Discrete Gravitational Dimensions

We study the physics of a single discrete gravitational extra dimension using the effective field theory for massive gravitons. We first consider a minimal discretization with 4D gravitons on the sites and nearest neighbor hopping terms. At the linear leve

HUTP-03/A015

DiscreteGravitationalDimensions

Nima andMatthewD.Schwartz

Je ersonPhysicalLaboratory,

HarvardUniversity,Cambridge,MA02138,USA

Westudythephysicsofasinglediscretegravitationalextradimensionusingthee ective eldtheoryformassivegravitons.We rstconsideraminimaldiscretizationwith4Dgravitonsonthesitesandnearestneighborhoppingterms.Atthelinearlevel,5Dcontinuumphysicsisrecoveredcorrectly,butatthenon-linearlevelthetheorybecomeshighlynon-localinthediscretedimension.ThereisapeculiarUV/IRconnection,wherethescaleofstronginteractionsathighenergiesisrelatedtotheradiusofthedimension.Thesenewe ectsformallyvanishinthelimitofzerolatticespacing,butdonotdosoquicklyenoughtoreproducethecontinuumphysicsconsistentlyinane ective eldtheoryuptothe5DPlanckscale.Nevertheless,thismodeldoesmakesenseasane ectivetheoryuptoenergiesparametricallyhigherthanthecompacti cationscale.Inordertohaveadiscretetheorythatappearslocalinthecontinuumlimit,thelatticeactionmusthaveinteractionsbetweendistantsites.Wespeculateontherelevanceoftheseobservationstotheconstructionof nitediscretetheoriesofgravityinfourdimensions.

arXiv:hep-th/

0302110v2 20 Mar 2003

Inarecentpaperatechniquewasintroducedforstudyinggravitationaltheoriesindiscrete“theoryspaces”Thesespacesarede nedbysites,withseparatefour-dimensionalmetrics,andtheirassociatedgeneralcoordinateinvariances,and“link elds”thatmapbetweensites.InanalogywiththeCallan-Coleman-Wess-Zuminoformalismforgaugetheories,thistech-niqueallowsustounderstande ective eldtheorieswithmultipleinteractingspintwo eldsinatransparentway.Inthesimplestcase[1],asinglelinkis“eaten”,leadingtoasinglemassivegraviton.Thephysicsofmassivegravi-tonsisqualitativelydi erentthanthatofmassivegaugebosons,owingtothepeculiarpropertiesofthescalarlon-gitudinalcomponentofthegraviton.Nevertheless,thereisasensiblee ectivetheorythatmakessensetoenergiesparametricallyabovethegravitonmass.Inthisletter,westudywhathappenswhenwestringtogethermanysitesandlinkstogeneratewhatlookslikeagravitationalextradimension.Suchmodelshavebeenconsideredbe-fore6,7,8,9,butnotatthelevelofaconsistente ective eldtheory.Giventhepeculiaritiesassociatedwithmassivegravity,weshouldexpectsurprises,andweindeedencounteranumberofthem.

Wewillbeginbyconsideringtheminimaldiscretiza-tions,withnearestneighborinteractions.Thediscretizeddimensioncanbetakentobeeitheracircleoraninterval:

containsapart

Ssite=

j

dxM

42

×

jj+1jj+1

gjM2m2(gµν gµν)(gαβ gαβ)

(gjµνgjαβ gjµαgjνβ)

(3)

U

TheUinSlinkstandsforUnitarygauge.Indeed,wecan

U

seethatagaugeischosenbecauseSlinkbreaksallbutonecopyofGC.

Theactionwehaveconstructedisjustanaivedis-cretizationofcompacti ed5DEinsteingravity.Toseethis,startwitha5DmetricGMN(x,z),withzthecom-pactextradimension.Ignoringtheradionandgravipho-tondegreesoffreedom(theydonota ectthefollowingdiscussion)itispossibletochooseagaugewherethemetrictakestheform

gµν(x,z)0

GMN=(4)

01

√ 5

3

Inthisgauge,the5DactiondxM5D

3gM5D

+

1

R4D[g]

Wewillconcentrateonthesimplercaseofanintervalfornow,andreturnbrie ytothecirclelateron.EachoftheNsitesisendowedwithafour-dimensionalmetricjgµν(x).Theaction

S=Ssite+Slink

(1)

a

jj+1

(gµν gµν)

(6)

We study the physics of a single discrete gravitational extra dimension using the effective field theory for massive gravitons. We first consider a minimal discretization with 4D gravitons on the sites and nearest neighbor hopping terms. At the linear leve

whichdirectlyproducesourunitarygaugeaction(3).Andwe

can

read

o

that

the

radiusR,latticespacinga,ande ective5DPlanckscaleM5Daregivenby:

R=Na,

a=m 1,

M5D=M2m

(7)

Now,asexplainedin[1],wecanrestoretheµbrokenGCsymmetriesin(3)byaddinglink eldsYItisusefultothinkofthelinkasj(x)betweenthesites.amapfromapointonsitejwithcoordinatexµj+1withcoordinateYµ

jtoapointonsitej(xj).Undergeneralcoordinatetransformationsgeneratedbyxj→fj(xj),thelink eldstransformas:

Yj→fj +11

Yj fj

(8)

whichallowsustocompare eldsonadjacentsites.

Theselink eldscanbeusedtoconstructobjectswhichtransformastensorsunderGCjandareinvariantunderGCj+1outofobjectswhichareinvariantunderGCjandtensorsunderGCj+1:

Gjµν+1(xj)≡ µYjα

νYβj+1jgαβ(Yj(xj))

(9)

Gjµν+1

(xj)canbethoughtofasapull-backofthemetriconsitej+1tositejusingthemapsYj.Thus,thelinkactionbecomes:

Slink= d4

xj

j

2

2N2

hn

+

n

Λ10

(15)

Where,expressed√intermsofthelowenergy4DPlanck

scaleMPl=

R5a2

1/10

(17)

Notethatbizarrely,theUVscaleatwhichthetheorybecomesstronglycoupleddependsonanIRscale,thesizeoftheextradimension!

Naturally,aswedecreasethelatticespacing,Λin-creases.However,foraconsistente ectivetheory,weshouldrequirethatallthestatesinthetheorybelighterthantheUVcuto Λ.Inparticular,theheaviestKKmode,ofmass～a 1,shouldbelighterthanΛ.This

We study the physics of a single discrete gravitational extra dimension using the effective field theory for massive gravitons. We first consider a minimal discretization with 4D gravitons on the sites and nearest neighbor hopping terms. At the linear leve

meansthatthelatticespacingcannotbedecreasedbe-yondacertainpoint,andthehighestUVcuto thethe-orycanpossiblyhaveis

Λmax～

1a min

～

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