Supersymmetry breaking in ISS coupled to gravity

We analyse the breakdown of supersymmetry in an ISS model in the presence of gravity, under the requirement that the cosmological constant vanishes dynamically. The gravitational backreaction is calculated in the metastable minimum and, in conjuction with

SupersymmetrybreakinginISScoupledtogravity

lak,O.J.Eyton-Williams

InstituteofTheoreticalPhysics,UniversityofWarsaw,00-681Warsaw,Poland

arXiv:0807.4120v3 [hep-th] 30 Jul 2008AbstractWeanalysethebreakdownofsupersymmetryinanISSmodelinthepresenceofgravity,undertherequire-mentthatthecosmologicalconstantvanishesdynamically.Thegravitationalbackreactioniscalculatedinthemetastableminimumand,inconjuctionwiththeconditionV=0,thisisshowntogeneratenon-zeroF-termsforthesquarks.Oncethesquarksarecoupledtothemessengersector,agaugemediationschemeisrealisedanditleadstoadistinctivesoftspectrum,withatwoorderofmagnitudesplitbetweenthegauginoandthesoftscalarmasses.1IntroductionInthisletterweanalysethemeta-stablepointofasimpleIntriligator,SeibergandShih(ISS)model,withintheframeworkofsupergravity.Thisallowsustocancelthecosmologicalconstant,whichweopttodobythesimplestpossiblemethod:addingaconstant,W0,tothesuperpotential.Thisissu cienttogenerateaphysicallyreasonablegravitinomassandbalancethenewnegativecontributionagainsttheoriginalpositivecontributioncomingfromtheISSpotential.Werecomputetheone-loope ectivepotentialinsupergravityandusethistocomputethegravitationalbackreactionontheglobalvacuum.Theperturbationsareshowntobesmall,asoneexpectsfromgravitationalcorrections,butnon-trivial.Themostinterestingdevelopmentisthatthemagneticquarksobtainnon-zero,butPlancksuppressed,F-terms.Hence,thereappeartwodistinctscalesinthesectorthatbreakssupersymmetry.Itisinterestingtocalculateinthissetuptherelativeimportanceofseveralmediationmechanisms,speci callyanomaly,gravityandgaugemediation.Wegiveorderofmagnitudeestimatesforthesoftmassesspectrumgeneratedbythesethreemechanismsandarguethatthespectrumcanhaveastrikinggapbetweenthegauginomassesandthesoftscalarmasses.ThisisreminiscentofsplitSUSYbutthesplitisnotallowedtobe

arbitrarilylargesinceitisconstrainedbytherequirementthatV=0inthemeta-stableminimum.

WenotethatsettingV=0attree-levelisclearlynotsu cienttoguaranteeitremainsclosetozerowhenloopcorrectionsareincluded.Asdiscussedindetailin[3,onegenericallyexpectsboththelogarithmiccorrectionspresentintheSUSYtheory(albeitwithgravitationallycorrectedmasses)andcontributionsoforderΛ2m23/2tobepresent,ifthetheoryiscut-o atΛ.However,asnotedin[4]anddiscussedfurtherinthiscontributionisdeterminedbythegeometryoftheK¨ahlerpotentialandthenumberofdegreesoffreedominthee ectivetheory,anditisperfectlypossibleforittovanish.Evenifitremains,itspresenceisnotnecessarilyparticularlydamaging,sinceitis xedbythesizeofm3/2.

Thepotentialcanbeparametrizedas(withMPsetto1andVlogcontribution):

(1 loop)V=VF+Vlog+(Z 3)m23/2

1(1 loop)denotingthelogarithmic1-loop(1)

We analyse the breakdown of supersymmetry in an ISS model in the presence of gravity, under the requirement that the cosmological constant vanishes dynamically. The gravitational backreaction is calculated in the metastable minimum and, in conjuction with

whereZis,atmost,anO(1)toO(Nf)parameterencapsulatingourignoranceaboutUVe ects.IfZ<0,theconditionV=0issatis edbyasmallerW0thanisrequiredtocancelthetree-levelpotential.Sinceweexpectthat|Z|～Λ2thisimpliesthat,aslongasΛ 1,W0willnotchangedramatically,andourresultswillbequalitativelyunchangedwithrespecttothecasewiththequadraticallydivergenttermomitted.Naturally,westillhavetore-tunetogetV=0,buttheloopcorrectiondoesnotincreasethedegreeoftuningrequired.Finally,ifZ 3itisclearthatthesemodelsbreakdownandthecosmologicalconstantcannotbetunedtozero.

Thebreakingofsupersymmetryisbestdescribedbythelowenergyvariablesinthemagnetictheory.Assuch,therelevantcut-o isthescaleatwhichthemagneticdescriptionisnolongervalid.Abovethisscalethe,SUSYpreserving,electricdescriptioniscorrectandhencetherearenofurthercontributionstoinherentlySUSYbreakinge ects.

Toobtaincontroloverloopcorrectionstothecosmologicalconstant,wemustpostulatethateithera)theUV’scontributiontoSUSYbreakingissub-dominanttotheIR’sandtheIRtheoryiscuto belowMP.orb)theK¨ahlergeometryconspireswiththenumberof avourstosuppresstheloope ects.

Whilethesecondoptionispossible(andonemighthopethattheapparentconspiracywillberesolvedbyaparticularstringconstruction)weprefertomakethesomewhatmilderassumptionofUVirrelevance.Wedo

notattempttoconstructadetailedUVsectorinthispaper,foranexamplewhereaKKLTmodel[6]isusedintheUVsee [7].Wereturntothisquestionbrie yinsection7.Essentially,wepostulatethatW=WISS+WUV, WUV=W0=0andthatFUV FΦ.

2GlobalISSreview

ISSshowedthatmeta-stableSUSYbreakingispossibleinawideclassofremarkablysimplemodels.OneoftheirmainexamplesissupersymmetricQCDwithNf avoursandNccolours.Ifoneliesinthefreemagneticrange,Nc<Nf<3

Nf,φ:(N),N=Nf Nc,thenumberofsquark avours

inthetheoryandwedenotethepartsofΦthatwilllaterobtainexpectationvaluesasfollows: magnetic Φ10.TheK¨ahlerpotentialiscanonical.Φ=0Φ0Consideringthetree-levelsuperpotentialinisolationone ndsthatthelowestenergystateisamodulispaceparametrisedby

φ000φ0 T= 0=µ2IN×N.,φ(3),φ=,φ0φΦ=cc00Φ00

SinceSUSYhastobebroken,thepotentialispositivede niteandisfoundtohaveanexpectationvalueofV=Ncµ4.Whentheone-loope ectsareincludedthemodulispaceisliftedand,asidefrom atdirectionsidenti edwithGoldstonebosons,auniqueminimumisfoundat:

Φ=0, 0=µIN×N.φ0=φcc(4)

Inadditiononemustincludethenon-perturbative,R-symmetryviolatingcontribution:

(N 3N) 1/NW=NhNfΛmfdet(Φ).(5)

NoticethattheexponentofΛm, (Nf 3N)= (3Nc 2Nf),isalwaysnegativeinthefreemagneticrange.Hencethecoe cientofthedeterminantgrowsasthecut-o shrinks.

2

We analyse the breakdown of supersymmetry in an ISS model in the presence of gravity, under the requirement that the cosmological constant vanishes dynamically. The gravitational backreaction is calculated in the metastable minimum and, in conjuction with

Sincethenon-perturbativepieceisR-symmetryviolatingitnecessarily[9]impliesthataSUSYpreservingminimummustexist,createdbythenon-perturbativepiece.InglobalSUSY1thismustbeatalowerenergythantheSUSYbreakingminimum.

2.1Anoteondynamicalscales

Wenowcalculatetherelationshipbetweenthedynamicalscales,ΛandΛm,oftheelectricandmagnetictheories,respectively.WemakeuseoftherelevantpartofthedictionarygiveninISS’spaperandthedualityrelation,givenby:

√(6)h= Λ

and

Λ3Nc NfΛm2Nf 3Nc Nf.=( 1)Nf NcΛ(7)

isadimensionalparameterinthemagnetictheory,relatedtotheelectricquarkmass,m0,andthewhereΛ = µ2magneticquarkmass,µthroughthefollowingrelation:Λ

situationcouldbeimprovedinSUGRAiftheSUSYpreservingpointalsohadW=0andtheSUSYbreakingpointV=0,butthisisdi culttoobtain,andnotthecasehere.In-fact,thedi erenceintheenergydensityisincreasedbythenegativecontributionsfromW=0.12KM=1The

We analyse the breakdown of supersymmetry in an ISS model in the presence of gravity, under the requirement that the cosmological constant vanishes dynamically. The gravitational backreaction is calculated in the metastable minimum and, in conjuction with

TheconstantcreatesanAdSminimumwithanegativeexpectationvalueequalinmagnitudetotheglobalISStheory’s,namelyVADS Ncµ4.However,thedi erencebetweenVintheAdSminimumandinthemetastableminimumisessentiallythesameasthethedi erencebetweenVintheSUSYminimumandinthemetastableintheglobalcase.Theheightofthebarrierisalsoessentiallythesameinbothcases.

FinallyournumericalstudiesdemonstratethatifW0～µ2thenΦ0getsexpectationvaluesoforder1,buttheexpectationvalueshrinksasW0→0,goingtozerointhatlimit.

4Onelooppotential

Sincetheinterplaybetweenthesupergravityandone-loope ectsissoimportanttoourresultsitisworthdiscussingthedetailsoftheone-loopcalculation,highlightingtheapproximationswehavemade.Firstofallwenotethatthemassmatrices,M,inthewell-knownformula:

1

64π2M2Vone loop=STrMlog4

isincontrasttoglobalISS,orindeedISSwithnoadditionalphysics,becauseinourcase Φ =0.When Φ =0theGoldstonebosonsaresolelylinearcombinationsofthemagneticquarks,butwhen Φ =0itcontributestothebreakingofSU(Nf)andhencetheGoldstonebosons.ThismeansthatanycontributionstotheGoldstonebosons’potentialthatviolatetheoriginalSU(Nf)symmetrygive(spurious)massestotheGoldstones.Henceoneshouldcomputethefullpotential,atleastto2ndorderinthe elds,toobtainreliableresults.

4This

4

We analyse the breakdown of supersymmetry in an ISS model in the presence of gravity, under the requirement that the cosmological constant vanishes dynamically. The gravitational backreaction is calculated in the metastable minimum and, in conjuction with

5Non-perturbativecontributions

Toestimatethevalueofthecut-o forwhichthenon-perturbativepiece(5)

dominates,anddestabilisesthepotential,wecalculatethesecondderivativeofthenon-perturbativecorrectiontothepotential,evaluatedat

2Vnon-pert.theminimumofthetree-level+one-loope ectivepotential.If,thenthenon- Φ2

perturbativepiecewilllikelydominateandthe eldswillrolltothesupersymmetricminimum.Thisallowsustoputroughlowerboundsonthecut-o ,suchthatthemeta-stablesolutionisstable.Thisisincontrasttotheglobalcaseinwhichthenon-perturbativee ectsvanishinthetree-level+one-loopminimum.Considering,forsimplicity’ssake,Nf=Nc+1:

2Vnon-pert.

Φ2ii

8π2～log(4) 1itisclearthatthenon-perturbativepiecewilldominate

andthe eldswillevolveintotheSUSYminimum.However,wecanseethat,evenifthecut-o hasthesameorderofmagnitudeasµ,thenon-perturbativepotentialalonewillnothaveasigni cante ect.ItissuppressedbythesmallexpectationvaluesofΦ0,givenbytheperturbativepotential.If Φ µthenon-perturbativepotentialcandominate,butthisisfarfromthecasehereandwecould ndnoexamplesintheliteraturewhereitwas.

Wealsonotethat,whiletherearenon-perturbativecontributionstotheone-loope ectivepotential,thesee ectsaregenericallysmall.Theywillonlyneedcloseconsiderationwhen Φ µandeveninthiscase,thenon-perturbativecontributionstothetree-levelwillbemoreimportant,exceptatsingularpointswheretheperturbativeexpansionbreaksdown.

Finally,thefullcalculationwithafullyrealisticnumberof avoursisnumericallyintractable,andweareonlyabletocomputeeverythingforthecasewhereNf=3andNc=2.Whilethiscasedoesnotcorrespondtoadualizedtheory,itdoescapturetheimportantlowenergyphenomena:therankconditionstillholdsandSUSYisstillbrokenspontaneously.However,thenon-perturbativecontributionsarequalitativelydi erent.Firstlythecoe cientisautomatically1,irrespectiveofthesizeofthecut-o ,andthedeterminantpieceislargerbyroughlyµ 2,sinceitcontainsonefewerpowerofΦij.Thismeansthatthenon-perturbativepiececancometodominate,eventhoughweknowthatitwouldbenegligibleintheNf=4,Nc=3case.Asaresult,wewereforcedtointroduceaconstant,Λnon-pert.,multiplyingthenon-perturbativepieceand ndthelargeststablevalue,beforewecouldbecertainthatthenon-perturbativee ectswereundercontrol.Morespeci cally

(N 3N) 1/NWnon pert→Wnon pert=Λnon-pert.NhNfΛmfdet(Φ)(12)

inwhatfollows.

6Numericalresults

Intable1therowΛnon-pert.isthecoe cientofthenon-perturbativepiece,introducedtocompensateforthemissingpowersofΦijthatwouldbepresentinthedeterminantforarealisticnumberof avours.Note,theF-termforΦ0isnotincludedsinceitisonlyshiftedbycorrectionsoforderµ3,soFΦ0=µ2+O(µ3).

Forphysicallyreasonablevaluesofµthise ectisnegligible.Also,Λ=10 2,(i.e.thestring/GUTscale)throughouttables1and3.Thereasonis,thisvalueofΛissu cientlysmallandensuresthatthequadratically

5

We analyse the breakdown of supersymmetry in an ISS model in the presence of gravity, under the requirement that the cosmological constant vanishes dynamically. The gravitational backreaction is calculated in the metastable minimum and, in conjuction with

Nf

1

µ

1.5210 14

φ0

Φ1 Fφ0/µ2 6.3310 12

1.0710 212.0010 61.5210 14 6.3310 12 1.0710 212.0010 41.5210 14 6.3310 12 1.0710 212.0010 21.5210 14 7.9510 12 1.0710 214.0010 21.5210 14 1.5710 11 1.0710 214.4010 21.5210 14 2.1010 11 1.0710 214.6010 21.5210 14 2.5610 11 1.0710 2

Table1:SolutionsforV=0,withthenon-perturbativepieceincluded.Thedependenceonthenon-perturbativecontributionisshowntobeverysmall,butasΛnon-pert.exceeds210 2itrapidlycomestodominate.Nf

1

µ

10 1

1.0610 14

4.3610 12

5.4610 3110 21.0810 14 4.4310 12 5.2710 3110 31.0910 14 4.5010 12 5.0910 3110 41.1110 14 4.5710 12 4.9310 3110 51.1310 14 4.6410 12 4.7710 3110 61.1510 14 4.7110 12 4.6310 31210 71.1610 14 4.7510 12 4.5310 3 φ0 Φ1 Fφ0/µ2

Table2:SolutionsforV=0wherethenon-perturbativeandone-loopquadraticpieceshavebeenneglected.ThesedatashowthelogarithmicdependenceofΦ0onthecut-o .

Nf

1

µ

m3/2

Φ0

FΦ1/µ29.9710 8 2.1010 14 1.5610 719.9710 7 1.9710 12 1.4210 619.9810 6 1.8610 10 1.3110 519.9810 5 1.7710 8 1.2010 4

We analyse the breakdown of supersymmetry in an ISS model in the presence of gravity, under the requirement that the cosmological constant vanishes dynamically. The gravitational backreaction is calculated in the metastable minimum and, in conjuction with

divergentloopcorrectiontothepotentialisnegligible,leavinguswiththelogarithmicpiecewhosesensitivitytotheactualvalueofthecut-o isveryweak.SincethesuperpotentialcanbewrittenW=WISS+W0,and WISS =0inthegloballimit,we ndthatm3/2=eKW0+O(µ3) W0.Henceweonlyincludem3/2inthetable.

Ournumericalstudiesshowedthatthenon-perturbativepieceisundercontrol,thatthereisonlyaverymild,logarithmiccut-o dependence,whensupergravitycorrectionsareaccountedforandthatthemainfeaturesofthemodel

are

independent

of

µ

,

assuming

µ MP.Theseresultscanbeseenin,respectively,tables1,2and3.√Fromthedataintable2wecanseethatforµ=

φ0 ～FΦ0

Φ0 M FΦ0F

Fφ0

4π

andthesoftscalarmasssquaredsareapproximately:4πW04πµ2(13)

m2=m2λ.

Inadditionwecanestimatethesoftmasscontributionsfromanomalymediation:

Ma=FAnom.βga/ga～

5We6As(14)αfaremessenger eldschargedunderthevisiblesectorgaugegroups.haveinmindanoperatorW Xfdiscussedpreviouslythegravitationale ectscomeinwithanadditionalpowerofµcomparedwiththeglobalSUSYterms,hencethenewF-termsare～µ3.

7

We analyse the breakdown of supersymmetry in an ISS model in the presence of gravity, under the requirement that the cosmological constant vanishes dynamically. The gravitational backreaction is calculated in the metastable minimum and, in conjuction with

i γ(m2)ijj=FAnom.2FAnom.2

16π2(16)

SinceFAnom.canatmostbeFΦ0(withoutpostulatinganothersourceofSUSYbreaking)itfollowsthat

thesecontributionsareofthesameorderofmagnitudeasthosegivenbygaugemediation.

Thisallowsustoestimatethesizeofthegravitinomass,basedontherequirementthatthegauginomassesbeinthevicinityofaTeVandthatthecosmologicalconstantbetunedtozero.Theserequirementsimplythatα2

α 8

10.Takingα=1

1/2

3andhencethegravitationalcontributiontothesoftscalarmasses

isautomaticallytwoordersofmagnitudelargerthantheothercontributions.

Asmentionedearlier,thegravitationalcontributiontothegauginomassesisnotnecessarilyorderm3/2.Forexampleinstringtheoriesonecan ndthatthegaugekineticfunctions,whoseexpectationvaluesspecifythegaugecouplingconstants,haveatreeleveldependenceonclosedstringmoduli[16]7.Sincepseudo-moduliareevidentlynotstringmoduli,thetree-levelgaugekineticfunctiondoesnothavetodependonthem.Hence,ifthestringmoduli(orany eldsthatappearinthegaugekineticfunction)donotcontributestronglytoSUSYbreaking8thenthegauginomasseswillreceiveanegligibletree-levelcontributionfromgravity.

Insummary,themodelsdiscussedheremaynaturallygenerate,viathegaugemediationchannel,aspectruminwhichthegauginomassesareloopsuppressedwithrespecttothesoftscalarmasses.Sincethegauginomassesreceiveunknownquantumcorrectionswecannotpredicttheprecisespectrum,whichdependsondetailsofthecompletemodel,thoughwedoexpecttoseeasplitinthespectrumofsoftmasses.Thisspectrumhasmuchincommonwiththeonepresentedintheearlyworkonanomalymediation.See,forexample,[20].

Inadditiontothis,onecanconsidermodifyingthemodel.Forexample,intheliteraturethereareexamples

[13,21,22,23]inwhichoperatorsareaddedtothesuperpotentialallowingthepseudo-modulusvevtogrowouttoaroundµ.Whilethisisclearlyaninterestinge ect,thedualtheoryresponsibleforgeneratingthisoperatorisasyetunknown.

8Conclusions

Insummary,ISSismorestablewithsupergravitycorrectionsthanwithout,assumingthatW0=0.ThepicturechangessomewhatwhenW0

control,oncetheone-loope ects=0,buthaveitbeenwasshownincluded.insection4thatthesupergravitye ectsaresmallandunder

Itwasthendemonstrated,insection5that,whilethenon-perturbativepieceisnon-zero,itisnecessarilysub-dominantforsmallvaluesofΦ.Hencethistermcanbeneglectedwhensupergravitye ectsaresubdominanttotheone-loope ects.Numericalanalysiscon rmedthis.

WealsoarguedthatonedoesnotneedtobeoverlyconcernedbycontributionoftheΛ2STr(M2)highlightedbytheworkof[3,5]sinceourcut-o is,oratleastcanbe,muchlowerthanthePlanckscale.Although,wemustrequirethatwhateverphysicstakesoverintheUV,ifithasalargecontributiontoSUSY-breaking,itmightnot(dependingonthesignofit’scontributiontothee ectivepotential)bepossibletosetthecosmologicalconstanttozero.Suchahighenergymodelwasconsideredin[5],butwedidnotattemptthiskindofconstructioninthispaper.Also,itisworthnotingthatitisconsistenttomakeuseofthesupergravitycorrectede ectivepotential,eventhoughourcut-o canbetakentobemanyordersofmagnitudebelow.Thisisbecausethenon-renormalisableoperatorspresentinSugraarenotgeneratedbytheintegrationoutofgravitational elds,andareinsteadrequiredtobepresentbysupersymmetryitself.

TheexplicitR-breakingintroducedbythepresenceofaconstantterminthesuperpotentialgeneratesnon-zerogauginomasses,throughgravitationallysuppressedinteractions(asonewouldexpect,sinceglobalSUSYisblindtothepresenceoftheconstant).

We analyse the breakdown of supersymmetry in an ISS model in the presence of gravity, under the requirement that the cosmological constant vanishes dynamically. The gravitational backreaction is calculated in the metastable minimum and, in conjuction with

Weshowed,insection7,thatintheabsenceofadditionalsupersymmetriccontributionstomessengermasses,thegaugemediationisonlypossibleifthemagneticquarkscoupletothemessenger elds.AllotherISS eldshaveoverlysmallscalarvevsandgiverisetoanunstablemessengersector.Thedirectconsequenceofwhichisthatthegravitinomassisquitelarge.ThisisthecasebecauseitisnotsetbythequarksF-terms,butthemuchlargermesonF-terms,throughtherequirementthatthecosmologicalconstantshouldvanishintheminimumofthee ectivepotential.ThisresultsinadirectconnectionbetweenR-symmetrybreakingandgauginomasses,withEq.(13)showingthattheyareproportionaltooneanother,withthecoe cientbeingdeterminedbythedetailsofthegaugegroup.

Acknowledgements

ThisworkwaspartiallysupportedbytheEC6thFrameworkProgrammeMRTN-CT-2006-035863,bythegrantMNiSWN20217631/3844andbyTOKProjectMTKD-CT-2005-029466.

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We analyse the breakdown of supersymmetry in an ISS model in the presence of gravity, under the requirement that the cosmological constant vanishes dynamically. The gravitational backreaction is calculated in the metastable minimum and, in conjuction with

Appendix

AOriginsofmassterms

Forsimplicityweneglectphasesinthefollowinganalysis.Henceallsymmetriese ectivelygofromU(N)→O(N)andthecountingofdegreesoffreedomre ectsthis.

Inthiscase,thetree-levelpotential,whenembeddedintosupergravity,hasameta-stableminimum.Thepositionoftheminimumisgivenbytheglobalresult,butwithsmallcorrectionstotheexpectationvalueof 0|2=µ2 1/2±1themagneticquarks|φ0|2=|φ

N2 N)aremasslessGoldstonebosonsandtheremainderhavemassesoforderµ,2NfN N2mesonswithmassoforderµ,(Nf N)2 1mesonswithmassoforderµ2andonemasslesspseudo-modulimeson.

Theoriginsofthesemassesareasfollows.Thequarksgettheirmassesfromtheirexpectationvalues,withφ0 0andviceversa,theo -diagonalelementsofΦ0obtainmassessolelyfromthesecondderivativegivingmasstoφ

ofeK(whichcontributesequallytoall elds),namely2V,whilethediagonalelementsgetmorecomplicatedcontributions.TheremainingelementsofΦgetthesamemassesasinglobalSUSY,butwithsmallcorrectionsfromtheSugracontributions.Theendresultofthisisthatthemassive eldsretainessentiallysamemassesasinglobalSUSYandallbutoneofthepseudo-moduli(whichremainszero)obtainmassesoftheorderofthecosmologicalconstant:2Ncµ4.ThisdemonstratesthatsupergravityservestoincreasethestabilityoftheISSminimum,asitreinforcesthestabilisinge ectscomingfromtheonelooppotential.2(2NfN

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