Dilatonic monopoles from (4+1)-dimensional vortices

Physics Letters B 534 (2002) 137–146

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Dilatonic monopoles from(4+ 1)-dimensional vortices

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, United Kingdom

Received 11 February 2002; accepted 14 March 2002

Editor: P.V. Landshoff

Abstract

We study spherically and axially symmetric monopoles of theSU(2) Einstein–Yang–Mills–Higgs-dilaton (EYMHD) systemwith a new coupling between the dilaton field and the covariant derivative of the Higgs field. This coupling arises in the studyof (4+ 1)-dimensional vortices in the Einstein–Yang–Mills (EYM) system. 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

In the 1920s, Kaluza and Klein studied the 5-dimensional version of the Einstein equations [1]by introducing a 5-dimensional metric tensor. Whenone dimension is compactified, the equations of 4-dimensional Einstein gravity plus Maxwell’s equa-tions are recovered. One of the new fields appearingin this model is the dilaton, a scalar companion ofthe metric tensor. In an analog way, this field arises inthe low energy effective action of superstring theoriesand is associated with the classical scale invariance ofthese models [2].

In recent years, a number of classical field theorymodels coupled to a dilaton have been studied. It wasfound that if the solutions ofSU(2) Yang–Mills–Higgs(YMH) theory, namely, the ’t Hooft–Polyakov mono-pole [3] and its higher winding number generalisations[4–6], are coupled to a massless dilaton [7,8], remark-

E-mail addresses: [email protected] (Y. Brihaye),[email protected] (B. Hartmann).

able similarities to the qualitative features of Einstein–Yang–Mills–Higgs (EYMH) monopoles [9,10] arise.Especially, it was observed that similarly to gravitythe dilaton can render attraction between like chargedmonopoles and thus bound multimonopole states arepossible.

Recently, the YMH system coupled to both gravityand the dilaton has been studied [8,11]. It was foundthat in the full Einstein–Yang–Mills–Higgs-dilaton(EYMHD) model, a simple relation between thett-component of the metric and the dilaton exists. TheAbelian solutions to which the configurations tend inthe limit of critical coupling are the extremal Einstein–Maxwell-dilaton (EMD) solutions. These have defi-nite expressions for the energy and the value of therr-component of the metric at the origin which onlydepend on the fundamental couplings.

Volkov argued recently [12] that if∂/∂x4 is asymmetry of the Einstein–Yang–Mills (EYM) systemin 4 + 1 dimensions, wherex4 is the coordinateassociated with the 5th dimensions, then the (4+ 1)-dimensional EYM system reduces effectively to a(3 + 1)-dimensional EYMHD system with a specific

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138 Y. Brihaye, B. Hartmann / Physics Letters B 534 (2002) 137–146

coupling between the dilaton field and the Higgsfield.

In this Letter, we study both the spherically andaxially symmetric solutions of the (3+1)-dimensionalEYMHD model deduced from the (4+1)-dimensionalEYM system. In Section 2, we give the EYMHDLagrangian and review how the specific couplingarises. In Sections 3 and 4, we give the ansatz andpresent our numerical results for the spherically andaxially symmetric solutions, respectively. In both thesesections emphasis is placed on the flat space limit andon the limit in which our system of equations reducesto the one in [12]. The summary and conclusions arepresented in Section 5.

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

The Lagrangian and the particular coupling of thedilaton field Ψ to the SU(2) gauge fieldsAµ

a andHiggs fieldsΦa (a = 1,2,3), respectively, arise ef-fectively from the Lagrangian of (4+ 1)-dimensionalEYM theory. If both the matter functions and the met-ric functions are independent onx4, the 5-dimensionalfields can be parametrized as follows (withM,N =0,1,2,3,4) [12]:

g(5)MN dxM dxN = e−ζ g(4)µν dx

µ dxν − e2ζ (dx4)2,

(1)µ,ν = 0,1,2,3

and

(2)AaM dxM = Aa

µ dxµ +Φa dx4, a = 1,2,3,

whereg(4)µν is the 4-dimensional metric tensor andζplays the role of the dilaton. Introducing a new cou-pling κ to study the influence of the dilaton systemati-cally, we setζ = 2κΨ and obtain the following actionof the effective 4-dimensional EYMHD theory:

S = SG + SM

(3)=∫

LG

√−g(4) d4x +

∫LM

√−g(4) d4x.

The gravity LagrangianLG is given by

(4)LG = 1

16πGR,

where G is Newton’s constant, while the matterLagrangianLM reads:

LM = −1

4e2κΨ F a

µνFµν,a − 1

2∂µΨ ∂µΨ

(5)− 1

2e−4κΨDµΦ

aDµΦa − e−2κΨ V(Φa

),

with Higgs potential

(6)V(Φa

) = λ

4

(ΦaΦa − v2)2

,

the non-Abelian field strength tensor

(7)Faµν = ∂µA

aν − ∂νA

aµ + eεabcA

bµA

cν,

and the covariant derivative of the Higgs field in theadjoint representation

(8)DµΦa = ∂µΦ

a + eεabcAbµΦ

c.

Here, e denotes the gauge field coupling constant,λ the Higgs field coupling constant andv the vacuumexpectation value of the Higgs field.

3. Spherically symmetric solutions

For the metric, the spherically symmetric ansatz inSchwarzschild-like coordinates reads [9]:

ds2 = g(4)µν dxµ dxν

= −A2(r)N(r) dt2 +N−1(r) dr2

(9)+ r2dθ2 + r2 sin2 θ d2ϕ,

with

(10)N(r) = 1− 2m(r)

r.

In these coordinates,m(∞) denotes the (dimension-ful) mass of the field configuration.

For the gauge and Higgs fields, we use the purelymagnetic hedgehog ansatz [3]

(11)Ara = At

a = 0,

Aθa = 1−K(r)

eeϕ

a,

(12)Aϕa = −1−K(r)

esinθeθ

a,

(13)Φa = vH(r)era.

Y. Brihaye, B. Hartmann / Physics Letters B 534 (2002) 137–146 139

The dilaton is a scalar field depending only onr

(14)Ψ = Ψ (r).

Inserting the ansatz into the Lagrangian and varyingwith respect to the matter fields yields the Euler–Lagrange equations, while variation with respect to themetric yields the Einstein equations.

With the introduction of dimensionless coordinatesand fields

x = evr, µ = evm,

(15)φ = Φ

v, ψ = Ψ

v.

The Lagrangian and the resulting set of differentialequations depend only on three dimensionless cou-pling constants,α, β andγ ,

α = √Gv = MW

eMPl, β =

√λ

e= MH√

2MW

,

(16)γ = κv = κMW

e,

whereMW = ev, MH = √2λv and MPl = 1/

√G.

With the rescalings (15) and (16), the dimensionlessmass of the solution is given byµ(∞)

α2 .With (15) and (16) the Euler–Lagrange equations

read:

(e2γψANK ′)′

(17)= A

(e2γψ K(K2 − 1)

x2 + e−4γψH 2K

),

(e−4γψx2ANH ′)′

(18)= AH(2e−4γψK2 + β2x2e−2γψ(

H 2 − 1)),

(19)

(x2ANψ ′)′ = 2γA

[e2γψ

(N(K ′)2 + (K2 − 1)2

2x2

)

− e−2γψ β2x2

4

(H 2 − 1

)2

− 2e−4γψ(

1

2N(H ′)2x2

+H 2K2)]

,

where the prime denotes the derivative with respectto x, while we use the following combination of the

Einstein equations

(20)Gtt = 2α2Ttt = −2α2A2NLM,

(21)gxxGxx − gttGtt = −4α2N∂LM

∂N

to obtain two differential equations for the two metricfunctions:

(22)

µ′ = α2(e2γψN(K ′)2 + 1

2Nx2(H ′)2e−4γψ

+ 1

2x2

(K2 − 1

)2e2γψ +K2H 2e−4γψ

+ β2

4x2(H 2 − 1

)2e−2γψ + 1

2Nx2(ψ ′)2

),

(23)

A′ = α2xA

(2(K ′)2

x2 e2γψ + e−4γψ(H ′)2 + (ψ ′)2).

Since we are looking for globally regular, finite energysolutions which are asymptotically flat, we impose thefollowing set of boundary conditions:

K(0)= 1, H(0)= 0,

(24)∂xψ|x=0 = 0, µ(0) = 0,

K(∞) = 0, H(∞) = 1,

(25)ψ(∞) = 0, A(∞) = 1.

3.1. Numerical results

We have restricted our numerical calculations forthe spherically as well as for the axially symmetricsolutions toβ = 0.

3.1.1. The α = 0 limitForα = 0, the gravitational field equations (22) and

(23) decouple from the rest of the system and we areleft with the YMHD model in flat space,N(x) ≡ 1 andA(x)≡ 1. In [7] it was observed that (in analogy to theEYMH system) the monopoles exist up to a maximalvalue of the dilaton couplingγ = γmax and from thereon a second branch of solutions tend to the Abeliansolution forγ → γcr < γmax with ψ(0) monotonicallydecreasing to−∞. Our numerical results indicate thatno suchγmax exists in the model studied here. We haveintegrated the equations forγ ∈ [0 : 10] and found thesolutions to exist for all these values ofγ . The profiles


Table 1

γ xK xH

0.1 2.2 1.81.0 3.3 2.15.0 11.2 5.1

10.0 22.4 8.9

of the functions suggest that forγ → ∞ the solutionstend to the vacuum solution withK(x)≡ 1,H(x)≡ 0andψ(x) ≡ 0. This is indicated by the fact that themass of the field configurations progressively tendsto zero for risingγ and that the matter fields areequal to their vacuum values on increasing intervalsof the coordinatex. To demonstrate this, in Table 1 wegive the values ofxK andxH , where the gauge fieldK(x) and the Higgs fieldH(x), respectively, reachthe value 0.5, i.e., K(xK) = 0.5 andH(xH) = 0.5.Moreover, we find thatψ(0) � 0 for all γ . Since forγ = 0, the BPS monopole solution is recovered, thecurve forψ(0) starts from zero atγ = 0. From thereit increases to a maximal value atγ ≈ 1.38 and thenslowly decreases to zero forγ → ∞. Together with(25) our results strongly suggest thatψ(x) tends tozero on the full intervalx ∈ [0 :∞[.

The difference between the model studied here andthe one studied in [7] is that in the standard case of theYang–Mills–Higgs-dilaton system, the equations areeffectively the equations of an Einstein–Yang–Mills–Higgs model with metric

ds2 = −e2κΨ (r) dt2 + e−2κΨ (r) dr2

(26)+ e−2κΨ (r)r2 dθ2 + e−2κΨ (r)r2 sin2 θ d2ϕ.

The components of the Einstein tensor then read:

Gtt = −κe4κΨ (r)

[2Ψ ′′ + 4

Ψ ′

r− κ(Ψ ′)2

],

(27)Grr = −Gθθ

r2= Gϕϕ

r2 sin2 θ= κ2(Ψ ′)2.

The following combination of the Einstein equations

Gtt −Gr

r −Gθθ −Gϕ

ϕ

(28)= 8πG(T tt − T r

r − T θθ − T ϕ

ϕ

)exactly gives the dilaton equation in the standard casewith γ 2 = κ2v2 ∝ Gv2. Equally, the Euler–Lagrangeequations for the gauge and Higgs field functions areobtained using the metric (26). Thus the dilaton can be

viewed as a metric field. Since we know that gravityleads to a critical value of the coupling constant, thisis also true in the standard Yang–Mills–Higgs-dilatoncase. A horizon forms forgtt → 0 which impliesΨ → −∞. This is exactly the limiting solution foundpreviously.

Now, for the equations studied here, this is notpossible. We cannot introduce a 4-dimensional metricwhich makes the equations in the limit ofA = N = 1reduce to a set of Einstein–Yang–Mills-equations.Thus, no critialγ can be expected on the basis of theabove argument. This is confirmed by our numericalresults.

The form of the Lagrangian (5) suggests that forγ

getting bigger and bigger, the only possibility to fulfillthe requirement of finite energy is thatψ(x) → 0 onthe full interval ofx.

3.1.2. The Volkov limit α2 = 3γ 2

In [12] only one fundamental coupling is given,the gravitational couplingG. Comparing the(4 + 1)-dimensional EYM system with the EYMHD systemstudied in this Letter, we conclude that for

(29)α2 = 3γ 2

our system of equations reduces to the one in [12].Studying the full system of equations, we were

particularly interested in reobtaining the results witha different numerical method1 and in studying someof the features of the solutions in greater detail.

Volkov observed a spiraling behaviour of the para-meters for the EYM vortices. As expected and shownin Fig. 1, we observe this feature as well. Our nu-merical results suggest that a number of branches ex-ist, on which the minimum of the metric functionN ,Nm, and the value of the metric functionA at the ori-gin,A(0), monotonically decrease. In the limit of crit-ical coupling, therr-component of the correspond-ing 5-dimensional metric tensor, expressed in 4 di-mensions through the functiong(x) := e2γψN(x) =e

2√3αψ

N(x) develops a double zero at a valuexm > 0of the dimensionless coordinatex [12]. Our numericalresults indicate thatxm ∈]0 : 0.082].

1 To integrate the equations, we used the differential equationsolver COLSYS which involves a Newton–Raphson method [13].


Fig. 1. The values of the minimum of the metric functionN , Nm,and of the metric functionA at the origin,A(0), are shown as afunction ofα for then = 1 solutions in the Volkov limitα2 = 3γ 2.The solid and dashed lined curves denote the values obtained in themodel studied here and the one studied in [11], respectively.

There exist several local maximal and minimalvalues ofα, α(ij)

max andα(ij)

min, respectively. Byα(ij)

max,minwe denote the value ofα, at which theith and thej thbranch join. We find:

α(12)max = 1.268, α

(23)min = 0.312,

(30)α(34)max = 0.419, α

(45)min = 0.395.

Apparently, the difference betweenαmax and its corre-spondingαmin decreases, i.e., the branches get smallerand smaller. We thus conjecture that the limiting solu-tion is reached for a value ofα close toα(45)

min = 0.395.Progressing on the branches, the qualitative behav-

iour of the functions changes [12]. This is shown forthe gauge field functionK(x) in Fig. 2. While for allvalues ofα on the first branch (as an example, weshowK(x) for α = 1.267 close toα(12)

max) and most ofthe values ofα on the second branch (see the profilefor α = 0.8) K(x) decreases monotonically from itsvalue at the originK(0) = 1, to its value at infinityK(∞) = 0, oscillations of the gauge field start to oc-cur on the second branch forα � 0.4185. Forα(23)

min =0.312 the local minimum and maximum, respectively,

Fig. 2. The gauge field functionK(x) is shown as function of thedimensionless coordinatex for the n = 1 solutions in the Volkovlimit α2 = 3γ 2 for the following five values ofα: 1 : α = 1.267

close toα(12)max, 2 : α = 0.8 (second branch), 3: α = 0.4185 (second

branch), 4: α(23)min = 0.312 and 5: α(45)

min = 0.395.

are already quite pronounced, while forα(45)min = 0.395,

the location of both the minimum and maximum hasmoved to smaller values ofx. We are convinced thatin analogy to [12] the number of oscillations increaseswhen proceeding on further branches. In Fig. 3, weshow the value of the dilaton functionψ at the ori-gin (multiplied by−1), −ψ(0), and the mass of thesolutions on the different branches as functions ofα.ψ(0) is equal to zero forα = 0 (since this impliesγ = 0), increases to a maximal value ofψ(0) = 0.174at α ≈ 1.0 and from there decreases first to zero atα = 1.220 on the second branch and further decreasesto a minimal value ofψ(0) = −5.274 atα = 0.314on the third branch. From there it starts to increase toψ(0) = −4.847 atα = 0.395 and then decreases againto ψ(0) = −7.164 atα(45)

min = 0.395.The functionψ(x) itself decreases monotonically

fromψ(0) to its valueψ(∞) = 0 for all values ofγ onthe first branch and forα � 1.24 on the second branch,staying positive for all values ofx. For α > 1.24 onthe second branch, a minimum (which has negativevalue) starts to form and dips down deeper to negativevalues when progressing on the branches. In the limitof critical couplingαcr, the dilaton functionψEMD of


Fig. 3. The value of the dilaton functionψ at the origin (multipliedby −1), −ψ(0), is shown as function ofα for the n = 1 solutionsin the Volkov limit α2 = 3γ 2. Also shown is the mass of thesesolutions.

the corresponding EMD solution [11]

(31)ψEMD =√

3

4αcrln

(1− X−

X

), X− =

(4

3

)1/4

where the dimensionless coordinatex is given in termsof X,

(32)x = αcrX

(1− X−

X

)1/4

is reached forx � xm, while it stays finite forx ∈[0 :xm[.

It has to be remarked here, though, that in thelimit of critical coupling, the gauge and Higgs fieldfunctionsK(x) andH(x) do not reach there Abelianvalues 0 and 1 atx = xm, respectively, but tend to thefixed point described in [12].

The mass of the solution stays close to one onthe first branch and increases monotonically on thesecond branch. On the third and fourth branch it differsonly little from the corresponding mass on the secondbranch and thus the three different branches can barelybe distinguished in Fig. 3. In the limit of criticalcouplingαcr, the mass tends to the massµEMD/α

2 of

the corresponding EMD solution:

(33)µ

α2 → µEMD

α2cr

=√

3

4

1

αcr.

Let us finally investigate the Einstein–Yang–Mills–Higgs (EYMH) system with the usual coupling of thedilaton considered in [11] in the Volkov limitα2 =3γ 2. This model does not contain the prefactor of thecovariant derivative involving the dilaton studied here.In the BPS limit (β = 0), the equation for the Higgsfield does not involve the dilaton field directly and viceversa, while in the model studied here it does evenfor β = 0 (see (18) and (19)). In Fig. 1, we show thevalues ofNm andA(0) for the model studied in [11]for the Volkov limitα2 = 3γ 2. The recalculation of theconfigurations for this specific relation betweenα andγ exactly confirms the results obtained in [11]. Thesolutions exist forα < αmax= 1.216, which fulfills thecondition (48) of [11]

(34)√α2

max+ γ 2 =√

4

3αmax≈ 1.4.

From there, on a second branch of solutions, theEinstein–Maxwell-dilaton (EMD) solution is reachedatαcr with the minimum of the metric functionN , Nm,tending to the valueN(x = 0) of the correspondingEMD solution:2

(35)Nm → NEMD(0) =(

γ 2

α2 + γ 2

)2

= 1

16.

This is demonstrated in Fig. 1.A(0) decreases tozero, whileNm > 0, indicating that the extremal EMDsolutions have a singularity atx = 0 not hidden by anhorizon.

4. Axially symmetric solutions

Since then = 1 monopole was shown to be theunique spherically symmetric solution inSU(2) YMHtheory [14], we need to impose an axially symmetricansatz (or one with even less symmetry) to constructhigher winding number solutions. The axially sym-metric ansatz for the metric in isotropic coordinates

2 In [11], Eq. (45) contains an error, since an overall square ismissing on the r.h.s. of that equation.


reads:

ds2 = −f dt2 + m

f

(dr2 + r2dθ2)

(36)+ l

fr2 sin2 θ dϕ2.

The functionsf , m and l now depend onr andθ . Ifl = m andf only depend onr, this metric reduces tothe spherically symmetric metric in isotropic coordi-nates and comparison with the metric in (9) yields thecoordinate transformation [15]:

(37)dr

r= 1√

N(r)

dr

r.

For the gauge fields we choose the purely magneticansatz [5]:

(38)Ata = 0, Ar

a = H1

ervϕ

a,

Aθa = 1−H2

evϕ

a,

(39)Aϕa = −n

esinθ

(H3vr

a + (1−H4)vθa).

while for the Higgs field, the ansatz reads [5,16]

(40)Φa = v(Φ1vr

a +Φ2vθa).

The vectors�vr , �vθ and�vϕ are given by:

�vr = (sinθ cosnϕ,sinθ sinnϕ,cosθ),

�vθ = (cosθ cosnϕ,cosθ sinnϕ,−sinθ),

(41)�vϕ = (−sinnϕ,cosnϕ,0).

The dilaton fieldΨ now depends onr andθ [8]:

(42)Ψ = Ψ (r, θ).

For H1 = H3 = Φ2 = 0, H2 = H4 = K(r), Φ1 =H(r), Ψ = Ψ (r) andn = 1, this ansatz reduces to thespherically symmetric ansatz in isotropic coordinates.

The Euler–Lagrange equations arise by varying theLagrangian with respect to the matter fields, whilewe use the following combinations of the Einsteinequations [17]

grr(Gµ

µ − 2Gtt

)

= 16πGm

f

(LM + gtt

∂LM

∂gtt− grr

∂LM

∂grr

(43)− gθθ∂LM

∂gθθ− gϕϕ

∂LM

∂gϕϕ

),

gr r(Gr

r +Gϕϕ

)

(44)= 16πGm

f

(LM − grr

∂LM

∂gr r− gϕϕ

∂LM

∂gϕϕ

),

gr r(Gr

r +Gθθ

)

(45)= 16πGm

f

(LM − grr

∂LM

∂gr r− gθθ

∂LM

∂gθθ

),

to obtain 3 differential equations for the metric func-tionsf , l andm, which are diagonal with respect to theset of derivatives (f,r,r ,m,r,r , l,r,r , l,θ,θ , l,r,θ ). With ananalog rescaling as in (15), the set of 10 partial differ-ential equations again depends only on the three fun-damental coupling constants introduced in (16).

At the origin, the boundary conditions read (with∂x = 1

ev∂r ):

∂xf (0, θ) = ∂xl(0, θ)

= ∂xm(0, θ) = 0,

(46)∂xψ(0, θ) = 0,

Hi(0, θ)= 0, i = 1,3,

Hi(0, θ)= 1, i = 2,4,

(47)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,

(48)ψ(∞, θ) = 0,

Hi(∞, θ) = 0, i = 1,2,3,4,

(49)φ1(∞, θ)= 1, φ2(∞, θ)= 0.

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

(50)H1 = H3 = φ2 = 0

and

∂θf = ∂θm = ∂θ l = ∂θH2

(51)= ∂θH4 = ∂θφ1 = ∂θψ = 0.


4.1. Numerical results

Constructing the axially symmetric solutions nu-merically,3 we were mainly interested whether boundmultimonopoles are possible in this model and if so,how the strength of attraction compares to that in [8].

4.1.1. The α = 0 limitSince gravity itself is known to be attractive, we

first studied the influence of the dilaton alone. Con-structing then= 2 solutions, we obtain the same qual-itative feature as for then = 1 case. The configurationsexist for all values ofγ , tending to the vacuum solu-tion H1 = H3 = φ1 = φ2 = ψ ≡ 0, H2 = H4 ≡ 1 forγ → ∞. Again, we find thatψ(0) � 0 and that thecurve forψ(0), starting from zero atγ = 0, developsa maximum atγ = 1.38 (which coincides with the cor-responding value forn = 1). From there it monotoni-cally decreases to zero forγ → ∞.

In Fig. 4, we show the difference between the massof then= 1 solution and the mass per winding numberof the n = 2 solution ;E = E(n = 1) − E(n =2)/2. The sign of;E indicates whether like chargedmonopoles attract or repel each other, the modulus of;E indicates the strength of attraction and repulsion,respectively. We find that for all values ofγ wehave studied, the monopoles are in an attractive phaseand that with increasingγ the strength of attractiongrows. For the model studied in [8], the maximal;E

was found to be;Emax ≈ 0.0082 atγ = 1.2. Fromthere;E decreases since the spherically as well asthe axially symmetric monopoles are tending to anessentially Abelian, spherically symmetric solution inthe limit of critical coupling. In the model studiedhere,;E is smaller for the same values ofγ (e.g.,we find that;E(γ = 1.2) ≈ 0.0069), but since nocritical coupling exists,;E grows with increasingγ .For, e.g.,γ = 5.0, we find;E ≈ 0.0167, which ismore that twice the maximal possible value for themodel in [8]. For large enough values ofγ , ;E shoulddecrease, because both the spherically and axiallysymmetric solution are tending to the vacuum. Sincethe absolute value of the mass is decreasing to zero,the same should happen for;E.

3 To solve the set of partial differential equations, we used thenumerical routine used and described, e.g., in [15].

Fig. 4. The difference between the mass of then = 1 solu-tion and the mass per winding number of then = 2 solution;E = E(n = 1) − E(n = 2)/2 is shown as function ofγ for thefirst (circles) and second (squares) branch arising in the EYMHDsystem in the Volkov limitα2 = 3γ 2. Also shown is;E for theYMHD solutions (triangles). Note that for the EYMHD solutions,theγ -axis represents a rescaledα-axis.

4.1.2. The Volkov limit α2 = 3γ 2

Studying then = 2 axially symmetric monopoles,we find the same qualitative features as in the spheri-cally symmetric case. We obtain:

(52)α(12)max = 1.279, α

(23)min = 0.295.

These values are slightly bigger and smaller, respec-tively, than the corresponding values for then = 1 so-lution. It has already been observed previously, thatthe branches for then = 2 solutions exist for highervalues of the maximal coupling [15]. We find here thatan analog holds true for the minimal coupling, namely,that then = 2 solutions exist for smaller minimal val-ues than in then = 1 case.

The behaviour ofψ(0) is completely analog to then = 1 case: starting from zero atα = 0, it increasesto a maximal value ofψ(0) ≈ 0.178 atα ≈ 1.0, thendecreases to zero atα ≈ 1.19 and reachesψ(0) ≈−5.324 atα(23)

min = 0.295. The dilaton function, whichhas only a weak angle-dependence, is positive andmonotonically decreasing on the first branch andstarts to form a negative valued minimum on thesecond branch. Proceeding on the second branch, the


solutions are less and less angle dependent and themaximum of the functionH1, H3 andΦ2 decreases.We did not manage to construct further brancheswith our numerical routine (since these branches arerather small), but believe that in analogy to then = 1solutions, then = 2 solutions develop a double zero,which is now (because of the choice of isotropiccoordinates) located atx = 0.

Comparing;E for the model studied here and forthat in [8], we find that the different values do notdeviate very much from each other. However, in themodel studied in [8] the parameter space, in whichsolutions exist, is limited by

√α2 + γ 2 ≈ 1.4. This

means that in the Volkov limit solutions only exist forα � αmax ≈ 1.21. Here, the solutions exist for biggervalues ofα = √

3γ and thus the maximum of;E isslightly bigger with;Emax ≈ 0.011. This is reachedon the first branch of solutions as demonstrated inFig. 4. The second branch starts from a value of;E

well below the corresponding one on the first branchand from there decreases. For all values ofγ = α/

√3,

the value of;E on the first branch exceeds the valueof ;E on the second branch.

5. Conclusion and summary

We have studied spherically and axially symmetricsolutions of SU(2) EYMHD theory. The specificcoupling between the dilaton and the Higgs field inthis model arises from a (4+ 1)-dimensional EYMmodel studied recently in [12]. We find that in the flatspace limit (α = 0), the solutions exist for all valuesof γ tending to the vacuum solution forγ → ∞.This contrasts to the situation in the usual YMHDmodel studied in [7], where the solutions only existfor γ � γmax tending to an Abelian solution in thelimit of critical coupling γcr. In both models, thedilatonic monopoles are in an attractive phase. Sinceno critical coupling exists in the model studied here,much stronger bound monopoles are possible.

Apparently, unlike the YMHD model with theusual coupling, the YMHD system studied here doesnot have analog features than the EYMH system.Moreover, the relation between thett-component ofthe metric and the dilaton found in [11], does not exist.

When in addition gravity is coupled, our equationsagree with those in [12] for a specific relation be-

tween the gravitational couplingα and the dilaton cou-pling γ , namelyα2 = 3γ 2. Studying the sphericallyand axially symmetric solutions of these equations inthis limit, we observe the same spiraling behaviour ofthe parameters found in [12]. Several branches existdiffering from the situation in the model with the usualdilaton coupling studied in [11], where maximally twobranches exist. Like in [8] the dilatonic monopoles canform bound multimonopole states.

In [11], non-Abelian black holes of the EYMHDmodel were studied giving further evidence that the“No-hair” conjecture does not hold in models in-volving non-Abelian fields. It would be interesting toconstruct the analogs in this system, especially theaxially symmetric non-Abelian black holes. Their ex-istence would show that Israel’s theorem cannot be ex-tended to EYMHD theory, like it has previously beenobserved in EYMH theory [15].

Finally, since in the low energy effective action ofstring theory, further corrections to the equations ofstandard physics, such as higher order corrections togravity in form of Gauss–Bonnet terms or new fieldslike the axion field arise, it would be interesting tostudy their influence on the solutions constructed here.

Acknowledgement

B.H. was supported by the EPSRC.

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Dilatonic monopoles from (4+1)-dimensional vortices