II-6. Particle Physics

RIKEN Accel. Prog. Rep. 54 (2021)

RIKEN Accel. Prog. Rep. 54 (2021)

Quarternion-spin-isospin model for the standard-model parameters
Y. Akiba ∗1
Table 1. Values of elementary action densities.

The standard model (SM) successully describes thenature, but it has at least 25 free parameters. In
the preceding article1) we reported empirical formulas with no free parameter for 24 SM parameters: 15
particle masses, four Cabbibo-Kobayashi-Masukawa(CKM) quark mixing parameters, three neutrino mixing angles, the ﬁne structure constant, and the strong
coupling constant.
The pattern of the formulas suggests that there isan algebraic model underpinning them and that thenumbers 6π and ϵ0 = 2 × (6π)−48 play key roles inthis model. The fact that ϵ0 agrees with the Hubble
constant H0 times the Planck time tpl , ϵ0 ≃ H0 tpl ,suggests that there is a relation between the mass of
the SM particles and the metric of spacetime.
In this article, we introduce the quarternion-spinisospin (QST) model, which explains these formulas
and the relation H0 tpl = ϵ0 . The QST model is so
named because it is based on operators that are products of unit quarternions and operators of spin andweak isospin.
The QST model is based on the following 64 operators that are introduced here as normalized primordialaction (NPA):(6π)2 ϵ0 iτ 3 (6π)2 ϵ0 i (6π)ϵ0 iτ 3 (6πϵ0 i) (6π)ϵ0 ϵ0 iτ 3 ϵ0 i ϵ03∗aL̂g3D2aL̂g2D
24∗aL̂GG
8aL̂uG
24∗bL̂cG
8bL̂tG
18bL̂mB
e4a13∗bµ4a−12a27a3∗cτ
4a−33∗a 1 + 4ν112a122bν2−4a42bν312a12212a12−2ν3′U15−12a2aU17 3 × (−4)−12b2bD−12a1u4b−8ac4b + 4c−24∗t4c−8bd−12a−12a−3∗s−12b3∗db4d63∗b27b 18aZ12∗b1
1
H−4−618cW
12∗b−18∗−27}{ I µ σ ν τ a ετ a εi εI c τ c εiτ 3 −εI c −εi,,,,,,, −ε, ε, i, 1 .
6π
6π 6π 6π6π
4π 2πThe QST model has ﬁve primary ansatzes.
(A1) The Planck time tpl = 5.3912 × 10−44 s is theminimum duration of time in nature. The time t is
an integer n in the unit of tpl , and the physics state
at t = n is represented as |n⟩. tpl is a fundamental
constant of nature, as are the speed of light in vacuumc and the Planck constant h̄.
(A2) The change of |n⟩ for tpl is represented by a setof 49 operators L̂EAD
we name as elementary action
pdensity (EAD). An EAD corresponds to a term of theSM Lagrangian.
(A3) An EAD is a sum of the equivalent vee productP48 (S48 , σ) of 48 NPAs,∑∑
NP AL̂EAD=
P48 (S48, σ) =∂ˆσ(p1 ) ∨ · · · ∨ ∂ˆσ(p48 ) ,p

(τ 1 , τ 2 , τ 3 ) → (τ 1 , τ 2 , τ 3 ), (τ 2 , τ 3 , τ 1 ), (τ 3 , τ 1 , τ 2 ).
(1)
(A5) The SM emerges as a continuous approximationof the QST model as each EAD becomes a term of theSM Lagrangian in the limit of tpl → 0.
We found 49 EADs that correspond to the elementary particles of the SM, which are summarized in Table 1. All of the 24 formulas of the parameters of thestandard model can be derived from the 49 EADs in
the table. The model also predicts that the CP violation in the neutrino sector is 100%. Thus, all of
the 25 free parameters of the standard model are derived from the QST model. In additon, the model predicts that the Hubble constant times the Planck timeis H0 tpl = 2 × (6π)−48 . We discuss the implications forgeneral relativity and cosmology in the next article.2)

NP A= {∂ˆp1 , · · · , ∂ˆp48 } is a set of 48 NPAs andwhere S48
σ is a permutation (k → σ(k), 1 ≤ k ≤ 48).
(A4) All 49 EADs are invariant for the following permutations:

References
1) Y. Akiba, in this report.
2) Y. Akiba, in this report.

(I 1 , I 2 , I 3 ) → (I 1 , I 2 , I 3 ), (I 2 , I 3 , I 1 ), (I 3 , I 1 , I 2 ),(σ 1 , σ 2 , σ 3 ) → (σ 1 , σ 2 , σ 3 ), (σ 2 , σ 3 , σ 1 ), (σ 3 , σ 1 , σ 2 ).
There are 12 EADs that are invariant for permutations:∗1RIKEN Nishina Center- 70 -1