II-6. Particle Physics

RIKEN Accel. Prog. Rep. 54 (2021)

RIKEN Accel. Prog. Rep. 00 (0)Empirical formulas for the standard-model parameters
Y. Akiba∗1We report empirical formulas for the parameters of
the standard-model. Table 1 lists the formulas for themass of the charged leptons (e, µ, τ ), three neutrinos(ν1 , ν2 , ν3 ), six quarks (u, c, t, d, s, b), and gauge bosons(W , Z), and Higgs boson (H). The formulas yield themasses in terms of the Planck mass

the mass values from the formulas and neutrino oscillation data. Table 3 lists the formulas for the CabibboKobayashi-Maskawa quark mixing parameters. Table 4 lists the formulas for the neutrino-mixing angles.
Table 5 lists the formulas for the ﬁne structure constant α and the strong coupling constant αs . These
formulas yield 24 of 25 free parameters of the standardmodel. The remaining one, the neutrino CP violation
angle δCP , has not been measured. The values calculated from the formulas are in good agreement withthe data. The one common constant in the mass formulas, ϵ0 = 2 × (6π)−48 , which agrees with the Hubbleconstant H0 times the Planck time tpl (ϵ0 ≃ H0 × tpl )within the accuracy of H0 , suggests that the particle
masses are related to the expansion of the universe.
A model to explain these formulae is reported in thenext article,1) and implications to gravity and cosmology are reported in the article appearing after that.2)

Mpl = 1.220910 ± 0.000029 × 1019 GeV.
The last column of the table presents the relative difc
ference |mcp /mm
p −1| of the calculated value mp and themmeasured value mp for particle p. Table 2 compares
Table 1. Formulas for the masses of the SM particles.
pformula (µp = mp /Mpl )e1 1/3ϵ12π 2 0µτ(ν3sd1+)

1 −1
2 1/21+ϵ3 0 (6π )1 −1
1/22ϵ01−6π)(1 −11/24πϵ01+6πν2b)−1

1 14 (6π)2)−1(3 1/3327 1
1−ϵ0+26π
4 (6π)2()−13 1
5 1
1/3
9πϵ0
1−+4 6π
4 (6π)2

ν1tcu(W2−1/4 1/4ϵ(8π 2 0H

21/2 1/4ϵ8π 2 0(1−3 19 1−
2 6π
4 (6π)2

3 1
9 11+−2 6π
2 (6π)2

Table 3. Formulas of the CKM matrix elements.

1.6 × 10−5()−1/2)−1/2)−1/2

5.5 × 10−3
3.8 × 10−2
3.9 × 10−3
5.6 × 10−3

2.2 × 10−2

1.6 × 10(16πs12−2( ((formula1
11−36πcalculated()(1+)(16π)−1 )1/2)−1 )1/2

1131+1−2π6π6π() ()()
1
1 −1
1 1/2
1−1+s1312π
6π6πs23

1.5 × 10−51.0 × 10−4

measured

calculated

measured

0.547

0.545 ± 0.016

0.729

0.714 ± 0.053

0.146

0.147 ± 0.003

Table 5. Formulas of the coupling constants α and αs (MZ )3.4 × 10−4−1

Table 2. Comparison of the calculated masses of neutrinos
with the neutrino oscillation data.

αs (MZ )Quantity
m22 − m21
m23 − m21

References∗1) )1/2

Table 4. Formulas of the neutrino-mixing matrix.

αCalculated
7.39 × 10−5 eV22.58 × 10−3 eV2formula1 −1
Vus1+0.22445 0.22452 ± 0.0044
6π( )1/221Vcb
0.04332 0.04214 ± 0.00076
36π
4 10.003753 0.00365 ± 0.00012
Vub3 (6π)2()
3 1 27 1
11++0.3497
0.355+0.012
η̄−0.011
2 6π 4 (6π)2 π

See Table 2

1 1
1 11++12 6π
12 (6π)2(5.2 × 10−5See Table 2

1/3(5.9 × 10−6See Table 28(6π)2 ϵ01/3
12ϵ01/38(6π)−2 ϵ0)−1(3 1
27 1
1/33(6π)ϵ01++2 6π
4 (6π)21/3ϵ0()
1 −1−1 1/3(6π) ϵ01+6π1 1/4Z
ϵ(8π 2 ) 0|mcp /mm
p − 1|

Measured−57.37+0.20
eV2−0.15 × 10−32.56 ± 0.04 × 10 eV2formula)
1 1 −1/2
44π 1 +√ ( 3 6π )121+4π6π(calculated

rel. error

137.0238

8.9 × 10−50.11851

3.5 × 10−31) Y. Akiba, in this report.
2) Y. Akiba, in this report.

RIKEN Nishina Center- 69 -1