What question did this study set out to answer?

The study aims to provide a structural explanation for the hierarchy of quark masses using fixed ratios and a unified relation.

June 15, 2026Open Access

Structural Origin of Quark Masses Hierarchy

Key Points

The study aims to provide a structural explanation for the hierarchy of quark masses using fixed ratios and a unified relation.
Developed a zero-parameter structural formulation of quark masses using ratios R and S.
Calculated and presented the six quark masses based on the derived formula without fitting parameters.
Provided a Python implementation for reproducibility.
Computed quark masses: m_u ≈ 2.145 MeV, m_d ≈ 4.648 MeV, m_c ≈ 1273.6 MeV, m_s ≈ 93.56 MeV, m_t ≈ 173.0 GeV, m_b ≈ 4.185 GeV.
The mass values align well with known reference values across all three generations of quarks.

Abstract

This work presents a zero-parameter structural formulation of the quark-mass hierarchy. Using two fixed rational ratios, =136, S=3124, six quark masses are represented by the unified relationqmₑ=Kq (3R²S) (8^uqR^{uq+1}S) ⁿ. resulting structural mass values areᵤ 2. 145\, MeV, d 4. 648\, MeV, c 1273. 6\, MeV, \ₛ 93. 56\, MeV, ₜ 173. 0\, GeV, b 4. 185\, GeV. values are compared directly with reference quark-mass values across all three generations. The work is intended as a compact numerical correspondence, not as a replacement for the Standard Model or its Yukawa mechanism. No individual quark-mass parameters are fitted, and no tuning is performed. A minimal Python implementation is provided for full reproducibility: https: //github. com/yasuotanakaresearch/zero-parameter-structure

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