The unitarity of the lepton mixing matrix is a critical assumption underlying the standard neutrino-mixing paradigm. However, many models seeking to explain the as-yet-unknown origin of neutrino masses predict deviations from unitarity in the mixing of the active neutrino states. Motivated by the prospect that future experiments may provide a precise measurement of the lepton mixing matrix, we revisit current constraints on unitarity violation from oscillation measurements and project how next-generation experiments will improve our current knowledge. With the next-generation data, the normalizations of all rows and columns of the lepton mixing matrix will be constrained to $\lesssim$10\% precision, with the $e$-row best measured at $\lesssim$1\% and the $τ$-row worst measured at ${\sim}10\%$ precision. The measurements of the mixing matrix elements themselves will be improved on average by a factor of $3$. We highlight the complementarity of DUNE, T2HK, JUNO, and IceCube Upgrade for these improvements, as well as the importance of $ν_τ$ appearance measurements and sterile neutrino searches for tests of leptonic unitarity.
We present a geometric description of lepton flavor mixing and CP violation in matter by using the language of leptonic unitarity triangles. The exact analytical relations for both sides and inner angles are established between every unitarity triangle in vacuum and its effective counterpart in matter. The typical shape evolution of six triangles with the terrestrial matter density is illustrated for a realistic long-baseline neutrino oscillation experiment.
We present a comprehensive analysis of leptonic unitarity triangles, using both current neutrino oscillation data and projections of next-generation oscillation measurements. Future experiments, sensitive to the degree of CP violation in the lepton sector, will enable the construction of precise triangles. We show how unitarity violation could manifest in the triangles and discuss how they serve as unitarity tests. We also propose the use of Jarlskog factors as a complementary means of probing unitarity. This analysis highlights the importance of testing the unitarity of the leptonic mixing ma…
We determine the elements of the leptonic mixing matrix, without assuming unitarity, combining data from neutrino oscillation experiments and weak decays. To that end, we first develop a formalism for studying neutrino oscillations in vacuum and matter when the leptonic mixing matrix is not unitary. To be conservative, only three light neutrino species are considered, whose propagation is generically affected by non-unitary effects. Precision improvements within future facilities are discussed as well.
We derive a new set of sum rules between the neutrino mass and mixing parameters in vacuum and their effective counterparts in matter. The moduli of nine genuine lepton mixing matrix elements can then be calculated in terms of the matter-corrected ones, and the latter can directly be determined from a variety of long-baseline neutrino oscillations. We show that it is possible to reconstruct the leptonic unitarity triangles and CP violation in a similar parametrization-independent way.
There are six leptonic unitarity triangles (LUTs) defined by six orthogonality conditions of the three-family lepton flavor mixing matrix in the complex plane. In the framework of the standard model or the minimal supersymmetric standard model, the evolutions of sides and inner angles of the six LUTs from a superhigh energy scale $Λ_{\rm H}^{}$ to the electroweak scale $Λ_{\rm EW}^{}$ due to the renormalization-group equation (RGE) running are derived in the integral form for both Dirac and Majorana neutrinos. Furthermore, the LUTs as an intuitively geometrical language are applied to the desc…
The recent IceCube observation of ultra-high-energy astrophysical neutrinos has begun the era of neutrino astronomy. In this work, using the unitarity of leptonic mixing matrix, we derive nontrivial unitarity constraints on the flavor composition of astrophysical neutrinos detected by IceCube. Applying leptonic unitarity triangles, we deduce these unitarity bounds from geometrical conditions, such as triangular inequalities. These new bounds generally hold for three flavor neutrinos, and are independent of any experimental input or the pattern of leptonic mixing. We apply our unitarity bounds …
Following analogy of the `db' triangle in the quark mixing case, we have explored the construction of the `ν_2.ν_3' leptonic unitarity triangle using the Maki-Nakagawa-Sakata matrix obtained by Bjorken {\it et al.} through generalization of the tribimaximal scenario. In particular, for the U_{e3} range 0.05-0.15, the existence of the leptonic unitarity triangle indicates a fairly good possibility of having non zero CP violation.
If leptonic unitarity is violated by new physics at an energy scale much lower than the electroweak scale, which we call low-scale unitarity violation, it has different characteristic features from those expected in unitarity violation at high-energy scales. They include maintaining flavor universality and absence of zero-distance flavor transition. We present a framework for testing such unitarity violation at low energies by neutrino oscillation experiments. Starting from the unitary 3 active plus $N$ (arbitrary integer) sterile neutrino model we show that by restricting the active-sterile a…
The non-unitarity of the leptonic mixing matrix is a generic signal of new physics aiming at the generation of the observed neutrino masses. We discuss the Minimal Unitarity Violation (MUV) scheme, an effective field theory framework which represents the class of extensions of the Standard Model (SM) by heavy neutral leptons, and discuss the present bounds on the non-unitarity parameters as well as estimates for the sensitivity of the Circular Electron Positron Collider (CEPC), based on the performance parameters from the preCDR.
Leptonic unitarity triangle (LUT) provides a geometric description of CP violations in the lepton-neutrino sector and is directly measurable in principle. In this work, we reveal that the angles in the LUT have definite physical meaning, and demonstrate the exact connection of the LUT to neutrino oscillations. For the first time, we prove that these leptonic angles act as phase shifts in neutrino oscillations, by shifting Δm^{2}L/2E to Δm^{2}L/2E + α, where (L, E, α) denote the baseline length, neutrino energy and corresponding angle of the LUT. Each LUT has three independent parameters and co…
The non-unitarity effects in the leptonic flavor mixing are regarded as one of the generic features of the type-I seesaw model. Therefore, we explore these effects in the TeV-scale type-I seesaw model, and show that there exist non-trivial correlations among the non-unitarity parameters, stemming from the typical flavor structure of the low-scale seesaw model. In general, it follows from analytical discussions and numerical results that all the six non-unitarity parameters are related to three model parameters, while the widely studied parameters η_{eτ} and η_{μτ} cannot be phenomenologically …
The non-unitarity of the effective leptonic mixing matrix at low energies is a generic signal of extensions of the Standard Model (SM) with extra fermionic singlet particles, i.e. sterile or right-handed neutrinos, to account for the observed neutrino masses. The low energy effects of such extensions can be described in a model-independent way by the Minimal Unitarity Violation (MUV) scheme, an effective field theory extension of the SM. We perform a global fit of the MUV scheme parameters to the present experimental data, which yields the up-to-date constraints on leptonic non-unitarity. Furt…
A low-energy non-unitary leptonic mixing matrix is a generic effect of some theories of new physics accounting for neutrino masses. We show how the extra CP-phases of a general non-unitary matrix allow for sizeable CP-asymmetries in the $ν_μ\to ν_τ$ channel. This CP-asymmetries turns out to be an excellent probe of such new physics.
In the light of the recent LHC data, we study precision tests sensitive to the violation of lepton universality, in particular the violation of unitarity in neutrino mixing. Keeping all data we find no satisfatory fit, even allowing for violations of unitarity in neutrino mixing. Leaving out sin$^2 θ_{\scriptsize \mbox{eff}}$ from the hadronic forward-backward asymmetry at LEP, we find a good fit to the data with some evidence of lepton universality violation at the $\mathcal{O}(10^{-3})$ level.
Given the best-fit results of six neutrino oscillation parameters, we plot the Dirac and Majorana unitarity triangles (UTs) of the 3\times 3 lepton flavor mixing matrix to show their real shapes for the first time. The connections of the Majorana UTs with neutrino-antineutrino oscillations and neutrino decays are explored, and the possibilities of right or isosceles UTs are discussed. In the neutrino mass limit of m_1 \to 0 or m_3 \to 0, which is allowed by current experimental data, we show how the six triangles formed by the effective Majorana neutrino masses \langle m\rangle_{αβ} (for α, β=…
Complete one-loop QED corrections to $D_s^+$ leptonic decays and impact on the CKM unitarity test
Chiba University（千叶大学）；Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University（名古屋大学小林-益川粒子宇宙起源研究所）；The University of Tokyo（东京大学）
We address the question of deviations from $3\times 3$ unitarity of the leptonic mixing matrix showing that, contrary to conventional wisdom, one may have significant deviations from unitarity in the framework of type I seesaw mechanism. In order for this scenario to be feasible, at least one of the heavy neutrinos must have a mass at the TeV scale, while the other two may have much larger masses. We present specific examples where deviations from $3\times 3$ unitarity are sufficiently small to conform to all the present stringent experimental bounds but are sufficiently large to have the pote…
We present a comprehensive study of the three-active plus $N$ sterile neutrino model as a framework for constraining leptonic unitarity violation induced at energy scales much lower than the electroweak scale. We formulate a perturbation theory with expansion in small unitarity violating matrix element $W$ while keeping (non-$W$ suppressed) matter effect to all orders. We show that under the same condition of sterile state masses $0.1\, \text{eV}^2 \lesssim m^2_{J} \lesssim (1-10)\, \text{GeV}^2$ as in vacuum, assuming typical accelerator based long-baseline neutrino oscillation experiment, on…
We show that a TeV scale inverse seesaw model for neutrino masses can be realized within the framework of a supersymmetric SO(10) model consistent with gauge coupling unification and observed neutrino masses and mixing. We present our expectations for non-unitarity effects in the leptonic mixing matrix some of which are observable at future neutrino factories as well as the next generation searches for lepton flavor violating processes such as μ--> e + γ. The model has TeV scale W_R and Z' bosons which are accessible at the Large Hadron Collider.
Leptonic unitarity triangle (LUT) provides fundamental means to geometrically describe CP violation in neutrino oscillation. In this work, we use LUT to present a new geometrical interpretation of the vacuum oscillation probability, and derive a compact new oscillation formula in terms of only 3 independent parameters of the corresponding LUT. Then, we systematically study matter effects in the geometrical formulation of neutrino oscillation with CP violation. Including nontrivial matter effects, we derive a very compact new oscillation formula by using the LUT formulation. We further demonstr…
The area of the unitarity triangle is a measure of CP-violation. We introduce the leptonic unitarity triangles and study their properties. We consider the possibility of reconstructing the unitarity triangle in future oscillation and non-oscillation experiments. A set of measurements is suggested which will, in principle, allow us to measure all sides of the triangle, and consequently to establish CP-violation. For different values of the CP-violating phase, $δ_D$, the required accuracy of measurements is estimated. The key elements of the method include determination of $|U_{e3}|$ and studies…
We study the linking between the CP violating phase of the lepton sectors and the unitarity triangle of the B0 meson system. Antusch, King, Malinsky and Spinrath have shown that the quark mass matrices with the negligible 1-3 mixing give an interesting relation between the phase of the quark mixing matrices and CP violating measure phi_2(alpha). This approach is extended by considering the SO(10) GUT including the Pati-Salam symmetry, which links the Yukawa matrices of the quark sector to the one of the lepton sector. We discuss the relation of the CP violating phases between both quark and le…
In the Next-to-Minimal Supersymmetric Standard Model with the inverse seesaw mechanism to generate neutrino masses, the lightest sneutrino may act as a feasible dark matter candidate in vast parameter space. In this case, the smallness of the leptonic unitarity violation and the recent XENON-1T experiment can limit the dark matter physics. In particular, they set upper bounds of the neutrino Yukawa couplings $λ_ν$ and $Y_ν$. We study such effects by encoding the constraints in a likelihood function and carrying out elaborated scans over the parameter space of the theory with the Nested Samplin…
A measurement of the ultrahigh-energy (UHE) cosmic neutrinos at a km^3-size neutrino telescope will open a new window to constrain the 3 x 3 neutrino mixing matrix V and probe possible new physics. We point out that it is in principle possible to examine the non-unitarity of V, which is naturally expected in a class of seesaw models with one or more TeV-scale Majorana neutrinos, by using neutrino telescopes. Considering the UHE neutrinos produced from the decays of charged pions arising from proton-proton and (or) proton-gamma collisions at a distant astrophysical source, we show that their fl…
We provide an $A_4$ based flavor symmetric scenario to accommodate the inverse seesaw mechanism for explaining light neutrino masses and mixing. We find that the lepton mixing, in particular the tri-bimaximal mixing pattern and its deviation through nonzero $θ_{13}$, is originated solely from the flavor structure of the lepton number violating contribution of the neutral lepton mass matrix. Here we discuss in detail how a nonzero value of $θ_{13}$ is correlated with the other parameters in the framework and its impact on the Dirac CP phase $δ$. We also analyze the non-unitarity effects on lept…
After the observation of non-zero $θ_{13}$ the goal has shifted to observe $CP$ violation in the leptonic sector. Neutrino oscillation experiments can, directly, probe the Dirac $CP$ phases. Alternatively, one can measure $CP$ violation in the leptonic sector using Leptonic Unitarity Quadrangle(LUQ). The existence of Standard Model (SM) gauge singlets - sterile neutrinos - will provide additional sources of $CP$ violation. We investigate the connection between neutrino survival probability and rephasing invariants of the $4\times4$ neutrino mixing matrix. In general, LUQ contain eight geometri…
The octant of the atmospheric mixing angle $θ_{23}$ is still undetermined by neutrino oscillation experiments. Long-baseline experiments like NO$ν$A, T2K and DUNE offer good prospects for determining the octant of $θ_{23}$. However, their capability to do so may be compromised by the possible presence of non-unitarity in the leptonic mixing matrix. In this paper, we study in detail the octant degeneracy with and without non-unitarity at the level of oscillation probabilities and events for these experiments. We also analyze their octant sensitivity and discovery reach and show that they are ha…
Neutrino Physics is a mature branch of science with all the three neutrino mixing angles and two mass squared differences determined with high precision. Inspite of several experimental verifications of neutrino oscillations and precise measurements of two mass squared differences and the three mixing angles, the unitarity of the leptonic mixing matrix is not yet established, leaving room for the presence of small non-unitarity effects. Deriving the bounds on these non-unitarity parameters from existing experimental constraints, on cLFV decays such as, $ μ\rightarrow eγ$, $ μ\rightarrow τγ$, $…
Common beliefs about unitarity are not reliable, and we do not know how to apply DGLAP evolution at small x. Together with the big discrepancy between the measurements of the total cross section at the Tevatron, a consequence is that the cross section at the LHC could be anywhere between 90 and 160 mb.
We perform realistic simulations of the current and future long baseline experiments such as T2K, NO$ν$A, DUNE and T2HK in order to determine their ultimate potential in probing neutrino oscillation parameters. We quantify the potential of these experiments to underpin the octant of the atmospheric angle $θ_{23}$ as well as the value and sign of the CP phase $δ_{CP}$.
We analyze the structure of the non-unitary leptonic mixing matrix in the inverse seesaw model with heavy singlets accessible at the LHC. In this model, unlike in the usual TeV seesaw scenarios, thelow-scale right-handed neutrinos do not suffer from naturalness issues. Underlying correlations among various parameters governing the non-unitarity effects are established, which leads to a considerable improvement of the generic non-unitarity bounds. In view of this, we study the discovery potential of the non-unitarity effects at future experiments, focusing on the sensitivity limits at a neutrin…
The discovery of leptonic CP violation is one of the primary goals of next-generation neutrino oscillation experiments, which is feasible due to the recent measurement of a relatively large leptonic mixing angle θ_{13}. We suggest two new working observables ΔA^{\rm m}_{αβ} \equiv \max[A^{\rm CP}_{αβ}(δ)] - \min[A^{\rm CP}_{αβ}(δ)] and ΔA^{\rm CP}_{αβ}(δ) \equiv A^{\rm CP}_{αβ}(δ) - A^{\rm CP}_{αβ}(0) to describe the CP-violating effects in long-baseline and atmospheric neutrino oscillation experiments. The former signifies the experimental sensitivity to the leptonic Dirac CP-violating phase …
In the presence of both direct and indirect unitarity violation in the lepton mixing matrix, we derive a complete set of series expansion formulas for neutrino oscillation probabilities in matter of constant density. Expansions in the mass hierarchy parameter $α\equiv Δm_{21}^{2} / Δm_{31}^{2}$ and those unitarity violation parameters $s^{2}_{ij}$ (for i = 1, 2, 3 and j = 4, 5, 6) up to the first order are studied in this paper. We analyse the accuracy of the analytical series expansion formulas in different regions of L / E. A detailed numerical analysis is also performed, of which the differ…
We investigate the relation between non-unitarity of the leptonic mixing matrix and leptogenesis. We discuss how all parameters of the canonical type-I seesaw mechanism can, in principle, be reconstructed from the neutrino mass matrix and the deviation of the effective low-energy leptonic mixing matrix from unitary. When the mass M' of the lightest right-handed neutrino is much lighter than the masses of the others, we show that its decay asymmetries within flavour-dependent leptogenesis can be expressed in terms of two contributions, one depending on the unique dimension five (d=5) operator g…
The possible existence of light and heavy sterile neutrinos may give rise to direct and indirect unitarity violation of the $3\times3$ lepton mixing matrix respectively. In the current work we study the potential of future accelerator neutrino facilities in probing the unitarity violation effects. Taking DUNE, T2HK and a low-energy Neutrino Factory (LENF) as working examples of future accelerator neutrino facilities, we study the distinct effects of direct and indirect unitarity violation on the discovery reach of the leptonic CP violation and precision measurements of $θ_{23}$ in the three ne…
Unitarity test of lepton mixing via energy dependence of neutrino oscillation
Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；Department of Physics, Facility of Engineering Science, Yokohama National University（横滨国立大学工学部物理系）
A realistic medium- or long-baseline neutrino experiment may suffer from terrestrial matter effects which are likely to contaminate the genuine T-violating asymmetry between ν_α\to ν_βand ν_β\to ν_αoscillations. With the help of the commutators of lepton mass matrices in matter, we show that this kind of contamination is negligible for a variety of experiments provided the neutrino beam energy E and the baseline length L satisfy the condition 10^{-7} (L/km)^2 (GeV/E) << 1. The same observation is true for the CP-violating term of the asymmetry between ν_α\to ν_βand \barν_α\to \barν_βoscillatio…
The historical discovery of neutrino oscillations using solar and atmospheric neutrinos, and subsequent accelerator and reactor studies, has brought neutrino physics to the precision era. We note that CP effects in oscillation phenomena could be difficult to extract in the presence of unitarity violation. As a result upcoming dedicated leptonic CP violation studies should take into account the non-unitarity of the lepton mixing matrix. Restricting non-unitarity will shed light on the seesaw scale, and thereby guide us towards the new physics responsible for neutrino mass generation.
The exploration of the Standard Model (SM) leptonic mixing has been led by the study of the neutrino (ν) oscillations phenomenon, whose discovery was acknowledged by the 2015 Nobel prize in physics. Half a century of experimental and theoretical effort has established and demonstrated consistency with the 3ν model and with the so far SM three family evidence. While no direct significant manifestation for physics beyond the Standard Model (BSM) has been found, the SM is known not to suffice to explain all today's observed phenomenology. In the forthcoming decade, most oscillation parameters are…
We present, both exactly and approximately, a complete set of mappings between the vacuum (or fundamental) leptonic mixing parameters and the effective ones in matter with non-standard neutrino interaction (NSI) effects included. Within the three-flavor neutrino framework and a constant matter density profile, a full set of sum rules is established, which enables us to reconstruct the moduli of the effective leptonic mixing matrix elements, in terms of the vacuum mixing parameters in order to reproduce the neutrino oscillation probabilities for future long-baseline experiments. Very compact, b…
In view of the latest T2K and MINOS observations regarding the mixing angle s_{13}, we have explored the possibility of existence of CP violation in the leptonic sector. Using hints from the construction of the 'db' unitarity triangle in the quark sector, we have made an attempt to construct the 'ν_1.ν_3' leptonic unitarity triangle, suggesting a good possibility of having non zero CP violation.
Oscillations of the Dirac neutrinos of three generations in vacuum are considered with allowance made for the effect of the CP-violating leptonic phase (analogue of the quark CP phase) in the lepton mixing matrix. The general formulas for the probabilities of neutrino transition from one sort to another in oscillations are obtained as functions of three mixing angles and the CP phase. It is found that the leptonic CP phase can, in principle, be reconstructed by measuring the oscillation-averaged probabilities of neutrino transition from one sort to another. The manifestation of the CP phase as…
We uncover an interesting phenomenon that neutrino flavor transformation in slowly varying matter density imitates almost exactly that of vacuum neutrino oscillation under suitably chosen experimental parameters. It allows us to have relatively large CP violating measure ΔP \equiv P(ν_μ \to ν_e) - P(\barν_μ \to \barν_e) which is essentially free from matter effect contamination. We utilize this phenomenon to design a low-energy long-baseline neutrino oscillation experiment to measure the leptonic CP violating phase.
We review the physics of purely leptonic decays of $π^\pm$, $K^\pm$, $D^{\pm}$, $D_s^\pm$, and $B^\pm$ pseudoscalar mesons. The measured decay rates are related to the product of the relevant weak-interaction-based CKM matrix element of the constituent quarks and a strong interaction parameter related to the overlap of the quark and antiquark wave-functions in the meson, called the decay constant $f_P$. The leptonic decay constants for $π^\pm$, $K^\pm$, $D^{\pm}$, $D_s^\pm$, and $B^\pm$ mesons can be obtained with controlled theoretical uncertainties and high precision from {\it ab initio} lat…
The recent measurements of the B_s meson mixing amplitude by CDF and of the leptonic branching fraction BR(B to tau nu) by Belle call for an upgraded analysis of the Unitarity Triangle in the Standard Model. Besides improving the previous constraints on the parameters of the CKM matrix, these new measurements, combined with the recent determinations of the angles alpha, beta and gamma from non-leptonic decays, allow, in the Standard Model, a quite accurate extraction of the values of the hadronic matrix elements relevant for K-Kbar and B_{s,d}-B_{s,d}bar mixing and of the leptonic decay consta…
The largest tau neutrino dataset to date is IceCube's atmospheric tau neutrino appearance dataset containing $>1,000$ tau neutrino and antineutrino events as determined by a fit to a standard three-flavor oscillation framework. On an event-by-event basis, however, it is impossible to know that any given event is a tau neutrino as they are identical to either an electron neutrino charged-current event or a neutral-current interaction of any active flavor. Nonetheless, we conclusively show that, using only the cascade sample even without knowledge of the oscillation parameters and without assumi…
We present results of a combined analysis in neutrino oscillations without unitarity assumption in the $3ν$ mixing picture. Constraints on neutrino mixing matrix elements are based on recent data from the reactor, solar and long-baseline accelerator neutrino oscillation experiments. The current data are consistent with the standard $3ν$ scheme. The precision on different matrix elements can be as good as a few percent at $3σ$ CL, and is mainly limited by the experimental statistical uncertainty. The $ν_e$ related elements are the most precisely measured among all sectors with the uncertainties…
If massive neutrinos are Majorana particles, then the lepton number should be violated in nature and neutrino-antineutrino oscillations $ν^{}_α\leftrightarrow \overlineν^{}_β$ (for $α, β= e, μ, τ$) will definitely take place. In the present paper, we study the properties of CP violation in neutrino-antineutrino oscillations with the non-unitary leptonic flavor mixing matrix, which is actually a natural prediction in the canonical seesaw model due to the mixing between light and heavy Majorana neutrinos. The oscillation probabilities $P(ν^{}_α\to \overlineν^{}_β)$ and $P(\overlineν^{}_α\to ν^{}…
According to different effects on neutrino oscillations, the unitarity violation in the MNSP matrix can be classified into the direct unitarity violation and the indirect unitarity violation which are induced by the existence of the light and the heavy sterile neutrinos respectively. Of which sub-eV sterile neutrinos are of most interesting. We study in this paper the possibility of searching for sub-eV sterile neutrinos in the precision reactor antineutrino oscillation experiments with three different baselines at around 500 m, 2 km and 60 km. We find that the antineutrino survival probabilit…
Lepton number charges might be the source of long range forces. If one accepts that neutrinos produced in the Sun do indeed oscillate while crossing the interior of the Sun, then the shift in the phase of the neutrino wavefunction caused by an hypothetical potential associated to the leptonic charge of the electrons in the Sun could affect the oscillation pattern beyond what is actually observed. We show that a "fine structure" constant $α_{L}$ in excess of $6.4 \times 10^{-54}$ is incompatible with present observational data. This bound is not valid for forces whose range is shorter than the …
The tau neutrino is the least well measured particle in the Standard Model. Most notably, the tau neutrino row of the lepton mixing matrix is quite poorly constrained when unitarity is not assumed. In this paper, we identify data sets involving tau neutrinos that improve our understanding of the tau neutrino part of the mixing matrix, in particular $ν_τ$ appearance in atmospheric neutrinos. We present new results on the elements of the tau row leveraging existing constraints on the electron and muon rows for the cases of unitarity violation, with and without kinematically accessible steriles. …
A general approach for construction of quark and lepton mass matrices is formulated. The hierarchy of quarks and charged leptons ("electrons") is large, it leads using the experimental values of mixing angles to the hierarchical mass matrix slightly deviating from one's suggested earlier by Stech and including naturally the CP-phase. The same method based on the rotation of generation numbers in the diagonal mass matrix is used in the electron-neutrino sector of theory, where neutrino mass matrix is determined by the Majorano see-saw approach. The hierarchy of neutrino masses, much smaller tha…
