# Koide formula

Geometrical interpretation of Koide formula (masses not in scale)

The Koide formula is an unexplained empirical equation discovered by Yoshio Koide in 1981. In its original form, it relates the masses of the three charged leptons; later authors have extended the relation to neutrinos, quarks, and other families of particles.

## Formula

The Koide formula is

${\displaystyle Q={\frac {m_{e}+m_{\mu }+m_{\tau }}{{\big (}{\sqrt {m_{e}}}+{\sqrt {m_{\mu }}}+{\sqrt {m_{\tau }}}{\big )}^{2}}}\approx 0.666661\approx {\frac {2}{3}},}$

where the masses of the electron, muon, and tau are measured respectively as me = 0.510998946(3) MeV/c2, mμ = 105.6583745(24) MeV/c2, and mτ = 1776.86(12) MeV/c2, and the digits in parentheses are the uncertainties in the last figures.[1] This gives Q = 0.666661(7).[2]

It is clear that no matter what particles are chosen to stand in place of the electron, muon, and tau, 13 ≤ Q < 1. The upper bound follows from the fact that the square roots are necessarily positive. The lower bound follows from the Cauchy–Bunyakovsky–Schwarz inequality, and also from the fact that the value ${\displaystyle {\frac {1}{3Q}}}$  can be interpreted as the squared cosine of the angle between the vector ${\displaystyle ({\sqrt {m_{e}}},{\sqrt {m_{\mu }}},{\sqrt {m_{\tau }}})}$  and the vector ${\displaystyle (1,1,1)}$  (see dot product).

The mystery is in the physical value. Not only is this result odd in that three apparently arbitrary numbers should give a simple fraction, but also that Q (in the case of electron, muon, and tau) is exactly halfway between the two extremes of other particle combinations: ​13 (should the three masses be equal) and 1 (should one mass dominate). The value of Q = ​23 corresponds to mτ = 1776.969 MeV/c2.

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 triplets of quarks depending on running masses.[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. Quark masses depend on the energy scale used to measure them, which makes an analysis more complicated.[7]:147

Taking the heaviest three quarks, charm (1.275 ± 0.03 GeV), bottom (4.180 ± 0.04 GeV) and top (173.0 ± 0.40 GeV), and without using their uncertainties gives the value cited by F. G. Cao (2012),[8]

${\displaystyle Q_{\text{heavy}}={\frac {m_{c}+m_{b}+m_{t}}{{\big (}{\sqrt {m_{c}}}+{\sqrt {m_{b}}}+{\sqrt {m_{t}}}{\big )}^{2}}}\approx 0.669\approx {\frac {2}{3}}}$

This was noticed by Rodejohann and Zhang in the first version of their 2011 article[9] but the observation was removed in the published version,[3] so the first published mention is in 2012 from Cao.[8]

Similarly, the masses of the lightest quarks, up (2.2 ± 0.4 MeV), down (4.7 ± 0.3 MeV), and strange (95.0 ± 4.0 MeV), without using their experimental uncertainties yield,

${\displaystyle Q_{\text{light}}={\frac {m_{u}+m_{d}+m_{s}}{{\big (}{\sqrt {m_{u}}}+{\sqrt {m_{d}}}+{\sqrt {m_{s}}}{\big )}^{2}}}\approx 0.56\approx {\frac {5}{9}}}$

a value also cited by Cao in the same paper.[8]

## 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).[10] 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".[11] However, the Japanese physicist Yukinari Sumino has proposed mechanisms to explain origins of the charged lepton spectrum as well as the Koide formula, e.g., by constructing an effective field theory in which a new gauge symmetry causes the pole masses to exactly satisfy the relation.[12] François 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.[13]

## References

1. ^ Amsler, C.; et al. (Particle Data Group) (2008). "Review of Particle Physics" (PDF). Physics Letters B. 667 (1–5): 1–6. Bibcode:2008PhLB..667....1A. doi:10.1016/j.physletb.2008.07.018.
2. ^ Since the uncertainties in me and mμ are much smaller than that in mτ, the uncertainty in Q was calculated as ${\displaystyle \Delta Q={\frac {\partial Q}{\partial m_{\tau }}}\Delta m_{\tau }}$ .
3. ^ a b 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. ^ Quadt, A., Top Quark Physics at Hadron Colliders (Berlin/Heidelberg: Springer, 2006), p. 147.
8. ^ a b c Cao, F. G. (2012). "Neutrino masses from lepton and quark mass relations and neutrino oscillations". Physical Review D. 85 (11): 113003. arXiv:1205.4068. Bibcode:2012PhRvD..85k3003C. doi:10.1103/PhysRevD.85.113003.
9. ^ Rodejohann, W.; Zhang, H. (2011). "Extension of an empirical charged lepton mass relation to the neutrino sector". arXiv:1101.5525 [hep-ph].
10. ^ Green, D., Cosmology with MATLAB (Singapore: World Scientific, 2016), p. 197.
11. ^ Motl, L. (16 January 2012). "Could the Koide formula be real?". The Reference Frame. Retrieved 2014-07-10.
12. ^ 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.
13. ^ Goffinet, F. (2008). A bottom-up approach to fermion masses (PDF) (PhD Thesis). Université catholique de Louvain.