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.

Contents

Formula

The Koide formula is:

Q = \frac{m_e %2B m_{\mu} %2B m_{\tau}}{(\sqrt{m_e}%2B\sqrt{m_{\mu}}%2B\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 me = 0.510998910(13) MeV/c2, mμ = 105.658367(4) MeV/c2, and mτ = 1,776.84(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.

While the original formula appeared in the context of preon models, other ways have been found to produce it, but in the whole this, it has never been explained nor understood.

Similar matches have been found for quarks depending on running masses, and for triplets of quarks not of the same flavour [3]

See also

Notes

  1. ^ C. Amsler et. al. (Particle Data Group) (2008, and 2009 partial update). "Review of Particle Physics – Leptons". Physics Letters B 667 (1-5): 1. Bibcode 2008PhLB..667....1P. doi:10.1016/j.physletb.2008.07.018. http://pdg.lbl.gov/2009/tables/rpp2009-sum-leptons.pdf. 
  2. ^ Since the uncertainties in me and mμ are much smaller than that in mτ, the uncertainty in Q was calculated as \scriptstyle{\Delta Q = \frac{\partial Q}{\partial m_\tau}\Delta m_\tau}.
  3. ^ Werner Rodejohann, He Zhang, Extension of an empirical charged lepton mass relation to the neutrino sector, Physics Letters B, Volume 698, Pages 152-156,

References

Further reading