P H Y S I C A L R E V I E W D V O L U M E 1 7 , N U M B E R 1 2 1 5 J U N E 1 9 7 8

Classical interactions of 't Hooft monopoles

Steven F. Magruder Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801

(Received 9 February 1978)

We find upper bounds on the interaction energies between two 't Hooft monopoles and between a monopole and an antimonopole. Our results may alter the confinement picture in the Georgi-Glashow model for certain small values of the Higgs mass parameter.

I . INTRODUCTION

In a famous paper,' Polyakov proposed that questions about confinement can be answered by isolating the relevant tunneling degrees of f ree- dom' in the path integral. For the (2 + 1) -dimen- sional Georgi-Glashow theory these degrees of freedom a r e the positions and charges of 't Hooft monopole^.^ By ignoring other degrees of f ree- dom, Polyakov transforms the path integral into the partition function for a gas of monopoles. In order to calculate this partition function we need to know the classical interactions between 't Hooft monopoles.

Several authors have addressed themselves to this q ~ e s t i o n , l * ~ * ~ but to our knowledge no rigorous results have been obtained. The most popular a$- proximation technique i s to use a singular gauge transformation to make the problem approximately linear in nature.''' This method leads to the p re - diction that 't Hooft monopoles interact by a Cou- lomb law like Abelian monopoles. ant on^ has ob- tained quite different results for the special case known a s the Prasad-Sommerfield limit.6 By a s - suming the asymptotic form for the two-monopole solution, Manton concludes that monopole pa i rs do not interact with a long range interaction, while monopole-antimonopole pa i rs interact with twice the usual Coulomb force.

In t h ~ s paper, we explicitly construct t r ia l con- figurations which can be used to find rigorous upper bounds on the interaction energies between two 't Hooft monopoles and between a monopole and a n antimonopole. Our results a r e valid for a rb i t ra ry separations of the monopoles.

11. THE MODEL

We a r e considering the Georgi-Glashow theory in two space and one time dimensions. The Eucli- dean Lagrangian for this theory i s given by

Here C j is the field strength tensor for the SU(2) ' gauge theory. The symbol Di denotes a covariant derivative. The rea l sca lar field 6 is a vector in the gauge grou2 space. The gauge potentials will be denoted by A ; . T o r ihe remaining part of the paper, we rescale A i , $J , L, and X to se t 17 and the unit of charge e both equal to 1.

We will sometimes be interested In the limit of this theory given by X-0. When we consider this limlt, we will keep the boundary condition

This defines the Prasad-Sommerfield limit. This i s the limit in which the Higgs mass vanishes, while the classical mass of the vector boson remains nonzero. '

In order to study the interactions of monopoles, we need a local definition of the monopole position and charge which i s consistent with the required asymptotic properties of magnetic flux and non- trivial homotopy class. Such a definition has been provided by Arafune, Freund, and G ~ e b e l . ~ Their definition i s given by the formula

where q , and X, a r e the charge and position of monopole a, and 6 is defined by

When the Higgs field satisfies (3 ) , we will say that there a r e monopoles with charge q, a t the positions X,. Note, however, that although this condition fixes the total magnetic f l u a t infinity, it does not determine the local distribution of mag- netic charge.

The "energy" of an assembly of monopoles i s defined a s the minimum of L subject to the con- straint (3). We call this quantity energy instead of action because we a r e thinking in t e rms of Poly- akov's classical monopole gas instead of the orig- inal quantum field theory.

3258 S T E V E N F . M A G R U D E R

111. MONOPOLE-ANTIMONOPOLE INTERACTION (5). F o r a finite L , we must have

To find the interaction energy between a non no pole -+ l im r 3 / 2 ~ i $(Y) = 0 .

and a n antimonopole, we mus t minimize L sub- ? - - (8)

ject to the constraint This gives us the asymptotic condition

(5) where A is a rb i t ra ry . In o r d e r to ensure that (9)

The interaction energy is then given by i s satisfied, and to keep our t r i a l configurations s imple, we choose configurations which satisfy

El = min(L) - 2E,, (6) -. A, = A ~ , $ x $ + A , $ , (10)

where E, i s the one-monopole self-energy. Note that the positivity of L gives us the lower bound where A is to be determined. This ansatz contains

the exact one-monopole solution as a special case.

E, a -2E,. (7) We define the variables a , p, and M by

We will find upper bounds for El by calculating L F=ll/l(cosa sin@, sinci sinp, cos@) . (11)

f o r t r i a l configurations which obey the constraint Using (10) and (ll), we can express L a s

This expression i s s t i l l too complicated for exact minimization. Our s t rategy f o r choosing a t r i a l configuration is to choose one which reduces to the known one-monopole solution3 near each mono- pole, and which is otherwise a s s imple a s pos- sible.

In the one-monopole solution, sat isf ies * a x ( Q x h g ) = ; j x ( ~ l n p ; a xB@). (13)

F o r our t r i a l configuration, we choose the s i m - plest solution of (131, namely

* A = O , (14)

s i$ ~ C Y x = Q( ~ / r , - l / r , ) . (15)

A simultaneous solution of (15) and (51, with the monopoles located on the z ax is i s given by

(a) a = @ ,

(bj COSP= case, - C O S ~ , + 1

In these equations, Y,, 8, and Y,, 8, a r e the radi i and polar angles measured f r o m the positions of the antimonopole a t z = 5 R and the nlonopole a t z = -+ R , respectively. The angle i s the azimu- thal angle. Since (16) resembles the one-monopole solution near each monopole, we choose a t r i a l configuration which sat isf ies (16).

The var iab les A and M a r e required by the finite- ness of L and the boundary condition (2) to satisfy

l i m A = l im hI= 1. +-+m

(17)

This asymptotic conditions i s , of course , sat isf ied

by the exact one-monopole solution. The approach to 1 i s exponential unless h vanishes. In this special c a s e M has the asymptotic f o r m 1 - l/r. In addition, fo r the finiteness of L , the quantities A and M must vanish a t each monopole. There- fore, the mos t natural f o r m s f o r A and M a r e p r o - ducts of the exact one-monopole solutions. We choose

(a ) A = G(YJG(Y,) , (b) lV= x ( r , ) X ( ~ ~ ) ,

where G(r i ) and ~ ( r , ) a r e the one-monopole solu- tions7 f o r A and 1'bl centered a t the position of mono- pole i .

The upper bound on EI which is obtained by the t r i a l configuration defined in (14), (16), and (18) can be est imated analytically in the limit of l a rge monopole-antimonopole separat ion R. [ T h i s es t i - mate i s found by a n inspection of the integrand in (12), while keeping in mind the known proper t ies of the functions G and X.]

F o r nonzero X , the resu l t i s

The superposition principle predicts equality in (19). F o r the c a s e h = 0 ( o r X << I/R << 1), our variational calculation gives

F o r this case , the superposition resul t i s ruled out. In Ref. 4, Manton predicted equality f o r (20) by assuming the asymptotic fo rm of the solution.

17 - C L A S S I C A L I N T E R A C T I O N S O F ' t H O O F T M O N O P O L E S 3259

In light of (19) and (20), we conjecture that our upper bounds represent good estimates for EI in the large-R region.

The reason for the drast ic change a t vanishing X can be explained a s follows. I? order to keep the energy finite, the magnitude of $ must vanish a t th_e position of each monopole. When X i s nonzero, I @ I approaches 1 exponentially a s you_move away from a monopole. In this case the ( 0 ~ ) ~ te rm in L has no effect on the long-range interactions2f the monopoles. However, when X vanJshes (D@)' is the only te rm in L which involves $. This t e rm resembles the (gauge covarianl) potential energy of an elastic medium which is deformed by the presence of the monopoles. This provides a long- range attractive force which is added to the mag- netic force. In the case of a monopole and an antimonopole, the elastic and magnetic forces work together. In the case of a monopole pair , the two forces cancel each other, a s we shall s ee in Sec. IV.

In the limit R - 0, the monopole and antimono- pole annihilate, giving the exact result

However, our t r ia l configuration does not r e - produce this result. A brief inspection of (12) r e - veals that our choice for M i s the source of the er ror . Therefore, we define a new t r ia l state which is the s ame a s the old t r ia l s tate except that M is given by

(The 1.5 was determined variationally.) The upper bounds on E, which we get from the combination of these two t r ia l s tates i s shown in Fig. 1 for several values of X.

IV. MONOPOLE-MONOPOLE INTERACTION

2 4 6 8 10 12 14

R

FIG. 1. Upper bound on interaction energy vs mon- opole-antimonopole separation. The lower bound on EI is equal to the y intercept of the upper bound.

Using (9) and the identity -e -. -

EijRDi @ . Fjk= a i (eiJk @ . F j k ) , we can derive the formula

where the last equality follows from (23). There- fore, we can express E, in the convenient form,

(The index i i s summed.) This form for E, im- mediately gives us the lower bound

EI 2 2 - 2 E o . (27)

To find the interaction energy between two mono- In order to get an upper bound on E , , we pro- poles we must minimize L subject to the constraint ceed just a s in the monopole-antimonopole case

except we use (26) to compute E,. In t e rms of the i ri jkai 6 a j 6 x a,~=4~6~(x-x,)+4~6~(x-x,) . variables a , 6 , X , A, and M, the expression for

(23) El becomes

The arguments which led to (15) and (17) now This configuration does not resemble the one- lead to monopole solution a t all. Consequently, we do

not expect it to give a very good upper bound for (a) X = O , large separations, where the one-monopole solu-

(b) a = 2 @ , tion should be approximately correc t near each

(29) monopole. (c) COSP = B ( C O S B ~ + C O S B ~ ) . In order to find an upper bound which i s a good

3260 S T E V E N F . M A G R U D E R 17 -

approximation to the exact E, a t l a r g e separat ions, we must find a t r i a l configuration which resembles the one-monopole solution near each monopole. Our f i r s t s t ep will be to choose the definition for A and iZI given in (18).

In o r d e r to avoid a choice f o r a! and p s i m i l a r to (29) and s t i l l satisfy the constraint (23), we mus t avoid cylindrical symmetry. Therefore, we put the monopoles on the x ax is a t x = &%R. F o r a finite V, we a r e then required to make sin@ vanish when e i ther sine, o r sine, vanishes. The con- s t ra in t s which we have mentioned s o f a r certainly do not determine a! and p uniquely, s o we t r i ed s e v e r a l forms. The one which gave the best bound w a s

The variable i s then partially determined by (13). A s imple solution which vanishes n e a r each monopole is given by

X I = &-' cose, - cosP + Gz cose, - cosp Y, sin8, r , sine, (31)

in a n obvious notation. The only trouble with this choice f o r A is that it develops a singularity in the limit R - 0. In o rder to eliminate this s ingular i ty , we make the choice

where Q is chosen to minimize E,. (In our calcu lations Q always fel l in the range 0.1-0.14.) The upper bound f o r E,(R) which is found through the t r i a l configuration defined in (18), (30), and (32) is shown in Fig. 2.

F o r asymptotically l a rge R and nonzero X , the

u O h f 4 k 6 8 9 1 o

R

F I G . 2 . Upper bound on interaction energy vs mon- opole-monopole separation. This energy is bounded below by 2 - 2 E o .

f o r m of the upper bound can be found by inspection of the integrand in (28) using our t r i a l configura- tion. The resu l t i s

Er I/R. (33)

Equality In (33) would be in agreement wlth the superposition principle. For the c a s e X = 0, the asymptotic f o r m 1s given by

The 1 / ~ ' law i s a resu l t of the cancellation of mag- netic fo rces agains elast ic fo rces . This cancella- tlon becomes quite obvious when one s tudies the asymptotic f o r m of (28). The lack of a Coulomb force was predicted in Manton's paper. Our r e - sul t rigorously ru les out a Coulomb force f o r the A = 0 case.

V. DISCUSSION

We have found upper and lower bounds on the interactions of 't Hooft monopoles. When the separat ion of the monopoles ( o r monopole and antimonopole) is l a rge compared with X - I , our resu l t s a r e consistent with the Coulomb interaction which is predicted by the superposition principle. At s m a l l separat ions, we find that the interaction energy remains ra ther small . In the special case X = 0, the Coulomb interaction is ruled out by our resu l t s f o r a rb i t ra r i ly l a rge separat ions. This can be understood a s the resul t of a n at t ract ive "elastic" interaction which i s just a s important a s the Coulomb interaction a t l a rge separat ions.

It i s interesting to speculate on what effects the non-Coulomb interactions might have on the prop- e r t i es of the monopole gas. When X-' i s s m a l l compared with the mean dis tance between mono- poles , we expect the superposition principle to be accurate . This i s the c a s e which Polyakov considers jn his demonstrat ion of confinement. When X-' i s l a rge , correct ions to the superposition principle may be important. Since the correct ions to the Coulomb interactions a r e generally a t t rac - tive in nature, it is c lea r that we cannot consider two-body interactions exclusively. This would make the gas appear unstable against collapse. Therefore we consider the general problem of minimizing (1) subject to the constraint (3) in the p resence of a finite density of monopoles and antimonopoles.

The effect of the constraint is m o r e t ransparent if we wr i te (1) in the f o r m

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The constraint (3) requires that 84 has a l / r singularity at the position of each monopole. Therefor:, we expect that the quantity minx [D, @(x) . D i $(x)] will increase a s the mono- pole density increases. When this te rm becomes larger tha2 Xq2, it will be energetically favor- able for 1 @ 1 to vanish. Once this has occurred, ><

it will be energetically favorable for the A fields to vanish also.

Assuming that the path integral for the Georgi- Glashow theory can still be approximated by the partition function for this monopole gas, we specu- late that the quantum theory will undergo a phase transition when X decreases to some small value. In the small X phase, the expectation value of the Higgs field vanishes, leading to a vanishing vector- boson mass and no confinement. This picture

agrees with the conventional wisdom for the case X = 0. In this case there i s no term in the Lagrang- ian which would be expected to generate a vector- boson mass.

ACKNOWLEDGMENTS

We wish to thank Professor S. J. Chang for sug- gesting the problem, for many useful conver- sations, and for a critical reading of the manu- script. We wish to thank Professor H.W. Wyld for providing us with his numerical results for the one-monopole configuration and for several useful conversations concerning the numerical work in this paper. This work was suppcrted by the National Science Foundation under Grant No. NSF MPS 73-05103.

'A. Polyakov, Nucl. Phys. H, 429 (1977). he importance of vacuum tunnelling for quantum

chromodynamics has been discussed by several authors. See for example C. Callan, R. Dashen, and D. Gross, Phys. Lett. =, 334 (1976); E, 375 (1977); G. 't Hooft, Phys. Rev. Lett. 37, 8 (1976); R. Jackiw and C. Rebbi, ibid. 31, 172 (1976).

3 ~ h e properties of 't Hooft monopoles a re explored in G. 't Hooft, Nucl. Phys. E, 276 (1974); A. Polyakov, Zh. Eksp. Teor. Fiz. Pis'ma Red. 20, 430 (1974)

[JETP Lett. 20, 194 (1974)l; Zh. Eksp. Teor. Fiz. 68, 1975 (1975) [Sov. Phys .4ETP 41, 988 (1975)] .

*N. Manton, Nucl. Phys. s, 525 (1977). 5 ~ . Arafune, P. Freund, and C. Goebel, J. Math: Phys. 16, 433 (1975).

' Z ~ r a s a d and C . Sommerfield, Phys. Rev. Lett. 35, 760 (1975).

he functions G / r and ,y have been computed numerical- ly in H . Wyld and R. Cutler, Phys. Rev. D l4, 1648 (1976).