Within the framework of three lepton families we have derived an exact and compact formula to describe the matter effect on neutrino mixing and CP violation. This model- and parametrization-independent result can be particularly useful for recasting the fu

RecentobservationoftheatmosphericandsolarneutrinoanomaliesbytheSuper-KamiokandeCollaborationhasprovidedrobustevidenceformassiveneutrinosthatmixamongdi erent avors[1] .ItopensaconvincingwindowfornewphysicsbeyondtheStandardModel,andhasimportantcosmologicalimplications.

Althoughthenon-acceleratorneutrinoexperimentshaveyieldedsomeimpressivecon-straintsontheparameterspaceofatmosphericandsolarneutrinooscillations,aprecisedeterminationofthelepton avormixinganglesandtheleptonicCP-violatingphase(s)hastorelyonanewgenerationofacceleratorexperimentswithverylongbaselines[4],includingthepossibleneutrinofactories[5].Insuchlong-baselineneutrinoexperimentstheearth-inducedmattere ects,whicharelikelytodeformtheneutrinooscillationbe-haviorsinvacuumandeventofakethegenuineCP-violatingasymmetries,mustbetakenintoaccount.Tosingleoutthe“true”theoryofleptonmassgenerationandCPviolationdependscruciallyuponhowaccuratelythefundamentalparametersoflepton avormix-ingcanbemeasuredanddisentangledfromthemattere ects.Itisthereforedesirabletoexploretheanalyticalrelationshipbetweenthegenuine avormixingmatrixandthematter-correctedonebeyondtheconventionaltwo- avorframework[6,7,8,9].Someat-temptsofthisnaturehavesofarbeenmade[7,10,11,12],buttheyaresubjecttospeci cassumptions,approximationsorparametrizationsandneedbesubstantiallyimproved.Inthisnotewepresentanexactandcompactformulatodescribethemattere ectonlepton avormixingandCPviolationwithintheframeworkofthreeleptonfamilies.Theresultiscompletelyindependentofthespeci cmodelsofneutrinomassesandthespeci cparametrizationsofneutrinomixing.Thereforeitwillbeparticularlyuseful,inthelongrun,torecastthefundamental avormixingmatrixfromtheprecisemeasurementsofneutrinooscillationsinavarietyoflong-baselineneutrinoexperiments.

Firstletusde nethe3×3lepton avormixingmatrixinvacuumtobeV.Itlinkstheneutrinomasseigenstates(ν1,ν2,ν3)totheneutrino avoreigenstates(νe,νµ,ντ):

IfneutrinosaremassiveDiracfermions,VcanbeparametrizedintermsofthreerotationanglesandoneCP-violatingphase.IfneutrinosareMajoranafermions,however,twoadditionalCP-violatingphasesareingeneralneededtofullyparametrizeV.ThestrengthofCPviolationinneutrinooscillations,nomatterwhetherneutrinosareoftheDiracorMajoranatype,dependsonlyuponauniversalparameterJ[13]:

Im VαiVβjVαjVβiVe1νe νµ = Vµ1Vτ1ντ Ve2Vµ2Vτ2ν1Ve3 Vµ3 ν2 .ν3Vτ3 (1) =J γ,k αβγ ijk,(2)

where(α,β,γ)and(i,j,k)runover(e,µ,τ)and(1,2,3),respectively.Inthespeci cmodelsoffermionmassgenerationVcanbederivedfromthemassmatricesofchargedleptonsandneutrinos[14].Totestsuchtheoreticalmodelsonehastocomparetheir