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

22OnemaycheckthatA=0leadstoλi=m2

iandDi=Ni,andthenUbecomestheunity

matrix.

ItshouldbenotedthattheunitarymatrixUtransformstheneutrinomasseigenstatesinmattertothoseinvacuum.Thereforethelepton avormixingmatrixinmatter,denotedbyVm,isaproductofthelepton avormixingmatrixinvacuum(V)andthematter-to-vacuumtransformationmatrixU:Vm=VU.TheelementsofVmturnouttobe

mVαi=Ni

DiVei λi m2

j VekVαk+λi m2

k VejVαj ,(9)

whereαrunsover(e,µ,τ)and(i,j,k)over(1,2,3)withi=j=k.ObviouslyA=0leads

mtoVαi=Vαi.Thisexactandcompactformulashowsclearlyhowthe avormixingmatrixinvacuumiscorrectedbythemattere ects.InstructiveanalyticalapproximationscanbemadeforEq.(9),oncethehierarchyofneutrinomassesisexperimentallyknownortheoreticallyassumed.

Theresultobtainedaboveisvalidforneutrinosinteractingwithmatter.Asforan-tineutrinos,thecorrespondingformulaforthe avormixingmatrixinmattercanstraight-forwardlybeobtainedfromEq.(9)throughthereplacementsV= V andA= A.

WiththehelpofEq.(9)wenowcalculatetheuniversalCP-violatingparameterinmatter,de nedasJmthrough

Im mmm m VαiVβjVαjVβi =Jm

γ,k αβγ ijk,(10)

where(α,β,γ)and(i,j,k)runover(e,µ,τ)and(1,2,3),respectively.TheunitarityofVmallowsustosimplifythelengthycalculationandarriveataninstructiverelationshipbetweenJmandJ:22 m2

21 m31 m32Jm=J