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 $\nu_1, \nu_2, \nu_3$ and $e^-, \mu^-, \tau^- $ as well as all quarks $u, c, t$ and $d, s, b$ of three generations, parametrized by three free constants $ \mu, \epsilon, \xi$ 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 $ \mu, \epsilon, \xi$. For charged leptons $\xi \simeq 0$ (but probably $\xi \neq 0$). If for them $\xi $ is put to be exactly 0, then $m_\tau = 1776.80$ MeV is predicted after the input of experimental $m_e$ and $m_\mu$ (the central value of experimental $m_\tau = 1776.99^{+0.29}_{-0.26}$ MeV corresponds to $ \xi =1.8\times 10^{-3}\neq 0$). For neutrinos $1/\xi \simeq 0$ (but $1/\xi \neq 0$ in the case of normal hierarchy $m^2_{\nu_1} \ll m^2_{\nu_2} \ll m^2_{\nu_3}$). If for neutrinos $1/\xi $ is conjectured to be exactly 0, then $(m_{\nu_1}, m_{\nu_2}, m_{\nu_3}) \sim (1.5, 1.2, 5.1)\times 10^{-2}$ eV are predicted after the input of experimental estimates $|m^2_{\nu_2} - m^2_{\nu_1}| \sim 8.0\times 10^{-5} {\rm eV}^2$ and $|m^2_{\nu_3} - m^2_{\nu_2}| \sim 2.4\times 10^{-3} {\rm eV}^2$. Thus, the mass ordering of neutrino states 1 and 2 is then inverted, while the position of state 3 is normal.