Lower Bound on U_{e3}^2 from Single and Double Bet

arXiv:hep-ph/0111269v225Dec2001hep-ph/0111269LowerBoundon|Ue3|2fromSingleandDoubleBetaDecayExperimentsHisakazuMinakata1,2∗andHiroakiSugiyama1†1DepartmentofPhysics,TokyoMetropolitanUniversity1-1Minami-Osawa,Hachioji,Tokyo192-0397,Japan2CenterforTheoreticalPhysics,MassachusettsInstituteofTechnologyCambridge,MA02139,USAAbstractWepointoutundertheassumptionofMajorananeutrinosthatalowerboundontheMNSmatrixelement|Ue3|2canbederivedbyusingconstraintimposedbyneutrinolessdoublebetadecayexperimentsandbypositivedetectionofneutrinomassbysinglebetadecayexperiments.Weshowthatthelowerboundexistsinanarrowregionoftheratiooftheobservablesinthesetwoexperiments,hmiββ/hmiβ.Itmeansthatoncetheneutrinomassisdetectedinthebound-sensitiveregiononemustsoonobservesignalinneutrinolessdoublebetadecayexperiments.14.60.Pq,26.65.+t,23.40.-sTypesetusingREVTEX∗E-mail:[e-mail]†E-mail:[e-mail],bynow,accumulatedevidencesintheatmospheric[1]andthesolarneutrino[2]observationsthatneutrinosdooscillate.Theexistenceofneutrinooscillationisfurtherstrengthenedbytheresultofthefirstlong-baselineman-madebeamexperimentK2K,inparticularbytheirlatestresult[3].Theyconstitutethefirstcompellingevidenceforphysicsbeyondthestandardmodelofparticlephysics.Inparticular,wehaveleanedthatanalmostmaximalmixingangleθ23isrequiredtoaccountfortheatmosphericneutrinoanomaly,whichisquiteunexpectedfromourexperienceinthequarksector.Whatisevenmoresurprizingtousisthat,accordingtothelatestglobalanalysesofthecurrentsolarneutrinodata[4–8],theangleθ12whichisresponsibleforthesolarneutrinooscillationislikelytobelargethoughmaynotnecessarilybemaximal.Itisinsharpcontrastwiththefactthattheremainingmixingangleθ13isconstrainedtobesmall,s213∼0.03,bythereactorexperiments[9].Giventhecurrentstatusofourunderstandingofthestructureofleptonflavormixingmatrix,theMNSmatrix[10],itwouldbeniceifthereareanyhintsonhowsmallistheangleθ13.∗Inthispaper,wetrytopursuesuchapossibilityandpointoutthatonecanderivealowerboundons213=|Ue3|2throughjointeffortsbydoublebetadecayexperimentsandbydirectmassdeterminationeitherbysinglebetadecayorbycosmologicalobservations.WeassumeinthispaperthatneutrinosareMajoranaparticletorelyontheboundimposedbydoublebetadecayexperiments.Beforegettingintothebussiness,letusbrieflysummarizeexistingknowledgeonhowθ13canbemeasured,orfurtherconstrained.Mostoptimistically,thenextgenerationlongbaselineexperiments,MINOS[12],JHF[13],andOPERA[14]willobserveνeappearanceeventsandmeasuretheangleθ13.Mostnotably,theJHFcanprobesin22θ13∼several×10−3initsphaseI[13].Ifrealized,alarge-volumereactorexperiment[15]canalsoprobethesimilar(toaslightlyshallower)region.Iftheangleissmallerthanthesensitivityregionoftheseexperiments,wehavetowaitforfuturesupermassivedetectorexperiments,utilizingeitherlowenergyconventionalsuperbeamsorneutrinofactoties.(Seee.g.,[16]forreferencescitedtherein.)Ifnatureissounkindastotunetheangleextremelysmall,sin22θ13≪10−5,thentheonlywaytodetectitseffectwouldbeviasupernovaneutrinos[17].∗Seee.g.,[11]forasummaryofremainingissuesinthreeflavormixingschemeofneutrinos.2II.CONSTRAINTFROMNEUTRINOLESSDOUBLEBETADECAYLetusstartbyexaminingconstraintfromdoublebetadecay.WeusethroughoutthispaperthestandardnotationoftheMNSmatrix:U=c12c13s12c13s13e−iδ−s12c23−c12s23s13eiδc12c23−s12s23s13eiδs23c13s12s23−c12c23s13eiδ−c12s23−s12c23s13eiδc23c13.(1)Usingthenotation,theobservableinneutrinolessdoublebetadecayexperimentscanbeexpressedashmiββ=3Xi=1miU2ei=m1c212c213e−iβ+m2s212c213e+iβ+m3s213ei(3γ−2δ),(2)wheremi(i=1,2,3)denoteneutrinomasseigenvalues,UeiaretheelementsinthefirstlowoftheMNSmatrix,andβandγaretheextraCP-violatingphasescharacteristictoMajorananeutrinos[18,19].Wehaveusedinthesecondlineof(2)theMajoranaphasesintheconventionof[20].Therehavebeenlargenumberofpapersquiterecentlywhichdevotedtoextractconstraintsfromneutrinolessdoublebetadecayexperiments[20–22].Wedefinetheneutrinomass-squareddifferenceasΔm2ij≡m2j−m2iinthispaper.Inthefollowinganalysis,wemustdistinguishthetwodifferentneutrinomasspatterns,thenormal(Δm2230)vs.inverted(Δm2230)masshierarchies.Weusetheconventionthatm3isthelargest(smallest)massinthenormal(inverted)masshierarchysothattheanglesθ12andθ23arealwaysresponsibleforthesolarandtheatmosphericneutrinooscillations,respectively.WethereforesometimesusethenotationsΔm223≡Δm2atmandΔm212≡Δm2⊙toemphasizethattheyareexperimentally(thelattertobe)measuredquantities.Becauseofthehierarchyofmassscales,Δm2⊙/Δm2atm≪1,Δm212canbemadealwayspositiveasfarasθ12istakeninitsfullrange[0,π/2][23].Inordertoderiveconstraintonmixingparametersweneedtheclassification.CaseA:m1c212c213e−iβ+m2s212c213e+iβ≥m3s213(3)CaseB:m1c212c213e−iβ+m2s212c213e+iβ≤m3s213(4)3BothtypesofmasshierarchiesareallowedinthecasesAandB.Foragivenexperimentalupperboundonhmiββ,onecanderivealower(upper)boundons213inthecaseA(B).Therefore,westartwiththecaseA.A.CaseAInthiscase,thelowerboundonhmiββcanbeobtainedasinthefollowingway;hmiββ≥c213(m1c212+m2s212)cosβ−i(m1c212−m2s212)sinβ−m3