The Koide’s Equation for the Sums of Masses in the Same Sectors

Abstract

Koide’s equation was discovered for masses in the same sectors.

1 Introduction

It is well-known that there are 3 basic elementary particles the masses of which are different although spins and charges are same. It is said that each particle belongs to a generation.

Koide gave an equation concerned with 3 generations of charged leptons [1].   

$({\Lambda }_1^2{}_{}+{\Lambda }_2^2+{\Lambda }_3^2)/{({\Lambda }_1+{\Lambda }_2+{\Lambda }_3)}^2=(1+2H^2)/3\equiv (1+K^2)/3;H^2=1/\mathrm{2.....}\left(1\right).$

Brannen introduced a mass formula by the field theory [2].

Where $\text{A}n=\sqrt{mn}$is the square root of mass. H and Δare parameters.

The Eq.(1) can be proved by Eq.(2). However vice versa is not possible. From the Eq.(2) we have [3]

2 Definitions and Calculations

We define the summation of masses of fermion particles in each sector.

The data are gained from Particle Data Group [4].

When we put the values (Mev./c²) in Eqs. (5)(6) and (7) into the left side of Eq.(1) we find K=1.00.

This value is the same as the one of Koide in Eq.(1). From Eq.(4) we have Δ = 0.05488 [3]_. (This last value is of course different from Δ = 2/9 = 0.222222 for charged lepton [2].)Using these values we gain by Eq.(2) the following values.

Λ1 = 38.58 (88.9%), Λ2 = 63.61 (97.2%), Λ3 = 424.85 (105.7%) [100×cal./experiment]

These values show that m1, m2 and m3 are approximately and systematically gained by Eq.(2).

The approximate values are due to the existence of quarks. The electric charges are different among m1,m2 and m3. However, the masses have no relations with the charges as mentioned in [5]. Therefore, we can discuss without any assumptions on this point.

3 Discussions

We may give an explanation as to the value K=1.00 for masses m1, m2 and m3 which is the same one for Koide’s equation. We assume that during evolution of the cosmos from the Big Bang a fermion proto-particle with the mass of summation m1+m2+m3 (Eq.(5),(6),(7))generated. Afterwards, the proto-particle separated into three composed fermion particles m1,m2 and m3 with different electric charges. This assumption may give some explanation as to why 3 kinds of particles exist. Each particle has had 3 generations. As a result except neutrino there exist 9 kinds of fermion particles in the present cosmos. This means that there is symmetry S3⊗S3 in the field including charged leptons, d-quarks and u-quarks. Because m1, m2 and m3 are composed of 3 particles with the same electric charges they may need some kind of gluon.

References

• [1]   Y. Koide, Lett. Nuovo Cimento, 34, 201 (1982); Phys. Lett. B, 120, 161 (1983); Phys. Rev. D Part. Fields, 28, 252 (1983).
• [2]   http://C.Brannen,httpbrannenworks.com/MASSES2.pdf,(2006).
• [3]   P.Zenczykowski,arXive.301.4143v2[hep-ph]21Feb (2013).
• [4]   K. A. Olive, et al. Particle Data Group, Chem. Phys., C38, 090001 (2014).
• [5]   H. Narumi, J. Compt, Chem.Jpn., 14, 179 (2015).