Narrow Results

We investigate the effect of CP violation in the leptonic sector. Due to the tiny neutrino masses its value is predicted to be very small and it is far beyond the experimental reach of the current experiments. Recently, the magic baseline experiment from CERN to INO (Indian Neutrino Observatory) with L = 7152 km has been proposed to get a sensitive limit on sin θ₁₃. We show that due to such magic baseline neutrino beam, it is possible to observe CP violation in the neutrino sector upto several percent for the beam energy between 1–10 GeV.

We propose a universal mixing hypothesis between quark and lepton sectors at high energy scale (probably GUT scale) where quark–lepton universality holds. Namely in the charged lepton diagonal base, all the other mass matrices for up and down quarks and neutrinos are diagonalized by the same unitary matrix except for the phase elements. Thanks to this hypothesis, the observed values of the Cabibbo–Kobayashi–Maskawa (CKM) quark mixing matrix and the mixing angles θ₁₂ and θ₂₃ in the Maki–Nakagawa–Sakata (MNS) lepton mixing matrix can predict the unknown magnitudes of the mixing angle θ₁₃ and of the CP violating Dirac phase δ in the MNS matrix. Their allowed regions are rather narrow, 0.036 < sin θ₁₃ < 0.048 and 6° < δ < 12°.

The Daya Bay oscillation has recently reported the precise measurement of θ₁₃≃8.8°±0.8° or θ₁₃ ≠0 at 5.2 σ level. The observed nonzero θ₁₃ can be accommodated by some general modifications to the democratic mixing matrix. Using such matrices we study the possibility of observing nonzero CP violation in the leptonic sector.

The recently observed lepton mixing angle θ₁₃ of the MNS mixing matrix is well incorporated in a universal mixing hypothesis between quark and lepton sectors. This hypothesis asserts that, in the charged lepton diagonal base, all other mass matrices for up- and down-type quarks and light neutrinos are diagonalized by the same unitary matrix except for the phase elements. It is expressed as V_CKM = U_MNS(δ′)^†PU_MNS(δ) for quark mixing matrix V_CKM and lepton mixing matrix U_MNS(δ) in the phenomenological level. Here P is a diagonal phase mass matrix. δ′ is a slightly different phase parameter from the Dirac CP-violating phase δ = 1.1π (best fit) in the MNS lepton mixing matrix.

The universality hypothesis for quark and lepton mixing matrices (CKM and MNS) is further developed. This hypothesis explains why the CKM is almost diagonal whereas the MNS is almost maximally mixed. If this hypothesis is true, the Dirac CP violating phase of the MNS mixing matrix is bounded around π or 0. Quark–lepton mass matrices which realize this hypothesis are constructed, showing simple power law relations among mass matrices for up-type, down-type quarks and neutrinos.

Averaged neutrino masses defined by are reanalyzed using up-to-date observed MNS parameters and neutrino masses by the neutrino oscillation experiments together with the cosmological constraint on neutrino masses. The values of 〈m_ν〉_ab are model-independently evaluated in terms of effective neutrino mass defined by which is observable in the single beta decay. We obtain lower bound for 〈m_ν〉_ee in the inverted hierarchy (IH) case, 17 meV ≤〈m_ν〉_ee and one for 〈m_ν〉_τμ in the normal hierarchy (NH) case, 5 meV≤〈m_ν〉_τμ. We also obtain that all the averaged masses 〈m_ν〉_ab have upper bounds which are at most 80 meV.

A simple form of neutrino mass matrix which has only two free parameters is proposed from a phenomenological point of view. Using this mass matrix, we succeed to reproduce all the observed values for the Maki–Nakagawa–Sakata (MNS) lepton mixing angles and the neutrino mass squared difference ratio. Our model also predicts δ_ν = 155° for the Dirac CP violating phase in the lepton sector and the effective neutrino mass 〈m 〉 = 6.3×10^-3eV in the neutrinoless double beta decay.

Leptogenesis is the most favorable mechanism for generating the observed baryon asymmetry of the universe (BAU) which implies CP violation in the high energy scale. The low energy leptonic CP violation is expected to be observed in the neutrino oscillations and 0ν2β decay experiments. Generally, it is not possible to connect both the CP violations. Here we revisit the issue of connecting the two in flavored leptogenesis scenario within the Type I seesaw in the light of recent neutrino oscillation and Planck data. With the recent precise measurements of θ₁₃ and BAU, we are able to find new correlations between the low and high energy CP violating phases when leptogenesis occurs at temperature between 10⁹ to 10¹² GeV and there is no contribution to CP violation from the heavy neutrino sector.

A bonus of the framed Standard Model (FSM), constructed initially to explain the mass and mixing patterns of quarks and leptons, is a solution (without axions) of the strong CP problem by cancelling the theta-angle term ${𝜃}_I\mathrm{Tr}(H^{\mu \nu }{H}_{\mu \nu }^{\ast })$ in colour^a by a chiral transformation on a quark zero mode which is inherent in FSM, and produces thereby a CP-violating phase in the CKM matrix similar in size to what is observed.¹ Extending here to flavour, one finds that there are two terms proportional to $\mathrm{Tr}(G^{\mu \nu }{G}_{\mu \nu }^{\ast })$: (a) in the action from flavour instantons with unknown coefficient, say ${𝜃}_I^{\prime }$, (b) induced by the above FSM solution to the strong CP-problem with therefore known coefficient ${𝜃}_C^{\prime }$. Both terms can be cancelled in the FSM by a chiral transformation on the lepton zero mode to give a Jarlskog invariant $J^{\prime }$ in the PMNS matrix for leptons of order $10^{-2}$, as is hinted by the experiment. But if, as suggested in Ref. 2, the term ${𝜃}_I^{\prime }$ is to be cancelled by a chiral transformation in the predicted hidden sector to solve the strong CP problem therein, leaving only the term ${𝜃}_C^{\prime }$ to be cancelled by the chiral transformation on leptons, then the following prediction results: $J^{\prime }\sim -0.012$$({\delta }_{\mathrm{\text{CP}}}^{\prime }\sim (1.11)\pi )$ which is (i) of the right order, (ii) of the right sign and (iii) in the range favoured by the present experiment. Together with the earlier result for quarks, this offers an attractive unified treatment of all known CP physics.

We use $F_{\mu \nu }$, $G_{\mu \nu }$ and $H_{\mu \nu }$ here to denote, respectively the $u(1)$, flavour $su(2)$ and colour $su(3)$ gauge fields.

We review some recent results on the connection between CP violation at low energies and Leptogenesis in the framework of specific flavour structures for the fundamental leptonic mass matrices with zero textures.