Remarks on the interaction between Born-Infeld solitons

arXiv:hep-th/0203106v3 16 Apr 2002DCPT-02/19

RemarksontheinteractionbetweenBorn-Infeldsolitons

YvesBrihaye∗

Facult´edesSciences,Universit´edeMons-Hainaut,B-7000Mons,Belgium

BettiHartmann†

DepartmentofMathematicalSciences,UniversityofDurham,DurhamDH13LE,U.K.

(February1,2008)

Abstract

WeconsidertheAbelianHiggsmodelaswellastheSU(2)Georgi-Glashow

modelinwhichthegaugefieldactionisreplacedbyanonlinearBorn-Infeld

action.Westudysolitonsolutionsarisinginthesemodels,namelythevortex

andmonopolesolutions,respectively.Weconstructformulaswhichprovide

goodapproximationsforthemassoftheBorn-Infelddeformedsolitonsusing

onlythedataoftheundeformedsolutions.Theresultsobtainedindicatethat

intheself-duallimit,theBorn-Infeldinteractionleadstoboundvortices,

whileformonopolesitgivesrisetorepulsion.

PACSnumbers:11.10Lm,11.27.+d,14.80.Hv

TypesetusingREVTEXI.INTRODUCTION

Inrecentyearsitbecameapparentthatwhenstudyinglowenergyeffectiveactionsof

stringtheory,thepartoftheLagrangiancontainingtheabelianMaxwellfieldstrengthtensor

anditsnon-abeliancounterpartinYang-Millsfieldtheories,respectively,mustbereplaced

byacorresponding(resp.abelianandnon-abelian)Born-Infeldterm.ThisideaisnotnewandwasfirstsuggestedbyBornandInfeldinthe1930s[1]togetridofsingularities

associatedwithpoint-likechargesinelectrodynamics.Asubtlepointinthegeneralisation

tonon-abelianBorn-Infeld(NBI)theoryishowtospecifythetraceoverthegaugegroup

generators.Fromtheviewpointofstringtheorythesymmetrizedtrace[2]seemsmore

favourablethantheordinarytrace.Ingeneral,theLagrangianinvolvingasymmetrizedtrace

isonlyknowninaperturbativeexpansion[3].Recently,however,anexplicitexpressionof

theSU(2)NBIactionforstatic,sphericallysymmetric,magneticconfigurationsinvolvinga

symmetrizedtracewasconstructed[4].Itwasfoundthatthequalitativeresultsareingood

agreementwiththeonesobtainedpreviously[5]usingtheordinarytrace.

Anumberofclassicalfieldtheorymodelhavebeenstudiedwithrespecttotheeffectsof

theBorn-Infeld(BI)interaction.ItwasfoundthatinSU(2)Yang-Millstheorythepresence

oftheBorn-Infeldtermleadstotheexistenceofparticle-likesolutions,socalled”glueballs”

[5],whichareabsentinthestandardcase(i.e.withoutBIinteraction).Thereasonforthisis

thatsimilartogravityintheEinstein-Yang-Millsmodel[6],theBorn-Infeldtermbreaksthe

scaleinvarianceandthusadmitsfiniteenergysolutions.Formodelsinwhichsoliton-type

solutionsalreadyexistinthestandardcase,itisofinteresttostudytheBIdeformation

oftheseconfigurations.ThishasbeendoneforanumberofmodelsincludingtheAbelian

Higgsmodel[7]andtheSU(2)Georgi-Glashowmodel[8].

Both,theAbelianHiggsmodel[9]andtheGeorgi-Glashowmodel[10]admitsoliton

solutionscharacterizedbyanintegeroftopologicaloriginandpossessaso-called”self-dual”

(BPS)limitforaspecificvalueoftheHiggsself-couplingconstant.Inthislimit,thesecond

orderEuler-Lagrangeequationscanbereplacedbyasetoffirstorder(”self-dual”)equations

andthemassofthen-solitonsolutionisgivenbyM(n)=nM(n=1)indicatingthatno

interactionbetweenthesolitonsexists[11].Therepulsionduetothelongrangegaugefieldis

exactlycompensatedbytheattractionoftheHiggsfieldwhich,becauseofitsmasslessness,

isalsolongrange.

AnaturalquestionarisinginthestudyoftheBorn-Infeldinteractioniswhetheritcan

leadtoattractionbetweenvorticesandmonopoles,respectively.Inthecaseofvortices,it

wasfoundthatintheself-duallimittheBorn-Infeldinteractionleadstoattraction[7].For

monopolesthequestionhasnotbeansweredyetandoneoftheaimsofthispaperisto

giveanindicationofwhethertheBorn-Infeldinteractionsimilartogravity[12]canleadto

attraction.

Herewepresentaformulawhichprovidesagoodapproximationfortheenergyofthe

BI-deformedsolitonsusingonlythedataofthestandardcase.InSectionII,westudythis

approximationfortheAbelianHiggsmodel,insectionIIIfortheGeorgi-Glashowmodel.

WegiveourconclusioninsectionIV.

2II.THEABELIANHIGGSMODEL

TheLagrangianoftheAbelianHiggsmodelreads[9]:

L=−1

2DµΦDµΦ∗−λ

e(n−P(ρ))dϕ(2)

Theself-dualcaseisgivenforα=e2

4FµνFµν−→β2󰀄

1−󰀋

2β2FµνFµν−1

1+1

16β4(Fµν˜Fµν)2≈

1+1

32β4[(FµνFµν)2+(Fµν˜Fµν)2]±O(β−6)(4)

Forβ2→∞,thestandardexpressionoftheU(1)theoryonthelhsof(3)isrecovered.In

thelimitofstaticandpurelymagneticsolutions,thepartcontainingthedualfieldstrength

tensor˜Fµνvanishes.

Inthispaper,weareinterestedinstatic,finiteenergysolutionsofthefullBIequations

withclassicalmassdenotedbyMbi(β2).Using(4),weobtainthefollowingapproximation

forthismass:

Mbi(β2)≈M0−1

β2∆bi(5)

Both,themassofthestandardsolitonM0andthefieldstrengthtensorsF(0)jkarecomputed

fromthestandardequations.Inotherwords,weapproximatethemassoftheBI-solitonby

theβ−2-correctionoftheBILagrangiandensityusingonlythesolutiondataofthestandard

U(1)theory.

Fortheself-dualcase,thefirstorderequationsread:

f′=Pf