Tripartite neutrino mass matrix

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Institute of theoretical physics

Almost degenerate neutrinos and bi-large lepton mixing

We observe that the {\it invariance} of neutrino mixing matrix under the simultaneous discrete transformations $\nu_1, \nu_2, \nu_3 \to -\nu_1, -\nu_2, \nu_3 $ and $\nu_e, \nu_\mu, \nu_\tau \to -\nu_e, \nu_\tau, \nu_\mu $ (neutrino "horizontal conjugation") {\it characterizes} (as a sufficient condition for it) the familiar bilarge form of neutrino mixing matrix, favored experimentally at present. Thus, the mass neutrinos $\nu_1, \nu_2, \nu_3 $ get a new quantum number, {\it covariant} with respect to their mixings into the flavor neutrinos $\nu_e, \nu_\mu, \nu_\tau $ (neutrino "horizontal parity" equal to -1, -1,1, respectively). The "horizontal parity" turns out to be embedded in a group structure consisting of some Hermitian and real $3\times 3$ matrices $\mu_1, \mu_2, \mu_3 $ and $\phi_1, \phi_2, \phi_3 $, forming pairs interconnected through neutrino mixings. They generate some discrete transformations of mass and flavor neutrinos, respectively, in such a way that the group relations $\mu_1 \mu_2 = \mu_3 $ (cyclic) and $\phi_1 \phi_2 = \phi_3 $ (cyclic) hold, while $\mu_a \mu_b = \mu_b \mu_a $ and $\phi_a \phi_b = \phi_b \phi_a $. Then, for instance, the $\mu_3$ matrix may be chosen equal to the "horizontal parity".

Is There a Dynamical Group Structure Behind the Bilarge Form of Neutrino Mixing Matrix

The well-known formal analogy between time and absolute temperature, existing on the quantum level, is considered as a profound duality relationship requiring some modifications in the conventional quantum dynamics. They consist of tiny deviations from uniform time run in the physical spacetime, as well as of tiny deviations from unitary time evolution characteristic for the conventional quantum theory. The first deviations are conjectured to be produced by local changes of total average particle number. Then, they imply the second deviations exerting in turn influence upon this particle number. Two examples of the second deviations are described: hypothetic tiny violation of optical theorem for particle scattering, and hypothetic slow variation of the average number of probe particles contained in a sample situated in proximity of a big accelerator (producing abundantly particles on a target).

Hypothetic Time--Temperature Duality as a Hint for Modifications in Quantum Dynamics?

In the first part of the note, we consider a neutrino texture, where the Dirac and righthanded Majorana masses are proportional. If the former are approximately proportional also to the charged lepton masses, then taking $\Delta m^2_{32}\sim3\times 10^{-3}$ eV$^2$ we estimate approximately that $\Delta m^2_{21}\sim{\cal O}(10^{-5}$ eV$^2)$, what is not very different from the recent KamLAND estimation $\Delta m^2_{21}\sim7\times 10^{-5}$ eV$^2$, consistent with the LMA solar solution. In the second part, we show generically that the {\it invariance} of neutrino mixing matrix under the simultaneous discrete transformations $\nu_e\to-\nu_e$, $\nu_\mu\to\nu_\tau$, $\nu_\tau\to\nu_\mu$ and $\nu_1\to-\nu_1$, $\nu_2\to-\nu_2$, $\nu_3\to\nu_3$ (neutrino "horizontal conjugation") {\it characterizes} the familiar bilarge form of mixing matrix, favored phenomenologically at present. Then, in the case of this form, the mass neutrinos $\nu_1$, $\nu_2$, $\nu_3$ get a new quantum number, {\it covariant} in their mixings (neutrino "horizontal parity", equal to $-1,-1, 1$, respectively). Conversely, such a covariance may be the {\it origin} of the bilarge mixing matrix. In Section 5, the "horizontal parity" is embedded in a group structure.

Bilarge Mixing Matrix and Its Invariance Under ``Horizontal Conjugation'' --- a New Discrete Transformation for Neutrinos

Hypothetic time -- temperature duality as a hint for modifications in quantum dynamics?

The author's idea of {\it algebraic compositeness} of fundamental particles, allowing to understand the existence in Nature of three fermion generations, is revisited. It is based on two postulates. i) For all fundamental particles of matter the Dirac square-root procedure $\sqrt{p^2}\to\Gamma^{(N)}\cdot p$ works, leading to a sequence $N=1,2,3,...$ of Dirac-type equations, where four Dirac-type matrices $\Gamma^{(N)}_\mu$ are embedded into a Clifford algebra {\it via} a Jacobi definition introducing four "centre-of-mass" and $(N-1)\times$four "relative" Dirac-type matrices. These define one "centre-of-mass" and $N-1$ "relative" Dirac bispinor indices. ii) The "centre-of-mass" Dirac bispinor index is coupled to the Standard Model gauge fields, while $N-1$ "relative" Dirac bispinor indices are all free undistinguishable physical objects obeying Fermi statistics with the Pauli principle which requires the full antisymmetry with respect to "relative" indices. This allows only for {\it three} $N = 1,3,5$ in the case of $N$ odd, implying the existence of {\it three and only three} generations of fundamental fermions.

Wojciech Krolikowski

Papers

Is There a Dynamical Group Structure Behind the Bilarge Form of Neutrino Mixing Matrix

Hypothetic Time--Temperature Duality as a Hint for Modifications in Quantum Dynamics?

Bilarge Mixing Matrix and Its Invariance Under ``Horizontal Conjugation'' --- a New Discrete Transformation for Neutrinos

Hypothetic time -- temperature duality as a hint for modifications in quantum dynamics?

Clifford Algebra Implying Three Fermion Generations Revisited