM
3 r2 + : : : ; g'1t =
4GJ
r2 sin2
4GJ
r2 cos2
+ : : : ; g'2t =
+ : : : : (4.13)
The behaviour of the metric functions at the origin is f = 1, b = b0 > 0, g(0) = 0, together
with W = W0 > 0, while F = , as in the probe limit.
0.9
0.8
b(r)
0
2
3
In gure 4, we display the pro les of the a typical spinning Skyrme star. The
corresponding distribution for energy and angular momentum densities look similar to those
shown in
gure 1 for the probe limit. The mass/angular momentum of solutions vs:
frequency are given as a limiting curve in
gure 11, for a particular value of . Note that
similar to the probe limit, both quantities monotonically increase with !. Also, in the limit
! ! 1, the solutions are still localized, without any special features, while they disappear
for ! > 1.
Finally we emphasise that all solitons in this work, gravitating or otherwise, possess
unit topological charge, as expected.
5
BHs with Skyrme hair
BH generalizations are generically found for any regular solitoniclike gravitating con
guration, at least for small values of the horizon radius rH . In this section we shall show this
trend remains true for the EinsteinSkyrme model discussed herein, as con rmed by our
numerical results.
5.1
5.1.1
Spherically BHs
The probe limit  Skyrmions on a Schwarzschild BH background
Similarly to the d = 4 case, it is useful to consider rst the probe limit and solve the Skyrme
equations on a d = 5 SchwarzschildTangherlini BH background [57]. The corresponding
line element is given by (4.1) with N (r) = 1
rH2 =r2 and (r) = 1, where rH > 0 the event
horizon radius. The approximate form of the solution close to the horizon reads
in terms of the \shooting" parameter f0, with
F (r) = f0 + f1(r
rH ) + O(r
rH )2;
f1 =
sin(f0)
0rH6 + 3 cos(f0) r
H 1 + 2 3 sin4 f0
4
2rH (rH4 1 + 6 3 sin2 f0)
:
(5.1)
(5.2)
1.5 0
1
2
radius rH , for Skyrme probe solutions on a Schwarzschild BH background.
The mass of the solutions is still computed from (3.9), with the corresponding curved
spacetime expressions and r = rH as a lower bound in the integral (the same holds for a
MP background).
with F (rH ) !
F (rH ) !
=2.
The results of the numerical integration are shown in gure 5 (note that the typical
pro le of the function F (r) is similar to that exhibited in the gravitating case, cf. gure 6).
One can see that the solutions exist up to a maximal horizon radius rH of the Schwarzschild
background, with a double branch structure for a range of rH . The solutions of the
fundamental branch (with label 1 in gure 5) terminate in the at spacetime solitons as rH ! 0,
in that limit. Along this branch, the mass of the solutions decreases with
increasing rH . The second branch (with label 2 in
ues again to rH ! 0, in which limit however, the mass of the solution M diverges, while
gure 5) starts at r(max) and
contin
H
We remark that the Skyrme solutions on a spacetime geometry with an event horizon
possess a noninteger topological charge,
B =
1 + 2 cos2 1
2
F (rH )
sin4 1
2
F (rH ) ;
(5.3)
belonging to the interval 1=2 < B
1. In fact, the above expression holds for all BH solutions in this work, including the rotating ones.
5.1.2
Including backreaction
The inclusion of gravity e ects is straightforward. The BH solutions are constructed within
the same ansatz used for solitons. They satisfy the following set of boundary conditions at
the horizon (which is located at r = rH , with 0 < rH 6 r < 1)
N (rH ) = 0 ;
(rH ) =
H > 0 ; F (rH ) = f0 :
(5.4)
2
F(r)
1
n
irzo
o
t
n
e
1 ev
0
0
10000
2
α
/
M
1000
0
m(r)
hair. Right panel: the values, at the horizon, of the Ricci scalar R and of the metric function
are
shown as functions of the horizon radius rH , for several values of .
100
50
0
0
3
2
H
)
r
(
F
α=0.1
α=0.05
(rH0.5
σ
0
0
H
functions of the horizon radius rH , for several values of .
The far eld behaviour is similar to that in the solitonic case. We note that these BHs
possess a Hawking temperature TH and a horizon area AH , which read
1
4
TH =
N 0(rH ) (rH ) ; AH = 2 2rH3 :
The pro le of a typical BH solution is shown in gure 6. The behaviour of the solutions
H
as a function of rH is presented in
gure 7, for several values of the coupling constant .
The properties of the spherically symmetric BHs can be summarized as follows. Starting
from any regular solution, i.e. a Skyrme star, with a given
and increasing the event
horizon radius, one
nds a
rst branch of solutions which extends to a maximal value
r(max). This maximal value decreases with increasing . This branch is the backreacting
counterpart of the corresponding one in the probe limit. The Hawking temperature and the
value of F (rH ) decrease along this branch, while the mass parameter increases; however,
the variations of the mass and of (rH ) are relatively small.
Extending backwards in rH , we nd a second branch of solutions. This second branch
extends up to a critical value of horizon radius r(cr) where an essential singularity seems to
occur. An understanding of the limiting solutions requires a reformulation of the problem
with a di erent coordinate system [58] which is beyond the scope of this work. Here we
note that the value of (rH ) on this branch decreases drastically and appears to vanish
as rH ! rH(cr). As a result, the Ricci scalar evaluated at the horizon strongly increases in
that limit. However, the mass remains nite, while the Hawking temperature goes to zero.
The pro le of the F function does not exhibit any special features in the limit, starting
always at some value F (rH ) >
=2. This special behaviour on the second branch can
partially be understood as a manifestation of the divergent behaviour we have noticed in
the probe limit.
We also mention that for the region of the parameter space where two di erent solutions
exist with the same mass, the event horizon area (i.e. the entropy) is always maximized
by the fundamental branch of the BH, see gure 7 (left panel). Thus we expect the upper
branch solutions to be always unstable.
Spinning BHs
The probe limit  Skyrmions on a MyersPerry BH background
The static hairy BHs we have just described possess rotating generalizations. However,
before considering solutions of the full EinsteinSkyrme system, it is again useful to consider
rst the probe limit and to solve the matter eld equations on a spinning BH background.
The corresponding BH is a d = 5 MyersPerry (MP) solution [59] with two equalmagnitude
angular momenta. Such a BH can be expressed as a particular case of the ansatz (4.12),
r
r
r
and it is parameterised in terms of the event horizon radius rH and the horizon angular
velocity
H , which are the control parameters in our numerical approach. For
completeness, we include the expression of quantities which enter the thermodynamics of a MP BH
3 r
8(1
1
2 rH q
2
H
2 r2 ) ;
H H
1
1
2 2H rH2 ;
2 r2
H H
1
J (MP ) = J (MP ) = J (MP ) =
A(HMP ) =
2
q
1
2 2r3
H
Here it is important to note the existence of a maximal size of the horizon radius rH for
a given value of horizon angular velocity
H . This corresponds to a zero temperature BH
with r(max) = 1=(p2 H ). There, the reduced angular momentum (3.10) approaches its
H
maximal value jMP = 1.
n
e
v
e
1.5
Ttt/4
r
j
3
together with the corresponding energy and angular momentum densities.
At in nity, the decay of the eld is still given by (3.16), such that F ! 0: Remarkably,
the assumption of existence of a power series expansion of F (r) as r ! rH implies that,
similar to other hairy rotating BH solutions [44, 66, 67], the synchronization condition
! =
H
near the horizon (with 2 = 0), which, up to order O(r
rH ) reads:
necessarily holds. This condition is also implied by the regularity of energy and angular
momentum densities as r ! rH . Then the function F (r) possess an approximate solution
F (r) = f0+
(1 rH2 !2)
4rH (2rH2 !2
1)[rH2 1 + 2 3(3 2rH2 !2) sin4 f0]
2 0 sin f0 + 1rH4 (rH2 !2 1) sin(2f0)+12 3(rH2 !2 1) cos f0 sin f05 (r rH ) ;
all higher order coe cients being determined by F (rH ) > 0.
The pro les of a typical solution on a given MP background are shown in gure 8.
The dependence of the properties of the solutions on the horizon size, as given by A(HMP )
and reduced angular momentum j(MP ), is shown in
gure 9. The basic picture found in
the static case is still valid here, with the existence of two branches of solutions for a given
BH background. The fundamental branch emerges from the at spacetime solitons, while
the mass and angular momentum of the solutions appear to diverge as the at spacetime
limit (rH ! 0) is approached, along the second branch.
We remark that no Skyrme solutions exist on a fast rotating MP background, i.e.
with j(MP ) close to unity. Also, one can see that the mass branches of spinning solutions
exhibit a `loop', when considered as a function of horizon properties (or reduced angular
(5.7)
(5.8)
momentum) of the MP background.
5.2.2
Including backreaction
The con gurations described in the last subsection survive when taking into account their
backreaction on the spacetime geometry. The spinning BHs are constructed within the
same ansatz as for the spinning solitons discussed in section 3.2.2 (in particular we set
M
400
200 0
ω=0.1
100
50
0
0
0.3
400
100
50
0
0
M =
J(S) +
M(S) ;
3
where M(S) and J(S) are the mass and angular momentum stored in the Skyrme eld
while their mass and angular momentum are read from the far eld expansion (4.13). As
usual, the temperature, horizon area and the global charges M; J are related through the
Smarr mass formula outside the horizon,
M(S) =
3 Z p
2
are shown as a function of horizon area and of the reduced MP angular momentum.
again g(r) = r2). However, they possess an horizon which is a squashed S3 sphere. The
horizon resides at the constant value of the radial coordinate r = rH > 0, and it is
characterized by f (rH ) = b(rH ) = 0. Restricting to nonextremal solutions, the following
expansion of the metric functions holds near the event horizon:
f (r) = f1(r rH ) + f2(r rH )2 + O(r rH )3; h(r) = hH + h2(r rH ) + O(r rH )2;
b(r) = b1(r rH ) + b2(r rH )2 + O(r rH )3; W (r) =
H + !1(r rH ) + O(r rH )2;
while F (r) = f0+f1(r rH )+: : : . For a given event horizon radius, the essential parameters
characterizing the event horizon are f1; b1, hH ,
H and !1 (with f1 > 0; b1 > 0), which
x all higher order coe cients in (5.9). The construction of the approximate nearhorizon
solution shows that, as expected, the synchronization condition (5.7) still holds in the
backreacting case.
As for a general MP BH, the (constant) horizon angular velocity
H is de ned in terms
of the Killing vector
solutions within the ansatz (4.12), the horizon angular velocities are equal, 1 =
2 =
H .
The Hawking temperature TH and the area AH of these BHs are xed by the near horizon
data in (5.9), with
TH =
p
b1f1
4
; AH = 2 2phH rH2 ;
(5.9)
(5.10)
(5.11)
gdrd d'1d'1 T
t
t
31 Tii ; J(S) =
Z p
gdrd d'1d'1T't i :
(5.12)
1
0.75
0.5
ω=0
ω=0.95
1
2
1/TH
3
panel: the (horizon areatemperature) diagram is shown for two sets of solutions.
In our approach, the input parameters are the coupling constant , the event
horizon radius rH and the horizon angular velocity
H (or equivalently, the
eld frequency
!). Physical quantities characterizing the solutions are then extracted from the numerical
solutions.
The pro les of a typical spinning BH with Skyrme hair are shown in gure 10 (left
panel). Some basic properties of the solutions are similar to those found in the probe limit.
For example, the scalar eld is always spatially localized within the vicinity of the horizon,
the distribution of mass and angular momenta densities being similar to that in gure 8.
The emerging global picture can be summarized as follows. For all values of the
parameters ; rH that we have considered, the static BHs are continuously deformed while
increasing gradually the parameter !. Similarly to the solitonic case, the solutions stop to
exist for ! > 1. Moreover, all BHs studied so far have a reduced angular momentum (3.10)
much smaller than one.
When taking instead a xed value of ! and varying the horizon parameter rH , our
results show that, for any 0 6 ! 6 1, a double branch structure of solutions exists,
characterized by two particular values of the horizon radius rH . The rst (or main) branch
exists for 0 6 rH 6 r(max), emerging from the corresponding (gravitating) Skyrme soliton
H
in the limit rH ! 0. The second branch exist for rH(c) 6 rH 6 r(max) approaching a critical
H
con guration as rH ! rH(c). This critical solution possesses nite global charges, a nonzero
horizon area while its Hawking temperature vanishes. However, it inherits the pathologies
of the static limit, e.g. the Ricci scalar appears to be unbounded on the horizon. The
dependence of the horizon area AH nf the (inverse of the) temperature TH is shown in
gure 10 (right panel), where we compare the results for static solutions with those for
BHs close to the maximal value of !. One notices that the horizon size remains nite as
the critical solution is approached.
In
gure 11 we exhibit the domain of existence of hairy BHs, in a M (!) (and J (!))
diagram for
= 0:01, the only value of the coupling constant we have investigated in a
systematic way. This domain, in the M ! diagram, has an almost rectangular shape, and
is delimited by four curves: the set of static BHs (! = 0), the set of Skyrme stars, the set
of maximal mass solutions, and nally the set of maximally rotating BHs with ! = 1.
H
B
H
critical HBHs
Skyrme Stars
0.5
ω
2/Jα1000
0
0
0.25
0.75
1
0.5
ω
critical HBHs
frequency diagram (left panel) and as an angular momentumfrequency diagram (right panel).
6
Conclusions
In this work we have considered an extension of the Skyrme model to ve spacetime
dimensions and investigated the basic properties of its codimension1 solutions.
Concerning the model, two salient properties of gravitating Skyrme systems in d = 3 +
1, can be used to motivate its study. The rst is that the solutions feature BHs hair and the
second is that in the gravity decoupling limit the solutions are topologically stable. In 3 + 1
dimensions, gravitating Skyrmions share the rst property with gravitating YangMills [60],
which support hair but in the
at spacetime limit disappear. The second property they
share with that gravitating YangMillsHiggs [61{63] system which supports topologically
stable monopoles. The latter persisit in the gravity decoupling limit, whence one notes the
closer similarity of gravitating Skyrmions with monopoles in 3 + 1 dimensions.
In spacetime dimensions higher than d = 3+1, the situation is rather more complicated
because of the restriction set by the Derrick scaling [9] requirement for the niteness of the
energy, as can be seen from the nonexistence of nite energy gravitating solutions of the
usual quadratic YangMills system in 4 + 1 dimensions [52]. In the YangMills (YM) case,
higher order YM densities, e.g. extended YM terms (eYM) like F (2p)2 must be included
to satisfy this requirement.10 In d = 6 and 7, this was done in [53] by adding11 the
F (
4
)2 eYM term to the usual F (
2
)2. Subsequently the solutions for the same model as
in [53] were constructed in [54] for d = 5, which displayed some peculiar features that we
have encountered in the d = 5 Skyrme case at hand. Not surprisingly the same holds for
gravitating Skyrme systems in dimensions greater than 4+1, namely including higher order
kinetic (Skyrme) terms, as we have done in this paper. But unlike in the YM case, where
the gravity decoupling solutions are topologically stable in even spacetime dimensions only,
in the Skyrme case they are stable in all dimensions, like in the monopole case.
10As found in [64, 65], the inclusion of a F (
4
)2 term in the YangMills action (which is optional in this
case) leads to a variety of new features also for d = 4, e.g. the existence of stable hairy nonAbelian BHs.
11In these dimensions, it is possible to add the F (6)2 eYM term too but that was eschewed.
More
interestingly in d = 6, there exist topologically stable eYM instantons so that those solutions persists in
the gravity decoupling limit. This feature persists in all even d = 2n > 4.
Concerning the explicit solutions described in this paper, we have considered both at
spacetime (Skyrmions) con gurations and selfgravitating solutions (Skyrme stars and BHs
with Skyrme hair). Overall, we have unveiled a rich and involved space of solutions.
In the spherically symmetric case, the pattern of the d = 4 solutions is recovered, with
a branch of gravitating Skyrmions emerging from the at space/Schwarzschild background
solutions. A secondary branch of solutions is also found, which, however, possesses a
di erent limiting behaviour than in the d = 4 case.
On the critical behaviour of spherically symmetric solutions as a function of
or rH ,
we remark that some features resemble the case of a higher dimensional gravitating
nonabelian system with higher derivatives terms in addition to the usual F (
2
)2 one [54]. The
clari cation of the critical behaviour therein has required a reformulation of the problem
with techniques bases on a
xed point analysis of nonlinear ODEs [58]. We expect that a
similar approach would help to clarify the critical behaviour of the gravitating solutions in
this work.
The spinning hairy BH we have reported in this paper are the
rst example of a
spinning BH with Skyrme hair, since the corresponding d = 3 + 1 solutions have not yet
been constructed. One salient feature of these rotating BHs with Skyrme hair is that
they possess a static limit. Whereas this is expected, since static BHs with Skyrme hair
are known in d = 3 + 1 spacetime dimensions, it contrasts in a qualitative way with the
behaviour of other BHs with scalar hair, namely Kerr BHs with scalar hair [44{46] or MP
BHs with scalar hair [66, 67]. It is therefore of some interest to expand on the comparison
between these two models, since they are both examples of BHs with scalar hair:
Skyrmions on Minkowski spacetime are topological solitons, in view of their asymptotic
boundary conditions. Qballs [42], on the other hand, which arise in models of
selfinteracting complex scalars elds, but with standard kinetic terms, are perhaps the
simplest example of a nontopological soliton. For the latter, the complex nature
of the scalar eld is crucial to satisfy Derrick's theorem, allowing for an underlying
harmonic time dependence of the scalar
eld but that vanishes at the level of all
physical quantities, thus yielding static or stationary lumps of energy.
Flat spacetime spinning Skyrmions have a static limit; thus they carry an arbitrarily
small angular momentum (for given topological charge). Spinning Qballs, on the
other hand, have their angular momentum quantised in terms of their charge [34, 35],
which in this case is a Noether charge, due to a U(
1
) global symmetry. Thus, they
have a minimum angular momentum, for given Noether charge, and the spinning
solutions are not continuously connected to the static ones.
When minimally coupled to Einstein's gravity, selfgravitating Skyrmions become
Skyrme stars. But the structure of the model to obtain these solutions remains
the same, namely the key higher order kinetic term is still mandatory. When
minimally coupled to Einstein's gravity, back reacting Qballs become boson stars. But
gravity can replace part of the key structure of the
at spacetime model: one can
get rid of the selfinteractions potential and keep only a mass term [68, 69], as the
HJEP1(207)3
nonlinearities of Einstein's gravity are su cient to counter balance the dispersive
nature of the scalar eld and create equilibrium boson stars.
Likewise, Skyrmions, Skyrme stars can rotate slowly and connect to the static limit,
whereas rotating boson stars form an in nite discrete set of families disconnected
from static boson stars [35, 70], for any model, with or without selfinteractions, in
any spacetime dimension.
Static Skyrme stars admit placing a BH horizon at their centre, yielding static BHs with
Skyrme hair, both in d = 4 [24, 26] and d = 5 (and likely in other dimensions).
HJEP1(207)3
Static boson stars do not admit placing a BH horizon at their centre, as shown by
the nohair theorem in [71].
In spite of all these di erences, the spinning BHs with Skyrme hair that we have found
in this paper rely on precisely the same condition that the Kerr BHs with scalar hair or MP
BHs with scalar hair, the latter being the hairy BH generalisation of spinning boson stars
in d = 4 and d = 5. This condition is the synchronisation of the phase angular velocity
of the Skyrme eld and the angular velocity of the horizon, eq. (5.7). This is yet another
example for the universality of this condition in obtaining spinning hairy BH solutions.
Finally, let us remark that possible avenues for future research include: i) the
investigation of stability of the considered con gurations (based on the analogy with the d = 4
case, we expect some of the gravitating solutions to be stable); ii) the construction of less
symmetric Skyrmions; the simplest case would be the (higher winding number) axially
symmetric solutions and con gurations with J1 6= J2. Solutions with discrete symmetry
only are also likely to exist in this model; iii) the construction of d > 5 generalizations: in
appendix B we present a general framework in this direction.
Acknowledgments
C.H and E.R. gratefully acknowledge support from the FCTIF programme. This work was
also partially supported by the H2020MSCARISE2015 Grant No. StronGrHEP690904,
the H2020MSCARISE2016 Grant No. FunFiCO777740, and by the CIDMA project
UID/MAT/04106/2013.
A
Solutions with a quartic term
The reported numerical results have 2 = 0. To test their generality, we have also studied
how the inclusion of a quartic term a ects the properties of the solutions, in a number
of cases.
Starting with the solutions in at spacetime background, we display in gure 12 the
mass M as a function of 2 for Skyrmions with several values of the frequency ! (note
that all solutions displayed in this appendix are for
0 =
1 =
3 = 1). As expected,
one can see that the presence of quartic term in the Skyrme action increases the mass of
the solutions, and for ! 6= 0, the same trend applies for the angular momentum. Observe
2
λ
2
4
6
HJEP1(207)3
Skyrmions, with di erent frequencies.
ω=0.5
ω=0
λ0=λ1=λ3=1
1
0.75
)
0
σ( 0.5
0.25
1000
2
α
/
M500
0
0
λ0=λ1=λ3=1
λ2=1
λ2=0
λ2=1
α
0.1
λ0=λ1=λ3=1
λ2=0
500
0
0
λ2=2
of the coupling constant , for Skyrme stars in our model with di erent values of 2
.
that the M (
2
)function is almost linear for small frequencies. It is also interesting to
mention that the ! = 0 Skyrmions with large enough values of 2 are well approximated
by the selfdual solution (3.2), the contribution of the quartic term dominating the system
in this case.
Turning now to gravitating solutions, we display in gure 13 families of spherically
symmetric Skyrme stars with several values of 2
. One can see that the picture found
in section 4.1 appears to be generic, with the occurrence of two branches of solutions in
terms of . Also, one notices the existence of a maximal value of , which decreases with
increasing
2, while the limiting behaviour on the second branch is similar to that found
for solutions without a quartic term.
Finally, the same conclusion is reached in the presence of an horizon, cf. gure 14,
where we show the (rH ; M ) and (rH ; (rH )) diagrams for spherically symmetric BHs with
Skyrme hair with a xed value of
and three values of 2
.
To summarize, the presence of a quartic term in the Skyrme action does not seem to
lead to new qualitative features, at least for the range of parameters considered.
0.75
0.25
α=0.025
λ1=3
0.1
r
H
H
λ1=1
BHs with Skyrme hair, as a function of the horizon radius for several values of 2 and a xed .
B
O(D + 1) Skyrme models on a RDEuclidean space
The Skyrme model [1, 2] in 3 + 1 (i.e., D = 3) dimensions12 is a nonlinear chiral eld theory
which supports topologically stable solitons in the static limit. These solitons, which are
called Skyrmions, describe baryons and nuclei. In its original formulation [1, 2], the model
is described by an SU(
2
) valued
eld U . The Skyrmions are stabilised by a topological
charge which is characterised by the homotopy class 3(SU(
2
)) = Z.
Alternatively, the chiral matrix U can be parametrised as U =
a a (a = 1; : : : ; 4)
and its inverse as U 1 = U y =
a ~a, where a and ~a are the chiral representations of the
algebra of SU(
2
). The scalar a is subject to the constraint a a = 1, such that it takes
its values on S3, the latter being parametrised by the angles parametrising the element U
of the group SU(
2
). The homotopy class in terms of a is now
3(S3) = Z. This is the
parametrisation that will be adopted here.
The parametrisation of the Skyrme scalar in terms of the chiral eld U is peculiar to
D = 3. Indeed in D = 2 the famous BelavinPolyakov vortices [72] of the O(
3
) sigma
model on R2 are parametrised by the S2 valued scalar13 subject to a a = 1 (a = 1; 2; 3),
pertaining to the homotopy class 2(S2) = Z.
In all dimensions D > 4, these low dimensional accidents are absent so Skyrme models
are de ned as the O(D + 1) sigma models on RD, described by the Skyrme scalar
a
a = 1; 2; : : : ; D + 1 subject to the constraint a a = 1.
j aj2 = 1 ;
a = 1; 2; : : : ; D + 1 ;
(B.1)
pertaining to the homotopy class D(SD) = Z.
namely in terms of the CP1
and CP1 sigma models via a = zy az.
12In this section, we shall take d = D + 1, i.e. D denotes the number of space dimensions.
13Indeed in this low dimensional case too, there is an alternative parametrisation of the \Skyrme scalar",
eld z (
= 1; 2) subject to zyz = 1, by virtue of the equivalence of the O(
3
)
one de nes the pform
which is the pfold product of
(
1
) =
i1; i2; : : : ; ip.
(p) = i1i2:::ip
a1a2:::ap d=ef [ai11 ia22 : : : iapp] ;
In this notation, the kinetic terms H
(p;D) are concisely de ned as
a, totally anitisymmetrised in the indices
i
H
(p;D) = j (p)j2
p=D
X
p=1
H
(D) =
pH
(p;D) + 0V ;
%~ = %(D) +
1
a 2
j j
and the topological charge density (up to normalisation) is
%(D)
' "i1i2:::iD "a1a2:::aD+1 ia11 ia22 : : : iaDD aD+1
aD aD+1 :
It is well known that (B.7) is essentially total divergence and hence quali es as a
topological charge density. To see this, subject the quantity
In any given dimension D, the energy density functional H(D) can be endowed with a
\potential" term, e.g. the \pion mass" type potential
V = 1
D+1 ;
and D possible \kinetic" terms H
(p;D), which are de ned as follows.
Employing the shorthand notation for the 1form
; i = 1; 2; : : : D ;
(B.2)
(B.3)
(B.4)
(B.5)
(B.6)
(B.7)
(B.8)
(B.9)
such that only the square of any `velocity' eld ia occurs in H
In this notation, the most general energy density functional in any dimension D is
to variations w.r.t. the scalar
eld
a, taking account of the Lagrange multiplier .
The result is 0 = 0, as expected from a density which is total divergence.
Alterna
tively, one can employ a parametrisation of
a that is compliant with the constraint
(B.1), e.g. when employing a particular Ansatz. In that case %(D) itself would take an
explicitly total divergence form [73].
To express the Bogomol'nyi inequalities of this system, we de ne the Hodge dual of
the (D
p)form
(D
p), which is the pform as
? (p) d=ef ? ia11::::::iapp =
1
p)! "i1:::ipip+1:::iD "a1:::apap+1:::aDaD+1 iapp++11::::::iaDD aD+1 :
For any given D, one can now state the D inequalities in terms of (B.4) and (B.8), each
labelled by p, as
2
(p)
(D 2p) ? (p)
> 0 ;
p = 1; 2; : : : D ;
is a constant with dimension L 1 compensating for the di erence between the
dimensions of (p) and ? (p), which are di erent in the general case D 6= 2p.
It is clear from the de nitions (B.4) and (B.8) that the crossterms in the inequalities
(B.9), for each p, is proportional to the topological charge density (B.7). It follows that
the p inequalities (B.9) lead to the required lower bound on the energy
H
(D) > %(D) =)
Z
RD H
(D) >
%(D) ;
Z
RD
(B.10)
provided of course that the potential (B.2) is by de nition positive de nite.
The best known examples of such topologically stable Skyrmions are the
BelavinPolyakov (selfdual) vortices [72] on R2, and the familiar Skyrmions [2] on R3. In the
D = 2 case the most general system (B.6) is that with all coe cients 0; 1; 2 present and
in the D = 3 case the most general one is that with all coe cients 0
; 1
; 2
; 3 present.
Of course, which terms must be retained in each case is governed by the requirement of
Derrick scaling.
Of the D inequalities (B.9) only one can be solved with power decaying solutions at
r ! 1 on RD, namely the one for which D = 2p when the dimensional constant
appear. Furthermore in this case the model (B.6) must consist exclusively of the p = D=2
does not
term in the sum, i.e. p = 0 for p 6= (D=2), otherwise the system will be overdetermined.
Such a model has solutions that saturate the topological lower bound (B.10). In all other
which includes Skyrmions in all odd Euclidean14 dimensions RD.
cases, when
p 6= 0 for p 6= (D=2), the solutions cannot saturate the lower bound (B.10),
In particular in the important case of R3, there exist no solutions to rstorder
(anti)selfduality equations saturating the lower bound. Exact solutions to the secondorder
equations can be constructed only numerically and not in closed form. However,
approximate solutions on R3 in closed form are known. For example the rational map ansatz [77],
and the AtiyahManton [78, 79] construction where the holonomy of the YangMills
instantons on R4 gives a good approximation for the Skyrmion on R3. This last approach
is extended to give an approximate construction for the Skyrmion on R7 by exploiting the
holonomy of the YangMills instantons on R8 in [80], which possible be extended to higher
The rstorder (anti)selfduality equations for the pSkyrme system H
(p;D) = j (p)j2
? (p)
(B.11)
can be solved in closed for subject to no symmetries only in the case D = 2. In that
case [72] the equations (B.11) reduce to the CauchyRiemann equations. In all higher
dimensions D = 2p > 2, only solutions of the system subject to radial symmetry are
known, and the form factors F (r) for all (p; D = 2p) are given by the same function (3.2).
14The (anti)selfduality equations resulting from (B.9) on SD for odd D, namely
(p)
(D 2p) ? (p) = 0
can be solved, since in that case the dimensional constant is absorbed by the radius of the sphere. See
e.g. [74{76].
It is interesting to remark here that this situation hold also in the case of the p hierarchy
of BPST instantons [81, 82] on R4p.
It is interesting to push the analogy between the selfduality equations of the
pYangMills systems on R4p and the pSkyrme systems on R2p. In both cases the spherically
symmetric solutions are described by the same radial function in all dimensions. In both
cases, these equations become more overdetermined with increasing dimension. In the case
of the pYM equations, axially symmetric solutions (where spherical symmetry was imposed
in the R4p 1 subspace of R4p) were found [83], but imposing less stringent symmetry
rendered the selfduality equations overdetermined [84]. It turns out that a similar situation
holds for the rstorder pSkyrme equations. In this case the only solutions in D = 2p
4 known are the radially symmetric ones, with the axially symmetric equations (when
spherical symmetry is imposed in the R2p 1 subspace of R2p) turn out to be overdetermined
(see appendix of ref [84]).
As a
nal remark, one notes that the rstorder selfduality equations of the
pYangMills system coincide with the rstorder selfduality equations of the pSkyrme equation
(3.1) with the replacement w(r) = cos F (r). But instanton and instantonantiinstanton
solutions in R4 are known in the case of p = 1 YangMills [51]. This raises the question
whether SkyrmionantiSkyrmion solutions may also exist for the pSkyrme model on R2p?
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
433 [INSPIRE].
(1984) 161 [INSPIRE].
[5] C.G. Callan Jr. and E. Witten, Monopole Catalysis of Skyrmion Decay, Nucl. Phys. B 239
[6] I. Floratos and B.M. A.G. Piette, Spherically symmetric solutions of the sixth order SU(N )
Skyrme models, J. Math. Phys. 42 (2001) 5580 [hepth/0109011] [INSPIRE].
[7] I. Floratos and B. Piette, Multiskyrmion solutions for the sixth order Skyrme model, Phys.
Rev. D 64 (2001) 045009 [hepth/0103126] [INSPIRE].
[8] S.B. Gudnason and M. Nitta, A higherorder Skyrme model, JHEP 09 (2017) 028
[arXiv:1705.03438] [INSPIRE].
Math. Phys. 5 (1964) 1252 [INSPIRE].
[9] G.H. Derrick, Comments on nonlinear wave equations as models for elementary particles, J.
[10] E. Braaten, S. Townsend and L. Carson, Novel Structure of Static MultiSoliton Solutions in
the Skyrme Model, Phys. Lett. B 235 (1990) 147 [INSPIRE].
[hepth/9702089] [INSPIRE].
416 (1998) 385 [hepth/9709221] [INSPIRE].
gravity, Phys. Rev. D 38 (1988) 3226 [INSPIRE].
268 (1991) 371 [INSPIRE].
Lett. B 271 (1991) 61 [INSPIRE].
their stability, Helv. Phys. Acta 66 (1993) 614 [INSPIRE].
[19] M. Heusler, N. Straumann and Z.h. Zhou, Selfgravitating solutions of the Skyrme model and
[INSPIRE].
[hepth/9207070] [INSPIRE].
[20] F. Canfora, N. Dimakis and A. Paliathanasis, Topologically nontrivial con gurations in the
4d Einsteinnonlinear model system, Phys. Rev. D 96 (2017) 025021 [arXiv:1707.02270]
[21] D. Kastor and J.H. Traschen, Horizons inside classical lumps, Phys. Rev. D 46 (1992) 5399
[22] C.A.R. Herdeiro and E. Radu, A new spin on black hole hair, Int. J. Mod. Phys. D 23
(2014) 1442014 [arXiv:1405.3696] [INSPIRE].
[23] C.A.R. Herdeiro and E. Radu, Asymptotically at black holes with scalar hair: a review, Int.
J. Mod. Phys. D 24 (2015) 1542014 [arXiv:1504.08209] [INSPIRE].
[24] H. Luckock and I. Moss, Black holes have Skyrmion hair, Phys. Lett. B 176 (1986) 341
[INSPIRE].
World Scienti c (1987).
[25] R. Ru ni and J.A. Wheeler, Introducing the black hole, Phys. Today 24 (1971) 30.
[26] H. Luckock, Black hole skyrmions, in String Theory, Quantum Cosmology and Quantum
Gravity, Integrable and Conformal Integrable Theories, H.J. De Vega and N. Sanches eds.,
[27] M.S. Volkov and D.V. Gal'tsov, Gravitating nonAbelian solitons and black holes with
YangMills elds, Phys. Rept. 319 (1999) 1 [hepth/9810070] [INSPIRE].
[28] C. Adam, O. Kichakova, Ya. Shnir and A. Wereszczynski, Hairy black holes in the general
Skyrme model, Phys. Rev. D 94 (2016) 024060 [arXiv:1605.07625] [INSPIRE].
[29] G. Dvali and A. Gu mann, Skyrmion Black Hole Hair: Conservation of Baryon Number by
Black Holes and Observable Manifestations, Nucl. Phys. B 913 (2016) 1001
[arXiv:1605.00543] [INSPIRE].
[30] S.B. Gudnason, M. Nitta and N. Sawado, Black hole Skyrmion in a generalized Skyrme
model, JHEP 09 (2016) 055 [arXiv:1605.07954] [INSPIRE].
[32] Ya. Shnir, Gravitating sphalerons in the Skyrme model, Phys. Rev. D 92 (2015) 085039
[33] R.A. Battye, S. Krusch and P.M. Sutcli e, Spinning skyrmions and the skyrme parameters,
Phys. Lett. B 626 (2005) 120 [hepth/0507279] [INSPIRE].
HJEP1(207)3
MottolaWipf model with sphaleron and instanton
elds, Phys. Lett. B 320 (1994) 294
[arXiv:1510.08735] [INSPIRE].
[arXiv:1508.06507] [INSPIRE].
[hepth/0205157] [INSPIRE].
064002 [grqc/0505143] [INSPIRE].
(2006) 213 [grqc/0608110] [INSPIRE].
[arXiv:0801.3471] [INSPIRE].
[INSPIRE].
[INSPIRE].
Phys. B 439 (1995) 205 [hepph/9410256] [INSPIRE].
D = 2,3, J. Math. Phys. 37 (1996) 2569 [INSPIRE].
[39] B.M.A.G. Piette, B.J. Schroers and W.J. Zakrzewski, Multisolitons in a twodimensional
Skyrme model, Z. Phys. C 65 (1995) 165 [hepth/9406160] [INSPIRE].
[40] B.M.A.G. Piette, B.J. Schroers and W.J. Zakrzewski, Dynamics of baby skyrmions, Nucl.
[41] K. Arthur, G. Roche, D.H. Tchrakian and Y.S. Yang, Skyrme models with selfdual limits:
(2003) R301 [arXiv:0801.0307] [INSPIRE].
221101 [arXiv:1403.2757] [INSPIRE].
[44] C.A.R. Herdeiro and E. Radu, Kerr black holes with scalar hair, Phys. Rev. Lett. 112 (2014)
[45] C.A.R. Herdeiro and E. Radu, Construction and physical properties of Kerr black holes with
scalar hair, Class. Quant. Grav. 32 (2015) 144001 [arXiv:1501.04319] [INSPIRE].
[46] C.A.R. Herdeiro, E. Radu and H. Runarsson, Kerr black holes with selfinteracting scalar
hair: hairier but not heavier, Phys. Rev. D 92 (2015) 084059 [arXiv:1509.02923] [INSPIRE].
[47] U. Ascher, J. Christiansen and R.D. Russell, A Collocation Solver for Mixed Order Systems
of Boundary Value Problems, Math. Comput. 33 (1979) 659 [INSPIRE].
[48] A.D. Jackson, A. Wirzba and N.S. Manton, New Skyrmion solutions on a three sphere, Nucl.
Phys. A 495 (1989) 499 [INSPIRE].
[49] B. Hartmann, B. Kleihaus, J. Kunz and M. List, Rotating Boson Stars in 5 Dimensions,
Phys. Rev. D 82 (2010) 084022 [arXiv:1008.3137] [INSPIRE].
the YangMills Equations, Phys. Lett. B 59 (1975) 85 [INSPIRE].
[50] A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Yu.S. Tyupkin, Pseudoparticle Solutions of
[51] E. Radu and D.H. Tchrakian, Selfdual instanton and nonselfdual instantonantiinstanton
[INSPIRE].
524 (2002) 369 [hepth/0103038] [INSPIRE].
[hepth/0202141] [INSPIRE].
generalizations of BartnickMcKinnon and colored black hole solutions in D = 5, Phys. Lett.
B 561 (2003) 161 [hepth/0212288] [INSPIRE].
[55] M. Heusler, No hair theorems and black holes with hair, Helv. Phys. Acta 69 (1996) 501
[grqc/9610019] [INSPIRE].
[56] M. Heusler and N. Straumann, Scaling arguments for the existence of static, spherically
symmetric solutions of selfgravitating systems, Class. Quantum. Grav. 9 (1992) 2177.
[57] F.R. Tangherlini, Schwarzschild eld in n dimensions and the dimensionality of space
problem, Nuovo Cim. 27 (1963) 636 [INSPIRE].
[58] P. Breitenlohner, D. Maison and D.H. Tchrakian, Regular solutions to higher order curvature
Einstein YangMills systems in higher dimensions, Class. Quant. Grav. 22 (2005) 5201
[grqc/0508027] [INSPIRE].
172 (1986) 304 [INSPIRE].
Phys. Rev. Lett. 61 (1988) 141 [INSPIRE].
45 (1992) 2751 [hepth/9112008] [INSPIRE].
383 (1992) 357 [INSPIRE].
B 442 (1995) 126 [grqc/9412039] [INSPIRE].
D 85 (2012) 084022 [arXiv:1111.0418] [INSPIRE].
[59] R.C. Myers and M.J. Perry, Black Holes in Higher Dimensional SpaceTimes, Annals Phys.
[60] R. Bartnik and J. Mckinnon, Particlelike solutions of the Einstein YangMills equations,
[61] K.M. Lee, V.P. Nair and E.J. Weinberg, Black holes in magnetic monopoles, Phys. Rev. D
[62] P. Breitenlohner, P. Forgacs and D. Maison, Gravitating monopole solutions, Nucl. Phys. B
[63] P. Breitenlohner, P. Forgacs and D. Maison, Gravitating monopole solutions. 2, Nucl. Phys.
[64] E. Radu and D.H. Tchrakian, Stable black hole solutions with nonAbelian elds, Phys. Rev.
[65] C.A.R. Herdeiro, V. Paturyan, E. Radu and D.H. Tchrakian, ReissnerNordstrom black holes
with nonAbelian hair, Phys. Lett. B 772 (2017) 63 [arXiv:1705.07979] [INSPIRE].
[66] Y. Brihaye, C.A.R. Herdeiro and E. Radu, MyersPerry black holes with scalar hair and a
mass gap, Phys. Lett. B 739 (2014) 1 [arXiv:1408.5581] [INSPIRE].
[67] C.A.R. Herdeiro, J. Kunz, E. Radu and B. Subagyo, MyersPerry black holes with scalar hair
and a mass gap: Unequal spins, Phys. Lett. B 748 (2015) 30 [arXiv:1505.02407] [INSPIRE].
[68] D.J. Kaup, KleinGordon Geon, Phys. Rev. 172 (1968) 1331 [INSPIRE].
[69] R. Ru ni and S. Bonazzola, Systems of selfgravitating particles in general relativity and the
concept of an equation of state, Phys. Rev. 187 (1969) 1767 [INSPIRE].
Quant. Grav. 14 (1997) 3131 [INSPIRE].
(2015) 375401 [arXiv:1505.05344] [INSPIRE].
181 (1986) 137 [INSPIRE].
Skyrme model, Phys. Rev. D 90 (2014) 085002 [arXiv:1406.4136] [INSPIRE].
Dimensions, JHEP 03 (2017) 076 [arXiv:1612.06957] [INSPIRE].
ibid. 100 (1985) 311] [INSPIRE].
[1] T.H.R. Skyrme , A Nonlinear eld theory , Proc. Roy. Soc. Lond. A 260 ( 1961 ) 127 [INSPIRE].
[2] T.H.R. Skyrme , A Uni ed Field Theory Of Mesons And Baryons, Nucl. Phys . 31 ( 1962 ) 556 .
[3] E. Witten , Global Aspects of Current Algebra, Nucl. Phys. B 223 ( 1983 ) 422 [INSPIRE].
[4] E. Witten , Current Algebra, Baryons and Quark Con nement , Nucl. Phys. B 223 ( 1983 ) [11] R.A. Battye and P.M. Sutcli e, Symmetric skyrmions , Phys. Rev. Lett . 79 ( 1997 ) 363 [12] R.A. Battye and P.M. Sutcli e, A Skyrme lattice with hexagonal symmetry , Phys. Lett. B [13] N.S. Manton and P. Sutcli e , Topological solitons, Cambridge University Press ( 2004 ).
[14] E. AyonBeato , F. Canfora and J. Zanelli , Analytic selfgravitating Skyrmions, cosmological bounces and AdS wormholes , Phys. Lett. B 752 ( 2016 ) 201 [arXiv: 1509 .02659] [INSPIRE].
[15] N.K. Glendenning , T. Kodama and F.R. Klinkhamer , Skyrme topological soliton coupled to [16] S. Droz , M. Heusler and N. Straumann , New black hole solutions with hair , Phys. Lett. B [17] P. Bizon and T. Chmaj , Gravitating skyrmions, Phys. Lett. B 297 ( 1992 ) 55 [INSPIRE].
[18] M. Heusler , S. Droz and N. Straumann , Stability analysis of selfgravitating skyrmions , Phys.
[34] M.S. Volkov and E. Wohnert , Spinning Q balls , Phys. Rev. D 66 ( 2002 ) 085003 [35] B. Kleihaus , J. Kunz and M. List , Rotating boson stars and Qballs , Phys. Rev. D 72 ( 2005 ) [36] T. Ioannidou , B. Kleihaus and J. Kunz , Spinning gravitating skyrmions , Phys. Lett. B 643 [37] R. Emparan and H.S. Reall , Black Holes in Higher Dimensions, Living Rev. Rel . 11 ( 2008 ) 6 [38] B.M. A.G. Piette , W.J. Zakrzewski , H.J.W. MuellerKirsten and D.H. Tchrakian , A Modi ed [42] S.R. Coleman , Q Balls , Nucl. Phys. B 262 ( 1985 ) 263 [Erratum ibid . B 269 ( 1986 ) 744] [43] F.E. Schunck and E.W. Mielke , General relativistic boson stars , Class. Quant. Grav . 20 solutions in D = 4 YangMills theory , Phys. Lett. B 636 ( 2006 ) 201 [ hep th/0603071] [52] M.S. Volkov , Gravitating YangMills vortices in 4+1 spacetime dimensions , Phys. Lett. B [53] Y. Brihaye , A. Chakrabarti and D.H. Tchrakian , Particlelike solutions to higher order curvature Einstein YangMills systems in ddimensions, Class . Quant. Grav. 20 ( 2003 ) 2765 [54] Y. Brihaye , A. Chakrabarti , B. Hartmann and D.H. Tchrakian , Higher order curvature [70] S. Yoshida and Y. Eriguchi , Rotating boson stars in general relativity , Phys. Rev. D 56 [71] I. Pena and D. Sudarsky , Do collapsed boson stars result in new types of black holes? , Class.
[72] A.M. Polyakov and A.A. Belavin , Metastable States of TwoDimensional Isotropic Ferromagnets , JETP Lett . 22 ( 1975 ) 245 [Pisma Zh . Eksp. Teor. Fiz . 22 ( 1975 ) 503] [73] D.H. Tchrakian , Higgs and Skyrme Chern Simons densities in all dimensions , J. Phys. A 48 [74] N.S. Manton and P.J. Ruback , Skyrmions in Flat Space and Curved Space, Phys . Lett. B [75] N.S. Manton , Geometry of Skyrmions, Commun. Math. Phys. 111 ( 1987 ) 469 [INSPIRE].
[77] C.J. Houghton , N.S. Manton and P.M. Sutcli e, Rational maps, monopoles and Skyrmions, [78] M.F. Atiyah and N.S. Manton , Skyrmions From Instantons, Phys. Lett. B 222 ( 1989 ) 438 [79] N.S. Manton and T.M. Samols , Skyrmions on S3 and H3 From Instantons , J. Phys. A 23 [80] A. Nakamula , S. Sasaki and K. Takesue , AtiyahManton Construction of Skyrmions in Eight