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UCB-PTH-07/03 LBNL-62350 SimpleScheme forGaugeMediation Hitoshi Murayama and Yasunori Nomura Department of Physics, University of California, Berkeley, CA 94720, USA and Theoretical PhysicsGroup, Lawrence BerkeleyNational Laboratory, Berkeley, CA94720, USA Wepresentasimpleschemeforconstructingmodelsthatachievesuccessfulgaugemediationofsupersymme- trybreaking. Inadditiontoourpreviouswork[1]thatproposeddrasticallysimplifiedmodelsusingmetastable vacuaofsupersymmetrybreakinginvector-liketheories,weshowtherearemanyothersuccessfulmodelsusing varioustypesofsupersymmetrybreakingmechanismsthatrelyonenhancedlow-energyU(1)Rsymmetries.In modelswheresupersymmetryisbrokenbyelementarysinglets,oneneedstoassumeU(1)Rviolatingeffectsare accidentallysmall,whileinmodelswherecompositefieldsbreaksupersymmetry, emergenceofapproximate low-energy U(1)R symmetries can beunderstood simplyon dimensional grounds. Even though thescheme stillrequiressomewhat smallparameterstosufficientlysuppressgravitymediation, wediscusstheirpossible originsdue to dimensional transmutation. Thescheme accommodates awide range of thegravitino massto 7 avoidcosmologicalproblems. 0 0 2 I. INTRODUCTION per can extend more generally to even wider classes of the- n ories. The low-energystructure of the models of Ref. [1] is a such that, while the entire superpotentialdoes not possess a J Despitemanynewideas,supersymmetryisstillregardedas 7 the prime candidate for physics beyond the standard model. U(1)R symmetry, terms relevant for supersymmetry break- ingpossessanaccidental(andapproximate)enhancedU(1) 2 If it exists at the TeV scale, it stabilizes the hierarchy be- R symmetry. InthemodelsofRef.[1],thisstructurearisesau- tweentheelectroweakandthePlanckscales,allowsforgauge 1 couplingunificationwith the minimalparticle content, hasa tomatically at low energies, since U(1)R violating effects in v thesupersymmetrybreakingsectorarisefromhigherdimen- natural candidate for the dark matter, and possibly connects 1 sionoperatorsandthusaresuppressedbypowersofthecutoff to string theory. On the other hand, having been around for 3 scale. In this paper we present many other models that are threedecades,itsdeficienciesarealsowellknown. Thesein- 2 assimpleasthoseinRef.[1],andhencethesimplicityofthe 1 cludepotentiallyexcessiveflavor-changingandCP-violating schemeisnotnecessarilytiedtothesupersymmetrybreaking 0 effects,cosmologicalgravitinoandmoduliproblems,andthe mechanismofRef.[4]onwhichthemodelsofRef.[1]were 7 lackofautomaticprotonlongevity.Inparticular,ithasbeena 0 nontrivialchallengeto break supersymmetryand mediate its based. / In addition, in this paper we also consider the possibil- h effecttothesupersymmetricstandardmodel(SSM)sectorin itythattheU(1) violatingtermsaresuppressed(orabsent) p aphenomenologicallysuccessfulmanner. R withoutobviouslow-energyreasons. Suchsuppressionsmay - p Gauge mediation of supersymmetry breaking [2, 3] is an arise throughaccidentallysmall parameters,as a propertyof e attractive solution to the phenomenological problems with string vacua, or for anthropic reasons. We also allow us to h supersymmetry. In particular, it naturally avoids excessive makeacertaindynamicalassumptiononthesignofaKa¨hler v: flavor-changing phenomena because gauge-mediated superpotential term that is not calculable due to strong interac- Xi symmetrybreakingeffectsareflavor universal. Onthe other tions. These relaxations of the requirements drastically en- hand,constructingexplicitandrealisticmodelsofgaugeme- hance a variety of possible theoretical constructions leading r diation has been a rather nontrivialchallenge that requiresa a tothestructuredescribedabove.Animportantkeytothesuc- fairamountofmodel-buildingefforts,andthisaspecthasbeen cessisthemasstermforthemessengers,whichissimplyone makingthescenarioappearasomewhatunlikelychoicebyna- ofthegenerictermsallowedbyallsymmetries. ture. Inorderforamodeltobeviable,severalconsistencycon- Inapreviouspaper[1],wehaveproposedadrasticallysim- ditionsneed to be met. Generic gravity-mediatedsupersym- plified class of models for gauge mediation of supersymme- metrybreakingmustbe sufficientlysmall to avoid excessive try breaking. The models have a supersymmetric SU(Nc), flavor-changingand CP-violating processes. The impact of SO(Nc)orSp(Nc)gaugetheorywithmassivequarks,mas- U(1)R violation, both at tree and loop levels, must be suffi- sivevector-likemessengerschargedunderthestandardmodel ciently small in the supersymmetry breaking sector to keep gaugegroup,andacompletelygeneralsuperpotentialamong the essential dynamics intact. In addition, one should be thesefields. Wehavefounditremarkablethatthissimpleand concerned about cosmological constraints on the gravitino, generalclass ofmodelscansuccessfullybreaksupersymme- moduli if any, the origin of the µ and µB terms, and so on. tryandgenerateaphenomenologicallydesiredformofsuper- Nonetheless, the framework we present here is sufficiently symmetry breaking masses, without any additional ingredi- generalandsimple thatwe expectmanymodelscan be con- ents. Thismakesusconjecturethatgaugemediationmaybe structedtoaddresstheseissues. Inparticular,theframework arathergenericphenomenoninthelandscapeofpossiblesu- accommodates a wide range of the gravitino mass, 1 eV < persymmetrictheories,which doesnotrequireanycontrived m <10GeV. ∼ 3/2 orartificialstructuresthatexistedinmanyofthepastmodels. The∼simplicity and the variety of the models presented Inthispaper,weshowthatthesuccessofthepreviouspa- in this paper revitalize interest in the gauge mediation sce- 2 nario,andmoregenerallyinweakscalesupersymmetry.They under which S, f and f¯carry the chargesof 2, 0 and 0, re- largelyeliminatetheconcernaboutweakscalesupersymme- spectively. In this case, the dynamics associated with (the try coming fromthe experimentalnon-observationof flavor- generationof) these terms can be responsible for the Ka¨hler changingorCP-violatingeffectsinadditiontotheonesinthe potential of Eq. (2). (Explicit examples for such dynamics standardmodel. will be presentedin later sections.) The U(1) symmetryis R The organization of the paper is as follows. In the next violatedbythelasttermofEq.(1),butitseffectontheKa¨hler sectionwedescribethebasicframework,andprovidegeneral potentialcanbe suppressedas longas M > κ2Λ/4π, aswe discussionsthatapplytovariousexplicitmodelspresentedin willseebelow. ∼ later sections. InSectionIIIwe presentclasses ofmodelsin Let us first discuss the model at tree level specified by whichthesupersymmetrybreakingfieldisanelementarysin- Eqs.(1,2). Thepotentialissimplygivenby glet. Thesemodelsuseaccidentalfeaturestoprovidetheap- proximateU(1)Rsymmetry.InSectionIVwepresentmodels V = µ2 κff¯2 1+ |S|2 +O |S|4 inwhichthesupersymmetrybreakingfieldarisesasacompos- − Λ2 Λ4 iintegfiseelcdt.oIrnatrhiseesseamuotodmelast,iUca(l1ly)Rasinanthaepspurpoexrismyamtemleotwry-ebnreeragky- (cid:12)(cid:12)+ κSf +(cid:12)(cid:12)M(cid:18)f 2+ κSf¯+M(cid:18)f¯2,(cid:19)(cid:19) (3) symmetry. Wepresentclassesofmodelsinwhichthesignof (cid:12) (cid:12) (cid:12) (cid:12) therelevantKa¨hlerpotentialtermcanbereliablycalculatedin whichhasagloba(cid:12)lsupersymm(cid:12)etric(cid:12)minimumat(cid:12) thelow-energyeffectivetheory,aswellasthoseinwhichthe M µ signisincalculable. Inparticular,aclassofmodelspresented S = , f =f¯= . (4) inSectionIVBenjoysthesamelevelofsuccessastheonein − κ κ1/2 Ref.[1]. InSectionVwediscussslightlydifferentclassesof This potential, however, also has a local supersymmetry models,whicharenonethelesscloselyrelatedtotheonespre- breakingminimumattheoriginoffieldspace,S =f =f¯= sentedinprevioussections. Finally,inSectionVIwediscuss 0, as longas M2 > κµ2. The masses forthe scalar compo- possiblewaystonaturallygeneratesmallparametersthatare nentsofSandthemessengersaregivenbym2 =µ4/Λ2and usedinthemodelsconstructedinSectionsIII–V.Conclusions S m2 = M2 κµ2, respectively. Note that in order for this aregiveninSectionVII. f ± pointtobeaminimum,itisimportantthatthesignofthesec- ondterminEq.(2)isnegative. Thiscanbeexplicitlyproven in some of the models presented in later sections, while in II. FRAMEWORK someothersthesignshouldbesimplyassumed. Thetunnelingrate fromthelocalminimumtothe truesu- In this section we present our basic framework. Explicit persymmetric minimum can be easily suppressed. To esti- modelswithinthisframeworkwillbegiveninlatersections. mateit,wecanbaseourdiscussionsonRef.[5],calculatinga semi-classicalfield theoretictunnelingratefora toytriangu- larpotential.Whiletheexpressionworkedouttherecannotbe A. Basicidea literallyappliedtoourcase,wecanstillapproximateourpo- tentialbyatriangularform,obtainingthedecayrateperunit The basic idea is very simple. We consider the following volumeΓ/V µ4e−B withB 8π2MΛ2/κ3/2µ3. SinceΛ superpotential is expectedto∼be > µ, the boun∼ceaction B can be easily of W = µ2S+κSff¯+Mff¯, (1) O(100)orlargerf∼orM > κ1/2µ. Tokeepthelifetimeofthe − localminimummuchlar∼gerthan the ageof the universe,we where S is a gauge singlet chiral superfield (elementary or needΓ/V H4,whereH 1.6 10−33eVisthepresent composite),andf,f¯aremessengers;µisthescaleofsuper- Hubblecon≪stant0. Theboun0d≃is the×strongestforM2 κµ2 symmetrybreaking,andκacouplingconstant. Theparame- (thelowestmessengerscale),buteventhentheconstra≈intson tersµ2,κandM canbetakenrealandpositivewithoutlossof theparametersarenotverystrongforΛ>M. generality. Forconcreteness,wetakethemessengerstobein At the local supersymmetry breaking∼minimum, the mes- 5+5∗representationsofSU(5)inwhichthestandardmodel sengersf,f¯have both supersymmetricand holomorphicsu- gaugegroupisembedded. persymmetrybreakingmasses We assume thatthe Ka¨hler potentialforS takes(approxi- mately)theform M =M +κ S M, (5) mess h i≈ S 4 S 6 K = S 2 | | +O | | , (2) and | | − 4Λ2 Λ4 (cid:18) (cid:19) expandedaroundtheoriginS = 0, whereΛ isamassscale. F =κ F =κµ2. (6) mess S h i (We assume a canonicalKa¨hler potentialfor the messengers forsimplicity.) ThisformoftheKa¨hlerpotentialisobtained, Here,wehaveassumedthattheexpectationvalueofS,which forinstance,ifthereisanapproximatelow-energyU(1)sym- isgeneratedbyU(1) violatingeffectsaswewillseebelow, R metry on S. Thissymmetry can be a U(1) symmetrypos- issmall. Theconditionsthatthisrequirementimposesonthe R sessed by the first two terms of the superpotential, Eq. (1), parametersofthetheorywillbediscussedshortly.Themasses 3 forthegauginosandscalarsintheSSMsectorarethengener- AnothersourceofU(1) violationcomesfromloopsofthe R atedbymessengerloops[2,3]andoforder messengers,whichdonotrespectU(1) becauseofthemass R term.TheseloopsgeneratethefollowingColeman–Weinberg g2 κµ2 effectivepotentialforS: m , (7) SUSY ≃ 16π2 M κ2µ4 κS ∆V where g represents generic standard model gauge coupling ≈ 16π2 F M (cid:18) (cid:19) constants.TakingthesemassestobeofO(100GeV 1TeV) correspondsto ∼ 5µ4 κ3(S+S†) κ4 (S2+S†2)+ , ≃ 16π2 M − 2M2 ··· (cid:26) (cid:27) κµ2 (13) 100TeV. (8) M ≈ where (x) is a real polynomial function with the coeffi- F Thegravitinomass,ontheotherhand,isgivenby cientsofO(1)uptosymmetryfactors. Inthesecondline,we haveshownthecoefficientsexplicitly,keepingonlythelead- F µ2 ingtermsinκµ2/M2 (anddroppinganirrelevantconstantin m h Si , (9) 3/2 ≈ M ≈ M ∆V), which correspondsto the correction to the Ka¨hler po- Pl Pl tentialoftheform∆K (1/16π2) (κ3/m)S 2(S+S†)+ where MPl 2.4 1018 GeV is the reduced Planck scale. (κ4/m2)S 4+(κ4/m2≈)S 2(S2+S{†2)+ | .|Theeffective ≃ × | | | | ···} Thus,requiringthatgravitymediationgivesonlysubdominant potentialofEq.(13)pullstheminimumatS =0towardsthe contributionstothescalarmasses,m <10GeV,wefind negativedirection,andreducesamass-squaredeigenvalueof 3/2 ∼ S fromµ4/Λ2. Yetfor µ<109.5GeV. (10) κ2 ∼ M > Λ, (14) 4π ∼ B. EffectsofU(1)R violation we find that these effects are parametrically suppressed and thestructureofthesupersymmetrybreakingsectorisnotsig- Theexistenceofthesupersymmetrybreakingminimumat nificantlymodified. Inparticular,thelocalminimumstaysat S =f =f¯=0canbeviewedasaresultoftheU(1) sym- smallS: S κ3Λ2/16π2M < min M/κ,Λ . Thecon- metrypossessedbythefirsttwotermsofEq.(1): R(SR) = 2, ditionfor|ahvoii|d≈ingtachyonicmes∼sengers{is } R(f) = R(f¯) = 0. This picture is corrected by U(1)R vi- M2 >κµ2. (15) olatingeffectscomingfromtheothersectorsand/ortermsin ∼ Note that the inequalities of Eqs. (14, 15) should be under- thetheory. OneoriginofU(1) violationarisesfromthesu- R perpotentialtermsS2 andS3,whicharethe(only)renormal- stoodthatorderonecoefficientsareomitted. izableterms,otherthanthoseinEq.(1),allowedbythegauge In general, the 4 parametersof the theory µ, κ, M and Λ symmetry.1 ThesetermscanbeautomaticallysuppressedifS arearbitrary,exceptthatweexpectΛ>µifthehigherdimen- siontermintheKa¨hlerpotentialofE∼q.(2)isinducedbythe isacompositefieldgeneratedatlowenergies(asinthemod- dynamicsgeneratingthefirst(two)term(s)ofthesuperpoten- elsofSectionIV)butingeneralmustbesuppressedforother tialofEq.(1). Byvaryingtheseparameters,awidevarietyof reasons if S is elementary (as in the models of Section III). Denotingtheextratermsas physicalpicturescanarise. Forµ2/Λ M,forexample,we ≫ canfirstintegrateouttheSscalar,whichismuchheavierthan M κ themessengers, and thenthelow-energytheorybelowtheS ∆W = SS2+ SS3, (11) 2 3 massappearsasthestandardgaugemediationmodel,withthe Lagrangiangivenby d2θ(M +θ2F )ff¯+h.c.Onthe mess mess constraintsontheparametersMS andκS areobtainedbyre- otherhand,intheoppositelimitofM µ2/Λ,wecanfirst quiringthattheresultingshiftofhSiissmallerthan≈Λ(for integrateoutthemesRsengersf andf¯. T≫hisgenerates“gaug- the expansion of Eq. (2) to be valid) and than ≈ M/κ (to ino mass operators” d2θS α α +h.c. as well as flavor avoidtachyonicmessengers): W W universal“scalar mass operators” d4θS†SΦ†Φ, where R Wα µ2 Mµ2 µ2 representstheSSMgaugefieldstrengthsuperfieldsandΦthe M <min , , κ < . (12) SSMmatterandHiggschiralsuperRfields.Thelow-energythe- | S|∼ (cid:26)Λ κΛ2 (cid:27) | S|∼ Λ2 orybelowthemessengermass,M,isthenasimplePolonyi- Note that these conditions are not very restrictive. This is type model– Eqs. (1, 2) with f and f¯set to zero – together because we use field space with small S, where there is a withtheseoperators,whichareresponsibleforthemassesof quadratic stabilizing potential for S arising from the second thegauginosandscalarsintheSSMsector. terminEq.(2). C. TheoriginofSandthescalesofthetheories 1 AlineartermofSintheKa¨hlerpotentialcanbeabsorbedintothedefini- The frameworkdescribed here representsa great simplifi- tionofthesuperpotentialbytheappropriateKa¨hlertransformation. cation in building models of gauge mediation. The only re- 4 quired aspect of model building is essentially to explain the O(0.01 0.1). This requires either small ζ, large η, small ∼ originofthe quarticterminEq.(2). Thereare manyclasses M ,oracombinationofthese: ∗ ofexplicitmodelsthatcanbeconstructedinthisframework, someofwhichwillbepresentedinSectionsIIIandIV.Inthe ζ M modelswhereS isanelementarysinglet(themodelsinSec- ∗ <O(10−4 10−3). (22) tion III), it must be assumed that the U(1) violating terms sηMPl ∼ ∼ R of Eq. (11) are suppressed without obvious low-energy rea- (A large value for η is obtained by generatingthe nonrenor- sons. Ontheotherhand,inthemodelswhereS isacompos- malizable coupling W Qnff¯ by integrating out heavy itefield(themodelsinSectionIV),thesetermsarenaturally ∼ fields below M in the theory.) In the case that S is a two- suppressed. SupposethattheS fieldconsistsofnelementary ∗ body composite, i.e. n = 2, this condition is satisfied sim- fields, S Qn/Λn−1 (n 2), where Q and Λ represent ∼ s ≥ s ply by having small mass parameters for elementary fields: genericconstituentsofS andthescale ofcompositeness,re- W mQQ with m M , which corresponds to having spectively. The parameters MS and κS in Eq. (11) are then sma∼llζ. ≪ ∗ suppressedas A largevarietyoftheoreticalconstructionsallowedinthis Λ2n−2 Λ3n−3 framework can lead to a wide range of the parameters µ, κ, MS ≈ Ms2n−3, κS ≈ Ms3n−3, (16) M and Λ. This implies in particular that the framework ac- ∗ ∗ commodates a wide range of the gravitino mass, 1 eV < respectively.Here,M isthecutoffscaleofthetheory. m < 10 GeV. The smallest gravitino mass is obtaine∼d ∗ 3/2 The S compositeness also suppresses the parameter κ in whenM∼2 κµ2 (100TeV)2 andκ O(1 4π). Sucha Eq. (1), weakening the transmission of gauge mediation ef- lightgravit≈inoisu≈sefultoavoidcosmolo≈gicalp∼roblemsasso- fects. Writingthefundamentalsuperpotential,whichreplaces ciatedwiththegravitino[6]. thefirsttwotermsofEq.(1),schematicallyas ζ η W Qn+ Qnff¯, (17) III. THEORIESWITHELEMENTARYSINGLETS ≈−Mn−3 Mn−1 ∗ ∗ wefind In this section we present classes of models in which the supersymmetrybreakingfieldS isanelementarysinglet. As ζΛn−1 ηΛn−1 µ2 = s , κ= s , (18) discussedintheprevioussection,thiscaserequiresaccidental Mn−3 Mn−1 ∗ ∗ suppressionsinU(1) violatingterms.Nonethelessitisquite R where we have defined the compositeness scale Λ by S = nontrivialthatsuccessfulgaugemediationisobtainedinvery s Qn/Λn−1. The requirement of Eq. (12) for preserving the simplemodelsoncesuchsuppressionsareassumed. s approximateU(1) symmetrywastohaveametastablemin- R imumaroundtheorigintojustifytheanalysis. Itrequiresan unexplainedsuppressionin M forelementaryS, whileitis A. Tree-levelsupersymmetrybreaking S easytosatisfyforcompositeS. Thegauge-mediatedcontributiontotheSSMsuperparticles AnobviouscandidateforproducingtherequiredKa¨hlerpo- isgivenby(seeEq.(7)) tential of Eq. (2) is the good-old O’Raifeartaigh model [7]. We replace the first term of Eq. (1) (and the second term of g2 ζηΛ2n−2 m s , (19) Eq.(2))by SUSY ≃ 16π2M2n−4M ∗ mess W = µ2S+λSX2+mXY, (23) where we have denoted the messenger mass explicitly as − M , leaving the possibility that the term with the S ex- mess where S, X and Y are singlet fields having the canonical pectation value contributes significantly in Eq. (5). We find that m is suppressed by (Λ /M )2n−2. The contribu- Ka¨hlerpotential(uptotermssuppressedbythecutoffscale). SUSY s ∗ Here, we simply assume that possible terms S2, S3, X2, tionfromgravitymediationis(seeEq.(9)) SXY, Y2 and SY2 are somehow suppressed. (The other m3/2 ≈ MζnΛ−sn3−M1 . (20) twerhmicshcXananbde Yforabrieddoednd.)byTahedpisacrraemteetZer2s µsy2m, λmeatnrdymundareer ∗ Pl takenrealandpositivewithoutlossofgenerality. DividingEq.(20)byEq.(19),andusingthestabilitycondition The superpotentialof Eq. (23) breaks supersymmetrydue Mm2ess >κµ2(seeEq.(15)),weobtain to the incompatibility between FS = 0 and FY = 0. For ∼ m2 > 2λµ2, the minimum is at X = Y = 0. The field m 16π2Mn−1M ζ M S is a flat direction at tree level, but is stabilized at the 3/2 ∗ mess >100 ∗ . (21) mSUSY ≈ g2 ηΛsn−1MPl ∼ sηMPl origin due to radiative corrections to the Ka¨hler potential. These corrections can be calculated most easily by comput- Inorderforthegauge-mediatedcontributiontodominateover ing the Coleman-Weinberg effective potential for S, arising the gravity-mediated one, we must have m /m < from loops of X and Y. The mass matrix of the X and Y 3/2 SUSY ∼ 5 fermionsinthebasis(ψ ,ψ )is Thecouplingsλ andλcanbetakenrealandpositivewithout X Y 5 lossofgenerality. Atquantumlevel,thetheoryconfineswith 2λS m thefollowingquantummodifiedmodulispace[9]: , (24) m 0 (cid:18) (cid:19) Pf(Q Q )=M M +M M =Λ4, (29) i j a a 6 6 s whilethatofthescalarsinthebasis(X,X†,Y,Y†)is whereΛ isthedynamicalscaleofSU(2)gaugeinteractions. s Because this constraint contradicts with the conditions for a m2+4λ2 S 2 2λµ2 2λmS† 0 | | − supersymmetricvacuum∂W/∂S =∂W/∂S =0,thetheory 2λµ2 m2+4λ2 S 2 0 2λmS a  − | | . breaks supersymmetry. Assuming λ > λ, the minimum is 2λmS 0 m2 0 5 at M = S = 0 and M = Λ2. We can thus eliminate  0 2λmS† 0 m2  a a 6 s  (25) M6 usingtheconstraintasM6 = (Λ4s −MaMa)1/2,andthe superpotentialofEq.(28)becomes The resulting Coleman-Weinberg potential can be expanded aroundtheoriginofS as W = λ S M λS(Λ4 M M )1/2, (30) − 5 a a− s− a a λ4µ4 3λ6µ4 ∆V = S 2 S 4+ , (26) wherewehavedenotedS6simplyasS.ThefieldSisaflatdi- 3π2m2| | − 10π2m4| | ··· rectionattreelevel.Wethusneedtoconsiderquantumeffects to find where the minimum is for S. For S Λ , the po- wherewehavedroppedanunimportantconstantandkeptonly ≫ s tentialgrowslogarithmicallywithS [10]. Thiscanbeshown theleadingtermsinλµ2/m2. Notethatsincethesuperpoten- explicitly because in this regime a weakly coupled descrip- tialofEq.(23)possessesaU(1) symmetryunderwhichS, R tionintermsofthefundamentalquarksQ isvalid,sothatthe X andY carrythechargesof2,0and2,respectively,thepo- i wavefunction renormalization factor Z can be reliably cal- tentialofEq.(26)isafunctiononlyof S 2. This, therefore, S | | culated. The potentialis V = Z−1(S)F 2, whichgrows correspondstotheKa¨hlerpotentialcorrectionsoftheformof eff S | S| forlargeS becauseoftheYukawacouplingλ. Eq.(2),withΛ2 =3π2m2/λ4. The behavior of the potential for small S is more subtle. To summarize, the complete superpotential of the model It was shown, however, in Ref. [11] that the behaviorof the presented here is given by the combination of Eqs. (1) and Ka¨hlerpotentialaroundthe originof S is indeedofthe type (23): in Eq. (2). The quartic correction due to strong coupling of W = µ2S+λSX2+mXY +κSff¯+Mff¯. (27) Qiisnotcalculable.YetnotingthatonlythecombinationλS − couplesto the strong sector, the contributionto the effective The other possible renormalizableterms mustbe suppressed Ka¨hlerpotentialofS comingfromstrongcouplingphysicsat as discussed in Section IIB. The Ka¨hler potential can be thescaleΛ′s ≈4πΛsisgivenby canonical. Λ′2 λS 2 K = s | | , (31) (4π)2 G Λ′2 (cid:18) s (cid:19) B. Dynamicalmodels where (x) is a polynomialfunctionwiththe coefficientsof G O(1)uptosymmetryfactors,andthefactorof4π isinserted Another class of models that reproduces the super- and usingnaivedimensionalanalysis[12]. Thequarticcorrection Ka¨hler potentials of Eqs. (1, 2) uses supersymmetry break- to the Ka¨hler potentialfor S is thereforeof O(λ4/16π2Λ′2) s ing theories of Ref. [8], based on quantum modified moduli fromthestrongsector. space. Consider an SU(2) gauge theory with four doublets Ontheotherhand,theColeman–WeinbergpotentialforS Qi andsix singletsSij = −Sji (i,j = 1,···,4). Itis con- due to loops of Ma givesthe quartic term of S in the effec- venienttoexploitthelocalequivalenceofSU(4)andSO(6) tive Ka¨hler potentialat O(λ2/Λ′2). Here, we have assumed groupsfor the flavor symmetry, and regardboth singletsSij thatλ andλareofthesameordserofmagnitude,andλ2/Λ′2 5 s and mesons Mij QiQj to be in the vector representation arises from the productof the one-loop factor, 1/16π2, four ≡ ofSO(6). Forthesakeofpresentation,weassumethatflavor couplingsofS,λ4,andtheinversesquareoftheM masses, a SO(6)isexplicitlybrokentoSO(5)bysuperpotentialinter- 1/(λΛ )2. Wethusfindthatforaperturbativevalueofλ,i.e. s actions, and refer to SO(5) vectors Sa, Ma (a = 1, ,5) λ<4π,thecalculablecorrectiondominatesovertheincalcu- andsingletsS6,M6.2 Thesuperpotential,whichreplac·e·s·the lab∼leoneinEq.(31). Indeed,onecanshowthatforλ5 λ first term of Eq. (1) and the second term of Eq. (2), is then the potential has a minimum at S = 0 with positive cu≥rva- givenby ture.3 With the renormalized λ , λ and Λ in Eq. (30), the 5 s W = λ S M λS M . (28) 5 a a 6 6 − − 3 The real parts of Ma become Nambu–Goldstone bosons of a sponta- neouslybrokenSO(6)symmetryandhencemasslessinthelimitλ5 → 2 ThemodelworksequallywellifSO(6)iscompletelybrokenbysuperpo- λ+0.Theirmasssquaredgoesnegativeforλ5<λ,andthenewminimum tentialinteractionsanalogoustoEq.(28). givesanonvanishingF componentforSa,insteadofS. 6 bosonshaveamassmatrix class is our previouswork [1] and its straightforwardgener- alizations based on the supersymmetry breaking mechanism m2 = ofRef.[4],wherethenegativequartictermintheKa¨hlerpo- B λ2Λ2 λλ Λ S 0 0 tentialoriginatesfromloopsoflightfields. However,thesuc- λλ5ΛsS† λ2−Λ2+5λ2sS 2 0 λ2Λ2 cessofourschemeisnotlimitedtothisclassofmodels. We − 50 s 5 s 0 | | λ2Λ2 λ−λ ΛsS† , also show other classes of models which enjoy comparable  0 −λ2Λ2s −λλ55ΛssS λ25−Λ2s+5 λs2|S|2 squucacretiscsetesr,mw.ithtree-levelordynamicaloriginofthenegative  (32) whilethefermions A. ModelsofRef.[1]andtheirstraightforwardvariations λ2Λ2 λλ Λ S† m2F = λλ5 Λs S λ2−Λ2+5λs2 S 2 . (33) Webeginbyreviewingaclassofmodelsconstructedinour (cid:18)− 5 s 5 s | | (cid:19) previousworkRef.[1].Strictlyspeaking,thesemodelsdonot Here,wehaveusedthefactthatthekinetictermsforM are reducetotheonegivenbyEqs.(1,2),sincethereareseveral a givenby K M†M /Λ2. The curvaturem2 at the origin, “S” fieldsthatcarrynonvanishingF-componentexpectation definedasV ≈=V a+ma2 Ss2+O(S 4),cantheSnbecalculated values,F . Thisslightlychangesthesituation. Forexample, 0 S| | | | S as turningonexpectationvaluesofthemessengerscannotabsorb alltheF ’s,soitdoesnotleadtoasupersymmetricminimum. S m2 = Nonetheless, the basic structure of the modelsis still that of S 5Λ2 λ2+λ2 (λ2+λ2)2 SectionII,andmanyoftheanalysesthereremainwithoutany 32πs2 (λ25−λ2)2lnλ52 λ2 +2λ2λ25ln 5eλ4 , essentialchanges. Atthequalitativelevel,eventheconstraint (cid:18) 5− 5 (cid:19) fromtunnelingcanpersist. Wesimplyhavetoreinterpretthe (34) tunnelingtothesupersymmetricminimumasthattoalower, phenomenologicallyunacceptableminimum,whichmayarise andwefindthatm2 0forallλ λ. Thisexplicitcalcula- tionconfirmsourpSow≥ercounting5,m≥2 λ2 F 2/16π2Λ2 byturningonmessengerexpectationvalues. S ≈ | S| s ≈ The models employ SU(N ), SO(N ) or Sp(N ) gauge λ4Λ2/16π2. c c c s theorieswithmassivevector-likequarks.Hereweconsideran The theory is not calculable for S Λ′/λ, and hence it in principle allows for a local minimu∼m thsere [13]. If there SU(Nc)gaugetheoryfordefiniteness,anddenotequarkand antiquark chiral superfields by Qi and Q¯i (i = 1, ,N ). pisheinndoemedenaolloogciaclamllyinaimccuemptaabtleSm∼iniΛm′su/mλ,.itInaltshoispproavpiedrewsae Wetakethenumberofquarkflavorstobeintheran·g·e·Ncf+ 1 N < 3N . Thetree-levelsuperpotentialinthissectoris havepickedtheminimumclosetotheoriginS 0,sincewe ≤ f 2 c ≈ givenby knowitexistsandthusisonafirmertheoreticalfooting. Tosummarize,thecompletesuperpotentialofthemodelis W =m Q¯iQj. (36) ij givenbythecombinationofEqs.(1)and(28): Weadoptthebasisinwhichthequarkmassmatrixisdiagonal, W =−λ5SaMa−λS6M6+κS6ff¯+Mff¯. (35) mij = −miδij with mi real and positive. We consider that all the masses are different to avoid (potentially) unwanted The other possible gauge-invariant, renormalizable terms Nambu–Goldstonebosons, andassume thatthey are ordered must be suppressed. Their coefficientsmust be smaller than asm >m > >m >0withoutlossofgenerality. foofrOd(immienn{sΛiosn,lMess})ofnoers.dimTehnesimonofduellorneedsuacnedsotofOth(e1/o1n6eπo2)f a Flooc1ralmmi i2n≪imu·mΛ··s,,wthheeNrfetheΛoryisbrtehaeksdysunpaemrsicyamlmsectarlye oonf s Eqs.(1,2)atlowenergies. Thecorrespondenceofthescales SU(N ) [4]. After integratingout the excitations of masses c is given by µ2 = λΛ2s and Λ ≃ 4πΛs/λ. Extensionsof the oforder(mΛs)1/2,therelevantdegreesoffreedomareSij = modeltoothergaugegroups,Sp(Nc)(Nc >1)andSU(Nc) Q¯iQj/Λs(i,j =Nf Nc+1, ,Nf)withthesuperpoten- (Nc >2),arestraightforward. tial − ··· W = m Λ Sii, (37) i s − IV. THEORIESWITHOUTELEMENTARYSINGLETS where we have assumed m m for simplicity. These de- i greesoffreedomobtainmasse∼soforder(mΛ )1/2/4πdueto Modelsintheprevioussectioncontainelementarysinglets s thecorrectionstotheKa¨hlerpotential. This,therefore,repro- S, so that the superpotential terms S2 and S3 must be sup- ducestheessentialstructureofEqs.(1,2). pressed“byhand”toobtaintheapproximateU(1) symme- R Thecompletesuperpotentialintheelectrictheoryisgiven tryinthesupersymmetrybreakingsector. Inthissection,we by the combination of the quark mass terms, Eq. (36), and presentmodelsthatdonotcontainanyfundamentalsinglets. generalinteractionsofthequarkswiththemessengers[1]: TheeffectivesingletS arisesasa compositefield at lowen- ergies, which allows for natural suppressions of the S2 and λ S3 terms in the low-energy effective superpotentials. One W =mijQ¯iQj + MijQ¯iQjff¯+Mff¯, (38) ∗ 7 whereM∗isthecutoffscaleofthetheory,andλij aredimen- ψ(16) H(10) X =ψψH Y =H2 sionless constants. The correspondence between the scales U(1)R −3 1 −5 2 of the present model and those in Section II is given by U(1)M −1 2 0 4 µ2 mΛ , κ λΛ /M and Λ 4π(mΛ )1/2, where s s ∗ s ≃ ≃ ≃ wehaveassumedmi mandλij λ. TABLEI:Global symmetriesof theSO(10) model intheabsence ∼ ∼ Wefinallycommentonanexampleofstraightforwardvari- ofasuperpotential. ationsofthemodelsreviewedabove.Intheabovemodels,the effective supersymmetry breaking fields are two-body com- positestates, Sii Q¯iQi, sothatthe supersymmetrybreak- ite states (n = 3 in the language of Section IIC). Accord- ∼ ingsuperpotentialofEq.(37)comesfromdimension-twoop- ingto the generaldiscussionsin Section IIC, the modelsre- eratorsintheultraviolet,Eq.(36).Wecan,however,alsocon- quiresmallcouplingsζ oranenhancementoftheoperators ij sidermodelsinwhichthesupersymmetrybreakingfieldsare ofEq.(41). n-body composite states with n > 2. Consider, for exam- ple, an SU(N ) gauge theory with N massless vector-like c f quarks, Qi and Q¯i (i = 1, ,Nf), and a massless adjoint B. SO(10)modelwithψ(16)andH(10) chiralsuperfieldX.4 Thesup·e·r·potentialofthetheoryisthen A general philosophy advocated in Ref. [1] is to discard λ W = 3TrX3−ζijQ¯iXQj. (39) a U(1)R symmetry altogether at the level of a fundamental theory. An approximate U(1) symmetry should then arise R For 1N + 1 N < 2N , this theory has a dual mag- in the low-energy effective theory as an accidental property netic2decscriptio≤n whfich is3infcrared free [14]. The dual the- of the supersymmetry breaking sector. Presumably the ear- ory is an SU(2N N ) gauge theory with N vector-like liest calculable model of supersymmetry breaking without a f c f quarks, qi and q¯i, −an adjoint, Y, and elementary singlets, U(1)R symmetry is an SO(10) gauge theory with two chi- Mij = Q¯iQj/Λ and Sij = Q¯iXQj/Λ2. The magnetic ral superfields, ψ(16) and H(10) [17]. This theory breaks s s theoryhasthesuperpotential supersymmetryunder the existence of an H mass term, and canberegardedasacontinuousdeformationofanincalcula- λ λ blemodelofsupersymmetrybreaking,SO(10)withasingle W = TrY3+ Mijq¯Yq +λSijq¯q ζ Λ2Sij. (40) −3 Λ i j i j− ij s 16[18],sincetheyholdthesameWittenindex[19]. Wecan s thususe thistheoryto constructa modelofgaugemediation (ThefirsttermisabsentforNf = 12Nc+1.) Here,wehave bycouplingittothemessengers,alongthelinesofRef.[1]. normalizedthefieldsq ,q¯,Y,Mij andSij tohavecanonical In the absence of a superpotential, the theory has global i i mass dimensions in the infrared, and we have taken Λ = symmetrieslistedinTableI.Thesesymmetriesareexplicitly el Λ Λ forsimplicity.5 brokenundertheexistenceofthemostgeneralrenormalizable mag s ≡ We find that the last two terms of Eq. (40) have the iden- superpotentialconsistentwiththegaugesymmetry: tical structure with the corresponding terms in the previous model.6 TheSijfieldscanthusservetheroleofthesupersym- W =λψψH mH2. (42) metrybreakingfields. ThestabilityofSij isensuredbyloops − 2 of the dual quarks q and q¯, and potentially unwanted light i i The general D-flat directions are parameterized by gauge- fieldsobtainmassesfromhigherdimensionoperatorsomitted invariantpolynomialsX = ψψH andY = H2. Atageneric inEq.(39). (Undertheexistenceofhigherdimensionopera- point in X-Y space, the gauge group is broken to SO(7),7 tors,anappropriatevacuummustbechoseninthedualmag- whosegauginocondensationgeneratesa nonperturbativesu- netictheory.)Togetherwiththecouplingstothemessengers perpotential η W = ij Q¯iXQjff¯, (41) 21/5 M∗2 W =cΛs , (43) np X2/5 this provides gauge mediation models in which the effective supersymmetry breaking fields are three-body compos- wherecisacalculableO(1)numericalcoefficient,andΛ the s dynamical scale of SO(10). Since the value of c is not im- portantintherestofthediscussions,wesetc=1bysuitably changingthenormalizationofΛ . s 4 Theabsenceofthemassesisnotcrucial. Theyjusthavetobesuppressed Themodeliscalculablewhenλ 1andm Λ . Inthis s ≪ ≪ sufficientlysothattheydonotaltertheessentialdynamics. limit, we canfirstignorethemassterm mH2/2andfinda 5 AsimilarsuperpotentialtoEq.(40)isobtainedforNf = 21(Nc+1)if − Ncisodd. Inthiscasetherelevantinfrareddegreesoffreedomareqi,q¯i andSij,sothatthefirsttwotermsofEq.(40)areabsent. Themodelalso works in this case, since a possible nonperturbative superpotential term (detSij)2S−1Mlk[15]isirrelevant. 7 This isanon-standard embedding, where SO(7)is embedded into the kl 6 Amorecomplicatedcasewithoutanaccidentallow-energyU(1)R sym- SO(8)subgroupofSO(10)afterweusethetrialitythatswitchesthevec- metrywasconsideredinRef.[16]. torrepresentationandoneoftheMajorana–Weylspinorrepresentations. 8 modulispaceofsupersymmetricvacua onefieldS significantlylighterthantherestoftheexcitations (suchasthe Y fieldabove),thenonecanwritea low-energy 2 5/7 effectivetheorythatcontainsonlyasinglecompositefieldS. X = Λ3, Y : arbitrary. (44) h i 5λ s By shifting the origin of S such that S = 0 at the mini- (cid:18) (cid:19) mum,thesuperpotentialcontainsalineahrtierm,andtheKa¨hler OnecanverifythattheU(1) anomaliesaresaturatedbythe potentialtakes genericallythe form of Eq. (2). We can then M compositeY alone. Aslongasλ 1andhence X Λ3 constructamodelofgaugemediationsimplybycouplingthe thetheoryisweaklycoupled,and≪theKa¨hlerpotehntiail≫forXs gaugeinvariantoperatorStothemessengerbilinearff¯inthe and Y can be worked out with the tree-levelapproximation. superpotential. Weuseasimilartechniquetothatin[20]. Theresultis A small variation of this picture is obtained, for exam- ple, in the SU(5)modelwith A(10), F(5), andtwo F¯(5∗) i K =x2+ 1 X + 1 Y 2. (45) (i = 1,2) [17]. This model can be viewed as a continuous √2x| | 4x2| | deformation of the incalculable model with only A and one F¯ [21],onceamasstermisgiventoapairofF andF¯. The Here,xistherealpositivesolutiontotheequation∂K/∂x= mostgeneralsuperpotentialofthemodelis 0: W =λAF¯ F¯ +λ′AAF mF¯ F, (49) 1 2 1 4x4 √2X x Y 2 =0, (46) − − | | −| | whilethenonperturbativesuperpotentialis whichcanbesolvedanalyticallyusingFerrari’smethod.With Λ6 X integratedoutalongthemodulispaceofEq.(44),thelow- Wnp = [(AF¯ F¯ )(As AF)]1/2, (50) energytheoryisonewithY alone,whoseKa¨hlerpotentialcan 1 2 beexpandedaroundtheoriginas whereΛ isthedynamicalscaleofSU(5).ThereisnoU(1) s R symmetry, but there is a globalSU(2) U(1) symmetry in 3 Y 2 Y 4 thismodel. Intermsofthegauge-invaria×ntpolynomialsX = K = X 2/3+ | | | | +O(Y 6). (47) 2|h i| 2 X 2/3 − 6 X 2 | | (1/2)(AF¯1F¯2),Y =(1/√2)(AAF)andSi =F¯iF,thetree- |h i| |h i| levelKa¨hlerpotentialcanbeworkedoutandexpandedinS i ItisguaranteedthatthisKa¨hlerpotentialdependsonlyonthe as combination Y 2becauseoftheU(1) invarianceofthethe- | | M S†S oryintheabsenceofthemassterm mY/2.Wethusfindthat K = 6(X + Y )2/3+ i i i the low-energytheory, characterize−d by the Ka¨hler potential | | | | 2(X + Y )2/3 | P| | | ofEq.(47)andthelinearsuperpotentialtermW = mY/2, ( S†S )2 hasanaccidentalU(1) symmetry,underwhichY −carriesa i i i +O(S6). (51) R −24(X + Y )2 i chargeof+2. P| | | | TherestofthediscussionreducestothegeneraloneinSec- TheglobalSU(2) U(1)invarianceofthetheoryguarantees × tionII.Notethatthenegativecoefficientforthequarticterm thatitdependsonS onlythroughthecombination S†S . i i i i in the Ka¨hler potential originates not from one-loop effects After minimizing the superpotential without the mass term asinthemodelsinSectionIVAbutratherfromthetree-level (m = 0), bothX andY arefixedandcan beintegPratedout. Ka¨hlerpotentialalongtheD-flatdirections.Correspondingly, The low-energytheory consists of S alone, which saturates i therearenootherlightfieldsinthetheorytogeneratethequar- theSU(2) U(1)anomalies.Giventhenegativequarticterm ticterm,whichmakesthemodelmoreeasilycompatiblewith in the Ka¨h×ler potential, the mass term breaks supersymme- cosmology.Thecouplingtothemessengersisgivenby try with a stable minimum at the origin S = 0. One can i verify that both S and S acquire positive squared masses. η 1 2 W = H2ff¯. (48) In fact, this model generalizes to SU(2k + 1) with an an- 2M ∗ tisymmetric tensor A, one fundamentalF and (2k 2) an- The correspondence of the scales can be worked out easily tifundamentals F¯i. With AkF and AF¯iF¯j terms in−the su- by canonically normalizing the Y field in Eq. (47): S perpotential and the nonperturbative superpotential Wnp Y/√2 X 1/3. It is given by µ2 mΛs/λ5/21, κ ≡ [(AkF)Pf(AF¯iF¯j)]−1/2, the low-energy theory is given ∝in ηΛs/λ|5h/21iM| ∗ andΛ Λs/λ5/21. Fo≃ranappropriaterang≃e terms of Si = F¯iF that match the anomalies of the global of the parameters, the≃superpotential of the model can be a Sp(k 1) U(1)symmetry.AmasstermmF¯ F wouldbreak 1 − × generic one compatible with the gauge symmetry, as in the supersymmetrywithastableminimumattheorigin.Thenthe modelsofRef.[1]. couplingtothemessengersF¯iFff¯/M∗ makesgaugemedia- In fact, the basic dynamics of the model just described is tionpossible. more general. Consider a modelof dynamicalsupersymme- try breaking in which some of the classical flat directions are lifted by superpotential interactions. By choosing these C. ModelswithincalculableKa¨hlerpotentials interactions appropriately, one can make expectation values of fields larger than the dynamical scale, and thus make the The first model we present here uses the supersymmetry modelcalculable. Now,ifthemodelallowsformakingonly breaking theory of Ref. [22], based on the phenomenon of 9 quantumsmooth-outofclassicalsingularitiesinmodulispace. U(1) andU(1)3 anomalies,so weexpectittohavea non- R R Consider an SU(2) gauge theory with a single chiral super- singular Ka¨hler potential at the origin. An introduction of a field Q in the I = 3/2 representation. The gauge invariant linearterminA4 wouldthenbreaksupersymmetry.Asinthe chiral operator in this theory is u = QQQQ, and we intro- SU(2)model,however,thequartictermintheKa¨hlerpoten- ducethefollowingtree-levelsuperpotential: tialisnotcalculable. We thushaveto assumethatits coeffi- cientisnegativeinordertousethistheory. ζ W = u, (52) −M YetanotherexampleisSO(N)theorieswithN 4vectors. ∗ − Theyhavetwoinequivalentbranches,onewithandtheother whereM isthecutoffscaleofthetheory,presumablyofor- ∗ without a dynamical superpotential [25]. All anomalies are der MPl, and ζ a dimensionless constant. Since u saturates saturatedbythemesonsMij = QiQj.9 Addingamassterm nontrivial’tHooftanomalymatchingconditions[23],weex- to just one of the flavors, the theory breaks supersymmetry. pect that u is the only low-energy degree of freedom. The AgainthequartictermintheKa¨hlerpotentialisnotcalculable Ka¨hlerpotentialforuisthengivenby andwe haveto simply assume thatitscoefficientis negative u2 tousethistheory. K =Λ2 | | , (53) sG Λ8 (cid:18) s (cid:19) whereΛ isthedynamicalscaleofSU(2)gaugeinteractions. s For u Λ4, (x) is expected to be a polynomial func- | | ≪ s G tion with the coefficientsofO(1) upto symmetryfactors— V. RELATEDMODELS theclassicalsingularityattheoriginofuissmoothedoutby quantumeffects. Denoting the field with the canonical dimension by S = Inthissectionwepresentmodelsthatdonotexactlyfallin u/Λ3,thelow-energysuper-andKa¨hlerpotentialsfor S thecategorydiscussedinSectionII.Wefirstpresentmodelsin Λ tasketheformgivenbythefirsttermofEq.(1)andE|q.|(2≪), whichthelow-energyeffectivetheoriescontainmorethanone s respectively. In the present theory, however, the sign of the field, S. In general, these theories have multiple composite quarticterminEq.(2)isincalculable,whileitmustbenega- fieldsXi (i = 1,2, )atlow energies,whicharestabilized ··· tiveinorderforthemodeltowork.Wethusmakeadynamical due to complicated Ka¨hler potentials. Models of gauge me- assumptionthatthesignofthistermisnegative. diation are then obtained by coupling the degree of freedom Thecompletesuperpotentialofthemodelisgivenby responsibleforsupersymmetrybreakingtothemessengersin the superpotential. In the case that the Ka¨hler potentials are ζ η W = u+ uff¯+Mff¯, (54) complicated,thelow-energyfieldsXicannotberegardedsim- −M∗ M∗3 plyasmultiplecopiesofanSfield,incontrastwiththecasein someofthepreviousmodelssuchastheonesinSectionIVA. where η is a dimensionless constant. The model reduces at low energies to that of Eqs. (1, 2), with µ2 = ζΛ3s/M∗, We then consider models that do not contain any degree κ = ηΛ3s/M∗3 and Λ ≃ Λs. In addition to the terms in of freedom which is significantly lighter than the dynamical Eq. (54), the most general terms consistent with the gauge scale. Whilethesemodelsaregenerallyincalculable,models symmetrymay present. The coefficientsofthese termsneed ofgaugemediationcanbeobtainedbycouplingthemessen- notbesuppressed,sincetheconstraintsofEq.(12)arealmost gerstoappropriatecompositeoperators. automaticallysatisfiedbecauseofthecompositenessofS. While the models discussed in this section do not have As discussed in Section IIC, the dimensionless coupling an identicallow-energystructure to those of Section II, they ζ must be small in order for the model to work (for η 1 ∼ sharemanyfeatures. Inparticular,thebasicconstructionsof andM M ;seeEq.(22).) Successfulparameterregions include∗,∼for exPalmple, Λ 1015 GeV, M 1018 GeV, themodelsarequitesimilar—wesimplyprepareasupersym- ζ 10−8,η 1,andMs ≃106GeV. ∗ ≃ metrybreakingmodelthathasastablesupersymmetrybreak- ≃ ≃ ≃ ingminimum(eitherglobalorlocal),andthencoupletheop- AnearlyidenticalanalysiscanbemadeonanSU(6)gauge erator responsible for supersymmetry breaking to the bilin- theory with a rank-three antisymmetric tensor Aijk. For earof(genericallymassive)messengers. Manyofthegeneral general D-flat configurations, the gauge group is broken to analysesinSectionIIalsopersist.Inparticular,ageneralcon- SU(3) SU(3), eachofwhichdevelopsagauginoconden- × straintonparametersinEq.(22)persists,despitethefactthat sate. Dependingon the relative phase between the two con- thepowersofζ appearinginthegauge-mediatedandgravity- densates,thenonperturbativesuperpotential mediatedcontributionsinEqs.(19,20)cannowbedifferent. Λ5 Here, ζ representsthe coefficientin frontof the operatorre- W = 1+( 1)1/3 s , (55) np − (A4)1/2 sponsible for supersymmetry breaking in the superpotential. WewillprovethisfactinSectionVA. (cid:0) (cid:1) canidenticallyvanish.8 ThecompositefieldA4 saturatesthe 9 Discrete anomalies are not matched because of a condensate 8 Foranalternativederivationofinequivalentbranches,see[24]. hQN−4WαWαi6=0;see[26]. 10 A. Modelswithmultiplelow-energyfields messengersaregivenby η ηΛ3 Aclassoftheoriesthatbreakssupersymmetrydynamically M =M + X M + s , (62) ischiralgaugetheorieswhichdonotpossessclassicalflatdi- mess M∗2h 1i≈ ζ3/7M∗2 rections, and in which global symmetries are spontaneously and broken[21,27,28]. Thesetheorieshavestablesupersymme- try breaking vacua, and one can construct models of gauge η ηζ3/7Λ4 mediationbycouplingthesetheoriesto(genericallymassive) F = F s, (63) messengersf,f¯. Herewepresentonesuchtheoryexplicitly, mess M∗2h X1i≈ M∗2 and analyze its relations to the class of models discussed in where we have omitted O(1) coefficients. The conditionfor SectionII. avoidingtachyonicmessengersis Consider an SU(3) SU(2) gauge theory with the mat- tercontentQ(3,2),U(×3∗,1),D(3∗,1)andL(1,2),withthe η1/2ζ3/14Λ2 tree-levelsuperpotentialW =ζQDL. Thistheorybreakssu- M > s. (64) mess M persymmetry at the vacuum with expectation values for the ∼ ∗ fieldsv Λ /ζ1/7 [28]. Here, Λ isthedynamicalscale of s s Aslongasthisconditionissatisfied,theminimumintheorig- ∼ SU(3),whichweassumetobelargerthanthatofSU(2).The inaltheorystaysasalocalsupersymmetry-breakingminimum vacuumenergyisgivenbyV ζ10/7Λ4. ∼ s inthetheorywiththemessengers. It is useful to analyze the theory in terms of the gauge- Theresultinggauge-mediatedcontributiontothescalarand invariant composite fields: X = QDL, X = QUL and 1 2 gauginomassesintheSSMsectorisoforder X = det(Q¯ Qj), where Q¯ (D,U) and j = 1,2 is 3 i i ≡ theSU(2)index. Includingnonperturbativeeffects,thelow- g2 ζ3/7ηΛ4 energyeffectivesuperpotentialis mSUSY ≈ 16π2M2M s , (65) ∗ mess Λ7 W =ζX +2 s. (56) while generic gravity-mediated contributions to the SSM- 1 X3 sectorscalars,arisingfromKa¨hlerpotentialtermsoftheform Q†QΦ†Φ/M2,U†UΦ†Φ/M2 andsoon,areoforder Forζ 1, expectationvaluesforthe fields aremuchlarger Pl Pl ≪ thanthedynamicalscale,v Λ ,sothattheKa¨hlerpotential ≫ s ζ5/7Λ2 is well approximated by the tree-level one. In terms of the m s, (66) compositefields,itisgivenby[20] 3/2 ≈ MPl A+Bx whereΦ representsmatterandHiggssuperfieldsintheSSM K =24 , (57) x2 sector. ToobtainmSUSY = O(100GeV 1TeV), weneed ∼ totake whereA=(X†X +X†X )/2,B =(X†X )1/2/3,and 1 1 2 2 3 3 ζ3/7ηΛ4 s 100TeV. (67) x=4√B cos 1arccos A . (58) M∗2Mmess ≈ 3 B3/2 (cid:18) (cid:19) The tunneling rate from the supersymmetry breaking mini- Byminimizingtheresultingscalarpotential,wefindthatthe mum to the true supersymmetric (runaway) minimum does minimumisat notgiveaverystrongconstraintontheparameters. FromtheexpressionsinEqs.(65,66),wefindthattheratio Λ3 Λ4 X 0.50 s , X =0, X 2.58 s , (59) ofgravity-togauge-mediatedcontributionsisgivenby h 1i≃ ζ3/7 h 2i h 3i≃ ζ4/7 m 16π2ζ2/7M2M with 3/2 ∗ mess m ≈ g2 ηΛ2M SUSY s Pl FX1 ≃−2.57ζ3/7Λ4s, FX2 =0, FX3 ≃3.42ζ2/7Λ(5s6,0) > 100 ζ1/2M∗ . (68) η1/2M where F represent the vacuum expectation values for the ∼ Pl Xi auxiliary componentsof chiral superfields X . We can thus i Notethatthisinequalityisidenticaltothecorrespondingone constructa modelofgaugemediationbycouplingX to the 1 in Section IIC, Eq. (21), despite the fact that the powers of messengersinthesuperpotential. ζ in Eqs. (64, 65, 66) are different from the corresponding Therelevantsuperpotentialforthemessengersis onesinSectionIIC.Therefore,thegeneralboundinEq.(22) η also applies to the present case. In particular, the coupling W = M2QDLff¯+Mff¯, (61) ζ must be suppressed for η 1 and M∗ MPl. An ex- ∗ ∼ ∼ ampleofphenomenologicallysuccessfulparameterregionsin whereM isthecutoffscaleofthetheory.Thesupersymmet- thepresentmodelisζ 10−8, η 1, Λ 1012−13 GeV, ∗ s ric andholomorphicsupersymmetrybreakingmassesforthe M 1018GeVandM≃ 105−6G≃eV. ≃ ∗ ≃ ≃