Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 51 (2018)

Empirical formulae of the masses of elementary particles
Y. Akiba∗1particle
formulacalculated(c)measured(m)|c/m − 1|
1/3
e1/(12π 2 )ϵ0 (1 + (1/4)(1/(6π)2 ))−1 Mpl
0.511002 MeV 0.510998946 ± 0.0000000031 MeV 5.9 × 10−61/3µ3/2ϵ0 (1 − 1/(2π) + 3/(4π)2 )−1 Mpl
105.6594 MeV 105.6583745 ± 0.0000024 MeV 9.6 × 10−61/32 −19πϵ0 (1 − 1/(8π) + (5/4)(1/(6π) )) Mpl
1.77684 GeV
1.77686 ± 0.12 GeV
1.9 × 10−5τ1/3t8 × (6π)2 ϵ0 Mpl
172.1 GeV
173.4 ± 0.75 GeV
7.5 × 10−31/3c12ϵ0 Mpl
1.24 GeV
1.28 ± 0.025 GeV(MS at mc )3.4 × 10−2−2 1/3u
8 × (6π) ϵ0 Mpl
2.07 MeV
2.15 ± 0.15 MeV (MS at 2 GeV) 3.8 × 10−2
1/3−1/12b3×2(6π)ϵ0 Mpl
4.33 GeV
4.18 ± 0.03 GeV(MS at mb )3.6 × 10−21/3sϵ0 Mpl
91.8 MeV
93.8 ± 2.4 MeV (MS at 2 GeV) 2.1 × 10−2−1 1/3d(6π) ϵ0 Mpl
4.87 MeV
4.7 ± 0.2 MeV (MS at 2 GeV) 3.6 × 10−22 1/42 −1Z
1/(8π )ϵ0 (1 + (1/11)(1/(2π) )) Mpl
91.1883 GeV
91.1876 ± 0.0021 GeV
7.4 × 10−6−1/4
2 1/4
2 −1W
2/(8π )ϵ0 (1 − (19/11)(1/(2π) )) Mpl 80.373 GeV
80.385 ± 0.015 GeV
1.9 × 10−41/2
2 1/4
2 −1H
2 /(8π )ϵ0 (1 + (14/11)(1/(2π) )) Mpl
125.22 GeV
125.1 ± 0.2 GeV
1.0 × 10−3I report emprical formulae of the masses of charged
leptons (e, µ, τ ), quarks (t, c, u, b, s, d), gauge bosons(Z, W ), and Higgs boson (H). The formulae yield themasses in terms of the Planck mass Mpl and a dimensionless constant ϵ0 = 2 × (6π)−48 . There is no adjustable parameter in the formulae.
The mass values calculated using the formulae arecompared with measured values. For the calculation,
the value of Planck mass from CODATA1) is used:
Mpl = 1.220910 ± 0.000029 × 1019 GeV.
The measured values of particle masses are taken from
PDG2016.2) The mass of a quark is dependent on thescale and the scheme. In the PDG review, the massof quarks other than the t quark are given in the MSscheme at µ = 2 GeV for the u, d, s quarks and atµ = mq for the b and c quarks. For the t quark, themeasured mass is considered to be the pole mass. Theformula for a quark is assumed to yield the MS massat the Z boson mass mZ . The ﬁrst-order renormalizedgroup equation (RGE) below is used to correct for themass value at mZ to the mass at the scale the PDGuses:)( ∫m(t)αs (t)
dt ,= exp −m0π
where t = log µ2 and αs (t) is the running QCD coupling constant at the scale µ. The value of αs (t) in thePDG2016 review is used for the calculation.
The formulae, the calculated values (c) using theformulae, the measured values (m), and diﬀerence|c/m − 1| are summarized in the table above. Thecalculation reproduce the measured mass values well.
The agreement is within the uncertainty of the measured mass or the Planck mass (2.4 × 10−5 ).
∗1RIKEN Nishina Center- 126 -

There is a pattern in the formulae. The formulae
can be summarized as
11/3Nl (6π)nl ϵ0 (1 + δl )−1 Mpl ,21/3mq = 2cq Nq (6π)nq ϵ0 Mpl ,cB
21/4ϵ (1 + δB )−1 Mpl ,mB =8π 2 0ml =where ml , mq , and mB are the masses of charged leptons, quarks, and bosons, respectively. Nl and Nq are
small positive integers, nl and nq are integers ranging
from −2 to 2, and δl and δB are small real numbers.
cb = −1/12 and cq = 0 for all other quarks, and cZ = 0,cW = −1/4, and cH = 1/2. The pattern suggests theexistence of a rule that determines the formulae. Note
1/3that all fermion masses are order of (6π)nf ϵ0 with−2 ≤ nf ≤ 2 and the boson masses are of the order
1/4of 1/8π 2 ϵ0 . Presumaby, this part is the main partof the mass, and (1 + δl ) and (1 + δB ) are correction
factors due to interactions.
Note that the value of ϵ0 is consistent with the product of the Hubble constant H0 and the Planck time
tpl = 1/Mpl :H0 × tpl = (1.211 ± 0.014) × 10−61
ϵ0 ≡ 2 × (6π)−48 = 1.220608 × 10−61
Here, the WMAP 9 year value of H0 is used. Thissuggests that the masses of elementary particles are
related to the expansion of space-time.
A theoretical model that can explain these formulae
is under development.
References
1) Rev. Mod. Phys. 88, 053009 (2016).
2) Review of Particle Physics, Chin. Phys. C 40, 100001(2016).