SU(3) Einstein-Yang-Mills Sphalerons and Black Holes

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Running Coupling in SU(3) Yang-Mills Theory

cal running coupling constant at small separation r. This requires control of the force from short to long distance in one simulation with a r L holding for the whole range of physical r involved. With L=a always limited to feasible lattice sizes like 32 or 48, compromises on the above conditions have to be accepted, and it is hardly possible to vary all scale ratios signi cantly to check for the stability of the results. State of the art calculations along these lines are reported in 1,2]. It has to be noted that the highest physical energies r 1 that can be reached here are below about 2 GeV, if one only stays a factor 2 : : : 3 away from the cuto energy. Cuto e ects are corrected semi-empirically using the lattice Coulomb propagator. While these are di cult and careful simulations, we nd it somewhat hard to assess the systematic errors in a completely convincing fashion. An alternative attempt to derive the coupling in QCD has been pioneered by the Fermilab group 3]. Here, in a quenched simulation, the spin averaged 1P-1S charmonium splitting is determined on a physically large lattice. Although this is a nice experimentally known scale with little sensitivity to the quark masses, also other masses could in principle be used here to set the scale. The point relevant in the present context is, that they extract from such a simulation the bare lattice coupling g0 together with the corresponding lattice spacing a in GeV. A perturbative method is then used to relate g0 to a physical coupling at a scale of the order of the cuto . The scale problem is clearly alleviated in comparison to the quark force method, as e ectively the cuto a 1 is identi ed with the high energy physical scale.

杨-米斯尔方程

杨-米斯尔方程
杨一米尔斯方程(Yang-Millsequation)是一个重要的微分方程，指杨一米尔斯作用量所确定的欧拉一拉格朗日方程。

杨一米尔斯方程也叫做杨—米尔斯理论。

杨氏理论是基于SU(N)组的一种量规理论，或者更普遍地说，是一个紧凑、半简单的李群。

杨振宁米尔斯理论旨在描述基本粒子的行为使用这些非阿贝尔李群和统一的核心的电磁和弱力(即U(1)×SU(2))以及量子色动力学理论的强力(基于SU(3))。

从而形成了对粒子物理标准模型理解的基础。

在一份私人信件中，沃尔夫冈，泡利在1953年提出了爱因斯坦的广义相对论的六维理论，将Kaluza、Klein、Fock等五维理论扩展到高维的内部空间。

然而，没有证据表明泡利发展了一个量子场的拉格朗日或它的量子化。

因为泡利发现他的理论“导致了一些非物质的阴影粒子”，他没有正式公布结果。

虽然保利没有发表他的六维理论，但他在苏黎世发表了两份关于它的演讲。

最近的研究表明，扩展的kaluza-klein理论一般不等同于杨斯-米尔斯理论，因为前者包含了额外的术语。

冀教版九年级Lesson 8 课件

Lead-in
The lifetime of Einstein
Lead-in
Albert Einstein
The greatest scientist of the 20th century and one of the greatest of all time.Among the several important discoveries Einstein made in his life, the greatest is the creation of his famous Theory of Relativity.
世纪
相对论
一生；终生
宇宙；万象
解决
New words
①可数名词 “聪明人”
e.g. She was one of the greatest minds of her generation.
②名词 “头脑；主意”
e.g. Are you quite clear in your own mind what you should do?
Unit 2 Great PeopleA Universe of Thought
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Particle-like solutions to the Yang--Mills-dilaton system in d=4+1 dimensions

a r X i v :h e p -t h /0611270v 3 20 F eb 2007Particle-like solutions to the Yang–Mills-dilatonsystem in d =4+1dimensionsEugen Radu †,Ya.Shnir ‡and D.H.Tchrakian †⋆†Department of Mathematical Physics,National University of Ireland Maynooth,Maynooth,Ireland ‡Institut f¨u r Physik,Universit¨a t Oldenburg,Postfach 2503D-26111Oldenburg,Germany ⋆School of Theoretical Physics –DIAS,10Burlington Road,Dublin 4,Ireland February 7,2008Abstract We construct static solutions to a SU (2)Yang–Mills (YM)dilaton model in 4+1dimensions subject to bi-azimuthal symmetry.The YM sector of the model consists of the usual YM term and the next higher order term of the YM hierarchy,which is required by the scaling condition for the existence of ﬁnite energy solutions.The basic features of two diﬀerent types of conﬁgurations are studied,corresponding to (multi)solitons with topological charge n 2,and soliton–antisoliton pairs with zero topological charge.1IntroductionMulti-instantons and composite instanton-antiinstanton bound states subject to bi-azimuthal symmetry were reported in a recent paper [1].These were constructed numerically,for the usual (p =1)SU (2)Yang-Mills (YM)system in 4Euclidean dimensions,the spherically symmetric special case being the usual BPST [2]instanton.In a work [3]unrelated to [1],regular and black hole static and spherically symmetric solutions to a Einstein–YM (EYM)system in 4+1dimensional spacetime were constructed numerically.The YM system in that model had gauge group SO (4),with the connection taking its values in (one of the two)chiral spinor representations of SO (4),namely in SU (2).Given that the solutions in [3]were static,i.e.that the YM ﬁeld is deﬁned on a 4dimensional Euclidean space,the SU (2)YM ﬁeld in [3]is the same one as that in [1].This is the relation between the two works [1]and [3],and our intention here is to exploit this relation.The present work serves two distinct purposes.Theﬁrst and main purpose is to pave the way for the construction of more general,non-spherically symmetric solutions to EYM systems in aﬁve dimensional spacetime.To our knowledge,no such results in EYM theory have appeared in the literature to date.Although considerable progress has been made in constructing asymptoticallyﬂat higher dimensional EYM solutions1all known conﬁgurations were subject to spherical symmetry.Our choice of a YM-dilaton(YMd) model is made because it has been shown that,at least in d=3+1dimensions,the classical solutions of this system mimic the corresponding EYM solutions[13],so the dilaton–YM exercise serves as a warmup for the considerably more complex gravitational problem. Our choice of a YMd model is made as an expedient in attempting the analysis of the corresponding EYM model,the last being of physical interest low energy eﬀective actions of string theory,descended from11dimensional supergravity.It is also a coincidence here,that these supergravity descended low energy eﬀective actions include the dilaton in addition to gravities and non Abelian matter.But here,the dilaton appears only as a substitute for gravity.A particular feature of the model to be introduced in the next section is that it features a term that is higher order in the YM curvature.As will be explained in section2,such terms are necessary to ensure that the solution yields aﬁnite mass.Such terms were employed in previous works[4,3,5,6,7,8]with precisely the same purpose.The physical justiﬁcation for introducing higher order YM terms,which goes hand in hand with the inclusion of higher order gravitational terms,is that these occur in the low energy eﬀective action of string theory[9].Thus in principle the choice of higher dimensional EYM models involves the selection of higher order terms in the gravitational and non Abelian curvatures,namely the Riemann and the YM curvatures,which are reparametrisation and gauge invariant. Because we are concerned withﬁnding classical solutions,we impose a pragmatic but important further restriction,namely that we consider only those Lagrange densities that are constructed from antisymmetrised2p curvature forms,and exclude all other powers of both Riemann and YM curvature2-forms.(In the gravitational case this results in the familiar Gauss–Bonnet type Lagrangians,while in the case of non Abelian matter to the YM hierarchy pointed out in footnote3below.)As a result,only velocity–squared ﬁelds appear in the Lagrangian,which is what is needed both for physical reasons and for solving the classicalﬁeld equations.In practice we add only the minimal number of such higher order terms that are necessitated by the requirements ofﬁnite mass.This criterion makes the inclusion of higher order gravitational terms unnecessary since we know from the(numerical)results of[4]that the qualitative properties of the classical solutions are insensitive to them.In addition to this argument based on numerical results there is an independent argument advanced at the end of section2of[6],based on the symmetries of the(higher order)gravitational terms,which in the absence of a dilaton dispenses with theeﬀectiveness of employing such terms.(Note here that the present YMd model is being used as a prototype for a EYM model,without an additional dilatonﬁeld.)This leaves one with higher order YM curvature terms only,whose status in the context of the string theory eﬀective action is complex and as yet not fully resolved.While YM terms up to F4 arise from(the non Abelian version of)the Born–Infeld action[10],it appears that this approach does not yield all the F6terms[11].Terms of order F6and higher can also be obtained by employing the constraints of(maximal)supersymmetry[12].The results of the various approaches are not identical.In this background,we restrict our considerations to terms in the YM hierarchy(see footnote3)only,in particular to theﬁrst two terms.Concerning our particular choice of4+1spacetime dimensions here,our reasons are: When imposing axial symmetry on a YMﬁeld in d=D+1dimensions the simplest way is,following[14],to impose spherical symmetry in the D−1dimensional subspace of the d spacelike dimensions.In this case the Chern-Pontryagin topological charge is ﬁxed by the boundary conditions imposed on theﬁrst polar angle,and no analogue of the vortex number appearing in the axially symmetric Ansatz for d=3[15]is featured[14]. Technically,the absence of a vortex number makes the numerical integration much harder. Imposing axial symmetry in turns in the x−y and z−u planes of D=4Euclidean space as in[1]on the other hand,features two(equal)vortex numbers,making the numerical work technically more accessible.It is our intention to use the particular bi-azimuthally symmetric Ansatz of[1]in D=4that has led us to restrict ourselves to d=4+1 dimensional spacetime.(Numerical work on implementing axial symmetry like in[14]is at present in active progress.)Of course,the exploitation of this type of symmetry is not restricted to4+1spacetime,but can be extended to any odd2q+1spacetime where q distinct azimuthal symmetries are imposed,but this in practice results in residual PDE’s of order three and higher for q≥3.The second and subsidiary aim of this work is to break the scale invariance of the usual YM system in D=4studied in[1],and the introduction of the dilatonﬁeld does just that.The question of instanton–antiinstanton bound states in a scale breaking model is an interesing enough matter in itself,presenting a second important motivation for this work.In Section2we present the model,impose the symmetry and state the boundary con-ditions,in successive subsections.The numerical results are presented in Section3,pre-senting both solutions with spherical and bi-azimuthal symmetry.We give our conclusions and remarks in theﬁnal section.2The modelThe model in5spacetime dimensions with coordinates x M=(x0,xµ)that we study here is described by the LagrangianL m=12·2!e2aφTr F2MN+τ2whereφis the dilatonﬁeld,F MN=∂M A N−∂N A M+[A M,A N]is the2-form YM cur-vature and F MNRS={F M[N,F RS]}is the4-form YM curvature consisting of the totally antisymmetrised product of two YM2-form YM curvatures.(The bracket[νρσ]implies cyclic symmetry.)τ1andτ2are dimensionful coupling strengths which will eventually be scaled out against the constant a in the exponent,which has the inverse dimension of the dilatonﬁeldφ.Similar to the d=3+1case,the form we choose for the coupling of the dilatonﬁeld to the nonabelian matter was found by requiring that a shiftφ→φ+φ0of the dilatonﬁeld to be compensated by a suitable rescaling of the coordinates.Let us give a brief justiﬁcation for the choice of the model(1).At the most basic level it is a YM–dilaton(YMd)model designed to simulate qualitatively a EYM model in d=5.Repacing the dilaton in(1)by a gravitational term is a physically relevant model, representing part of a low energy eﬀective action in d=5.Adding gravitational terms to (1)as it stands is a EYMd model,which is just as physically relevant.The YM system,which scales as L−4,in d=4+1supports static solitons,namely the BPST instantons in d=4+0dimensions.When the usual Einstein–Hilbert gravity, which scales as L−2,is added to the YM term,the soliton collapses because of the(Derrick) scaling mismatch.To compensate for this scaling mismatch,a term scaling as L−ν,with ν≥5must be added.If one is to restrict to positive deﬁnite terms2,νwill be even,and the most economical choice isν=6.A typical such term would be Tr(F∧DX)2,where X is a scalarﬁeld,e.g.a Higgs or sigma–modelﬁeld.This necessitates the introduction of a completely new(scalar)ﬁeld which unlike the dilaton is not directly recognised as a constituent of a low energy eﬀective action.For this reason we eschew this choice,and restrict our attention instead to systems featuring only YM(and eventually YMd)ﬁelds. The most economical choice then is to compensate with the YM term Tr(F∧F)2,scaling withν=4.We note,ﬁnally,that adding a(positive or negative)cosmological constant does not remedy the scaling mismatch since these terms do not scale at all.Indeed,in all higher dimensional EYM cases studied,withΛ=0[4,3],Λ<0[6]andΛ>0[7],the mass turns out to be inﬁnite when higher order YM terms are not employed.The YM sector of the action density in4+1dimensions employed here,is that one used in[3],namely the superposed p=1and p=2members of the YM hierarchy3.The dilaton breaks the scale invariance of the usual p=1YM system,and a simple Derrick-type scaling argument shows that noﬁnite mass/energy solution can exist ifτ2=0,i.e. the p=2term in(1)is necessary.The YM and dilatonﬁeld equations readτ1Dµ e2aφFµν +12π2 2e2aφˆL1+6e6aφˆL2 .(3)5AµAνAρAσ scaling as L−5isa possibility,albeit a considerably harder problem technically,and is at present under active consideration.3The YM hierarchy labeled by the integer p was introduced in[16]in the context of self-dual solutions in4p Euclidean dimensions,but superpositions of various p members were employed ubiquitously since.In(3)we have used the notationˆL 1=τ12·4!Tr F2MNRS.(4)2.1Imposition of symmetry and residual actionIn the YM connection A M=(A0,Aµ),we choose the temporal component A0=0to vanish and the spacelike components Aµis subjected to two successive axial symmetries,described in[1].We denote the Euclidean four dimensional coordinates as xµ=(x,y;z,u)≡(xα;x i), withα=1,2and i=3,4,and use the following parametrisationxα=r sinθˆxα≡ρˆxα,x i=r cosθˆx i≡σˆx i,(5) where r2=|xµ|2=|xα|2+|x i|2,with the unit vectors appearing in(5)parametrised as ˆxα=(cosϕ1,sinϕ1),ˆx i=(cosϕ2,sinϕ2),with0≤θ≤πρ Σαβˆxβ+φm2A ab iΣab,Aρ=−1index a in(9)-(10)runs only over three values,and reassigning the values of the index i=1,2,the analogues of(7)and(8)contract to giveA i= χ4+n2ρ (εˆx)i(εn(2))jΣj3+A34σˆx i n(2)jΣj3,(11)Aρ=A34σn(2)jΣjm,(12) exhibiting the Abelian connection A34σanalogous to the Abelian connection A34ρappearing in(7)and the isodoublet function(χ3,χ4).The corresponding axially symmetric decomposition ofΦin(10)isΦ=ξ1n(2)jΣj4+ξ2Σ34.(13)In(11)-(12)and(13)we have used the unit vector n(2)i=(cos n2ϕ2,sin n2ϕ2),with vorticity integer n2.Theﬁnal stage of symmetry imposition is to treat the two azimuthal symmetries imposed in the x−y and the z−u planes on the same footing,leading to the equality of the two vortex numbers,n1=n2≡n.Denoting the residual functions(A34ρ,A34σ)=(aρaσ,),(χ3,χ4)=χA,(ξ1,ξ2)=ξA, and regarding(aρ,aσ)as an Abelian connection on the quater plane deﬁned by(ρ,σ),the residual action densities can be expressed exclusively in terms of the SO(2)curvaturefρσ=∂ρaσ−∂σaρ(14) and the covariant derivativesDρχA=∂ρχA+aρ(εχ)A,DσχA=∂σχA+aσ(εχ)A,(15)DρξA=∂ρξA+aρ(εξ)A,DσξA=∂σξA+aσ(εξ)A.The residual two dimensional YM action densities descending from the p=1and the p=2 termsˆL1andˆL2deﬁned by(4)are,respectively,L1=τ1σ |DρχA|2+|DσχA|2 +σρσ(εABχAξB)2,L2=τ2Since our numerical constructions will be carried out using the coordinates(r,θ)we display (16)also asL1=τ1cosθ |D rχA|2+1sinθ |D rξA|2+1r sinθcosθ(εABχAξB)2(18)L2=τ2g L m= ∞0dr π/20dθ 1r2φ2,θ)+(e2aφL1+e6aφL2) ,(20) and equals the total action of solutions,viewed as solitons in a d=4Euclidean space.2.2Boundary conditionsTo obtain regular solutions withﬁnite energy density we impose at the origin(r=0)the boundary conditionsa r=0,aθ=0,χA= 0−n2 ,ξA= 0−n1 ,(21)which are requested by the analyticity of the YM ansatz,and∂rφ|r=0=0for the dilaton ﬁeld.In order toﬁndﬁnite mass solutions,we impose at inﬁnitya r=0,aθ=−2m,χA=(−1)m+1n2 sin2mθcos2mθ ,ξA=−n1 sin2mθcos2mθ ,φ=0,(22)m being a positive integer.Similar considerations lead to the following boundary conditions on theρandσaxes:a r=1n1∂θξ1,χ1=0,ξ1=0,∂θχ2=0,ξ2=−n1,∂θφ=0,(23)forθ=0,anda r=1n2∂θχ1,χ1=0,ξ1=0,χ2=−n2,∂θξ2=0,∂θφ=0,(24)forθ=π/2,respectively.2.3Topological chargeIn our normalisation,the topological charge is deﬁned as q =12εµν 14 εµν∂µ(χA D νξA −ξA D νχA )d 2x.(27)The integration in (26)iscarriedoutover the 2dimensional space x µ=(x ρ,x σ).As expected this is a total divergence expressed by (27).Using Stokes’theorem,the two dimensional integral of (27)reduces to the one dimen-sional line integralq =12[1−(−1)m ]n 1n 2.(29)3Numerical resultsApart from the coupling constants τ1and τ2the model contains also the dilaton constant a .Dimensionless quantities are obtained by rescalingφ→φ/a,r →r (τ2/τ1)1/4,(30)This reveals the existence of one fundamental parameter which gives the strength of the dilaton-nonabelian interactionα2=a 2τ3/21/τ1/22,(31)which is a feature present also in the EYM case [3].We use this rescaling to set τ1=1,τ2=1/3in the numerical computation,without any loss of generality.One can see that the limit α→0can be approached in two ways and two diﬀerent branches of solutions may exist.The ﬁrst limit corresponds to a pure p =1YM theory with vanishing dilaton and p =2YM terms,the solutions here replicating the (multi-)instantons and composite instanton-antiinstanton bound states discussed in [1].The other possibility corresponds to a ﬁnite value of the dilaton coupling a as τ1→0.Thus,the second limitingconﬁguration is a solution of the truncated p=2YM system interacting with the dilaton, with no p=1YM term.We have studied YMd solutions with m=1,2.From our knowledge of the tolopogical charges(29),the m=1solutions will describe(multi)solitons and the m=2solutions, soliton-antisoliton conﬁgurations.Also,to simplify the general picture we set n1=n2=n in the boundary conditions(21)-(24).The spherically symmetric solutions are found by using a standard diﬀerential equations solver.The numerical calculations in the bi-azimuthally symmetric case were performed with the software package CADSOL,based on the Newton-Raphson method[17].In this case,theﬁeld equations areﬁrst discretized on a nonequidistant grid and the resulting system is solved iteratively until convergence is achieved.In this scheme,a new radial variable x=r/(1+r)is introduced which maps the semi-inﬁnite region[0,∞)to the closed region[0,1].As will be described below,solutions exist for certain ranges of the parameterα.It turns out that m=1solutions with all n and m=2solutions with n=1exist for a range ofαstarting from aα→0limit,but do not persist all the way up to the second α→0limit.(However,the way the solutions approach the limitα→0depends on m.) By contrast weﬁnd that m=2solutions with all n>1,exist for allαbetween the two limits.3.1m=1conﬁgurationsn=1spherically symmetric solutionsIn the spherically symmetric limit,which case we shall analyse numericallyﬁrst,the angular dependence of these functions isﬁxed and the only remaining independent function depends on the variable r.The independent function in this case is aθ=w(r)−1,with the remaining fuctions(a r,χA,ξA)given bya r=0,χ1=−ξ1=1r )+9e6φτ2(w2−1)2r4) ′=2e2φw(w2−1)r2 .(33)The asymptotic solutions to these functions can be systematically constructed in both regions,near the origin and for r≫1.The small r expansion isw(r)=1−br2+O(r4),φ=φ0+4α2(τ1with b,φ0two real parameters,while as r→∞weﬁndw(r)=±1−4φ127r6+O(1/r8),φ=φ127r6+O(1/r8).(35)We numerically integrate the Eqs.(33)with the above set of boundary conditions forτ1=1,τ2=1/3and varyingα.The picture we found is very similar to that found for the EYM system[3],the dilaton coupling constant playing the role of the Newton constant. First,for a givenα,solutions with the right asymptotics exist for a single value of the ”shooting”parameter b which enters the expansion(34).Forαsmall enough,a branch of solutions smoothly emerges from the BPST conﬁguration[2].Whenαincreases,the mass M and the absolute value of the dilaton function at the origin increase,as indicated in Figure1.These solutions exist up to a maximal valueαmax≃0.36928of the parameterα.As in the corresponding gravitating case[3],we found another branch of solutions inthe intervalα∈[αcr(1),αmax]withα2cr(1)≃0.2653.On this second branch of solutions, bothφ(0)and M continue to increase but stayﬁnite.However,a third branch of solutions exists forα∈[0.2653,0.2652],on which the two quantities increase further.A fourth branch of solutions has also been found,with a correspondingαcr(3)≃0.2642.The mass M,the value of the dilatonﬁeld at the originφ(0)and the initial(shooting)parameter b increase along these branches.Further branches of solutions,exhibiting more oscillations aroundα≃0.264are very likely to exist but their study is a diﬃcult numerical problem. This critical behaviour is described as a conicalﬁxed point in the analytic analysis in[5]. Therefore we conclude that,as in the spherically symmetric gravitating case[3],the limit τ1=0is not approached for solutions with m=1,n=1.As a general feature,all solutions discussed here present only one node in the gauge function w(r).As in the higher dimensional EYM models discussed in[4,5],no multinode solutions were found.n>1Solutions with bi-azimuthal symmetry with nontrivial dependence on both r andθare found for(n1,n2)=1subject to the boundary conditions(21)-(24).We have studied solutions for m=1with2≤n≤5.The general features of the m=1solutions are the same for all n>1.Also,as seen in(29),the m=1conﬁgurations carry a topological charge q=n2.The corresponding solutions of the F2MN model are self-dual and have been considered already in[18],[19](for a diﬀerent parametrization of the gaugeﬁeld,however).These solutions are constructed by starting with the known spherically symmetric con-ﬁguration and increasing the winding number n in small steps.The iterations converge, and repeating the procedure one obtains in this way solutions for arbitrary n.The physical values of n are integers.The typical numerical error for the functions is estimated to be of the order of10−3or lower.Any spherically symmetric conﬁguration appears to result in generalisations with higher winding numbers n.Moreover,the branch structure noticed for the m=1,n=1case seems to be retained by the higher winding number m=1solutions.Again,theﬁrst branchof solutions exists up to a maximal value ofα,where another branch emerges,extending backwards inα.We managed to construct higher winding number n counterparts of the ﬁrst two branches of spherically symmetric solutions.The mass M and the absolute value of the dilaton function at the origin increase along these branches,as shown in Figure2. Note that the value of the dilaton function at the origin exhibited in theﬁgures is actually φ(r=0,θ=0),restricting toθ=0.This restriction is reasonable since for all solutions with bi-azimuthal symmetry discussed in this paper,the dilaton function at r=0presents almost no dependence on the angleθ.We expect that the oscillatory pattern ofφ(0)arising from the conicalﬁxed point observed above for the spherically symmetric n=1solutions,will also be discovered for the n>1solutions here,but their construction is a diﬃcult numerical problem beyond the scope of the present work.In Figure3we present the gauge functions,the dilaton,and the topological charge density1̺=parametrised by the eﬀective coupling constantα,while the latter has no such parameter. As will be described below,m=2n=1solutions exist for a certain range ofα,and thisrange excludes the limiting case where the contribution to the action of the dilaton term and the p=2YM term in(1)disappear,i.e.a F2MN model.Weﬁnd that in the limitα→0resulting from a→0,cf.(31),no solutions of thistype exist.However in the limitα→0corresponding to aﬁnite value of the dilaton coupling a asτ1→0,such solutions exist.This limiting conﬁguration is then a solution of the truncated system consisting of the dilaton term and p=2YM term F2MNRS,whichdominate.Its characteristic feature is that for this conﬁguration both nodes of the eﬀective Higgsﬁelds|χ|and|ξ|merge on theθ=π/4axis.A family of solutions of the model(1) emerges from this conﬁguration.Asαincreases,the nodes move towards the symmetry axes,ρandσ,respectively,forming two identical vortex rings whose radii slowly decrease while the separation of both rings from the origin increase.At the critical valueαcr≃0.265 the node structure of the conﬁguration changes,both vortex rings shrink to zero size and two isolated nodes appear on each symmetry axis.This structure is known for the usual YM system in d=4+0[1],indeed,increasing ofαalong this branch can be associated with increasing of the couplingτ1w.r.t.τ2as the dilaton coupling a remainsﬁxed;then the term F2MN becomes leading.The maximum of the action density however is still located onθ=π/4axis.Another similarity with the instanton-antiinstanton solution of the d=4+0p=1YM theory is that the gauge functions a r,aθas well the dilaton functionφof the n=1,m=2 solutions also are almostθ-independent.Along this branch the mass of the solutions grows with increasingαsince with increasing couplingτ1the contribution of the term F2MN also increases.As the eﬀective coupling increases further beyondαcr the relative distance between the nodes increases,one lump moving towards the origin while the other one moves in the opposite direction.Along this branch both the value of the dilatonﬁeld at the origin|φ(0)| and mass of conﬁguration M increase asαincreases.This branch extends up to a maximal valueα(1)max≃0.311beyond which the dilaton coupling becomes too strong for the static conﬁguration to persist.The second branch,whose energy is higher,extends backwards up toα(2)max≃0.279.Along this branch both|φ(0)|and the mass of the conﬁguration continue to increase asαdecreases.Also the separation between the nodes decreases and both nodes invert direction of the motion,moving toward each other along this branch. In Figure5we present the values of the dilaton function at the originφ(0)and the total mass(rescaled byα2)of these conﬁgurations as functions ofα.n=2This conﬁguration also resides in the topologically trivial sector and can be considered as consisting of two solitons of charges n=±2.Then the interaction between the nonabelian matterﬁelds becomes stronger than in the case of unit charge constituents and the expected pattern of possible branches of solutions is diﬀerent from the n=1case above.Indeed,the n=2,m=2solutions show a diﬀerent dependence on the coupling con-stantα,with two branches of solutions.The lower branch emerges from the correspondingsolution in pure p=1YM theory with vanishing dilaton and p=2YM terms,replicating the corresponding solution in[1].The variation of the eﬀective coupling along this branchis associated with the decrease ofτ1,atﬁxedτ2andﬁxed dilaton coupling a.The secondbranch emerges from a solution of the p=2YM-dilaton system,the unrescaled mass M diverging in this limit,with the rescaled mass Mα2vanishing as seen from Figure5a.Atthe maximal valueαmax≃0.2372this branch bifurcates with the lower YM branch.For larger values ofα,the dilaton coupling becomes too strong for the static conﬁgurations to persist.Thus for0≤α<αmax we notice the existence of(at least)two distinct solutionsfor the same value of coupling constant.For the same value ofα,the mass of the second branch solution is larger that that ofthe corresponding lower branch conﬁguration(s).One should also notice the existence ofa curious backbending of the lower branch for0.193<α<0.218.Four distinct solutions exist in this case for the same value ofα(three of them located on the lower branch), distinguished by the value of the mass and the dilatonﬁeld at the origin.This pattern is illustrated in Figure5.Again,observation of the positions and structure of the nodes of the eﬀective scalarﬁelds allows us to better understand the behaviour of the solutions.For lower branch solutions with small values ofαthere are two(double)nodes of theﬁelds|χ|and|ξ|on the ρandσsymmetry axes respectively.The locations of nodes correspond to the locations of the two individual constituents and the action density distribution posesses two distinct maxima on theθ=π/4axis.The distance between these nodes changes only slightly along the lower mass branch.The backbending inαobserved in this case is reﬂected also for in the relative positions of the nodes.At the maximal value ofα,the inner node is located atρ(1)0=σ(1)0≃2.97and the outer node is located atρ(2)0=σ(2)0≃4.18.Along the upper branch,asαslightly decreases belowαmax,the inner node inverts direction of its movement toward the outer node which still moves inwards.Thus,both nodes on the symmetry axis rapidly approach each other and merge forming a two vortex ring solution asα≃0.2355.The action density then has a single maximum onθ=π/4 axis.Asαdecreases further both nodes move away from the symmetry axis and their positions do not coincide with the location of the maximum of the action density.Further decreasingαresults in the increase of the radii of the two rings around the symmetry axis, and in the limitα→0the rings touch each other on theθ=π/4axis.In Figure4we give three dimensional plots of the modulus of the eﬀective Higgsﬁeld ξfor the n=m=2upper branch vortex solution atα=0.20and the n=m=2lower branch double node solution at the same value ofα.The action density as given by(1)is also plotted atα=0.20both for the upper and for the lower branches.The numerical calculations indicate the possibility that the solutions of the fundamentalYM branch,namely the branch on which the p=1YM term dominates,are not unique. It is possible that higher linking number conﬁgurations with higher masses might exist. This possibility will be explored elsewhere.。

Unit3Albert Einstein爱因斯坦

Achievements
• • • • • • I. General relativity 广义相对论 II. Special relativity 狭义相对论 III. Brownlan effect 布朗运动 IV. Photoelectric effect 光电效应 V. E=mc^2 E=mc^2 VI. Einstein field equation 爱因斯坦场方程 • VII. Bose-Einstein statistics 波色-爱因斯坦统计
Albert Einstein died in Nineteen-FiftyFive. He was seventy-six years old.
THE END
Einstein’s famous sayings
1. Imagination is more important than knowledge. 想象力比知识更重要。 2. Try not to become a man of success, but rather try to become a man of value. 试着不去做一个成功的人，而去做一个有价值的人。 3.Anyone who has never made a mistake has never tried angthing new. 一个人从未犯错是因为他从不尝试新鲜事物。 4.Weakness of attitude becomes weakness of character. 态度上的弱点会变成性格上的弱点。 5. Life is like riding a bicycle. To keep your balance you must keep moving. 人生就像骑单车。想保持平衡就得往前走。 6.Logic will get you from A to B. Imagination will take you everywhere. 逻辑会把你从A带到B，想象力能带你去任何地方。

物理学家：托马斯·杨

生平简介科学成就趣闻轶事一、生平简介托马斯·杨（Thomax Young,1773—1829年）英国医生兼物理学家，光的波动说的奠基人之一。

1773年6月13日生于萨默塞特郡的米菲尔顿。

他从小就有神童之称，兴趣十分广泛。

后来进入伦敦的圣巴塞罗缪医学院学医，21岁时，即以他的第一篇医学论文成为英国皇家学会会员。

为了进一步深造，他到爱丁堡和剑桥继续学习，后来又到德国哥廷根去留学。

在那里，他受到一些德国自然哲学家的影响，开始怀疑起光的微粒说。

1801年进行了著名的杨氏干涉实验，为光的波动说的复兴奠定了基础。

1829年5月10日杨氏在伦敦逝世。

二、科学成就1．著名的杨氏干涉实验，为光的波动说奠定一基础。

杨氏干涉实验的巧妙之处在于，他让通过一个小针孔S0的一束光，再通过两个小针孔S1和S2，变成两束光。

这样的两束光因为来自同一光源，所以它们是相干的。

结果表明，在光屏上果然看见了明暗相间的干涉图样。

后来，又以狭缝代替针孔，进行了双缝干涉实验，得到了更明亮的干涉条纹。

在他之前，不少人曾进行过光的干涉实验。

由于他们是用两个独立的非相干光源发出的两束光迭加，因此，这些实验都失败了。

他用这个实验首先引入干涉概念论证了波动说，又利用波动说解释了牛顿环的成因和薄膜的彩色。

1801年他引入叠加原理，把惠更斯的波动理论和牛顿的色彩理论结合起来，成功地解释了规则光栅产生的色彩现象。

1803年，他又用波动理论解释了障碍物影子具有彩色毛边的现象。

1820年他用比较完善的波动理论对光的偏振作出了比较满意的解释，认为只要承认光波是横波，必然会产生偏振现象。

2．对人眼感知颜色的研究，建立三原色原理他还第一个测量了7种颜色光的波长。

他曾从生理角度说明了人眼的色盲现象；他还建立了三原色原理，指出一切色彩都可以从红、绿、蓝这三种原色的不同比例的混和而得到。

3．对弹性力学的研究托马斯·杨对弹性力学很有研究，特别是对胡克定律和弹性模量。

The string solution in SU(2) Yang-Mills-Higgs theory

a rXiv:h ep-th/966124v12J un1996The string solution in SU(2)Yang-Mills-Higgs theory V.D.Dzhunushaliev ∗and A.A.Fomin Theoretical physics department,the Kyrgyz State National University,720024,Bishkek,Kyrgyzstan Abstract The tube solutions in Yang -Mills -Higgs theory are received,in which the Higgs ﬁeld has the negative energy density.This solutions make up the discrete spectrum numered by two integer and have the ﬁnite linear energy density.Ignoring its transverse size,such ﬁeld conﬁguration is the rest inﬁnity straight string.PACS number:03.65.Pm;11.17.-w At the end of 50-th years W.Heisenberg has been investigate the non-linear spinor matter theory (see,for example,[1],[2]).It is supposed that on the basis one or another nonlinear spinor equation the basic parameters of the elementary particles existing at that time will be derived:masses,charges and so on.The mathematical essence of this theory lies in the fact that the nonlinear spinor Heisenberg equation (HE)(or in the simpler case the nonlinear boson equation like nonlinear Schr¨o dinger equation)has the discrete spectrum of the solutions having physical meaning (possesing,for example,the ﬁnite energy).This solutions give the mass spectrum in clas-sical region even.This gave hope that after quantization more or less likely mass spectrum and the charges of the elementary particles would be derive.Now the string can to arise in Dirac theory with the massive vector ﬁeld A µby interaction 2magnetic charges with opposite sign [3].At present timethe investigations continue along this line and explore not1-dimensional ob-ject(string)stretched between quarks(see,for example,[4])but3-dimensional (tube)ﬁlled byﬁeld(see,for example,[5],[6]).So,for example,a tube of the chromodynamicalﬁeld and its properties in[5]is considered.But this consideration is phenomenological because a question on the reason of the ﬁeld pinching isn’t aﬀected,also a question on theﬁeld distribution in the tube isn’t analyzed.In this article we shown that the Yang-Millsﬁeld interacted with Higgs scalarﬁeld is conﬁned in tube.In this case the Higgsﬁeld have the negative energy density.In[2]it is showed that the nonlinear Klein-Gordon and Heisenberg equations have the regular solutions.They are the spherical symmetric par-ticlelike solutions numered by integer,i.e.they form discrete spectrum with the corresponding energy value.One would expect(and this will be showed below)that we have in axial symmetric case as well as in spherical-sym-metric case the physical interesting(string)solutions withﬁnite energy per unith length.Finally,we present some qualitative argument in favour of the existence suchﬁeld conﬁgurations(tube,string)according[4].In QCD vacuumﬁeld taken external pressure on the gluon tube.Diameter of such tube will be deﬁned from equilibrium condition between external pressure of the vacuum ﬁeld and internal pressure of the gluonﬁeld in tube.It can be evaluate by minimizing the energy density of such tube which is the diﬀerence between the positive energy density of the chromodynamicﬁeld and negative energy density of vacuumﬁeld in QCD.This diameter R0after corresponding cal-culations is equal:ΦR0=F aµνFµνa−14g2where a =1,2,3is SU (2)colour index;µ,ν=0,1,2,3are spacetime indexes;F aµν=∂µA aµ−∂νA aµ+ǫabc A bµA aνis the strength tensor of the SU (2)gauge ﬁeld;F µν=F aµνt a ,t a are generators of the SU (2)gauge group;D µΦ=(∂µ+A µ)Φ;V (Φ)=λ(Φ+Φ−4η2)/32;g,η,λare constant;Φis an isodoublet of the Higgs scalar ﬁeld;The Yang -Mills -Higgs equations system look by following form in this model:D µF µνa =(−γ)−1/2∂µ (−γ)1/2F µνa +ǫabc A bµF µνc =g 2∂Φ+,(4)where γis the metrical tensor determinant.We seek the string solution in the following form:the gauge potential A aµand the isodoublet of the scalar ﬁeld Φwe chosen in cylindrical coordinate system (z,r,θ)as :A 1t =2ηf (r ),(5)A 2z =2ηv (r ),(6)A 3θ=2ηrw (r ),(7)Φ= 2ηϕ(r )0 (8)By substituting Eq’s(5-8)in Eq’s(3-4)we receive the following equations system:f ′′+f ′x =v 4 −f 2+w2 −g 2ϕ2 ,(10)w ′′+w ′x 2=w 4 −f 2+v 2 −g 2ϕ2,(11)ϕ′′+ϕ′λis introduced;(′)means thederivative with respect to x ;and the following renaming are made:g 2λ−1/2→3g2,f(x)λ−1/2→f(x),v(x)λ−1/2→v(x),w(x)λ−1/2→w(x).We will study this system by the numerical tools.In this article we investigate the easiest case v=f=0.Thus system(9-12)look as:w′′+w′x2=−g2wϕ2,(13)ϕ′′+ϕ′2+···,(15) w=w0x+w3x32ϕ0 1−ϕ20 ,(17) w3=−3w(x)≈1√x−Cg2x2,(22)where integers m and n enumerate the knot number ofϕ(x)and w(x)func-tions respectively.According to this we shall denote the boundary value ϕ(0)and parameter g in the following manner:ϕ∗mn and g∗mn.The result of numerical calculations on Fig.1,2are displayed(w1=0.1).The asymptotic behaviour of theϕmn(x)and w mn(x)functions as in(21)-(22)results in that the energy density of thisﬁelds drop to zero as exponent on the inﬁnity and this means that this tube has theﬁnite energy per unit. It is easy to show that aﬂux of colour”magnetic”ﬁeld H z across the plane z=const isﬁnite.Thus we can to speak that the Yang-Mills-Higgs theory have the tube solution if the Higgsﬁeld have the negative energy density.It is notice that this solutions are not topological nontrivial thread.Ignoring the transversal size of obtained tube we receive the rested boson string withﬁnite linear energy density.References5[1]Nonlinear quantumﬁeld theory.Ed.D.D.Ivanenko,Moskow,IL,p.464,1959.[2]R.Finkelstein,R.LeLevier,M.Ruderman,Phys.Rev.,83,326(1951).[3]Nambu Y.,Phys.Rev.1974,D10,p.4262.[4]Bars I.,Hanson A.J.,Phys.Rev.,1976,D13,p.1744.[5]Nussinov S.,Phys.Rev.D.1994,v.50,N5,p.3167.[6]Olson C.,Olsson M.G.,Dan LaCourse,Phys.Rev.D.1994,v.49,N9,p.4675.[7]Barbashow B.M.,Nesterenko W.W.Relativistic string model inhadronic physics,Moskow,Energoatomizdat,p.179,1987.[8]V.D.Dzhunushaliev,Superconductivity:physics,chemistry,technique,v.7,N5,767,1994.6。

Exact solutions in the Yang-Mills-Wong theory

1 The
2
of gauge invariant operators obeys the so-called factorization relation, and quantum ﬂuctuations disappear [8]. Thus QCD becomes a classical theory as N → ∞. We suggest that the large-N YMW theory is intimately related to the classical limit of QCD. Note, however, that the conﬁnement problem is out of the question now. Indeed, it is conceivable that quarks constituting a hadron experience an attractive constant force originating from a term of the potential Aµ which linearly rises with distance between the quarks, and such a behavior of Aµ is to provide the area law for the Wilson loop functional [9]. Are we correct in interpreting the area law as the evidence of the constant attractive force? As will be shown, an exact classical solution Aµ with the linearly rising term actually exists. Although this term contributes to the ﬁeld strength, it produces no force. The general reason for such a surprising result is the conformal invariance. The linearly rising term violates the scale invariance. While such a violation being allowable for the gauge quantities Aµ and Fµν , it cannot be tolerated for observables. One may expect a dimensional parameter, measuring a gap in the energy spectrum and violating the scale symmetry, to emerge only upon quantization leading to anomalies. Meanwhile exact classical solutions are crucial in learning the symmetry of the vacuum. One believes two phases of the strong interacting matter to exist, hot and cold, which must be distinguished by their symmetry. At high temperatures, the asymptotical freedom dominates, hence the conventional SU(3)c symmetry is inherent in the hot phase. On the ˇ cki [10] developed an exhaustive phenomenological classiﬁother hand, Ne’eman and Sijaˇ cation of hadrons on the basis of inﬁnite-dimensional unitary representations of SL(4, R), which hints that SL(4, R) is the cold phase symmetry. Where does this SL(4, R) come from? Coleman [11] argued that the symmetry of the vacuum is the symmetry of world. Given the vacuum invariant under SL(4, R), excitations about it possess the same symmetry. Since the symmetry of the gluon vacuum is nothing but the symmetry of the background ﬁeld, the responsibility for SL(4, R) rests with the background described by a certain solution of the QCD equations in the classical limit. It is the background generated by quarks in hadrons that provides the SL(4, R) relief for gluon excitations. We will ﬁnd two classes of exact retarded solutions to the classical Yang-Mills (YM) equations. Solutions of the ﬁrst class, invariant under SU(N ), appear to be related to the background in the hot phase. Solutions of the second class might be treated as the background generated by bound quarks in the cold phase. These solutions are complex valued with respect to the Lie algebra su(N ), but one can convert them to the real form to yield the invariance under SL(N, R) or its subgroups. In particular, the background generated by any three-quark cluster is invariant under SL(4, R), and that generated by any two-quark cluster is invariant under SL(3, R). ˇ cki operates in spacetime while the present Notice that SL(4, R) of Ne’eman and Sijaˇ SL(4, R) acts in the color space. However, we attempt to interweave two arenas by reference to that color degrees of freedom may be convertible into spin degrees of freedom, the fact discovered by Jackiw and Rebbi, and Hasenfratz and ’t Hooft [12]. The paper is organized as follows. Section II outlines the general formalism of the YMW theory. The next section is devoted to a justiﬁcation of the Ansatz whereby we seek exact retarded solutions of the YM equations with the source composed of several arbitrarily moving quarks. Finding such solutions is traced by the simplest example of the single-quark source, Sec. IV. Properties of the background generated by two-quark sources 3

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a rXiv:h ep-th/95453v11A pr1995SU(3)Einstein-Yang-Mills Sphalerons and Black Holes Burkhard Kleihaus 1,Jutta Kunz 1,2and Abha Sood 11Fachbereich Physik,Universit¨a t Oldenburg,Postfach 2503D-26111Oldenburg,Germany 2Instituut voor Theoretische Fysica,Rijksuniversiteit te Utrecht NL-3508TA Utrecht,The Netherlands February 1,2008Abstract In the SU(3)Einstein-Yang-Mills system sequences of static spherically sym-metric regular solutions and black hole solutions exist for both the SU(2)and the SO(3)embedding.We construct the lowest regular solutions of the SO(3)embedding,missed previously,and the corresponding black holes.The SO(3)solutions are classiﬁed according to their boundary conditions and the numberof nodes of the matter functions.Both,the regular and the black hole solutions are unstable.Utrecht-Preprint THU-95/81IntroductionThe SU(2)Einstein-Yang-Mills system possesses a sequence of static spherically sym-metric regular particle-like solutions[1],which are unstable[2,3].The n-th solution of the sequence has n nodes and2n unstable modes[4,5].The lowest solution has been interpreted in analogy to the electroweak sphaleron[6]as the top of a barrier between vacua[3].Beside the regular solutions the SU(2)Einstein-Yang-Mills system possesses static spherically symmetric black hole solutions.There are in fact two diﬀerent types of black hole solutions having the same mass.These are the Schwarzschild black holes with vanishing gaugeﬁelds and the SU(2)coloured black holes[7,8,9].Like the regular solutions the coloured black hole solutions are unstable[10,11,5].Thus for a certain range of masses the system possesses two distinct types of black holes,providing a counterexample to the“no-hair conjecture”for black holes,unless the coloured black holes are discarded as a counterexample because of their instability.Here we consider static spherically symmetric regular solutions and black holes of the SU(3)Einstein-Yang-Mills system(with vanishing time component of the gauge ﬁeld).Such solutions are obtained for both,the SU(2)embedding and the SO(3)em-bedding.The SU(2)embedding reproduces the known SU(2)solutions,while the SO(3) embedding leads to new interesting solutions.Missing the lowest regular solutions,sev-eral regular SO(3)solutions have been found previously by K¨u nzle[12],but he did not succeed in obtaining the corresponding black holes.We here construct theﬁrst few solutions of a new class of regular SO(3)solutions. These solutions include the lowest regular SO(3)solution,which we have obtainedﬁrst as a limiting solution of the SU(3)Einstein-Skyrme system[13](in analogy to the SU(2) case[14]).We classify the SO(3)solutions according to their boundary conditions and the number of nodes of the matter functions.Analogous to the SU(2)Einstein-Yang-Mills system asymptoticallyﬂat SO(3)black hole solutions emerge from the regular solutions by requiring regularity at aﬁnite event horizon.We construct the black hole solutions corresponding to theﬁrst few solutions with the least nodes.The stability of regular and black hole solutions of arbitrary gauge groups has been studied recently[15].We apply the theorems of Ref.[15]to demonstrate the instability of both regular and black hole SO(3)solutions.2SU(3)Einstein-Yang-Mills Equations of Motion We consider the SU(3)Einstein-Yang-Mills actionS=S G+S M= L G√−gd4x(1)withL G=12Tr(FµνFµν),(3) whereFµν=∂µAν−∂νAµ−ie[Aµ,Aν],(4)Aµ=1r.(7) Generalized spherical symmetry for the gaugeﬁeld is realized by embedding the SU(2) or the SO(3)generators T i in SU(3).In the SU(2)-embedding T=12re( e r× τ)i,(8)with the SU(2)Pauli matrices τ=(τ1,τ2,τ3).In the SO(3)-embedding T=(λ7,−λ5,λ2), and the corresponding ansatz for the gaugeﬁeld with vanishing time component is A0=0,A i=2−K(r)2re ( e r× Λ)i, e r· Λ)+,(9)where[,]+denotes the anticommutator,and Λ=(λ7,−λ5,λ2).The SU(2)-embedding,eq.(8),leads to the well studied SU(2)Einstein-Yang-Mills equations[1,7,8,9].To obtain the SU(3)Einstein-Yang-Mills equations for the SO(3)-embedding,eq.(9),we also employ the tt and rr components of the Einstein equations, yielding for the metric functionsµ′=N(K′2+H′2)+12A′=√√4πG)r,and the prime indicates the deriva-tive with respect to x.For the matterﬁeld functions we obtain the equations1(ANK′)′=AH H2+7K2−4 .(14)4x2With help of eq.(11)the metric function A can be eliminated from the matterﬁeld equations.Note,that the equations are symmetric with respect to an interchange of the functions K(x)and H(x),and to the transformations K(x)→−K(x),and H(x)→−H(x),yielding degenerate solutions.Comparing the equations of the SO(3)embedding to those of the SU(2)embed-ding[1]shows that to each SU(2)solution there corresponds a scaled SO(3)solution. Deﬁning x=2˜x,andµ=2˜µthe functions K(x)=2w(˜x),H(x)=0satisfy the SO(3) equations with coordinate x,when the function w satisﬁes the SU(2)equations with coordinate˜x.Thus these SO(3)solutions have precisely double the mass of their SU(2) counterparts.3Regular SolutionsLet usﬁrst consider the regular solutions of the SU(3)Einstein-Yang-Mills system. Requiring asymptoticallyﬂat solutions implies that the metric functions A andµboth approach a constant at inﬁnity,and that the matter functions approach a vacuum conﬁguration of the gaugeﬁeld.We here adoptA(∞)=1,(15) thusﬁxing the time coordinate,andK(∞)=±2,H(∞)=0,(16)K(∞)=0,H(∞)=±2.(17) At the origin regularity of the solutions requiresµ(0)=0,(18)and the gaugeﬁeld functions must satisfyK(0)=±2,H(0)=0,(19)K(0)=0,H(0)=±2.(20) Because of the symmetries of the SO(3)Einstein-Yang-Mills equations it is suﬃcient to study solutions with K(0)=2and H(0)=0.The other boundary conditions lead to degenerate solutions.In the following we present some numerical results for the regular solutions of the SO(3)embedding.In Table1we show the mass˜µ=µ/2of the lowest SO(3)solutions.Their ADM mass ism ADM=µ(∞)√e.(21)We observe,that the two lowest solutions,missed in the previous analysis by K¨u nzle [12],have a smaller mass than the lowest scaled SU(2)solution.To compare with the SO(3)solutions found by K¨u nzle[12]we note,that his func-tions u1and u2are related to the functions K and H as followsu1(x)=K(x)+H(x)2,(23) with the boundary conditions at the origin u1(0)=u2(0)=1,and at inﬁnity u1(∞)=±1and u2(∞)=±1.Let us adopt the classiﬁcation of the solutions with respect to their boundary con-ditions at inﬁnity,the nodes(n1,n2)of the functions(u1,u2)[12],and the total number of nodes n=n1+n2.We see in Table1,that the lowest SO(3)solution has the node structure(0,1),i.e.n=1.In contrast,the lowest scaled SU(2)solution,being the lowest SO(3)solution found by Kuenzle[12],has the node structure(1,1),i.e.n=2. Naturally,the mass of the SO(3)solution with one node only is lower than the mass of the scaled SU(2)solution with n=2.But there is a second SO(3)solution with a lower mass.This solution has the node structure(0,2),i.e.a total of two nodes like the low-est scaled SU(2)solution.Evidently,the whole class of solutions with node structure (0,n)has been missed before[12].This class contains the lowest SO(3)solution,and for a given total number of nodes,these new solutions appear to be lowest.Table1further gives the coeﬃcientsβ1andβ2for the numerical integration with the shooting method[12]u1(˜x)=1+β1˜x2+β2˜x3+...,(24)u2(˜x)=1+β1˜x2−β2˜x3+ (25)In Figs.1-4we show the lowest SO(3)solution.It is obtained independently in the limit of vanishing coupling constant on the unstable upper branch of the Einstein-Skyrme system[13].The excited solutions and further details will be given elsewhere [16].The instability of the regular solutions follows from Theorem1of Ref.[15].There the instability of the solutions of K¨u nzle[12]was demonstrated.Weﬁnd that the theorem applies also to the new class of solutions with node structure(0,n),including the lowest mass solution(whereα=1[15]as well).4Black Hole SolutionsWe now turn to the black hole solutions of the SU(3)Einstein-Yang-Mills system. Imposing again the condition of asymptoticﬂatness,the black hole solutions satisfy the same boundary conditions at inﬁnity as the regular solutions.The existence of a regular event horizon at x H requiresµ(x H)=x H4x2K K2+7H2−4 ,(27)N′H′=12+µout√e=µ(∞)√e,(29)as a function of the horizon x H.For x H→0the black hole solutions approach the regular solutions.With increasing horizon x H these black hole solutions keep their identity in terms of the boundary conditions and the node structure.Only the second solution with node structure(1,1)(#4of Table1)disappears.This solution has the same boundary conditions and node structure as the lowest scaled SU(2)solution(#3of Table1),but a slightly higher mass.It in fact merges into the scaled SU(2)solution at a horizon of x H=1.7146,leaving a unique black hole solution with node structure(1,1). (The same feature holds for the two solutions with node structure(2,2)of Ref.[12]. The solution with higher mass merges into the solution with lower mass,again a scaled SU(2)solution,at a horizon of x H=1.3745.)Note,that the order of the solutionschanges from the order of the regular solutions as the horizon increases.For instance, beyond x H=0.703solution#5has a lower mass than solution#3.In Table2we present some properties of the black hole solutions with a horizon x H=1,emerging from theﬁrstﬁve regular solutions of Table1,using again the notation of Ref.[12].The table contains the values of the functions u1and u2at the horizon,needed for a numerical shooting procedure.The families of SO(3)black hole solutions change continuously as a function of the horizon x H.As examples we show the radial functions for the lowest SO(3)black hole solutions for the horizons x H=1,2,3,4in Figs.(1)-(4).Further details of these solutions and the excited SO(3)Einstein-Yang-Mills black holes will be given elsewhere [16].The instability of the black hole solutions follows from Theorem2of Ref.[15].We ﬁnd that the theorem applies to all black hole solutions constructed(whereα=1[15] as well).5ConclusionThe SU(3)Einstein-Yang-Mills system possesses a sequence of regular spherically sym-metric solutions based on the SO(3)embedding,besides the well-studied sequence based on the SU(2)embedding[1].The SO(3)solutions can be labelled according to their node structure with two integers(n1,n2)and the total number of nodes n.The low-est solution has node structure(0,1)and n=1.Theﬁrst excited solution has node structure(0,2),while the second excited solution,the lowest scaled SU(2)solution,has node structure(1,1).The third excited solution[12]has the same node structure as the lowest scaled SU(2)solution,being only slightly higher in mass.The next solution then has n=3,again with node structure(0,n),suggesting that this class of solutions has the lowest mass for a given total number of nodes.The regular SU(2)solutions are known to be unstable[2,3],the solution with n nodes has2n unstable modes[4,5].The SO(3)solutions are unstable as well,since Theorem1of Ref.[15]applies.It is an interesting open problem to study the number of unstable modes andﬁnd a relation to the number of nodes.The lowest SU(2)solution has been interpreted in analogy to the electroweak sphaleron[6]as the top of a barrier between vacua[3].Furthermore,like the elec-troweak sphaleron[17,18],the gravitating sphaleron also possesses a fermion zero mode[19]and gives rise to level-crossing[19,20].It appears to be interesting to study fermions also in the background of the lowest SO(3)solution.Corresponding to each regular SO(3)solution there exist black hole solutions.These solutions keep their identity in terms of the node structure,for arbitrary horizon.If there are several solutions with the same structure of nodes,solutions may disappearby merging with the lowest solution of a given node structure,as is for instance the case for the lowest scaled SU(2)solution and its excitation with node structure(1,1).The SU(2)black holes are known to be unstable[10,11,5].The SO(3)black hole solutions are unstable as well,since Theorem2of Ref.[15]applies.It is an interesting open problem,especially with respect to the bifurcations,to study the number of unstable modes of the SO(3)black holes.Since the SU(3)Einstein-Yang-Mills system also contains Schwarzschild black holes, there are then many static,neutral black hole solutions(including the SU(2)black holes)for a given mass,enlarging the counterexample to the“no-hair conjecture”.But only the Schwarzschild solution is stable.The coloured black holes are all unstable.Charged SU(3)black hole solutions have been considered previously[21],and SU(2)×U(1) solutions have been constructed[21].Here a natural extension is to consider chargedSO(3)black hole solutions.AcknowledgementWe gratefully acknowledge discussions with M.Volkov.References[1]R.Bartnik,and J.McKinnon,Particlelike solutions of the Einstein-Yang-Millsequations,Phys.Rev.Lett.61(1988)141.[2]N.Straumann,and Z.H.Zhou,Instability of the Bartnik-McKinnon solutions ofthe Einstein-Yang-Mills equations,Phys.Lett.B237(1990)353.[3]D.V.Gal’tsov,and M.S.Volkov,Sphalerons in Einstein-Yang-Mills theory,Phys.Lett.B273(1991)255.[4]vrelashvili,and D.Maison,A remark on the instability of the Bartnik-McKinnon solutions,Phys.Lett.B343(1995)214.[5]M.S.Volkov,O.Brodbeck,vrelashvili,and N.Straumann,The numberof sphaleron instabilities of the Bartnik-McKinnon solitons and nonabelian blackholes,preprint ZU-TH-3-95,hep-th/9502045.[6]F.R.Klinkhamer,and N.S.Manton,A saddle-point solution in the Weinberg-Salam theory,Phys.Rev.D30(1984)2212.[7]M.S.Volkov,and D.V.Galt’sov,Black holes in Einstein-Yang-Mills theory,Sov.J.Nucl.Phys.51(1990)747.[8]on,Colored black holes,Phys.Rev.Lett.64(1990)2844.[9]H.P.K¨u nzle and A.K.M.Masoud-ul-Alam,Spherically symmetric static SU(2)Einstein-Yang-Millsﬁelds,J.Math.Phys.31(1990)928[10]N.Straumann,and Z.H.Zhou,Instability of colored black hole solutions,Phys.Lett.B243(1990)33.[11]M.S.Volkov,and D.V.Gal’tsov,Odd-parity negative modes of Einstein-Yang-Mills black holes and sphalerons,Phys.Lett.B341(1995)279.[12]H.P.K¨u nzle,Analysis of the static spherically symmetric SU(n)-Einstein-Yang-Mills equations,Comm.Math.Phys.162(1994)371.[13]B.Kleihaus,J.Kunz,and A.Sood,SU(3)Einstein-Skyrme solitons and blackholes,Utrecht preprint THU-95/6,hep-th/9503087.[14]on,and T.Chmaj,Gravitating skyrmions,Phys.Lett.B297(1992)55.[15]O.Brodbeck,and N.Straumann,Instability proof for Einstein-Yang-Mills soli-tons and black holes with arbitrary gauge groups,ZU-TH-38-94,gr-qc/9411058.[16]B.Kleihaus,J.Kunz,and A.Sood,in preparation.[17]J.Boguta,and J.Kunz,Hadroids and sphalerons,Phys.Lett.B154(1985)407.[18]J.Kunz,and Y.Brihaye,Fermions in the background of the sphaleron barrier,Phys.Lett.B304(1993)141.[19]G.W.Gibbons,and A.R.Steif,Anomalous fermion production in gravitationalcollapse,Phys.Lett.B314(1993)13.[20]M.S.Volkov,Einstein-Yang-Mills sphalerons and level crossing,Phys.Lett.B334(1994)40.[21]D.V.Gal’tsov,and M.S.Volkov,Charged non-abelian SU(3)Einstein-Yang-Millsblack holes,Phys.Lett.B274(1992)173.˜µ(∞)u1u2β1β2u1(∞)u2(∞)123scaled SU(2)4K¨u nzle56K¨u nzleTable1:Properties of the lowest regular SO(3)solutions are given in the notation of Ref.[12].The ADM mass is obtained from the second column via Eq.(21)with µ(∞)=2˜µ(∞),the third and forth column give the number of nodes of the functions u1and u2deﬁned in Eqs.(22)-(23),theﬁfth and sixth column provide the expansion coeﬃcients of these functions as deﬁned in Eqs.(24)-(25),and the seventh and eighth column give the values of the functions u1and u2at inﬁnity.˜µ(∞)u1u2u1(˜x H)u2(˜x H)u1(∞)u2(∞)123scaled SU(2)45Table2:Properties of the lowest SO(3)black hole solutions are given for the horizon x H=1=2˜x H following the classiﬁcation of Table1.The ADM mass is obtained from the second column via Eq.(21)withµ(∞)=2˜µ(∞),the third and forth column give the number of nodes of the functions u1and u2deﬁned in Eqs.(22)-(23),theﬁfth and sixth column provide the value of these functions at the horizon,and the seventh and eighth column give the values of the functions u1and u2at inﬁnity.Figure1:The function K(x)is shown for the regular solution and for the black hole solutions with horizons x H=1,2,3and4as a function of x.Figure2:The function H(x)is shown for the regular solution and for the black hole solutions with horizons x H=1,2,3and4as a function of x.Figure3:The functionµ(x)is shown for the regular solution and for the black hole solutions with horizons x H=1,2,3and4as a function of x.Figure4:The function A(x)is shown for the regular solution and for the black hole solutions with horizons x H=1,2,3and4as a function of x.Figure5:The mass fraction outside the horizon,µout,of the SO(3)black hole solutions of Table2is shown as a function of the horizon x H.When the mass fraction within the horizon,x H/2,is added,the ADM mass is obtained.。