Physics Letters B 528 (2002) 288–294

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Bound monopoles in the presence of a dilaton

Yves Brihayea, Betti Hartmannb

a Faculté des Sciences, Université de Mons-Hainaut, B-7000 Mons, Belgiumb Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, UK

Received 23 November 2001; accepted 17 January 2002

Editor: P.V. Landshoff

Abstract

We study axially symmetric monopoles of both the SU(2) Yang–Mills–Higgs dilaton (YMHD) as well as of the SU(2)Einstein-Yang–Mills–Higgs dilaton (EYMHD) system. We find that equally to gravity, the presence of the dilaton field canrender an attractive phase. We also study the influence of a massive dilaton on the attractive phase in the YMHD system. 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

The Georgi–Glashow model with SU(2) gaugegroup constitutes the simplest non-abelian gauge fieldtheory in which topological solitons exist: magneticmonopoles [1,2]. It consists of an SU(2) Yang–Millstheory coupled to a Higgs triplet in the adjoint rep-resentation of the group and is spontaneously brokenby a Higgs potential (we refer to it as to the YMHmodel in the following). The solutions are character-ized by their winding numbern, which arises due totopological arguments and is proportional to the mag-netic charge of the configuration. The solution withunit topological chargen = 1 can be constructed witha spherically symmetric ansatz of the fields. Since thiswas found to be the unique spherically symmetric so-lutions [3], the field configurations corresponding tohigher values of the topological chargen > 1 (themultimonopoles) involve at most axial symmetry andlead to systems of partial differential equations [4,5].One feature of multimonopoles is their instability: forgeneric values of the coupling constants of the theory

the long-range repulsion due to the gauge fields cannotbe overcome by the short-range attraction due to theHiggs field. Only in the so-called BPS (Bogomol’nyi–Prasad–Sommerfield) limit [3,6] in which the Higgsfield is massless and therefore long-range, the two in-teractions exactly compensate [7,8]. The spatial com-ponents of the stress-energy tensor were shown to van-ish [9] and thus systems of non-interacting monopolesexist.

A few years ago, the YMH model was cou-pled to Einstein gravity [10] (resulting in a the-ory labelled EYMH) and the spherically symmetricgravitating monopoles with unit topological chargewere constructed. Also studied were the correspond-ing non-abelian black holes solutions, which violatethe “no-hair” conjecture. Quite recently [11], it wasdemonstrated that bound states of gravitating multi-monopoles exist in the EYMH model. Indeed, solvingthe equations for numerous values of the coupling con-stants, it was shown that two phases exist. For smallvalues of the Higgs coupling constant, there exists aphase for which the binding energy of the 2-monopole

0370-2693/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0370-2693(02)01202-9

Y. Brihaye, B. Hartmann / Physics Letters B 528 (2002) 288–294 289

and the 3-monopole is negative, leading to classicalsolutions bounded by gravity.

On the other hand, it was pointed out [12], thatthe coupling of the YMH system to a dilaton field(labelled YMHD) renders regular classical solutionsthat share many properties with that of the EYMHmodel.

It is, therefore, natural to check if the couplingto a dilaton field can also lead to systems of boundmonopoles. The aim of this paper is to study this ques-tion by analyzing the equations of the full EYMHDmodel incorporating both gravitation and a dilatonfield. Our numerical integration of the equationsstrongly indicate that the analogy between the EYMHand the YMHD models persist also for the multimono-pole solutions.

In Section 2 we specify the model and its differentcomponents, the axially symmetric ansatz and the rele-vant rescaling. The boundary conditions are presentedin Section 3. The numerical solutions and their rele-vant features are discussed in Section 4. In particular,we study the effect of the dilaton on both the solutionsin flat and curved space and also briefly discuss theimplications of a massive dilaton.

2. SU(2) (Einstein–)Yang–Mills–Higgs dilatontheory

The action of the Yang–Mills–Higgs dilaton(YMHD) theory reads:

(1)S = SM =∫

LM

√−g d4x,

while for the Einstein–Yang–Mills–Higgs dilaton(EYMHD) theory an additional term from the gravityLagrangian arises:

(2)

S = SG + SM =∫

LG

√−g d4x +∫

LM

√−g d4x.

The gravity Lagrangian is given by:

(3)LG = 1

16πGR,

whereG is Newton‘s constant.The matter Lagrangian is given in terms of the

gauge fieldAaµ, the dilaton fieldΨ and the Higgs field

Φa (a = 1,2,3):

LM = −1

4e2κΨ F a

µνFµν,a − 1

2∂µΨ ∂µΨ

(4)

− 1

2DµΦaDµΦa − e−2κΨ V

(Φa

) − 1

2m2Ψ 2,

wherem denotes the mass of the dilaton field and

(5)V(Φa

) = λ

4

(ΦaΦa − v2)2

.

The field strength tensor is given by:

(6)Faµν = ∂µAa

ν − ∂νAaµ + eεabcA

bµAc

ν,

and the covariant derivative of the in the adjointrepresentation given Higgs field reads:

(7)DµΦa = ∂µΦa + eεabcAbµΦc.

e denotes the gauge field coupling,κ the dilaton cou-pling, λ the Higgs field coupling andv the vacuumexpectation value of the Higgs field.

3. Axially symmetric ansatz

For the metric, the axially symmetric ansatz inisotropic coordinates reads:

(8)

ds2 = −f dt2 + m

f

(dr2 + r2dθ2) + l

fr2 sin2 θ dϕ2,

wheref , m andl are functions ofr andθ only. In thespecial case of the YMHD systemm(r, θ) = l(r, θ) =f (r, θ) = 1.

The ansatz for the purely magnetic gauge field is

Aµdxµ = 1

2Aa

µτa dxµ

= 1

2er

[τnφ (H1dr + (1− H2)r dθ)

(9)− n(τnr H3 + τn

θ (1− H4))r sinθ dϕ

],

and for the Higgs field the ansatz reads

(10)Φ = Φaτa = (Φ1τ

nr + Φ2τ

nθ

),

where the matter field functionsH1, H2, H3, H4, Φ1andΦ2 depend only onr andθ . The symbolsτn

r , τnθ

andτnφ denote the dot products of the cartesian vector

290 Y. Brihaye, B. Hartmann / Physics Letters B 528 (2002) 288–294

of Pauli matrices,�τ = (τ1, τ2, τ3), with the spatialunit vectors

�enr = (sinθ cosnϕ,sinθ sinnϕ,cosθ),

�enθ = (cosθ cosnϕ,cosθ sinnϕ,−sinθ),

(11)�enφ = (−sinnϕ,cosnϕ,0),

respectively. Here, the topological chargen enters theansatz for the fields.

The dilaton fieldΨ is a scalar field depending onr, θ :

(12)Ψ = Ψ (r, θ).

3.1. Rescaling

With the introduction of the dimensionless radialcoordinatex and rescaling of the Higgs field, thedilaton field and the dilaton mass, respectively:

x ≡ rev, φ = Φ

v, ψ = Ψ

v,

(13)Mdil = m

ev,

the set of partial differential equations depends onlyon three fundamental coupling constants:

(14)α = √4πGv, β =

√λ

e, γ = vκ,

whereα = 0 in the YMHD system.

3.2. Mass of the solution

In the case of a massive dilaton (Mdil �= 0), the massof the solutionµ can be obtained from integratingthe Lagrangian density (4). ForMdil = 0, however,simple relations between the mass of the solution andthe derivative of the corresponding function at infinityexist. In the YMHD system the mass is given in termsof the derivative of the dilaton field at infinity [12]

(15)µ = 1

γlim

x→∞x2∂xψ,

while in the EYMHD system it is given in terms of thederivative of the metric functionf at infinity

(16)µ = 1

2α2 limx→∞x2∂xf.

The massµab of the corresponding abelian solutionsis given in the EYMHD system by:

(17)µab = (α2 + γ 2)−1/2

with α = 0 in the limit of the YMHD system.

4. Boundary conditions

We look for regular, static, finite energy solutionsthat are asymptotically flat. The requirement of regu-larity leads to the following boundary conditions at theorigin:

∂xf (0, θ) = ∂xl(0, θ) = ∂xm(0, θ) = 0,

(18)∂xψ(0, θ) = 0,

Hi(0, θ) = 0, i = 1,3,

Hi(0, θ) = 1, i = 2,4,

(19)φi(0, θ) = 0, i = 1,2.

At infinity, the requirement for finite energy andasymptotically flat solutions leads to the boundaryconditions:

f (∞, θ) = l(∞, θ) = m(∞, θ) = 1,

(20)ψ(∞, θ) = 0,

Hi(∞, θ) = 0, i = 1,2,3,4,

(21)φ1(∞, θ) = 1, φ2(∞, θ) = 0.

In addition, boundary conditions on the symmetryaxes (theρ- andz-axes) have to be fulfilled. On bothaxes:

(22)H1 = H3 = φ2 = 0

and

∂θf = ∂θm = ∂θ l = ∂θH2 = ∂θH4

(23)= ∂θφ1 = ∂θψ = 0.

5. Numerical results

Subject to the above boundary conditions, we havesolved the system of partial differential equationsnumerically.

Y. Brihaye, B. Hartmann / Physics Letters B 528 (2002) 288–294 291

5.1. Monopoles in YMHD theory

It was noted recently [11] that in a certain parame-ter range of the coupling constants an attractive phaseexists in the EYMH system. Inspired by the observa-tion that the monopoles in the YMHD system sharemany features with the monopoles in the EYMH sys-tem [12], we first studied the (multi)monopoles ofthe YMHD system in the limit of vanishing dilatonmass. We find that in the BPS limit (β = 0) there ex-ists an attractive phase for all values ofγ > 0. Thisis in close analogy to the EYMH system, where at-traction between the BPS monopoles exists for allα > 0. Indeed, the plot of the energy per windingnumber overγ in the YMHD system looks simi-lar than Fig. 3 of [11] whenα is interchanged withγ and “Reissner–Nordström (RN)” is interchangedwith “Einstein–Maxwell dilaton (EMD)”. Moreover,we find that when comparing the quantity

(24)4E = E(n = 1)

1− E(n = 2)

2

for the monopoles in the EYMH system for a specificvalue of α̃ with that of the monopoles in the YMHDsystem for a value ofγ = α̃, the two values equal eachother (at least within our numerical accuracy).

We were also interested in the implications of amassive dilaton. The massive dilaton was previouslyconsidered only for the spherically symmetric solu-tions [13]. We studied the influence of the dilatonmassMdil on the attractive phase. Since now a massis involved, the dilaton field decays exponentially—contrasted to a power law decay in the massless case—and the relation (15) between the derivative of the dila-ton field at infinity and the mass of the solution is nolonger valid. Our numerical results are shown in Fig. 1,where we present the difference between the mass(per winding number) of then = 1 solution and themass per winding number of then = 2 solution4E =E(n=1)

1 − E(n=2)2 . Clearly, the attraction is lost for

Mdil > M̂dil(γ ). We find thatM̂dil(γ = 0.5) ≈ 0.040andM̂dil(γ = 1.0) ≈ 0.028, respectively. Our numeri-cal results suggest further that forMdil → ∞ the value4E turns to zero indicating that the monopoles arenon-interacting in this limit. This is demonstrated inTable 1 forγ = 1.0:

This result can be understood considering that forMdil → ∞ the dilaton functionψ(x, θ) has to turn to

Fig. 1. The quantity4E = E(n=1)1 − E(n=2)

2 is shown as functionof the dilaton massMdil for two different values ofγ in the YMHDsystem (α = 0).

Table 1

Mdil 4E

0.0 0.007370.01 0.003720.1 −0.013901.0 −0.01380

10.0 −0.00023

zero on the full intervalx ∈ [0 : ∞[ for all θ . Thusfor the case studied here (β = 0), the BPS limit ofthe YMH system is recovered forMdil → ∞. Ournumerical results strongly support this interpretation.We find that with increasingMdil the dilaton fieldtends more and more to the trivial solutionψ(x, θ) = 0and that the mass tends to one, which (in our rescaledvariables) is just the mass of the BPS solution in theYMH system.

5.2. Monopoles in EYMHD theory

Here, we only considered the case ofMdil = 0.We first studied the influence of the dilaton field onthe attraction between like monopoles in the limit ofvanishing Higgs coupling (BPS limit). In Fig. 2, weshow the difference between the mass per windingnumber of then = 1 and then = 2 solution4E =

292 Y. Brihaye, B. Hartmann / Physics Letters B 528 (2002) 288–294

Fig. 2. The quantity4E = E(n=1)1 − E(n=2)

2 is shown as functionof γ for three different values ofα, including α = 0.0, whichrepresents the YHMD system.

E(n=1)1 − E(n=2)

2 for a fixed α and varyingγ . Forα = 0 the limit γ = 0 represents the BPS limit of theYMH theory. The monopoles are non-interacting forβ = 0 and therefore the energy per winding numberis equal for all (multi)monopole solutions of differenttopological sectors. Since the YMHD-monopoles inthe BPS limit reside in an attractive phase for allγ �= 0, 4E should be positive, which indeed, isdemonstrated in Fig. 2. Forα = 0.5 the attractionbetween like monopoles in the EYMHD system isbigger than in the pure EYMH system (γ = 0) for allγ �= 0. The curve reaches a maximum of the differenceat some valueγ and from there, the difference getssmaller. This can be understood from the fact thatfor rising γ , the solutions tend to the EMD solutionswhich have mass per winding number equal for alln.The curve forα = 1.0 shows the same behaviour apartfrom the fact, that now for biggerγ the attractiongets smaller than in theγ = 0 case. This is due tothe fact, that in the pure EYMH system, the attractionhas nearly reached its maximum atα = 1.0 and thatnow inclusion of the dilaton field very soon makes thesolution tend to a EMD solution.

To study the influence of the dilaton on the mono-pole solutions forβ �= 0, we followed [11] and deter-minedγeq(β). This is—for a fixedβ—the value ofγ

Fig. 3.γeq is shown as a function ofβ for three different values of

α. Also shown isγ n=1max . The attractive phase exists for parameters

values above theγeq curve and below the correspondingγ n=1max line.

for which the mass of then = 1 solution is equal to themass per winding number of then = 2 solution. Forγ < γeq, the mass per winding number of then = 2 isbigger than the mass of then = 1 solution which im-plies that the monopoles are repelling, while forγ >

γeq it is smaller leading to an attractive phase. Becauseglobally regular solutions exist only forγ � γ n

max(β)

[14], the attractive phase is limited in parameter spaceby theγ n=1

max curve. (Sinceγ n=2max (β) > γ n=1

max (β) for thevalues ofβ for which the attractive phase exists, themasses of then = 1 andn = 2 solution can only becompared forγ � γ n=1

max .) For β = β̂ , the two curvesmeet and no attractive phase is possible forβ > β̂ . Inthe EYMH system it was found that̂β ≈ 0.21 [11]. InFig. 3, the values ofγeq andγ n=1

max are shown for threedifferent values ofα. α = 0 represents the YMHD sys-tem and theγ n=1

max - and γeq-curves look similar thanthe αn=1

max- andαeq-curves of [11]. This again under-lines the close analogy of the EYMH system and theYMHD system.

Comparing the three curves, we find that the valuesof bothγ n=1

max andγeq drop to smaller values ofγ forfixed β and increasingα. This results in the fact thatthe value ofβ̂ seems to be independent onα. For α1

Y. Brihaye, B. Hartmann / Physics Letters B 528 (2002) 288–294 293

the attractive phase is thus obtained for smaller valuesof γ then in theα2 case, ifα1 > α2. It does not seemto exceedβ > 0.21 for any value ofα though.

6. Summary and concluding remarks

We have studied axially symmetric dilatonic mono-poles in flat and curved space. In the limit of vanishinggravitational coupling and vanishing dilaton mass, wefind that the presence of the dilaton field can render anattractive phase similar to gravity. The close analogybetween the EYMH and the YMHD system observedin [12] thus persists for the multimonopoles. Whenthe dilaton field is massive, the attraction between themonopoles in the BPS limit of the YMHD systemis lost for Mdil > M̂dil(γ ) and the monopoles arerepelling. ForMdil → ∞, the dilaton function has toturn to the trivial solution (to fulfill the requirementof finite energy). The dilaton decouples from the fieldequations and the pure YMH system, in which the BPSmonopoles are known to be non-interacting, is left.

When in the BPS limit both gravitation and the(massless) dilaton are coupled to the monopoles,the value of4E = E(n=1)

1 − E(n=2)2 —indicating the

strength of attraction—first increases from its value inthe EYMH system with increasingγ . 4E reaches amaximum at a value ofγ depending onα and fromthere decreases.

In the non-BPS limit, the (massless) dilaton field isable to overcome the long-range repulsion of the gaugefields in a similar way than gravity. We find that theattractive phase is limited in parameter space and thatthe value ofβ for which the attractive phase is lost isindependent onα.

While the n = 1 monopole is stable due to thepreservation of the topological charge, the stability ofn > 1 monopoles is not obvious since it might be pos-sible that they decay into singly charged monopolesthereby preserving the total topological charge. Weconjecture that the monopoles are stable as long asthey reside in the attractive phase.

We have studied axially symmetric monopolesfor n = 2 here. However, it was observed that forn � 3 BPS monopoles with discrete symmetries exist[15]. Since in the BPS limit the energy per windingnumber is equal for all configurations (independenton the actual structure), it would be interesting to

construct these solutions in the (E)YMHD system.Only then it could be decided, which configurationis the one of lowest energy for a given topologicalsector. Moreover, computing the energy per windingnumber of these solutions, the influence of gravityand the dilaton field, respectively, on monopoleswith discrete symmetries could be investigated. Ithas to be pointed out though, that up to now noexplicit ansatz exists which would allow a numericalconstruction of these configurations. In the BPS limit,the monopoles fulfill a first order differential equation,the Bogomol’nyi equation. Since this equation isintegrable, mathematical techniques such as twistormethods [16] are available. In the non-BPS limitthough, the full system of second-order differentialequations has to be solved. This is a difficult numericaltask and remains a challenge for the future. However,in analogy to the soliton solutions in the Skyrmemodel [17], we conjecture that the actual minimalenergy configurations of the system studied here haverather discrete than axial symmetry forn � 3. Sinceour study of the binding energy ofn = 3 axialmultimonopoles shows that the domain of parameterspace wheren = 3 bound solutions exist varies onlylittle from the n = 2 case, it is very likely that (atleast in the region of parameter space we have studied)the discrete symmetry solutions are even more bindedthan the axial ones we have constructed.

Acknowledgements

One of us (B.H.) wants to thank the Belgium FNRSfor financial support. We gratefully acknowledge dis-cussions with J. Kunz and B. Kleihaus.

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