Eur. Phys. J. C (2025) 85: 1082
https://doi.org/10.1140/epjc/s10052-025-14771-0

Regular Article - Theoretical Physics

Universal pattern in fermion mass ratios and implications for flavor models

VS Asset Management, LLC, 06807, Cos Cob, CT, USA

^a samir@vsasset.com

Received: 10 July 2025
Accepted: 9 September 2025
Published online: 30 September 2025

Abstract

We show that a single analytic expression using only the labels that distinguish Standard Model mass eigenstates—the ordering index $d\in \{1,2,3\}$ and electric charge $q$—fits all fermion mass ratios with high precision and stable $\mathcal{O}(1)$ parameters. The functional form $ln(m_1/m_2)=d_1^{{\zeta }_1}{d}_2^{{\zeta }_2}{\kappa }^{[(d_1|q_1{|)}^{\gamma (\mu )}-(d_2|q_2{|)}^{\gamma (\mu )}]}$ reproduces all 30 independent quark ratios from $M_Z$ to ${10}^{12}\text{GeV}$ with ${\chi }^2/\text{dof}\lesssim {10}^{-6}$ using per-scale fits (24 parameters), fixed-$\kappa$ fits (18), or global fits (5–6). Parameters remain near $\kappa \simeq 2.3$, ${\zeta }_1\simeq 1.16$, ${\zeta }_2\simeq -0.8$, $\gamma \simeq 1.1$. Three orthogonal checks establish specificity and robustness: (i) a 120-permutation test where only the physical mapping of ratios to $(d,|q|)$ fits; (ii) a 720-basis scan in which the adjacent-ratio basis is uniquely preferred by AIC/BIC and many alternatives fail to converge; (iii) alternative functional forms are decisively disfavored by AIC/BIC. The same form, with natural parameters, describes charged-lepton and neutrino ratios; for $q=0$ the charge term vanishes, leaving pure $d^{\zeta }$ scaling. We sketch three minimal mechanisms—modified Froggatt–Nielsen, partial compositeness, and a Randall–Sundrum overlap—that generate the additive $d$–$|q|$ dependence at the exponent level and reduce to the neutrino limit. The pattern predicts percent-level deviations from standard running near ${10}^{14}\text{GeV}$ and, with $\text{meV}$-scale anchors, $\Sigma m_{\nu }\simeq 0.06\text{eV}$. To our knowledge, no previous empirical relation achieves simultaneous cross-sector consistency with such parameter stability; the result offers a compact target for flavor model building.