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Koide formula

This unexplained relation was discovered by Yoshio Koide in 1981, and relates the masses of the three leptons so well that it predicted the mass of the tau lepton.

Let

$Q = \frac{m_e + m_{\mu} + m_{\tau}}{(\sqrt{m_e}+\sqrt{m_{\mu}}+\sqrt{m_{\tau}})^2}$

It is clear that 1 / 3 < Q < 1 from the definition. 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 cosine squared of the angle between the vector $(\sqrt{m_e},\sqrt{m_{\mu}},\sqrt{m_{\tau}})$ and (1,1,1).

The mystery is in the physical value. The masses of the electron, muon, and tau lepton are measured respectively as $m_e = 0.511\ \rm{MeV}/c^2,\ m_{\mu}=105.7\ \rm{MeV}/c^2,\ m_{\tau} = 1777\ \rm{MeV}/c^2$, which gives

$Q = \frac{2}{3} \pm 0.01 %$

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 result has never been explained or understood.

References

Phys. Rev. D 28, 252–254 (1983)

Physics Letters B, Volume 120, Issues 1-3 , 6 January 1983, Pages 161-165