A universal shape of empirical mass formula for all leptons and quarks

Abstract

A specific universal shape of empirical mass formula is proposed for all leptons $ν_1, ν_2, ν_3$ and $e^-, μ^-, τ^- $ as well as all quarks $u, c, t$ and $d, s, b$ of three generations, parametrized by three free constants $ μ, ε, ξ$ assuming four different triplets of values. Four such triplets of parameter values are determined or estimated from the present data. Mass spectra in the four cases are related to each other by shifting the triplet of parameters $ μ, ε, ξ$. For charged leptons $ξ\simeq 0$ (but probably $ξ\neq 0$). If for them $ξ$ is put to be exactly 0, then $m_τ= 1776.80$ MeV is predicted after the input of experimental $m_e$ and $m_μ$ (the central value of experimental $m_τ= 1776.99^{+0.29}_{-0.26}$ MeV corresponds to $ ξ=1.8\times 10^{-3}\neq 0$). For neutrinos $1/ξ\simeq 0$ (but $1/ξ\neq 0$ in the case of normal hierarchy $m^2_{ν_1} \ll m^2_{ν_2} \ll m^2_{ν_3}$). If for neutrinos $1/ξ$ is conjectured to be exactly 0, then $(m_{ν_1}, m_{ν_2}, m_{ν_3}) \sim (1.5, 1.2, 5.1)\times 10^{-2}$ eV are predicted after the input of experimental estimates $|m^2_{ν_2} - m^2_{ν_1}| \sim 8.0\times 10^{-5} {\rm eV}^2$ and $|m^2_{ν_3} - m^2_{ν_2}| \sim 2.4\times 10^{-3} {\rm eV}^2$. Thus, the mass ordering of neutrino states 1 and 2 is then inverted, while the position of state 3 is normal.