Koide formula

The Koide formula is an unexplained relation discovered by Yoshio Koide in 1981. It relates the masses of the three charged leptons so well that it predicted the mass of the tau.

Formula

The Koide formula is:

Q={\frac {m_{e}+m_{\mu }+m_{\tau }}{({\sqrt {m_{e}}}+{\sqrt {m_{\mu }}}+{\sqrt {m_{\tau }}})^{2}}}\approx {\frac {2}{3}}

It is clear that 1⁄3 < Q < 1. The superior bound follows if we assume that the square roots can not be negative. R. Foot remarked that 1⁄3Q can be interpreted as the squared cosine of the angle between the vector

({\sqrt {m_{e}}},{\sqrt {m_{\mu }}},{\sqrt {m_{\tau }}})

and the vector

(1,1,1).

The mystery is in the physical value. The masses of the electron, muon, and tau are measured respectively as m_e = 0.510998910(13) MeV/c², m_μ = 105.658367(4) MeV/c², and m_τ = 1776.84(17) MeV/c², where the digits in parentheses are the uncertainties in the last figures.^[1] This gives Q = 0.666659(10).^[2] Not only is this result odd in that three apparently random numbers should give a simple fraction, but also that Q is exactly halfway between the two extremes of 1⁄3 and 1.

This has never been explained nor understood.

Notes

^ C. Amsler et. al. (Particle Data Group) (2008, and 2009 partial update). "Review of Particle Physics – Leptons" (PDF). Physics Letters B. 667 (1–5): 1. Bibcode:2008PhLB..667....1P. doi:10.1016/j.physletb.2008.07.018. {{cite journal}}: Check date values in: |year= (help)
^ Since the uncertainties in m_e and m_μ are much smaller than that in m_τ, the uncertainty in Q was calculated as $\scriptstyle {\Delta Q={\frac {\partial Q}{\partial m_{\tau }}}\Delta m_{\tau }}$ .

References

C. Amsler; et al. (2008). "Review of Particle Physics". Physics Letters B. 667 (1–5): 1. Bibcode:2008PhLB..667....1P. doi:10.1016/j.physletb.2008.07.018. {{cite journal}}: Explicit use of et al. in: |author= (help)
R. Foot (1994). "A note on Koide's lepton mass relation". arXiv:hep-ph/9402242. {{cite arXiv}}: |class= ignored (help)
Y. Koide (1983). "New view of quark and lepton mass hierarchy". Physical Review D. 28 (1): 252–254. Bibcode:1983PhRvD..28..252K. doi:10.1103/PhysRevD.28.252.

Y. Koide (1984). "Erratum: New view of quark and lepton mass hierarchy". Physical Review D. 29 (7): 1544. Bibcode:1984PhRvD..29Q1544K. doi:10.1103/PhysRevD.29.1544.

Y. Koide (1983). "A fermion-boson composite model of quarks and leptons". Physics Letters B. 120 (1–3): 161–165. Bibcode:1983PhLB..120..161K. doi:10.1016/0370-2693(83)90644-5.
Y. Koide (2000). "Quark and lepton mass matrices with a cyclic permutation invariant form". arXiv:hep-ph/0005137. {{cite arXiv}}: |class= ignored (help)
Y. Koide (2005). "Challenge to the mystery of the charged lepton mass". arXiv:hep-ph/0506247. {{cite arXiv}}: |class= ignored (help)
S. Oneda; Y. Koide (1991). Asymptotic symmetry and its implication in elementary particle physics. World Scientific. ISBN 981-02-0498-1.

Koide formula

Formula

See also

Notes

References

Further reading