Supersymmetric Q-balls and boson stars in (d + 1) dimensions Betti Hartmann∗ and Ju¨rgen Riedel† School of Engineering and Science, Jacobs University, 28725 Bremen, Germany October 2, 2012 2 1 0 2 Abstract p e We construct supersymmetric Q-balls and boson stars in (d+1) dimensions. These non-topological S solitons are solutions of a scalar field model with global U(1) symmetry and a scalar field potential that 9 appearsingauge-mediatedsupersymmetry(SUSY)breakingintheminimalsupersymmetricextensionofthe 2 StandardModel(MSSM).Weareinterestedinboththeasymptotically flataswellasintheasymptotically Anti-de Sitter (AdS) solutions. In particular, we show that for our choice of the potential gravitating, ] asymptotically flat boson stars exist in (2+1) dimensions. Weobserve that the behaviourof the mass and h chargeoftheasymptoticallyflatsolutionsattheapproachofthemaximalfrequencydependsstronglyonthe t - numberof spatial dimensions. Fortheasymptotically AdSsolutions, themodelon theconformal boundary p can be interpreted as describing d-dimensional condensates of scalar glueballs. e h [ PACS Numbers: 04.40.-b, 11.25.Tq 1 v 6 1 Introduction 9 0 A number of non-linear field theories possess solitonic-like solutions. These have broad applications in many 0 branches of physics and constitute localized, globally regular structures with finite energy. Topologicalsolitons . 0 [1] possess a conserved topological charge that is connected to the existence of non-contractible loops in the 1 theory. Non-topologicalsolitons[2,3]ontheotherhandappearinmodelswithsymmetriesandpossessalocally 2 conserved Noether current and a globally conserved Noether charge. An example of such a non-topological 1 : soliton is the Q-ball [4] and its generalizationin curved space-time, the boson star [5, 6, 7, 8, 9, 10]. These are v solutionsofmodelswithself-interactingcomplexscalarfieldsandtheconservedNoetherchargeQisthenrelated i X to the globalphase invarianceof the theory and is directly proportionalto the frequency ofthe harmonictime- r dependence. Q can e.g. be interpreted as particle number [2]. As such, these solutions have been constructed a in (3+1)-dimensional models with non-renormalizable Φ6-potential [11, 12, 13], but also in supersymmetric extensions to the Standard Model (SM) [14]. In the latter case, several scalar fields interact via complicated potentials. It was shown that cubic interaction terms that result from Yukawa couplings in the superpotential and supersymmetry (SUSY) breaking terms lead to the existence of Q-balls with non-vanishing baryon or lepton number or electric charge. These supersymmetric Q-balls have been considered as possible candidates for baryonic dark matter [15] and their astrophysical implications have been discussed [16]. In [17], these objects have been constructed numerically using the exact form of a scalar potential that results from gauge- mediated SUSY breaking. However, this potential is non-differentiable at the SUSY breaking scale. In [18] a differentiable approximation of this potential was suggested and the properties of the corresponding Q-balls have been investigated. Most models of a quantum theory of gravity need more than (3+1) dimensions and as such, it is surely of interest to investigate the properties of soliton solutions in higher dimensions. The first ∗[email protected] †[email protected] 1 study of Q-balls in higher dimensional space-time has been done in [19]. However, only a linearized version of the Lagrangianandequationsofmotiondepending onlyinzerothorderonthe ratiobetweenthe typicalenergy scale and the Planck mass has been used. In this case, an analytical solution can be given, however,the model does not capture the non-linear phenomena such as e.g. the behaviour of the mass and charge at the maximal frequency. A similar study has been done in [20] for (2+1) dimensions. Q-balls and boson star solutions of the full system of coupled non-linear equations in (4+1)-dimensionalasymptotically flat space-time have been investigated in [21]. Interestingly, it was found that the behaviour of the mass and charge at the approach of the maximal possible frequency is different for d = 4 as compared to d = 3. This was related to a scaling behaviour of the solutions at this critical approach and different dimensions of the spatial integrals. Spinning generalisations of these solutions can also be constructed [11, 12, 13, 21, 22, 23]. These solutions possess a quantised angular momentum that is an integer multiple of the Noether charge. Topological and non-topological solitons in Anti-de Sitter (AdS) space-time have been investigated inten- sively recently. The interest in these objects is related to the AdS/CFT correspondence [24, 25] which states that a gravity theory in a d-dimensional Anti-de Sitter (AdS) space–time is equivalent to a Conformal Field Theory(CFT)onthe(d 1)-dimensionalboundaryofAdS.Interestingly,thisisaweak-strongcouplingduality − that can be used to describe strongly coupledQuantum Field Theories with the help of weakly coupled gravity theories. This has been applied to a modeling of high temperature superconductivity with the help of classical blackholeandsolitonsolutionsinAdS [26,27,28]. Thebasic modelsuse ascalarfieldcoupledto aU(1)gauge field and the observation that close to the horizon of the black hole the effective mass of the scalar field can become negative with masses below the Breitenlohner–Freedmanbound [29] such that the scalar field becomes unstable and possesses a non–vanishing value on and close to the horizon of the black hole. When computing the conductivities it turns out that the formation of a scalar field on a charged black hole corresponds to a phase transition from a conductor to a superconductor. However, insulator/superconductor phase transitions also play an important role in high temperature superconductivity and as such models including solitons have been suggested that describe this phenomenon [30, 31, 32]. The AdS soliton is related to the black hole by a double Wick rotation with one of the coordinates compactified to a circle and has originally been suggested to describe a confining vacuum in the dual gauge theory [33, 34] since it possesses a mass gap. For solutions with Ricci-flat horizons there is a phase transition between the AdS black hole and the AdS soliton [35] which was interpreted as a confining/deconfining phase transition in the dual gauge theory. Note that this is different for blackholes inglobalAdSwhere the black holedecaysto globalAdS space-timewhenloweringthe temperature [36]. Inthe limitofvanishinggaugecouplingthesolitonsolutionscorrespondtoplanarbosonstarsinAdSspace- time. Since the scalar field is uncharged the interpretation in terms of insulators/superconductors is difficult in this case. However, since the AdS/CFT correspondence connects strongly coupled CFTs to weakly coupled gravity theories the prototype example of a strongly coupled field theory comes to mind - Quantum Chromo- dynamics(QCD).As suchthe planarbosonstarsinAdS havebeeninterpretedasBose-Einsteincondensatesof glueballs. Glueballs are color-neutral bound states of gluons predicted by QCD and the scalar glueball (which is also the lightest possible glueball) is predicted to have a mass of 1-2 GeV (see e.g. [37] for an overview on experimental results). Since these glueballs appear due to non-linear interactions and as such cannot be described by a perturbative approachit is very difficult to make predictions within the frameworkof Quantum FieldTheory. However,holographicmethods havebeenappliedtomakepredictionsaboutglueballspectra(see e.g. [38] and reference therein). Non-spinning boson stars in (d+1)-dimensional AdS space-time have been studied before using a massive scalarfield without self-interaction[39] and in (3+1) dimensions with an exponential self-interactionpotential [40]. Spinning solutions in (2+1) and (3+1) dimensions have been constructed in [23] and [41], respectively. Inthispaper,weareinterestedinQ-ballsandbosonstarsinbothasymptoticallyflataswellasasymptotically AdS space-time with (d+1) dimensions. We use an exponentialscalar field potential alreadyemployed in [40]. We will consider first the asymptotically flat case generalising some of the results obtained in [21] to higher dimensions and then also consider asymptotically AdS solutions. Our paper is organised as follows: In Section 2 we give the model, equations of motion and boundary conditions. In Section 3, we present our numerical results and conclude in Section 4. The Appendix 1 and 2 contain results on the generalisation of an exact solution first found in [41] to (d+1) dimensions and on the 2 existence of asymptotically flat boson stars in (2+1) dimensions, respectively. 2 The model In the following we will study non-spinning Q-balls and boson stars in a (d+1)-dimensional Anti-de Sitter (AdS) space time. The action S reads R 2Λ 1 S = √ gdd+1x − + + ddx√ hK (1) m Z − (cid:18)16πG L (cid:19) 8πG Z − d+1 d+1 where R is the Ricci scalar, G denotes the (d+1)-dimensional Newton’s constant, Λ is the negative cosmo- d+1 logical constant related to the AdS radius ℓ by Λ = d(d 1)/(2ℓ2). The second term on the right hand side − − of (1) is the Gibbons-Hawking surface term [42] with h the induced metric and K the trace of the extrinsic curvature on the AdS boundary. is the matter Lagrangiangiven by m L = ∂ Φ∂MΦ∗ U(Φ) , M =0,1,....,d , (2) m M L − − | | where Φ denotes a complex scalar field and we choose the metric to have mainly positive signature. U(Φ) is | | the potential Φ2 U(Φ)=m2η2 1 exp | | . (3) | | susy(cid:18) − (cid:18)−η2 (cid:19)(cid:19) susy This potential is motivated by supersymmetric extensions of the Standard Model [17, 18]. Here η is a susy parameter such that η2/(d−1) corresponds to the energy scale below which supersymmetry is broken, while m susy denotes the scalar boson mass. The coupled system of ordinary differential equations is then given by the Einstein equations G +Λg =8πG T , M,N =0,1,..,d (4) MN MN d+1 MN with the energy-momentum tensor ∂ T = g 2 L MN MNL− ∂gMN 1 = g gKL(∂ Φ∗∂ Φ+∂ Φ∗∂ Φ)+U(Φ) +∂ Φ∗∂ Φ+∂ Φ∗∂ Φ (5) MN K L L K M N N M − (cid:20)2 (cid:21) and the Klein-Gordon equation ∂U (cid:3) Φ=0 . (6) (cid:18) − ∂ Φ2(cid:19) | | The matter Lagrangian (2) is invariant under the global U(1) transformation m L Φ Φeiχ . (7) → As such the locally conserved Noether current jM, M =0,1,..,d associated to this symmetry is given by i jM = Φ∗∂MΦ Φ∂MΦ∗ with jM =0 . (8) −2 − ;M (cid:0) (cid:1) The globally conserved Noether charge Q of the system then reads Q= ddx√ gj0 . (9) −Z − 3 2.1 Ansatz and Equations For the metric we use the following Ansatz in spherical Schwarzschild-like coordinates 1 ds2 =−A2(r)N(r)dt2 + N(r)dr2+r2dΩ2d−1 , (10) where 2n(r) 2Λ N(r)=1 r2 (11) − rd−2 − (d 1)d − and dΩ2 is the line element of a (d 1)-dimensional unit sphere. Note that the gravitational constant is d−1 − chosen such that the d=2 case has no Newtonian limit [43]. As such the metric function n(r) is well-behaved also in the d=2 case. Note that for weak and static gravitational fields g (1+2ψ(r)), where ψ(r) is the tt ∼− Newtonian potential. Then the behaviour of ψ(r) in d=2 which is ψ(r) ln(r) would imply the divergence ∼− of n(r) if that limit would exist within our Ansatz. For the complex scalar field, we use a stationary Ansatz that contains a periodic dependence of the time- coordinate t: Φ(t,r)=eiωtφ(r) , (12) where ω is a constant and denotes the frequency. In order to be able to use dimensionless quantities we introduce the following rescalings r r , ω mω , ℓ ℓ/m , φ η φ , n n/md−2 (13) susy → m → → → → and find that the equations depend only on the dimensionless coupling constants η2 κ=8πG η2 =8π susy , (14) d+1 susy Md−1 pl,d+1 where M is the (d+1)-dimensional Planck mass. Note that with these rescalings the scalar boson mass pl,d+1 m m becomes equal to unity. In these rescaled variables and coupling constants the coupled system of B ≡ non-linear ordinary differential equations reads rd−1 ω2φ2 n′ =κ Nφ′2+U(φ)+ , (15) 2 (cid:18) A2N(cid:19) ω2φ2 A′ =κr Aφ′2+ , (16) (cid:18) AN2(cid:19) rd−1ANφ′ ′ =rd−1A 1∂U ω2φ . (17) (cid:18)2 ∂φ − NA2(cid:19) (cid:0) (cid:1) Theseequationshavetobesolvednumericallysubjecttoappropriateboundaryconditions. Wewanttoconstruct globally regular solutions with finite energy. At the origin we hence require ′ φ(0)=0 , n(0)=0 , (18) while we chooseA( )=1 (any other choice wouldjust resultin a rescalingofthe time coordinate). Moreover, ∞ while the scalar field function falls of exponentially for Λ=0 with 1 φ(r >>1) exp 1 ω2r +... (19) ∼ rd−21 (cid:16)−p − (cid:17) it falls of power-law for Λ<0 with φ d d2 φ(r >>1)= ∆ , ∆= +ℓ2 . (20) r∆ 2 ±r 4 4 Whensolvingtheequationsnumerically,wewillchooseasfourthboundaryconditionφ( )=0forΛ=0,while ∞ for Λ<0 we will choose the fall-off given in (20). φ is then a constant that has to be determined numerically ∆ and which can be interpreted via the AdS/CFT correspondence as the value of the condensate of glueballs in the dual theory living on the d-dimensional boundary of global AdS. The explicit expression for the Noether charge reads ∞ 2πd/2 ωφ2 Q= dr rd−1 . (21) Γ(d/2)Z AN 0 For κ=0, we can determine the mass M fromthe behaviourof the metric function n(r) at infinity. This reads 6 [39] n(r 1)=M +n r2∆+d+.... , (22) 1 ≫ where n is a constant that depends on ℓ. 1 For κ=0 we haveA 1 and n 0. Then the mass M correspondsto the integralof the energy density T0 ≡ ≡ 0 and reads ∞ 2πd/2 ω2φ2 M = dr rd−1 Nφ′2+ +U(φ) . (23) Γ(d/2)Z (cid:18) N (cid:19) 0 Note that this expression is perfectly finite in AdS. While N contains a term r2 the fall-off of the scalar ∝ function φ guarantees that M has a finite value. Using the expression for the charge Q (21), we can give a relation between the charge and the mass ∞ 2πd/2 M =ωQ+ dr rd−1 Nφ′2+U(φ) . (24) Γ(d/2)Z (cid:0) (cid:1) 0 For Λ = 0 it was found [18] that in the “thin-wall” approximation which corresponds to ω 0 the mass and ∼ charge are related as follows M d+1 . (25) ωQ ∼ d This has been used to draw conclusions on the stability of the Q-balls. 3 Numerical results The solutions to the coupled system of nonlinear differential equations are only known numerically. We have solved these equations using the ODE solver COLSYS [44]. The solutions have relative errors on the order of 10−6 10−10. − 3.1 Q-balls We have first studied the case κ=0. This corresponds to Q-balls in a Minkowski (Λ =0) or AdS background (Λ < 0), respectively. In this case, the Einstein equations decouple from the system and n(r) 0, while ≡ A(r) 1. ≡ It is known that Q-balls in d = 3 exist on a limited interval of the frequency ω [ω : ω ]. For Λ = 0 min max ∈ the mass and charge diverge at both boundaries [12, 13], while for Λ=0 this is still true at ω , but now the min 6 two quantities tend to zero at ω [40]. This is related to the fact that in the limit ω ω the scalar field max max → function tends to zero everywhere. In Minkowski space-time this still leads to an infinite value of the integral sincethespace-timeisinfinite,howeverAdSspace-timeactsasaconfiningboxandassuchtheintegralbecomes zero. 5 3.1.1 Λ=0 Q-balls with an exponential interaction potential of the form (3) have been studied in [18, 40] in (3 + 1) dimensions. Here, we extend these results to d=3 and show that the behaviour of the mass M and the charge 6 Q depend crucially on the space dimension d. Our results for the mass M and charge Q in dependence on ω are given in Fig.1 for d = 2,3,4,5,6. We observe that the mass and charge diverge at ω = ω for d 3, while for d = 2 they tend to finite values. max ≥ Moreover, since the potential and effective potential U := ω2φ2 U(φ) do not depend on d the arguments eff − employed in [11] can also be used here, such that ω 1 and ω 0 do not depend on d. This is clearly max min ≡ ≡ see in Fig.1. 6 e+06 L 1e+0 1 = 0.0 2d = 0.0 3d M1e+021e+04 ========w = −−−−− 000000001.........000111110 45623456dddddddd Q1e+021e+04 ==========wL = −−−−− 00000000001...........00000111110 2345623456dddddddddd 0 0 0 0 + + e e 1 0.4 0.6 0.8 1.0 1.2 1.4 1 0.2 0.4 0.6 0.8 1.0 1.2 1.4 w w (a) M overω (b) Qoverω Figure1: ThevalueofthemassM (left)andthechargeQ(right)oftheQ-ballsindependenceonthefrequency ω in Minkowski space-time (Λ=0) and AdS space-time (Λ= 0.1) for different values of d. − In order to get an idea about the stability of these objects we can compare the mass M with that of Q free scalarbosonsofmassm m. Due toourrescalingsm 1andthemassofQfreescalarbosonisjustequalto B ≡ ≡ Q. Any solutionwith M <Q wouldhence be stable to decay into Q free bosons. Our results ford=2,3,4,5,6 areshownin Fig.2. Ind=3 itwasfound [12,13,40] thatthere existtwosolutions withdifferentM for agiven charge Q, one of which is stable to the decay into Q free bosons, while the other is unstable. Here we observe that this is also true for d > 3: a stable branch exists up to a maximal value of the charge and then extends backwards to form a second branch that for some critical value of the charge becomes unstable. We observe that the bigger d the bigger is the value of the charge at which the solutions become unstable. On the other hand, for d=2 we find that the solutions are always stable with respect to the decay into Q free bosons. The fact that Q-balls are stable for small values of ω was pointed out already in [18]. In this so-called “thin-wall limit” the Q-balls fulfill the relation M ((d+1)/d)ωQ. Since ω is small M <Q in this limit and the Q-balls ≃ are stable with respect to a decay to Q free bosons. On the other hand for the “thick-wall limit” it is more difficult to make statement, but the results in [18] again agree with our numerical findings. 3.1.2 Λ=0 6 This case correspondsto Q-balls in a fixed AdS backgroundand has been studied for the exponential potential and d = 3 in [40]. Our results are shown in Fig.2 for Λ = 0.1 and d = 2,3,4,5,6. Similar to d = 3 the − value of the mass M and charge Q tend to infinity at ω = 0 independent of d. On the other hand, the min mass and charge tend to zero at ω . Moreover, we find that ω decreases with increasing d having the max max largest value for d = 2. For Λ = 0 and κ = 0 it is known that in the d = 3 case and a particular choice of 6 potential exact solutions to the scalar field equation exist [41]. We show in Appendix 1 that this generalizes to d dimensions. While the potential necessary to obtain this result is not of the form chosen in this paper, 6 6 +0 80 2d 450 3d e 1 40 300 20 200 4 20 40 60 100 200 300 400 L 0 + = 0.0 2d e = 0.0 3d M1 4d = 0.0 4d 3000 === ( M00..=00Q 56)dd 1e+02 15001500 2500 4000 20000 5d 180000 6d 00 16000 16000 19000 22000 140000140000 170000 200000 + e 11e+00 1e+02 1e+04 1e+06 Q Figure 2: We show the value of the mass M of the Q-balls in dependence on their charge Q for different values of d in Minkowski space-time. The small subplots show the behaviour close to the minimal value of Q. however, our numerical results for ω agree quite well with the analytic expression given by ω = ∆/ℓ. max max Thisis relatedto the factthatforω ω the functionφ(r) tends tozeroeverywhere. Hence allhigherorder max → terms in the potential become negligible and the Ansatz made in [41] gives a good result. Ournumericalresultsforω independenceonΛanddareshowninFig.3togetherwiththe valueof∆/ℓ. max As is apparent from this figure the analytical result agrees quite well with our numerical values. Moreover, we observe as expected from the analytical result that ω increases with decreasing Λ and decreases with max increasing d. InFig.4weshowthevalueofthemassM independenceonQford=2,3,4,5,6andΛ= 0.1. Verysimilar − to d = 3 the Q-balls have mass M larger than Q and are hence unstable, while for sufficiently large Q they become stable with respect to this decay. The value of Q=Q at which this transition happens depends on crit Λ and it was found that Q increases with decreasing Λ [40]. We find that the number of spatial dimensions crit d also has an influence on the value of Q . We find that Q increases with increasing d. crit crit It is also known that radially excited Q-ball solutions exist which possess a number k N of zeros in the ∈ scalar field function. In Fig.5(a) we show the mass M as function of ω and as function of Q, respectively for Λ= 0.1 andd=3,4 and k=0,1,2. We observethat for fixed d the value of ω increaseswith the increase max − of k. Moreover,the bigger k the bigger is the difference between ω for d=3 and d=4. The dependence of max the mass M on Q shown in Fig.5(b) indicates that all solutions with nodes are unstable to decay into Q free bosons. This is not surprising since these can be seen as excited Q-balls in AdS space-time. As pointed out in [31] the field theory on the boundary of AdS describes condensates of scalar glueballs. Thiswasfurther investigatedin[40], whereQ-ballsin(3+1)-dimensionalasymptoticallyglobalAdS havebeen studied. In Fig.6 we show our results for different values of Λ and d. Apparently, the expectation value of the dual operator < O >1/∆, which corresponds to the value of the condensate of scalar glueballs decreases for increasing d when fixing φ(0). This is related to the fact that the scalar field can spread into more dimensions when d is increased and hence less condensate is collected. Furthermore, the value of the condensate increases with decreasing Λ. This is connected to the fact that the “AdS box” decreases in size for decreasing Λ and as such the value of the condensate becomes bigger. 7 0 0 2. 1.285 68dd 2. == −−00..01L1 max1.61.8 1.2651.275−0.1010 −0.1014 −0.1018 max1.61.8 1.321.341.36 L = −0.1 ==== −−−−0000....50151 ( (a(aannnaaalylyltyticticiacaal)l)l) w 1.4 ===f (2460ddd) = 0 w 1.4 3.0 3.2 3.4 === 1 820ddd (analytical) 1.2 = 4d (analytical) = 6d (analytical) 1.2 == 180dd ( (aannaalylytitcicaal)l) 1.0 −0.10 −0.15 −0.20 −0.25 −0.30 −0.35 −0.40 −0.45 3 4 5 6 7 8 9 10 L d+1 (a) ωmax overΛ (b) ωmax overd Figure 3: The value of ω in dependence on Λ (left) and in dependence on d (right). Though we plot d here max as a continuous parameter, we should only read of the value for d N. We also give the value of ∆/ℓ and find ∈ that it gives a good approximation to our numerical data. 8 0 + e 1 2d 3d 06 3000 3000 + 1e 1500 1500 L 1500 2500 4000 1500 2500 4000 = −0.1 2d = −0.1 3d 4 = −0.1 4d M1e+0 3000 4d === −−(M00..=11Q 56)dd 1e+02 15001500 2500 4000 3000 5d 3000 6d 0 1500 1500 0 1500 2500 4000 1500 2500 4000 + e 11e+00 1e+02 1e+04 1e+06 1e+08 Q Figure 4: We show the value of the mass M of the Q-balls in dependence on their charge Q for different values of d in AdS space-time with Λ= 0.1. − 3.2 Boson stars We now discuss the case κ=0. This correspondsto bosonstarsin an asymptotically flat(Λ=0) orasymptot- 6 ically AdS space-time (Λ<0), respectively. 3.2.1 Λ=0 As pointed out in [39], boson stars in (2 + 1)-dimensional, asymptotically flat space-time do not exist for massive scalar fields without self-interaction. In the Appendix 2 we show that this is different in our case and 8 5 0 0 + 0 e 0 1 0 1 4 0 0 + 0 e 0 1 1 3 M0 L & k M+0 10 === −−−000...111 &&& 012 444ddd 1e L & k 10 === −−−000...111 &&& 012 333ddd 1e+02 ====== −−−−−−000000......111111 &&&&&& 012012 444333dddddd 1 0 1 + e 0.5 1.0 1.5 2.0 11e+01 1e+02 1e+03 1e+04 1e+05 w Q (a) M overω (b) M overQ Figure 5: The value of the mass M of the Q-balls in dependence on ω (left) and in dependence on the charge Q (right) in AdS space-time for different values of d and number of nodes k of the scalar field function. 0 2 0. L = −0.1 2d = −0.1 3d 5 = −0.1 4d 1 = −0.1 5d 0. == −−00..11 67dd 1>D == −−00..55 23dd O0 = −0.5 4d <0.1 == −−00..55 56dd = −0.5 7d 5 0 0. 0 0 0. 0 5 10 15 20 f (0) Figure 6: We show the expectation value of the dual operator on the AdS boundary < O >1/∆ corresponding to the value of the condensate of scalar glueballs in dependence on φ(0) for different values of Λ and d. that gravitating, asymptotically flat boson star solutions in (2+1) do exist in our model with an exponential self-interaction potential. We have also studied the dependence of the mass and charge on the frequency ω. Our results are show in Fig.7forthe mass. Thecurveslookqualitativelysimilarforthe chargeQ,this iswhy wedon’tshowthemhere. We observe that the behaviour at ω depends crucially on the number of spatial dimensions d. For d = 3 max the mass and charge tend to zero, while for d = 4 they tend to a finite value. This has already been observed before and is confirmed with our type of potential. For d = 5 we find that now the mass and charge tend to infinityattheapproachofω . Wehaveintegrateduptovaluesofthemassandchargeof107 andbelievethat max approachingω evencloser,these values wouldfurther increase. This can be understood using the argument max 9 0 00 500 3d 5 200 50 0.95 0.98 1.01 00 4d M5 6000 2000 0.995 0.998 1.001 k 50 ==== 0000....000001015 5 54 5d4ddd 6000 5d === 000...0000105 53 3d2dd 2000 0 =w = 0 1.0.01 2d 0.95 0.98 1.01 1 0.2 0.4 0.6 0.8 1.0 1.2 w Figure7: The value ofthe massM ofthe bosonstarsin dependence onthe frequencyω for Λ=0 anddifferent values of d and κ. The small subfigures show the behaviour of M at the approach of ω for d=3,4,5 (from max top to bottom). Note that the curvesfor the chargeQ look qualitatively very similar, this is why we don’t give them here. employed for d=4 in [21]. As noticed in this latter paper, the scalar field function and radial coordinate show a scaling behaviour that is equal in d=3 and d=4. We find that this is also the case here and generalizes to d>4 such that the behaviour is φ(r) φˆ(rˆ), n 0, A 1 for ω ω with max → → → → 1/2 φˆ=φ/φ , rˆ= φ κ1/2 r (26) 0 0 (cid:16) (cid:17) with φ a constant that tends to zero in the limit ω ω . Now this implies e.g. for the charge Q (the 0 max → argument works similarly for the mass M): ∞ ∞ Q= 2πd/2 ω dr rd−1 φ2 Q= 2πd/2 ω φ2−d/2κ−d/4 drˆrˆd−1 φˆ2 . (27) Γ(d/2) Z −→ Γ(d/2) max 0 Z 0 0 For d 3 this tends obviously to zero, for d = 4 this becomes constant, and for d 5 this tends to infinity, ≤ ≥ respectively for φ 0. In addition to this we observe that the approachto ω is not smooth in d=5. This 0 max → is shown in Fig. 8. For d = 3 and d = 4 the mass tends smoothly to zero and a finite value, respectively. For d = 5 we observe that the mass tends to a finite value on a lower branch of solutions, but that close to ω max new branches of solutions exist. These are quite small in extend and in fact are barely noticeable for d>5. As such, a second branch of solutions extends backwards from ω down to a critical value of ω and then bends max backwards to tend to infinity. Our conclusion hence is that while these solutions can exist for arbitrarily large values of the mass (and charge)there exists a mass gap in which solutions are not allowed. Furthermore, there isasmallintervalofω inwhichuptothreesolutionswithdifferentmassesexist. Tounderstandthispattern,we plotthethreesolutionsforω closetoω inFig.9. Weobservethatforthe samevalueofω the threesolutions max are distinguished by the value of φ(0) with φ(0) decreasing from the first to the third branch. Moreover, the solution spreads out more and more over r. On the first branch, the solution is still quite localized around the origin, while it becomes very delocalised on the third branch. 10