3 January 2002

Physics Letters B 524 (2002) 227–232www.elsevier.com/locate/npe

The Born–Infeld sphaleron

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 31 October 2001; received in revised form 20 November 2001; accepted 21 November 2001Editor: P.V. Landshoff

Abstract

We study theSU(2) electroweak model in which the standard Yang–Mills coupling is supplemented by a Born–Infeld term.The deformation of the sphaleron and bisphaleron solutions due to the Born–Infeld term is investigated and new branches ofsolutions are exhibited. Especially, we find a new branch of solutions connecting the Born–Infeld sphaleron to the first solutionof the Kerner–Gal’tsov series. 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

While in the pure Yang–Mills (YM) theory noparticle-like, finite-energy solutions are possible, theymight exist as soon as scale invariance breaking termsappear in the Lagrangian. These can be of differenttypes. The possibility discovered first is the couplingof the YM system to a scalar Higgs field. This breaksthe scale invariance and particle-like solutions exist:

(a) monopoles [1], when the Higgs field is given inthe adjoint representation ofSU(2);

(b) sphalerons [2], when the Higgs field is given in thefundamental representation ofSU(2).

While these solutions were studied extensively sincetheir discovery, the coupling of gravity to YM theorieswas rather neglected. It was believed that gravity wasto weak to have important effects on these solutions.The interest only rose when Bartnik and McKinnon[3] found a particle-like, finite-energy solution of thecoupled Einstein–YM system.

E-mail address: betti.hartmann@durham.ac.uk (B. Hartmann).

Recently, it was discovered that superstring the-ory induces important modifications for the standardquadratic YM action. Likely the relevant effective ac-tion is of the Born–Infeld type [4] and, naturally,a scale invariance breaking term is involved. Thiswas exploited to construct classical “glueballs” in theSU(2) Born–Infeld theory [5].

Considering the classical equations of non-abelianfield theories coupled to various other fields leads—depending on the coupling constants—in general to arich pattern of solutions involving several branches re-lated by bifurcations and/or connected by catastrophe-like cusps [6–8].

In this Letter, we investigate the effect of a Born–Infeld term on the sphaleron and bisphaleron solutionsarising in theSU(2) part of the electroweak model andwe find that these solutions are smoothly deformed upto a critical value of the Born–Infeld parameterβ2

BI .From there, another branch of solution exists whichconnects the sphaleron branch to the first solution ofthe series obtained in [5], while the bisphaleron branchbifurcates with the sphaleron branch at a value ofβ2

BIdepending on the Higgs mass.

0370-2693/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0370-2693(01)01381-8

228 Y. Brihaye, B. Hartmann / Physics Letters B 524 (2002) 227–232

The model, the notations and the ansatz are given inSection 2, we discuss some special limiting solutionsin Section 3 and present and discuss our numericalresults in Section 4. Finally, in Section 5 we commenton the stability of the solutions.

2. SU(2) Born–Infeld–Yang–Mills–Higgs(BIYMH) theory

Neglecting theU(1) part (for technical reasons),we consider the Lagrangian of theSU(2) part ofthe electroweak model in which the term of theLagrangian containing the field strength of the Yang–Mills (YM) fields is replaced by the correspondingBorn–Infeld (BI) term. The Lagrangian density reads:

(1)

L = β2BI(1−R)+Dµφ

+Dµφ

− λ

4

(φ+φ − v2

2

)2

,

(2)

R =(

1+ 1

2β2BI

FaµνF

aµν

− 1

16β4BI

(FaµνF̃

aµν)2)1/2

,

where βBI is the coupling of the BI term and hasdimensionL−2. In the limit β2

BI → ∞, the standardSU(2) electroweak Lagrangian is recovered. We studyclassical, spherically symmetric, static solutions withfinite energyE(β2

BI, v2, λ). A suitable rescaling of the

space variable�x leads to a rescaling of the energy asfollows:

(3)E(β2

BI, v2, λ

)= vE

(β2

BI

v4 ,1, λ

)(4)=√

βBI E

(1,

v2

βBI, λ

).

We are using the standard spherically symmetricansatz of [6,7]

Aa0 = 0,

Aai = 1− fA(r)

grεaij x̂j + fB(r)

gr(δia − x̂i x̂a)

+ fC(r)

grx̂i x̂a,

(5)φ = v√2

[H(r)+ iK(r)(x̂aσa)

](0

1

),

where fA(r), fB(r), fC(r), H(r) and K(r) arefunctions of the radial coordinater only and x̂a ≡ra/r. It is well known (see, e.g., [6,7,9]) that thisansatz is plagued with a residual gauge symmetry,which we will fix here by imposing the axial gauge:

(6)xiAi = 0 ⇒ fC(r) = 0.

Since we consider static solutions here and due tothe choiceAa

0 = 0, the term including the dual fieldstrength tensor in (2) vanishes. The masses of thegauge boson and Higgs boson, respectively, are givenby:

(7)MW = 1

2gv, MH = √

λv.

With the rescaling

(8)x =MWr,

the energyE depends only on two fundamentalcouplings:

β2 ≡ β2BI

(MW)4= 16

β2BI

g4v4,

(9)ε ≡ 1

2

(MH

MW

)2

= 2λ

g2

and is given as 1-dimensional integral over the energydensity E (the prime denotes the derivative withrespect tox):

(10)E(β2, ε

)= 4π

g2 MW

∫dx E,

E = β2x2

(√√√√1+ 2

f ′2A + f ′2

B + (f 2A+f 2

B−1x

)2β2x2 − 1

)+ (

K(fA + 1)+HFB)2 + (

H(FA − 1)−KfB)2

(11)+ 2x2(H ′2 +K ′2)+ εx2(H 2 +K2 − 1)2.

The corresponding Euler–Lagrange equations canbe obtained in a straightforward way and have to besolved with respect to specific boundary conditionswhich are necessary for the solution to be regular andof finite energy. In [6,7] it was found that two possiblesets of boundary conditions exist. After an algebra,it can be shown that these two sets of conditionsalso hold when the Yang–Mills part of the action isreplaced by the Born–Infeld term. The first of thesesets was used to construct the solution of [2]. Two

Y. Brihaye, B. Hartmann / Physics Letters B 524 (2002) 227–232 229

of the radial functions vanish identically:fB(x) =H(x)= 0 and the other two have to obey:

fA(0)= 1, fA(∞)= −1,

(12)K(0)= 0, K(∞)= 1.

For the second set, all functions are nontrivial and theirboundary conditions at the origin read:

fA(0)= 1, fB(0)= 0,

(13)H ′(0)= 0, K(0)= 0,

while in the limit x → ∞, the functions have toapproach constants in the following way:

limx→∞

(fA(x)+ ifB(x)

)= exp(2iπq),

(14)limx→∞

(H(x)+ iK(x)

)= exp(iπq).

The solutions of this second type are thus character-ized by a real constantq ∈ [0,1[ . This parameter hasto be determined numerically and depends onε andβ .

3. Limiting solutions

Two parameter limits of the model studied here areof special interest and have been studied previously:

(a) β2BI = ∞. In this limit, the standardSU(2) elec-

troweak Lagrangian is obtained. Classical, finite-energy solutions were found for the two differentsets of boundary conditions discussed above. Thecase in whichfB andH vanish is the sphalerondiscovered by Klinkhamer and Manton [2]. It hasenergy of the form

(15)Esp = MW

αWI (ε), αW = 4π

g2 ,

whereI (ε) has to be determined numerically. Itwas found [2,10] that

I (0)� 3.04, I (0.5)� 3.64,

(16)I (∞)� 5.41.

For sufficiently large values ofε (i.e., ε > εcr ≈72), solutions of the second type appear [6,7]: thebisphalerons which occur as a pair of solutionsrelated by a parity operation. Their classical en-ergy is lower than the energy of the sphaleron(e.g.,Ibi(ε = ∞)≈ 5.07). In the limitε → εcr , the

functions describing the bisphaleron converge in auniform way to the functions of the correspondingsphaleron.

(b) v = 0. In this case, the Higgs field decouplesand we are left with the pureSU(2) Born–Infeldtheory which was studied recently in [5]. Whilethe pure YM system does not admit finite-energysolutions, the presence of the BI term breaks thescale invariance and finite-energy solutions, so-called “glueballs”, exist. These are indexed by thenumbern of nodes of the functionf (n)

A (r). The

first element of the sequence,f(1)A (r), has exactly

one node and with the convention adopted in [5]the energy (in units of 4π ) is given by

(17)EKG =E(1,0,0)≡M.

It was found [5] that, for the first solution of thesequence,M � 1.13559.

4. Numerical results

We have studied numerically the classical equationsfor several values of the parametersβ2

BI , v, λ. Bytaking advantage of the scaling (3) and (4), we fixedv = 1 without loosing generality. These solutionsare therefore essentially characterized by the classicalenergyE(β2

BI,1, λ) =E(β2, ε).

4.1. BI-sphaleron

Starting from the sphaleron solution (β2 = ∞), weconstructed a branch of smoothly deformed solutionsup to a critical valueβ2 = β2

c along which the energyslightly decreases.β2

c depends onε and is increasingfor increasingε which is demonstrated in Table 1.

No solution seems to exist forβ2 < β2c , but we

obtain strong numerical evidence that a second branchof solution exist forβ2 ∈ [∞, β2

c ]. For fixedβ2, theenergy of the solution of this second branch is higherthan the energy of the one of the first branch. Wetherefore refer to these branches as to the lower and

Table 1

ε 0.5 1 2 10 100 200

β2c 26.5 37.4 39.7 65.0 110.0 121.0

230 Y. Brihaye, B. Hartmann / Physics Letters B 524 (2002) 227–232

upper branch, respectively. For the upper branch, wefind

(18)limβ2→∞

E(β2, ε

)= ∞.

Fig. 1. The energyE is shown as a function of 1/β2 for ε = 0.5.

Fig. 2. The ratioE(1/β2)/β1/2 is shown as a function of 1/β2

for ε = 0.5. The horizontal solid line represents the energy of then= 1 KG solution in units of 4π with M � 1.135.

This can be understood as follows: on the lowerbranch,β2 → ∞ corresponds tov fixed and nonequalzero withβ2

BI → ∞, while on the upper branch thislimit corresponds toβ2

BI fixed andv → 0. Because ofthe special choice of the coordinates (8), the solutionon the upper branch shrinks to zero (see also Fig. 4),while its energy tends to infinity forβ2 → ∞. With anappropriate rescalingE(β2, ε) → (β)−1/2E(β2, ε),the mass of then = 1 Kerner–Gal’tsov (KG) solutionis recovered:

(19)M = limβ→∞β−1/2E

(β2, ε

).

For ε = 0.5, the energy of the solutions on bothbranches before the rescaling is illustrated in Fig. 1,the energy after the rescaling in Fig. 2.

In Fig. 3 we show the valuex0 of the radialcoordinatex, for whichf (1)

A attains its node. Clearly,this value tends to zero forβ2 → ∞ on the upperbranch. As already mentioned, this is due to the factthat the rescaled variablex shrinks to zero forv → 0.Also shown in Fig. 3 is the contributionEHiggs ofthe Higgs field energy to the total energy of the

Fig. 3. The valuex0 for which f(1)A

(x0) = 0 as well as thecontributionEHiggs of the Higgs field energy to the total energy of

the solution are plotted as functions of 1/β2 for ε = 0.5. The labelsL andU , respectively, refer to the lower and upper branch.

Y. Brihaye, B. Hartmann / Physics Letters B 524 (2002) 227–232 231

Fig. 4. The profiles of the solutions(f (1)A ,K) as function of

y = x/x0 are shown for several values ofβ2 and ε = 0.5. For

comparison, the profile of the functionf (1)A

of the n = 1 KGsolution is also shown.K(y)≡ 0 for the KG solution.

solution. While on the lower branch it stays finite forall values ofβ2, it tends to zero on the upper branchfor β2 → ∞. This supports the interpretation that onthe upper branch the Higgs field is trivial forβ2 = ∞.

Finally, we demonstrate the convergence of then = 1 solution to the corresponding Kerner–Gal’tsovsolution forε = 0.5 in Fig. 4. The profiles of the gaugefield and Higgs field functions are shown on the lowerbranch forβ2 = 100 and a valueβ2 = 27 close to thecritical β2

c . Clearly, the functions tend to that of theKG solution on the upper branch for increasingβ2. Inthe limitβ2 → ∞, fA tends to the gauge field functionof the KG solution, while the Higgs field functionKtends to zero on the full interval[0,∞[ .

4.2. BI-bisphaleron

For sufficiently high value of the Higgs-bosonmass (i.e., forε > 72), bisphaleron solutions canbe constructed. As in the case of the sphaleron,bisphalerons are smoothly deformed by the Born–Infeld parameter. By studying these solutions in detailfor ε = 100 andε = 200 and varyingβ2, we observed

Fig. 5. The valueH(0) ∝ |φ(0)| characterizing the bisphaleronsolution is shown as a function of 1/β2 for two values ofε.

that like the BI-sphalerons the deformed bisphaleronsexist up to a critical valueβ2 = β̃2

c . For ε = 100,the BI-bisphaleron branch merges into the lower BI-sphaleron branch atβ2 = β̃2

c . For still higher valuesof ε the pattern is slightly different. Another branchof BI-bisphaleron exist on the intervalβ ∈ [β̃2

c , β̂2]

and the solutions on this branch merge into theupper BI-sphaleron branch in the limitβ2 → β̂2.In both cases, all radial functions characterizing theBI-bisphaleron converge in an uniform way to theradial functions associated with the corresponding BI-sphaleron. These various phenomena are illustratedin Fig. 5 where the valueH(0) ∝ |φ(0)| is shownas function of 1/β2 for ε = 100 andε = 200. Forε = 100, this quantity clearly tends to zero forβ̃2

c =β̂2 ≈ 113.6. For ε = 200, we find thatβ̃2

c ≈ 108.8(with H(0) �= 0), while the bisphaleron bifurcateswith the sphaleron (withH(0) tending to zero) atβ̂2 ≈ 164. We further observed that for fixedβ2 andfor both lower and upper branches, the energy ofthe BI-bisphaleron is close, although lower, than theenergy of the BI-sphaleron. In the limitβ2 → β̂2, thetwo upper branches (if any) merge into a single onecorresponding to the BI-sphaleron upper branch.

232 Y. Brihaye, B. Hartmann / Physics Letters B 524 (2002) 227–232

5. Stability

Let us finally discuss the (in)stability of our solu-tions. When two branches of solutions do exist termi-nating into a catastrophe spike, like the ones shownin Figs. 1 and 2, it is widely believed that the numberof negative modes is constant along each branch andthat the number of negative modes on the branch withhigher energy exceeds the number of negative modeson the branch with lower energy by one unit. The rea-soning is based on catastrophe theory [12] and wasdemonstrated to hold in the context of classical solu-tions in various models (e.g., [8,11]).

For ε < 72, the lower branch corresponds to thedeformed electroweak sphaleron and thus possesses asingle direction of instability. Therefore, the solutionson the lowest branch likely have one direction ofinstability while the solutions on the upper branchpossess two. This is not in contradiction with theresult of [5] because these solutions, when embeddedinto a field theory with extra fields (here the Higgsfield), likely acquire extra unstable modes due to thesupplementary degrees of freedom.

For the values ofε > 72 we have considered here,the bisphaleron possesses one direction of instabilityon the lower branch while the sphaleron has two.On its upper branch (i.e., for̂β2 � β2 � β̃2

c ), theBI-bisphaleron (respectively BI-sphaleron) possessestwo (respectively three) directions of instability. Forβ2 � β̂2, only the upper branch of the BI-sphaleronexists and it likely possesses two unstable modes,similarly to the caseε < 72.

Finally, for ε > 9500, the situation is still moreinvolved [6,7]. Nevertheless, the BI-bisphaleron pos-sesses a single direction of instability on its lowerbranch and the above conclusions should apply, too.

6. Summary

We have studied the Born–Infeld deformation oftheSU(2) electroweak sphaleron and bisphaleron. Wefind that the solutions exist up to a critical value ofthe Born–Infeld couplingβ2

c (respectivelyβ̃2c ) which

depends on the Higgs mass parameter.The fact that classical solutions of Born–Infeld like

gauge field theories cease to exist at a critical value of

the parameterβ2 was observed in several models withabelian [13] and non-abelian [14,15] gauge groups.Especially for the case of BI-vortices, it was shownrecently [16] that the magnetic fieldB(0) of thesolution becomes infinite when the critical valueβ2

c

is approached.The situation here is slightly different. Indeed, the

solutions exist only forβ2 > β2c , but we constructed

numerically a second branch of solutions existing upto β2 = ∞. There the solution tends to the first so-lution of the KG series and—after an appropriaterescaling—the energy on this upper branch reachesthe mass of then = 1 KG solution in the limitβ2 → ∞. When BI-bisphaleron are present, the pat-tern is qualitatively similar: the BI-bisphaleron branchbifurcates from one of the BI-sphaleron branches (up-per or lower, depending of the value ofε). No BI-bisphalerons exist as an upper branch for sufficientlyhigh values ofβ2.

Sphaleron solutions can also be studied in modelswhere the Born–Infeld term is incorporated by using asymmetrized trace. The result of [17] suggests though,that no qualitative changes of the critical patternshould occur.

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