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Neutrino Mixing and CP Violation in Matter

Within the framework of three lepton families I present a transparent analytical relationship between the neutrino mixing and CP-violating parameters in vacuum and thos in matter. Such a model- and parametrization-independent result will be particularly useful to recast the fundamental lepton flavor mixing matrix from the future long-baseline neutrino experiments.

Today strong evidence, that neutrinos are massive and lepton flavors are mixed, has been accumulated from a variety of neutrino experiments. The mixing matrix of three different lepton families may in general consist of non-removable complex phases, leading to CP or T violation. Leptonic CP violation can manifest itself in neutrino oscillations. The best way to observe CP-and T-violating effects is to carry out the long-baseline neutrino experiments. In such experiments the earthinduced matter effects, which are likely to deform the neutrino oscillation behaviors in vacuum and fake the genuine CP-violating signals, must be taken into account. To single out the "true" theory of lepton mass generation and CP violation depends crucially upon how accurately the fundamental parameters of lepton flavor mixing can be measured and disentangled from the matter effects. It is therefore desirable to explore the most transparent analytical relationship between the genuine flavor mixing matrix and the matter-corrected one.
In this talk I present an exact and compact formula to describe the matter effect on lepton flavor mixing and CP violation within the framework of three lepton families [1]. The result is completely independent of the specific models of neutrino masses and the specific parametrizations of neutrino mixing. Therefore it will be particularly useful, in the long run, to recast the fundamental flavor mixing matrix from the precise measurements of neutrino oscillations in a variety of long-baseline neutrino experiments.
In vacuum the 3 × 3 lepton flavor mixing matrix V links the neutrino mass eigenstates (ν 1 , ν 2 , ν 3 ) to the neutrino flavor eigenstates (ν e , ν µ , ν τ ). If neutrinos are massive Dirac fermions, V can be parametrized in terms of three rotation angles and one CP-violating phase. If neutrinos are Majorana fermions, however, two additional CP-violating phases are in general needed to fully parametrize V . The strength of CP violation in neutrino oscillations, no matter whether neutrinos are of the Dirac or Majorana type, depends only upon a universal parameter J [2]: where (α, β, γ) and (i, j, k) run over (e, µ, τ ) and (1,2,3), respectively. In the specific models of fermion mass generation V can be derived from the mass matrices of charged leptons and neutrinos [3]. To test such theoretical models one has to compare their predictions for V with the experimental data of neutrino oscillations. The latter may in most cases be involved in the potential matter effects and must be carefully handled.
In the flavor basis where the charged lepton mass matrix M l is diagonal (i.e., M l = Diag{m e , m µ , m τ }), the effective Hamiltonian responsible for neutrinos propagating in matter can be written as Here m i (for i = 1, 2, 3) denote neutrino masses, A = 2 √ 2G F N e E describes the charged-current contribution to the ν e e − forward scattering, N e is the background density of electrons, and E stands for the neutrino beam energy. The neutralcurrent contributions, which are universal for ν e , ν µ and ν τ neutrinos, lead only to an overall unobservable phase and have been neglected. One can diagonalize Φ m ν with a unitary transformation: where λ i denote the effective mass-squared eigenvalues of three neutrinos in matter. Explicitly we have where x, y and z are given by [4] x = ∆m 2 21 + ∆m 2 31 + A , Note that the unitary matrix V m is just the lepton flavor mixing matrix in matter. After a lengthy calculation we arrive at the elements of V m as [1] where α runs over (e, µ, τ ) and (i, j, k) over (1, 2, 3) with i = j = k, and Obviously A = 0 leads to V m αi = V αi . This exact and compact formula shows clearly how the flavor mixing matrix in vacuum is corrected by the matter effects. Instructive analytical approximations can be made for Eq. (5), once the hierarchy of neutrino masses is experimentally known or theoretically assumed [5].
If . Following the calculations in Ref. [2], we arrive at Such an elegant relation has already been observed in Ref. [6]. It indicates that the matter contamination to CP-and T-violating observables, which must be dependent upon J m , is in general unavoidable. Note that both J m and λ i are complicated functions of the matter parameter A. The results obtained above are valid for neutrinos interacting with matter. As for antineutrinos, the matter effects arise from the charged-current contribution to the "ν e e + forward scattering". The corresponding formulas for antineutrino mixing can straightforwardly be obtained from Eqs. (5) and (7) through the replacements V =⇒ V * and A =⇒ −A.
The matter-corrected flavor mixing and CP-violating parameters can be determined from a variety of long-baseline neutrino experiments. We calculate the conversion probabilities of ν α (orν α ) to ν β (orν β ) neutrinos in matter and obtain where (α, β) run over (e, µ), ; and ∆ ij ≡ 1.27(λ i − λ j )L/E with L the distance between the production and interaction points of ν α (in unit of km) and E the neutrino beam energy (in unit of GeV). P m (ν β → ν α ) and P m (ν β →ν α ) can be read off from Eq.  Majorana phases, can be completely determined by four rephasing-invariant quantities (e.g., four independent |V m αi | or three independent |V m αi | plus J m ). As the solar and atmospheric neutrino oscillations in vacuum are essentially associated with the elements in the first row and the third column of V , it is favored to choose |V e1 |, |V e2 |, |V µ3 | and J as the four basic parameters. To be specific we take |V e1 | = 0.816, |V e2 | = 0.571, |V µ3 | = 0.640, and J = ±0.020 for neutrinos and antineutrinos. Such a choice is consistent with the CHOOZ experiment [7], the large-angle MSW solution to the solar neutrino problem, and a nearly maximal mixing in the atmospheric neutrino oscillation [8]. The relevant neutrino mass-squared differences are typically taken to be ∆m 2 21 = 5 · 10 −5 eV 2 and ∆m 2 31 = 3 · 10 −3 eV 2 . Then we compute the ratios |V m αi |/|V αi | and J m /J as functions of the matter parameter A in the range 10 −7 eV 2 ≤ A ≤ 10 −2 eV 2 . The numerical results are shown in Figs. 1 to 4.
We observe that matter effects can be significant for the elements in the first and the second columns of V , if A ≥ 10 −5 eV 2 . In comparison, the magnitudes of |V e3 |, |V µ3 | and |V τ 3 | may be drastically enhanced or suppressed only for A > 10 −3 eV 2 . The neutrinos are relatively more sensitive to the matter effects than the antineutrinos.
The magnitude of J m decreases, when the matter effect becomes significant (e.g., A ≥ 10 −4 eV 2 ). However, this does not imply that the CP-or T-violating asymmetries in realistic long-baseline neutrino oscillations would be smaller than their values in vacuum. Large matter effects can significantly modify the frequencies of neutrino oscillations and thus enhance or suppress the genuine signals of CP or T    violation.
If the earth-induced matter effects can well be controlled, it is possible to recast the fundamental flavor mixing matrix V from a variety of measurements of neutrino oscillations. Such a goal is expected to be reached in the neutrino factories [9,10].
I would like to thank H. Fritzsch and A. Thomas for supporting my p articipation in this interesting conference.