Leptonic CP violation phases using an ansatz for t

arXiv:hep-ph/0402176v230Aug2004PP-04-024SU4252-790LeptonicCPviolationphasesusinganansatzfortheneutrinomassmatrixandapplicationtoleptogenesisSalahNasri∗DepartmentofPhysics,UniversityofMaryland,CollegePark,MD20742-4111,USAJosephSchechter†DepartmentofPhysics,SyracuseUniversity,Syracuse,NY13244-1130,USASherifMoussa‡DepartmentofMathematics,FacultyofScience,AinShamsUniversity,Egypt(Dated:February2004)WefurtherstudythepreviouslyproposedAnsatz,Tr(Mν)=0,foraprediagonallightMajoranatypeneutrinomassmatrix.IfCPviolationisneglectedthisenablesonetousetheexistingdataonsquaredmassdifferencestoestimate(uptoadiscreteambiguity)theneutrinomassesthemselves.IfitisassumedthatonlytheconventionalCPphaseispresent,theAnsatzenablesustoestimatethisphaseinadditiontoallthreemasses.IfitisassumedthatonlythetwoMajoranaCPphasesarepresent,theAnsatzenablesustopresentaoneparameterfamilyofsolutionsforthemassesandphases.Thisenablesustoobtainasimple“global”viewofleptonnumberviolationeffects.FurthermoreusinganSO(10)motivationfortheAnsatzsuggestsanamusingtoy(clone)modelinwhichtheheavyneutrinoshavethesamemixingpatternandmassratiosasthelightones.Inthiscaseonlytheiroverallmassscaleisnotknown(althoughitisconstrainedbytheinitialmotivation).Usingthistoymodelwemakearoughestimateofthemagnitudeofthebaryontophotonratioinducedbytheleptogenesismechanism.SolutionsclosetotheCPconservingcasesseemtobefavored.PACSnumbers:I.INTRODUCTIONRemarkably,therecentKamLAND[1],SNO[2]andK2K[3]experimentshaveaddedsomuchtotheresultsobtainedfromearliersolarneutrino,atmosphericneutrinoandacceleratorexperiments[4]thatourknowledgeabouttheneutrinomassesandpresumedleptonmixingmatrixisalmostasgreatasourknowledgeofthecorrespondingquantitiesinthequarksector.StillthereisanuncertaintyabouttheinterpretationduetotheresultsoftheLSNDexperiment[5].However,thisexperimentwillbecheckedsoonbytheminiBoonecollaborationsoonecanwaitforconfirmationbeforeconsideringwhetherthereisreallyaproblemwiththeusualpictureofthreemassiveneutrinos.Inanyevent,thereisastrongpresumptionthatthisknowledgewillplayanimportantroleingoingbeyondthestandardmodelofelectroweakinteractions.Onedetailis,ofcourse,lackingcomparedtothequarkcase.Sincetheneutrinooscillationexperimentsmeasureonlythedifferencesoftheneutrinosquaredmasses,theneutrinomassesthemselvesarenotknown.Accordingtothelatestanalysis[6]thebestfittothesedifferencesis:m22−m21=6.9×10−5eV2,|m23−m22|=2.6×10−3eV2.(1)Now,thereisasimplecomplementaryAnsatzforthe3x3neutrinomassmatrix,Mνwhich,withsomeassumptions,enablesonetoobtaintheneutrinomassesthemselvesfromEq.(1);itrequires:Tr(Mν)=0.(2)∗Electronicaddress:[e-mail]†Electronicaddress:[e-mail]‡Electronicaddress:[e-mail]2ItshouldberemarkedthatMνistoberegardedastheprediagonalneutrinomassmatrix.Furthermore,intherelationm1+m2+m3=0whichevidentlyresultsiftheneutrinomassmatrixistakentoberealsymmetric,theindividualmassesmaybeeitherpositiveornegative.Thenegativemassescanbeconvertedtopositiveonesbyaddingappropriatefactorsofiinthediagonalizingmatrix.Eq.(2)wasmotivatedin[7]fromthegrandunifiedmodel,SO(10)[8]andin[9]bynotingthatitwouldholdifMνisthecommutatoroftwoothermatrices,asmayoccurincertainmodels.IfCPviolationisneglectedthereareessentiallytwopossiblesolutionsoftheAnsatz:eitherm1andm2havethesamesignandareapproximatelyequaltoeachotherandto−m3/2orelsem1andm2havetheoppositesignandareapproximatelyequaltoeachotherinmagnitudebutmuchlargerthanthemagnitudeofm3.InthepresentpaperwewilltakethepointofviewthattheAnsatz,Eq.(2),ismotivatedfromSO(10).However,theanalysisisofcoursenotdependentonthemotivation.TheSO(10)motivationarisesfromtheobservationthatEq.(2)is,althoughitseemsatfirstdifferent,essentiallythesameasthecharacteristicpredictionofgrandunification:mb=rmτ,(3)relatingthemassofthebquarkwiththemassofthethetaulepton(r≈3takesaccountoftherunningofmassesfromthegrandunificationscaletothelowenergyhadronicscaleofabout1GeV[10]).NotethatinSO(10)theneutrinomassmatrixtakestheform:Mν=ML−MTDM−1HMD,(4)whereML,MHandMDarerespectivelythemassmatricesofthelightneutrinos,heavyneutrinosandheavy-lightmixing(or“Diracmatrix”).Tostartwith,Mνisanarbitrarysymmetricmatrix.IfitisrealwehaveCPinvariance.Generallythesecond,seesaw[11]termisconsideredtodominate.However,asexplainedin[7],thepresentmodelisbasedontheassumptionthatthefirsttermdominates.Thatmightnotbeunreasonablesincearoughorderofmagnitudeestimateforthesecondtermwouldbem2t/1017orabout3×10−4eV.(Thequantity1017includesafactorr2≈10).Thusthesecondtermcouldbenegligibleifneutrinomassesareappreciablygreaterthanthisvalue.In[7]thecomplementaryAnsatzwasmainlystudiedforthecaseofrealMν.HerewewillbeprimarilyinterestedinthemoregeneralcomplexcasewhichallowsfornonzeroCPviolation.Furthermore,theinputsquaredmassdifferenceswerenottakentobeverysimilartothoseinEq.(1)butwerebasedonaleastsquaresfit[12]ofmanydifferentexperimentsincludingLSND.HereweshalladoptthemoreconventionalvaluesgiveninEq.(1).ArelatedanalysisofEq.(2)wasrecentlymadein[13].Foranunderstandingoftheinterestingleptogenesismechanism[14]ofbaryogenesisitisimp