SUSY Moose Runs and Hops An extra dimension from a broken deformed CFT

arXiv:hep-th/0607031v2 19 Jul 2006SUSYMooseRunsandHops:

AnextradimensionfromabrokendeformedCFT

JoshuaErlichandJongAnlyTan1

1DepartmentofPhysics,CollegeofWilliamandMary,Williamsburg,Virginia23187-8795

Wefindaclassoffourdimensionaldeformedconformalfieldtheorieswhichappear

extradimensionalwhentheirgaugesymmetriesarespontaneouslybroken.The

theoriesaresupersymmetricmoosemodelswhichflowtointeractingconformalfixed

pointsatlowenergies,deformedbysuperpotentials.Usinga-maximizationwegive

strongnonperturbativeevidencethatthehoppingtermsintheresultinglatticized

actionarerelevantdeformationsofthefixedpointtheories.Thesetheorieshavean

intricatestructureofRGflowsbetweenconformalfixedpoints.Ourresultssuggest

thatatthestablefixedpointseachofthebulkgaugecouplingsandsuperpotential

hoppingtermsisturnedon,infavoroftheextradimensionalinterpretationofthe

theory.However,wearguethatthehigherdimensionalgaugecouplingisgenerically

smallcomparedtothesizeoftheextradimension.Inthepresenceofabrane

thetopologyoftheextradimensionisdetermineddynamicallyanddependsonthe

numbersofcolorsandbulkandbraneflavors,whichsuggestsphenomenological

applications.TheRGflowsbetweenfixedpointsinthesetheoriesprovideaclassof

testsofCardy’sconjectureda-theorem.

I.INTRODUCTION

Thepossibilityofextraspatialdimensionsbeyondthethreewhichwecommonlyobserve

hasprovidedaprovocativeparadigmformodelbuildinginparticlephysicsandgravity

inrecentyears[1,2,3].However,duetothenon-renormalizabilityofhigher-dimensional

gaugetheoriestheyarenecessarilylowenergyeffectivetheories.Anultravioletcompletion

ofhigher-dimensionaltheorieswasprovidedbythedeconstructionapproach[4,5].Inade-

constructedextradimension,theextradimensionappearsasalatticeof(3+1)-dimensional

gaugetheoriesconnectedtooneanotherthroughlinkfieldstransforminginthebifundamen-

talrepresentationofthegaugegroupsatneighboringlatticesites.Fornon-supersymmetric2

gaugetheoriestheUVcompletionmaybeprovidedbyanasymptoticallyfreegaugethe-

oryinwhichfermionscondensetoprovidethelinkfieldsasnonlinearsigma-modelfields.

Alternatively,thelinkfieldscanarisefromfundamentalscalarfieldswhichobtainvacuum

expectationvaluesleavinganeffectivetheorywhichappearshigherdimensionalbelowsome

scale.However,scalarfieldssuffertheusualproblemthattheirexpectationvaluesarenat-

urallyoforderthecutoffscale,andsupersymmetry(SUSY)isthereforefavoredtorestore

naturalnessinthislinearsigmamodelapproach.

Inthispaperwestudyaclassofdeconstructedfive-dimensional(5D)supersymmetric

gaugetheories.Thelatticizedhigher-dimensionaltheoriescontainhoppingtermsinthe

Lagrangianwhicharethelatticizedkinetictermscontainingderivativesintheextradimen-

sions.Thescalarlinkfieldsobtainvacuumexpectationvalues(VEVs)whichbreakthe

productgaugesymmetrypartlyorcompletely.Thespectrumofgaugebosonsmimicsthe

Kaluza-Klein(KK)spectrumoftheextradimensionuptoascalewhichdependsonthe

sizeoftheextradimensionRandthelatticespacingl.Thescaleabovewhichthespectrum

ceasestobehigher-dimensionalisroughlyΛKK≡R2/lforaflatextradimension.Above

thatscalethetheoryisa4Dgaugetheorywithk∼R/lfactorsofthe5Dgaugegroup.There

aretwootherdimensionfulscalestokeepinmind:thedynamicalstrongcouplingscaleof

each4DgaugegroupfactorΛQCD,whichweassumeforsimplicitytobethesameforeachof

thegaugegroupsinthedeconstructedtheory;andtheultimatecutoffofthedeconstructed

theoryΛ4,whichmustbemuchhigherthanΛKKandΛQCDforthedeconstructedtheory

toprovideaUVcompletionofthe5Dtheory.

Flavorsinthebulkoftheextradimensionareaddedtothedeconstructedtheoryby

replicatingtheflavorsateachlatticesiteandincludingahoppingsuperpotential[6].Flavors

ona3-branearemodeledbylocalizingthoseflavorstoasinglelatticesite.Themoose(or

quiver)diagramrepresentingthedeconstructedtheoryonacirclewithNfbulkflavorsand

Nbflavorslocalizedtoa3-braneisgiveninFIG.1a.Themoosediagramforatheoryon

anintervalwithextraflavorsononeboundaryisshowninFIG.1b.Nodesinthemoose

diagramcorrespondtoSU(N)gaugegroups,andlinescorrespondtofieldstransforming

inthefundamentalrepresentationofthegaugegroupsatthenodesonwhichthoselines

end.Wedonotdistinguishfundamentalandantifundamentalrepresentationsinthemoose

diagramsasitisconventionaltoassignlinksbetweennodesthebifundamentalrepresentation(3

NfN

f

NfNfNffN + Nb

fN + NbNfNfNf

(b)(a)

FIG.1:a)Circularmoosewithbrane.b)Intervalmoosewithbrane.

Thehoppingtermsintheactionforbulkflavorsarereproducedbyasuperpotential,as

wereviewbelow.Intheabsenceofthesuperpotentialthemodelsthatwestudyhavea

largenumberofinteractingconformalfixedpoints.Ateachfixedpointsomecombination

oftheindependentgaugecouplingsisnonvanishing.Weusethea-maximizationprocedure

ofIntriligatorandWecht[7]tocalculatethedimensionsofoperatorsatthesefixedpoints.

Dependingonthenumbersofcolorsandbulkandbraneflavorswedeterminewhichofthe

conformalfixedpointsarestabletogaugingoftheungaugedsymmetrygroupsatthosefixed

points,andwearguethatgenericallyallofthebulkgaugecouplingsarenonvanishingat

thestablefixedpoints.

Wewillshowthatforthebranelessmoosethehoppingsuperpotentialiscomposedof

relevantoperatorsintheneighborhoodofthefixedpoints,sothatthoseoperatorswill

genericallyhavenonvanishingcouplingsintheinfraredasrequiredfortheextradimensional

interpretationofthemoosemodelnearitsstablefixedpoint.Furthermore,wewilldiscover

thatthetopologyofthedeconstructedextradimensiondependsonthenumbersofcolors

andbulkandbraneflavors.

A.TheModel

FordefinitenesswewillmostlyconsiderthecircularmoosemodelwithNfbulkflavors

andNbbraneflavors,asinFIG.1a.Thegaugegroupsandmattercontentaresummarized