Higher-order Skyrme hair of black holes

Journal of High Energy Physics, May 2018

Abstract Higher-order derivative terms are considered as replacement for the Skyrme term in an Einstein-Skyrme-like model in order to pinpoint which properties are necessary for a black hole to possess stable static scalar hair. We find two new models able to support stable black hole hair in the limit of the Skyrme term being turned off. They contain 8 and 12 derivatives, respectively, and are roughly the Skyrme-term squared and the so-called BPS-Skyrme-term squared. In the twelfth-order model we find that the lower branches, which are normally unstable, become stable in the limit where the Skyrme term is turned off. We check this claim with a linear stability analysis. Finally, we find for a certain range of the gravitational coupling and horizon radius, that the twelfth-order model contains 4 solutions as opposed to 2. More surprisingly, the lowest part of the would-be unstable branch turns out to be the stable one of the 4 solutions.

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Higher-order Skyrme hair of black holes

HJE Higher-order Skyrme hair of black holes Sven Bjarke Gudnason 0 1 3 Muneto Nitta 0 1 2 0 Hiyoshi 4-1-1 , Yokohama, Kanagawa 223-8521 , 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 Higher-order derivative terms are considered as replacement for the Skyrme term in an Einstein-Skyrme-like 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 Skyrme-term squared and the so-called BPS-Skyrme-term squared. In the twelfth-order 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 twelfth-order model contains 4 solutions as opposed to 2. More surprisingly, the lowest part of the would-be 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 higher-order 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 non-vacuum con guration that surrounds said BH. An attempt at answering this question in the negatory is made by the weak no-hair conjecture: for a scalar eld coupled to Einstein gravity, all BHs fall in the Kerr-Newman family of solutions, i.e. being characterized by their mass, charge and spin. The rst stable counterexample was provided in the Einstein-Skyrme 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 BPS-Skyrme 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 sixth-order derivative term which we shall call the BPS-Skyrme 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 BPS-Skyrme model [18, 19].1 The BPS-Skyrme model consists of the sextic BPS-Skyrme term and an appropriate potential term. The standard Skyrme model, in comparison, contains a kinetic term and the fourth-order 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 at-space limit of the model, the BPS-Skyrme term is able to stabilize Skyrmions. In ref. [28] we constructed a family of higher-order 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 BPS-Skyrme 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 BPS-Skyrme 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 higher-order 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 non-positive 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 baby-Skyrme 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 non-Kahler and odd dimensional. So far, the only available supersymmetric extension of Skyrme-like models consists solely of a four-derivative 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 above-mentioned minimal Lagrangians. Let us brie y mention some further activities in the eld of Skyrme-type BH hair. The BH Skyrme hair was extended from the Schwarzschild case to spacetimes which are asymptotically anti-de 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 BPS-Skyrme 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 BPS-Skyrme 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 Einstein-Skyrme model have also been studied [44]. A lower-dimensional 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 3-space with RP 3 without the point at in nity and contemplates a 2-valued 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 BPS-Skyrme 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 BPS-Skyrme 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 Schwarzschild-type 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 2n-th 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 one-parameter family of models (which we will denote { 3 { model (with = 13 ) can be thought of as the kinetic term and the Skyrme-term squared, while the 2 + 12 model is the kinetic term and the BPS-Skyrme-term squared. As known in the literature, usually for the Skyrme-type 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 Arnowitt-Deser-Misner (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 higher-derivative 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 higher-order 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 energy-momentum 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 i-th 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 semi-de 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 BPS-Skyrme 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 four-dimensional 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 stress-energy tensor corresponding to the above Lagrangian densities where we used the chain rule to express the stress-energy tensor in terms of C for which we get C(1) = 2h1i ; C(2) = 4h2i + 4h1ih1i ; ghri = rhri ; Finally, we can write down the stress-energy 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 1-Skyrmion 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 four-vector 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 stress-energy 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 once-derived 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 stress-energy 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 time-time component of the energy-momentum tensors (2.69){(2.84), only the kinetic term, the Skyrme term and the eighth-order term have a nonvanishing contribution to the right-hand 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 higher-order 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 higher-order 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 sub-Lagrangian 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 sixth-order 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 at-space 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 counter-intuitive, 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 clear-cut 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 Skyrme-term 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 BPS-Skyrme term | that is, an eighth-order derivative term. The minimal construction of ref. [28] limits the terms such that there are no powers of a derivative in the i-th 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 eighth-order 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 Skyrme-term 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 Skyrme-term 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 Skyrme-term 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 well-de 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 arc-like 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 arc-like 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 time-dependent 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 time-dependent 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 Sturm-Liouville problem with a nontrivial (non-constant) weight or density function u + 6wCf 2 C U e C !2 : The right-hand side of the above equation is the weight function of the Sturm-Liouville 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 Sturm-Liouville 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 Sturm-Liouville 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 Sturm-Liouville problem (5.23) directly using a numerical nite di erence method. The reason why the kernel and weight functions are di erent in our higher-order model as compared to the standard Skyrme model is due to the linearization of a higher-order 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 higher-order 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 twelfth-order 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 twelfth-order derivative term. In order to do so, we need to rescale the Lagrangian such that the free parameter is the coe cient of the twelfth-order 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 twelfth-order term coe cient instead of Skyrme-term 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 Skyrme-type 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 higher-derivative 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 Skyrme-like models with terms containing six to twelve derivatives. The higher-derivative terms considered here are not the most generic ones, but the so-called minimal terms constructed in ref. [28]. The main motivation was to get a better understanding of the criteria for when a Schwarzschild-type BH can support scalar hair. Indeed in refs. [14, 15], it was shown that although the Skyrme term can support BH hair, the sextic BPS-Skyrme term cannot. In this paper we have checked 3 further models, i.e. a model with a kinetic term and a 2n-th order term, n = 4; 5; 6. The 2 + 8 model is a one-parameter family of models and it turned out that it can support BH hair as long as the model does not purely consist of the BPS-Skyrme term times the standard kinetic term. One of the possibilities that are stable, is the Skyrme-term 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 BPS-Skyrme 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 BPS-Skyrme 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 Skyrme-term 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 Skyrme-term 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 Sturm-Liouville 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 Sturm-Liouville 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 twelfth-order 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 twelfth-order 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 Skyrme-term coe cient. A peculiar observation about the limit of the Skyrme-term 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 Skyrme-term 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 horizon-radius limit for these models. Finally, in the 2+12 model which has BH hair stabilized by a twelfth-order 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 BPS-Skyrme 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 BPS-Skyrme 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 BPS-Skyrme model can be rewritten as a non-barotropic 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 non-gravitating cases that the 2 + 4 + 6 model with potential terms di erent from the standard pion mass term admit non-spherical Skyrmions [22, 23, 55{57]. It is an open question whether such potentials can give rise to non-spherical 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. non-minimal Lagrangians. The rst class of more general Lagrangians will contain only a second-order 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 higher-order 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 sixth-order Lagrangian was relaxed to contain more general terms than the BPS-Skyrme 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 Grant-in-Aid 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) Grant-in-Aid for Scienti c Research (KAKENHI Grant No. 16H03984) and by the MEXT-Supported Program for the Strategic Research Foundation at Private Universities \Topological Science" (Grant No. S1511006). The TSC-computer 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 (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [INSPIRE]. [1] H. Luckock and I. Moss, Black holes have skyrmion hair, Phys. Lett. B 176 (1986) 341 [2] N.K. Glendenning, T. Kodama and F.R. Klinkhamer, Skyrme topological soliton coupled to gravity, Phys. Rev. D 38 (1988) 3226 [INSPIRE]. 268 (1991) 371 [INSPIRE]. Lett. B 271 (1991) 61 [INSPIRE]. Phys. Lett. B 285 (1992) 21 [INSPIRE]. [3] S. Droz, M. Heusler and N. Straumann, New black hole solutions with hair, Phys. Lett. B [4] M. Heusler, S. Droz and N. Straumann, Stability analysis of selfgravitating skyrmions, Phys. [5] M. Heusler, S. Droz and N. Straumann, Linear stability of Einstein Skyrme black holes, [7] M.S. Volkov and D.V. Gal'tsov, Gravitating non-Abelian solitons and black holes with Yang-Mills elds, Phys. Rept. 319 (1999) 1 [hep-th/9810070] [INSPIRE]. [8] N. Shiiki and N. Sawado, Black holes with Skyrme hair, gr-qc/0501025 [INSPIRE]. [arXiv:1510.08735] [INSPIRE]. [11] E. Witten, Global aspects of current algebra, Nucl. Phys. B 223 (1983) 422 [INSPIRE]. [12] E. Witten, Current algebra, baryons and quark con nement, Nucl. Phys. B 223 (1983) 433 [13] S.B. Gudnason, M. Nitta and N. Sawado, Gravitating BPS skyrmions, JHEP 12 (2015) 013 [14] 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]. [15] S.B. Gudnason, M. Nitta and N. Sawado, Black hole skyrmion in a generalized Skyrme model, JHEP 09 (2016) 055 [arXiv:1605.07954] [INSPIRE]. [16] G.S. Adkins and C.R. Nappi, Stabilization of chiral solitons via vector mesons, Phys. Lett. B 137 (1984) 251 [INSPIRE]. skyrmion, Phys. Lett. B 154 (1985) 101 [INSPIRE]. [17] A. Jackson, A.D. Jackson, A.S. Goldhaber, G.E. Brown and L.C. Castillejo, A modi ed [18] C. Adam, J. Sanchez-Guillen and A. Wereszczynski, A Skyrme-type proposal for baryonic matter, Phys. Lett. B 691 (2010) 105 [arXiv:1001.4544] [INSPIRE]. [19] C. Adam, J. Sanchez-Guillen and A. Wereszczynski, A BPS Skyrme model and baryons at large Nc, Phys. Rev. D 82 (2010) 085015 [arXiv:1007.1567] [INSPIRE]. [20] S.B. Gudnason, M. Nitta and S. Sasaki, A supersymmetric Skyrme model, JHEP 02 (2016) 074 [arXiv:1512.07557] [INSPIRE]. model, JHEP 01 (2017) 014 [arXiv:1608.03526] [INSPIRE]. [21] S.B. Gudnason, M. Nitta and S. Sasaki, Topological solitons in the supersymmetric Skyrme [22] S.B. Gudnason and M. Nitta, Incarnations of skyrmions, Phys. Rev. D 90 (2014) 085007 [arXiv:1407.7210] [INSPIRE]. (2015) 085040 [arXiv:1502.06596] [INSPIRE]. [23] S.B. Gudnason and M. Nitta, Fractional skyrmions and their molecules, Phys. Rev. D 91 [24] C. Adam, M. Haberichter, T. Romanczukiewicz and A. Wereszczynski, Radial vibrations of BPS skyrmions, Phys. Rev. D 94 (2016) 096013 [arXiv:1607.04286] [INSPIRE]. [26] S.B. Gudnason, B. Zhang and N. Ma, Generalized Skyrme model with the loosely bound potential, Phys. Rev. D 94 (2016) 125004 [arXiv:1609.01591] [INSPIRE]. [27] C. Adam, M. Haberichter, T. Romanczukiewicz and A. Wereszczynski, Roper resonances and quasi-normal modes of skyrmions, JHEP 03 (2018) 023 [arXiv:1710.00837] [INSPIRE]. [28] S.B. Gudnason and M. Nitta, A higher-order Skyrme model, JHEP 09 (2017) 028 [arXiv:1705.03438] [INSPIRE]. (1991) 885 [INSPIRE]. [29] L. Marleau, The Skyrme model and higher order terms, Phys. Lett. B 235 (1990) 141 [30] L. Marleau, Modifying the Skyrme model: pion mass and higher derivatives, Phys. Rev. D 43 [31] L. Marleau, All orders skyrmions, Phys. Rev. D 45 (1992) 1776 [INSPIRE]. [32] N. Shiiki and N. Sawado, Black hole skyrmions with negative cosmological constant, Phys. Rev. D 71 (2005) 104031 [gr-qc/0502107] [INSPIRE]. [33] N. Shiiki and N. Sawado, Regular and black hole solutions in the Einstein-Skyrme theory with negative cosmological constant, Class. Quant. Grav. 22 (2005) 3561 [gr-qc/0503123] [INSPIRE]. [34] I. Perapechka and Y. Shnir, Generalized skyrmions and hairy black holes in asymptotically AdS4 spacetime, Phys. Rev. D 95 (2017) 025024 [arXiv:1612.01914] [INSPIRE]. [35] Y. Brihaye and T. Delsate, Skyrmion and Skyrme-black holes in de Sitter spacetime, Mod. Phys. Lett. A 21 (2006) 2043 [hep-th/0512339] [INSPIRE]. [36] S. Zajac, Late-time evolution of the gravitating skyrmion, Acta Phys. Polon. B 40 (2009) [37] S. Zajac, Late-time tails of self-gravitating skyrmions, Acta Phys. Polon. B 42 (2011) 249 1617 [arXiv:0906.4322] [INSPIRE]. [arXiv:1001.4818] [INSPIRE]. Rev. D 70 (2004) 114023 [hep-ph/0409054] [INSPIRE]. 1442 [gr-qc/0307115] [INSPIRE]. [38] N. Shiiki, N. Sawado and S. Oryu, Collective quantisation of a gravitating skyrmion, Phys. [39] N. Sawado and N. Shiiki, Axially symmetric black hole skyrmions, eConf C 0306234 (2003) [40] N. Sawado, N. Shiiki, K.-I. Maeda and T. Torii, Regular and black hole skyrmions with axisymmetry, Gen. Rel. Grav. 36 (2004) 1361 [gr-qc/0401020] [INSPIRE]. [41] H. Sato, N. Sawado and N. Shiiki, Collective quantization of axially symmetric gravitating B = 2 skyrmion, Phys. Rev. D 75 (2007) 014011 [hep-th/0609196] [INSPIRE]. [42] T. Ioannidou, B. Kleihaus and J. Kunz, Spinning gravitating skyrmions, Phys. Lett. B 643 (2006) 213 [gr-qc/0608110] [INSPIRE]. [43] I. Perapechka and Y. Shnir, Spinning gravitating skyrmions in a generalized Einstein-Skyrme model, Phys. Rev. D 96 (2017) 125006 [arXiv:1710.06334] [INSPIRE]. [44] Ya. Shnir, Gravitating sphalerons in the Skyrme model, Phys. Rev. D 92 (2015) 085039 [arXiv:1508.06507] [INSPIRE]. gravitating perfect uid solitons in (2 + 1) dimensions, arXiv:1802.07278 [INSPIRE]. holes with Skyrme hair in [arXiv:1710.03833] [INSPIRE]. arXiv:1803.09736 [INSPIRE]. HJEP05(218)7 neutron stars, Phys. Lett. B 742 (2015) 136 [arXiv:1407.3799] [INSPIRE]. the Bogomol'nyi-Prasad-Sommer eld Skyrme model: mean- eld limit versus full eld theory, Phys. Rev. C 92 (2015) 025802 [arXiv:1503.03095] [INSPIRE]. [arXiv:1605.00543] [INSPIRE]. Phys. Lett. B 768 (2017) 274 [arXiv:1611.09370] [INSPIRE]. of the BPS Skyrme model, Phys. Rev. D 90 (2014) 045003 [arXiv:1405.2927] [INSPIRE]. chemical potential and in-medium properties of BPS skyrmions, Phys. Rev. D 91 (2015) 125037 [arXiv:1504.05185] [INSPIRE]. [arXiv:1403.1245] [INSPIRE]. model, Phys. Rev. D 91 (2015) 045027 [arXiv:1410.8407] [INSPIRE]. (2016) 025008 [arXiv:1606.00336] [INSPIRE]. [6] P. Bizon and T. Chmaj , Gravitating skyrmions, Phys. Lett. B 297 ( 1992 ) 55 [INSPIRE]. [9] T.H.R. Skyrme , A nonlinear eld theory , Proc. Roy. Soc. Lond. A 260 ( 1961 ) 127 [INSPIRE]. [10] T.H.R. Skyrme , A uni ed eld theory of mesons and baryons , Nucl. Phys . 31 ( 1962 ) 556 nucleon in the Skyrme model , Int. J. Mod. Phys. E 25 ( 2016 ) 1650097 [arXiv: 1608 .00979] [45] C. Adam , T. Romanczukiewicz , M. Wachla and A. Wereszczynski , Exactly solvable [46] M. Wachla , Gravitating gauged BPS baby skyrmions , arXiv: 1803 .10690 [INSPIRE]. [47] Y. Brihaye , C. Herdeiro , E. Radu and D.H. Tchrakian , Skyrmions, Skyrme stars and black ve spacetime dimension , JHEP 11 ( 2017 ) 037 [48] F.R. Klinkhamer and J.M. Queiruga , Antigravity from a spacetime defect , [49] C. Adam , C. Naya , J. Sanchez-Guillen, R. Vazquez and A. Wereszczynski , BPS skyrmions as [50] C. Adam , C. Naya , J. Sanchez-Guillen, R. Vazquez and A. Wereszczynski , Neutron stars in [55] S.B. Gudnason and M. Nitta , Domain wall skyrmions , Phys. Rev. D 89 ( 2014 ) 085022


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Sven Bjarke Gudnason, Muneto Nitta. Higher-order Skyrme hair of black holes, Journal of High Energy Physics, 2018, 71, DOI: 10.1007/JHEP05(2018)071