View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server SNUTP-94-73 CALT-68-1944 TOPOLOGY OF A NONTOPOLOGICAL MAGNETIC MONOPOLE Cho onkyu Lee Department of Physics and Center for Theoretical Physics Seoul National University, Seoul 151-742, Korea 12 Piljin Yi California Institute of Technology, Pasadena, CA 91125, U.S.A. ABSTRACT Certain nontop ological magnetic monop oles, recently found by Lee and Wein- b erg, are reinterpreted as top ological solitons of a non-Ab elian gauged Higgs mo del. Our study makes the nature of the Lee-Weinb erg monop oles more transparent, esp ecially with regard to their singularity structure. Submitted to Physics Letters B. HEP-TH-9408163 1 e-mail address: [email protected] 2 Address after Sept. 1, 1994: Physics Department, Columbia University, New York, NY 10027, U.S.A. When Dirac [1] founded the theory of magnetic monop oles in 1931, the monop ole was not something that p eople could not live without. Things changed a great deal in the seventies when 't Ho oft and Polyakov [2] showed that magnetic monop oles inevitably o ccur as solitons of sp ontaneously broken non-Ab elian gauge theories; such as all grand uni ed theories where an internal semi-simple gauge symmetry is sp ontaneously broken to U (1). Their existence is understo o d in terms of the nontrivial top ology of the vacuum manifold, and as such the non-Ab elian nature of the original gauge group plays a crucial role. In particular, the Dirac quantization rule is naturally enforced by the underlying non-Ab elian structure. Recently,however, Lee and Weinb erg [3] constructed a new class of nite-energy mag- netic monop oles in the context of a purely Ab elian gauge theory. Amazingly enough, the corresp onding U (1) p otential is simply that of a p oint Dirac monop ole (with the monop ole strength satisfying the Dirac quantization rule), yet the total energy is rendered nite by intro ducing a charged vector eld of p ositive gyromagnetic ratio and by ne-tuning a quartic self-interaction thereof. For more general values of the couplings, this theory to- gether with Einstein gravitywas found to pro duce new magnetically charged black hole solutions with hair [3]. One might b e tempted to conclude from this that the existence of the magnetic monop oles do es not require an underlying non-Ab elian structure, let alone a nontrivial top ology of the vacuum manifold. We b elieve this is a bit premature, and is misleading as far as this particular mo del is concerned. As p ointed out by Lee and Weinb erg [3], their U (1) theory may b e regarded as a gauge- xed version of the usual SO(3) Higgs mo del at some sp ecial values of the couplings. An imp ortant question to ask here is whether there exists such a hidden structure at other values of the couplings as well. In this letter, we will show that there is indeed a hidden non-Ab elian gauge symmetry for general values of the couplings, and that subsequently the integer-charged monop oles of Lee and Weinb erg may b e regarded as top ological solitons asso ciated with certain nonrenormalizable defor- mations of the SO(3) Higgs mo del. Adopting the radial gauge rather than the unitary gauge, we found that the apparent Dirac string disapp ears as usual, while the singularity at the origin still needs to b e examined. An imp ortantbypro duct of our study is a top o- logical understanding of the Dirac quantization rule for the integer-charged Lee-Weinb erg monop oles. 1 The Ab elian mo del of Ref.[3] consists of a U (1) electromagnetic p otential A ,acharged vector eld W , and a real scalar . The Lagrange densitywas chosen to have the form 1 1 g 2 L = F F jD W D W j + H F H H 4 2 4 4 1 2 2 @ @ m ()jW j V (); (1) 2 where F = @ A @ A and D W = @ W + ie A W . The magnetic moment H is given by the antisymmetric pro duct ie (W W W W ) so that g is the gyromagnetic ratio of the charged vector. Here, g and are p ositive constants, while m()vanishes at = 0 but is equal to m 6= 0 when is at its (nontrivial) vacuum value. As mentioned W ab ove, when g =2, = 1 and m()= e, this is nothing but the unitary gauge version of the sp ontaneously broken su(2) gauge theory of Ref.[2], and thus renormalizable. But for generic values of g and , the theory is nonrenormalizable. The ansatz for the unit-charged, spherically symmetric, Lee-Weinb erg monop ole can b e written most succinctly in term of the di erential forms A A dx and W W dx : 1 iu(r) p = (r ); A = (cos 1) d'; W = [ exp i' ][ d + i sin d']: (2) e e 2 u and are radial fuctions to b e determined by the eld equations and appropriate b oundary conditions. Note that the electromagnetic p otential is simply that of a p oint- like Dirac monop ole and the eld con guration app ears singular at origin, in addition to the presumably harmless Dirac string along the negative z -axis. Akey observation in 2 Ref.[3] is that, if the relationship 4 = g holds, the total energy of the resulting solution can b e actually nite despite this singular b ehaviour at origin. In addition to such a unit- charged solution, the Dirac quantization rule allows all integral and half-integral magnetic charges. Let us turn to the question of the hidden non-Ab elian structure. Consider the following a nonrenormalizable SO(3) gauge theory with the gauge connection 1-form B =(B dx ) T a a and a triplet Higgs = T . a g 2 1 1 a a 0 G G + F H H H L = 4 4 4 1 1 a a 2 2 a a a a ^ ^ (D ) (D ) (m (jj) e )(D ) ) (D V (jj) (3) 2 2 e a a @ + G is the non-Ab elian gauge eld strength asso ciated with B , while D abc b c a a ^ B and =jj. In addition, we de ned two gauge-invariantantisymmetric 2 3 tensor F and H as follows, 1 a a a b c ^ ^ ^ ^ F H G ; H (D ) (D ) : (4) abc e a 3 a 3 ^ Now in the unitary gauge = ,wemay identify with jj, A with B , and W p 1 2 with (B + iB )= 2. This results in the following reduction formulae: F ) F ; H ) H ; 3 G ) F H ; p 1 2 G + iG ) 2(D W D W ); a a 2 ^ ^ (D ) (D ) ) 2e W W ; a a 2 2 (D ) (D ) ) @ @ +2e W W : a a 0 Expanding the G G term, one easily notices that the SO(3) invariant Lagrangian L reduces to L of the apparently Ab elian theory of Lee and Weinb erg. According to the standard argument [4], all con gurations with a regular and nite asymptotic b ehaviour may b e classi ed in terms of a top ological winding numb er: I 1 i a b c ^ ^ ^ n = dS @ @ ; (5) ij k abc j k 8 where the integral is over the asymptotic two-sphere. Note that n is an integer in general, and the magnetic monop ole strength is related to this number by g = 4 n=e [5]. mag netic The unit-charged case (n = 1) is of sp ecial interest, b ecause the corresp onding solution has the spherical symmetry.For example, if g =2, = 1 and m = e, the n = 1 solution is given by the usual 't Ho oft-Polyakov monop ole. Actually, the ansatz (2) for the sp erically symmetric Lee-Weinb erg monop ole is exactly the same as that of the 't Ho oft-Polyakov monop ole but written in the unitary gauge. Written in the so-called radial gauge, the ansatz for the 't Ho oft-Polyakov monop ole has the following well-known hedgehog con guration: b c x dx abc a a a [1 u(r)]: (6) =^x (r); B = 2 er 3 F was previously used by `t Ho oft [2] to representphysical electromagnetic elds. 3 Gauge transforming to the unitary gauge, 1 a i'T i T i'T 3 2 3 x^ T =T with e e e ; (7) a 3 a lengthy but straightforward calculation shows that the hedgehog con guration (6) is indeed gauge-equivalent to the ansatz (2). This clearly demonstrates that the spheri- cally symmetric unit-charged Lee-Weinb erg monop ole has the top ological winding num- ber n = 1, and that the Dirac string apparent in (2) is sup er uous. Similar pro cedures for n>1 non-Ab elian con gurations should also lead to all integer-charged Lee-Weinb erg 4 monop oles automatically, although, due to the lack of the spherical symmetry, the ex- plicit forms of the necessary gauge transformations are more dicult to nd. Now that the Dirac string disapp eared by virtue of the radial gauge, let us concentrate on the p ossible singular structure at origin r = 0. Generalizing to include dyons [6] while a maintaining n =1,wemay mo dify the hedgehog con guration (6) to have nontrivial B . t b c x dx abc a a a a [1 u(r)] x^ v(r)dt: (8) =^x (r); B = 2 er On such con gurations, the total energy is given by the following functional of u, , and v .