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HEP-TH-9408163 Address after Sept. 1, 1994: e-mail address: piljin@theory 2 1 ein- T-68-1944 SNUTP-94-73 CAL provided byCERNDocumentServer a y Lee and W e brought toyouby al Physics b erg monop oles more etic y structure. ein or GNETIC MONOPOLE tly found b CORE etters B. oul 151-742, Kor CT 12 gy, Pasadena, CA 91125, U.S.A. Piljin Yi ABSTRA chnolo d to Physics L Cho onkyu Lee e es the nature of the Lee-W Submitte oul National University, Se Se terpreted as top ological solitons of a non-Ab elian gauged Higgs artment of Physics and Center for The top ological magnetic monop oles, recen t, esp ecially with regard to their singularit Dep Our study mak California Institute of T transparen b erg, are rein mo del. Certain non TOPOLOGY OF A NONTOPOLOGICAL MA

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

L = F F jD W D W j + H F H H

4 2 4 4

2 2

@ @ m ()jW j V (); (1)

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

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)

= (r ); A = (cos 1) d'; W = [ exp i' ][ d + i sin d']: (2)

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

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

nonrenormalizable SO(3) gauge theory with the gauge connection 1-form B =(B dx ) T

and a triplet Higgs = T .

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 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

tensor F and H as follows,

a a a b c

^ ^ ^ ^

F H G ; H (D ) (D ) : (4)

abc

a 3 a 3

Now in the unitary gauge = ,wemay identify with jj, A with B , and W

1 2

with (B + iB )= 2. This results in the following reduction formulae:

F ) F ;

H ) H ;

G ) F H ;

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 a b c

^ ^ ^

n = dS @ @ ; (5)

ij k abc j k

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 =

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

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

maintaining n =1,wemay mo dify the hedgehog con guration (6) to have nontrivial B .

b c

x dx

abc

a a a a

[1 u(r)] x^ v(r)dt: (8) =^x (r); B =

On such con gurations, the total energy is given by the following functional of u, , and

v .

2 2 2

1 1 1 du dv d

+ + E = dx

2 2

e r dr 2 dr 2 dr

1 u

4 2 2 2 2

+ [ u gu +1]+ [m +e v ]+V() : (9)

2 4 2 2

2e r e r

For the integrand, weevaluated T , the energy-density asso ciated with L , on the con-

a 3 2

guration (8) using such radial gauge results as F = x =er and H = u(r ) F .

ij ij a ij ij

Clearly, for the con guration (8) to b e regular at origin, we need to have u(0) = 1,

v (0) = 0, and (0) = 0, while, for a nite total energy,wemust also require

4 2

u (0) gu (0)+1=0: (10)

Only if this last condition is met, the integrand of the energy functional E do es not have

any r singularity near origin,

What ab out half-integer-charged monop oles of Lee and Weinb erg [3]? The top ological quantization

(5) clearly dictates that it is not p ossible to remove the Dirac string of such U (1) monop oles by lifting the

con guration to the underlying SO(3) and by p erforming a non-Ab elian gauge transformation. Instead,

the lifted Dirac string will b e carrying a Z ux that is again undetectable. Here, Z corresp onds to the

2 2

fundamental group of SO(3). 4

Do es there exist an everywhere regular nite energy solution of the form (8)? The

answer is no in general, such a solution b eing p ossible only when g = 2 and = 1. This

can b e easily seen as follows. To b e a static solution to the eld equations, the con gu-

ration must extremize the energy-functional. Supp oseu (r )issuch a regular nite energy

solution, and consider a small variation u(r) around it such that u(0) = 0. Varying E ,

wenow observe that the only contributions that may lead to a small-r singularityof r

are given by the following variation,

Z Z

1 1

3 4 2 3 3

A dx [ u gu +1] ) A = dx [4u 2gu ] u + : (11)

2 4 2 4

2e r 2e r

2 3

Cho osing u(r)=r+O(r ), we nd 4 u (0) 2g u (0)=42g = 0 as a necessary

condition foru (r ) to satisfy the equation of motion. But the nite energy condition (10)

requires g = h + 1 for such a regular solution, which is not compatible with 4 2g =0

unless g = 2 and =1.Thus the existence of a regular nite-energy monop ole solution

demands the sp ecial values of vector couplings, g = 2 and = 1, appropriate for the

renormalizable eld theory.

However, we do obtain a nite-energy solution if wecho ose not to insist on the every-

where regularity of the con guration (8). The nite-energy monop ole of Lee and Weinb erg

b elongs to this class. Instead of demanding u(0) = 1, let u(r ) approach a nite constant

C as r ! 0 but still let v (0) = (0) = 0. With such a singular b oundary condition, the

nite-energy requirement and the stationary argumentabove translate into the following

two conditions:

4 2 3

C gC +1 = 0; 4C 2gC =0: (12)

This in turn leads to the Lee-Weinb erg condition 4 = g with g>0aswell as to the

b oundary condition u (0) = g=2 =2=g [3]. More careful study of such a singular nite-

energy monop ole may b e carried out with the help of the following equations of motion

for these radial functions,

g u d u

2 2 2 2

+( =(m +e v )u u )

2 2

dr 2 r

2 2 2

d 2 d u dm dV

+ =

2 2 2

dr r dr e r d d

2 2

d v 2 dv 2u

+ v =0

2 2

dr r dr r 5

For instance, a linearization of this nonlinear system near origin predicts that the solutions

are in general non-analytic at r =0. The same technique repro duces the well-known

analytic b ehaviour, u(r )=1+O(r ) and =1+O(r), of the 't Ho oft-Polyakov monop oles

when the couplings are those of the renormalizable theory.

The condition 4 = g that is necessary for a nite-energy solution to exist can b e

also recognized on the basis of the energy-functional E itself. The only part of E that

is not necessarily p ositive de nite is the quantity A of (11). But the integrand of A is

quadratic in u (r ) with co ecients and g , and subsequently the energy functional E is

2 2

b ounded from b elow if and only if 4 g . In particular when 4 = g , it is easy to see

that there exists a nite energy con guration of typ e (8) (but not necessarily a solution

yet). In that case, wemay secure a lo cal minimum of E ,thus a nite-energy solution,by

2 2

suitably adjusting u(r ), (r ) and v (r ). If 4>g (4

are of in nite p ositive (negative) energy, but nevertheless physically signi cant if coupled

to gravity [3].

In this letter, we presented a new interpretation of the integer-charged Lee-Weinb erg

monop oles in terms of a top ological structure arising in a nonrenormalizable extension

of the 't Ho oft-Polyakov monop ole mo del. In the unitary gauge where the top ological

nature is obscured, the monop oles are riddled with Dirac strings, alb eit harmless, and

the singularity structures of the underlying non-Ab elian elds are less transparent. We

reexamined the core of the unit-charged monop ole from the radial gauge viewp oint and

repro duced some of the results in Ref.[3].

An obvious follow-up question to ask is: would it always b e p ossible to provide

a non-Ab elian gauge theoretic interpretation for Lee-Weinb erg-typ e monop oles, when

the theory involves several charged vector elds? We nd this unlikely. For example,

consider the following generalization of Lee-Weinb erg mo del with the N charged vector

(n) (n)

elds W ;n=1;:::;N and the corresp onding magnetic moment tensors H :

1 1

2 (n) (n)

L = j D W F F jD W

4 2

X X

1 1

(n) (n) (m)

+ g H F H H + : (13)

n nm

4 4

n n;m

We are grateful to E. Weinb erg for suggesting this. 6

As long as the relationship 4 = g g holds, an ansatz, entirely analogous to (2) of the

nm n m

N = 1 case, leads to a spherically symmetric nite-energy monop ole with the b oundary

(n) 2

condition satisfying g [u (0)] = 2. But it is rather dicult to imagine why there

n=1

should exist non-Ab elian interpretations for all such generalized theories, sp eci cally for

all N and for all values of the vector couplings. On the other hand, what one may exp ect

to b e true is that the resulting nontop olgical monop oles are generically singular at origin

despite the nite total energy and that, just as wehave observed in N = 1 case ab ove,

the complete regularity is recovered only when the relevantvector coupling structure

is renormalizable, p ossibly corresp onding to a sp ontaneously broken non-Ab elian gauge

theory.

Acknowledgment

We are very grateful to K. Lee and E. Weinb erg for sharing their insight and also for

critical comments on this work. C.L. enjoyed useful conversations with V. P. Nair, and

P.Y. would like to thank H-K Lo for an illuminating discussion on the top ological charge

and the Dirac quantization. The work of C.L. is supp orted in part by the Korean Science

and Engineering Foundation (through the Center for Theoretical Physics at SNU) and

the Ministry of Education, Republic of Korea. The work of P.Y. is supp orted in part by

D.O.E. Grant No. DE-FG03-92-ER4070 1. 7

References

[1] P. A. M. Dirac, Pro c. Roy. So c. (London) Ser. A133 (1931) 60.

[2] G. 't Ho oft, Nuc. Phys. B79 (1974) 276; A. M. Polyakov, JETP Lett. 20 (1974) 194.

[3] K. Lee and E. Weinb erg, Nontopological magnetic monopoles and new magnetical ly

charged black holes, CU-TP-634, 1994.

[4] See, for example, S. Coleman, in New phenomena in subnuclear physics, Pro c. 1975

International Scho ol of Subnuclear Physics, ed. A Zichichi, Plenum, New York.

[5] J. Arafune, P.Freund and C. Go eb el, J. Math. Phys. 16 (1975) 433.

[6] B. Julia and A. Zee, Phys. Rev. D11 (1975) 2227. 8