Some simple mixing and mass matrices for neutrinos

Abstract

We argue that the accumulated neutrino data, including recent results from KamLAND and K2K, point to a neutrino mixing matrix with (V 11 , V 21 , V 31 ;. We propose some simple neutrino mass matrices which predict such a mixing matrix.In this brief note, we suggest that the accumulating neutrino data [1][2][3], including the recent results from KamLAND [4] and K2K [5], point to a relatively simple neutrino mass matrix. The data can be explained by oscillations between three active neutrinos with the atmospheric neutrino and K2K data explained by oscillation between the muon and the tauon neutrinos, and the solar neutrino and KamLAND data explained by the oscillation between the electron and muon neutrinos. Following standard convention, let us denote the neutrinos current eigenstates, coupled to the charged leptons by the W bosons, by ν α (α = e, µ, τ ) and the neutrino mass eigenstates by ν i (i = 1, 2, 3). We will take the neutrinos to be Majorana as seems likely [6], and thus the neutrino mass matrix M αβ is symmetric in the basis of the current eigenstates. We will also for simplicity assume CP conservation so that M is real. Thus, M is diagonalized by an orthogonal transformation

Viability of exact tri-bimaximal, golden-ratio and bimaximal mixing patterns and renormalization-group running effects

“…We have θ 23 = 45 • and θ 13 = 0 for all three mixing patterns in question, while θ 12 = arctan(1/ √ 2) ≈ 35.3 • for TBM [2][3][4][5], θ 12 = 45 • for BM [6][7][8] and θ 12 = arctan[2/(1 + √ 5)] ≈ 31.7 • for GR [9][10][11]. 1 For quite a long time, these constant mixing patterns had been regarded as good candidates for the leptonic flavor mixing matrix, as they all agreed with the experimental data fairly well.…”

“…where the evolution functions I 0 and I α (for α = e, µ, τ ) are defined as 5) and M Λ ν is the neutrino mass matrix at the high-energy scale Λ. Note that a superscript "Λ" will be attached to the quantity at the high-energy scale Λ, so as to distinguish it from its counterpart at the low-energy scale λ.…”

“…Two cases of neutrino mass hierarchy are also distinguished, in the assumption that the hierarchy patterns at the low-and high-energy scales are matched, i.e., they are both in NH, or both in IH. 5 The neutrino parameters at the low-energy scale λ are then confronted to the latest global-fit results from ref. [53].…”

“…In table 3, we also list the pull for each observable y i , with its definition given by 5) where the theoretical prediction is referred to that at the best-fit point. We then see that only a fair χ 2 fit is reached for TBM in the NH case, and θ 12 and θ 23 indeed have large pulls, confirming our previous analytical findings.…”

Neutrino mass matrix with approximate flavor symmetry

“…Note added: It has been called to my attention by A.Zee that the matrix (9) in this paper is similar to M 0 with y = −1, given by X.G.He and A.Zee [16]. Our motivation is rather different.…”