Physics:Koide formula

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Short description: Unexplained empirical equation in particle physics

The Koide formula is an unexplained empirical equation discovered by Yoshio Koide in 1981. In its original form, it is not fully empirical but a set of guesses for a model for masses of quarks and leptons, as well as CKM angles. From this model it survives the observation about the masses of the three charged leptons; later authors have extended the relation to neutrinos, quarks, and other families of particles.[1]:64–66

Formula

The Koide formula is

[math]\displaystyle{ Q = \frac{\; m_e + m_\mu + m_\tau \;}{\left(\,\sqrt{m_e\,} + \sqrt{m_\mu\,} + \sqrt{m_\tau\,} \,\right)^2} = 0.666661(7) \approx \frac{\,2\,}{3}~, }[/math]

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; the digits in parentheses are the uncertainties in the last digits.[2] This gives Q = 0.666661(7).[3]

No matter what masses are chosen to stand in place of the electron, muon, and tau, 1/3 ≤ Q < 1 . The upper bound follows from the fact that the square roots are necessarily positive, and the lower bound follows from the Cauchy–Bunyakovsky–Schwarz inequality. The experimentally determined value, 2/3, lies at the center of the mathematically allowed range. But note that removing the requirement of positive roots it is possible to fit an extra tuple in the quark sector (the one with strange, charm and bottom).

The mystery is in the physical value. Not only is the result peculiar, in that three ostensibly arbitrary numbers give a simple fraction, but also in that in the case of electron, muon, and tau, Q is exactly halfway between the two extremes of all possible combinations: 1/3 (if the three masses were equal) and 1 (if one mass dominates). Importantly, the relation holds regardless of which unit is used to express the masses.

Robert Foot also interpreted the Koide formula as a geometrical relation, in which the value [math]\displaystyle{ \frac{1}{\,3\,Q\,} }[/math] is the squared cosine of the angle between the vector [math]\displaystyle{ \left[\,\sqrt{m_e\,}, \sqrt{m_\mu\,}, \sqrt{m_\tau\,}\,\right] }[/math] and the vector [math]\displaystyle{ [1, 1, 1] }[/math] (see dot product).[4] That angle is almost exactly 45 degrees: [math]\displaystyle{ \theta = (45.000 \pm 0.001)^\circ~. }[/math][4]

When the formula is assumed to hold exactly (Q = 2/3), it may be used to predict the tau mass from the (more precisely known) electron and muon masses; that prediction is mτ = 1776.969 MeV/c2.[5] Please note that solving the Koide formula can also predict the third particle mass to be around 3.37 MeV/c2.

While the original formula arose in the context of preon models, other ways have been found to derive 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.[6][7][8] 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.[9]

Speculative extension

Carl Brannen has proposed[5] the lepton masses are given by the squares of the eigenvalues of a circulant matrix with real eigenvalues, corresponding to the relation

[math]\displaystyle{ \sqrt{\,m_n\;} = \mu \left[\,1 + 2 \eta \cos\left( \delta + \frac{\,2\pi\,}{3}\cdot n \right) \,\right] ~,~ }[/math] for n = 0, 1, 2, ...

which can be fit to experimental data with η2 = 0.500003(23) (corresponding to the Koide relation) and phase δ = 0.2222220(19), which is almost exactly 2/9 . However, the experimental data are in conflict with simultaneous equality of η2 = 1/2 and δ = 2/9 .[5]

This kind of relation has also been proposed for the quark families, with phases equal to low-energy values 2/27 = 2/9 × 1/3 and 4/27 = 2/9 × 2/3, hinting at a relation with the charge of the particle family (1/3 and 2/3 for quarks vs. 3/3 = 1 for the leptons, where   1/3 × 2/3 × 3/3 ≈ δ ).[10]

Origins

The original derivation [11] postulates [math]\displaystyle{ m_{e_i} \propto (z_0 + z_i)^2 }[/math] with the conditions

[math]\displaystyle{ z_1+z_2+z_3=0 }[/math]
[math]\displaystyle{ \frac 13(z_1^2+z_2^2+z_3^2)=z_0^2 }[/math]

from which the formula follows. Besides, masses for neutrinos and down quarks were postulated to be proportional to [math]\displaystyle{ z_i^2 }[/math] while masses for up quarks were postulated to be [math]\displaystyle{ \propto (z_0+2 z_i)^2 }[/math]

The published model[12] justifies the first condition as part of a symmetry breaking scheme, and the second one as a "flavor charge" for preons in the interaction that causes this symmetry breaking.

Note that in matrix form with [math]\displaystyle{ M= A A^+, A=Z_0+Z }[/math] the equations are simply [math]\displaystyle{ \text{Tr } Z = 0, \text{Tr }Z_0^2 = \text{Tr }Z^2 }[/math]

Similar formulae

There are similar formulae which relate other masses. Quark masses depend on the energy scale used to measure them, which makes an analysis more complicated.[13]

Taking the heaviest three quarks, charm (1.275 ± 0.03 GeV), bottom (4.180 ± 0.04 GeV) and top (173.0 ± 0.40 GeV), regardless of their uncertainties, one arrives at the value cited by F. G. Cao (2012):[14]

[math]\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}. }[/math]

This was noticed by Rodejohann and Zhang in the first version of their 2011 article,[15] but the observation was removed in the published version,[6] so the first published mention is in 2012 from Cao.[14]

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

[math]\displaystyle{ Q_\text{light} = \frac{m_u + m_d + m_s}{\big(\sqrt{m_u} + \sqrt{m_d} + \sqrt{m_s}\big)^2} \approx 0.57 \approx \frac{5}{9}, }[/math]

a value also cited by Cao in the same article.[14]

Note that an older article, H. Harari, et al.,[16] calculates theoretical values for up, down and strange quarks, coincidentally matching the later Koide formula, albeit with a massless up-quark.

[math]\displaystyle{ Q_\text{light} = \frac{0 + m_d + m_s}{\big(\sqrt{0} + \sqrt{m_d} + \sqrt{m_s}\big)^2} }[/math]

Running of particle masses

In quantum field theory, quantities like coupling constant and mass "run" with the energy scale.[17]:151–152 That is, their value depends on the energy scale at which the observation occurs, in a way described by a renormalization group equation (RGE).[18] 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".[19]

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.[20] Koide has published his opinions concerning Sumino's model.[21][22] François Goffinet's doctoral thesis gives a discussion on pole masses and how the Koide formula can be reformulated to avoid needing the square roots of masses.[23]

As solutions to a cubic equation

A cubic equation usually arises in symmetry breaking when solving for the Higgs vacuum, and is a natural object when considering three generations of particles. This involves finding the eigenvalues of a 3x3 mass matrix. For this case, a characteristic polynomial

[math]\displaystyle{ 4m^3 -24 n^2 m^2+9n(n^3-4)m-9 }[/math]

with roots [math]\displaystyle{ m_i }[/math] as real and positive. Let [math]\displaystyle{ m=x^2 }[/math] and it can be factorised as

[math]\displaystyle{ (2x^3-6nx^2+3n^2x-3)(2x^3+6nx^2+3n^2x+3) }[/math]

Since from elementary symmetric polynomials, one has [math]\displaystyle{ x_1+x_2+x_3=3n }[/math] and [math]\displaystyle{ 2(x_1 x_2+x_2 x_3 + x_3 x_1) = 3n^2 }[/math], the subsequent cubic equation for [math]\displaystyle{ x }[/math] has solutions that fit a formula of Koide-type for any value of [math]\displaystyle{ n }[/math] since,

[math]\displaystyle{ \frac{\; m_1 + m_2 + m_3 \;}{\left(\,\sqrt{m_1\,} + \sqrt{m_2\,} + \sqrt{m_3\,} \,\right)^2} = 1- {2( x_1 x_2+x_2 x_3 + x_3 x_1) \over (x_1+x_2+x_3)^2 } = 1 - { 3n^2 \over (3 n)^2 }=\frac 23 }[/math]

For the relativistic case, Goffinet's dissertation presented a similar method to build a polynomial with only even powers of [math]\displaystyle{ m }[/math].

For the charged lepton triple, the value of [math]\displaystyle{ n }[/math] needed is extremely close to the integer [math]\displaystyle{ 3 }[/math].

Higgs mechanism

One explanation for the Koide formula would be a Higgs particle with [math]\displaystyle{ U(3) }[/math] flavour charge [math]\displaystyle{ \Phi^{a\overline{b}} }[/math] given by:

[math]\displaystyle{ V(\Phi) = (2 Tr(\Phi)^2 - 3 Tr(\Phi^2))^2 }[/math]

with the charged lepton mass terms given by [math]\displaystyle{ \overline{\psi}\Phi^2 \psi }[/math].[24] Such a potential is minimised when the masses fit the Koide formula. Although it doesn't give the mass scale which would have to be given by additional terms of the potential. Thus the Koide formula may suggest existence of additional scalar particles beyond the Standard Model Higgs boson.

See also

References

  1. ↑ Zenczykowski, P., Elementary Particles And Emergent Phase Space (Singapore: World Scientific, 2014), pp. 64–66.
  2. ↑ Amsler, C. (2008). "Review of Particle Physics". Physics Letters B 667 (1–5): 1–6. doi:10.1016/j.physletb.2008.07.018. PMID 10020536. Bibcode: 2008PhLB..667....1A. https://cds.cern.ch/record/1481544/files/PhysRevD.86.010001.pdf. 
  3. ↑ Since the uncertainties in me and mμ are much smaller than that in mτ, the uncertainty in Q was calculated as [math]\displaystyle{ \Delta Q = \frac{\,\partial\,Q\,}{\,\partial \,m_\tau\,}\,\Delta m_\tau ~.~ }[/math]
  4. ↑ 4.0 4.1 Foot, R. (1994-02-07). "A note on Koide's lepton mass relation". arXiv:hep-ph/9402242.
  5. ↑ 5.0 5.1 5.2 Brannen, Carl A. (May 2, 2006). "The lepton masses". http://brannenworks.com/MASSWS2.pdf. 
  6. ↑ 6.0 6.1 Rodejohann, W.; Zhang, H. (2011). "Extension of an empirical charged lepton mass relation to the neutrino sector". Physics Letters B 698 (2): 152–156. doi:10.1016/j.physletb.2011.03.007. Bibcode: 2011PhLB..698..152R. 
  7. ↑ Rosen, G. (2007). "Heuristic development of a Dirac-Goldhaber model for lepton and quark structure". Modern Physics Letters A 22 (4): 283–288. doi:10.1142/S0217732307022621. Bibcode: 2007MPLA...22..283R. https://www.researchgate.net/publication/228650747. 
  8. ↑ Kartavtsev, A. (2011). "A remark on the Koide relation for quarks". arXiv:1111.0480 [hep-ph].
  9. ↑ Rivero, A. (2011). "A new Koide tuple: Strange-charm-bottom". arXiv:1111.7232 [hep-ph].
  10. ↑ Zenczykowski, Piotr (2012-12-26). "Remark on Koide's Z3-symmetric parametrization of quark masses". Physical Review D 86 (11): 117303. doi:10.1103/PhysRevD.86.117303. ISSN 1550-7998. Bibcode: 2012PhRvD..86k7303Z. 
  11. ↑ Koide, Yoshio (1981), Quark and lepton masses speculated from a Subquark model 
  12. ↑ Koide, Y. (1983). "A fermion-boson composite model of quarks and leptons". Physics Letters B 120 (1–3): 161–165. doi:10.1016/0370-2693(83)90644-5. Bibcode: 1983PhLB..120..161K. 
  13. ↑ Quadt, A., Top Quark Physics at Hadron Colliders (Berlin/Heidelberg: Springer, 2006), p. 147.
  14. ↑ 14.0 14.1 14.2 Cao, F. G. (2012). "Neutrino masses from lepton and quark mass relations and neutrino oscillations". Physical Review D 85 (11): 113003. doi:10.1103/PhysRevD.85.113003. Bibcode: 2012PhRvD..85k3003C. 
  15. ↑ Rodejohann, W.; Zhang, H. (2011). "Extension of an empirical charged lepton mass relation to the neutrino sector". arXiv:1101.5525 [hep-ph].
  16. ↑ Harari, Haim; Haut, Hervé; Weyers, Jacques (1978). "Quark masses and cabibbo angles"]. Physics Letters B 78 (4): 459–461. doi:10.1016/0370-2693(78)90485-9. Bibcode: 1978PhLB...78..459H. https://cds.cern.ch/record/870700/files/c78-08-23-p611.pdf. 
  17. ↑ Álvarez-Gaumé, L., & Vázquez-Mozo, M. A., An Invitation to Quantum Field Theory (Berlin/Heidelberg: Springer, 2012), pp. 151–152.
  18. ↑ Green, D., Cosmology with MATLAB (Singapore: World Scientific, 2016), p. 197.
  19. ↑ Motl, L. (16 January 2012). "Could the Koide formula be real?". The Reference Frame. https://web.archive.org/web/20210802185927/http://motls.blogspot.com/2012/01/could-koide-formula-be-real.html. Retrieved 2023-12-21. 
  20. ↑ 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. doi:10.1088/1126-6708/2009/05/075. Bibcode: 2009JHEP...05..075S. 
  21. ↑ Koide, Yoshio (2017). "Sumino Model and My Personal View". arXiv:1701.01921 [hep-ph].
  22. ↑ Koide, Yoshio (2018). "What Physics Does The Charged Lepton Mass Relation Tell Us?". arXiv:1809.00425 [hep-ph].
  23. ↑ Goffinet, F. (2008). A bottom-up approach to fermion masses (PDF) (PhD dissertation). Université catholique de Louvain.
  24. ↑ Koide, Yoshio (1989). Charged lepton mass sum rule from U(3) family Higgs potential model. https://inspirehep.net/literature/285940. 

Further reading

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