Skyrmions, Skyrme stars and black holes with Skyrme hair in five spacetime dimension

Journal of High Energy Physics, Nov 2017

We consider a class of generalizations of the Skyrme model to five spacetime dimensions (d = 5), which is defined in terms of an O(5) sigma model. A special ansatz for the Skyrme field allows angular momentum to be present and equations of motion with a radial dependence only. Using it, we obtain: 1) everywhere regular solutions describing localised energy lumps (Skyrmions); 2) Self-gravitating, asymptotically flat, everywhere non-singular solitonic solutions (Skyrme stars), upon minimally coupling the model to Einstein’s gravity; 3) both static and spinning black holes with Skyrme hair, the latter with rotation in two orthogonal planes, with both angular momenta of equal magnitude. In the absence of gravity we present an analytic solution that satisfies a BPS-type bound and explore numerically some of the non-BPS solutions. In the presence of gravity, we contrast the solutions to this model with solutions to a complex scalar field model, namely boson stars and black holes with synchronised hair. Remarkably, even though the two models present key differences, and in particular the Skyrme model allows static hairy black holes, when introducing rotation, the synchronisation condition becomes mandatory, providing further evidence for its generality in obtaining rotating hairy black holes.

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Skyrmions, Skyrme stars and black holes with Skyrme hair in five spacetime dimension

HJE Skyrmions, Skyrme stars and black holes with Skyrme hair in ve spacetime dimension Yves Brihaye 0 2 4 Carlos Herdeiro 0 2 Eugen Radu 0 2 D.H. Tchrakian 0 1 2 3 Belgium 0 2 Maynooth 0 2 Ireland 0 2 0 10 Burlington Road, Dublin 4 , Ireland 1 Department of Computer Science, Maynooth University 2 Campus de Santiago , 3810-183 Aveiro , Portugal 3 School of Theoretical Physics 4 Physique-Mathematique, Universite de Mons-Hainaut 5 Dublin Institute for Advanced Studies We consider a class of generalizations of the Skyrme model to ve spacetime dimensions (d = 5), which is de ned in terms of an O(5) sigma model. A special ansatz for the Skyrme eld allows angular momentum to be present and equations of motion with a radial dependence only. Using it, we obtain: 1) everywhere regular solutions describing localised energy lumps (Skyrmions); 2) Self-gravitating, asymptotically non-singular solitonic solutions (Skyrme stars), upon minimally coupling the model to Einstein's gravity; 3) both static and spinning black holes with Skyrme hair, the latter with rotation in two orthogonal planes, with both angular momenta of equal magnitude. In the absence of gravity we present an analytic solution that satis es a BPS-type bound and explore numerically some of the non-BPS solutions. In the presence of gravity, we contrast the solutions to this model with solutions to a complex scalar eld model, namely boson stars and black holes with synchronised hair. Remarkably, even though the two models present key di erences, and in particular the Skyrme model allows static hairy black holes, when introducing rotation, the synchronisation condition becomes mandatory, providing further evidence for its generality in obtaining rotating hairy black holes. Black Holes; Classical Theories of Gravity 1 Introduction and motivation 2 The O(5) Skyrme model in d = 4 + 1 dimensions 2.1 A BPS bound and the topological charge 2.2 Coupling to gravity 2.3 A codimension-1 Skyrme eld ansatz 3 Flat spacetime Skyrmions 3.1 3.2 A spherical BPS Skyrmion in the quartic model The general model 3.2.1 3.2.2 3.2.3 The e ective action and densities A virial identity and scaling The solutions 4 Skyrme stars 4.1 Spherical stars 4.2 Spinning stars 5 BHs with Skyrme hair 5.1 Spherically BHs 5.2 Spinning BHs 5.1.1 5.1.2 The probe limit | Skyrmions on a Schwarzschild BH background Including backreaction 5.2.1 The probe limit | Skyrmions on a Myers-Perry BH background 5.2.2 Including backreaction 6 Conclusions A Solutions with a quartic term B O(D + 1) Skyrme models on a RD-Euclidean space 1 Introduction and motivation The Skyrme model has more than fty years of history, starting from Skyrme's original construction and its basic solution, with unit baryon number [1, 2]. It provided the very rst explicit example of solitons in a relativistic non-linear eld theory in d = 3 + 1 spacetime dimensions. Such solutions, dubbed Skyrmions, have found interesting applications, e.g. as an e ective description of low energy QCD [3, 4] and on the issue of proton decay [5]. { 1 { In its simplest version, the Skyrme model is described by a set of four scalars f ag, a = 1; ::; 4, satisfying the sigma-model constraint a a = 1, with a target space1 S3 SU( 2 ) and a Lagrangian density possessing a global O( 4 ) symmetry. In addition to a standard (quadratic) kinetic term and a potential, the model contains an extra (quartic) term which is fourth order in derivatives2 allowing it to circumvent Derrick's no-go theorem for nite energy lump-like solutions in eld theory [9]. These Skyrmions possess some topological properties, being characterized by the homotopy class 3(SU( 2 )) = Z. Moreover, the energy functional of the Skyrme model has a Bogomol'nyi-type bound, in terms of the topological charge B associated with the homotopy, which is identi ed as the baryon number. The solutions of the Skyrme model have been studied intensively over the last decades. can be found in the monograph [13]. Skyrmions persist after taking their gravitational backreaction into account, within the Skyrme model minimally coupled to Einstein's gravity (even including a cosmological constant, e.g. [14]). The properties of such, hereafter Skyrme stars, have been considered in [15{20]. Moreover, and following a generic rule in gravitating solitons [21{23], these starlike con gurations are compatible with the addition of a (small) horizon at their center. In the Skyrme model case, this construction was carried out by Luckock and Moss [24], and provided the rst physically relevant counter-example to the no-hair conjecture [25]. The construction in [24] was performed in the probe limit, i.e. a Skyrme test eld on a Schwarzschild black hole (BH) background, but subsequent work included the backreaction [26]. This results in BHs with (primary) Skyrme hair, some of the solutions being stable against spherical linear perturbations (a review of these aspects can be found in [27]; see also [28{32] for more recent work). Most of the known Skyrme-model solutions are static, but spinning generalizations have been constructed. Spinning at spacetime Skyrmions were considered by Battye, Krusch and Sutcli e [33]. In contrast to the (conceptually simpler) spinning Q-ball solutions [34, 35], for spinning Skyrmions the angular momentum J is a continuous parameter that can be arbitrarily small, so that they can rotate slowly and rotating solutions are continuously connected to static ones. The e ects of gravity on these spinning Skyrmions has been studied by Ioannidou, Kleihaus and Kunz in [36], revealing, in particular, a number of con gurations which do not have a at space limit. No rotating BHs with Skyrme hair, however, have been constructed so far, presumably due to the complexity of the numerical problem. As we shall see, the complexity just mentioned can be considerably alleviated by considering the generalisation of the Skyrme model to higher (odd) spacetime dimensions. 1Usually the SU( 2 ) group element U is employed, which in terms of the real eld a is given by U = a a, with a = (i i; 1I), and U 1 = U y = a ~a, with ~a = ( i i; 1I), where i are the standard Pauli matrices. 2Higher derivative terms, up to the allowed sextic kinetic term, can also be included as corrections [6, 7] and are generally expected, see e.g. the recent work [8]. { 2 { Moreover, in recent years, the interest in eld theory solutions in d 6= 4 increased signi cantly. A recurrent lesson has been that well known results in d = 4 physics do not have a simple extension to higher dimensions. For example, the BH solutions in d > 5 models of gravity are less constrained, with a variety of allowed horizon topologies [37]. In the Skyrme case, however, the only other dimensions considered so far in the literature is d = 3 [38{41], even though (gravitating) Skyrmions should also have d > 5 generalizations. In this paper we shall consider a higher dimensional Skyrme model with the main goal of testing the existence of rotating BHs with Skyrme hair, but also to understand how the spacetime dimension a ects standard results for Skyrme physics. The Skyrme system to be addressed is a generalisation of the usual d = 4 Skyrme model, containing higher derivative HJEP1(207)3 terms in addition to the standard ones. This can be done, in principle, in all (including even) dimensions; for concreteness we restrict our attention to the d = 4 + 1 case, where we have carried out a detailed numerical study. A technical advantage of this case is the possibility to consider con gurations with two equal angular momenta, for which there is a symmetry enhancement. As such, the problem results in a system of ordinary di erential equations (ODEs) which are easier to study. We shall also contrast the Skyrme model with the better known and conceptually simpler model of a complex scalar eld, for which one also nds at spacetime eld theory solutions (Q-balls [34, 35, 42]), gravitating solitons (boson stars [43]) and hairy BHs (Kerr BHs with scalar hair [44{46]). In particular for the latter there are no static hairy BHs. But in agreement with the latter, rotating BHs with Skyrme hair also require a synchronisation condition, as described below. The paper is organized as follows. In section 2 we introduce the model together with a codimension-1 Skyrme eld ansatz. The non-gravitating and gravitating solitons of the model | Skyrmions and Skyrme stars | are discussed in sections 3 and 4. BHs with Skyrme hair are presented in section 5. In sections 3{5 we study both static and rotating solutions. We give our conclusions and remarks in section 6. Numerical solutions with a quartic term in the Skyrme action are reported in appendix A. Appendix B contains a discussion of the Skyrme model for a general number of spacetime dimensions. Conventions and numerical method. Throughout the paper, mid alphabet latin let ters i; j; : : : label spacetime coordinates, running from 1 to 5 (with x 5 = t); early latin letters, a; b; : : : label the internal indices of the scalar eld multiplet. As standard, we use Einstein's summation convention, but to alleviate notation, no distinction is made between covariant and contravariant internal indices. The background of the theory is Minkowski spacetime, where the spatial R4 is written in terms of bi-polar spherical coordinates, ds2 = dr2 + r2(d 2 + sin2 d'12 + cos2 d'22) dt2 ; (1.1) where 2 [0; =2] is a polar angle interpolating between the two orthogonal 2-planes and ('1; '2) 2 [0; 2 ] are azimuthal coordinates, one in each 2-plane. r and t denote the radial and time coordinate, 0 6 r < 1 and 1 < t < 1, respectively. { 3 { For most of the solutions the numerical integration was carried out using a standard shooting method. In this approach, we evaluate the initial conditions at r = 10 5 (or r = rH +10 5) for global tolerance 10 14, adjusting for shooting parameters and integrating towards r ! 1. The spinning gravitating solutions were found by using a professional software package [47]. This solver employs a collocation method for boundary-value ODEs and a damped Newton method of quasi-linearization. A linearized problem is solved at each iteration step, by using a spline collocation at Gaussian points. An adaptive mesh selection procedure is also used, such that the equations are solved on a sequence of meshes until the successful stopping criterion is reached. 2 The O(5) Skyrme model in d = 4 + 1 dimensions a The Skyrme model can be generalised to an arbitrary number of dimension | cf. appendix B. Restricting to d = 4 + 1 spacetime dimensions, the Skyrme model is de ned P in terms of the O(5) sigma model real elds f ag, a = 1; : : : ; 5, satisfying the constraint a a = 1. It proves useful to introduce the following notation for all the allowed kinetic terms, quadratic, quartic, sextic and octic: HJEP1(207)3 in terms of which the kinetic terms are written: a ( 1 ) = i ab ( 2 ) = ij abc ( 3 ) = ijk abcd ( 4 ) = ijkl iajbkc ld + jakbcl id + kli j ; abc d abcd abcd i1j1k1l1 i2j2k2l2 gi1i2 gj1j2 gk1k2 gl1l2 : Observe that a;i = agij , ab;ij = j kablgikgjl, etc., where gij is the metric tensor of the ve dimensional background geometry. We shall always include in the action the quadratic term F 2. Then, for most solutions in this paper, as will be justi ed in section 3.2.2 by a virial identity, only the sextic term F 6 is also needed, so we will eschew the octic terms, which do not bring any qualitative features to the solutions. We shall likewise drop the quartic term F 4, though in appendix A we verify that its inclusion does not change the general features. The sole exception to this pattern is the BPS solution of section 2.1 which relies only on the quartic term. The Lagrangian density of the model considered throughout is LS = 2 F where V is the Skyrme potential whose explicit form will be discussed later and i > 0 are coupling constants. Observe these are dimensionful constants: [ 0] = length 5, [ 1] = { 4 { (2.1) (2.2) for rotating solutions. equations length 3, [ 2] = length 1 and [ 3] = length. Inclusion of the potential term is mandatory From the Lagrangian (2.2) it follows that the scalars a satisfy the Euler-Lagrange da d a 2 1 ri ia + 8 2 i r b;k j jakb;i + 9 3 bc;jkri iajbkc + 0 = 0 : (2.3) A BPS bound and the topological charge The model (2.2) with 0 = 1 = 3 = 0, i.e. the action IS = possesses some special properties, provided the spacetime geometry is ultra-static (gtt = 1) and the Skyrme ansatz has no time dependence. Then the resulting system lives e ectively on a four dimensional space with Euclidean signature (which, however, can be curved), being conformal invariant. After de ning the two-form Hodge dual of iajb as ? iajb = p g aba1b1c iji1j15 a1b1;i1j1 c ; we can state the Bogomol'nyi inequality which implies that the mass-energy of the model is bounded from bellow a1a2 i1i2 ? a1a2 2 i1i2 > 0 ; M > 2 4 B ; where B is the topological charge B = Z d x 4 p g T ; with while the topological current is M = exist,3 being solutions of the 1st order equations As we shall see in the next section, self-dual solutions saturating the above bound It is clear that the bound (2.7) holds as well for the general Lagrangian (2.2), in which case it can never be saturated, since the contribution of the supplementary terms is always positive. 3This contrasts with the Skyrmions on R3 where no BPS solitons exist, while they do on S3 [48]. iajb = ? iajb : { 5 { (2.5) (2.6) (2.7) (2.8) (2.9) (2.10) (2.11) where LS is the Lagrangian density (2.2) for the Skyrme sector and G is Newton's constant. Variation of (2.12) w.r.t. the metric tensor leads to the Einstein equations where the energy-momemtum tensor is in terms of the contributions of the distinct terms in (2.2), which read Rij 1 2 gij = 8 G Tij ; Tij = 0Ti(j0) + 1Ti(j1) + 2Ti(j2) + 3Ti(j3) ; Ti(j0) = Ti(j1) = i j a a gij V ( a); 1 2 gij F 2; Ti(j2) = iakb1 jakb2 gk1k2 Ti(j3) = 1 6 1 4 gij F 4; ik1l1 jakbc2l2 gk1k2 gl1l2 abc 1 6 gij F 1 + i 2 = sin F (r) sin ei('1 !t) ; 3 + i 4 = sin F (r) cos ei('2 !t) ; 5 = cos F (r) : As usual, for a given ansatz, the gravity equations (2.13) are solved together with the matter equations (2.3), subject to some physical requirements (e.g.asymptotic atness and niteness of the total mass). 2.3 A codimension-1 Skyrme eld ansatz The O(5) solutions in this work are constructed within a Skyrme elds ansatz in terms of a single function F (r): Here, w > 0 is an input parameter | the frequency of the elds. The corresponding expression of the topological charge density is: For the Skyrme potential we shall take the usual `pion mass'-type which is a natural generalization of that used in the d = 4 model. A remarkable feature of the ansatz (2.16), rst suggested in [49], albeit for a complex doublet rather than a Skyrme eld, is that for any geometry in this work the angular dependence is factorized in a consistent way, and the Skyrme equations (2.3) reduce to a single ODE for the function F (r). { 6 { (2.12) (2.13) (2.14) (2.15) (2.16) (2.17) (2.18) Flat spacetime Skyrmions We start by considering solutions in the probe limit, i.e. we solve the Skyrme equations on a xed spacetime background. Apart from being technically simpler, we shall nd that they possess already a number of basic properties of the corresponding gravitating generalizations. A spherical BPS Skyrmion in the quartic model In the simplest spherically symmetric case and a at background, the model with only a quartic term (2.4) allows a simple analytical solution, found with the ansatz (2.16) (with ! = 0). Then the rst order eqs. (2.6) reduce to4 Restricting to the plus sign,5 the solution of the above equation reads with r0 > 0 an arbitrary parameter. This is an everywhere regular con guration, with F (r) interpolating between (at r = 0) and zero (at r = 1). Its energy density is F 0 sin F r = 0 : F (r) = 2 arctan r0 r ; (r) = Ttt = 96 2r04 ; while all other components of Tij vanish. The total mass of solution is It would be interesting to investigate the existence of static non-selfdual solutions of the simple model (2.4). They are expected to exist, in analogy with instanton-anti-instanton solutions to Yang-Mills theory [51]. 3.2 3.2.1 The general model The e ective action and densities The simple quartic model (2.4) is too restrictive. Indeed, the self-duality eqs. (2.11) cannot be satis ed for gtt 6= 1 or ! 6= 0. The situation in this restrictive case is the same as what occurs in the Einstein-Yang-Mills system [52], where the solutions of the usual Yang-Mills (YM) model F ( 2 )2 do not survive when considering their backreaction on the spacetime geometry. This is because the scaling requirement is violated. In that case, this defect was remedied by adding the higher-order YM term F ( 4 )2 and regular gravitating YM solutions were constructed for d = 6; 7 spacetime dimension [53]. Subsequently, the case d = 5 was considered as well [54], where also BH solutions were constructed. 4It is interesting to note that the solution (3.2) is related to the radially symmetric BPST instanton [50] described by the form factor w(r), via w(r) = cos F . 5The minus sign solution reads F (r) = 2 arctan(r0=r) and possesses similar properties. { 7 { Despite the fact that this describes an e ective one-dimensional system, the con gurations with ! 6= 0 are not spherically symmetric6 and carry an angular momentum density, j, j T't 1 sin2 = T't 2 cos2 = ! sin2 F 1 + 2 2 F 02 + gTtt = 2 2 dr r3 ; J1 = J2 = J = Z d x 4 p gT't 1;2 = 2 0 S = where For the Skyrme system, the F 4 density scales as L 4, while the usual gravity scales as L 2. Thus, as in the YM case, it is necessary to add higher-order terms. In the general case with all i 6= 0 and a Skyrme ansatz given by (2.16), one can show that the equation for F (r) can also be derived from the e ective action drLe ; with Le = r3 0(1 Following d = 5 BH physics conventions, we de ne also the reduced angular momentum of a spinning con guration as 3.2.2 A virial identity and scaling The form (3.5) of the reduced allows the derivation of an useful Derrick-type virial relation. Let us assume the existence of a globally regular solution F (r), with suitable boundary 6This contrasts with the complex scalar elds model in [49], which possesses spherically symmetric Q-ball solutions supported by the harmonic time dependence of the elds. j 27 J 2 8 M 3 : { 8 { conditions at the origin and at in nity. Then each member of the 1-parameter family F ( r) assumes the same boundary values at r = 0 and r = 1 and the action S(F ) must have a critical point at = 1, i.e. [dS =d ] =1 = 0. This results in the following virial identity satis ed by the nite energy solutions of the eld equations sin4 F r4 The positivity of all terms in the previous relation shows that the existence of d = 5 Skyrmions can be attributed to a balance between: the attractive interaction provided by the usual kinetic term in the Skyrme action together with the potential (left hand side terms); and a repulsive interaction provided by the sextic term, plus the centrifugal force in the rotating case (right hand side terms). Also, one can see that, as anticipated above, the contribution of the quadratic term occurs with an overall w2-factor only, and is not mandatory for the existence of solutions. The Skyrme Lagrangian (2.2) contains four input parameters i. However, the constant multiplying the quadratic term can be taken as an overall factor for the Skyrme action. Also, the equation for F is invariant under the transformation r ! r, 0= 1 ! 2= 1 ! 2=( 2 1), 3= 1 ! 3=( 4 1) together with w ! w= , which can be used to x the value of one of the constants 0; 2 or 3 . Then following the d = 4 case, we de ne a characteristic length L and mass MS of the Skyrmion system as given by the constants multiplying the quadratic and potential terms, with L r 1 0 ; MS 2 1 ; 0 (3.12) the numerical results being obtained in units set by L and MS. However, to avoid cluttering the output with a complicated dependence of L; MS we shall ignore these factors in the displayed numerical results. The problem still contains two free constants which multiply the quartic and sextic terms in the Skyrme action. Moreover, in the presence of gravity, one extra parameter occurs. Therefore, the determination of the domain of existence of the solutions would be a lengthy task. In this work, in order to simplify the picture, we have chosen to solve a model without the quartic term and with a unit value for the parameter multiplying the sextic term. Thus, all reported numerical results below are found for the following choice of the coupling constants 0 = 1 = 3 = 1; 2 = 0 : (3.13) To provide evidence that this choice does not restrict the generality of our results, we present, in appendix A, solutions including the presence of a quartic term in the action, 2 6= 0, which indeed does not appear to a ect the qualitative properties of the solutions. { 9 { The function F (r) satis es a non-enlightening second order di erential equation which we shall not display here (its ! = 0 limit can be read o by setting N (r) = (r) = 1 in (4.8)). This equation does not seem to possess exact solutions and it is solved numerically with the boundary conditions7 F (0) = regularity requirements. and F ( 1 ) = 0, which follow from nite energy and The asymptotics of F (r) can be systematically constructed in both regions, near the origin and for large r. The corresponding expression for small r is F (r) = + f1r + f3r3 + O(r5); with f3 = f1 0 + 1(2f12 + !2) + 6f14( 2 12[ 1 + 6f12( 2 + f12 3)] 3!2) ; in terms of a single undetermined parameter f1 < 0. One notice that the energy density at r = 0 is nonzero, with while the angular momentum density j, given by (3.7), vanishes as O(r2). An approximate solution valid for large r can be written in terms of the modi ed Bessel function of the second kind, K, F (r) c r " K 2; r r 0 1 !2 # c r 2 e r q 01 !2 r3=2 + : : : ; (3.16) (with c a constant), which shows the existence of an upper bound on the scalar eld frequency, ! p 0= 1 = 1=L, the solutions becoming delocalized for larger values of !. Therefore, similar to other examples of spinning scalar solitons (e.g. [34, 35]), the presence of a potential term in the action, 0 6= 0, is a pre-requisite for the existence of nite mass solutions. Note, however, that we have found numerical evidence for the existence of static solitons with V = 0, which decay as 1=r3 at in nity. Solutions interpolating between (3.14) and (3.16) are easily constructed | gure 1. In our approach, the control \shooting" parameter is f1 which enters the near origin expansion (3.14) | see gure 1, inset of right panel. For a given frequency !, a single nodeless solution is found for a special value of f1. The pro les of the energy density and of the angular momentum density of typical solutions are shown in gure 1, including the static one, which has ! = 0. In gure 2 we display the total mass and angular momentum of the spinning Skyrmions as a function of their frequency. One can see that both quantities increase monotonically with !, with a smooth ! ! 1 limit which maximizes the values of mass and angular momentum. Also, at least for the considered values of the coupling constants, the Skyrmions are never fast spinning objects, with a reduced angular momentum j always much smaller than one. 7Solutions with F (r) interpolating between k (with k > 1) at r = 0 and F ( 1 ) = 0 do also exist. However, they are more massive and likely to be unstable. 4.5 2 ω=0.75 ω=0.25 1 -1.2 1 f-1.4 0 2 r background are shown together with their energy density. Right panel: the distribution of the angular momentum density for spinning Skyrmions. The inset shows the values of the parameter f1 which enters the small r expression of the solutions. ) r ( F ω=0.95 ω=0 60 T t -T30 0 0 ω=0 Skyrmions on a at spacetime background are shown as a function of their frequency !. The above solutions possess gravitating generalizations, which are found by solving the Skyrme equation (2.3) together with the Einstein equations (2.13). A suitable metric for spherically symmetric con gurations reads +r2(d 2 +sin2 d'12 +cos2 d'22) N (r) 2(r)dt2 ; where N (r) 1 the function m(r) being related to the local mass-energy density up to some overall factor. For static, spherically symmetric solutions, the scalar ansatz is still given by (2.16) with m(r) r2 ; (4.1) Ls = r 3 This form of the system allows us to derive, following [55, 56] a generalization of the at spacetime virial identity (3.11). Following the same reasoning as in section 3.2.2, we nd that the nite energy solutions satisfy the integral relation which clearly shows that nontrivial gravitating solutions with nite mass cannot exists in a model without the sextic term. Indeed, in that case the right hand side would vanish and all terms in the integrand of the left hand side of (4.4) would either vanish or be strictly positive, making the equality impossible for a non-trivial con guration. Observe that turning on gravity adds an extra attractive term, in addition to those provided by the quadratic and potential terms. By using the same dimensionless radial coordinate and rescaling as in the nongravitating case (together with m ! m=L2), one nds that the gravitating system possess one extra dimensionless coupling constant 2 = 4 G 13=2= 10=2 : Then the Einstein equations used in the numerics reduce to (recall that we set 0 = 1 = 1 and 2 = 0, 3 = 1) The equations of the model can also be derived from the reduced Lagrangian: and where (4.2) (4.3) HJEP1(207)3 (4.5) (4.6) (4.7) (4.8) m0 = 0 = 3 r 2 + 3 together with an extra constrain equation. The function F (r) satis es the 2nd order sin F N 1 + 6 2 sinr22 F + 6 3 sinr44 F 1 + 1 r2 sin2 F P1 F(r) where m(r) 1 α/ panel: the total mass M , the control parameter f1 and the value of the metric function origin are shown as a function of coupling constant, , for spherically symmetric Skyrme stars. P1 = 3 2 sin(2F ) rF 0 rN 0 + N 3 + ; r 0 P2 = cos(3F ) + cos F (2N r2F 02 P3 = cos(3F ) + cos F (8N r2F 02 1) + 2rF 0 sin F rN 0 + N 1 + ; (4.9) 1) + 4rF 0 sin F rN 0 + N 1 + : r 0 r 0 These equations are solved by imposing the boundary conditions8 at the origin (r = 0), while at in nity, the solutions satisfy N (0) = 1; (0) = 0 > 0; F (0) = ; N ! 1; ! 1; F ! 0 : The properties of the spherically symmetric Skyrme stars can be summarized as follows. For all studied cases, m(r), (r), and F (r) are monotonic functions of r,9 the pro le of a typical solution being presented in gure 3 (left panel). For small values of there is a fundamental branch of solutions that reduces to the at space Skyrmion as increases, the mass parameter M decreases, as well as the value (0). The solutions exist up to a maximal value max of the parameter . At the same time, the absolute value of the \shooting" parameter f1 increases with . We found that a secondary branch of solutions emerges at max, extending backwards in branch, both (0) and f1 decrease as | gure 3 (right panel). Along this second decreases, while the value of the physical mass ! 0. When 8An approximate form of the solutions compatible with these conditions can easily be constructed. For example, the small r expansion contains two essential parameters, F (0) and (0). 9Excited solutions with nodes in the pro le of F , which would therefore be a non-monotonic function, are likely to exist as well. (4.10) (4.11) M= 2 strongly increases. Some understanding of the limiting behaviour can be obtained by noticing that the ! 0 limit can be approached in two di erent ways, as G ! 0 ( at space, rst branch) or as 1 ! 0 (second branch). Then we conjecture that the limiting solution on the upper branch corresponds to a gravitating model without the F 2-term in the Skyrme Lagrangian. d = 5 rotating spacetimes generically possess two independent angular momenta. Here, however, we focus on con gurations with equal-magnitude angular momenta which are compatible with the symmetries of the matter energy-momentum tensor we have chosen. A suitable metric ansatz reads (note the existence of a residual gauge freedom which will HJEP1(207)3 ds2 = + g(r)d 2 + h(r) sin2 [d'1 W (r)dt]2 + h(r) cos2 [d'2 W (r)dt]2 (4.12) h(r)] sin2 cos2 (d'1 d'2)2 b(r)dt2 : For such solutions the isometry group is enhanced from Rt U( 1 )2 to Rt U( 2 ), where Rt denotes the time translation. This symmetry enhancement allows factorizing the angular dependence and thus leads to ordinary di erential equations. The angular momentum and energy densities are given by j = = T't1 sin2 (! = W ) b(r) T't2 cos2 The complete ansatz, (4.12) and (2.16), can be proven to be consistent, and, as a result, the Einstein-Skyrme equations reduce to a set of ve ODEs (in the numerics, we x the metric gauge by taking g(r) = r2). We seek asymptotically at solutions, subject to the following boundary conditions as r ! 1: f = b = g(r)=r2 = 1 and F = W = 0. The total (ADM) mass M and angular momenta J1 = J2 = J , are read o from the asymptotic behaviour of the metric functions, gtt = 1 + 8GM 3 r2 + : : : ; g'1t = 4GJ r2 sin2 4GJ r2 cos2 + : : : ; g'2t = + : : : : (4.13) The behaviour of the metric functions at the origin is f = 1, b = b0 > 0, g(0) = 0, together with W = W0 > 0, while F = , as in the probe limit. 0.9 0.8 b(r) 0 2 3 In gure 4, we display the pro les of the a typical spinning Skyrme star. The corresponding distribution for energy and angular momentum densities look similar to those shown in gure 1 for the probe limit. The mass/angular momentum of solutions vs: frequency are given as a limiting curve in gure 11, for a particular value of . Note that similar to the probe limit, both quantities monotonically increase with !. Also, in the limit ! ! 1, the solutions are still localized, without any special features, while they disappear for ! > 1. Finally we emphasise that all solitons in this work, gravitating or otherwise, possess unit topological charge, as expected. 5 BHs with Skyrme hair BH generalizations are generically found for any regular solitonic-like gravitating con guration, at least for small values of the horizon radius rH . In this section we shall show this trend remains true for the Einstein-Skyrme model discussed herein, as con rmed by our numerical results. 5.1 5.1.1 Spherically BHs The probe limit | Skyrmions on a Schwarzschild BH background Similarly to the d = 4 case, it is useful to consider rst the probe limit and solve the Skyrme equations on a d = 5 Schwarzschild-Tangherlini BH background [57]. The corresponding line element is given by (4.1) with N (r) = 1 rH2 =r2 and (r) = 1, where rH > 0 the event horizon radius. The approximate form of the solution close to the horizon reads in terms of the \shooting" parameter f0, with F (r) = f0 + f1(r rH ) + O(r rH )2; f1 = sin(f0) 0rH6 + 3 cos(f0) r H 1 + 2 3 sin4 f0 4 2rH (rH4 1 + 6 3 sin2 f0) : (5.1) (5.2) 1.5 0 1 2 radius rH , for Skyrme probe solutions on a Schwarzschild BH background. The mass of the solutions is still computed from (3.9), with the corresponding curved spacetime expressions and r = rH as a lower bound in the integral (the same holds for a MP background). with F (rH ) ! F (rH ) ! =2. The results of the numerical integration are shown in gure 5 (note that the typical pro le of the function F (r) is similar to that exhibited in the gravitating case, cf. gure 6). One can see that the solutions exist up to a maximal horizon radius rH of the Schwarzschild background, with a double branch structure for a range of rH . The solutions of the fundamental branch (with label 1 in gure 5) terminate in the at spacetime solitons as rH ! 0, in that limit. Along this branch, the mass of the solutions decreases with increasing rH . The second branch (with label 2 in ues again to rH ! 0, in which limit however, the mass of the solution M diverges, while gure 5) starts at r(max) and contin H We remark that the Skyrme solutions on a spacetime geometry with an event horizon possess a non-integer topological charge, B = 1 + 2 cos2 1 2 F (rH ) sin4 1 2 F (rH ) ; (5.3) belonging to the interval 1=2 < B 1. In fact, the above expression holds for all BH solutions in this work, including the rotating ones. 5.1.2 Including backreaction The inclusion of gravity e ects is straightforward. The BH solutions are constructed within the same ansatz used for solitons. They satisfy the following set of boundary conditions at the horizon (which is located at r = rH , with 0 < rH 6 r < 1) N (rH ) = 0 ; (rH ) = H > 0 ; F (rH ) = f0 : (5.4) 2 F(r) 1 n irzo o t n e 1 ev 0 0 10000 2 α / M 1000 0 m(r) hair. Right panel: the values, at the horizon, of the Ricci scalar R and of the metric function are shown as functions of the horizon radius rH , for several values of . 100 50 0 0 3 2 H ) r ( F α=0.1 α=0.05 (rH0.5 σ 0 0 H functions of the horizon radius rH , for several values of . The far eld behaviour is similar to that in the solitonic case. We note that these BHs possess a Hawking temperature TH and a horizon area AH , which read 1 4 TH = N 0(rH ) (rH ) ; AH = 2 2rH3 : The pro le of a typical BH solution is shown in gure 6. The behaviour of the solutions H as a function of rH is presented in gure 7, for several values of the coupling constant . The properties of the spherically symmetric BHs can be summarized as follows. Starting from any regular solution, i.e. a Skyrme star, with a given and increasing the event horizon radius, one nds a rst branch of solutions which extends to a maximal value r(max). This maximal value decreases with increasing . This branch is the backreacting counterpart of the corresponding one in the probe limit. The Hawking temperature and the value of F (rH ) decrease along this branch, while the mass parameter increases; however, the variations of the mass and of (rH ) are relatively small. Extending backwards in rH , we nd a second branch of solutions. This second branch extends up to a critical value of horizon radius r(cr) where an essential singularity seems to occur. An understanding of the limiting solutions requires a reformulation of the problem with a di erent coordinate system [58] which is beyond the scope of this work. Here we note that the value of (rH ) on this branch decreases drastically and appears to vanish as rH ! rH(cr). As a result, the Ricci scalar evaluated at the horizon strongly increases in that limit. However, the mass remains nite, while the Hawking temperature goes to zero. The pro le of the F -function does not exhibit any special features in the limit, starting always at some value F (rH ) > =2. This special behaviour on the second branch can partially be understood as a manifestation of the divergent behaviour we have noticed in the probe limit. We also mention that for the region of the parameter space where two di erent solutions exist with the same mass, the event horizon area (i.e. the entropy) is always maximized by the fundamental branch of the BH, see gure 7 (left panel). Thus we expect the upper branch solutions to be always unstable. Spinning BHs The probe limit | Skyrmions on a Myers-Perry BH background The static hairy BHs we have just described possess rotating generalizations. However, before considering solutions of the full Einstein-Skyrme system, it is again useful to consider rst the probe limit and to solve the matter eld equations on a spinning BH background. The corresponding BH is a d = 5 Myers-Perry (MP) solution [59] with two equal-magnitude angular momenta. Such a BH can be expressed as a particular case of the ansatz (4.12), r r r and it is parameterised in terms of the event horizon radius rH and the horizon angular velocity H , which are the control parameters in our numerical approach. For completeness, we include the expression of quantities which enter the thermodynamics of a MP BH 3 r 8(1 1 2 rH q 2 H 2 r2 ) ; H H 1 1 2 2H rH2 ; 2 r2 H H 1 J (MP ) = J (MP ) = J (MP ) = A(HMP ) = 2 q 1 2 2r3 H Here it is important to note the existence of a maximal size of the horizon radius rH for a given value of horizon angular velocity H . This corresponds to a zero temperature BH with r(max) = 1=(p2 H ). There, the reduced angular momentum (3.10) approaches its H maximal value jMP = 1. n e v e 1.5 -Ttt/4 r j 3 together with the corresponding energy and angular momentum densities. At in nity, the decay of the eld is still given by (3.16), such that F ! 0: Remarkably, the assumption of existence of a power series expansion of F (r) as r ! rH implies that, similar to other hairy rotating BH solutions [44, 66, 67], the synchronization condition ! = H near the horizon (with 2 = 0), which, up to order O(r rH ) reads: necessarily holds. This condition is also implied by the regularity of energy and angular momentum densities as r ! rH . Then the function F (r) possess an approximate solution F (r) = f0+ (1 rH2 !2) 4rH (2rH2 !2 1)[rH2 1 + 2 3(3 2rH2 !2) sin4 f0] 2 0 sin f0 + 1rH4 (rH2 !2 1) sin(2f0)+12 3(rH2 !2 1) cos f0 sin f05 (r rH ) ; all higher order coe cients being determined by F (rH ) > 0. The pro les of a typical solution on a given MP background are shown in gure 8. The dependence of the properties of the solutions on the horizon size, as given by A(HMP ) and reduced angular momentum j(MP ), is shown in gure 9. The basic picture found in the static case is still valid here, with the existence of two branches of solutions for a given BH background. The fundamental branch emerges from the at spacetime solitons, while the mass and angular momentum of the solutions appear to diverge as the at spacetime limit (rH ! 0) is approached, along the second branch. We remark that no Skyrme solutions exist on a fast rotating MP background, i.e. with j(MP ) close to unity. Also, one can see that the mass branches of spinning solutions exhibit a `loop', when considered as a function of horizon properties (or reduced angular (5.7) (5.8) momentum) of the MP background. 5.2.2 Including backreaction The con gurations described in the last subsection survive when taking into account their backreaction on the spacetime geometry. The spinning BHs are constructed within the same ansatz as for the spinning solitons discussed in section 3.2.2 (in particular we set M 400 200 0 ω=0.1 100 50 0 0 0.3 400 100 50 0 0 M = J(S) + M(S) ; 3 where M(S) and J(S) are the mass and angular momentum stored in the Skyrme eld while their mass and angular momentum are read from the far eld expansion (4.13). As usual, the temperature, horizon area and the global charges M; J are related through the Smarr mass formula outside the horizon, M(S) = 3 Z p 2 are shown as a function of horizon area and of the reduced MP angular momentum. again g(r) = r2). However, they possess an horizon which is a squashed S3 sphere. The horizon resides at the constant value of the radial coordinate r = rH > 0, and it is characterized by f (rH ) = b(rH ) = 0. Restricting to nonextremal solutions, the following expansion of the metric functions holds near the event horizon: f (r) = f1(r rH ) + f2(r rH )2 + O(r rH )3; h(r) = hH + h2(r rH ) + O(r rH )2; b(r) = b1(r rH ) + b2(r rH )2 + O(r rH )3; W (r) = H + !1(r rH ) + O(r rH )2; while F (r) = f0+f1(r rH )+: : : . For a given event horizon radius, the essential parameters characterizing the event horizon are f1; b1, hH , H and !1 (with f1 > 0; b1 > 0), which x all higher order coe cients in (5.9). The construction of the approximate near-horizon solution shows that, as expected, the synchronization condition (5.7) still holds in the backreacting case. As for a general MP BH, the (constant) horizon angular velocity H is de ned in terms of the Killing vector solutions within the ansatz (4.12), the horizon angular velocities are equal, 1 = 2 = H . The Hawking temperature TH and the area AH of these BHs are xed by the near horizon data in (5.9), with TH = p b1f1 4 ; AH = 2 2phH rH2 ; (5.9) (5.10) (5.11) gdrd d'1d'1 T t t 31 Tii ; J(S) = Z p gdrd d'1d'1T't i : (5.12) 1 0.75 0.5 ω=0 ω=0.95 1 2 1/TH 3 panel: the (horizon area-temperature) diagram is shown for two sets of solutions. In our approach, the input parameters are the coupling constant , the event horizon radius rH and the horizon angular velocity H (or equivalently, the eld frequency !). Physical quantities characterizing the solutions are then extracted from the numerical solutions. The pro les of a typical spinning BH with Skyrme hair are shown in gure 10 (left panel). Some basic properties of the solutions are similar to those found in the probe limit. For example, the scalar eld is always spatially localized within the vicinity of the horizon, the distribution of mass and angular momenta densities being similar to that in gure 8. The emerging global picture can be summarized as follows. For all values of the parameters ; rH that we have considered, the static BHs are continuously deformed while increasing gradually the parameter !. Similarly to the solitonic case, the solutions stop to exist for ! > 1. Moreover, all BHs studied so far have a reduced angular momentum (3.10) much smaller than one. When taking instead a xed value of ! and varying the horizon parameter rH , our results show that, for any 0 6 ! 6 1, a double branch structure of solutions exists, characterized by two particular values of the horizon radius rH . The rst (or main) branch exists for 0 6 rH 6 r(max), emerging from the corresponding (gravitating) Skyrme soliton H in the limit rH ! 0. The second branch exist for rH(c) 6 rH 6 r(max) approaching a critical H con guration as rH ! rH(c). This critical solution possesses nite global charges, a nonzero horizon area while its Hawking temperature vanishes. However, it inherits the pathologies of the static limit, e.g. the Ricci scalar appears to be unbounded on the horizon. The dependence of the horizon area AH nf the (inverse of the) temperature TH is shown in gure 10 (right panel), where we compare the results for static solutions with those for BHs close to the maximal value of !. One notices that the horizon size remains nite as the critical solution is approached. In gure 11 we exhibit the domain of existence of hairy BHs, in a M (!) (and J (!)) diagram for = 0:01, the only value of the coupling constant we have investigated in a systematic way. This domain, in the M -! diagram, has an almost rectangular shape, and is delimited by four curves: the set of static BHs (! = 0), the set of Skyrme stars, the set of maximal mass solutions, and nally the set of maximally rotating BHs with ! = 1. H B H critical HBHs Skyrme Stars 0.5 ω 2/Jα1000 0 0 0.25 0.75 1 0.5 ω critical HBHs frequency diagram (left panel) and as an angular momentum-frequency diagram (right panel). 6 Conclusions In this work we have considered an extension of the Skyrme model to ve spacetime dimensions and investigated the basic properties of its codimension-1 solutions. Concerning the model, two salient properties of gravitating Skyrme systems in d = 3 + 1, can be used to motivate its study. The rst is that the solutions feature BHs hair and the second is that in the gravity decoupling limit the solutions are topologically stable. In 3 + 1 dimensions, gravitating Skyrmions share the rst property with gravitating Yang-Mills [60], which support hair but in the at spacetime limit disappear. The second property they share with that gravitating Yang-Mills-Higgs [61{63] system which supports topologically stable monopoles. The latter persisit in the gravity decoupling limit, whence one notes the closer similarity of gravitating Skyrmions with monopoles in 3 + 1 dimensions. In spacetime dimensions higher than d = 3+1, the situation is rather more complicated because of the restriction set by the Derrick scaling [9] requirement for the niteness of the energy, as can be seen from the nonexistence of nite energy gravitating solutions of the usual quadratic Yang-Mills system in 4 + 1 dimensions [52]. In the Yang-Mills (YM) case, higher order YM densities, e.g. extended YM terms (eYM) like F (2p)2 must be included to satisfy this requirement.10 In d = 6 and 7, this was done in [53] by adding11 the F ( 4 )2 eYM term to the usual F ( 2 )2. Subsequently the solutions for the same model as in [53] were constructed in [54] for d = 5, which displayed some peculiar features that we have encountered in the d = 5 Skyrme case at hand. Not surprisingly the same holds for gravitating Skyrme systems in dimensions greater than 4+1, namely including higher order kinetic (Skyrme) terms, as we have done in this paper. But unlike in the YM case, where the gravity decoupling solutions are topologically stable in even spacetime dimensions only, in the Skyrme case they are stable in all dimensions, like in the monopole case. 10As found in [64, 65], the inclusion of a F ( 4 )2 term in the Yang-Mills action (which is optional in this case) leads to a variety of new features also for d = 4, e.g. the existence of stable hairy non-Abelian BHs. 11In these dimensions, it is possible to add the F (6)2 eYM term too but that was eschewed. More interestingly in d = 6, there exist topologically stable eYM instantons so that those solutions persists in the gravity decoupling limit. This feature persists in all even d = 2n > 4. Concerning the explicit solutions described in this paper, we have considered both at spacetime (Skyrmions) con gurations and self-gravitating solutions (Skyrme stars and BHs with Skyrme hair). Overall, we have unveiled a rich and involved space of solutions. In the spherically symmetric case, the pattern of the d = 4 solutions is recovered, with a branch of gravitating Skyrmions emerging from the at space/Schwarzschild background solutions. A secondary branch of solutions is also found, which, however, possesses a di erent limiting behaviour than in the d = 4 case. On the critical behaviour of spherically symmetric solutions as a function of or rH , we remark that some features resemble the case of a higher dimensional gravitating nonabelian system with higher derivatives terms in addition to the usual F ( 2 )2 one [54]. The clari cation of the critical behaviour therein has required a reformulation of the problem with techniques bases on a xed point analysis of nonlinear ODEs [58]. We expect that a similar approach would help to clarify the critical behaviour of the gravitating solutions in this work. The spinning hairy BH we have reported in this paper are the rst example of a spinning BH with Skyrme hair, since the corresponding d = 3 + 1 solutions have not yet been constructed. One salient feature of these rotating BHs with Skyrme hair is that they possess a static limit. Whereas this is expected, since static BHs with Skyrme hair are known in d = 3 + 1 spacetime dimensions, it contrasts in a qualitative way with the behaviour of other BHs with scalar hair, namely Kerr BHs with scalar hair [44{46] or MP BHs with scalar hair [66, 67]. It is therefore of some interest to expand on the comparison between these two models, since they are both examples of BHs with scalar hair: Skyrmions on Minkowski spacetime are topological solitons, in view of their asymptotic boundary conditions. Q-balls [42], on the other hand, which arise in models of selfinteracting complex scalars elds, but with standard kinetic terms, are perhaps the simplest example of a non-topological soliton. For the latter, the complex nature of the scalar eld is crucial to satisfy Derrick's theorem, allowing for an underlying harmonic time dependence of the scalar eld but that vanishes at the level of all physical quantities, thus yielding static or stationary lumps of energy. Flat spacetime spinning Skyrmions have a static limit; thus they carry an arbitrarily small angular momentum (for given topological charge). Spinning Q-balls, on the other hand, have their angular momentum quantised in terms of their charge [34, 35], which in this case is a Noether charge, due to a U( 1 ) global symmetry. Thus, they have a minimum angular momentum, for given Noether charge, and the spinning solutions are not continuously connected to the static ones. When minimally coupled to Einstein's gravity, self-gravitating Skyrmions become Skyrme stars. But the structure of the model to obtain these solutions remains the same, namely the key higher order kinetic term is still mandatory. When minimally coupled to Einstein's gravity, back reacting Q-balls become boson stars. But gravity can replace part of the key structure of the at spacetime model: one can get rid of the self-interactions potential and keep only a mass term [68, 69], as the HJEP1(207)3 non-linearities of Einstein's gravity are su cient to counter balance the dispersive nature of the scalar eld and create equilibrium boson stars. Likewise, Skyrmions, Skyrme stars can rotate slowly and connect to the static limit, whereas rotating boson stars form an in nite discrete set of families disconnected from static boson stars [35, 70], for any model, with or without self-interactions, in any spacetime dimension. Static Skyrme stars admit placing a BH horizon at their centre, yielding static BHs with Skyrme hair, both in d = 4 [24, 26] and d = 5 (and likely in other dimensions). HJEP1(207)3 Static boson stars do not admit placing a BH horizon at their centre, as shown by the no-hair theorem in [71]. In spite of all these di erences, the spinning BHs with Skyrme hair that we have found in this paper rely on precisely the same condition that the Kerr BHs with scalar hair or MP BHs with scalar hair, the latter being the hairy BH generalisation of spinning boson stars in d = 4 and d = 5. This condition is the synchronisation of the phase angular velocity of the Skyrme eld and the angular velocity of the horizon, eq. (5.7). This is yet another example for the universality of this condition in obtaining spinning hairy BH solutions. Finally, let us remark that possible avenues for future research include: i) the investigation of stability of the considered con gurations (based on the analogy with the d = 4 case, we expect some of the gravitating solutions to be stable); ii) the construction of less symmetric Skyrmions; the simplest case would be the (higher winding number) axially symmetric solutions and con gurations with J1 6= J2. Solutions with discrete symmetry only are also likely to exist in this model; iii) the construction of d > 5 generalizations: in appendix B we present a general framework in this direction. Acknowledgments C.H and E.R. gratefully acknowledge support from the FCT-IF programme. This work was also partially supported by the H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904, the H2020-MSCA-RISE-2016 Grant No. FunFiCO-777740, and by the CIDMA project UID/MAT/04106/2013. A Solutions with a quartic term The reported numerical results have 2 = 0. To test their generality, we have also studied how the inclusion of a quartic term a ects the properties of the solutions, in a number of cases. Starting with the solutions in at spacetime background, we display in gure 12 the mass M as a function of 2 for Skyrmions with several values of the frequency ! (note that all solutions displayed in this appendix are for 0 = 1 = 3 = 1). As expected, one can see that the presence of quartic term in the Skyrme action increases the mass of the solutions, and for ! 6= 0, the same trend applies for the angular momentum. Observe 2 λ 2 4 6 HJEP1(207)3 Skyrmions, with di erent frequencies. ω=0.5 ω=0 λ0=λ1=λ3=1 1 0.75 ) 0 σ( 0.5 0.25 1000 2 α / M500 0 0 λ0=λ1=λ3=1 λ2=1 λ2=0 λ2=1 α 0.1 λ0=λ1=λ3=1 λ2=0 500 0 0 λ2=2 of the coupling constant , for Skyrme stars in our model with di erent values of 2 . that the M ( 2 )-function is almost linear for small frequencies. It is also interesting to mention that the ! = 0 Skyrmions with large enough values of 2 are well approximated by the self-dual solution (3.2), the contribution of the quartic term dominating the system in this case. Turning now to gravitating solutions, we display in gure 13 families of spherically symmetric Skyrme stars with several values of 2 . One can see that the picture found in section 4.1 appears to be generic, with the occurrence of two branches of solutions in terms of . Also, one notices the existence of a maximal value of , which decreases with increasing 2, while the limiting behaviour on the second branch is similar to that found for solutions without a quartic term. Finally, the same conclusion is reached in the presence of an horizon, cf. gure 14, where we show the (rH ; M ) and (rH ; (rH )) diagrams for spherically symmetric BHs with Skyrme hair with a xed value of and three values of 2 . To summarize, the presence of a quartic term in the Skyrme action does not seem to lead to new qualitative features, at least for the range of parameters considered. 0.75 0.25 α=0.025 λ1=3 0.1 r H H λ1=1 BHs with Skyrme hair, as a function of the horizon radius for several values of 2 and a xed . B O(D + 1) Skyrme models on a RD-Euclidean space The Skyrme model [1, 2] in 3 + 1 (i.e., D = 3) dimensions12 is a nonlinear chiral eld theory which supports topologically stable solitons in the static limit. These solitons, which are called Skyrmions, describe baryons and nuclei. In its original formulation [1, 2], the model is described by an SU( 2 ) valued eld U . The Skyrmions are stabilised by a topological charge which is characterised by the homotopy class 3(SU( 2 )) = Z. Alternatively, the chiral matrix U can be parametrised as U = a a (a = 1; : : : ; 4) and its inverse as U 1 = U y = a ~a, where a and ~a are the chiral representations of the algebra of SU( 2 ). The scalar a is subject to the constraint a a = 1, such that it takes its values on S3, the latter being parametrised by the angles parametrising the element U of the group SU( 2 ). The homotopy class in terms of a is now 3(S3) = Z. This is the parametrisation that will be adopted here. The parametrisation of the Skyrme scalar in terms of the chiral eld U is peculiar to D = 3. Indeed in D = 2 the famous Belavin-Polyakov vortices [72] of the O( 3 ) sigma model on R2 are parametrised by the S2 valued scalar13 subject to a a = 1 (a = 1; 2; 3), pertaining to the homotopy class 2(S2) = Z. In all dimensions D > 4, these low dimensional accidents are absent so Skyrme models are de ned as the O(D + 1) sigma models on RD, described by the Skyrme scalar a a = 1; 2; : : : ; D + 1 subject to the constraint a a = 1. j aj2 = 1 ; a = 1; 2; : : : ; D + 1 ; (B.1) pertaining to the homotopy class D(SD) = Z. namely in terms of the CP1 and CP1 sigma models via a = zy az. 12In this section, we shall take d = D + 1, i.e. D denotes the number of space dimensions. 13Indeed in this low dimensional case too, there is an alternative parametrisation of the \Skyrme scalar", eld z ( = 1; 2) subject to zyz = 1, by virtue of the equivalence of the O( 3 ) one de nes the p-form which is the p-fold product of ( 1 ) = i1; i2; : : : ; ip. (p) = i1i2:::ip a1a2:::ap d=ef [ai11 ia22 : : : iapp] ; In this notation, the kinetic terms H (p;D) are concisely de ned as a, totally anitisymmetrised in the indices i H (p;D) = j (p)j2 p=D X p=1 H (D) = pH (p;D) + 0V ; %~ = %(D) + 1 a 2 j j and the topological charge density (up to normalisation) is %(D) ' "i1i2:::iD "a1a2:::aD+1 ia11 ia22 : : : iaDD aD+1 aD aD+1 : It is well known that (B.7) is essentially total divergence and hence quali es as a topological charge density. To see this, subject the quantity In any given dimension D, the energy density functional H(D) can be endowed with a \potential" term, e.g. the \pion mass" type potential V = 1 D+1 ; and D possible \kinetic" terms H (p;D), which are de ned as follows. Employing the shorthand notation for the 1-form ; i = 1; 2; : : : D ; (B.2) (B.3) (B.4) (B.5) (B.6) (B.7) (B.8) (B.9) such that only the square of any `velocity' eld ia occurs in H In this notation, the most general energy density functional in any dimension D is to variations w.r.t. the scalar eld a, taking account of the Lagrange multiplier . The result is 0 = 0, as expected from a density which is total divergence. Alterna tively, one can employ a parametrisation of a that is compliant with the constraint (B.1), e.g. when employing a particular Ansatz. In that case %(D) itself would take an explicitly total divergence form [73]. To express the Bogomol'nyi inequalities of this system, we de ne the Hodge dual of the (D p)-form (D p), which is the p-form as ? (p) d=ef ? ia11::::::iapp = 1 p)! "i1:::ipip+1:::iD "a1:::apap+1:::aDaD+1 iapp++11::::::iaDD aD+1 : For any given D, one can now state the D inequalities in terms of (B.4) and (B.8), each labelled by p, as 2 (p) (D 2p) ? (p) > 0 ; p = 1; 2; : : : D ; is a constant with dimension L 1 compensating for the di erence between the dimensions of (p) and ? (p), which are di erent in the general case D 6= 2p. It is clear from the de nitions (B.4) and (B.8) that the cross-terms in the inequalities (B.9), for each p, is proportional to the topological charge density (B.7). It follows that the p inequalities (B.9) lead to the required lower bound on the energy H (D) > %(D) =) Z RD H (D) > %(D) ; Z RD (B.10) provided of course that the potential (B.2) is by de nition positive de nite. The best known examples of such topologically stable Skyrmions are the BelavinPolyakov (self-dual) vortices [72] on R2, and the familiar Skyrmions [2] on R3. In the D = 2 case the most general system (B.6) is that with all coe cients 0; 1; 2 present and in the D = 3 case the most general one is that with all coe cients 0 ; 1 ; 2 ; 3 present. Of course, which terms must be retained in each case is governed by the requirement of Derrick scaling. Of the D inequalities (B.9) only one can be solved with power decaying solutions at r ! 1 on RD, namely the one for which D = 2p when the dimensional constant appear. Furthermore in this case the model (B.6) must consist exclusively of the p = D=2 does not term in the sum, i.e. p = 0 for p 6= (D=2), otherwise the system will be overdetermined. Such a model has solutions that saturate the topological lower bound (B.10). In all other which includes Skyrmions in all odd Euclidean14 dimensions RD. cases, when p 6= 0 for p 6= (D=2), the solutions cannot saturate the lower bound (B.10), In particular in the important case of R3, there exist no solutions to rst-order (anti)self-duality equations saturating the lower bound. Exact solutions to the second-order equations can be constructed only numerically and not in closed form. However, approximate solutions on R3 in closed form are known. For example the rational map ansatz [77], and the Atiyah-Manton [78, 79] construction where the holonomy of the Yang-Mills instantons on R4 gives a good approximation for the Skyrmion on R3. This last approach is extended to give an approximate construction for the Skyrmion on R7 by exploiting the holonomy of the Yang-Mills instantons on R8 in [80], which possible be extended to higher The rst-order (anti-)self-duality equations for the p-Skyrme system H (p;D) = j (p)j2 ? (p) (B.11) can be solved in closed for subject to no symmetries only in the case D = 2. In that case [72] the equations (B.11) reduce to the Cauchy-Riemann equations. In all higher dimensions D = 2p > 2, only solutions of the system subject to radial symmetry are known, and the form factors F (r) for all (p; D = 2p) are given by the same function (3.2). 14The (anti-)self-duality equations resulting from (B.9) on SD for odd D, namely (p) (D 2p) ? 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Yves Brihaye, Carlos Herdeiro, Eugen Radu, D.H. Tchrakian. Skyrmions, Skyrme stars and black holes with Skyrme hair in five spacetime dimension, Journal of High Energy Physics, 2017, 37, DOI: 10.1007/JHEP11(2017)037