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 = 1,776.84(17) MeV/c², where the digits in parentheses are the uncertainties in the last figures. This gives Q = 0.666659(10). 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 (should the three masses be equal) and 1 (should one mass dominate).

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

Running of particle masses

In quantum field theory, quantities like coupling constant and mass "run" with the energy scale. That is, their value depends on the energy scale at which the observation occurs, in a way described by a renormalization group (RG) equation. One usually expects relationships between such quantities to be simple at high energies (where some symmetry is unbroken) but not at low energies, where the RG flow will have produced complicated deviations from the high energy relation. The Koide relation is exact (within experimental error) for the pole masses, which are low-energy quantities defined at different energy scales. For this reason, many physicists regard the relation as "numerology" (e.g.). However, the Japanese physicist Yukinari Sumino has constructed an effective field theory in which a new gauge symmetry causes the pole masses to exactly satisfy the relation.

See also

• CKM matrix
• Clifford algebra
• Generation
• Higgs mechanism
• Higgsless model
• PMNS matrix
• Preon
• Quark–lepton complementarity
• Seesaw mechanism
• Technicolor

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. 
• R. Foot (1994). "A note on Koide's lepton mass relation". arXiv:hep-ph/9402242 [hep-ph]. 
• 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 [hep-ph]. 
• Y. Koide (2005). "Challenge to the mystery of the charged lepton mass". arXiv:hep-ph/0506247 [hep-ph]. 
• S. Oneda; Y. Koide (1991). Asymptotic symmetry and its implication in elementary particle physics. World Scientific. ISBN 981-02-0498-1. http://books.google.com/books?isbn=9810204981. 

Further reading

• A. Rivero, A. Gsponer (2005). "The strange formula of Dr. Koide". arXiv:hep-ph/0505220 [hep-ph]. 
• N. Li; B.-Q. Ma (2006). "Energy scale independence for quark and lepton masses". arXiv:hep-ph/0601031 [hep-ph]. 
• C. Brannen (2010), "Spin Path Integrals and Generations", Foundations of Physics, Bibcode 2010FoPh...40.1681B, doi:10.1007/s10701-010-9465-8, http://www.brannenworks.com/Gravity/spinpath.pdf  (See the article's references links to "The lepton masses" and "Recent results from the MINOS experiment".)
• Wolfram Alpha, http://www.wolframalpha.com/input/?i=solve+%280.510998910+%2B+105.658369+%2B+m%29%2F%28sqrt%280.510998910%29+%2B+sqrt%28105.658369%29+%2B+sqrt%28m%29%29%5E2+%3D+2%2F3  (Solves for the predicted tau mass from the Koide formula.)