Higherorder Skyrme hair of black holes
HJE
Higherorder Skyrme hair of black holes
Sven Bjarke Gudnason 0 1 3
Muneto Nitta 0 1 2
0 Hiyoshi 411 , Yokohama, Kanagawa 2238521 , Japan
1 Lanzhou 730000 , China
2 Department of Physics, and Research and Education Center for Natural Sciences, Keio University
3 Institute of Modern Physics, Chinese Academy of Sciences
Higherorder derivative terms are considered as replacement for the Skyrme term in an EinsteinSkyrmelike model in order to pinpoint which properties are necessary for a black hole to possess stable static scalar hair. We nd two new models able to support stable black hole hair in the limit of the Skyrme term being turned o . They contain 8 and 12 derivatives, respectively, and are roughly the Skyrmeterm squared and the socalled BPSSkyrmeterm squared. In the twelfthorder model we nd that the lower branches, which are normally unstable, become stable in the limit where the Skyrme term is turned o . We check this claim with a linear stability analysis. Finally, we nd for a certain range of the gravitational coupling and horizon radius, that the twelfthorder model contains 4 solutions as opposed to 2. More surprisingly, the lowest part of the wouldbe unstable branch turns out to be the stable one of the 4 solutions.
Black Holes; Solitons Monopoles and Instantons; E ective Field Theories

2
3
4
5
6
7
8
1
1 Introduction
Black hole Skyrme hair
Discussion and conclusion
Introduction
Lagrangian components of a higherorder Skyrme model
Linear stability of the 2 + 4 + 12 model
Taking the Skyrme model limit of the 2 + 4 + 12 model
Black holes pose interesting questions about fundamental physics, such as quantum
information and whether quantum theory is truly unitary at all scales. A simpler question is
whether a black hole (BH) is characterized just by global charges at in nity or it has the
ability to possess hair. That is, can the BH stably couple to elds of the standard model
yielding a nonvacuum con guration that surrounds said BH. An attempt at answering
this question in the negatory is made by the weak nohair conjecture: for a scalar eld
coupled to Einstein gravity, all BHs fall in the KerrNewman family of solutions, i.e. being
characterized by their mass, charge and spin.
The rst stable counterexample was provided in the EinsteinSkyrme model [1{8]. In
some sense a counterexample in a speci c scalar eld theory is not too exciting. However,
the Skyrme model [9, 10] is believed to be an e ective
eld theory describing QCD at
low energies, at least in the limit of a large number of colors [11, 12], where baryons are
identi ed with solitons  called the Skyrmions. This means that at low energies, the
presence of e ective operators induced by QCD would potentially be able to give rise to
stable BH hair.
{ 1 {
Recently, a program of trying to understand in more detail whether the Skyrme term
plays a special role in stabilizing BH hair has been pursued. It started with a simple
observation stating that the BPSSkyrme model, which consists of a sextic derivative term
and a potential, cannot sustain stable BH hair [13]. This was later proved analytically in
ref. [14]. The sixthorder derivative term which we shall call the BPSSkyrme term, was
known for long time as it is induced by integrating out the ! meson [16, 17]. This term later
caught much attention due to the existence of a submodel with an energy bound that is
saturable, hence the name BPSSkyrme model [18, 19].1 The BPSSkyrme model consists
of the sextic BPSSkyrme term and an appropriate potential term. The standard Skyrme
model, in comparison, contains a kinetic term and the fourthorder Skyrme term [9, 10];
the latter term can be viewed as a curvature term. Finally, a model sometimes called
the generalized Skyrme model, contains the standard Skyrme model as well as the
BPSSkyrme term (see, e.g. refs. [22{27] for recent papers). The latter two models can have
potentials as well. In refs. [14, 15] it was shown numerically that in the generalized Skyrme
model, the BH hair ceases to exist in the limit where the Skyrme term is turned o . This
was somewhat surprising because in the gravitating case and in the atspace limit of the
model, the BPSSkyrme term is able to stabilize Skyrmions.
In ref. [28] we constructed a family of higherorder derivative terms, which could be
used exactly for testing the BH hair stability.
These terms are not the most generic
terms possible, but are constructed with the same concept of minimality that underlies the
Skyrme term and the BPSSkyrme term. First of all, all the terms are constructed out of
the strain tensor, which implies that only one derivative acts on each
eld component in
the Lagrangian density. This guarantees a second order equation of motion, which however
is highly nonlinear in single derivatives. The above mentioned minimality for the kinetic,
the Skyrme and the BPSSkyrme term, can be described as each term possessing at most 2
derivatives in the same spatial direction. This is achieved by means of antisymmetrization.
The same minimality is then applied to the terms with eight, ten and twelve derivatives.
Now the minimal number of derivatives in one spatial direction cannot be lower than 4. By
means of antisymmetrization, all terms with 6 or more derivatives in one spatial direction
are eliminated and the outcome is what we denoted the minimal Lagrangians.
Another attempt at constructing higherorder Lagrangians was done by Marleau, who
found a recipe for constructing terms that for the spherically symmetric hedgehog gave
very simple Lagrangian densities [29{31]. The construction yields Lagrangian densities for
the hedgehog with exactly 2 radial derivatives. This implies that there must be 2(n
1)
angular derivatives in a term with 2n derivatives. Unfortunately, for n > 3 that yielded
Lagrangian densities with nonpositive de nite static energies. In particular, in ref. [28] we
showed that although the latter construction gives stable radial equations of motion, the
angular directions contain instabilities which can be triggered by a babySkyrme string in
the baryon number 0 sector. For a BH hair con guration it is even worse, as the expansion
of the Einstein equation yields a rst derivative of the pro le function at the horizon which
1The name may suggest that the model can easily be supersymmetrized, but that is not the case due to
the target space being nonKahler and odd dimensional. So far, the only available supersymmetric extension
of Skyrmelike models consists solely of a fourderivative term, with the target space complexi ed [20, 21].
{ 2 {
is positive (from fh .
); hence no solutions exist. Thus the instabilities present in the
construction are immediately seen by the gravitational background. This obstacle lead us
to construct the abovementioned minimal Lagrangians.
Let us brie y mention some further activities in the eld of Skyrmetype BH hair.
The BH Skyrme hair was extended from the Schwarzschild case to spacetimes which are
asymptotically antide Sitter (AdS) [32{34] and de Sitter (dS) [35]. In particular, ref. [34]
extended the result of refs. [14, 15] to AdS spacetime, i.e. that the sextic BPSSkyrme term
is not able to support stable BH hair.
Gravitating Skyrmions have also been considered in the literature, see e.g. [
4, 6, 7, 36,
37
]. Collective quantization of their zero modes have also been studied [38]. The spherically
symmetric case of the gravitating Skyrmions has been extended to a higher charged axially
symmetric case [39, 40] and it was again quantized [41]. Axially symmetric BH hair was
also constructed in ref. [40]. Spinning gravitating Skyrmions were considered [42, 43] and
it was further found that the BPSSkyrme model does not possess spinning Skyrmions [43]
 neither in a curved nor in a at background. More exotic con gurations like the
gravitating axisymmetric sphalerons in the EinsteinSkyrme model have also been studied [44].
A lowerdimensional example was considered, where exactly solvable gravitating
babySkyrmions were found in 2 + 1 dimensions [45], see ref. [46] for a gauged version thereof.
In 5 dimensions, a generalization of the Skyrme model to O(5) also possesses solitons and
in particular, solitonic hair of BHs [47] with and without spin. An exotic study in this
direction involves nonstandard boundary conditions for the metric, identi es 3space with
RP 3 without the point at in nity and contemplates a
2valued Skyrmion that can give
rise to a negative gravitational mass, thus antigravity [48]. Neutron stars and in particular
the equation of state, which is of crucial importance in that subject, have been studied in
the BPSSkyrme model coupled to gravity [49, 50]. Due to the simplicity of the model,
the solutions could be compared to the mean eld approximation [50], which interestingly
showed some deviations. Unfortunately, since neutron stars are expected to possess some
amount of spin, which is acquired during a star's core collapse, the BPSSkyrme model
cannot quite be a good approximation due to the instability found in ref. [43]. In some
limit it may give reasonable answers, nevertheless.
Another interesting idea was proposed by Dvali and Gu mann, stating that the baryon
number may actually be conserved by BHs and become Skyrmionic hair once swallowed
by the BH [51, 52]. They also proposed that there may be ways for an observer outside
the horizon to measure the number of swallowed baryons, although they rely on certain
assumptions.
In this paper, we will focus on what terms are able to sustain stable static hair on a
Schwarzschildtype BH. We will only consider the above mentioned minimal Lagrangians as
components in the full Lagrangian and leave other possibilities for future work. Speci cally,
the new models that we consider have a kinetic term, the Skyrme term as well as a 2nth
order derivative term, with n = 4; 5; 6. For brevity, we shall call them the 2 + 4 + 2n
models. The idea is then to slowly turn o the Skyrme term, the limit will be denoted
the 2 + 2n model. We nd that only the 2 + 8 and the 2 + 12 models are stable. In the
case of the 2 + 8 model, there is a oneparameter family of models (which we will denote
{ 3 {
model (with
= 13 ) can be thought of as the kinetic term and the Skyrmeterm squared,
while the 2 + 12 model is the kinetic term and the BPSSkyrmeterm squared.
As known in the literature, usually for the Skyrmetype BH hair, there are two branches
of solutions; one upper branch (in the value of the pro le function at the horizon) and one
lower branch. The lower branch in the standard Skyrme model is found to be unstable;
it has a higher ArnowittDeserMisner (ADM) mass and this is backed up by a linear
perturbation analysis, which shows that the lower branch has one negative eigenvalue in
the perturbation spectrum [4, 5, 8]. In the 2 + 12 model, we nd a new behavior, where the
ADM mass of the lower branch crosses over that of the upper branch and thus becomes the
stable one. To this end, we carry out a linear perturbation analysis to check the eigenvalue
spectrum, and indeed the lower branches have regions where they contain only positive
eigenvalues. Further studies in this direction, however, is needed in order to determine full
nonlinear stability. Finally, in the 2+12 model, we nd a range of the gravitational coupling
where not 2, but 4 solutions exist. That is, the lower branch ceases to be single valued in the
gravitational coupling. The surprising result is that the lower part of the lower branch 
farthest away from the upper branch  turns out to be the one with the lowest ADM mass.
The paper is organized as follows. In the next section, we will set up the notation of the
higherderivative terms and construct the component Lagrangians that we will use for the
model mentioned above. Section 3 then couples our generic model Lagrangian with Einstein
gravity and in section 4 the explicit Einstein equations, equations of motion and boundary
conditions are written down for each model and
nally, the numerical results will be
presented. Linear stability of the 2+4+12 model will then be analyzed in section 5. Due to the
surprising spectrum of eigenvalues of the perturbations, we check the Skyrme model limit in
section 6. The models with BH hair surviving without the presence of the Skyrme term are
then studied as functions of the gravitational coupling in section 7. Finally, section 8
concludes with a summary and discussion of our results, as well as an outlook on future work.
2
Lagrangian components of a higherorder Skyrme model
The class of models we will study in this paper is built out of several component Lagrangians
with di erent numbers of derivatives. In this section, we will set up the framework for
the component Lagrangians and their corresponding energymomentum tensors. The
2nth order Lagrangians that we consider in this paper are those proposed in ref. [28] and
were termed minimal Lagrangians. They are minimal in the sense that they contain the
smallest possible number of derivatives in the ith direction. There are of course many
more possibilities for terms with 2n derivatives, which we will not consider here.
{ 4 {
The minimal Lagrangians of order 2n are [28],
where we have de ned the following building blocks
h1i;
3
h i
4
h i
as well as the quantities
and the O(4) invariants are written neatly as
Y g p+1jr p n p n p :
Here, r = 1; : : : ; 6, n
@ n, and the modulo function in the rst index, p + 1jr (meaning
p + 1 mod r), simply ensures that the index r+1 is just 1. The coe cients c2n
0 in the
Lagrangians in eqs. (2.1){(2.6) all have to be positive semide nite, whereas the coe cients
a can take any real values. In fact, we will show shortly that the coe cients a are irrelevant
for the Lagrangian formulation of the theory.
The scalar elds n(x) = (n0; n1; n2; n3) are related to the chiral Lagrangian eld U (x) 2
SU(2) as
U = n012 + ina a;
{ 5 {
1
2
D
e
Tr[L L ] = n
n = 66Ve BBB
6
2
4
00
2
1
2
2
2
3
1
3
A
C
CC Ve T77 ;
7
5
for which the invariants (2.16) simply read
Here, 1;2;3 are the eigenvalues of the strain tensor and Ve is the corresponding
diagonalization matrix.
Inserting the above relation into I4;5;6 of eqs. (2.13){(2.15) and using the de nitions
in eqs. (2.7){(2.12), it follows that they vanish identically
hri =
21r + 22r + 32r:
Using this, we can simplify the Lagrangians L4;5;6 to
with a being the Pauli matrices and a = 1; 2; 3 is summed over. The eld U transforms
as U ! gLU gRy with gL;R 2 SU(2)L;R and hence the symmetry group is SU(2)L SU(2)R,
which is spontaneously broken to the diagonal SU(2)L+R (or it is also simultaneously broken
explicitly by a pion mass term, which however we will not consider in this paper).
The
rst three Lagrangians (eqs. (2.1){(2.3)) are well known. The
rst, L2, is the
standard kinetic term, the second, L4, is the Skyrme term and the third, L6, is the
BPSSkyrme term [18]. The physical interpretation of the three terms is that the kinetic or
Dirichlet term accounts for the kinetic energy, the Skyrme term measures the curvature on
the O(4) target space and
nally, the BPSSkyrme term acts like a perfect uid term [53].
We will now show that I4;5;6 vanish identically. In order to do this, we will utilize the
eigenvalues of the fourdimensional strain tensor, de ned by using the left invariant form
L
HJEP05(218)7
(2.18)
(2.19)
(2.20)
(2.21)
(2.22)
(2.23)
(2.24)
Since we are interested in black holes, we need the stressenergy tensor corresponding
to the above Lagrangian densities
where we used the chain rule to express the stressenergy tensor in terms of
C
for which we get
C(1) = 2h1i ;
C(2) = 4h2i + 4h1ih1i ;
ghri = rhri ;
Finally, we can write down the stressenergy tensor T(2n) for each component Lagrangian,
L2n, as
(2.25)
(2.26)
(2.27)
(2.28)
(2.29)
(2.30)
(2.31)
(2.32)
(2.33)
(2.34)
(2.36)
(2.37)
(2.38)
(2.39)
(2.40)
(2.41)
(2.42)
HJEP05(218)7
T(2) = c2C(1) + g L2;
T(4) = c24C(2) + g L4;
T(6) = c36C(3) + g L6;
T(8) =
U = 12 cosf(r) + ixa a
r sinf(r);
{ 7 {
We will now specialize to the case with spherical symmetry, which is relevant for the
1Skyrmion sector. Thus we can assume the hedgehog Ansatz for the Skyrme eld
with a pro le function f (r) satisfying the boundary condition f (r ! 1) ! 0, which in
terms of the fourvector n reads
n = sin f (r) sin cos ; sin f (r) sin sin ; sin f (r) cos ; cos f (r) :
(2.43)
In this paper, we will choose a metric tensor of the form 1 C(r)
which is compatible with a spherically symmetric Schwarzschild black hole.
We can now plug in the hedgehog Ansatz and metric to the component Lagrangians and
corresponding stressenergy tensors. First we note that the invariants have an astonishingly
simple form for the hedgehog
hsi = Csfr2s +
2 sin2s(f )
r2s
;
throughout the paper.
metric
where fr
@rf is the radial derivative of f . We will use this notation for derivatives
We will rst evaluate the building blocks Cs with the hedgehog and Schwarzschild
and the identities Is are readily checked to vanish
C1 = Cfr2 +
C2 =
C3 =
C4 =
C5 =
C6 =
r2
;
Cfr2 +
Cfr2;
4 sin2(f )
3 sin4(f )
r2
r4
r4
r6
r8
4 sin4(f ) C2fr4 +
10 sin6(f ) C2fr4 +
3 sin8(f ) C2f 4;
r
2 sin4(f )
r4
;
8 sin6(f )
r6
5 sin8(f )
r8
Cfr2;
Cfr2;
It is convenient to note that the invariant derived with respect to the inverse metric, has a
very simple form once we plug in the hedgehog Ansatz and Schwarzschild metric. Due to
spherical symmetry and diagonal metric, they remain diagonal and their components read
Let us now evaluate all the oncederived building blocks with respect to the inverse metric,
sin4(f )
r4
C2fr4 +
(s) = 0 due to the static Ansatz while all C
(s) = sin2( )C(s) due to spherical
Finally, we can evaluate the stressenergy tensors with the hedgehog Ansatz and
Schwarzschild metric giving
Tr(r8) = 3(2c8j4;4 + c8j4;2;2)
T(8) = (2c8j4;4 + c8j4;2;2)
sin4(f)Cfr4 + 2c8j4;2;2
r4
sin6(f)fr2 c8j4;4
r6
siCn8r(8f);
sin4(f)C2fr4 + 4c8j4;2;2
r2
sin6(f)Cfr2 + 3c8j4;4
r4
sinr86(f);
Tt(t10) = c10j4;4;2N2Csin6(f) 2C2fr4 + sin2(f)Cfr2 ;
r6 r2
Tr(r10) = c10j4;4;2
sin8(f)Cfr4;
sin8(f)C2fr4;
and due to spherical symmetry, we have that T = sin2( )T .
(2.69)
(2.70)
(2.71)
(2.72)
(2.73)
(2.74)
(2.75)
(2.76)
(2.77)
(2.78)
(2.79)
(2.80)
(2.81)
(2.82)
(2.83)
(2.84)
(2.85)
(2.86)
HJEP05(218)7
Once we have chosen the Lagrangian, L, out of our set of Lagrangians and
xed the
constants, we are ready to solve the Einstein equation
G
= R
R = 8 GT ;
eshed out as
2
1 r2Crr + rCr +
rCNr +
N
3r2CrNr +
r2CNrr = 8 GT ;
N
for the metric in eq. (2.44). After taking suitable linear combinations, we can write them as
1
r
1
r
Cr +
1
r2
C
r
N
1
1
2
g
1
r2
1
C
2N
N 2CCr +
N 2C
N 2C2 = 8 GTtt;
1
r2
+
Cr +
rC
rN
2Nr = 8 GTrr;
= 8 G
Nr = 4 G
NrT2tCt ;
rTtt
N 2C2 + 4 GrTrr;
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
2One may naively think that a contribution can come from N =N , but the combination T
+Ttt=(N 2C2)
does not contain negative powers of C even though each term separately does. Thus the Einstein equation
is not singular at the horizon.
which with two boundary conditions determine N; C and hence the metric.
Finally, we also need the equation of motion coming from varying the matter
Lagrangian
1
The black hole horizon is de ned as C(rh) = 0, where rh is the horizon radius. The other
boundary conditions we need to impose, is that the pro le function goes to zero at spatial
in nity, f (1) = 0, and the correct value of the rst derivative at the horizon radius, fr(rh).
The latter can be derived by taking the r ! rh limit of the rst Einstein equation and the
equation of motion, yielding
lim
r!rh
1
Cr
+
1
rh
lim
N (rh) r!rh C(r)
Ttt(r)
;
where limr!rh C(r) = 0. Since the metric function C accompanies the radial derivative
squared, the limit is relatively simple. Indeed, by looking at the timetime component of
the energymomentum tensors (2.69){(2.84), only the kinetic term, the Skyrme term and
the eighthorder term have a nonvanishing contribution to the righthand side of eq. (3.8).
We can rewrite eq. (3.9) by using the fact that fr is necessarily accompanied by at
least a factor of C. Therefore, the rst term can only give a nonzero contribution when
the radial derivative hits a factor of C.2 Thus we can write
lim
r!rh
where we have used eq. (3.6).
In our calculations, we will use a variable simply related to C, namely C = 1 2m
i.e. m and it has the physical interpretation as a gravitational mass function; more precisely,
limr!1 m = m(1) = mADM is the ADM mass. It is a good observable to distinguish stable
and unstable branches of solutions by (without turning to a linear stability analysis).
As the inverse metric that accompanies the double derivative of the pro le function,
frr, is grr = C, the equation of motion is singular exactly at the horizon. For this reason,
we will start all calculations a tiny step r from the horizon: i.e. at r = rh + r and
extrapolate the values of the elds linearly using the derivatives of f and .
It is well known that the topological charge or topological degree is only a full integer
when there is no BH horizon. That is, part of the topological charge is swallowed by the
BH. In this paper we only consider the spherically symmetric hedgehog, for which the
topological charge outside the horizon is less than unity. In particular, we have
B =
1 Z
2 2 outside
horizon
d3x p
gp
gtt
r
C3 =
3
rh
dr sin2(f )fr =
2fh
sin 2fh
2
;
(3.12)
f (rh) is the value of the pro le function at the horizon. B < 1 for fh < ,
which is the case for all the BH hair solutions we nd in this paper. Note that since the
rst derivative of B with respect to fh is zero at fh = , the charge is close to one for a
range of fh . .
4
The 2 + 4 + 2n model
hair persists.
4.1
General setup
The model is de ned as
This model is based on the standard black hole Skyrme hair with an added higherorder
term, which is higher order than the Skyrme term. The purpose of this model is to construct
a stable modi ed black hole hair, and then slowly turn o the Skyrme term to see if the
L = L2 + L4 + L2n +
R
16 G
;
where n = 3; 4; 5; 6 speci es the higherorder term added to the normal black hole Skyrme
hair, R is the Ricci scalar and G is Newton's constant.
We will now switch to dimensionless units
where a sets the (inverse) length scale and has mass dimension 1. Hence, we can write the
Lagrangian as
where we have de ned
equations:
which we can invert to
i.e. the e ective (dimensionless) gravitational coupling, while M0 will turn out to set the
mass scale for the soliton hair, as we will show shortly.
We will use the freedom of choosing the scales to remove c2 and c2n from the dynamical
Cf 2 +
for n = 3; 4; 5; 6.
+
+
1
:
;
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
; (4.11)
(4.12)
HJEP05(218)7
where we have de ned
which can be veri ed to be a dimensionless parameter of the model. The dynamical
equations now only depend on the e ective gravitational coupling , and the coupling .
Finally, we will show that the soliton hair in the rescaled units has a mass given by
M0 times a dimensionless integral (number):
The mass units of the subLagrangian constants are [c2n] = 4
2n, and thus it can readily
be veri ed that both M0 and a have mass dimension one: [M0] = [a] = 1 (as they should).
Hence, we can now write the rescaled Lagrangian as
Z 1
rh
Z 1
h
d
=
and as a check, we can see that the Einstein equations are now dimensionless
G
and the suitable linear combinations in eqs. (3.5){(3.6) read
c2 L2( ) +
c4 L4( ) +
c2n L2n( ) ;
1
c2n
+
+
It will be useful in the next subsections, to have dimensionless expressions for the
boundary conditions. Let us start with eq. (3.8):
C
=
2
1
h
" 2 sin2 f
+
whereas the matter equation of motion is slightly more involved. We will use the
expression (3.10) and write it in dimensionless units as
The boundary conditions at the black hole horizon can now be written as the following
expansion
=
2
h + (
f = fh + (
h) ( h) + O (
h)f ( h) + O (
h)2 ;
h)2 ;
2C f +
4 sin2(f )C f
2
2 sin 2f
2
2 sin2(f ) sin 2f
4
lim
lim
C = 1
C ( h)
c2n
+
2
:
For the numerical calculations, we will use the dimensionless mass function, , given via
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
(4.18)
f ( h) is the shooting parameter and provides nonperturbative information
about the soliton hair.
( h) is extracted from eq. (4.13) by using that C
f ( h), i.e. the derivative of the pro le function of the Skyrme hair at the horizon, is isolated
2
2 .
in eq. (4.14) and C is eliminated by insertion of eq. (4.13).
The last observable that we will use here, is the Hawking temperature
TH =
N ( h)C ( h) :
4
C ( h) is almost a local quantity; it can be found by just knowing the value of the pro le
at the horizon, fh. This is not so with N ( h); this is a nonlocal quantity and it must be
obtained by integrating N from in nity down to the horizon radius, h, using eq. (4.12),
where we have used the boundary condition N (1) = 1. The Hawking temperature de ned
above is dimensionless; to convert to the dimensionful temperature one should multiply by
a yielding aTH .
In the following subsections, we will study each case of n = 3; 4; 5; 6 in turn.
4.2
The 2 + 4 + 6 model
This model was already studied in ref. [15], and the result was that the sixthorder
BPSSkyrme term cannot stabilize solitonic black hole hair. For completeness, we will review the
results here and provide a few new
gures. This will help facilitate a comparison between
the characteristics of this model and the other ones.
Let us rst complete the Einstein equations (4.11){(4.12), C + 1
C
while the equation of motion for the pro le function is
Finally, the boundary conditions (4.13){(4.14) can be written as
=
h +
2
sin2(fh) 2 +
f = fh + 3h sin(2fh)
6
2
h
4
2h sin2 fh
4h + 2
2
sin2 fh
2
h
2h +
sin4 fh:
(
We are now ready to present the numerical results for this model. Figure 1 shows the
following quantities: the pro le function at the horizon (the shooting parameter) fh, the
derivative of the pro le function at the horizon f ( h), the ADM mass (1) and
nally,
the Hawking temperature TH . They are all plotted as functions of the horizon radius,
h. These quantities are used in many gures in this paper and we will henceforth refer
to them as the standard quantities. The stable branches are indicated with black solid
lines and the red dashed lines display the unstable branches. The two branches meet in all
the gures at the bifurcation point, which determines the largest possible black hole size
that can support the black hole hair for the given parameters. The numbers in the gures
indicate the values of
for the di erent branches. From
gure 1(c) it is easy to verify that
the unstable branches all have larger ADM mass, for all values of the horizon radius, h,
than their corresponding stable branches do.
A feature of the 2 + 4 + 6 model, which is not present in the standard Skyrme model
(the 2 + 4 model), is that the unstable branches do not continue all the way back up to the
atspace limit ( h ! 0) solution [14, 15]. We can see two things that happen right about
where the unstable branches cease to exist; gure 1(b) shows that (minus) the derivative
of the pro le function at the horizon,
f ( h), increases drastically and
gure 1(d) shows
that the temperature drops drastically, right before the branches cease to exist.
The most important point of gure 1 is that as
is turned o
(sent to zero), even
the stable branches tend to not exist. In order to show it more explicitly, we provide the
same gures, but plotted as functions of
in gure 2. It is clear from
gure 2(a), that as
= 0:
(4.20)
4
h)2 ;
h)2 ;
(4.19)
2.8
100
ρ
(
fρ
−10−1
100
10
H 1
T
0.1
0.01
ρ
h
0.7
0.8
2 + 4 + 6 model: (a) the value of the pro le function at the horizon, fh; (b) the derivative of the
pro le function at the horizon, f ( h); (c) the ADM mass, (1); (d) the Hawking temperature
TH , all as functions of the size of the black hole, i.e. the horizon radius, h. The numbers on the
gures indicate the di erent values of
tends to zero, only parts of the branches with very small horizon radii, h remain. In
the limit of
! 0, no branch remains and the Skyrme hair ceases to exist. Figure 2(b)
shows what happens to the derivative of the pro le function at the horizon as
is turned
o . A bit counterintuitive, perhaps, the derivative actually tends to zero. The ADM mass
gets smaller as
! 0, which signals that the amount of black hole hair decreases, see
gure 2(c). It is not completely clearcut what happens to the Hawking temperature in the
limit of
! 0, gure 2(d). It is clear that only the branches with very small horizon radii,
h, persist for small . It seems that the bifurcation point moves slightly up as the horizon
radius is decreased. In any case, for small h, the unstable branches possess temperature
curves that go drastically fast towards zero, until they cease to exist.
The gures 1 and 2 provide solid evidence for the fact that the BH hair does not exist
in the
! 0 limit.
.
0
100
10−1
0.0020.03
0.04
0.12
0.06
0.12
0.13
(c)
β
2 + 4 + 6 model: (a) the value of the pro le function at the horizon, fh; (b) the derivative of the
pro le function at the horizon, f ( h); (c) the ADM mass, (1); (d) the Hawking temperature TH ,
all as functions of the Skyrmeterm coe cient, . The numbers on the gures indicate the di erent
values of h = 0:01; 0:02; : : : ; 0:16.
4.3
The 2 + 4 + 8 model
This model contains the rst term in increasing order of derivatives after the sextic term,
also known as the BPSSkyrme term  that is, an eighthorder derivative term. The
minimal construction of ref. [28] limits the terms such that there are no powers of a derivative
in the ith direction higher than four. This leaves us with two independent terms, as we
will see and the coe cients in ref. [28] are called c8j4;4 and c8j4;2;2 due to their composition
in terms of eigenvalues, see eq. (2.19). We can intuitively think of the two terms as the
quadratic part of the Skyrme term squared and the cross terms, respectively, see ref. [28]
for details. The term with coe cient c8j4;2;2 also has the interpretation as being the
BPSSkyrme term multiplied by the kinetic (Dirichlet) term. Furthermore, for c8j4;2;2 = 2c8j4;4
the two terms combine to be the Skyrme term squared.
The rescaled Lagrangian density (4.7) divides the term L8 by a factor of c8. Since we
have two independent coe cients, let us de ne
2 [0; 1] interpolates between the two terms. Here
we will be interested in the following three possibilities. The two terms separately is one
way to disentangle their e ect, therefore we will consider
= 0 and
= 1. Furthermore,
we will consider
= 13 because it corresponds to the Skyrme term squared.
The
= 0 term is composed by one eigenvalue of the strain tensor in eq. (2.18) to the
fourth power, multiplied by the two other (nonzero) eigenvalues squared, i.e. 41 22 23 and
then the product is symmetrized, yielding
41 42 + 14 43 + 24 43:
sin4 f
3
+
= 13 term, which also corresponds to the Skyrme term squared, reads
These three cases should be enough to determine which parts of the eighthorder term are
essential for stabilizing the BH hair.
We are now ready to complete the Einstein equations (4.11){(4.12), C + 1
C
= 2
= 1 term, on the other hand, is composed by only two eigenvalues of the strain
tensor in eq. (2.18), both to the fourth power, i.e. 41 42 and then the product is again
symmetrized over all 3 eigenvalues, yielding
and the equation of motion for the pro le function reads
Cf
+
100
10−1
)
unstable branch
0.2
ρ
h
.
3
ρ
h
(b)
unstable branch
(d)
= 0: (a) the value of the pro le function at the horizon, fh; (b) the
derivative of the pro le function at the horizon, f ( h); (c) the ADM mass, (1); (d) the Hawking
temperature TH , all as functions of the size of the black hole, i.e. the horizon radius, h. The
numbers on the gures indicate the di erent values of
Finally, we need the boundary conditions (4.13){(4.14), which can be written as
2
6
h
=
h + sin2(fh) 2 +
f = fh + 5h sin(2fh)
8
4
6h sin2 fh
sin2 fh +
2
h
6h +
sin6 fh
6
h
4h sin2 fh + 2 sin6 fh
6h + 2
2
4h sin2 fh + 2(1
) sin6 fh
4h sin4 fh
2
sin8 fh:
(
(
h)2 ;
= 0. The numerical results for this case are shown in
gure 3 and
gure 4, where the standard quantities are shown as functions of the horizon
radius, h, and as functions of the Skyrmeterm coe cient, , respectively.
Qualitatively, everything is very similar to the 2 + 4 + 6 model. That is, the branches
shrink as
tends to zero; in particular, the bifurcation point, which represents the largest
possible black hole possessing BH hair, moves to smaller and smaller values of the horizon
f 2.4
2.8
2.6
2.2
2
1.8
0.01
0.02
0.5
β
(d)
.
0
6
7
8
0.5
β
n
o
i
t
if
b
ρ
rc 0.14
a
u
0.18
0.16
0.12
0.1
0.08
10−1
10−2
upper branch
lower branch
upper branch
Upper (black solid lines) and lower (red dashed lines) branches of solutions in the
2 + 4 + 12 model: (a) the value of the pro le function at the horizon, fh; (b) the derivative of the
pro le function at the horizon, f ( h); (c) the ADM
mass,
(1); (d) the Hawking temperature TH ,
all as functions of the Skyrmeterm coe
cient,
. The numbers on the
gures indicate the di erent
values of
h = 0:01; 0:02; : : : ; 0:19.
0
0.2
0.4
0.6
0.8
1
β
The horizon radius at the bifurcation point (maximal BH that supports hair) as a
function of the Skyrmeterm coe
cient
. The minimum is around
h = 0:04.
2.6
2.2
2
1.8
1.6
h
0.30.32
0.34
ρ
(
H
T
10
1
0.1
0.01
(b)
ρh
(d)
in the 2 + 4 + 12 model: (a) the value of the pro le function at the horizon, fh; (b) the derivative of
the pro le function at the horizon, f ( h); (c) the ADM mass, (1); (d) the Hawking temperature
TH , all as functions of the size of the black hole, i.e. the horizon radius, h. The numbers on the
gures indicate the di erent values of
= 0:04; 0:06; : : : ; 0:44.
h . 0:1, the upper branches do not have a bifurcation point at any nite
and thus have
a wellde ned
! 0 limit. The last branch with a bifurcation point in gure 11(a) is the
h = 0:1 branch and said point is around
= 0:25. This indicates that crit is probably
between 0:2 and 0:25.
with
the
All the lower branches depicted in
gure 11 are connected to the upper branches
crit and therefore do not exhibit the peculiar behavior seen in
gure 10. The
derivative of the pro le function at the horizon goes up slightly, but remains of order one in
! 0 limit, see gure 11(b) and the Hawking temperature remains basically constant
in the limit, see gure 11(d). This applies to the upper branches. Since the lower branches
possess somewhat more complicated behavior around and after the phase transition, they
are not depicted in gure 11.
We can see from
gure 10(a) that the bifurcation point moves to smaller BH radii as
decreases from 1 to about 0:04 and then it turns around and continues to increase as
0.04
0.036
ρ
h
ρh
(d)
2.6
ρ
(
fρ
−
1
10
H
T
1
h
0.04
0.032
in the 2 + 4 + 12 model: (a) the value of the pro le function at the horizon, fh; (b) the derivative of
the pro le function at the horizon, f ( h); (c) the ADM mass, (1); (d) the Hawking temperature
TH , all as functions of the size of the black hole, i.e. the horizon radius, h. The numbers on the
gures indicate the di erent values of
= 0; 0:004; 0:008; : : : ; 0:04. The upper and lower
= 0
branches are shown in orange and dark green colors, respectively.
the second series of solutions with
point is decreasing in h, as
in h.
goes to zero. The bifurcation point, which corresponds to the largest BH size possessing
hair, is plotted in gure 12.
We will now make a more detailed plot of
branches as functions of h and in order to
avoid clutter, we will make the rst series of solutions with
2 [0; 0:04], see gure 14. In
2 [0:04; 0:44], see gure 13, and
gure 13 the bifurcation
decreases, and in gure 14 the bifurcation point is increasing
The physics in gure 13 is very versatile; three di erent situations arise. The classi
cation is based on the ADM masses, see gure 13(c). For
0:44 the ADM mass of the
lower branch is always above that of the upper branch. This makes lower branches of
solutions unstable and everything is as usual. In a range of
2 (0:44; 0:12) (approximately),
the lower branch has a lower ADM mass than that of the upper branch for a nite range
branches cross and the lower branch becomes the unstable branch for h < ?
in BH sizes h 2 [ ?h; birfurcation]. At h =
?h the ADM masses of the lower and upper
h. Finally, for
2 [0; 0:12] the lower branch possesses a lower ADM mass than that of the upper branch
for its entire domain of existence. The last branch, i.e. the smallest value of
for which the
ADM mass of the lower branch crosses that of the upper branch is
= 0:14. For
= 0:12,
the branch terminates just about where the ADM mass of the lower branch would cross
over that of the upper branch. For
< 0:12, the lower branches behave drastically di erent
than those with
0:12. In the case of the ADM mass, instead of bending down, going
up and then continuing back with a constant ADM mass as h decreases (for
0:12), it
just goes downwards in an arclike shape until the branch terminates, see gure 13(c).
This same transition point 
0:12  also has an impact on fh, see gure 13(a).
Indeed, the lower branches keep a relatively high value (
2) of the pro le function at the
horizon. For
< 0:12, approximately, the lower branches experience a paradigm shift and
they continue downwards in an arclike shape until they terminate. The derivative of the
pro le function at the horizon,
f ( h), also experiences the same paradigm shift around
0:12, see gure 13(b). For
0:12, the derivatives go upwards until the lower branches
terminate whereas for
< 0:12, the derivatives go below that of the upper branches and
only shortly before they terminate, they start to go upwards. Finally, this \paradigm shift"
can also be seen in the Hawking temperature, see gure 13(d). For
0:12 the end point
of the lower branches moves downwards in
h in a monotonic fashion as
is decreased,
whereas for
< 0:12 the behavior is di erent. Instead, the temperature is raised a bit and
the branches move in a higher level, see gure 13(d).
Figure 13 is the most interesting one as it contains the \paradigm shifting" behavior
and possible a phase transition. Figure 14, however, is the remaining details for
which  as we have mentioned above  is separated from
gure 13 in order to avoid
clutter (overlapping curves) due to the fact that the bifurcation point turns around and
starts moving up again, see gure 12.
Indeed the behavior of the curves in gure 14 is practically monotonic and the most
interesting fact is that the
! 0 limit exists. The catch in this model is that the lower
branch possesses a lower ADM mass than its corresponding upper branch. In order to
verify the stability and be able to claim which of them is the stable branch of solutions,
we will perform a linear stability analysis in the next section.
5
Linear stability of the 2 + 4 + 12 model
In all models, except the 2 + 4 + 12 model, the unstable branches had lower values of the
pro le function at the horizon, fh, and a higher ADM mass. Hence, we have not made a
detailed stability analysis for those cases. The 2 + 4 + 12 model is di erent. In this model,
the there is a critical value of
for which the upper (in fh) branches have a higher ADM
mass and hence are unstable (or at best metastable). For
<
crit this is the case and the
lower (in fh) branches start to have a lower ADM mass for a nite range in h than their
corresponding upper ones. In the limit of
! 0, the lower branch retains a lower ADM
mass than the upper branch throughout its entire domain of existence.
HJEP05(218)7
Our claim is that there is a phase transition and the upper branch becomes unstable
(or metastable) for a range in BH sizes (horizon radii). In order to back up this claim, we
will here make a linear stability analysis of the model in question.
Starting with the Lagrangian of the 2 + 4 + 12 model and turning on time dependence
of the pro le function f ( ) ! f ( ; ), we can write the matter part of the action as
Smatter = 4
Z 1
h
d e
"
u
1
e2 C
f
2
Cf 2
f
2
Cf 2
1
e2 C
2
#
v ;
where we have de ned the functions
v
2 + 2 sin2 f;
;
log N;
and the dimensionless time coordinate
at, as well as
which will be convenient shortly.
Turning on time dependence of the metric functions ( ) !
( ; ) and C( ) ! C( ; )
does not alter the metric part of the usual two combinations of the Einstein equations
C + 1
C = 2
uC
f 2 +
+ v + w
C2f 4
2u
f 2 +
4w
Cf 4
f 4
e4 C3
;
but it gives and additional equation from the
component of the Einstein equations
C
C
u + 2wC
f f
:
Finally, we also need the full timedependent equation of motion for the pro le function
e uCf
wf f
uf f
2e C
2
1
2
e uf Cf 2
wf
2e3 C2
e wf C2f 4 + e
wf f 2f 2
(5.9)
f
2
e C
2
where uf
u + uf f ), and similarly for the other functions.
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
(5.7)
(5.8)
w
e3 C2
f
3
1
2
e vf = 0;
Armed with the full timedependent system of equations, it is now easy to write down
the linearized perturbations. Hence, let us de ne
f ( ; )
( ; )
C( ; )
f ( ) + f 0( ; );
( ) + 0( ; );
C( ) + C0( ; );
f = f 0 ;
whose linearization reads
Next, we will linearize eq. (5.7): 2
Ce 0 + 2 e vf f 0
wf C2f 4f 0
2wCf 4C0
4wC2f 3f 0 :
(5.16)
0 = uf f 2f 0 + 2uf f 0 + 2wf Cf 4f 0 + 2wC0f 4 + 8wCf 3f 0 ;
and insert this as well as the equation of motion (5.9) evaluated on the background soliton
e wC2f 3f 0 ;
which when integrated yields
C0 = 4
u + 2wCf 2 Cf f 0 + e q~( ):
where f , N and C are the solutions of the soliton background while f 0, N 0 and C0 are
uctuation
elds about the background soliton. The linearization greatly simpli es especially
the high powers of the time derivatives since
and hence any power larger than one will eliminate the term.
function, C0, by integrating eq. (5.8),
Following refs. [4, 5, 8], we start by determining the perturbation of the radial metric
C0 = 4
u + 2wCf 2 Cf f 0 + q( );
where u means that u(f ) is evaluated with the background eld: u(f ), and q( ) is an
integration constant. We will now prove that the integration constant vanishes. In order
to do so, we will rst combine the two Einstein equations (5.6) and (5.7) to get
C + 1
C
C
w
e2 C
2#
;
(5.10)
(5.11)
(5.12)
(5.13)
(5.14)
(5.15)
(5.17)
(5.18)
(5.19)
Comparing the above equation with eq. (5.14), we can conclude that the integration
constant cannot be time dependent. Using now that the appropriate boundary conditions for
the uctuations are that all uctuations vanish at spatial in nity, we can nally conclude
that q = q~ = 0.
Finally, we need the linearized equation of motion for the perturbation of the pro le
function, f 0. After some massage, we can write the full perturbation in terms of that of
the pro le function as
where the potential is
1
2
e2 uwC2f 4 !
1
2
1
e C
f 0 ;
1
2
u + 6wCf 2 Cf 0
U f 0 = u + 2wCf 2
U
e uf Cf
e wf C2f 3
e vff +
e uff Cf 2 +
e wff C2f 4
4 e uuf Cf 3
8 e wuf C2f 5
8 e uwf C2f 5
16 e wwf C3f 7
16 2
e
2
2
1 u3Cf 4 + 3u2wC2f 6 + 6uw2C3f 8 + 4w3C4f 10 ;
where we have used the equation of motion for the background eld, f , to eliminate vf .
We will now set
f 0( ; ) = ( )ei! ;
(5.20)
(5.21)
(5.22)
(5.23)
e2 u2Cf 2 !
e2 w2C3f 6 !
HJEP05(218)7
for which eq. (5.20) is a regular SturmLiouville problem with a nontrivial (nonconstant)
weight or density function
u + 6wCf 2 C
U
e C
!2 :
The righthand side of the above equation is the weight function of the SturmLiouville
problem and !2
Physically, if !2 < 0 then ! is imaginary and the perturbation mode signals an instability
at the linear level. Full nonlinear stability requires the absence of linear instabilities,
although that is generally not su cient for claiming nonlinear stability. In this paper, we
2 R is the eigenvalue and by SturmLiouville theory it has to be real.
will content ourselves with the above linear stability analysis.
In the case of the standard Skyrme model, which corresponds to the case of w = 0,
the stability is considerably simpler and the SturmLiouville problem can be transformed
into a free eigenvalue problem with a complicated potential by setting
= = u and using
tortoise coordinates de ned by dx = d =(e C) [8]. The latter transformation is able to
p
simplify the problem
each other.
considerably because the weight function, r(x), and the kernel function, p(x), are equal to
In our case with w 6= 0, the kernel and the weight functions are di erent and we have
not been able to nd a suitable transformation to simplify the problem. We will therefore
solve the SturmLiouville problem (5.23) directly using a numerical nite di erence method.
The reason why the kernel and weight functions are di erent in our higherorder model
as compared to the standard Skyrme model is due to the linearization of a higherorder
q(x) =
!2r(x) ;
(5.24)
2 1 0.031
ω0.0308
lower branch
2 + 4 + 12 model. The numbers on the gures indicate the di erent values of
= 0; 0:1; : : : ; 1. The
upper and lower
= 0 branches are shown in orange and dark green colors, respectively.
derivative operator on a static background. That is, the linearization of f 4 yields 4f 3f 0 ,
whereas the linearization of f 4 vanishes because of the static background. The higherorder
term still gives a contribution to the time derivative of the
uctuation, but it comes from
a cross term, f 2f 2 and hence does not give the same factor as the radial derivative of the
uctuation does.
We are now ready to present the numerical results for the lowest eigenvalue, !12, of
the perturbation of the pro le function, f 0, shown in gure 15. If we start with the upper
branches, they start o at a small horizon radius, h, with a positive lowest eigenvalue (!12 >
0), which decreases as the horizon radius is increased. We would expect the eigenvalue to
go towards zero and at the bifurcation point meet a negative eigenvalue coming from below
zero. This does not happen in this model. If we start with the small values of
the behavior is as follows. Instead the lowest eigenvalue of the upper branch goes upwards
and meets that of the lower branch at the bifurcation point. The lower branch hence does
not have a negative mode and the lowest eigenvalue of the uctuation for the lower branch
is higher than that of the upper branch.
Now, when
is increased to above about 0:5, the lower branch has a hybrid
behavior. Instead of having a negative lowest eigenvalue of the
uctuation, it emanates from
bifurcation point over that of the upper branch  stays there for a nite range in horizon
radius  and then turns negative shortly before the branch terminates at a smaller horizon
radius, see gure 15. There are only quite few data points with a negative eigenvalue of
the uctuation, but enough to conclude that the lower branch develops an unstable mode
for large enough
& 0:5.
The fact that the lower branch (in fh) does not have an unstable mode over its entire
domain of existence is indeed a surprise. Since the analysis carried out here is limited
to a linear stability analysis, this does not rule out a nonlinear instability. Physically,
the absence of an unstable mode in the
uctuation spectrum could be interpreted as the
lower branch being a local minimum or vice versa. In order to determine which local
minimum is the global minimum of solution space, we would still turn to the ADM mass.
0.18
0.16
0
0.08 lower branch
Our explanation at this point is simply that the highly nonlinear nature of our model has
created a situation where the two branches both become local minima. Except for very
particular points in parameter space, only one of them is a global minimum.
With the above discussion in mind, we will now make a phase diagram of the 2 + 4 + 12
model, see gure 16. The expected features are the radius of the bifurcation point (black
upper curve) and the point where the lower branch terminates (red lower curve). The
crossover from where the lower branch has a lower ADM mass happens at
. 0:5 and the
curve decreases in
as the horizon radius, h, gets smaller (orange dashed curve).
Since it is puzzling that the lower branches do not possess the expected unstable modes,
we will consider the limit of turning o
the twelfthorder term (but keeping the Skyrme
term) in the 2 + 4 + 12 model in the next section.
6
Taking the Skyrme model limit of the 2 + 4 + 12 model
In this section we will take the limit of the 2 + 4 + 12 model becoming the standard Skyrme
model; i.e. turning o
the twelfthorder derivative term. In order to do so, we need to
rescale the Lagrangian such that the free parameter is the coe cient of the twelfthorder
term and not the Skyrme term.
The model is de ned as
L = L2 + L4 + L12 +
=
c2C1
c4
2 C2
R
16 G
c912 C32 +
R
16 G
;
where the Cs are de ned in eqs. (2.7){(2.9). Similarly to section 4, we will now switch to
dimensionless units,
ar, for which we get
L = a3M0
c2
aM0 C1
ac4
2M0 C2
9M0
a9c12 C32 +
R
4
;
(6.1)
(6.2)
where the e ective (dimensionless) gravitational coupling is still given by eq. (4.4) and M0
is the mass scale of the soliton. Instead of letting c4 be the free parameter, we will x the
coe cient of L4 and let c12 be the free parameter. This is done by
which gives
which both have mass dimension 1 as they must.
The rescaled Lagrangian is now written as
where the new parameter, is de ned as
In the limit of
! 0, this model becomes the Skyrme model.
The soliton mass then reads
M = 4 M0
Z 1
d
2N
C1 +
1
2 C2 +
9 C32 :
The dimensionless Einstein equations read C + 1
C
Cf 2 +
and the equation of motion becomes
sin8(f )C2f 4 #
sin 2f
2 2
N Cf
N
sin(2f )Cf 2
sin 2f
2
2
6 sin6(f ) sin(2f )C2f 4
(6.3)
(6.4)
(6.5)
(6.6)
(6.7)
;
(6.8)
HJEP05(218)7
c
4
+ C f +
2
Cf
3Cf
6Cf
= 0: (6.9)
Finally, the boundary conditions are simply given by eqs. (4.40){(4.42) with
= 1 (they
do not depend on ).
The numerical results for this model, i.e. the 2 + 4 + 12 model with variable
instead of
(variable twelfthorder term coe cient instead of Skyrmeterm coe cient), are shown in
gure 17. The standard quantities are displayed in the usual four panels. We are interested
in what happens in the
! 0 limit, for which the model becomes the standard Skyrme
model (coupled to gravity). First we note that varying
in the range [0:1; 1] has little
e ect on both the upper and lower branches, see gure 17(a). The approach from the
0 0
0 0
2 .
0 0
050 .01
00
00
.
10
.001 .005 .01 1
1
0.5
0.3
0.2
0.1
0 0.00002
0.0
00
01
2.8
ρ
(
ρ
f−101
0
102
101
100
TH10−1
10−2
10−3
00
.
00
.
ρh
(d)
upper branch, η = 0
upper branch
lower branch
lower branch, η = 0
0 .00 .00
0 0
001 002
.0000 .000
0
50 1
2 + 4 + 12 model, where instead of decreasing , we will send to zero: (a) the value of the pro le
function at the horizon, fh; (b) the derivative of the pro le function at the horizon, f ( h); (c) the
ADM mass, (1); (d) the Hawking temperature TH , all as functions of the size of the black hole,
i.e. the horizon radius, h. The numbers on the gures indicate the di erent values of
2
10 5, 10 4, 0:001, 0:01, 0:05, 0:1, 0:2, 0:3, 0:5, 1. The upper and lower
= 0, 10 5
,
= 0 branches
are shown in orange and dark green colors, respectively.
branch with
1 to the Skyrme model is almost logarithmic in . The
= 0 (i.e. the
standard Skyrme model limit) branches of solutions are distinct from the nonzero
ones
by the fact that the lower branch of the
= 0 returns smoothly to f
in the limit of
vanishing horizon radius, h ! 0, see the dark green dashed curve in gure 17(a). Once a
tiny nonzero
is turned on, the lower branch does not return all the way when h ! 0, but
terminates at a nite horizon radius. The derivative of the pro le function at the horizon
also behaves di erently for the lower
= 0 branch as compared to the nonzero
branches,
see
gure 17(b). The lower
whereas the lower nonzero
= 0 branch remains almost constant in the h ! 0 limit,
branches raise up (in negative values) about two orders of
magnitude before they terminate at a
nite horizon radius. A
nal distinction between
the lower
= 0 branch and the nonzero ones is seen in
gure 17(d), where the Hawking
2 ω1 0.031
lower branch
05
00
.
010
0 .0000 .0000
01 02
0.01
0.05
0.1
0.2
0.3
0.5
1
0
−0.2
−0.4
−0.6
−1
−1.2
−1.4
−1.6 0
2 ω1−0.8
upper branch, η = 0
upper branch
lower branch
10 5, 10 4, 0:001, 0:01, 0:05, 0:1, 0:2, 0:3, 0:5, 1. The upper and lower
gures indicate the di erent values of
= 0,
= 0
branches are shown in orange and dark green colors, respectively.
temperature goes smoothly back from the bifurcation point in the h ! 0 limit. The lower
nonzero
branches on the other hand, drop very drastically and suddenly in temperature
where they terminate at a nite horizon radius.
The ADM masses for all the lower branches in this model are larger than all those
of the upper branches, which is the expected (from the standard Skyrme model scenario)
behavior, see
gure 17(c). So far, except for the distinct behavior of the
= 0 branch,
everything seems in line with the results for the 2 + 4 + 6 model studied in refs. [14, 15].
The expectation from the standard Skyrme model is that the lower branches have larger
ADM masses than the upper branches, and they also possess a negative lowest eigenvalue
in the linear uctuation spectrum  signaling an instability at the linear level.
Now we will consider the lowest eigenvalue for this model, see
gure 18. Although
all the ADM masses for the lower branches are higher, they do not all have a negative
eigenvalue in their linear uctuation spectrum over the entire range of horizon radii. If we
start with the
= 0 branch, everything is as expected. The upper branch has a positive
eigenvalue which drops suddenly near the bifurcation point (see gure 18(a)) and the lower
branch has one single negative mode all the way in
h and it only increases near the
bifurcation point to meet with that of the upper branch. If we turn on a very tiny
in
the range [10 5; 10 4], the trend continues as just described. Then for
around 10 3 a
transition occurs and the eigenvalue for the upper branch no longer drops to zero near the
bifurcation point. Instead the lower branch now possesses only (linearly) stable modes for
a nite range in h from a nite value larger than where the branch terminates, up to the
bifurcation point. For
= 10 3, the upper branch still has the largest eigenvalue compared
to the lower branch, but that quickly changes and for
& 0:01, there is a
nite range in
h where the eigenvalue of the lower branch is larger than that of the upper branch. After
this range, the eigenvalue of the lower branch drops suddenly to negative values where it
remains until the branch terminates at a nite horizon radius, see gure 18(b).
In this section, we will study the dependence of the existing models with
= 0 on the
e ective gravitational coupling (4.4). These models are thus new Skyrmetype models that
possess BH hair without having the Skyrme term component in the Lagrangian density.
The 2 + 2n model is the
! 0 limit of the 2 + 4 + 2n model of section 4. Only two
models survive in the limit, namely the 2 + 4 + 8 model with
6= 0 and the 2 + 4 + 12
model. The
6= 0 condition in the 2 + 4 + 8 model corresponds to having a nonvanishing
and
c8j4;4 term in the Lagrangian density. Here we will consider the 2 + 8 model with
= 1
= 13 as well as the 2 + 12 model. The
= 13 case of the 2 + 8 model corresponds
to the higherderivative term being the Skyrme term squared, whereas the
= 1 case is
the three curvatures of the Skyrme term individually squared without the corresponding
cross terms. After rescaling, and in the case of the 2 + 8 model, xing , the e ective
gravitational coupling is the only parameter of the model. As the
= 0 branches for xed
= 0:01 have been studied already in section 4, here we will consider only the standard
quantities as functions of , for a few di erent BH sizes (di erent horizon radii). The
equations of motion and the boundary conditions for the 2 + 2n models with n = 4 and
n = 6 are simply given in section 4 with
= 0. Thus we will not repeat them here.
In gure 19, the standard quantities are shown for the
= 1 case of the 2 + 8 model
as functions of the gravitational coupling . The di erent branches correspond to di erent
horizon radii and as expected, the branch with the smallest horizon radius has the largest
value of the pro le function, fh. As the horizon radius is increased, the branches move
downwards in fh, gure 19(a), as expected, since the branches will move downwards in fh to
meet their respective bifurcation point. The upper branches are called stable here, as they
everywhere have lower ADM mass than the corresponding lower branches, see gure 19(c).
We have not carried out a linear perturbation analysis for this case, as we expect the lower
branches to be unstable, if not linearly, then at best metastable. Although the ADM mass
increases roughly linearly with the gravitational coupling,
(see gure 19(c)), the Hawking
temperature remains almost constant as
is varied, for the stable branches. We made an
extensive search for the unstable or lower branch for h = 0:1, but were unable to nd any
solutions  both for large and small values of fh. Finally, let us mention that we have
found that the unstable branches for h = 0:4; 0:5 continue all the way to
! 0.
For completeness, we have made the same plots for the
= 13 case in gure 20. Because
these plots are quite similar to those of gure 19, let us just mention the di erences. Indeed
the quantitative behavior is the same, but we have been able to nd an unstable branch
with horizon radius h = 0:1. Both the unstable branches h = 0:1 and h = 0:2 end at
a nite horizon radius, while those with h = 0:3; 0:4 continue back in the limit of
Again the ADM masses suggest that the lower branches are unstable, see gure 20(c).
! 0.
Finally, we will consider the last model, i.e. the 2 + 12 model, which only has the
gravitational coupling, , as a parameter (after rescaling of the length and energy units).
The result is shown in gure 21. This model possesses a more complicated branch structure
than the two avors of the 2 + 8 model. The upper branches behave as expected; they
HJEP05(218)7
0.5
0.5
0.03
0.03
α
0.4
0.3
stable branch
unstable branch
0.4
0.3
0.2
0.1
stable branch
unstable branch
103
)h102
ρ
(
fρ
−
101
100 0
1
H
T
fh
2.8
2.6
μ 0.25
0
0.01
0.02
0.04
0.05
0.06
0.5
0.02
0.03
0.4
0.4
0.3
0.04
stable branch
unstable branch
0.04
0.05
0.06
0
0.01
0.02
0.04
0.05
0.06
0.1 0
0.01
= 1: (a) the value of the pro le function at the horizon, fh; (b) the derivative of
the pro le function at the horizon, f ( h); (c) the ADM mass, (1); (d) the Hawking temperature
TH , all as functions of the gravitational coupling, . The numbers on the gures indicate the
di erent values of h = 0:1; 0:2; : : : ; 0:5.
start from above in fh with small horizon radius and move downwards as h is increased.
They have little dependence on , expect near their bifurcation point. The lower branches,
however are far more complicated. The lower branch with h = 0:9 (brown dashed line) is
the only one depicted which is single valued in . The lower branches with h = 0:7 (green
dashed line) and h = 0:5 (yellow dashed line) are still continuous, but not single valued
in . Surprisingly, the lower part of the h = 0:5 lower branch exists up till quite large
' 0:04, before it sharply turns back, see gure 21(a). That part of the unstable branch
has in fact a smaller ADM mass than the upper branch, see the lower yellow dashed line in
gure 21(c). As the horizon radius is decreased, the lower branches become discontinuous.
Indeed, the h = 0:3 lower branch has an upper part connected to the upper branch and
a disconnected lower part, see gure 21(a). Finally, the h = 0:1 lower branch only exists
very close to the bifurcation point, i.e. for quite large values of . We did not nd a
disconnected lower part for this value of the horizon radius.
0.4
0.3
0.2
0.1
0.3
0.2
0.1
stable branch
unstable branch
0.4
0.03
2.6
2.2
2
μ 0.25
10
)
h
ρ
(
ρ
f− 1
TH 0.1
0.03
α
0.2
0.2
0.1
stable branch
unstable branch
stable branch
unstable branch
0.01
0.02
0.04
0.05
0.06
= 13 : (a) the value of the pro le function at the horizon, fh; (b) the derivative of
the pro le function at the horizon, f ( h); (c) the ADM mass, (1); (d) the Hawking temperature
TH , all as functions of the gravitational coupling, . The numbers on the gures indicate the
di erent values of h = 0:1; 0:2; 0:3; 0:4.
As mentioned above, the ADM masses interestingly show that the lower parts of the
lower branches have lower ADM masses than their corresponding upper branches, which
would make them the stable branches, gure 21(c). Note, however, that these lower parts do
not exist all the way up to the bifurcation point, so for values of
close the bifurcation point,
the upper branch would be the stable one. We have not carried out a linear perturbation
analysis for this case, see comments in the next section.
It is interesting to see what happens to the derivative of the pro le function at the
horizon for the lower parts of the lower branches, which according to the ADM masses are
the stable ones, see gure 21(b). Let us consider the h = 0:5 lower branch (yellow dashed
line). The unstable part of the lower branch has a higher derivative at the horizon and about
where this branch becomes the stable one, the derivative drops below that of the upper
branch, see gure 21(b). The lower parts of the lower branches, which are the stable ones for
those values of , do in fact all have smaller derivatives of the pro le function at the horizon.
0.05
h = 0.01
h = 0.03
h = 0.05
h = 0.07
lower branch, ρh = 0.09
α
)h 20
ρ
(
f−ρ 15
30
10
0
101
100
10−1
TH10−2
10−3
10−4
10−5 0
α
(d)
2 + 12 model: (a) the value of the pro le function at the horizon, fh; (b) the derivative of the pro le
function at the horizon, f ( h); (c) the ADM mass, (1); (d) the Hawking temperature TH , all as
functions of the gravitational coupling, . The numbers on the gures indicate the di erent values
of h = 0:01; 0:03; : : : ; 0:09. The lower branches are colored di erently for each horizon radius, see
the legend in panel (a).
Finally, we will show the Hawking temperature for this model. This calculation turned
out to be the hardest one in the paper and we needed to use over 1 billion lattice points
for our integrator to get convergence for the temperature. The interesting result from this
calculation is that the lower parts (in fh) of the lower branches, which are the stable ones
for the corresponding values of , have higher Hawking temperature compared with the
upper parts of the same branches, see gure 21(d).
8
Discussion and conclusion
In this paper, we investigated a number of Skyrmelike models with terms containing six
to twelve derivatives. The higherderivative terms considered here are not the most generic
ones, but the socalled minimal terms constructed in ref. [28]. The main motivation was to
get a better understanding of the criteria for when a Schwarzschildtype BH can support
scalar hair. Indeed in refs. [14, 15], it was shown that although the Skyrme term can
support BH hair, the sextic BPSSkyrme term cannot. In this paper we have checked 3
further models, i.e. a model with a kinetic term and a 2nth order term, n = 4; 5; 6. The
2 + 8 model is a oneparameter family of models and it turned out that it can support
BH hair as long as the model does not purely consist of the BPSSkyrme term times the
standard kinetic term. One of the possibilities that are stable, is the Skyrmeterm squared.
The 2 + 10 model has turned out not to be able to sustain BH hair. Finally, the 2 + 12
model is basically the kinetic term and the BPSSkyrme term squared and surprisingly it
does possess a stable BH hair. The BH hair comes in two branches, one upper branch (in
the pro le function at the horizon, fh) which is typically stable, and one lower branch (in
fh) which is typically unstable (see below, however).
A feature already seen in the generalized Skyrme model coupled to Einstein gravity,
is that the unstable branches, for sizable coupling to the BPSSkyrme term, end at a nite
BH horizon radius simply because the temperature approaches zero. This can be viewed
as the BH approaching an extremal BH state or more pragmatically as the derivative of
the eld pro le at the BH horizon blowing up. This feature has turned out to be shared by
the other models that do not possess BH hair in the limit where the Skyrme term is turned
o ; in particular, those are the 2 + 4 + 6, the 2 + 4 + 8 (
= 0), and 2 + 4 + 10 models.
For the standard Skyrme model coupled to Einstein gravity, the upper and the lower
branches of solutions correspond to the stable and unstable solutions. This was checked in
refs. [4, 5] with a linear perturbation analysis which showed that the lower branch contains
a single negative frequency of the perturbation modes of the linearized system. This system
contains implicitly both the linear perturbations of the metric as well as of the Skyrme eld
radial pro le. The metric perturbations are then eliminated, yielding a single eld master
equation. Evidence for the instability of the lower branches was already seen by calculating
their respective ADM masses, for which the lower branches always possessed the higher
ADM masses compared to the upper (stable) branches. However, in the 2 + 4 + 12 model
the situation turned out to be somewhat more intricate. Indeed, when the Skyrme term is
slowly turned o , the lower branch switched to become the one with a lower ADM mass.
For an intermediate range of the Skyrmeterm coe cient, the lower branch possesses a
lower ADM mass compared to the upper branch, for a
nite range from the bifurcation
point down to a critical radius where the ADM masses cross over (details have been shown
in gure 13). Finally, in the limit of a vanishing Skyrmeterm coe cient, the lower branches
of solutions remain the ones with the lowest ADM masses. Those lower branches, however,
still terminate at a nite horizon radius. This behavior is mapped out in the phase diagram
of gure 16. To this end, we have carried out a linear perturbation analysis of the BH hair
system analogous to that of refs. [4, 5]. It turned out, however, that the problem at hand
is more complicated due to the higher nonlinearity of the problem, which causes the kernel
and the weight function of the resulting SturmLiouville problem to di er. Consequently,
the master equation for the perturbations of the 2 + 4 + 12 model cannot be written as
a Schrodinger equation and the full SturmLiouville problem needed to be solved. The
result of this analysis was consistent with the naive conclusion from the ADM masses,
namely, the lower branches become the stable ones in the limit of the Skyrme term being
turned o in the 2 + 4 + 12 model. This result is surprising. As a double check we have
tried turning o the twelfthorder term and thus returning to the standard Skyrme model,
and indeed, the lowest eigenvalue of the perturbations returned to the standard behavior.
That is, the upper branch has only positive eigenvalues and the lower branch has a single
negative eigenvalue. Once both the Skyrme term and the twelfthorder term are turned on
with sizable (order one) coe cients, the eigenvalue possesses a hybrid behavior; for small
horizon radii the lowest eigenvalue is negative, but it turns positive at a
nite radius and
crosses over that of the upper branch until they meet at the bifurcation point.
Since the models under study in this paper are highly nonlinear problems, the linearized
perturbation analysis may not su ce. Indeed, as a future investigation, full nonlinear
stability should be considered seriously in order to understand the situation of the 2+4+12
model in the limit of a small or vanishing Skyrmeterm coe cient.
A peculiar observation about the limit of the Skyrmeterm coe cient being turned
o in the 2 + 4 + 12 model, is that the explanation for the lower branches terminating
at a nite horizon radius until now was that the Hawking temperature would go to zero,
or equivalently the derivative of the pro le function at the horizon would blow up. In
this case, however, the lower branches still terminate at a
nite radius, but the reason
switches from being the derivative of the pro le function blowing up (i.e. equal to the
Hawking temperature dropping to zero) to the derivative going to zero. That also has as
consequence that the solutions cease to exist, but the Hawking temperature remains nite.
To complicate the situation with the termination of the lower branches, in the 2 + 4 + 8
model (for any value of , the Skyrmeterm coe cient) the lower branches terminate at a
nite horizon radius, but with a
nite Hawking temperature and an apparently
nite rst
derivative of the pro le function at the horizon. Further studies are needed to conclude
the fate of the small horizonradius limit for these models.
Finally, in the 2+12 model which has BH hair stabilized by a twelfthorder term and no
Skyrme term, a certain range of the gravitational coupling exists for which there are not 2
but 4 solutions with the same horizon radius, h. According to their ADM masses, the lower
part of the lower branch is the stable one. The upper branch may possess metastability
and the upper part of the lower branch may be either unstable or metastable.
Throughout this paper, we have turned o
a potential in order to keep the analysis
as clean as possible. Although we have not checked, we think that most, if not all, results
will be qualitatively similar if a potential would be turned on; in particular a standard
pion mass term. The absence of BH hair in the 2 + 4 + 6 model without a mass term
or other potential is thus con rmed; ref. [14] carried out all their numerical calculations
with the pion mass term present. Ref. [14] also gives a physical explanation for the lack
of BH hair in the model with only the BPSSkyrme term, which claims that the pressure
becomes negative at the horizon due to the potential. This explanation cannot cover the
case considered here, where we have excluded the potential altogether. It may be that the
zero pressure from the BPSSkyrme term is not su cient for preventing a collapse of the
hair. Further studies are needed for a conclusion on this issue.
energy tensor of the model (excluding the kinetic term) can be rewritten in the form of
a perfect uid and whether this may be a criteria for whether a BH can possess stable
hair or not. In ref. [54], it was shown that the BPSSkyrme model can be rewritten as a
nonbarotropic perfect uid using the Eulerian formulation of a relativistic uid. The uid
element velocity was identi ed with the baryon charge current. Not all models considered
here can be written in terms of just the baryon charge current or baryon charge density.
Although some can, others cannot and some are hybrids of the latter two options.
It is known in nongravitating cases that the 2 + 4 + 6 model with potential terms
di erent from the standard pion mass term admit nonspherical Skyrmions [22, 23, 55{57].
It is an open question whether such potentials can give rise to nonspherical BH Skyrmions,
other than axially symmetric ones.
An obvious generalization of this study is to consider more general terms with the
same number of derivatives; i.e. nonminimal Lagrangians. The rst class of more
general Lagrangians will contain only a secondorder equation of motion (albeit nonlinear in
derivative terms), but with more than four derivatives in the same direction, for the sixth,
eighth, tenth and twelfth order terms. A further generalization is to include more than one
derivative acting on the same eld, yielding a higherorder equation of motion. We avoided
such a complication here and in ref. [28], because in general it could give rise to an
Ostrogradsky instability. A
rst interesting question would be, if the sixthorder Lagrangian
was relaxed to contain more general terms than the BPSSkyrme term, would it be able to
sustain stable BH hair. We will leave this question for future work.
Acknowledgments
The work of S. B. G. is supported by the National Natural Science Foundation of China
(Grant No. 11675223). The work of M. N. is supported in part by a GrantinAid for
Scienti c Research on Innovative Areas \Topological Materials Science" (KAKENHI Grant
No. 15H05855) from the Ministry of Education, Culture, Sports, Science (MEXT) of Japan.
The work of M. N. is also supported in part by the Japan Society for the Promotion of
Science (JSPS) GrantinAid for Scienti c Research (KAKENHI Grant No. 16H03984) and by
the MEXTSupported Program for the Strategic Research Foundation at Private
Universities \Topological Science" (Grant No. S1511006). The TSCcomputer of the \Topological
Science" project in Keio university was used for some of the numerical calculations.
Open Access.
This article is distributed under the terms of the Creative Commons Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
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