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PublishedforSISSAbySpringerReceived:July21,2010Accepted:August7,2010Published:August30,2010BeyondMFVinfamilysymmetrytheoriesoffermionmassesZygmuntLalak,aStefanPokorskia,1andGrahamG.
Rossb,caPhysicsDepartment,UniversityofWarsaw,InstituteofTheoreticalPhysics,Ul.
Hoza69,PL-00681Warsaw,PolandbCERN,1211Geneva23,SwitzerlandcRudolfPeierlsCentreforTheoreticalPhysics,UniversityofOxford,1KebleRoad,Oxford,OX13NP,U.
K.
E-mail:Zygmunt.
Lalak@fuw.
edu.
pl,Stefan.
Pokorski@fuw.
edu.
pl,g.
ross1@physics.
ox.
ac.
ukAbstract:MinimalFlavourViolation(MFV)postulatesthattheonlysourceofavourchangingneutralcurrentsandCPviolation,asintheStandardModel,istheCKMmatrix.
HoweveritdoesnotaddresstheoriginoffermionmassesandmixingandmodelsthatdousuallyhaveastructurethatgoeswellbeyondtheMFVframework.
Inthispaperwecom-paretheMFVpredictionswiththoseobtainedinmodelsbasedonspontaneouslybroken(horizontal)familysymmetries,bothAbelianandnon-Abelian.
ThegenericsuppressionofavourchangingprocessesinthesemodelsturnsouttobeweakerthanintheMFVhy-pothesis.
Despitethis,inthesupersymmetriccase,thesuppressionmaystillbeconsistentwithasolutiontothehierarchyproblem,withmassesofsuperpartnersbelow1TeV.
AcomparisonofFCNCandCPviolationinprocessesinvolvingavarietyofdierentfamilyquantumnumbersshouldbeabletodistinguishbetweenvariousfamilysymmetrymodelsandmodelssatisfyingtheMFVhypothesis.
Keywords:BeyondStandardModel,QuarkMassesandSMParameters,SupergravityModelsArXivePrint:1006.
23751HansFischerSeniorFellow,InstituteforAdvancedStudies,TechnicalUniversity,Munich,GermanyOpenAccessdoi:10.
1007/JHEP08(2010)129Contents1Introduction22Minimalavourviolationandbeyond32.
1MFV32.
2Themessengerscale42.
3BeyondMFV53Familysymmetrymodels63.
1Abelianfamilysymmetry73.
2Non-Abelianfamilysymmetry94ComparisonofMFVandfamilysymmetrymodels.
Dimension3quarkbilinearoperators95ComparisonofMFVandfamilysymmetrymodels.
Dimension6fourquarkoperators145.
1Factorisationofoperators145.
1.
1Fi=1,Fj=1,i=joperators145.
1.
2Fi=2,Fj=2,i=joperators145.
1.
3Non-factorisableoperators155.
2Determinationofthecoecientsofthedimension6operators156SUSY176.
1IdenticationofthescaleΛ176.
2SUSYGIMsuppression186.
3FactorisationofoperatorsinSUSY207Comparisonwithexperiment207.
1Experimentalboundsonthesquarkmasses207.
2Familysymmetrypredictionforsoftmasses227.
2.
1Contributionfromnon-degeneracyofsquarkmassesandD-terms227.
2.
2Contributionsfromodiagonalsquarkmass248Aterms259Summaryandconclusions26AFamilysymmetrymodels28–1–1IntroductionTheStandardModel(SM)providesanaccuratedescriptionofallthepresentlyavailableexperimentaldataforavourchangingneutralcurrent(FCNC)andCPviolatingprocesses.
TheirprecisionisalreadygoodenoughtoleaveroomonlyforsmallcorrectionsfromthephysicsbeyondtheSM(BSM).
Thus,ifthescaleofnewphysicsisO(1TeV),asisrelevantforthehierachyproblem,itsavourstructuremustbestronglyconstrained.
AninterestinghypothesisthatisconsistentwiththepresentconstraintsisthatthephysicsbeyondtheStandardModelsatisestheprincipleofMinimalFlavourViolation(MFV):accordingtoittheonlysourceofFCNCandCPviolation,asintheSM,istheCKMmatrix.
TheMFVconjecturecanbeimplementedinsomeconcreteBSMtheories.
Forinstance,itissatisedintheMSSMwithuniversalsoftscalarmassesandcoecientsofthetrilinearsofttermsproportionaltotheassociatedYukawacouplings.
ThenewFCNCandCPviolatingeectsarethensmallenoughtobeconsistentwiththedataevenforsquarkmasseswellbelow1TeV.
HoweveriftheMSSMisextendedtoincludeaspontaneouslybrokenfamilysymmetryMFVisviolatedevenif,beforespontaneousfamilysymmetrybreaking,thesoftscalarmassesareuniversalandthecoecientsofthetrilinearsofttermsareproportionaltotheassociatedYukawacouplings.
ThephenomenologicalimplicationsoftheMFVhypothesiscanbeinvestigatedinanelegantmodelindependentwaybyusinganeectiveeldtheoryapproach(EFT)[1].
InthisframeworktheSMlagrangianissupplementedbyallhigherdimensionoperatorsconsistentwiththeMFVhypothesis,builtusingtheYukawacouplingstreatedasspurionelds.
ThepotentialdeviationsofthedatafromtheSMpredictionsarethenparametrizedintermsoffewfreeparameterssuchastheinverse(messenger)massscaleassociatedwiththehigherdimensionoperators,withtheiravourstructurexedbythestructureoftheCKMmatrices.
TheMFVhypothesisreliesonthephenomenologicalknowledgeoftheCKMmatrixandimplicitlyassumesthattheeventualtheoryoffermionmassesisconsistentwithit.
However,thismaynotbethecase.
Indeed,explicittheoriesoffermionmassesandmix-ingusuallyviolatetheMFVhypothesisanditisthepurposeofthepresentpapertoinvestigatethisproblem.
OurlaboratorywillbeFroggatt-Nielsen-likemodels,[2–9],withspontaneouslybrokenfamilysymmetriesandfamiloneld(s)whosevacuumexpectationvalues(vevs)determinetheYukawacouplings(foranearlierdiscussiononthepossibleviolationoftheMFVhypothesisinmodelswithbrokenfamilysymmetriessee[11,12]and[13]).
In[13]adetailedphenomenologicalanalysishasbeenperformedfortheMSSMwithsomeAbelianandnon-Abelian[9]familysymmetries.
1Following[1]wewillanalysethiscaseusingtheSMEFTapproach.
HorizontalsymmetriesmustthenbeimposedonthehigherdimensionoperatorsoftheeectiveSMandthefamiloneldscanbeusedintheirconstructionasspurionelds.
Althoughtheeectiveeldtheoryapproachisquitegeneral,caremustbetakenwheninterpretingtheboundsonthemessengermassscalebecausetheinterpretationdoesdependonthenatureofthenewphysics.
Thisoccursifthereismorethanonescaleassociated1Seealso[10].
–2–withBSMphysics.
WeshallillustratethisproblemwithadetaileddiscussionoftheSUSYcaseinwhichtherearetwobasicscales,theSUSYbreakingscaleandthefamilymessengerscale.
InthiscaseitisusefultoapplytheEFTapproachabovetheSUSYbreakingscaleinthemannersuggestedin[14,15],andweextendourfamilysymmetryanalysistocoverthisapproachtoo.
WerstreviewtheMFVhypothesisfortheSMviewedasanEFT.
WethenconstructtheanalogoushigherdimensionoperatorsinFroggatt-Nielsenliketheoriesusingthespu-riontechniquegeneralizedtothiscase(foranearlierdiscussionoftheuseofthespuriontechniquebeyondMFVsee[1]andinmodelswithfamilysymmetriessee[11]).
Weillus-tratetheexpectationbycomparingtheboundsontheeectivemessengerscaleobtainedinMFVandinavarietyoffamilysymmetrymodelsthathavebeenproposedtoexplaintheobservedpatternoffermionmassesandmixings.
InthesecondpartofthepaperwediscusstheproblemoftheinterpretationoftheeectivemessengerscaleinsupersymmetricmodelsandextendouranalysistoanEFTdescriptionabovetheSUSYbreakingscale.
Inthispaperwewillconsideronlyavourchangingprocessesoriginatinginthequarksector.
2Minimalavourviolationandbeyond2.
1MFVTheSMfermionsconsistofthreefamilieswithtwoSU(2)Ldoublets(QLandLL)andthreeSU(2)Lsinglets(UR,DRandER).
Eachoftheseeldsisatripletinavourspace.
ThelargestgroupofunitaryeldtransformationsthatcommuteswiththegaugegroupisU(3)5.
ThiscanbedecomposedasGF=SU(3)3qSU(3)2lU(1)5,(2.
1)whereSU(3)3q=SU(3)QLSU(3)URSU(3)DR,SU(3)2l=SU(3)LLSU(3)ER.
ThesymmetryisbrokenbytheYukawainteractions,L=QLYDDRH+QLYUURHc+LLYEERH+h.
c.
,(2.
2)whereHc=iτ2Hand=v2/2.
TreatingtheYukawacouplingmatrixasspurioneldstransformingasYU(3,3,1)SU(3)3q,YD(3,1,3)SU(3)3q,YE(3,3)SU(3)2l(2.
3)thefullLagrangianhasanSU(3)5invariantform.
MFVpostulatesthattheonlysourceofGFbreakingaretheYukawaspurionsandparameterisesthehigherdimensionavourviolatingtermsbyusingthemtoconstructthemostgeneralSU(3)5invariantsetofhigherdimensionoperatorsthatmakeupthefulleectiveeldtheory.
Theleadingtermsarethedimension6operatorsgivenintable1.
Following[1]itisconvenienttowritethemintermsofproductsoftwo-fermionoperatorswhichseparatelyshouldbeSU(3)5invariant,becausetheavourstructureofalltheoper-atorsoftable1isdeterminedbytheavourstructureofthesetwofermionoperators.
In–3–FlavourviolatingΛMFV(inTeV)dimensionsixoperator+O0=12(QLλFCγQL)26.
45.
0OF1=HDRλdλFCσνQLFν9.
312.
4OG1=HDRλdλFCσνTaQLGaν2.
63.
5O1=(QLλFCγQL)(LLγLL)3.
12.
7O2=(QLλFCγτaQL)(LLγτaLL)3.
43.
0OH1=(QLλFCγQL)(HiDH)1.
61.
6Oq5=(QLλFCγQL)(DRγDR)1Table1.
Boundsonthesuppressionscaleofthedimension6operatorsintheMFVscenario.
TheSMisextendedbyaddingavour-violatingdimension-sixoperatorswithcoecient±1/Λ2MFV(+ordenotetheirconstructiveordestructiveinterferencewiththeSMamplitude).
D'Ambrosioetal.
[1]reporttheboundsat99%CLonΛMFV,inTeV,forthesingleoperator(inthemostrepresentativecases).
particularthefourfermionoperatorsfactoriseintotheproductoftwofermionoperators.
AsweshalldiscussthisfactorisationdoesnotalwaysapplybeyondMFV.
DuetothesmallnessofthedownquarkYukawacouplingsthedominantoperatorsdisplayedintable1haveexternaldownquarksforwhichtheupYukawacouplingsareresponsiblefortheavourchangingterms.
Theleadingtwo-fermionoperatorsfromwhichonemaydeterminetheMFVpredictionsfortheoperatorsoftable1areQLYuYuQL,DRYdYuYuQL.
(2.
4)TheavourstructureoftheseoperatorsisdeterminedbytheavourstructureofYukawamatrices.
Intheelectroweakbasiswherethedown-typequarksaremasseigenstates(EWDD),toaverygoodapproximation,itisdeterminedbytheentriesproportionaltothe"square"ofthetopquarkYukawacoupling:(YuYu)ijwherei,jareavourindices.
InthisframeλFC=(YuYu)ij≈λ2tU3iU3jwherethematrixUistheCKMmatrix.
TherelativemagnitudeofvariousFCNCeectsisdeterminedbytheorderofmagnitudeofthemixinganglesandtheirabsolutevaluesdependinadditionontheratiosofthecouplingsofthoseoperatorsoverthe(unknown)scaleofnewphysicsthathasbeenintegratedout.
Forthesakeofeasyreference,intable1wequotetheboundsonthesuppressionscaleΛfromref.
[1],obtainedbyusingthemeasuredvaluesofthemixingangles.
Here,thescaleΛisdenedasaneectivescale,withtheoperatorcouplingequalto1.
Ifthenewphysicscontributese.
g.
onlyatthelooplevel,theboundonitsactualphysicalscaleislowerbyfactorα.
2.
2ThemessengerscaleThedimension6operatorsoftable1appearintheeectiveLagrangianmultipliedbyafactor1/Λ2thathasthedimensionoftwoinversepowersofmass.
Thisfactorarisesdue–4–tothepropagatorofthemessengerstatethatisresponsibleforgeneratingtheoperatorandthathasbeenintegratedoutwhenconstructingtheeectiveLagrangianrelevantatenergyscaleslessthanthemessengermass.
Inphenomenologicalstudiesthelowerlimitonthisfactorisdeterminedandgivesanestimateofthepossiblescaleofnewphysics.
Howeversomecareisneededininterpretingthislimitbecausetheremaybemorethanonemessengerscaleinvolved.
Inparticular,inarealisticextensionoftheStandardModelthereusuallyexistsamechanismeasingthehierarchyproblem,withanassociatedmassscaleΛh.
Thisrolecouldbeplayedbysupersymmetrywiththecharacteristicscaleofmasssplittingsinsupermultiplets,MSUSY,orbyastronglycoupledgaugetheorywiththeconnementscaleΛconforbythemassscaleofKaluza-KleinstatesinRandall-Sundrummodels.
Thesectorresponsiblefortheavourviolationhasitsowncharacteristicscale,whichweshallcallthefamilymessengerscaleMwhichcanbelargerthanΛhorcoincidewithit.
TheeectiveLagrangianisrelevantatenergyscaleslessthanthemessengermassMandlessthanΛh.
Dependingonthedetailsofthetheory,thesuppressionfactorcouldbeoneofthefollowing:1/Λ2h,1/(ΛhM),1/M2.
IfMΛh,operatorssuppressedbyonlytherstfactorwillbethemostimportant.
WewillreturntoadetaileddiscussionoftheidenticationofΛinsupersymmetricmodelsinsection5.
2.
3BeyondMFVMFVisbasedontheveryrestrictiveassumptionthattheYukawacouplingsaretheonlysourceofavoursymmetrybreaking.
Thisassumptionisnotvalidformany(most)oftheattemptstobuildatheoryoffermionmassesandmixingandsoitisofinteresttodevelopaformalismcapableofdescribingsuchmodelsandhighlightingthemaindiscrepanciestobeexpectedfromMFV.
Considerthecaseofthetwofermionoperatorsjustdiscussed.
ThemostgeneralsetofnontrivialSU(3)3qrepresentationsofthetwofermionoperatorsthatcanbemadeupofquarksandantiquarksis(3,3,1),(3,3,1),(3,1,3),(3,1,3),(1,3,3),(1,3,3),(8,1,1),(1,8,1),(1,1,8)(2.
5)InMFV,c.
f.
equation(2.
4),thefundamentalYukawacouplingstransformas(3,3,1)and(3,1,3)andthesemustbecombinedwiththequarkbilinearstoformSU(3)3qinvariantscorrespondingtothedimension6fourfermionoperatorsoftable1.
Howeverinmod-elsoffermionmasstheremaybespurions,combinationsoffundamentalfamiloneldswithnon-vanishingvacuumexpectationvalues(vevs),withdierentSU(3)3qtransforma-tionpropertiestothoseoftheYukawacouplings.
Thisthenleadstonewpossibilitiesfortheconstructionoffourquarkoperators.
Forexampleinreference[11],theeectoffunda-mentalspurionstransformingas(8,1,1)wasstudiedindetail.
However,asstressedbelow,buildingSU(3)3qinvariantcombinationsoffourquarkoperatorsandfamiloneldstypicallyinvolvefamiloncombinations,i.
e.
spurions,transforminginallpossibleSU(3)3qrepresen-tations,notnecessarilywithcorrelatedmagnitude,inamannerthatdoesnotcorrespondtobuildingfourfermionoperatorsstartingfromasinglefundamentalspurion.
Animportantconsequenceofthisisthatfamilysymmetriesoftenrequirefewerin-sertionsofthefamiloneldsthanwouldbeexpectedinMFV.
Forexampletoconstruct–5–the(8+1,1,1)representationinMFVrequirestwoYukawaspurioninsertions,QLYuYuQLinvolvingLRandRLcouplingsatthemessengerlevelbutcanbedirectlyconstructedfromfamiloneldsinamannernotinvolvingtheRHsector.
3FamilysymmetrymodelsInthispaperweshallbeconcernedwiththedeparturesfromMFVtobeexpectedinmodelsoffermionmassesandmixingsbasedonspontaneouslybrokenfamilysymmetries.
Awidevarietyoffamilysymmetrieshavebeenconsidered,varyingfromoneormoreAbelianfamilysymmetriesortheirdiscretesubgroupstonon-Abeliansymmetriesordiscretenon-Abeliansymmetries.
Suchmodelshavebeenshowntobeabletogeneratethehierarchicalstructureofquarkmassesandmixingangles.
ToillustratetheimplicationsforFCNCwewillconsideravarietyofrepresentativemodels.
Thersttwomodels[36]haveasingleAbelianfamilysymmetryfactorandasinglefamiloneldwhosevacuumexpectationvalue(vev)spontaneouslybreaksthesymmetry.
Thethird(supersymmetric)model[24,25]alsohasasingleAbelianfactorbuthastwofamiloneldsthatacquireequalvevsalongaD-atdirection.
InadditiontheHiggseldcancarryachargeunderthesymmetry.
ThemodelgeneratesatexturezerothatleadstoaprecisepredictionfortheCabbiboangleinexcellentagreementwithexperiment.
Thefourthmodel[37]involvestwoAbelianfactors.
Unlikealltheothermodelsconsideredhere,inthecurrentquarkbasis,thedominanto-diagonaltermgeneratingtheCabibboanglecomesfromtheup-andnotthedown-quarkmassmatrix.
ThefthmodelinvolvesaNon-Abelianfamilysymmetryandthemodelwasdevelopedtodescribebothquark,chargedleptonandneutrinomassesandmixing.
Thegroupisadiscretenon-AbeliansubgroupofSU(3)familysymmetry,thediscretesubgroupchosenbecauseitleadstoneartri-bi-maximalneutrinomixinginagreementwithexperimentalmeasurements.
HoweverthestructureofthelowdimensiontermsisdeterminedbytheSU(3)symmetryandsoforthediscussionhereitdoesnotmatterthatonlyadiscretesubgroupisunbroken.
Finallyweconsideramodel[39]withthreeAbelianfactorsbasedonthestructurefoundinF-theorystringmodels[38]inwhichthefamilysymmetryisasubgroupoftheunderlyingE(8)stringsymmetry.
InthistheemergenceofthreeAbelianfactorsisnaturalandunlikethepreviousmodelsthechargesofthefermionsarestronglyconstrainedbytheE(8)symmetry.
ForthecasethatthesymmetryisAbelian,alltheindependentSU(3)3qrepresentationsofspurionsbilinearinthefermioneldsaregeneratedatafundamentallevel.
Asubsetofthedimension6fourfermionoperatorsarealsofundamentalandcannotbebuiltfromthetwofermionoperators,i.
e.
theydonotfactorise.
Asweshalldiscussthisleadstoapotentialenhancementofavourviolation.
Forthecasethesymmetryisnon-Abelian,asforMFVonlyarestrictedsetoffundamentalspurionrepresentationsbilinearinthefermioneldsarepresentandthedimension6operatorsmaybebuiltfromthem.
OnemayworryaboutthepossibleeectsofGoldstonemodesresultingfromthespon-taneousbreakingofthefamilysymmetry.
Forthecasethatthefamilysymmetryisalocalgaugesymmetrythefamilonsprovidethelongtitudinalcomponentofthefamilygaugeboson.
Ifthesymmetrybreakingscaleislargethesebosonswillnotappearintheeective–6–1.
QLXQLLQL2.
DRXDRRDR3.
URXURRUR4.
QLXDLRDR5.
QLXULRURTable2.
Flavourchangingdimension3operatorsintheStandardModel.
TheassociatedLorentzandcolourstructureisnotshown.
lowenergylagrangianandtheireectwillbenegligible.
ForthecasethefamilysymmetryisadiscretesymmetrytherearenoGoldstonemodesandthefamilonscanbeveryheavy.
Inwhatfollowswewillnotconsiderthecasethatthefamilysymmetryisglobalandsowewillnotdiscussthepossibleeectsofmasslessfamilons.
Westartwithadiscussionofthequarkbilinearoperatorsrelevanttothestructureofquarkmassesandtotheconstructionofhigherdimensionoperators.
Inthenextsectionweextendtheanalysistothedimension6operatorsrelevanttoavourchangingprocesses.
Thesetofdimension3operatorsthatviolateavouraregivenintable2.
Inthiswehavesuppressedthefamilyindexso,forexample,QLXQLLQL=QiLXQLL,ijQL,jfori,j=1,2,3.
Asdiscussedabove,forthecaseofMFVonlytherstandthefourthoperatorsareneededtoconstructthedimension6avourchangingoperators,theremainingonesgivenegligiblecontributionsduetothesmallnessofthedownquarkYukawacouplings.
Howeverforfamilysymmetriesalloperatorscanbesignicant.
Weturnnowtoadiscussionofthemagnitudeofthecoecients,X,oftheseoperators.
3.
1AbelianfamilysymmetryConsideraU(1)familysymmetry.
UptocoecientsoforderunitytheelementsoftheYukawamatricesaregivenintermsofthefamilychargesoffermionsdenedasqifortheavourcomponentsoftheleft-handeddoubletQL,anduianddifortheavourcomponentsofthe(left-handed)quarksingleteldsUcandDc,thechargeconjugateoftheright-handedavourtripletsURandDR,respectively.
WerstconsidertheholomorphiccaseinwhichthesymmetryisspontaneouslybrokenviafamilonscarryingonlyonesignofU(1)charge.
Forasinglefamilon,θ,withU(1)chargeequal+1theYukawamatrixofcouplingshastheform(theU(1)chargeoftheHiggsdoubletistakentobezero)2QLYUURHc=QiLajiθMuj+qiURjHcifuj+qi≥0,otherwise=0,(3.
1)whereajiarecoecientsoforderunityandθnowdenotesthefamilonvev.
Notethatthisstructureappliestothesuperpotential(F-terms)insupersymmetrictheoriesbecause2Weworkinthecanonicalbasisforthekineticterms.
Therotationfromanon-canonicalbasistothecanonicalonedoesnotchangeourconsiderations,see[37,40].
–7–supersymmetrydoesnotallowtermsinvolvingtheconjugateofthechiralsuperelds.
Non-supersymmetrictheoriesdonothavethisrestrictionsoforthemthenon-holomorphicformdiscussedbelowapplies.
ThesameistrueofD-termsinsupersymmetrictheories.
Giventhisweturntothestructureinthenon-holomorphiccase.
InsupersymmetrictheoriesthisappliestoF-termstooforthecasetherearefamiloneldswiththesamechargebutofbothsign.
ThisisverycommoninsupersymmetricmodelswherethefamilysymmetrybreakingfamiloneldsθandθwithU(1)charges+1and-1acquireequalvevsalongaD-atdirection.
Wearedenotingthiscommonvevbyθ.
Asjustmentionedthenon-holomorphiccasealsoappliestotheD-termsandtonon-supersymmetrictheoriesbecauseinthemthesymmetriesallowtermsinvolvingthefamilonoritsconjugate.
InallthesecasestheYukawacouplingstaketheformQLYUURHc=QiLajiθM|uj+qi|URjHc(3.
2)Toavoidunnecessaryduplicationofformulawewillusethenotation|uj+qi|todenoteboththecasesofequations(3.
1)andequation(3.
2).
InpracticetheholomorphicformisonlyrelevanttotheformofthefermionmassmatrixinSUSYtheories;thenon-holomorphicformappliestotheoperatorcoecientsinallcases.
WeassigntothecombinationajiθMm|uj+qi|transformationruleasfor(3,3,1)underSU(3)3q.
Onecanregardthe3x3matrixofthecoecientsajiasaspurioneldtransformingas(3,3,1)underSU(3)3qandthefactorsΦiL=(θ/M)qiandΦiu=(θ/M)uiasU(1)spurionswhicharesingletsundertheavourgroup.
3ItisnotationallyconvenienttowritethisasajiθM|uj+qi|=ajiΦiLΦUj≡ΦLaLUΦU(3.
3)whereΦL=((θ/M)q1,(θ/M)q2,(θ/M)q3),(Φu=(θ/M)u1,.
.
.
.
)andthemodulusintheexpo-nentistobetakenforthecombinedcharges.
IntermsofthefamilonsthequarkYukawalagrangianreadsLY=(QLΦLaLDΦDDR)H+(QLΦLaLUΦUUR)Hc+h.
c.
(3.
4)Thequarkbilinearsinthesetermscorrespondtotheoperators4and5oftable2.
Theremainingoperatorscanbeconstructedinananalogousway,withthehelpoffamilonsandhorizontalandavoursymmetriesgiving:QLΦLaLLΦLQL,URΦUaUUΦUUR,DRΦDaDDΦDDR.
(3.
5)wherethematricesofO(1)coecientsaIJarenotrelatedandtransformas(8,1,1),(1,8,1)and(1,1,8),respectivelyforI,J=LL,UU,DD.
TheaboveanalysisisreadilyextendedtothecasethatthefamilysymmetryisU(1)L*U(1)R.
InthiscaseΦL=((θL/ML)qL1,(θL/ML)qL2,(θL/ML)qL3)andΦR=((θR/MR)qR1,(θR/MR)qR2,(θR/MR)qR3)wherewehaveallowedfordierentmessengerscalesassociatedwiththefamiloneldsbreakingtheleftandrightU(1)symmetries.
3WethankA.
Weilerforausefuldiscussionofthispoint.
–8–3.
2Non-AbelianfamilysymmetryTherehasbeenaproliferationofmodelsbasedonnon-Abeliansymmetriesdrivenbythepossibilitythattheycanexplainthenearbi-tri-maximalmixingobservedintheleptonsectorthroughneutrinooscillationexperiments.
Itisonlythroughanon-AbelianstructurethatYukawacouplingstodierentfamiliescanberelatedincludingtheO(1)coecientsandthisisneededtogeneratebi-tri-maximalmixing.
Althoughthemotivationcomesfromtheleptonsectoritisnaturaltotrytoextendthesymmetrytoincludequarksandforthisreasonweincludeadiscussionofnon-Abelianfamilysymmetrieshere.
AgainthefamilysymmetrymustbechosentobeasubgroupofSU(3)3q.
Hereweconsiderthesimplecasethatthenon-AbelianfamilygroupisthediagonalSU(3)subgrouporadiscretesubgroupofit.
Thesymmetryisbrokenbyfamiloneldsinadeniterepresentationofthesymmetry.
ForthecasethattheLHandchargeconjugateRHeldsareinthetripletrepresentationtheyacquireavevtheformΦ=(c1,c2,c3)whereciareconstants.
ThiseldmustbeusedtobuildtheYukawacouplingsandthehigherdimensionoperators.
ThusthequarkLagrangianmaycontaintermsoftheformLY=αDQLΦΦDRH/M2D+αUQLΦΦURHc/M2U+h.
c.
,(3.
6)wherewehaveallowedfordierentmessengermassesinthedownandtheupsector.
TheparametersαU,Dare(familyindependent)constantsandtherelativemagnitudeoftheYukawamatrixelementsissetbytheconstantsinΦ.
Inpractice,inordertogeneratetheobservedmassesandmixingangles,severalfamiloneldsarenecessary.
TheremainingtwoquarkoperatorsareconstructedinasimilarmannerandhavetheformαLLQLΦΦQL/M2D,αUUURΦΦUR/M2U,αDDDRΦΦDR/M2D.
(3.
7)InwhatfollowswewillcomparethepredictionofMFVwithfourrepresentativefamilymodels.
Thestructureofthesemodelsisgiveninappendix1.
4ComparisonofMFVandfamilysymmetrymodels.
Dimension3quarkbilinearoperatorsInthissectionwecomparethepredictionsforthemagnitudeoftheFCNCeectsbasedontheMFVconjecturewiththosetobegenericallyexpectedinmodelswithfamilysymme-tries.
Forthecasetheoperatorsoftable2involvedownquarksitisnecessarytoworkintheelectroweakbasiswithdiagonaldownquarkYukawamatrix(EWDD),asusedinthebeginningofthissectiontodiscusstheMFVresults.
Forthecasetheoperatorsoftable2involvingupquarksitisnecessarytotransformtothebasisinwhichtheupquarksarediagonalbeforeestimatingthecoecients.
Asmentionedabove,theseoperatorsarenegligibleintheMFVcaseduetothesmallnessofthedownYukawacouplingsbutmaybesignicantinthefamilysymmetrycase.
Themasseigenstate(primed)basisisobtainedbyrotatingrightandleftelds,D′R=VDDR,U′R=VUUR,QL=Q′LSd.
(4.
1)–9–whereSd,VU,Dareunitarymatrices.
IntheEWDDbasisthedownYukawacouplingsarediagonalsotheLagrangianhastheformL=Q′LYDdD′RH+Q′LSdSuYUdU′RHc+h.
c.
,(4.
2)wherethesubscriptddenotesdiagonalmatrices.
TheCKMmatrixisU=SuSd.
FromthispointonweworkintheEWDDbasisbutdroptheprimes.
TodeterminethematricesSu,dwemustdiagonalisetheassociatedmassmatrices.
Followingfromequation(3.
4)theupanddownmassmatriceshavetheformMuij∝θMU|qi+uj|,MDij∝θMD|qi+dj|,(4.
3)wherewehaveallowedfordierentexpansionparametersintheupanddownsectors.
ThersttwoandthefourthAbelianfamilysymmetryexamplespresentedinappendix1havethesameexpansionparameterintheupanddownsectors,MU=MD.
ThethirdAbelianexampleandthenon-Abelianexamplebothallowfordierentexpansionparameters.
WewritethemassmatricesintheformM=m3m112′1m23′2′31.
Forthemodelsconsideredherethematrixcanbewritteninleadingorderofpowersof=θ/ManduptocoecientsofO(1)asM=m311/m221/m213231m1000m200011′1/m2′2′1/m21′3′2′31,whereiand′iaresmallanddeterminedbypowersof(seebelow).
Themiaretheratiosofthetwolightmasseigenvaluestothethirdgenerationmass.
ThisleadstoSu,d≈1u,d1mu,d2u,d2u,d1mu,d21u,d3u,d2u,d31.
(4.
4)ForthecaseoftheU(1)models(modelsI,IIamdIII),expressingSintermsofthehorizontalU(1)chargesonegetsSu,d≈1|q1+d2||q2+d2|u,d|q1q3|u,d|q1+d2||q2+d2|u,d1|q2q3|u,d|q1q3|u,d|q2q3|u,d1,(4.
5)whereu,d=θ/MU,DandformodelIIIthechargeshouldbeevaluatedsettingω=0.
ItisstraightforwardtodetermineSfortheremainingmodelsandtheresultinallhavethe–10–formSu,d≈1u,d3u,du,d12u,d3u,d2u,d1.
(4.
6)NotethattheSu,disdeterminedentirelybythechargesoftheleft-handeddoubletelds.
FortheU(1)modelstheunitarymatricesneededtogotothemassbasisofthesingletquarksaregivenbyVD≈1|q2+d1||q2+d2|d|d1d3|d|q2+d1||q2+d2|d1|d2d3|d|d1d3|d|d2d3|d1(4.
7)andVUisgivenbythesameformwithuinsteadofd.
ItisagainstraightforwardtodetermineVfortheremainingmodels.
ForallthemodelstheresultingmixingmatricesaregivenintermsoftheYukawacouplingslistedintheappendixbyVD≈1YD,21/YD,22YD,31YD,21/YD,221YD,32YD,31YD,321(4.
8)andtheequivalentformforVU.
Wearenowreadytoanalyzethefamilysymmetryimplicationsforthemagnitudeofthedimension3twofermionavourchangingoperators.
Considerrsttherstoperatorintable2withdownquarksastheexternalquarks.
InMFVXQLLisYuYutransformedtoEWDDand,fortheU(1)modelsconsideredhere,isgivenbyXQ,MFVLLij=λ2tU3iU3jλ2t|qiq3|+|qjq3|.
(4.
9)TransformationtoEWDDoftherelevantrstavonoperatorofequation(3.
5)givesQLΦLaLLΦLQL→QLSd(ΦLaLLΦL)SdQL.
(4.
10)sotheequivalentcouplingfortheU(1)familysymmetrycaseisgivenbyXQ,U(1)LLij=Sd(ΦLaLLΦL)Sdij.
(4.
11)ForthersttwoU(1)modelsofappendix1qi>0andthusXQ,U(1)LLij≈|qLiqLj|.
Foriorjequalto3themagnitudeisthesameforMFVandfortheU(1)familysymmetry.
Howeverthereisadierenceforij=12.
Wehave(forλt=1)XQ,MFVLL12∝|q1|+|q2|,XQ,U(1)LL12∝|q1q2|.
(4.
12)ForthemodelIIIthesituationisdierentsincethecontributionfromSdinequa-tion(4.
11)isgovernedbythecharge|q1||q2|=1whilethecontributionfromΦLisgovernedbythecharge|q1q2|=5.
InthiscaseXQ,U(1)LL12isdominatedbySd12andisofthesameorderasforthersttwomodels.
Howeverif,intheabsenceoffamilysymmetry–11–breaking,theinteractionsarefamilyblindthereisaGIMcancellationthateliminatesthiscontributionfromSd.
Thisisclearfromequation(4.
11)becausethefamilyblindassump-tionrequiresaLL11=aLL22andthenthecontributiontothe(1,2)matrixelementcancelsbetweentheSdandSdcontribution.
InwhatfollowswewilltaketheextremefamilyblindcaseinourestimatesofthepossiblesuppressionfromfamilysymmetriessoformodelIIItoowehaveXQ,U(1)LLij≈|qLiqLj|.
TheanalysisisreadilyextendedtotheU(1)*U(1)modelIVandtheresultsaresummarisedintable3.
Notethatinthiscaseitisnotnecessarytoassumethefamilyblindassumptioninthedownsectorbeforespontaneousbreakingbecause,inthismodel,therotationsneededtodiagonalisethedownquarksectorareverysmall.
Howeverwehaveassumedtheupsectorisfamilyblindwhencomputingtheupquarkoperatorsuppressionfactorsgiveninthetable.
Inthetabletheoperatorchargesparelisted;theassociatedoperatorcoecientsaregivenby|p|.
AsimilarnotationisusedinthecaseoftheF-theorymodelsinvolvingthreeAbelianfactors.
InthiscaseXDLRandXULRdonotappearexceptincombinationwithaHiggseldssuppressedbyvevs.
Weshallreturntoamoredetaileddiscussionofthefamilyblindassumptioninthesupersymmetriccontext.
Inthemodelbasedonanon-Abeliansymmetryitisapredictionofthesymmetrythat,intheabsenceoffamilysymmetrybreaking,theinteractionsarefamilyblindandsoXQ,U(1)LLijisgivenentirelybythespurioncontribution.
Thesymmetriesofthemodel[24,25]limitthespurioncombinationstoΦ3Φ3,Φ23Φ23,Φ23Φ123andΦ123Φ123andthisleadstothesuppressionfactorsshowninthefthcolumnoftable3.
Asasecondexampleconsiderthefourthoperatorintable2.
InMFVtheleadingtermtransformingas(3,1,3)isYdYuYusointheMFVtheoperatormatrixofcoecientXD,MFVRLisgivenbyYdXQ,MFVLLij.
ForthersttwoAbelianU(1)modelswehaveQLΦLaLDΦDDR→QLSd(ΦLaLDΦD)VdDRk,pQLi|qiqk|+|qk+dp|+|dpdj|DRj,(4.
13)sotheequivalentcouplingisXD,U(1)LRij≈k,p|qiqk|+|qk+dp|+|dpdj|.
(4.
14)ForthethirdmodeltheresulttakesadierentformduetotheappearanceofnegativechargesthatchangetheformofSdandVD.
Forthenon-AbelianmodelthestructureisthesameasthatfortherstoperatorconsideredabovebecausetheLHandchargeconjugateRHstateshavethesametransformationpropertyunderthefamilysymmetry.
Analexampleisgivenbythesecondoperatoroftable2.
IthasMFVstructureDRYdYuYuYDDR→DRλdλFCλdDR.
(4.
15)andfortheU(1)modelsisDR(ΦDaDDΦD)DR→DRVD(ΦDaDDΦD)VDDRDRi|didj|DRj,(4.
16)–12–XQLLij=ΦLiΦLjMIMIIMIIIU(1)2NAFtheoryMFV(12)115(3,1)53(2,0,0)55(13)333(3,0)33(1,1,5)33(23)222(0,1)22(1,1,5)22XDRRij=ΦDiΦDjMIMIIMIIIU(1)2NAFtheoryMFV(12)115(5,3)113(2,0,0)55(λdλs)(13)115(1,1)33(2,2,5)53(λdλb)(23)000(4,2)82(0,2,5)42(λsλb)XURRij=ΦUiΦUjMIMIIMIIIU(1)2NAFtheoryMFV(12)115(2,2)63(2,0,0)5(13)335(1,2)53(1,1,5)5(23)220(1,0)12(1,1,5)4XDLRij=ΦLiΦDjMIMIIMIIIU(1)2NAFtheoryMFV(12)343+w(7,1)93(2,0,3)5(λs)(13)343+w(3,1)53(2,2,2)3(λb)(23)232+w(0,2)42(0,2,2)2(λb)λd454(2,2)64λs232(4,0)42λb010(0,1)20XULRij=ΦLiΦUjMIMIIMIIIU(1)2NAFtheoryMFV(12)553+w(4,0)43(2,0,0)(13)333+w(3,0)33(1,1,5)(23)222+w(0,1)22(0,0,0)λu668(2,2)64λc442(1,1)32λt000(0,0)00Table3.
Chargestructureofthedimension3operatorsoftable2.
Thecoecientoftheoperatorisgivenby|p|wherepisthecharge.
FortheU(1)2modelthecoecientis|p1|+2|p2|.
where,asbefore,wehaveassumedfortheModelIIIthat,intheabsenceoffamilysymmetrybreaking,theinteractionsarefamilyblind.
Intable3welisttheresultingchargesassociatedwiththevariousmatrixelementsofthedimension3operatorsgivenintable2.
Therst5columnsgivethechargestructureoftheoperatorcoecientsXforthemodelsintroducedinappendix1.
Theassociatedoperatorcoecientsaresimplygivenbypwherepisthemodulusofthischarge.
FortheU(1)2modelwealsoshowinparenthesistheunderlyingcoecientintermsofthetwoexpansionfactors.
ForcomparisonweshowtheequivalentchargesfortheMFV.
InparenthesiswegivetheYukawacouplingfactorthatmustalsobeincludedwhenthe–13–externalquarksarenotLHdownquarks;thesearesosmallthattheoperatorisusuallydroppedinMFV.
5ComparisonofMFVandfamilysymmetrymodels.
Dimension6fourquarkoperatorsOfcourseinphenomenologicalstudiesitisthedimension6avourchangingoperatorsofthetypeshownintable1thatarerelevant.
5.
1FactorisationofoperatorsWestartwithadiscussionofwhichdimension6operatorsfactoriseinthesensethattheyaredeterminedbythecoecientsofthedimension3bilinearoperatorsdiscussedinthelastsection.
NotethatthefactorisationappliestoalloperatorsforthecaseofMFV.
5.
1.
1Fi=1,Fj=1,i=joperatorsInournotation,Fi=±1meansachangebyoneunitoftheithavour,forinstancetheoperator(b.
.
.
.
s)annihilatesaquarksandcreatesaquarkb,soF2=1,F3=+1.
TheseoperatorsincludeOF1,G1,l1,l2,H1,q5oftable1togetherwithrelatedoperatorsinvolvingupquarks.
Fortheoperatorsinvolvingonlytwoquarksitisobviousthattheavourchangingcomponentcomesfromthequarkbilinearoperatorandsothedimension6coecientisdeterminedbyequivalentcoecientofthedimension3operator.
Thisclassofoperatoralsoinvolvesoperatorsinvolvingfourquarks,suchasOq5thathavefamilychangeonlyinonefactor.
IntheAbelianfamilymodels,uptoO(1)factors,theoperatorcoecientisdeterminedbytheoverallsumoftheU(1)charges.
FortheoperatorOq5thechargesofthesecondbilinearfactorsumtozeroandthecoecientisdeterminedbytherstquarkbilinearoperatoralone.
FortheoperatorrelatedtoOq5byaFierztransformationtheoverallchargeclearlyremainsthesameandsoitscoecientisalsodeterminedbytheavourchangingquarkbilinearoperatorformedwhenFierztransformingbacktotheformofOq5.
Thesameconclusionappliestotheotherfourquarkoperatorsofthistype.
Forthenon-AbelianfamilysymmetrythestructureissomewhatdierentbecausethenumberoffamiloninsertionsmaychangefortheoperatorsrelatedbyFierztransfor-mationsiftheFierztransformationresultsintwoquarkbilinearfactorseachofwhichinvolvesavourchange.
Inthiscasetheleadingtermcorrespondstotheorderingoftheoperatorwithavourchangeinasinglebilinearfactorandthisfactoralonedeterminesthecoecient.
5.
1.
2Fi=2,Fj=2,i=joperatorsAnexampleofthisclassofdimension6fourquarkoperatorisgivenbytheoperatorO0.
Sinceitinvolvesthesquareofadimension3twoquarkoperatorthecoecientisdeterminedbythesquareofthecoecientsofthequarkbilinearoperator.
AgainthisfactorisationisonlyuptoO(1)factors.
ForthisclassofoperatorFierztransformationdoesnotaectthisstructure.
–14–FlavourviolatingΛ/ΛMFVdimensionsixoperatorEx.
1Ex.
2Ex.
3U(1)2N-AFO0=12(QLXQLLQL)2441121OF1=HDRXDLRσνQLFνx2x3/2x2xx2x2OG1=HDRXDLRσνTaQLGaνx2x3/2x2xx2x2O1=(QLXQLLγQL)(LLγLL)221111O2=(QLXQLLγτaQL)(LLγτaLL)221111OH1=(QLXQLLγQL)(HiDH)221111Oq5=(QLXQLLγQL)(DRγDR)221111Table4.
Boundsonthesuppressionscaleofthefamiloninducedoperators.
TheSMisextendedbyaddingavour-violatingdimension-sixoperatorswithcoecient1/Λ2.
HerewereporttheboundsonΛforthefamilysymmetrymodelsintermsoftheboundsonΛMFVforMFVgivenintable1.
Herex=(mt/mb)1/2.
Theboundscomefromtheavourchangingoperatorsinvolvingthersttwofamilies.
5.
1.
3Non-factorisableoperatorsThereareseveraltypesofoperatorthatdonotfactorise.
AnexampleistheFi=2,Fj=1,Fk=1,i=j=koperators.
SuppressingtheLorentzstructure,anexampleofthesedimension6fourquarkoperatorsisgivenby(QL1QL3)(QL2QL3).
HereF3=2,F1=F2=+1.
Dependingontheparticularformofthefamilysymmetrythecoecientsoftheseoperatorsmaynotfactoriseintotheproductofanycombinationofthequarkbilinearpairsthatmakeuptheoperator.
Furtherexamplesofnon-factorisingop-eratorsareQLiURjQLkDRlXijkl,QLiQLjURkURlYijkl1andQLiQLjDRkDRlYijkl2withfam-ilychangeinbothofthefactors.
5.
2Determinationofthecoecientsofthedimension6operatorsFortheFi=1,Fj=1,i=joperatorsthedimension6operatorcoecientsaregivenbythecoecientassociatedwiththeappropriateavourchangingdimension3twoquarkoperator.
Asdiscussedabovethisisdeterminedbyxwherexisthemodulusoftheassociatedchargelistedintable3.
OneexceptiontothisrulearethecoecientsoftheoperatorsOF1andOG1intheModelIIIwherethehorizontalchargeωoftheHiggseldhastobetakenintoaccount.
FortheAbelianfamilysymmetriesthesecoecientsaredetermineduptoanO(1)factorbutinthecaseofthenon-Abelianfamilysymmetrytherelativemagnitudeofthecoecientsatagivenpowerofxaredetermined.
ThefactorisableFi=2,Fj=1,Fk=1,i=j=koperatorcoecientisgivenbytheproductoftheappropriateavourchangingdimension3twoquarkoperator.
Usingthistheresultingboundsonthescaleofnewphysicscomingfromtheoperatorslistedintable1areshownintable4forthemodelsofappendix1relativetotheMFVvaluegivenintable1.
NotethattheseboundscomefromtheoperatorsinvolvingthedownandstrangequarksthataredominantintheMFVcase.
Sincex≈1itmaybeseenthat–15–ComponentMIMIIMIIIU(1)2NAF1.
X121289|62w|(11,1)13652.
X211289|62w|(6,2)10653.
X322345|2+w|(1,2)5674.
X213167|8w|(2,5)1263Table5.
CoecientsXijklofdimension6four-fermionoperatorsoftheformQLiURjQLkDRl.
Thecoecientoftheoperatorisgivenbypwherepisthemodulusofthecharge.
allmodelsexcepttheU(1)*U(1)modelrequirealargermediatorsuppressionscaletokeeptheFCNCassociatedwiththeoperatorsOF1andOG1withinpresentbounds.
ThereasonforthisisthatonlytheU(1)*U(1)modelhas,inthecurrentquarkbasis,verysmallmixingbetweenthersttwofamiliesinthedownquarkmassmatrix,theCabibboanglebeinggeneratedfromthemixingintheupquarksector.
Thephysicalinterpretationofthemediatorsuppressionscaledependsonthemicro-scopicphysicsthathasbeenintegratedout.
InparticularinsupersymmetricmodelsitmayberelatedtothesupersymmetrybreakingscaleandinsomecasestheboundsonFCNCmaybediculttoreconcilewithSUSYsolvingthehierarchyproblem.
InthenextsectionweshalldiscusstheidenticationofthemediatorscaleforthecaseoftheMinimalSupersymmetricStandardModel(theMSSM)andinsection7considertheFCNCtestsinSUSYmodelsinmoredetail.
AsnotedabovetheU(1)*U(1)modelillustratesthefactthatfamilysymmetrymodelscangiveapproximatelythesameexpectationforthetable1operatorcoecientsasMFV.
Inthiscaseonemustturntotheotherpossibleoperatorsinvolvingthethirdgenerationtodistinguishthem.
Weemphasisedabovethat,incontrasttotheMFVcase,theoperatorsappearingintable1maynotbetheonlyonescontributingsignicantlytoavourchangingprocessesinthefamilysymmetrymodels.
Forthefactorisingoperatorsitiseasytousetable3todeterminethecoecientsoftheremainingoperators.
Forexampleforavourchanginginvolvingthelightquarks,the(1,2)sector,therstthreedimension3operatorsoftable2allhavethesameorderofcoecientsforthefamilymodelsconsidered.
ThisistobecomparedtoMFVinwhichonlytherstoperatorissignicantc.
f.
table1.
ThesecondandthirdoperatorshaveadierentLorentzstructureandconsequentlytheimplicationsforthephenomenologicalimportanceofthedimension6operatorsinvolvingthemmaybesignicantlydierentfromthoseinvolvingtherstoperatoroftable2.
ItisbeyondthescopeofthispapertoperformacompleteanalysisoftheFCNCeectsfollowingfromtheseterms.
Howeverinsection7wewillconsiderthephenomenologicalimplicationsofalltheoperatorsoftable2forthecaseofsupersymmetricmodels.
Finallyweturntothenon-factorisingoperatorsoftheformFi=2,Fj=1,Fk=1,i=j=k.
Therearemanypossibleoperatorsofthistypebecauseonecancombinethedierentdimension3bilinearoperatorsinmanyways.
Intables5and6weillustratethefamilysymmetrypredictionforthecoecientsoftheseoperators–16–ComponentMIMIIMIIIU(1)2NAF1.
Y121222210(6,2)108102.
Y12132448(6,1)88103.
Y12312222(0,1)2664.
Y21312223(6,1)8810Table6.
CoecientsYijkl2ofdimension6four-fermionoperatorsoftheformQL,iQL,jDR,kDR,l.
Thecoecientoftheoperatorisgivenbypwherepisthemodulusofthecharge.
byjusttwoexamples.
FortheAbelianfamilysymmetriesthecoecientofthedimension6fourquarkoperatorisgivenbythefactorpwherepisthemodulusoftheoverallchargeoftheoperator.
Forthenon-Abeliansymmetrythecoecientisdeterminedbyidentify-ingtheproductoffamiloneldsneededforagivenoperator,chosenfromtheallowedsetlistedabove.
Oneseesaverywiderangeofcoecientsandlowsuppressioninmanycases.
Moreoverthepredictedcoecientsdiersignicantlybetweenmodelssothetheobserva-tionofaspecicpatternofavourchangingprocesseswouldprovidestrongevidenceforaparticularfamilysymmetry.
6SUSYTheanalysishassofarconsideredtheeectiveeldtheoryrelevantatenergyscalesbelowthemassofthenewstatesresponsibleforgeneratingtheavourchangingoperators.
ItisimportanttostressthattheanalysisisquitegeneralandcoversallpossibilitiesforBeyondtheStandardModelphysics.
However,asdiscussedabove,theinterpretationofthemeaningoftheinversemassscalecharacterisingtheboundontheoperatorrequiresadiscussionoftheunderlyingphysicsorigin.
Inthissectionwediscussthecasethatthehierarchyproblemissolvedbylow-energysupersymmetrybutallowtheavoursymmetrybreakingscaletobemuchhigher.
6.
1IdenticationofthescaleΛSincetherearetwofundamentalscalesitisnecessarytodeterminethescale,orcombinationofscales,thatisrelevanttotheboundonthescale,Λ,oftable1.
Toanswerthisitisnecessarytoconsidertheleadingavourchangingoperatorsinthesupersymmetrictheoryabovethesupersymmetrybreakingscale,MSUSY,butbelowtheavoursymmetrybreakingscale,M.
Sincethequarksandleptonshavescalarpartnerstherearenewoperatorsthatmayviolateavourandtheleadingoneshavealowerdimensionthanthedimension6operatorsbuiltofSMstatesalone.
TheSUSYoperatorsgeneratetheSMdimension6operatorsbut,asweshalldiscuss,ΛshouldnotbeinterpretedastheavourchangingscaleiftheunderlyingSUSYoperatorshavedimension(7.
3)where=gf(7.
4)Followingfromthisonehasδd12LL≈m2cdLSd11Sd21+csLSd12Sd22+cbLSd13Sd23(7.
5)wherem2istheaveragesquarkmasssquared.
Similarexpressionsareobtainedfortheotherδs.
Asdiscussedin[21–25]themagnitudeoftheD-termisproportionalto(m2φm2φ)wherem2iarethesoftsupersymmetrybreakingmassessquaredofthefamilonelds.
Ifthisfactorisoforderm2oneseesthattheexpectationisthatδd12LLisoforder.
Intable7weseethat,formqij=350GeVthephenomenologicalupperboundsontheLLandRRδ'sareatmostoftheorderof2,andtheproduct√LL*RRinthe(1,2)isboundedby4Thus,attherstsighttheD-termcontributionisobyafactor3comparedtotheexperimentalboundsfoundassumingmqij=350GeV.
However,thesepredictionsarevalidatthescaleMofthefamilysymmetrybreakingandbeforecomparingthemtotheexperimentalboundsoneshouldcorrectthemusingtherenormalisationgrouprunningtodeterminethematlowscaleswheretheexperimentalboundsapply.
Thedominantrenormalisationeectsareavourblindstronginteractioncontributionstothediagonalsquarkmassentriescomingfromtermsproportionaltothegluinomass.
Theseeectsdependstronglyontheratiox0=m21/2/m20wherem1/2andm0arethegluinoandsquarkmassesatthescaleM[26].
ForaroughestimateofsucheectsintherunningdownfromtheGUTscaleonecanuseapproximateformulaemg≈3m1/2andm2q≈m20+6m21/2.
First,weseethatx=1impliesx0=1/3andveryweakgluinorenormalisationeects.
Thesquarkmassof350GeVcorrespondsthentom0=200GeVandm1/2=120GeV.
Next,wecanaskforwhatvaluesofx0wecangainatleastfactor3,tomakethepredictionsconsistentwiththeexperimentalbound.
Neglectingthesmallrenormalisationoftheo-diagonalentries,onendsconsistencyform1/2/m0=7.
For350GeVsquarksthisimpliesm0=20GeVandm1/2=140GeV.
Forthisvalueofx0largervaluesofm1/2arealsocomfortable.
Forinstance,form1/2=300Gevwegetmg=900GeVandmq=800GeV,consistentwithlowne-tuning[28,29].
Asanalexample,forx0≈1thevaluesoftheδ'sarerenormalisedintherunningdownfromtheGUTscaleto1TeVbyafactoroforder0.
1andtobringtheresultintoagreementwiththeboundsrequiresthesquarksofthersttwogenerationsofabout15TeV.
SuchalargemassintroducesalargenetuningimplyingthatSUSYdoesnotsolvethelittlehierarchyproblem.
Thisdiscussionnicelyillustratestheinterplay–23–betweentheFCNCeectsandthesoftsupersymmetrybreakingspectruminmodelswithfamilysymmetries[24,25,30,31].
ThemagnitudeoftheD-termcanbemuchsmalleralsoforotherreasons[24,25].
OnepossibilityinsupergravitymediatedSUSYbreakingoccursinfamilysymmetrymodelssuchasmodelIIIwithconjugatepairsoffamilonsφandφ.
Inthiscasethefactor(m2φm2φ)vanishesfordegeneratefamilonseliminatingtheD-termcontribution.
SuchdegeneracycanresultiftheunderlyingSUSYbreakingeldisdominantlythedilatonthatcouplesuniversally.
ForthecaseofgaugemediatedsupersymmetrybreakingthesoftfamilonmassesareautomaticallymuchsmallerthatthesoftsquarkmassesbecausetheyareSMgaugesingletsandtheircouplingtothegaugemediationsectorisviatheircouplingtothequark,introducinganadditionalloopfactorinthemasssquaredcalculation.
Suchafactorisexpectedtorenderthiscontributionsubdominant.
FinallyitmaybethatthefamilysymmetriesarediscreteratherthancontinuousandinthiscasethereisnoD-termtoworryabout.
Althoughourdiscussionhasbeeninthecontextofcontinuoussymmetriestheymayalsoapplytotheirdiscretesubgroups.
TobespecictheresultsareunchangedfortheZNsubgroupofU(1)providedtheoperatorchargesarenotgreaterthanN/2givingalowerboundonN.
7.
2.
2ContributionsfromodiagonalsquarkmassConsidertheboundscomingfromtheLLterms.
ForthemthesquarkmasstermsintheLagrangianhavetheformmqijqiqj|qjqi|correspondingto(δqij)LL=|qjqi|.
Thisisthesuppressionassociatedwiththedimension3,ΦLiΦLj,operatorslistedintable3.
Theotherentriesoftable3immediatelygivetheremainingsuppressionfactorsassociatedwiththeother(δqij)MM.
AsfortheD-terms,thesepredictionsarevalidatthescaleMofthefamilysymmetrybreakingandoneshouldcorrectthemusingtherenormalisationgrouprunningtodeterminethematlowscaleswheretheexperimentalboundsapply.
Thepreviousdiscussionremainsvalidinthiscase,too.
Thus,thecoecientstakenfromtable3shouldberescaledbyafactordependingonthevalueoftheratiox0ofthesoftmassesatthescaleM,beforecomparingthemwiththephenomenologicalboundsoftable7.
Asmentionedearlier,formqij=350GeVthephenomenologicalupperboundsontheproduct√LL*RRinthe(1,2)sectoris4.
Acomparisonoftable3withtable7showsthatonlythistermrequiresspecialattentioninsomeofthemodelsforanaveragesquarkmassof350GeV.
InModelsIandII,thesuppressionfactorforthe√LL*RRinthe(1,2)sectorisonlysoweneedeitherlargeenoughvalueoftheratiox0orheaviersquarksorboth,asdiscussedintheprevioussubsection.
The(1,3)and(2,3)sectorsarestillsafeevenforlight3rdgenerationsquarks.
ModelsIII,U(1)2,thenon-AbelianmodelandtheF-theorymodelhavesuppressionfactorsof5,8,3and5respectivelyforthe√LL*RRinthe(1,2)sector.
AllowingforamodestsuppressionduetorunningofO(0.
1),correspondingtox0=1,evenforalightsquarksectorwithmassesofO(350GeV)theyaresafelywithinthepresentbounds.
ItisinterestingthatModelIIIpredictsanunsuppressedRRinsertioninthe(2,3)sectorandanimprovedphenomenologicalboundseparatelyonthisinsertionwouldbeveryinteresting.
–24–qij(δqij)LRd122*1044d130.
08d230.
012d114.
7*1066u119.
3*1066u120.
022Table8.
Thephenomenologicalupperboundsonchirality-mixing(δqij)LR,whereq=u,dtakenfromthesummaryofIsidorietal.
[16,17].
Theconstraintsaregivenformq=1TeVandforx=mg/mq=1.
Itisassumedthatthephasescouldsuppresstheimaginarypartsbyafactor0.
3.
Theconstraintsonδd12,13,δu12,δd23andδqiiarebasedonrefs.
[18–20]and[35]respectively(withtherelationbetweentheneutronandquarkEDMsasin[42]).
Tosummarise,supersymmetricfamilysymmetrymodelsoffermionmassgenericallyviolatetheMFVhypothesis.
However,theyoerabroadspectrumofpossibilities,frombeingconsistentintheFCNCsectorwiththepresentexperimentalboundswithnocon-straintsonthesoftsupersymmetrybreakingparameterstorequiringspecialpatternofSUSYbreaking.
Variousmodelspredict"signicant"departuresfromtheMFVbutonlyinalimitednumberofprocessesinvolvingheavyquarkssuggestingasystematicallystudyofallFCNCdatamayrevealdeviationsfromMFV.
8AtermsWeturnnowtotheAtermsthatenterinthetrilinearscalarquarkcouplingsAqijHqqLiqRjwhereHq,q=u,daretheqtypeHiggsbosonsandvq=Hq.
Thesetermsgiverisetochirality-mixing(δqij)LR=vqAqij|SCKMm2qijsquarkmassinsertionsintheSCKMbasis,whereq=u,dandmqijistheaveragesquarkmassdenedabove.
Intable8wegivethecurrentboundsonthesechiralitymixingmasses[16,17],seealso[27].
Inthetablewealsoexpresstheδ'sintermsoftheexpansionparameter.
TodeterminetheimplicationsoftheseboundsforthefamilysymmetrymodelsnotethatinthemAqijaresuppressedbythesamepowersofastheYukawacouplingsYqijgiveninappendix1.
Insuchmodels,Aqij=AqijYqijwherethecoecientsAqijaregivenbyanoverallmassscalefactormultipliedbyO(1)constants.
Rotatedtotheappropriatebasis(inthecaseoftheoperatorsinvolvingdsquarkstheSCKMbasisandtheEWDDbasisareequivalent)Aqij|SCKM∝SdAqVdij.
Inallexamplesofchargeassignmentsconsideredinthispaper,theo-diagonalAqij|SCKMarealsosuppressedbythesamepowersofastheYukawacouplingsYqijgiveninappendix1.
Assumingforthemomentthattheconstantofproportionalityistheaveragesquarkmassthechirality-mixing(δqij)LR∝Yqijvq/mqij.
Comparingwiththefactorsofappendix1andtakingintoaccountthatvq/mqijMP:YU=653542321YD=43332211ExampleII.
AsecondU(1)holomorphicexample[36]hasthechargeassignement:qL1,2,3:(3,2,0)dc1,2,3:(2,1,1)uc1,2,3:(3,2,0)(A.
2)ThisgivesthefollowingYukawamatrices–28–U(1)U(1)′Q130Q201Q300D112D241D301U112U210U300Table9.
ChargesintheU(1)2model.
YU=653542321YD=5444332ExampleIII.
Thethirdexampleisanon-holomorphicmodelthathasnotpreviouslybeendiscussed.
InadditiontohavingthegoodpredictionforVcb=O(ms/Mb)italsohasa(1,1)texturezerogivingtherelationVus=O(ms/md).
Inthiscasetherearetwofamilonelds,θ,θ,withcharges±1andequalvevstoensureD-atness.
TheHiggseldshavechargeωandthequarkchargesareqL1,2,3:(3+w,2+w,w)dc1,2,3:(5,0,0)uc1,2,3:(5,0,0)(A.
3)wherewisafreeparameter.
ItgivesthefollowingYukawamatrices:YU,D=8u,d3u,d3u,d3u,d2u,d2u,d5u,d11whereu,d=MU,Dandwehaveallowedfordierentmessengermassesintheupandthedownsectors.
ExampleIV:AU(1)*U(1)′model.
Thechargesaredenedintable9,seealso[37].
TheexpansionparameterfortheU(1)is1andfortheU(1)′itis2.
Weshallassume(after[37])that1,and22.
Theresultingmassmatricesare–29–U(1)U(1)′U(3)′′Q1112Q2112Q3003D1111D2111D3114U1112U2112U3003θ13115θ14115θ53130θ54130Table10.
ChargesintheU(1)3F-theorymodel.
YU=64373251YD=695744562.
ExampleV:ANon-Abelianmodel.
ThefamilysymmetryisSU(3),underwhichthequarkstransformasfollows(see[24,25]):QL3,DR,UR3.
(A.
4)ThefamilonstransformasfollowsΦu,d33,Φ233,Φ1233,(A.
5)expectationvaluesoftheform:Φu,d3/MU,D=(0,0,1),Φ23/MU,D=(0,1,1)*u,d,Φ123/MU,D=(1,1,1)*(u,d)2,(A.
6)whered=0.
15,u=0.
05(d)2.
TheallowedYukawacouplingsinvolvingthesefamilonsarerestrictedbyadditionalfamilyindependentsymmetries.
FortheLLandRRoperatorsthesesymmetriesrequirethefamiloneldsonlyappearinpairsinvolvingthesamefamiloneld.
FortheLRtermsthefamiloneldsappearinthecombinationsφ123φ23,φ23φ23andφ3φ3withthecorrespondingmassmatricesgivenbyYU,D=03u,d3u,d3u,d2u,d2u,d3u,d2u,d1–30–wherewehaveallowedfordierentmessengermassesintheupandthedownsectors.
ExampleVI:AnF-theorymodel.
RecentlytherehasbeenconsiderableinterestinF-theorystringmodelsandtheirimplicationsforfermionmasses.
SuchmodelscanhaveAbelianfamilysymmetries.
ThesesymmetriesandthechargesofthemattereldsunderthesesymmetriesarestronglyconstrainedbytheunderlyingE(8)symmetryoftheassoci-atedstringtheory[38].
ToillustratethestructurethatcanemergeweincludehereanF-theorymodel[39]withanunderlyingSU(5)GUTsymmetry.
InthismodelthereisaU(1)3familysymmetry,asubgroupoftheSU(5)⊥subgroupofE(8)(SU(5)*SU(5)⊥E(8))whenaZ2monodromyisimposed.
Thechargesofthequarksunderthesesymmetriesaregivenintable10.
Alsoshownarethechargesofthefamiloneldsbreakingthesesymmetries.
Therearefourfamilonelds,θ13,θ14,θ53,θ54andtheyacquirevevsofO(2,3,2,3)respectively.
TheYukawacouplingshavetheformYU=653532321YD=033322001.
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