A Technical But Interesting Paper On Fermion Mass Ratios

A new paper looks at fundamental fermion mass ratios from the perspective of something similar to an extended Koide's rule approach.

We revisit the "three generations" problem and the pattern of charged-fermion masses from the vantage of octonionic and Clifford algebra structures. Working with the exceptional Jordan algebra J3(OC) (right-handed flavor) and the symmetric cube of SU(3) (left-handed charge frame), we show that a single minimal ladder in the symmetric cube, together with the Dynkin Z2 swap (the A2 diagram flip), leads to closed-form expressions for the square-root mass ratios of all three charged families. The universal Jordan spectrum (q - delta, q, q + delta) with a theoretically derived delta squared = 3/8 fixes the endpoint contrasts; fixed Clebsch factors (2, 1, 1) ensure rung cancellation ("edge universality") so that adjacent ratios depend only on which edge is taken. The down ladder determines one step, its Dynkin reflection gives the lepton ladder, and choosing the other outward leg from the middle yields the up sector.

From the same inputs we obtain compact CKM "root-sum rules": with one 1-2 phase and a mild 2-3 cross-family normalization, the framework reproduces the Cabibbo angle and Vcb and provides leading predictions for Vub and Vtd/Vts. We perform apples-to-apples phenomenology (common scheme/scale) and find consistency with current determinations within quoted uncertainties. Conceptually, rank-1 idempotents (points of the octonionic projective plane), fixed symmetric-cube Clebsches, and the Dynkin swap together account for why electric charge is generation-blind while masses follow the observed hierarchies, and they furnish clear, falsifiable mass-ratio relations beyond the Standard Model.

Tejinder P. Singh, "Fermion mass ratios from the exceptional Jordan algebra" arXiv:2508.10131 (August 13, 2025) (90 pages).

Another interesting paper develops a relationship between the mixing ratios of the unitary triangle in the CKM matrix and the CP violating phase of that matrix. The abstract below deviates from my usual editing conventions to preserve the details of superscripts and subscripts in the notion without a lot of extra editing work that is prone to human error:

In this letter, we obtain a rephasing invariant formula for the CP phase in the Kobayashi--Maskawa parameterization δKM=arg[−VuddetVCKM/VusVubVcdVtd]. General perturbative expansion of the formula and observed value δKM≃π/2 reveal that the phase difference of the 1-2 mixings ei(ρd12−ρu12) is close to maximal for sufficiently small 1-3 quark mixings su,d13. Moreover, by combining this result with another formula for the CP phase δPDG in the PDG parameterization, we derived an exact sum rule δPDG+δKM=π−α+γ which relating the phases and the angles α,β,γ of the unitarity triangle.

Masaki J. S. Yang, "Rephasing Invariant Formula for the CP Phase in the Kobayashi-Maskawa Parametrization and the Exact Sum Rule with the Unitarity Triangle δPDG + δKM = π −α +γ" arXiv:2508.10249 (August 14, 2025) (6 pages).