Cancellations in Neutrinoless Double Beta Decay an

arXiv:hep-ph/0008044v419Dec2000DO–TH00/12hep-ph/0008044CancellationsinNeutrinolessDoubleBetaDecayandtheNeutrinoMassMatrixWernerRodejohann∗Lehrstuhlf¨urTheoretischePhysikIII,Universit¨atDortmund,Otto–HahnStr.4,44221Dortmund,GermanyAbstractInadegenerateschemewithmassm0acompleteanalysisoftheallowedrangeoftheeffectiveelectronneutrinoMajoranamasshmiisperformed.Specialatten-tionispaidtoeffectsofcancellationscausedeitherbyintrinsicCPparitiesoftheeigenstates(CPinvariance)orbycomplexmixingmatrixelements(CPviolation).Weinvestigateallpossibilitiesandgiveineachcaseconstraintsonthephases,therelativeCPparitiesortheneutrinomassscale.Asolarmixinganglesin22θsmallerthan0.7jeopardizesthedegeneratemassscheme.Akeyvalueofhmi/m0isidenti-fied,whichisindependentonthesolarsolutionandwouldruleoutcertainschemes.AlsoitwouldanswerthequestionregardingthepresenceofCPviolation.Evenifatotalneutrinomassscaleandaneffectivemassismeasured,thevalueofthephasesorparitiesisnotfixed,unlessinsomespecialcases.Theresultinguncertaintyintheothermassmatrixelementsisatleastofthesameorderthantheonestemmingfromnuclearmatrixelementscalculations.Keywords:NeutrinoOscillation,DoubleBetaDecay,MassiveNeutrinos,MajoranaNeutrinosPACS:14.60.Pq,23.40.-s∗Emailaddress:[e-mail]1IntroductionInthelightofrecentimpressiveexperimentalevidenceonneutrinooscillations[1]thenextfundamentalquestiontobeansweredistheoneregardingtheneutrinocharacter.FromthetheoreticalsideMajorananeutrinosarefavored,sincetheypopoutofalmosteveryGUTandareforexampletheresultoftheveryattractivesee–saw–mechanism[2].Inthiscase,heavyneutrinosarepredicted.ExperimentalinformationontheexistenceofMajorananeutrinosmightcomefromneutrinolessdoublebetadecay(0νββ)[3]orfromproductionofheavyMajorananeutrinosatcolliders,seee.g.[4]andreferencestherein.Themoststringentlimithowevercomesfromthefirstprocess,investigatingtheeffectiveMajoranamassoftheelectronneutrino,withacurrentboundof[5]hmi∼0.2eV.(1)Plansexisttobuildexperimentsexploringregionsuptohmi≃0.002eV[6].Thevariationwithinafactorofroughly3betweendifferentcalculationsoftherequirednuclearmatrixelementshastobekeptinmind.Resultsofoscillationexperimentscanbeusedtorestrictthevalueofhmiindifferentmassschemesandforthedifferentsolutionsofthesolarneutrinoproblem[7,8,9,10,11,12,13,14,15,16,17].Typicalkeyscalesforhmiare0.1and0.005eV,thuslyingintherangeofcurrentandforthcomingprojects.However,hmihasaformwhichincludespossibilitiesforthecontributionstocanceleachother,namelyviaCP–violatingphasesorviatheintrinsicCP–paritiesofthemassstates,whichexistinthecaseofCPconservation.Theseeffectswereincludedinmostoftheabovegivenreferencestogetthemaximalandminimalvaluesofhmi.In[15]constraintsonCP–violatingphasesweregivenbyusingagraphicalrepresentationforthecomplexmassmatrixelements,althoughwithoutapplyingnumericalvalues.NumericalstudiesregardingtheMajoranaphasesweregivenin[8,10].Someoverlapwiththeseworksexists,however,wegivemanyplotsandstatements,whicharenewtotheliteratureandadoptanapproach,whichallowstoinvestigatedifferentsituationsregardingmeasurementsofhmiand/orm0.Herem0denotesthecommonmassscaleinadegeneratemassscheme.Amongthetopicsdiscussedareaclarificationofthekindofcancellation,i.e.adistinctionbetweenthe2possibilities.Whenstartingfromadegenerateschemeandinagivensituationthemaximalallowedm0iscomparabletothescaleimpliedbytheatmosphericanomaly,sothedegenerateschemeisruledoutandthederivedmasslimitholdsforthelargestofthe3masseigenstates.Wewillfindthatonlyinveryspecialcasesdefinitestatementsaboutthephasesorparitiescanbemadeandthatalonefromthisfactthereisalargeuncertaintyinthevaluesoftheothermassmatrixentries.Thisuncertaintyisrangingfrom2tofactorsaround20.Theotherentriesareclearlyneededtodistinguishbetweendifferentdiscussedmodels.Theonlyinformationaboutthephasesorparitiescancomefromneutrinolessdoublebetadecay:Allothermassmatrixentriesareimpossibletomeasuredirectly,sincetherespectivebranchingratiosorcrosssectionstheygovernarewaybeyondexperimentalaccess[14].Apartfromtheobviousaspectthatfundamentalparametersofamodelneedtobemeasured,otherinterestingapplicationofthephasesexist:In[18]itwasfoundthattheMajoranaphasesplayacrucialroleinthestabilityofthemixinganglesagainstquantumcorrections.Ref.2[19]findsthattheirvalueshaveinfluenceonthemagnitudeofthelepton–asymmetryintheuniverse,whichcanbemaderesponsiblefortheobservedbaryon–asymmetry.Thus,thepreciseknowledgeofallmixingparametersintheleptonsectoriscertainlyimportant.Thepaperisorganizedasfollows:InSection2wepresentthegeneralframeworkandbasicformulaeforCPviolationandconservation,respectively.Section3seesadiscussionoftheconnectionbetweenoscillationandhmiinhierarchicalschemes,whereasSec.4concentratesonthedegeneratescheme.Somespecialmixingmatricesandthegeneralcasewithexperimentallyfavoredvaluesarediscussed.Foreachcaseconstraintsonthephases,paritiesandm0aregiven.ThepaperisconcludedinSec.5withasummaryofourresults.2FormalismFlavoreigenstatesνα(α=e,μ,τ)areconnectedtomasseigenstatesνi(i=1,2,3)viaamixingmatrix,i.e.να=Sαiνi.Apropertreatmentofthisissuecanbefoun