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arXiv:hep-ph/9502418v11Mar1995hep-ph/9502418,WIS-95/7/Mar-PHLeptonMassMatrixModelsYuvalGrossmanandYosefNirDepartmentofParticlePhysicsWeizmannInstituteofScience,Rehovot76100,IsraelThesmallnessandhierarchyinfermionparameterscouldbetheresultofselectionrulesduetoanAbelianhorizontalsymmetrybrokenbyasmallparameter.Whenappliedtotheleptonsector,thenforalargeclassofmodels,anumberofinterestingorderofmagnituderelationsarise:withij,m(νi)/m(νj)∼sin2θij;m(ℓ−i)/m(ℓ−j)∼sinθij;m(νi)/m(νj)∼m2(ℓ−i)/m2(ℓ−j);m(νe)∼m(νμ)∼m(ντ).Therelationsbetweenneutrinomassesandmixingsmaybecomeexactifthehorizontalsymmetrytogetherwithholomor-phyinducecertainzeroentriesintheleptonmassmatrices.Afullhighenergytheoryislikelytoincludescalarswithflavorchangingcouplingsandheavyleptonsinvectorrepre-sentations;however,themassesoftheseparticlesaretooheavytobedirectlyobservedinexperiment.Indirectevidenceforthehorizontalsymmetrymayarisefromothersectorsofthetheory:non-degeneratesleptonsareallowedasthesymmetryalignsleptonandsleptonmassmatrices;lightleptoquarksareallowedasthesymmetrycanmaketheircouplingsdiagonalandchiral.1.IntroductionOnlythreeparametersoftheleptonsectorareunambiguouslyexperimentallydeter-mined.Thesearethethreechargedleptonmasses[1]:me=0.5110MeV,mμ=105.7MeV,mτ=1777MeV.(1.1)Theunambiguousinformationonneutrinomassesconsistsofonlyupperbounds[1]:mνe≤5.1eV,mνμ≤0.16MeV,mντ≤31MeV.(1.2)However,varioustheoreticalargumentsandobservationalpuzzlessuggestthatneutrinosmayhavenon-vanishingmasses.Forexample,ifντistobeanimportantcomponentinthedarkmatterthenmντ=O(10eV).(1.3)IfthesolarneutrinoproblemissolvedbytheMSWmechanismwithνe−νμoscillationsthen[2]mostlikely,m2νμ−m2νe∼6×10−6eV2,sin22θ12∼7×10−3.(1.4)Inanycase,theleptonparameters(certainlythechargedleptonmassesandverylikelyalsomixinganglesandneutrinomasses)aresmallandhierarchical.Thesmallnessandhierarchyarethemostpuzzlingfeaturesoffermionparameters.Theysuggestthatthereshouldexistamorefundamentaltheorywherethehierarchyisgeneratedinanaturalway.Oncesuchaframeworkisfound,weshouldinvestigateitspredictionsandfindwaysinwhichitcanbeexperimentallytested.Suchatask,basedonthewealthofinformationonthequarksector,wasrecentlytakeninrefs.[3][4][5].Thesmallnessandhierarchyofthequarkmassesandmixinganglescanbenaturallygen-eratedintheorieswhich,atlowenergy,aredescribedeffectivelybyanAbelianhorizontalsymmetrythatisexplicitlybrokenbyasmallparameter.Thepossibleevidenceforsuchasymmetrydividesintothreetypes:(i)Numerology:thesymmetryleadstoorderofmagnituderelationsamongthevariousparameters,e.g.|Vub|∼|VusVcb|.1(ii)Directevidence:thefullhighenergytheoryislikelytorequiretheexistenceofnewparticles,e.g.scalarswithflavorchangingcouplingsandquarksinvectorrepresenta-tions.(iii)Indirectevidence:thesymmetryhasimplicationsforothersectorsofthetheory,e.g.thesquarkspectrum[4],theHiggspotential[5],leptoquarkcouplings[6]andbaryonnumberviolation[7][8].Theconclusionofref.[5]wasthatitistheindirectevidencethatismostlikelytorevealtheexistenceofthehorizontalsymmetrymechanism.Inthisworkweextendthestudyofrefs.[3][4][5][6][8]totheleptonsector.(Forrecent,relatedwork,seerefs.[9][10].)Eventhoughthereismuchlessexperimentalinformationintheleptonsector,weareabletofindinterestingpredictionsthatareindependentofdetailsoftheleptonspectrum.Thepaperisorganizedasfollows:insection2wepresentthetheoreticallowenergyframeworkinwhichwework,namelytheleptonandscalarfields,thesymmetryandtheselectionrules.Insection3westudyfeaturesoftheleptonparametersthatgenerallyfollowfromtheselectionrulesandareindependentofanyspecificphenomenologicalinput.Insection4wemakesomeassumptionsaboutneutrinomassesandmixingandpresentexplicitmodelswheretheresultinghierarchyinleptonparametersarisesnaturally.Insection5weinvestigatethepossibilityofexactrelationsamongleptonparameters.Section6studiestheembeddingofthelowenergyeffectivemodelinafullhighenergytheoryandtheconstraintsontherelevantenergyscales.Insection7weinvestigatewhetherlepton–sleptonalignmentcouldsolvetheproblemofunacceptablylargeleptonflavorchangingprocesseswithoutsleptondegeneracy.Insection8weinvestigatetheroleofhorizontalsymmetriesinallowingforlightleptoquarks.Insection9,thepossibilityofanunbrokenhorizontalsymmetryisexplored.Ourconclusionsaresummarizedinsection10.2.TheTheoreticalFrameworkWeworkintheframeworkofsupersymmetricAbelianhorizontalsymmetrythathasbeenrecentlyinvestigatedinrefs.[3][4][5].WeassumethatthelowenergyspectrumconsistsofthefieldsoftheminimalsupersymmetricStandardModel.Theleptonand2HiggssupermultipletscarrythefollowingSU(2)L×U(1)Yquantumnumbers:Li(2)−1/2,¯ℓi(1)+1,φu(2)+1/2,φd(2)−1/2(2.1)wherei=1,2,3isagenerationindex.TheLifieldsaredistinguishedfromtheφdfieldbyRparity(Rp):inLi,thefermioncomponentsareRpeven,whileinφdthescalarcomponentisRpeven.Eachofthesupermultipletsin(2.1)carriesachargeunderanAbelianhorizontalsymmetryH.Formostofourdiscussion,itmakesnodifferencewhetherHislocalorglobal,continuousordiscrete.(ForadiscreteZN,weusuallyassumethatNislargeenoughthattheYukawasectorhasaneffectiveU(1)symmetry.)Hisexplicitlybrokenbyasmallparameterλtowhic