Two-Point Functions of Chiral Fields at One Loop in Type II

We compute the two-point functions for chiral matter states in toroidal intersecting D6-brane models. In particular, we provide the techniques to calculate Moebius strip diagrams including the worldsheet instanton contribution.

Two-PointFunctionsofChiralFieldsatOneLoopin

TypeII

arXiv:0805.1874v1 [hep-th] 13 May 2008

K.Benakli andM.D.Goodsell

LaboratoiredePhysiqueTh´eoriqueetHautesEnergies1,Tour24-25,5eme´etage,Boite

126,4placeJussieu,F-75252ParisCedex05France

Abstract

Wecomputethetwo-pointfunctionsforchiralmatterstatesintoroidalintersectingD6-branemodels.Inparticular,weprovidethetechniquestocalculateM¨obiusstripdiagramsincludingtheworldsheetinstantoncontribution.

1Introduction

Today’shighenergyphysicsfacesopenquestionssuchastheconstructionofaquan-tumtheoryofgravity,explainingbaryogenesis,therotationcurvesofgalaxies,theexpansionoftheuniverse,orunderstandingthegaugehierarchyproblem...ThesemotivateinvestigationsofmanyextensionsoftheStandardModelofparticlephysics,including,forinstance,supersymmetricandhigherdimensionalones.Thecorrespond-inge ective eldtheories ndinstringtheoryauniqueframeworkforanultravioletcompletion.There,thequantum eldtheorypeturbativeexpansionisreplacedbyasumoverworld-sheetsurfaces,eachorderbeingultraviolet niteinsupersymmetricvacua.

Becausetheycharacterisethee ectivelowenergytheory,thelowestdimensioncorrelationfunctionsareofspecialinterest.Itisimportant,forinstance,tounder-standtheirdependenceonthedi erentdataofthestringcompacti cation.Ofthese,thetwo-pointfunctionofscalarsφibelongingtochiralmultipletsΦigivesinformationabouttheK¨ahlermetric:

φi(k)

Φ

i

.(1.1)

Thecomputationofthesetwopointfunctionsisthesubjectofthiswork.

λ

Thechiral eldshaveacubicsuperpotentialW ijk

kbenakli@lpthe.jussieu.frgoodsell@lpthe.jussieu.fr1

Unit´eMixtedeRechercheduCNRSUMR7589

We compute the two-point functions for chiral matter states in toroidal intersecting D6-brane models. In particular, we provide the techniques to calculate Moebius strip diagrams including the worldsheet instanton contribution.

typeIIAitmayonlydependupontheK¨ahlerandopenstringmoduli.Theoneloopcorrelatorcanbesplitinto eldtheorypartandthresholdcorrection

(1)GΦI,

1ΦjGΦk,

16π2

GΦi,

16π2

,(1.4)

whereYijkisthephysicalYukawacoupling.Justasthresholdcorrectionsmodifythetree-levelgaugecouplings,socorrectionstotheK¨ahlerpotentialchangetheYukawacouplings(seeforexample[1])ofwhichtheoneloopcorrectionsmaybesigni cant.Moreover,theycanteachusabouttheexpectede ectsofsupersymmetrybreakingwhenmediatedbymoduli elds.

Below,wewillcarryoutthesecomputationsforthecaseoftoroidalorbifold/orientifoldcompacti cationfactorisableonthreetwotoriT2oftypeIIstrings(see[2]forreviewsofIIBmodelbuilding,and[3]forIIA,and[4]forrecentworkinthisarea).Thesemodelsprovideanexceptionallaboratoryinthisrespectastheyhaveasimplegeometricalpictureandtheyallowanexplicitcomputationthroughconformal eldtheorytechniques[5,6,7].Thechiralstatesareidenti edwiththemasslessmodesoftwotypesofopenstrings:(i)openstringswithbothendsonthesamestackofbranes;and(ii)openstringswithoneendonabranestackaandthesecondendona

κoneachtorusκ.di erentoneb,thetwostacksintersectingatnon-vanishinganglesθab

Toourknowledgetheone-looptwopointfunctionshavebeencomputedonlyfortheformercase(i)in[8,9,10].Thiswasrelatedtothecalculationofcorrectionstoclosedstringmoduli,asalsoin[11,12,13].Moreover,becauseoftheperseapplicationofmoduli eldsforthedeterminationofcoupling“constants”,supersymmetrybreakingorin ationarymodels,previousworksfocusedonthecontributionsfromtheN=2sectorsoforbifoldswherethemodulidependenceappears.Howeverthestatesoftype(ii)localisedatbraneintersectionsplayanimportantrole,asforexample,usuallythematterandtheHiggs eldsinphenomenologicalconsiderationsaremostoftenidenti edwiththem.Webelieveitisusefultoprovidethetoolsandexpressionstocomputeexplicitlytheone-looptwopointfunctionsforthesestates.Thecomputationproceedsinadi http://doc.guandang.netingthetechniquesof[14]fortwistoperators,thecomputationofthetree-levelcorrelationsofboundarychangingoperatorshavebeenintroducedforopenstringsin[15],successivelyused, rstwithfourinsertionswithorthogonalbraneintersectionsin[16],thenforgenericangles,theworldsheetinstantoncontributionswasfoundin[17]beforethefullCFTcomputationwasperformedin[18,19,20],and nallygenericN-pointfunctionswerecalculatedin[21].Later,buildingonthegeneralmethodsof[22],one-loopdiagramswithboundarychangingoperatorshavebeenconstructed[23,24].

We compute the two-point functions for chiral matter states in toroidal intersecting D6-brane models. In particular, we provide the techniques to calculate Moebius strip diagrams including the worldsheet instanton contribution.

Amplitudesinvolvingboundarychangingoperatorsaresensitivetothecompact-i cationspacedata(asthevacuumexpectationvaluesofthemoduli)astheyaresuppressedbycontributionsoftheworld-sheetinstantons.Fortree-levelamplitudesthisdependenceappearexplicitelyforthreeandhigherpointscorrelations.However,athigherloops,itisalreadypresentatthetwo-pointfunction.PartofthisdependenceisduetotheYukawacouplingsinequation(1.3),butitisinterestingtoinvestigatetheremainingmoduli-dependentparts.Forthecaseoftheannulusamplitude,somestepsincomputingthetwo-pointfunctionhavebeentakeninthisdirectionin[24].Althoughthemethodwasoutlined,thecomputationinthecasewheretheamplitudeinvolvedthreeboundarieswasnotperformed.Moreover,wedidnot ndasimilarcomputationfortheM¨obiusstripavailableintheliterature.Wewillpresentheretherelevanttechniquesandapplythemtogetexplicitresults.

Thepaperisorganisedasfollows.Insection2,wewillstartbycomputingtheone-looptwopointfunctioninthecaseofopenstringswithbothendsonthesamestackofD-braneswhenanorbifoldactionleadstoanN=1masslesstheory.Thiswillallowustoco …… 此处隐藏：19070字，全部文档内容请下载后查看。喜欢就下载吧 ……