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

productionandinteractionpointsofνα(inunitofkm)andEtheneutrinobeamenergy(inunitofGeV).Theprobabilitiesofνβ→ναandν¯β→ν¯αtransitionscanbereado m= J m,respectively.fromEq.(11)withthereplacementsJm= JmandJOnecanthende netheCP-andT-violatingasymmetriesas

ACP=Pm(να→νβ) Pm(¯να→ν¯β)

.(13)Pm(να→νβ)+Pm(νβ→να)

NotethatAT=ACPholdsinvacuum(i.e.,A=0),asaconsequenceofCPTinvariance.Anydiscrepancybetweenthesetwoobservableswillde nitelymeasurethemattere ectsinlong-baselineneutrinoexperiments.

Finallyletusgiveanumericalillustrationofthematter-inducedcorrectionstothe avormixingmatrixandCP(orT)violationinvacuum.TheelementsofVm,excepttheMajoranaphases,canbecompletelydeterminedbyfourrephasing-invariantquanti-

mmties(e.g.,fourindependent|Vαi|orthreeindependent|Vαi|plusJm).Asthesolarand

atmosphericneutrinooscillationsinvacuumareessentiallyassociatedwiththeelementsinthe rstrowandthethirdcolumnofV,itisfavoredtochoose|Ve1|,|Ve2|,|Vµ3|andJasthefourbasicparameters.Forillustrationwetake|Ve1|=0.816,|Ve2|=0.571,|Vµ3|=0.640,andJ=±0.020forneutrinosandantineutrinos§.Suchachoiceisconsis-tentwiththeCHOOZexperiment[17],thelarge-angleMSWsolutiontothesolarneutrinoproblem[18],andanearlymaximalmixingintheatmosphericneutrinooscillation.The

5relevantneutrinomass-squareddi erencesaretypicallytakentobe m2eV2

21=5·10

3and m2eV2[1,18].TocalculatetheCP-andT-violatingasymmetriesin31=3·10

thelong-baselineneutrinoexperiments,weassumeaconstantearthdensitypro leandtakeA=2.28·10 4eV2E/[GeV][12].WealsochoosethebaselinelengthL=732km,ingtheseinputsaswellasEqs.(9)–(12),we rsttake(α,β)=(e,µ)andcomputetheasymmetriesACPandATchangingwithEintherange2GeV≤E≤30GeV.Thenwecomputetheratiosm|Vαi|/|Vαi|andJm/JasfunctionsofthematterparameterA,insteadofE,intherange10 7eV2≤A≤10 2eV2.ThenumericalresultsareshowninFigs.1and2.

Weobservethatmattere ectscanbesigni cantfortheelementsinthe rstandthesecondcolumnsofV,ifA≥10 5eV2.Incomparison,themagnitudesof|Ve3|,|Vµ3|and|Vτ3|maybedrasticallyenhancedorsuppressedonlyforA>10 3eV2.Theneutrinosarerelativelymoresensitivetothemattere ectsthantheantineutrinos.

ThemagnitudeofJmdecreases,whenthemattere ectbecomessigni cant(e.g.,A≥10 4eV2).However,rgemattere ectscansigni cantlymodifythefrequenciesofneutrinooscillations