We found some general exact static multimonopole solutions that satisfy the first order Bogomol'nyi equations and possess infinite energy. These multimonopole solutions can be categorized into two classes, namely the A2 and B2 solutions

Khai, Ming Wong

Teh, Rosy

English

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EXACT MULTIMONOPOLESOLUTIONS OF THEYANG-MILLS-HIGGS THEORY*Rosy Tentand Khai-Ming Wong+School of Physics, Universiti Sains Malaysia11800 USM Penang, MalaysiaJune 2003Abs1;ractWe found some general exact static multimonopole solutions that satisfythe first order Bogomol'nyi equations and possess infinite energy. Thesemultimonopole solutions can be categorized into two classes, namely the A2and B2 solutions. The A2 solution is a multimonopole solution with all themagnetic charges superimposed at the origin. The B2 solution correspondsto a configuration of even number of isolated I-monopoles located on a circle,symmetrically about the z-axis.1 INTRODUCTIONThe SU(2) Yang-Mills-Higgs theory, with the Higgs field in the adjoint represen-tation, can possess both the magnetic monopole and multimonopole solutions.The 't Hooft-Polyakov magnetic monopole was discovered [2] in the mid-seventies.Solutions of a unit magnetic charge are spherically symmetric [2, 3]. However,multimonopole solutions possess at most axial symmetry [4]. In the limit of van-ishing Higgs potential, monopole and multimonopole solutions had been shownto exist. Solutions that satisfy the Bogomol'nyi condition or the Bogomol'nyi-Prasad-Sommerfield (BPS) limit have minimal energies.In this paper, we used the extended and generalized ansatz of re£.[l]. Wework on the SU(2) Yang-Mills-Higgs model with a vanishing Higgs potential. Thescalar Higgs field in our work is then taken to have no mass or self-interaction.*Contributed paper presented at the "Persidangan Fizik Kebangsaan", (PERFIK 03), August.15-17,2003, Shahzan Inn, Bukit Fraser, Pahang.te-mail: rosyteh@usm.mytSpeaker; e-mail: khaimingl@hotmail.com1"UNTUK KEGUNAAN JAWATANKUASA PENYELIDlKAN UNIVERSITITANDATANGAN PENGERUSIJAWATANKUASA PENYELIDlKANPUSAT PENGAJIAN SAINS FIZIK5,g .284-6/We found that the SU(2) Yang-Mills-Higgs theory do possess some exact staticmultimonopole solutions. They satisfy the first order Bogomol'nyi equations andpossess infinite energies at the origin. These multimonopole solutions are catego-rized into two classes, namely the A2 and B2 solutions. The two classes possessvery different characteristic which we shall discuss in later section. The A2 solu-tion represents configuration with all the magnetic charges superimposed at theorigin. There are no zeros of Higgs field at finite r. The B2 solution correspondsto even number of equally spaced 1-monopoles located on a circle in the equatorialplane.The SU(2) Yang-Mills-Higgs model consist of the Yang-Mills vector fields A~and the Higgs scalar field <pa in 3+1 dimensions. The index a is the SU (2) internalspace index. For a given a, <pa is a scalar whereas A~ is a vector under the Lorentztransformation. The SU(2) Yang-Mills-Higgs Lagrangian in 3+1 dimensions isL = _!Fa FaJ.tv + !DJ.t<PaD . (<pawa _ f-L2)2 (1)4J.tv 2 J.ta 4 >.'where f-L is the mass of the Higgs field, and (3 is the strength of the Higgs potential.The vacuum expectation value of the Higgs field is then f-L/"';>'" The Lagrangianfrom Eq.(l) is invariant under the set of independent SU(2) gauge transformationsat each space-time point.The covariant derivative of the Higgs field iswhere A~ is the gauge potential and the gauge field strength tensor isF:v = 0J.tA~ - ovA~ + EabcAtA~.(2)(3)(6)The gauge field coupling constant 9 is set to one here. The metric used is 9J.tv =(- + ++). The SU(2) group indices a, b, c run from 1 to 3 whereas the spatialindices f-L, 1/, a run from 0 to 3 in Minkowski space.The equations of motion obtained from Eq.(l) areDJ.t Fa = oJ.tFa + EabcAbJ.t FC = Eabc<pbD <pc (4)J.tV J.tV J.tV v ,DJ.t DJ.twa = O. (5)We will examine only the static solutions with Ag = O. The conserved energy ofthe static system which is obtained from the Lagrangian isE = Jd3x (!B?-B?- +!D'a D·a + !>.(<paa _ f-L2 )2) .2~t 2 t t 4 >.The indices i, j and k are purely spatial indices where i, j, k run from 1 to 3. Thisenergy vanishes when the gauge potential, Af is zero or a pure gauge, waa = f.-L2 / >.and Dia = O. The tensor -(7)2introduced by 't Hooft [2] can be identified with the electromagnetic field tensor.This tensor can also be written in a more transparent form [4](8)where A~ = <j>aA~, the unit vector <j>a = (l)a /!(l)1 and the Higgs field magnitude1(l)1= V(l)a(l)a. The Abelian electric field is Ei = FOi and the Abelian magneticfield is Bi = - ~CijkFjk' The topological magnetic current k~ [4] is defined to bek 1 ov,T..a Op,T..b OU,T..C~ = Srr c/-LVPU Cabc ':l!'. ':l!' ':l!' , (9)(13)and the corresponding conserved topological magnetic charge isM - Jd3xko = 8~JCijkCabcOi (<j>aOj<j>bOk<j>c) d3x- 8~fd2(Ji (CijkCabc<j>aOj <j>bOk <j>c)- 4~ f d2(Ji Bi. (10)2 THE MULTIMONOPOLE SOLUTIONSIn this paper, the Higgs field is taken to have no mass and hence the self-interactionvanishes. The magnitude of the Higgs field vanishes as l/r at infinity. It is inthis limit that we are able to obtain explicit and exact multimonopole solutions.These solutions are solved from the Yang-Mills-Higgs equations by using both thesecond order Euler-Lagrange equations and the first order Bogomol'nyi equationsBf ± D/pa = 0 with the positive sign. There is no solution to the Bogomol'nyiequations with the negative sign. A multimonopoleof magnetic charge M withall its magnetic charges superimposed at one point in space is denoted by a M-monopole.We used the ansatzof Ref.[l] with the gauge fields and the Higgs field givenbyA~ - -~VJ(r) (oaJy~ + JyaOJL ) + ~R(e) (JyafJL + faJy/-L)+ ;G(e, ep) (fae~ - eafJL ) , (11)(l)a = (l)I fa + (l)2 ea + (l)3 Jya. (12)with(l)I = ~VJ(r), (l)2 = ~R(e), (l)3 = ~G(e, ep). The spherical coordinate orthonor-mal unit vectors are defined by .fa _ sine cos¢ 6f + sin e sin ¢ 6~ + cos e6~,ea _ cos e cos ¢ 6f + cos e sin ¢ 6~ - sin e 6~,epa _ _ sin¢ 61+ cos¢ 6~,3With the ansatz, Eqs.(ll) and (12), the equations of motion (4) and (5) canbe simplified and reduced to four ordinary differential equations of first order:G+ Gcot B= 0, Gr/J csc(J +G2 = _b2 csc2 B, (14)nl/ + 7jJ- 7jJ2 = -p, (15)R+RcotB - R2=p - b2csC2B, (16)where p and b2 are arbitrary constants. Prime means the partial derivative %r' dotmeans the partial derivative go ' superscript ¢ means the partial derivative ttl>.The two classes of solutions obtained from Eq.(13) to Eq.(15) are the A2 andB2 solutions. The A2 solution isR = tan B+(m + 2) cos B, G = (m +2) csc e tan(m + 2)B,(m + 1) - mr2m+1'Ij; = 1 +r2m+1 . ' p = m(m + 1), b = -,-{m + 2). (17)The B2 solution isR = (m + 1) cot B, G = (m +1) csc Btan(m + l)B,(m + 1) - mr2m+1'Ij;= 1+r2m+1 .. , p=m(m+l),b=m+l. (18)The parameter m in Eq.(17) and (18) is a positive integer.The ansatz, Eqs.(ll) and (12), has vanishing gauge potential, AIL - <t>aA~.Hence the Abelian electric field is zero and the Abelian magnetic field is indepen-dent of the gauge field A~. To calculate the Abelian magnetic field B i , we rewritethe Higgs field of Eq.(12) from the spherical coordinate system to the Cartesiancoordinate system,where<1> a _ <1>1 fa + <1>2 ea + <1>3 J;a_ <l?1 6a1 + <1>2 6a2 + <1>3 6a3 (19)<1>1 - sin Bcos ¢ <1>1 + cos Bcos ¢ <1>2 - sin ¢ <1>3 = Iwl cos a sin {32 - sin Bsin ¢ <1>1 + cos Bsin ¢ <1>2 + cos ¢ <1>3 = I<1>Icos a cos {3<1>3 - cosB <1>1 - sine <1>2 = 1<1>1 sina. (20)The Higgs field unit vector is then simplified to4a = cos a sin{3 6a1 + cos a cos {3 6a2 + sin a c5a3 (21)The Abelian magnetic field is found to beB- _ 1 {osinaoj3_ OSinaoj3}~.t - r2 sin e oe o¢ o¢ oB rt+_1_ {a sin a a{3 _ asin a O(3} e.. Tsin e o¢ or or o¢ t+~ {a sin a oj3 _ asin a Oj3} J...r or oB oB or 'f't (22)4where. _ 'I/J cos e - R sin e (3 _ _ '" _ -1 .( 'I/J sin e +R cos e)sma- -J'l/J2+R2+G2' -i 'fI,i- tan G .The Abelian field magnetic flux isn 471"M =f d20"iBi- JBi (r2 sin e)ri de d¢,The magnetic charge M is given byM = ~ {27r r (8 sin a 8(3 _ 8 sin a 8(3) ded¢471" Jo Jo 8e 8¢ 8¢ 8e(23)(24)(25)The magnetic charges of the A2 solution at the origin can be exactly integratedto· be m + 3. The net magnetic charges of the configuration when r tends to infinitycan be obtained by using approximate integration and it is found also to be (m+3).Hence the A2 solution consists of a (m+3)-monopole sitting at the origin of thecoordinate axes. Therefore, when m = 0, we have a3-monopole at the origin andwhen m = 1, we obtain a configuration of 4-monopole at the origin.The B2 solution is different from the A2 solution in that the positive mag-netic charges are not superimposed at one point in space. The B2 solution hasisolated I-monopoles located on a circle in the equatorial plane with rotationalsymmetry about the z-axis. The I-monopoles here are finitely separated and theseI-monopoles are located at the zeros of the Higgs field. There is no multimonopoleat the origin.The case when m = 0 is a special case of the B2 solution, the zeros of Higgsfield are located at infinity and there are no zeros of Higgs field at finite r. Theconfiguration has no monopole at finite r. For m = 1, there are 2(m + 1) equallyseparated I-monopoles. Hence when m = 1, we have a configuration offourisolated I-monopoles, each being located at the four zeros of the Higgs field. Whenm = 2, we have six isolated I-monopoles at the six zeros of the Higgs field, and soon.3 COMMENTS AND OUTLOOKSThe B2 solution with m = 0 does not have any zero of Higgs field at finite r.Hence instead of the two I-monopole solution, we only obtain a solution that hasno monopoles at finite r. The B2 solution contains only even number of finitelyseparated I-monopoles. Hence it will be interesting if one can find solutions withodd number of isolated I-monopoles to complement the B2 solution.The two classes of solutions in this paper possess singularity at the origin.This singularity either corresponds to multimonopole as in the A2 solution or no5monopole at all as in the B2 solution. The zeros of the Higgs field correspond toa I-monopole in the B2 solution. In the A2 solution, there are no zeros of Higgsfield at finite r.A third multimonopole solution is the C solution which will be reported sep-arately. Work can still be done to find different solutions to the Riccati equationof Eq.(14).4 ACKNOWLEDGEMENTThe author would like to thank Universiti Sains Malaysia for the short term grant(Account No: 304/PFIZIK/634039).References[1] Rosy Teh, 2002, J. Fiz. Mal. 23: 196; 2001, Int. J. Mod. Phys. A 16: 3479.[2] G.'t Hooft, 1974, Nucl. Phy. B 79: 276; A.M. Polyakov, 1975, Sov. Phys. -JETP 41: 988; 1975, Phys. Lett. B 59: 82; 1974, JETP Lett. 20: 194.[3] M.K. Prasad and C.M. Sommerfield, 1975, Phys. Rev. Lett. 35: 760; E.B.Bogomol'nyi and M.S. Marinov , 1976, Sov. J. Nucl. Phys. 23: 357.[4] C. Rebbi and P. Rossi, 1980, Phys. Rev. D 22: 2010; R.B. Ward, 1981,Commun. Math. Phys. 79: 317; P. Forgacs, Z. Horvarth and L. Palla, 1981,Phys. Lett. B 99: 232; 1981, Nucl. Phys. B 192: 141; M.K. Prasad, 1981,Math. Phys. 80: 137; M.K. Prasad and P. Rossi, 1981, Phys Rev. D 24: 2182.6

Exact Multimonopole Solutions Of The Yang-Mills-Higgs Theory.

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Exact Multimonopole Solutions Of The Yang-Mills-Higgs Theory.

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