# Koide formula

The Koide formula is an unexplained empirical equation 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 13 < Q < 1. The superior bound follows if we assume that the square roots can not be negative. By Cauchy-Schwarz 13Q 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 = 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 13 (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 (both by Sumino and by Koide, see references below). As a whole, however, understanding remains incomplete.

Similar matches have been found for quarks depending on running masses, and for triplets of quarks not of the same flavour.[3][4][5] With alternating quarks, chaining Koide equations for consecutive triplets, it is possible to reach a result of 173.263947(6) GeV for the mass of the top quark.[6]

## Similar Formulae

There are similar empirical formulae which relate other masses. For example the bottom and top quark masses of approximately 5 GeV and 174 GeV satisfy:

$Q = \frac{m_B + m_{T}}{(\sqrt{m_B}+\sqrt{m_{T}})^2} \approx \frac{3}{4}.$

Where again the fraction 3/4 is exactly in the middle of 12 < Q < 1, although the masses of these quarks are known less accurately.

Quark masses depend on the energy scale used to measure them, which makes an analysis more complicated.

Taking the heaviest three quarks, charm (1290 MeV), bottom (4370 MeV) and top (174100 MeV), gives a much closer match:

$Q = \frac{m_C + m_{B} + m_{T}}{(\sqrt{m_C}+\sqrt{m_{B}}+\sqrt{m_{T}})^2} \approx \frac{2}{3}.$

It is possible to get exactly 2/3 within the experimental uncertainties of the masses (as of 2015). This was noticed by R&Z in the first version of their paper[7] but the observation was removed in the published version, so the first published mention is from F. G. Cao.[8]

The masses of the lightest quarks (up, down, strange) are not known well enough to test the formula but if up is taken equal to zero; in such case the formula -including the masslessness of up quark- reduces to one of Harari-Haut-Weyers,[9] known well before Koide.

## 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 equation (RGE). 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.[10]). 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.[11] Goffinet's doctoral thesis gives a discussion on pole masses and how the Koide formula can be reformulated without taking the square roots of masses.[12]

## References

1. ^ Amsler, C. (Particle Data Group) et al. (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. Check date values in: |date= (help)
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. ^ Rodejohann, W.; Zhang, H. (2011). "Extension of an empirical charged lepton mass relation to the neutrino sector". Physics Letters B 698 (2): 152–156. arXiv:1101.5525. Bibcode:2011PhLB..698..152R. doi:10.1016/j.physletb.2011.03.007.
4. ^ Rosen, G. (2007). "Heuristic development of a Dirac-Goldhaber model for lepton and quark structure" (PDF). Modern Physics Letters A 22 (4): 283–288. Bibcode:2007MPLA...22..283R. doi:10.1142/S0217732307022621.
5. ^ Kartavtsev, A. (2011). "A remark on the Koide relation for quarks". arXiv:1111.0480 [hep-ph].
6. ^ Rivero, A. (2011). "A new Koide tuple: Strange-charm-bottom". arXiv:1111.7232 [hep-ph].
7. ^ Rodejohann, W.; Zhang, H. (2011). "Extension of an empirical charged lepton mass relation to the neutrino sector". Physics Letters B 698 (2): 152–156. arXiv:1101.5525. Bibcode:2011PhLB..698..152R. doi:10.1016/j.physletb.2011.03.007.
8. ^ Cao, F. G. (2012). "neutrino masses from lepton and quark mass relations...". Phys Rev D 85: 113003.
9. ^ Harari, H.; Haut, H.; Weyers, J. (1978). "Quark Masses And Cabibbo Angles". Physics Letters B 78: 459.
10. ^ Motl, L. (16 January 2012). "Could the Koide formula be real?". The Reference Frame. Retrieved 2014-07-10.
11. ^ Sumino, Y. (2009). "Family Gauge Symmetry as an Origin of Koide's Mass Formula and Charged Lepton Spectrum". Journal of High Energy Physics 2009 (5): 75. arXiv:0812.2103. Bibcode:2009JHEP...05..075S. doi:10.1088/1126-6708/2009/05/075.
12. ^ Goffinet, F. (2008). A bottom-up approach to fermion masses (PDF) (PhD Thesis). Université catholique de Louvain.