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Letters B 783 (2018) 193–199

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Physics Letters B

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Solitons in a cavity for the Einstein-SU(2) Non-linear Sigma Model and Skyrme model ∗ Alex Giacomini a, Marcela Lagos b, , Julio Oliva b, Aldo Vera b a Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia, Chile b Departamento de Física, Universidad de Concepción, Casilla, 160-C, Concepción, Chile a r t i c l e i n f o a b s t r a c t

Article history: In this work, taking advantage of the Generalized Hedgehog Ansatz, we construct new self-gravitating Received 29 May 2018 solitons in a cavity with mirror-like boundary conditions for the SU(2) Non-linear Sigma Model and Received in revised form 14 June 2018 Skyrme model. For spherically symmetric , we are able to reduce the system to three Accepted 15 June 2018 independent equations that are numerically integrated. There are two branches of well-behaved solutions. Available online 21 June 2018 The first branch is defined for arbitrary values of the Skyrme coupling and therefore also leads to a Editor: M. Cveticˇ gravitating soliton in the Non-linear Sigma Model, while the second branch exists only for non-vanishing Skyrme coupling. The solutions are static and in the first branch are characterized by two integration constants that correspond to the frequency of the of the Skyrme field and the value of the Skyrme profile at the origin, while in the second branch the latter is the unique parameter characterizing the solutions. These parameters determine the size of the cavity, the at the boundary of the cavity, the energy of the scalar field and the charge associated to a U (1) global symmetry. We also show that within this ansatz, assuming analyticity of the fields, there are no spherically symmetric hole solutions. © 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction where 1 is the 2 × 2identity matrix, ti =−iσi the SU(2) genera- tors, σi being the Pauli matrices and Non-linear Sigma models appear in many contexts, as for ex- 0 = μ i =i μ 0 2 + i = ample to describe the dynamics of Goldstone bosons [1], in con- Y cos α(x ), Y n sin α(x ), (Y ) Yi Y 1 , (2) densed matter systems [2], in [3], as well as being with a generalized radial unit vector the building blocks of classical theory. In the case of mesons, it can be shown that the low energy dynamics can be n1 = cos (xμ) sin F (xμ), n2 = sin (xμ) sin F (xμ), correctly described by a Non-linear Sigma Model for SU(2). In n3 = cos F (xμ). (3) such low energy processes, the mesons can be seen as Goldstone bosons. In flat , the inclusion of the Skyrme term allows Here α,  and F are arbitrary functions of the space– co- to construct static regular solitons with finite energy, which de- ordinates. This ansatz was originally introduced in the context scribe [4]. In the latter scenario the ansatz for the SU(2) of the Gribov problem in regions with non-trivial [5],  ˆ  and has been shown to provide a very fruitful arena to construct group element is given by Usol = exp(iF(r)τ · x), with τ the SU(2) generators. A more general ansatz is defined by the Generalized new solutions of the theory. In reference [6], the compatibility of this ansatz on the Einstein–Skyrme theory was thoroughly ex- Hedgehog Ansatz, which includes Usol as a particular case, and is plored considering a space–time which is a warped product of a defined by two-dimensional space–time with an Euclidean constant curvature ± manifold. Also, within this ansatz, a novel non-linear superposi- U 1 = Y 01 ± Y it , (1) i tion law was found in [7]for the Skyrme theory, which was latter 2 extended to the curved geometry of AdS2 × S in reference [8]. Even more, the ansatz allows for exact solitons with a kink pro- * Corresponding author. E-mail addresses: [email protected] (A. Giacomini), [email protected] file [9]. Asymptotically AdS and bouncing cosmologies (M. Lagos), [email protected] (J. Oliva), [email protected] (A. Vera). with self-gravitating Skyrmions were constructed in [10]as well as https://doi.org/10.1016/j.physletb.2018.06.036 0370-2693/© 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 194 A. Giacomini et al. / Physics Letters B 783 (2018) 193–199

μ other time dependent cosmological solutions with non-vanishing  = 0 , ∇μ∇ α = 0 , (7) topological charge [11]. Also within the context of the general- ∇μ∇ν ∇ ∇ = ized hedgehog ansatz, for the SU(2) Non-linear Sigma Model, ( ) μ ν 0 , (8) topologically non-trivial gravitating solutions were constructed in μ ν (∇ ∇ α)∇μα∇ν = 0 . (9) [12] which cannot decay on the trivial due to topological obstructions and, more recently, planar asymptotically AdS hairy Even though these equations may seem too restrictive, we will black hole solutions were found in [13]. show below that they are compatible with the existence of soli- In this paper we will explore a new family of solutions within tonic solutions in a cavity. the Generalized Hedgehog Ansatz which describe spherically sym- With this, the Einstein and Skyrme equations reduce to metric, static configurations in a cavity. By imposing mirror-like boundary condition for the matter field we numerically construct Eμν = Gμν − κ Tμν = 0 , (10) new self-gravitating solitons for the SU(2) Skyrme model and Non-linear Sigma Model. In Section 2 we introduce the General- with ized Hedgehog Ansatz. In Section 3 we reduce the system to three  non-linear equations and argue that in order to have configura- = ∇ ∇ + 2 ∇ ∇ + 2 tions with finite energy it is necessary to introduce a mirror at Tμν ( μα)( να) sin α( μ)( ν) λ sin α a finite proper distance from the origin. Section 4 is devoted to   the numerical integration of the system that leads to two well- 2 2 × (∇) (∇μα)(∇να) + (∇α) (∇μ)(∇ν) behaved branches. The first branch is well behaved for arbitrary values of the coupling constant of the Skyrme term λ, while the   1 2 2 2 2 2 2 second leads to well-behaved solutions only for non-vanishing λ. − gμν (∇α) + sin α(∇) + λ sin α(∇) (∇α) , Section 5 contain the conclusions and further comments as well 2 as the proof that, within this ansatz, there are no black holes sup- (11) ported by an analytic Skyrme field. supplemented by    2. The SU(2) Einstein–Skyrme and Einstein-Nonlinear Sigma 1 2 μ 2 2 Model α − sin(2α)(∇) + λ (∇μα)∇ sin α(∇) 2

In this paper we will be concerned with the gravitating 1 + sin2 α(∇)2(α) − sin(2α)(∇α)2(∇)2 = 0 . (12) Einstein–Skyrme model as well as with the Einstein-Non-linear 2 Sigma Model systems. The action is given by     These equations can also be obtained from the effective action √ R K λ [ ]= 4 − + μ + μν    I g, U d x g Tr A Aμ Fμν F , (4) √ 2κ 4 8 R ρ 2 ρ Ieff = −g − 2 ∂ρα∂ α + sin (α)∂ρ∂  −1 2κ where R is the Ricci scalar, Aμ := U ∇μU and Fμν =[Aμ, Aν].  i Here U is a scalar field valued in SU(2) and therefore Aμ = A ti . λ μ + sin2(α)(∇α)2(∇)2 , (13) We work in the mostly plus signature, Greek and Latin indices 2 run over spacetime and the algebra, respectively. Hereafter with- out loosing generality we set K = 1. provided the constraints (7)–(9)are fulfilled. Einstein equations The field equations for this theory are the Einstein equations (10) and the Skyrme equation (12)are obtained from the varia- tion of Ieff w.r.t. the metric and the scalar α, respectively, and the E = − = μν Gμν Tμν 0 , (5) equation for  is trivially satisfied after imposing the constraints with energy-momentum tensor given by (7)–(9). The effective action, as well as the constraints, are invari-  ant under the global transformation 1 1 α λ αβ Tμν =− Tr Aμ Aν − gμν Aα A + (g Fμα Fνβ 2 2 4 = =  δ(1)α 0 ,δ(1)  , (14) 1 αβ − gμν Fαβ F ) , where  is a parameter. The symmetry transformation δ allows 4 (1) to construct a locally conserved current which, when integrated satisfying the dominant [14], and the Skyrme within the cavity, leads to a finite conserved charge. equations 3. The system and its finite energy solutions μ λ μ ν ∇ Aμ + ∇ [A , Fμν]=0. (6) 4 We consider a static spherically symmetric space–time metric We will consider the generalized hedgehog ansatz (2) and (3)with F xμ = π . The functions α and  of the ansatz (2) and (3)are 2 dr2 scalar functions: α describes the energy profile of the configura- ds2 =−f (r)dt2 + + r2 dθ 2 + sin2 θdφ2 , (15) tion while  describes its orientation in isospin space. One can h(r) check that the above ansatz has vanishing charge, thus and the following dependence for the matter fields we are within the pionic sector. The group manifold of SU(2) is 3 the three-sphere S , and our ansatz turns on the field along the α = α(r),  = ωt , (16) ⊂ S3 submanifold. The advantage of the Generalized Hedgehog Ansatz is given by the fact that the Skyrme equations reduce to a where ω is a frequency, leading to a time independent energy- single equation provided [6], momentum tensor and therefore the physical quantities for this A. Giacomini et al. / Physics Letters B 783 (2018) 193–199 195 configuration are static.1 For this ansatz, the constraint equations solitons as well as black holes, which are unstable, and in the pre- (7)–(9)are automatically fulfilled and the Einstein–Skyrme sys- vious references there have been observed dynamical in tem reduces to three independent equations. We work with the both directions in different regimes.2 equations Ett and Err (with Eμν defined in (10)), as well as (12). Introducing for simplicity u(r) = sin α(r), and setting 2κ = 1one 4. The new solitons obtains the following non-linear system The requirement of having a regular center at r = 0leads to  r2ω2u2 u2 − λhu 2 − 1 two soliton branches. The first corresponds to a branch analytic in the Skyrme coupling, with   + f 2(u2 − 1)(rh − 1) + h(2u2 − r2u 2 − 2 = 0 , (17) 1 f (r) = f + u2ω2r2  1 0 0 f 2(1 − u2) + h 2(u2 − 1) + r2u 2  3  2 4 2 2 − + 2 − 2 2 + 6 2 4 u0ω 6 f0 (2u0 1) f0u0(7 19u0)λω 6u0λ ω 4 − 2 4 +  + 2 2  2 −  − 2 = + r r rω u 2hf u (rλω hu 2hf rω ) 0 , (18) 180 f ( f − u2λω2)2  0 0 0 − 2rω2 fu(u2 − 1)2 + (u2 − 1) 2 f 2(1 + h) − 4λω2 fhu2 + O 6  r , (21.1)   + r2λω4u4 u + 2r(λω2 − f ) fhuu 2 u2ω2 h (r) = 1 − 0 r2 − 2 − 2 2 −  = 1 2rfh(u 1) λω u f u 0 . (19)  6 f0  2 4 2 2 2 2 2 6 2 4 u ω f (2 + u ) − 2 f0u (2 + u )λω + 3u λ ω It is worth pointing out that the parameter ω can be absorbed + 0 0 0 0 0 0 r4 90 f 2( f − u2λω2)2 in the field equations by rescaling the radial coordinate as well as 0 0 0 the Skyrme coupling in the form r → r¯ = ωr, λ → λ¯ = ω2λ. While + O r6 , (21.2) in the Non-linear Sigma Model this transformation reduces the number of independent parameters to be provided before numeri- 2 − 2 u0(u0 1)ω 2 cal integration, in the presence of the Skyrme term the freedom in u1(r) = u0 + r ¯ 6( f − u2λω2) ω is mapped to the freedom to choose the value of λ. 0 0  2 4 2 2 2 2 2 6 2 2 Asymptotic flatness requires as a necessary condition (not nec- u0(u − 1)ω f (3 − 7u ) + 5 f0u (1 + u )λω − 6u λ ω − 0 0 0 0 0 0 r4 essarily sufficient) that the matter fields have to vanish at infinity. 2 2 3 360 f0( f0 − u λω ) Therefore, assuming that α goes to zero as r goes to infinity, on an 0 asymptotically flat space–time, equation (17)reduces to + O r6 . (21.3)

  rα (r) + 2α (r) + rω2α(r) = 0 . (20) The second branch, non-analytic in λ, is given by

Consistently, this equation is equivalent to the equation for the ra- 11u2ω2 4u (20u2 − 21)ω2 = 2 2 + 0 2 + 0 0 3 + O 4 dial profile of a massless scalar field in –time, and f2(r) λu0ω r r r , 30 − 2 admits the following asymptotic behavior α(r) → cos(ωr)/r as r 225 5λ(1 u0) goes to infinity. (22.1) It is a straightforward computation to show that this asymptotic 7 4(6 − 5u2) behavior is not compatible with having a finite . If we want 2 0 3 4 h2(r) =1 − r + r + O r , (22.2) to construct gravitating solitons in this sector of the Generalized 30λ 3 2 75λ 2 u0 5(1 − u ) Hedgehog Ansatz for the Skyrme model, it turns out to be neces- 0 sary to confine the system into a cavity. In what follows we do so, 2 1 − u 3 − 10u2 by imposing mirror-like boundary conditions for the profile at 0 0 2 α u2(r) =u0 − √ r + r a finite value of the radial coordinate r = rm, i.e. we impose the 5λ 150λu0 boundary condition α(rm) = 0. As shown in [21], the introduction − 2 + 4 261 1890u0 2125u0 3 4 of a finite box also allows to construct examples of Skyrmions– + r + O r . (22.3) 3 2 2 − 2 anti-Skyrmions bound states, as well as time crystals. 38250λ u0 5(1 u0) A similar situation occurs for the Einstein–Maxwell system cou- pled to a massless charged scalar (see e.g. appendix A of [15]). The The latter solution is intrinsic to the presence of the Skyrme term. asymptotic behavior of the scalar field is not compatible with the These two branches define the data at the origin which, after requirement of asymptotic flatness and finite mass for solitons and numerical integration, will determine the data at the mirror lo- = black holes, and one is therefore forced to enclose the system into cated at r rm. a cavity. This system has been particularly fruitful for the study Note that for the first branch, for a given value of the Skyrme of the non-linear of the superradiant instability due to coupling, the free parameters are f0, u0 and ω. Normalizing the the of a scalar field including a mass term [16,17] time coordinate t to coincide with the of a geodesic = as well as self-interaction [18], leading to the formation of hairy observer located at the origin sets f0 1. This region is shared by black holes [19]. The system in a cavity allows for the existence of all the configurations and one can therefore compare their physical parameters in a consistent manner. We therefore fix f0 = 1in this

1 The kinetic term for the Non-linear Sigma Model, (∂α)2 + sin2 α(∂)2, is   −2 2 As shown in [20], a negative cosmological constant provides a setup to naturally mapped to 1 +||2/4 |∂|2 with  = ρ exp (iχ) via the transformation  = χ implement an effective cavity, such that the superradiant instability in the spheri- 4−ρ2 and α = arccos . This makes explicit the fact that  is a phase that, accord- 4+ρ2 cally symmetric charged case leads to a hairy black hole even for a massless scalar, ing to our ansatz, rotates in time at a frequency ω with respect to the coordinate obtaining results that are qualitatively similar to those of the system enclosed in a time t. cavity. 196 A. Giacomini et al. / Physics Letters B 783 (2018) 193–199

Fig. 1. Metric functions as well as the Skyrme field for different values of u0 and frequency for the Branch 1. The cavity wall is located at the first zero of the Skyrmion profile. We have set the Skyrme coupling λ = 1. branch. For the second branch, the value of the −gtt component and of the metric at the origin is not a free parameter any more and is rm   = 2 2 2 fixed by f2(0) λu ω . We can still normalize the time coordinate r λ 0 =− +  2 2 to coincide with the proper time of a geodesic observer located at Q 16πω dr 1 α h(r) sin(α) . (24) f (r)h(r) 2 → 2 2 0 the origin by introducing the scaling t t/ λu0ω . In this man- ner, the parameter ω is absorbed from all the functions and the Below, we present the results of the integration for each branch. rotation of the phase is locked in terms of the Skyrme coupling and the value of the scalar at the origin as  = ωt →  = t/ λu2. 4.1. Branch 1: analytic in λ 0 Equivalently, this is accomplished if we directly set ω = 1/ λu2 0 For the first branch, the free parameters are u0 and ω. Fig. 1 for the integration of this branch. In this manner, for a given value shows the functions integrated from the system for four differ- of the Skyrme coupling, u0 is the unique parameter characterizing ent combinations of the frequency and the strength of the Skyrme the solutions in the second branch. field at the origin. The mirror is located at the first zero of the For the numerical integration we proceed as follows: We fix black curve which represents the field u(r) = arcsin α(r). The de- the coordinate t to be the proper time of an observer at the origin, pendence of the radius of the mirror r = rm as a function of which leaves us with two (ω and u0) and one (u0) free param- u0 and the frequency is depicted in Fig. 2. The radius of the eter for the first and second branch, respectively. Then, for both mirror is an increasing function of the value of the field at the − branches, we integrate the system (17)–(19)from a regulator  ∼ 0 origin and increases with ω 1. The later is expected from the outwards, using the initial conditions for radial integration that asymptotic behavior since the periodicity of the zeros of the field ∼ come from expansion (21) and (22)up to order O(r8). We lo- α(r) cos(ωr)/r is locked in terms of the time periodicity of the = cate the radius of the mirror at the first zero of the Skyrme field phase  ωt. Fig. 2 also shows that one could locate the mirror u(r). Finally, the functions obtained after the integration are used at an arbitrarily large proper distance from the origin as u0 ap- proaches to 1, notwithstanding as seen in the left panel of Fig. 3 to compute the energy and the U (1) charge which are respectively as u → 1the energy and the charge diverge, as expected from given by 0 the asymptotic analysis, therefore only mirrors located at a finite

rm proper distance from the origin are compatible with having finite energy and charge. As can be seen from the right panel of Fig. 3, M =−4π T t r2dr = 8πr (1 − h(r )) , (23) t m m for all the solitons obtained here the U (1) charge is larger than the 0 energy of the configuration. A. Giacomini et al. / Physics Letters B 783 (2018) 193–199 197

Fig. 2. This figure depicts the radius of the mirror r = rm as an increasing function of the value of the Skyrme field at the origin for different values of ω (with λ = 1) for the Branch 1.

4.2. Branch non-analytic in λ

As mentioned above, the second branch is non-analytic in the Skyrme coupling and it is characterized by a unique integration constant u0 after one sets the time coordinate to coincide with the proper time of a geodesic observer located at the origin. Fig. 4 depicts the behavior of the metric functions as well as the Skyrme profile for different values of the latter at the origin. Upper left panel of Fig. 5 shows the behavior of the radius of the mirror as a function of the amplitude of the Skyrme field at the origin. Again, the radius diverges as u0 approaches 1but as shown in the upper Fig. 4. Metric functions as well as the Skyrme field of the second branch, for differ- right Fig. 5 the mass and charge would diverge in that case. For ent values of u0. The lapse function has been set to 1at the origin which locks the small values of the mass and charge the curves seem to overlap. frequency of the phase of the Skyrme field. Lower panel of Fig. 5 shows that indeed there is a critical value for u0 above which the charge surpasses the value of the mass, while been included which is located at the first zero of the Skyrme pro- below this critical value the mass is larger than the charge. In that file, and we studied the behavior of the mass and U (1) charge as figure we have included the curve Q = M only for reference. a function of the location of the boundary. The conserved charges for the different cases can be compared since all these configura- 5. Conclusions and further comments tions share the region located at the origin which allows to define a common normalization for the globally timelike Killing vector ∂t . In this paper we have constructed new solutions of the The regularity of the solutions at the origin imply the existence of Einstein–Skyrme model for SU(2) group. We make use of the Gen- two branches of solutions, and while the first branch exists for any eralized Hedgehog Ansatz turning on the fields along the S2 ⊂ S3 value of the Skyrme coupling, the existence of the second branch is submanifold. In the absence of the Skyrme term the system effec- intrinsic to the presence of the term introduced by Skyrme to sta- tively reduces to a Non-linear Sigma Model on S2. A cavity has bilize the solitons. After normalizing the time coordinate in order

Fig. 3. Left panel: The energy and the U (1) charge of the solitons for two values of the frequency and a variety of values of the field at the origin for Branch 1. As expected, the charges diverge as u0 → 1. Right panel: shows the behavior of Q vs M for the solitons in Branch 1. For all the cases Q > M as can be seen by comparing with the red curve (M = Q ) that has been included only as a reference. We have considered 100 values of u0 in each curve, and its value increases in the range 0 < u0 < 1as one departs from the origin. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.) 198 A. Giacomini et al. / Physics Letters B 783 (2018) 193–199

Fig. 5. Upper left panel: This figure shows that the radius of the mirror is an increasing function of the value of the Skyrme field at the origin for the non-analytic branch and it diverges as u0 → 1. Upper right panel: Mass and charge as a function of the amplitude of the Skyrme field at the origin. Both diverge as the radius of the mirror goes to infinity. Lower panel: Q vs M. There is a critical value above which the charge is greater than the mass (the curve Q = M has been included only for reference). Here we have set λ = 1. it to coincide with the proper time of a geodesic observer located non-trivial, black hole solutions. Therefore, the boson con- at the origin one is left with solutions parameterized by two con- structed in this work cannot decay into a hairy black hole with 4 stants (u0, ω) in the first branch and by a single constant u0 in the same symmetries, because such black hole does not exist. It the second branch. In the former case we observe that the charge is interesting to note that the author of reference [24] constructed is always greater than the mass, while in the latter the charge is boson solutions in the Non-linear Sigma Model case by adding larger than the mass only above a critical value of the mass which a designed self-interacting potential to (13)without further con- induces a lower critical value for the amplitude of the Skyrme field straint.5 The presence of the self-interaction allows to construct at the origin. configurations of finite mass even when the boundary of the cavity One might be tempted to construct black holes in a cavity with is located at an infinite proper distance from the origin. It would non-vanishing Skyrme profile in this system. Assuming the exis- be interesting to include the self-interaction also for a finite cavity. tence of a regular horizon located at r = r+, as well as assuming analyticity for u(r) at the horizon, one can show that the field Acknowledgements equations have two branches. In the first branch u(r) = 0, which implies U equals the identity of SU(2), and the expansions of We thank Fabrizio Canfora for proposing this project as well functions f and g reconstruct the Schwarzschild solutions. The as for collaboration on its early stages. M.L. and A.V. appreciates second branch leads to a near horizon expansion of the form the support of CONICYT Fellowship 21141229 and 21151067, re- 2 spectively. J.O. thanks Andrés Anabalón, Adolfo Cisterna and Carlos u(r) = u1(r − r+) + O((r − r+) ), (25.1) Herdeiro for enlightening discussions. A.V. thanks Gustavo Álvarez −1 2 g(r) = r+ (r − r+) + O((r − r+) ), (25.2) for useful discussions. This work was also funded by FONDECYT 2 2 grant 1181047 and CONICYT Grant DPI20140053 (J.O. and A.G.) f (r) =−r+ω (r − r+) + O((r − r+) ), (25.3) and FONDECYT Grant 1150246 (A.G.). which is not consistent with the structure of an horizon.3 This shows that, within the ansatz here considered, there are no 4 Boson stars with a local SU(2) symmetry in the context of Einstein–Yang–Mills are constructed in [23]. 3 Note that this is compatible with the well-known result of Luckock and Moss 5 This Lagrangian can also be seen as a member of the bi-scalar extension of [22]since their ansatz differs from ours. In fact our ansatz for the group element is Horndeski theories [25]. For recent constructions of boson stars in such setup see time dependent. e.g. [26]and [27]. A. Giacomini et al. / Physics Letters B 783 (2018) 193–199 199

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