What question did this study set out to answer?

This research aims to establish a parameter-free geometric Hamiltonian model that accurately represents charged-lepton mass ratios and analyze the Koide relation's scheme dependence.

January 24, 2026Open Access

A parameter-free geometric Hamiltonian reproducing charged-lepton mass ratios at the subpercent level and clarifying the scheme dependence of the Koide relation

Key Points

This research aims to establish a parameter-free geometric Hamiltonian model that accurately represents charged-lepton mass ratios and analyze the Koide relation's scheme dependence.
Introduced a 3×3 Hermitian geometric Hamiltonian with fixed constants for calculations.
Calculated lepton mass ratios using the geometric model and compared to PDG values.
Conducted a one-loop MS̄ renormalization-group analysis to explore the Koide relation's dependence on mass schemes.
Achieved charged-lepton mass ratios at subpercent accuracy compared to PDG values.
Reported mμ/me ratio at 208.06 with a 0.62% error and mτ/me at 3506.78 with 0.85% error.
Koide relation verified with high precision across different mass schemes, showing sharp scheme dependence.

Abstract

A parameter-free geometric Hamiltonian reproducing charged-lepton mass ratios at the subpercent level Abstract We report a simple, parameter-free 3×3 Hermitian "geometric Hamiltonian" whose eigenvalues reproduce the charged-lepton mass ratios (mμ/me, mτ/me) at the subpercent level using only two fixed constants: ti/tr = 180/π (radian-to-degree conversion) v/|ti| = √15 (Standard Model fermion count per generation) Key Results Observable Theory (Geometric) PDG 2024 (pole) Error mμ/me 208.06 206.77 0.62% mτ/me 3506.78 3477.37 0.85% Koide ratio 0.66694 0.66667 0.04% All predictions within 1% of PDG 2024 with zero adjustable parameters. Physical Interpretation 180/π ≈ 57.3: Phase space ↔ angle space conversion; cost of 90° rotation in topology √15 ≈ 3.87: √(fermions per generation) where 15 = 12 quarks + 3 leptons Scheme Dependence of the Koide Relation We further show with a Standard Model one-loop MS̄ renormalization-group analysis that the Koide relation K ≡ (me + mμ + mτ)/(√me + √mμ + √mτ)² ≃ 2/3 is sharply scheme dependent: Pole masses: |K − 2/3| = 2.20 × 10⁻⁶ ✓ MS̄ running masses (μ = MZ): |K − 2/3| = 1.27 × 10⁻³ Companion Work This geometric Hamiltonian template is extended to quark flavor mixing (CKM matrix) in a companion paper, demonstrating that the same geometric constants organize both mass eigenvalues and mixing eigenvectors. Keywords charged-lepton masses, Koide formula, geometric Hamiltonian, Standard Model, mass hierarchy, fermion generations, renormalization group Code Availability The complete numerical implementation (SM one-loop RGE, pole-to-MS̄ conversion, and figure generation) is available as a Google Colab notebook. Citation If you use this work, please cite: Iizumi, M. (2025). A parameter-free geometric Hamiltonian reproducing charged-lepton mass ratios at the subpercent level and clarifying the scheme dependence of the Koide relation. Zenodo. https://doi.org/10.5281/zenodo.18337936 License This work is licensed under CC BY 4.0.

A parameter-free geometric Hamiltonian reproducing charged-lepton mass ratios at the subpercent level and clarifying the scheme dependence of the Koide relation

Key Points

Abstract

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