A study of 3+N neutrino scenarios through singular values Kamil Porwit, Wojciech Flieger, Janusz Gluza Division of Field Theory and Elementary Particle Physics, University of Silesia, Katowice, Poland Aim of studies ▶Investigation of the experimentally determined 3 × 3 neutrino mixing matrix from point of view of matrix theory, ▶confrontation of 3 + N scenarios with experimental data, ▶determination of a number of sterile neutrinos by analyzing singular values of three dimensional mixing matrices, ▶construction of the complete unitary mixing matrix via the procedure of matrix dilation. Common parametrizations of non-unitary mixing matrices [1] ▶QR decomposition: V = (I −α)U = TU where α is lower triangular matrix and U is unitary. ▶Polar decomposition: V ′ = (I −η)U′ where η is a Hermitian matrix and U′ is unitary. Present status [1] ▶Current upper bounds on the α matrix and corresponding T matrix for three diﬀerent neutrino mass scenarios: m > 246 GeV αee 1.3 · 10−3 αµµ 2.2 · 10−4 αττ 2.8 · 10−3 |αµe| 6.8 · 10−4 |ατe| 2.7 · 10−3 |ατµ| 1.2 · 10−3 α ≤    0.00130 0 0 |0.00068| 0.00022 0 |0.00270| |0.00120| 0.0028    ∆m2 ≳100 eV 2 αee 2.4 · 10−2 αµµ 2.2 · 10−2 αττ 1.0 · 10−1 |αµe| 2.5 · 10−2 |ατe| 6.9 · 10−2 |ατµ| 1.2 · 10−2 α ≤    0.024 0 0 |0.025| 0.022 0 |0.069| |0.012| 0.100    ∆m2 ∼0.1 −1 eV 2 αee 1.0 · 10−2 αµµ 1.4 · 10−2 αττ 1.0 · 10−1 |αµe| 1.7 · 10−2 |ατe| 4.5 · 10−2 |ατµ| 5.3 · 10−2 α ≤    0.010 0 0 |0.017| 0.014 0 |0.045| |0.053| 0.100    Matrix theory: singular values [2] Singular values σi of a given matrix A are positive square roots of the eigenvalues λi of the matrix AA′ σi(A) = q λi(AA†), i = 1, 2, 3, ... Properties: ▶operator norm (spectral norm): ∥A∥:= sup ∥x∥=1 ∥Ax∥= σmax(A) ▶contractions: ∥A∥≤1 General beyond the Standard Model (BSM) scenario [3] Extensions from three neutrinos setting: V →W =  V Vlh Vhl Vhh  ▶matrix W is unitary, ▶operator norm is bounded by unity: ∥V ∥≤1 Any deviation of singular value of V from unity →BSM scenario. Matrix dilation: Minimal number of additional sterile neutrinos is determined by the number of singular values strictly less than one. Strategy: 3 + N scenario studies ▶We study scenarios with additional neutrinos by constructing T matrices with prescribed singular values. ▶Number of singular values strictly less than one completely ﬁx the minimal number n of additional neutrinos: σ(T) = {σ1, σ2, σ3}, n = number of σi < 1 ▶Conditions given by constrains on α matrix are met through the random generation of T entries within region given by ﬁxed singular values. ▶To ﬁnd non-diagonal cases we involve the idea of an inverse eigenvalue problem, i.e., we construct matrix with prescribed singular values [4]. Results: 3 + 1 ▶m > 246 GeV σ(T) = {1.000, 1.000, 0.992} T =   0.9987 0 0 0.0007 0.9988 0 0.0027 0.0012 0.9972   ▶∆m2 ≳100 eV 2 σ(T) = {1.000, 1.000, 0.899} T =   1.000 0 0 0.001 1.000 0 0.012 0.012 0.900   ▶∆m2 ∼0.1 −1 eV 2 σ(T) = {1.000, 1.000, 0.888} T =   0.994 0 0 0.013 0.993 0 0.045 0.053 0.900   Results: 3 + 2 ▶m > 246 GeV σ(T) = {1.000, 0.999, 0.999} T =    0.998702 0 0 0.000476 + 0.000471i 0.999880 0 0.000252 + 0.000139i 0.001013 −0.000198i 0.998797   |T| =    0.99870 0 0 0.00067 0.9998 0 0.00028 0.0010 0.9988    ▶∆m2 ≳100 eV 2 σ(T) = {1.000, 0.970, 0.969} T =    0.976753 0 0 0.024451 + 0.000184i 0.991755 0 0.005957 −0.000192i 0.010959 −0.000386i 0.970301   |T| =    0.977 0 0 0.025 0.992 0 0.006 0.011 0.970    ▶∆m2 ∼0.1 −1 eV 2 σ(T) = {1.000, 0.980, 0.979} T =    0.990114 0 0 0.016262 + 0.000113i 0.986144 0 0.011641 −0.000160i 0.010193 −0.000068i 0.982614   |T| =    0.990 0 0 0.016 0.986 0 0.012 0.010 0.983    Conclusions ▶Based on Weyl inequalities, errors of T entries are estimated on: 0.003 [3], ▶considering error all mass scenarios allows the addition of 1 and 2 additional neutrinos. However in the case of 3 + 2 and m > 246 GeV result is not decisive, ▶singular values and inverse eigenvalue problem are