Explorations of two empirical formulae for fermion masses

TL;DR

This paper investigates two empirical mass formulae for leptons and quarks, analyzing their validity and implications using current experimental data, and finds that the formulae hold under certain conditions with small deviations.

Contribution

The study evaluates the applicability of extended Koide formulae to lepton and quark masses, highlighting conditions for their validity and insensitivity to energy scale variations.

Findings

$K_l=2/3$ depends on tau mass uncertainty

$K_q$ is stable across energy scales from GeV to $10^{12}$ GeV

Discrepancy in $K_q$ is about 5% above 2/3

Abstract

Two empirical formulae for the lepton and quark masses (i.e. Kartavtsev's extended Koide formulae), $K_l=(\sum_l m_l)/(\sum_l\sqrt{m_l})^2=2/3$ and $K_q=(\sum_q m_q)/(\sum_q\sqrt{m_q})^2=2/3$, are explored in this paper. For the lepton sector, we show that $K_l=2/3$, only if the uncertainty of the tauon mass is relaxed to about $2\sigma$ confidence level, and the neutrino masses can consequently be extracted with the current experimental data. For the quark sector, the extended Koide formula should only be applied to the running quark masses, and $K_q$ is found to be rather insensitive to the renormalization effects in a large range of energy scales from GeV to $10^{12}$ GeV. However, $K_q$ is always slightly larger than $2/3$, but the discrepancy is merely about $5\%$.