Fate of the Born-Infeld solution in string theory

G.Karatheodoris,A.PinzulandA.Stern

arXiv:hep-th/0211033v5 1 Feb 2003DepartmentofPhysics,UniversityofAlabama,

Tuscaloosa,Alabama387,USA

ABSTRACT

WearguethattheBorn-InfeldsolutionontheD9−braneisunstableunderinclusionofderivativecorrectionstoBorn-Infeldtheorycomingfromstringtheory.Morespecifically,wefindnoelectrostaticsolutionstothefirstordercorrectedBorn-InfeldtheoryontheD9−branewhichgiveafinitevaluefortheLagrangian.

1

TheBorn-Infeld(BI)nonlineardescriptionofelectrodynamicsisofcurrentinterestduetoitsroleasaneffectiveactionforDp-branes.[1],[2]Oneofitsmainfeaturesistheexistenceofamaximumallowedvalue|E|maxforelectrostaticfields.∗Inthe30′s,BornandInfeldpromotedthistheorybecauseithasasphericallysymmetricsolutionwithfiniteclassicalself-energy.[3]ItagreeswiththeCoulumbsolutionatlargedistances,buthasafinitelimitfortheradialcomponentoftheelectricfieldattheorigin.Thislimitisjust|E|max.Asaresult,thevectorfieldissingularattheorigin.Equivalently,thesolutionisthusdefinedonaEuclideanmanifoldwithonepoint(correspondingtothe‘source’)removed,despitethefactthattheenergydensityiswellbehavedatthepoint.Thetheorycontainsonedimensionfulparameter(namely,|E|max),whichwasdeterminedbyBornandInfeldafterfixingtheclassicalself-energywiththeelectronmass.FortheDp-branethedimensionfulparameteristhestringtension,andsoanyBI-typesolution(orBIon[4])appearingthereshouldcorrespondtoachargedobjectwithcharacteristicmassatthestringtension.ThefactthatenergeticsadmitsvectorfieldsthatarenoteverywheredefinedindicatesthatBItheorycannotbeacompletedescription.[5]Moreover,instringtheory,theBIeffectiveactionisonlyvalidforslowlyvaryingfields,i.e.itisthelowestorderterminthederivativeexpansionforthefulleffectiveDp-braneaction.Sincederivativesofthefieldsarenot‘small’intheinterioroftheBIsolution,thevalidityofsuchasolutioninstringtheorycanbequestioned.ItisthereforeofinteresttoknowwhetherornotanaloguesoftheBIsolutionssurviveforthefulleffectiveDp-braneaction.AderivativeexpansionhasbeenrecentlycarriedouttoobtainlowestorderstringcorrectionstotheBIactionforthespace-fillingD9-brane.[6],[7]Withtherestrictiontoelectrostaticconfigurations,wefindthattheBIsolutionontheD9-braneisunstablewiththeinclusionofsuchcorrections,indicatingthatsuchsingularfieldconfigurationsmaynotfollowfromthefulleffectiveaction.Thereasonisbasicallyduetotheresultthatderivativecorrectionsmakeitdifficultfortheelectrostaticfieldtoattainitsmaximumvalue|E|max.HereweonlyallowforfieldconfigurationsthatleadtoafinitevaluefortheLagrangian.†Thisisareasonablerequirementfromthepointofviewofthepathintegral,whereoneexpectstorecovertheclassicalsolutionsintheWKBapproximation.Moreover,wefindnonontrivialelectrostaticsolutionsontheD9-braneD9-braneassociatedwithafiniteLagrangian.

Thesituationhereisincontrasttoskyrmionphysics,wheretherearenonontrivialsolutionstothezerothordereffectiveactionforQCD.Higherorderderivativecorrections,liketheSkyrmeterm,arenecessarytostabilizetheskyrmion.Ontheotherhand,BIonsappearatlowestorder,butbecomeunstableuponincludingthenextorderelectrostaticcorrections.Thereremainsthepossibility,however,ofstabilizingtheBIonwiththeinclusionofotherdegreesoffreedom.Forexample,ifweallowformagneticeffectsonemightfindthatthehigherordercorrectionsgiveamagneticdipolemomenttotheBIon.Anotherpossibilitywhichisofcurrentinterest,concernsBI-typesolutionsthatappearafterdimensionalreduction.Inthis

case,theBIactionisreplacedbytheDirac-Born-Infeldaction(DBI),containingdegreesoffreedomassociatedwiththetransversemodesofthebrane.ClassicalsolutionstotheDBIactionwerefoundin[4],[8],[9],andtheyrepresentfundamentalstringsattachedtobranes.‡Itisofinteresttoexaminehowsuchsolutionsareaffectedbyderivativecorrections[6].(Onesetofsolutions(theBPSsolutions)werefoundtobeunaffectedtoallorders[10].)Wehopetoaddresstheseissuesinfutureworks.

WebeginwithareviewoftheBorn-Infeldelectrostatics.TheBIactionisexpressedintermsofthedeterminantofthematrixwithelements

hµν=ηµν+(2πα′)Fµν,

(1)

whereFµν=∂µAν−∂νAµisthefieldstrengthandoneassumestheflatmetricηµν.OntheD9−braneitisgivenby

SBI=

(0)

1

(0)

−det[hµν],(2)

wheredis9.IntheabsenceofmagneticfieldsLBIsimplifiesto

󰀈

(0)

LBI=1−

.(5)rd−1

TheintegrationconstantQisthecharge.(5)approachestheCoulombsolutionwhenr→∞.Whenr→0,θ→π

AsnosuchinterpretationispossiblefortheBIonontheD9-brane,itisconvenientthatwefindittobeunstable.

‡

3

=ΠiF0i+

󰀈

−det[hµν](hi0−h0i),

(7)

µ

arethemomentaconjugatetoAiandhµνhνρ=δρ.Asusual,themomentumconjugatetoA0isconstrainedtobezero.Intheabsenceofmagneticfields,(7)reducesto

󰀡π=

󰀡Π

󰀈

1+󰀡π2−1−(2πα′)∂iπiA0

=

1i󰀡−1−(2πα′)∂iπA0,

1−f

2whereweintegratedbyparts.ThecoefficientofA0givestheGausslawconstraint.remainingtermscanbeusedtoidentifytheself-energyoftheBIsolution

d−1

E(0)BI

=

Ω(4π2α′)5gs

󰀆

d10

x󰀄1−

󰀈4

∆

󰀋󰀇

,

∆=hµνhρσhαβhγδ(SνραβSσµγδ−2SβγνρSδασµ),

whereκ=

(2πα′)2

(9)

The

(12)

undersuchdiffeos.§WeavoidthiscomplicationandworkinCartesiancoordinatesxi.For

󰀡(󰀡󰀡(󰀡electrostaticfieldsEx)=fx)/(2πα′),

Sijk0=−Sij0k=∂i∂jfk+

󰀡21−f

fℓ

(∂ifk∂jfℓ−∂ifℓ∂jfk)

(13)

Thenexploitingthesymmetryoftheindiceswecanwrite

−1

2

SijkℓSijkℓ+

1

tanθ

󰀂

H′2+

r4d−1

cos4θ

cos3θ

󰀉

d−1H′2+

κ

󰀉d−1′rH

(Q−rd−1tanθ)cos2θ=2

−d−1

cos3θ

2

WenotethatbyrescalingrandQwecansetκequaltounity.Also(17)isinvariantunderQ→−Qandθ(r)→−θ(r),andthisgivesaprescriptionformappinganypossiblecharged

󰀂󰀃󰀇

rd−5sinθ126−53d+85r2H2+4cos2θ(5+d−3r2H2)+cos4θ(d−2−r2H2)(17)

solutiontotheanti-solution.Thelefthandsideof(17)vanishesfortheoriginalBIsolution(5),whiletherighthandsiderepresentsderivativecorrections.Substituting(5)intothelefthandsidegivesavanishinglysmallcorrectionasr→∞,butitissingularforr→0.TheBIsolutionthereforecannotbetrustedneartheorigin.

Duetothepresenceofhighordertimederivativesinthecorrectiontermsintheaction(12),thecomputationoftheHamiltonianforthesystem,andconsequentlythecorrectiontotheelectrostaticenergy(10),isproblematic.Forthisreasonweonlyrequiresolutionsto(17)tobeassociatedwithafinitevaluefortheLagrangian(15)insteadoftheenergy.SincetheBI(actually,Coulumb)solutionisvalidatlarger,

θ(r)→

Q

(19)

whereH′→−(logǫ)′′andweagainsetκequaltoone.Takingd=9andQ>0,onehasthefollowingsolutionneartheorigin

󰀉

732r45

,asr→0(20)ǫ(r)→

6

ǫ3

󰀋′′󰀄󰀇

+3rd−1H′2−(d−1)8(rd−3H)′+rd−5[13−7d+12r2H2],

ItiseasytocheckthatitgivesafinitecontributiontotheLagrangianasr→0.(Forthisoneonlyneedsthatǫ(r)goestozeroslowerthanr5/3.)Ontheotherhand,afternumericallyintegratingthissolutionstartingfromsomeinitialvaluer0toincreasingvaluesofrwefindthatforanyvalueofQ,ǫgoesquicklytoπ/2or−π/2,andsotheLagrangiandensityispoorlybehavedforlargevaluesofr.BelowweplottheresultsforQ=1:

1200

1.5705

0.000050.00010.000150.00021.56951.5691.5685

1000800600400200

0.000050.00010.000150.0002fig.3θ(r)vs.r

fig.4r8(LBI+LBI)vs.r

(0)(0)

Fromfig.3,θ′>0totherightoftheminimum,andsonomatchwithfig.1ispossiblethere.Totheleftoftheminimum,θ′tendsto−∞asr→0accordingto(20),whilefromfig.1itappearstovanishinthelimit.Tomakemattersworse,thegraphsweobtainforθneartheoriginarehighlysensitivetotheinitialvaluer0oftheintegration,andthispersistsforallvaluesofQ.Wehavecheckedthatthisisnotduetotheneglectofhigherordertermsin(20).Sonotonlyistherenoagreementbetweenthetwonumericalintegrationprocedures,thevalidityofthesolutionneartheoriginisquestionable.Wearethusunabletofindanysphericallysymmetricchargedsolutionsto(17)consistentwiththerequirementofafiniteLagrangian.

WealsofindnoaxiallysymmetricchargedsolutionswithfiniteLagrangiandensityevery-where.Again,theycorrespondtod<9,andatzerothorderwerecharacterizedbya(9−d)

󰀡issingular.Using(18)wecanonceagainnumerically-dimensionalEuclideansurfacewheref

integratetosmallvaluesofr.Theconclusionsarethesameasford=9.ForQ>0,θ(r)tendstothelimitofπ/2asr→0,andthecorrespondingradialLagrangiandensitydivergesasr→0(fasterthan1/r).ForQ>0andd≥6,thesolutionto(19)neartheoriginhastheform󰀉d−5

5Cr

ǫ(r)→,asr→0,(21)whereC=20008,33324,51836ford=6,7,8,respectively.Aswiththecased=9,itis

consistentwiththerequirementofafiniteLagrangianforsmallr,buttheLagrangiandensitydivergesafternumericallyintegratingtoincreasinglarger,andtheproblemofthesensitivitytotheinitialvaluer0persists.Soagainwefindnoagreementbetweenthetwonumericalintegrationprocedures.Ford<6,wefindnosolutionsto(19)neartheoriginsatisfyingtherequirementofafiniteLagrangian.

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Abovewefoundnegativeresultsforallconfigurationsassociatedwithsomenonzerovalueforthecharge.AlthoughtherearenoQ=0solitonsolutionsofthezerothorderBorn-Infeldsystem,thisneednotbeaprioritrueathigherorders.Inthiscase,θ(r)foranysphericallysymmetricsolutionwouldnotfalloffasapowerasr→∞,butrather

󰀄󰀇−αr

e

θ(r)→Acosαr+Bsinαr

[8]C.G.CallanandJ.M.Maldacena,Nucl.Phys.B513,198(1998).[9]P.S.Howe,N.D.LambertandP.C.West,Nucl.Phys.B515,203(1998).[10]L.Thorlacius,Phys.Rev.Lett.80,1588(1998).

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