Koide formula

The Koide formula is

: $Q\; =\; \backslash frac\{m\_\backslash text\{e\}\; +\; m\_\backslash mu\; +\; m\_\backslash tau\}\{\backslash left(\; \backslash sqrt\{\; m\_\backslash text\{e\}\; \}\; +\; \backslash sqrt\{\; m\_\backslash mu\; \}\; +\; \backslash sqrt\{\; m\_\backslash tau\; \}\; \backslash right)^2\}\; =\; \backslash frac\{\; 2\; \}\{\; 3\; \}\; ,$

where the masses of the electron, muon, and tau are measured respectively as {{nobr|{{mvar|m}}{{sub|e}} {{=}} {{val|0.51099895000|(15)|ul=MeV/c2}}}}, {{nobr|{{mvar|m}}{{sub|{{math|μ}}}} {{=}} {{val|105.6583755|(23)|u=MeV/c2}}}}, and {{nobr|{{mvar|m}}{{sub|{{math|τ}}}} {{=}} {{val|1776.93|(09)|u=MeV/c2}}}}; the digits in parentheses are the uncertainties in the last digits.

{{cite journal

|last1=Navas |first1=S.

|display-authors=etal

|collaboration=Particle Data Group

|year=2024

|title=Review of Particle Physics

|journal=Phys. Rev. D

|volume=110 |pages=030001

|doi=10.1103/PhysRevD.110.030001

|hdl=20.500.11850/695340

|hdl-access=free

}}

This gives {{math| Q {{=}} {{val|0.66666446|(508)}}}}.{{efn|

Since the uncertainties in {{math|m{{sub|e}}}} and {{math|m{{sub|μ}}}} are much smaller than that in {{math|m{{sub|τ}}}}, the uncertainty in {{mvar|Q}} was calculated as $\backslash Delta\; Q\; =\; \backslash frac\{\; \backslash partial\; Q\; \}\{\; \backslash partial\; m\_\backslash tau\; \}\; \backslash Delta\; m\_\backslash tau\; .$

}}

No matter what masses are chosen to stand in place of the electron, muon, and tau, the ratio {{math| Q}} is constrained to {{math| {{sfrac| 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, {{math|{{sfrac| 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, {{mvar|Q}} is exactly halfway between the two extremes of all possible combinations: {{math|{{sfrac| 1 | 3 }}}} (if the three masses were equal) and {{math|1}} (if one mass dwarfs the other two). {{mvar|Q}} is a dimensionless quantity, so the relation holds regardless of which unit is used to express the magnitudes of the masses.

Robert Foot also interpreted the Koide formula as a geometrical relation, in which the value $\backslash textstyle\; \backslash frac\{\; 1\; \}\{3\; Q\}$ is the squared cosine of the angle between the vector $[\; \backslash sqrt\{\; m\_\backslash text\{e\}\; \},\; \backslash sqrt\{\; m\_\backslash mu\; \},\; \backslash sqrt\{\; m\_\backslash tau\; \}\; ]$ and the vector {{math|{{bracket|1, 1, 1}}}} (see Dot product).

{{cite arXiv

|last=Foot |first=R.

|date=1994-02-07 |df=dmy-all

|title=A note on Koide's lepton mass relation

|eprint=hep-ph/9402242

}}

That angle is almost exactly 45 degrees: $\backslash theta\; =\; 45.000^\backslash circ\; \backslash pm\; 0.001^\backslash circ\; .$

When the formula is assumed to hold exactly ({{math|Q {{=}} {{sfrac|2|3}}}}), it may be used to predict the tau mass from the (more precisely known) electron and muon masses; that prediction is {{math|m{{sub|τ}} {{=}} {{val|1776.969|u=MeV/c2}}}}.

{{cite web

|last=Brannen |first=Carl A.

|date=2 May 2006

|title=The lepton masses

|website=Brannen's personal website

|url=https://brannenworks.com/MASSWS2.pdf

|access-date=18 Oct 2020

}}

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.

{{cite journal

|first1=W. |last1=Rodejohann

|first2=H. |last2=Zhang

|year=2011

|title=Extension of an empirical charged lepton mass relation to the neutrino sector

|journal=Physics Letters B

|volume=698 |issue=2 |pages=152–156

|arxiv=1101.5525 |bibcode=2011PhLB..698..152R

|doi=10.1016/j.physletb.2011.03.007 |s2cid=59445811

}}

{{cite journal

|last=Rosen |first=G.

|year=2007

|title=Heuristic development of a Dirac-Goldhaber model for lepton and quark structure

|journal=Modern Physics Letters A

|volume=22 |issue=4 |pages=283–288

|bibcode=2007MPLA...22..283R |doi=10.1142/S0217732307022621

|url=https://www.researchgate.net/publication/228650747

}}

{{cite arXiv

|last=Kartavtsev |first=A.

|year=2011

|title=A remark on the Koide relation for quarks

|eprint=1111.0480 |class=hep-ph

}}

With alternating quarks, chaining Koide equations for consecutive triplets, it is possible to reach a result of {{val|173.263947|(6)|u=GeV/c2}} for the mass of the top quark.

{{cite arXiv

|last1=Rivero |first1=A.

|year=2011

|title=A new Koide tuple: Strange-charm-bottom

|eprint=1111.7232 |class=hep-ph

}}

The Koide relation exhibits permutation symmetry among the three charged lepton masses $m\_\backslash text\{e\}$, $m\_\backslash mu$, and $m\_\backslash tau$.

{{cite thesis

|last=Goffinet |first=F.

|year=2008

|title=A bottom-up approach to fermion masses

|degree=PhD

|publisher=Université catholique de Louvain

|place=Louvain, FR

|url=https://inspirehep.net/literature/1327002

}}

This means that the value of $Q$ remains unchanged under any interchange of these masses. Since the relation depends on the sum of the masses and the sum of their square roots, any permutation of $m\_\backslash text\{e\}$, $m\_\backslash mu$, and $m\_\backslash tau$ leaves $Q$ invariant:

Q = \frac{m_\text{e} + m_\mu + m_\tau}{\left( \sqrt{m_\text{e}} + \sqrt{m_\mu} + \sqrt{m_\tau} \right)^2} = \frac{m_{\sigma(\text{e})} + m_{\sigma(\mu)} + m_{\sigma(\tau)}}{\left( \sqrt{m_{\sigma(\text{e})}} + \sqrt{m_{\sigma(\mu)}} + \sqrt{m_{\sigma(\tau)}} \right)^2}

for any permutation $\backslash sigma$ of $\backslash \{\; \backslash text\{e\},\; \backslash mu,\; \backslash tau\; \backslash \}$.

The Koide relation is scale invariant; that is, multiplying each mass by a common constant $\backslash lambda$ does not affect the value of $Q$. Let $m\text{'}\_i\; =\; \backslash lambda\; m\_i$ for $i\; =\; \backslash text\{e\},\; \backslash mu,\; \backslash tau$. Then:

\begin{align}

Q' &= \frac{m'_\text{e} + m'_\mu + m'_\tau}{\left( \sqrt{m'_\text{e}} + \sqrt{m'_\mu} + \sqrt{m'_\tau} \right)^2} \\

&= \frac{\lambda m_\text{e} + \lambda m_\mu + \lambda m_\tau}{\left( \sqrt{\lambda m_\text{e}} + \sqrt{\lambda m_\mu} + \sqrt{\lambda m_\tau} \right)^2} \\

&= \frac{\lambda (m_\text{e} + m_\mu + m_\tau)}{\left( \sqrt{\lambda} (\sqrt{m_\text{e}} + \sqrt{m_\mu} + \sqrt{m_\tau}) \right)^2} \\

&= \frac{\lambda (m_\text{e} + m_\mu + m_\tau)}{\lambda \left( \sqrt{m_\text{e}} + \sqrt{m_\mu} + \sqrt{m_\tau} \right)^2} \\

&= \frac{m_\text{e} + m_\mu + m_\tau}{\left( \sqrt{m_\text{e}} + \sqrt{m_\mu} + \sqrt{m_\tau} \right)^2} \\

&= Q

\end{align}

Therefore, $Q$ remains unchanged under scaling of the masses by a common factor.

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

: $\backslash sqrt\{\backslash ,m\_n\backslash ;\}\; =\; \backslash mu\; \backslash left[1\; +\; 2\; \backslash eta\; \backslash cos\backslash left(\; \backslash delta\; +\; \backslash frac\{\backslash ,2\backslash pi\backslash ,\}\{3\}\backslash cdot\; n\; \backslash right)\; \backslash right]\; ~,~$ for {{mvar|n}} = 0, 1, 2, ...

which can be fit to experimental data with {{mvar|η}}{{sup|2}} = 0.500003(23) (corresponding to the Koide relation) and phase {{mvar|δ}} = 0.2222220(19), which is almost exactly {{sfrac|2|9}} . However, the experimental data are in conflict with simultaneous equality of {{mvar|η}}{{sup|2}} = {{sfrac|1|2}} and {{mvar|δ}} = {{sfrac|2|9}} .

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

{{cite journal

|last=Zenczykowski |first=Piotr

|date=2012-12-26

|title=Remark on Koide's Z3-symmetric parametrization of quark masses

|journal=Physical Review D

|volume=86 |issue=11 |page=117303

|doi=10.1103/PhysRevD.86.117303 |arxiv=1210.4125

|bibcode=2012PhRvD..86k7303Z |s2cid=119189170 |issn=1550-7998

}}

The original derivation

{{citation

|last=Koide |first=Y. |author-link=Yoshio Koide

|year=1981

|title=Quark and lepton masses speculated from a subquark model

}}

postulates $m\_\{e\_i\}\; \backslash propto\; (z\_0\; +\; z\_i)^2$ with the conditions

: $z\_1\; +\; z\_2\; +\; z\_3\; =\; 0$

: $\backslash tfrac\{\; 1\; \}\{\; 3\; \}\; (z\_1^2+z\_2^2+z\_3^2)\; =\; z\_0^2$

from which the formula follows. Besides, masses for neutrinos and down quarks were postulated to be proportional to $z\_i^2$ while masses for up quarks were postulated to be $\backslash propto\; (\; z\_0\; +\; 2\; z\_i\; )^2\; ~.$

The published model

{{cite journal

|last=Koide |first=Y. |author-link=Yoshio Koide

|year=1983

|title=A fermion-boson composite model of quarks and leptons

|journal=Physics Letters B

|volume=120 |issue=1–3 |pages=161–165

|bibcode = 1983PhLB..120..161K

|doi=10.1016/0370-2693(83)90644-5

}}

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 $M\; =\; A\backslash \; A^\backslash dagger$ and $A\; =\; Z\_0\; +\; Z$ the equations are simply $\backslash operatorname\{tr\}\; Z\; =\; 0$ and $\backslash operatorname\{tr\}\; Z\_0^2\; =\; \backslash operatorname\{tr\}\; Z^2\; .$

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.Quadt, A., Top Quark Physics at Hadron Colliders (Berlin/Heidelberg: Springer, 2006), [https://books.google.com/books?id=UpKMo682Yq0C&pg=PA147&redir_esc=y#v=onepage&q&f=false p. 147].

Taking the heaviest three quarks, charm ({{val|1.275|0.03|u=GeV/c2}}), bottom ({{val|4.180|0.04|u=GeV/c2}}) and top ({{val|173.0|0.40|u=GeV/c2}}), regardless of their uncertainties, one arrives at the value cited by F. G. Cao (2012):

: $Q\_\backslash text\{heavy\}\; =\; \backslash frac\{m\_\backslash text\{c\}\; +\; m\_\backslash text\{b\}\; +\; m\_\backslash text\{t\}\}\{\backslash left(\backslash sqrt\{m\_\backslash text\{c\}\}\; +\; \backslash sqrt\{m\_\backslash text\{b\}\}\; +\; \backslash sqrt\{m\_\backslash text\{t\}\}\backslash right)^2\}\; \backslash approx\; 0.669\; \backslash approx\; \backslash frac\{2\}\{3\}.$

This was noticed by Rodejohann and Zhang in the preprint of their 2011 article,

{{cite arXiv

|last1=Rodejohann |first1=W.

|last2=Zhang |first2=H.

|year=2011

|title=Extension of an empirical charged lepton mass relation to the neutrino sector

|arxiv=1101.5525

|class=hep-ph

}} but the observation was removed in the published version, so the first published mention is in 2012 from Cao.

{{cite journal

|last1=Cao |first1=F. G.

|year=2012

|title=Neutrino masses from lepton and quark mass relations and neutrino oscillations

|journal=Physical Review D

|volume=85 |issue=11 |page=113003

|arxiv=1205.4068

|bibcode=2012PhRvD..85k3003C

|doi=10.1103/PhysRevD.85.113003

|s2cid=118565032

}}

The relation

: $Q\_\backslash text\{middle\}\; =\; \backslash frac\{m\_\backslash text\{s\}\; +\; m\_\backslash text\{c\}\; +\; m\_\backslash text\{b\}\}\{\backslash left(-\; \backslash sqrt\{m\_\backslash text\{s\}\}\; +\; \backslash sqrt\{m\_\backslash text\{c\}\}\; +\; \backslash sqrt\{m\_\backslash text\{b\}\}\; \backslash right)^2\}\; \backslash approx\; 0.675$

is published as part of the analysis of Rivero,

{{cite journal

|last1=Rivero |first1=A.

|year=2024

|title=An interpretation of scalars in SO(32)

|journal=European Physical Journal C

|volume=84 |issue=10

|page=1058

|arxiv=2407.05397

|doi=10.1140/epjc/s10052-024-13368-3

}} who notes (footnote 3 in the reference) that an increase of the value for charm mass makes both equations, heavy and middle, exact.

The masses of the lightest quarks, up ({{val|2.2|0.4|u=MeV/c2}}), down ({{val|4.7|0.3|u=MeV/c2}}), and strange ({{val|95.0|4.0|u=MeV/c2}}), without using their experimental uncertainties, yield

: $Q\_\backslash text\{light\}\; =\; \backslash frac\{m\_\backslash text\{u\}\; +\; m\_\backslash text\{d\}\; +\; m\_\backslash text\{s\}\}\{\backslash left(\backslash sqrt\{m\_\backslash text\{u\}\}\; +\; \backslash sqrt\{m\_\backslash text\{d\}\}\; +\; \backslash sqrt\{m\_\backslash text\{s\}\}\backslash right)^2\}\; \backslash approx\; 0.57,$

a value also cited by Cao in the same article. An older article, H. Harari, et al.,{{cite journal |title=Quark masses and Cabibbo angles]|url=https://cds.cern.ch/record/870700/files/c78-08-23-p611.pdf|journal=Physics Letters B|volume=78 |issue=4 |year=1978 |doi=10.1016/0370-2693(78)90485-9|last1=Harari|first1=Haim|last2=Haut|first2=Hervé|last3=Weyers|first3=Jacques|pages=459–461|bibcode=1978PhLB...78..459H}} calculates theoretical values for up, down and strange quarks, coincidentally matching the later Koide formula, albeit with a massless up-quark.

: $Q\_\backslash text\{light\}\; =\; \backslash frac\{0\; +\; m\_\backslash text\{d\}\; +\; m\_\backslash text\{s\}\}\{\backslash left(\backslash sqrt\{0\}\; +\; \backslash sqrt\{m\_\backslash text\{d\}\}\; +\; \backslash sqrt\{m\_\backslash text\{s\}\}\backslash right)^2\}$

This could be considered the first appearance of a Koide-type formula in the literature.

In quantum field theory, quantities like coupling constant and mass "run" with the energy scale.

{{cite book

|author1-link=Luis Álvarez-Gaumé |last1=Álvarez-Gaumé |first1=L.

|last2=Vázquez-Mozo |first2=M. A.

|title=An Invitation to Quantum Field Theory

|place=Berlin, DE / Heidelberg, DE

|publisher=Springer

|year=2012

|pages=[https://books.google.com/books?id=0oNj1AdbFq8C&pg=PA151&redir_esc=y#v=onepage&q&f=false 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).

{{cite book

|last=Green |first=D.

|title=Cosmology with MATLAB

|place=Singapore

|publisher=World Scientific

|year=2016

|page=[https://books.google.com/books?id=7mhIDQAAQBAJ&pg=PA197&redir_esc=y#v=onepage&q&f=false 197]

}}

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".

{{cite web

|last=Motl |first=L. |author-link=Luboš Motl

|date=16 January 2012

|title=Could the Koide formula be real?

|series=The Reference Frame |type=blog

|website=blogspot.com

|url=https://motls.blogspot.com/2012/01/could-koide-formula-be-real.html

|access-date=2023-12-21 |df=dmy-all |url-status=dead

|archive-url=https://web.archive.org/web/20210802185927/http://motls.blogspot.com/2012/01/could-koide-formula-be-real.html

|archive-date=2021-08-02

}}

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 with a new gauge symmetry that causes the pole masses to exactly satisfy the relation.

{{cite journal

|last=Sumino |first=Y. |author-link=Yukinari Sumino

|year=2009

|title=Family gauge symmetry as an origin of Koide's mass formula and charged lepton spectrum

|journal=Journal of High Energy Physics

|volume=2009 |issue=5 |pages=75

|arxiv=0812.2103 |bibcode=2009JHEP...05..075S

|doi=10.1088/1126-6708/2009/05/075 |s2cid=14238049

}}

Koide has published his opinions concerning Sumino's model.

{{cite arXiv

|last=Koide |first=Yoshio |author-link=Yoshio Koide

|year=2017

|title=Sumino model and my personal view

|class=hep-ph |eprint=1701.01921

}}

{{cite arXiv

|last=Koide |first=Yoshio |author-link=Yoshio Koide

|year=2018

|title=What physics does the charged lepton mass relation tell us?

|class=hep-ph |eprint=1809.00425

}}

François Goffinet's doctoral thesis gives a discussion on pole masses and how the Koide formula can be reformulated to avoid using square roots for the masses.

{{cite thesis

|last=Goffinet |first=F.

|year=2008

|title=A bottom-up approach to fermion masses

|degree=PhD

|publisher=Université catholique de Louvain

|place=Louvain, FR

|url=https://inspirehep.net/literature/1327002

}}

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 3 × 3 mass matrix.

For this example, consider a characteristic polynomial

: $4\; m^3\; -\; 24\; n^2\; m^2\; +\; 9\; n\; (n^3\; -\; 4)\; m\; -\; 9${{citation needed|date=February 2023}}

with roots $m\_j\; :\; j\; =\; 1,\; 2,\; 3\; ,$ that must be real and positive.

To derive the Koide relation, let $m\; \backslash equiv\; x^2$ and the resulting polynomial can be factored into

: $(\; 2\; x^3\; -\; 6\; n\; x^2\; +\; 3\; n^2x\; -\; 3\; )(\; 2\; x^3\; +\; 6\; n\; x^2\; +\; 3\; n^2\; x\; +\; 3\; )$

or

: $4\; (\; x^3\; -\; 3\; n\; x^2\; +\; \backslash tfrac\{\; 3\; \}\{\; 2\; \}\; n^2x\; -\; \backslash tfrac\{\; 3\; \}\{\; 2\; \}\; )(\; x^3\; +\; 3\; n\; x^2\; +\; \backslash tfrac\{\; 3\; \}\{\; 2\; \}\; n^2\; x\; +\; \backslash tfrac\{\; 3\; \}\{\; 2\; \}\; )$

The elementary symmetric polynomials of the roots must reproduce the corresponding coefficients from the polynomial that they solve, so $~~\; x\_1\; +\; x\_2\; +\; x\_3\; =\; \backslash pm\; 3\; n\; ~~$ and $~~\; x\_1\; x\_2\; +\; x\_2\; x\_3\; +\; x\_3\; x\_1\; =\; +\; \backslash tfrac\{\; 3\; \}\{\; 2\; \}\; n^2\; ~.$ Taking the ratio of these symmetric polynomials, but squaring the first so we divide out the unknown parameter $n\; ,$ we get a Koide-type formula: Regardless of the value of $n\; ,$ the solutions to the cubic equation for $x$ must satisfy

: $\backslash frac\{\; 2\; (\; x\_1\; x\_2\; +\; x\_2\; x\_3\; +\; x\_3\; x\_1)\; \}\{~\; (\; x\_1\; +\; x\_2\; +\; x\_3\; )^2\; \}\; =\; \backslash frac\{\; (3\; n^2)\; \}\{~\; (\backslash pm\; 3\; n)^2\; \}\; =\; \backslash frac\{\; 1\; \}\{\; 3\; \}$

so

: $1\; -\; \backslash frac\{\; 2\; x\_1\; x\_2\; +\; 2\; x\_2\; x\_3\; +\; 2\; x\_3\; x\_1\; \}\{\; (\; x\_1\; +\; x\_2\; +\; x\_3\; )^2\; \}\; =\; 1\; -\; \backslash frac\{\; 1\; \}\{\; 3\; \}\; =\; \backslash frac\{\; 2\; \}\{\; 3\; \}\; ~.$

and

: $1\; -\; \backslash frac\{\; 2\; x\_1\; x\_2\; +\; 2\; x\_2\; x\_3\; +\; 2\; x\_3\; x\_1\; \}\{~\; (\; x\_1\; +\; x\_2\; +\; x\_3\; )^2\; \}\; =\; \backslash frac\{\; (\; x\_1\; +\; x\_2\; +\; x\_3\; )^2\; -\; 2\; x\_1\; x\_2\; -\; 2\; x\_2\; x\_3\; -\; 2\; x\_3\; x\_1\; \}\{~\; (\; x\_1\; +\; x\_2\; +\; x\_3\; )^2\; \}\; =\; \backslash frac\{\; x\_1^2\; +\; x\_2^2\; +\; x\_3^2\; \}\{~\; (\; x\_1\; +\; x\_2\; +\; x\_3\; )^2\; \}\; .$

Converting back to $\backslash sqrt\{\; m\; \}\; =\; x$

: $\backslash frac\{\; m\_1\; +\; m\_2\; +\; m\_3\; \}\{\; \backslash left(\; \backslash sqrt\{\; m\_1\; \}\; +\; \backslash sqrt\{\; m\_2\; \}\; +\; \backslash sqrt\{\; m\_3\; \}\; \backslash right)^2\; \}\; =\; \backslash frac\{\; 2\; \}\{\; 3\; \}\; .$

For the relativistic case, Goffinet's dissertation presented a similar method to build a polynomial with only even powers of $m\; .$

Koide proposed that an explanation for the formula could be a Higgs particle with $\backslash mathrm\{U\}(3)$ flavour charge $\backslash Phi^\{a\backslash overline\{b\}\}$ given by:

: $V(\backslash Phi)\; =\; \backslash left[\; 2\; \backslash left[tr(\backslash Phi)\backslash right]^2\; -\; 3\; tr(\backslash Phi^2)\; \backslash right]^2$

with the charged lepton mass terms given by $\backslash overline\{\backslash psi\}\; \backslash Phi^2\; \backslash psi\; .${{cite journal |last=Koide |first=Yoshio |year=1990 |title=Charged lepton mass sum rule from U(3) family Higgs potential model |journal=Modern Physics Letters A |volume=5 |issue=28 |pages=2319–2324 |doi=10.1142/S0217732390002663 |bibcode=1990MPLA....5.2319K |url=https://inspirehep.net/literature/285940}} Such a potential is minimised when the masses fit the Koide formula. Minimising does not give the mass scale, which would have to be given by additional terms of the potential, so the Koide formula might indicate existence of additional scalar particles beyond the Standard Model's Higgs boson.

In fact one such Higgs potential would be precisely $V(\backslash Phi)\; =\; \backslash det[(\backslash Phi-\backslash sqrt\{m\_\backslash text\{e\}\})]^2\; +\; \backslash det[(\backslash Phi-\backslash sqrt\{m\_\backslash mu\})]^2\; +\; \backslash det[(\backslash Phi-\backslash sqrt\{m\_\backslash tau\})]^2$ which when expanded out the determinant in terms of traces would simplify using the Koide relations.

{{notelist}}

{{columns-list|colwidth=15em|

• Anomalous magnetic dipole moment
• CKM matrix
• Clifford algebra
• Generation
• Higgs mechanism
• Alternatives to the Standard Higgs Model
• PMNS matrix
• Quark–lepton complementarity
• Seesaw mechanism
• Technicolor

}}

{{reflist|25em}}

• {{cite journal

|last=Koide |first=Y.

|year=1983

|title=New view of quark and lepton mass hierarchy

|journal=Physical Review D

|volume=28 |issue=1 |pages=252–254

|bibcode = 1983PhRvD..28..252K

|doi=10.1103/PhysRevD.28.252

}}

:* {{cite journal

|last1=Koide |first1=Y.

|year=1984

|title=Erratum: New view of quark and lepton mass hierarchy

|journal=Physical Review D

|volume=29 |issue=7 |pages=1544

|bibcode = 1984PhRvD..29Q1544K

|doi=10.1103/PhysRevD.29.1544

|doi-access=free

}}

• {{cite book

|last1=Oneda |first1=S.

|last2=Koide |first2=Y.

|title=Asymptotic symmetry and its implication in elementary particle physics

|year=1991

|publisher=World Scientific

|url=https://books.google.com/books?isbn=9810204981

|isbn=978-981-02-0498-3

}}

• {{cite journal

|last=Koide

|first=Y.

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|title=Quark and Lepton Mass Matrices with a Cyclic Permutation Invariant Form

|url=https://core.ac.uk/download/pdf/25285108.pdf

|journal=Physics

|arxiv=hep-ph/0005137

|bibcode=2000hep.ph....5137K

|access-date=2021-08-26

|archive-date=2022-10-23

|archive-url=https://web.archive.org/web/20221023043219/https://core.ac.uk/download/pdf/25285108.pdf

|url-status=dead

}}

• {{cite arXiv

|last=Koide |first=Y.

|year=2005

|title=Challenge to the mystery of the charged lepton mass

|eprint=hep-ph/0506247

}}

• {{cite journal

|last1=Li |first1=N.

|last2=Ma |first2=B.-Q.

|year=2006

|title=Energy scale independence for quark and lepton masses

|arxiv=hep-ph/0601031

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|volume=73

|issue=1

|pages=013009

|journal=Physical Review D

|bibcode=2006PhRvD..73a3009L|s2cid=2624370

}}

• {{cite journal

|last=Brannen |first=C.

|year=2010

|title=Spin Path Integrals and Generations

|url=https://www.brannenworks.com/Gravity/spinpath.pdf

|journal=Foundations of Physics

|volume=40 |issue= 11|pages=1681–1699

|arxiv=1006.3114

|bibcode=2010FoPh...40.1681B

|doi=10.1007/s10701-010-9465-8

|citeseerx=10.1.1.749.3756

|s2cid=11007648

}} (See the article's references links to "The lepton masses" and "Recent results from the MINOS experiment".)

• {{Commons category-inline}}
• [https://www.wolframalpha.com/input?i=solve++%280.510998910+%2B+105.658369+%2B+m+%29%2F%28+sqrt+0.510998910+%2B+sqrt+105.658369+%2B+sqrt+m+%29%5E2+%3D++2%2F3 Wolfram Alpha], link solves for the predicted tau mass from the Koide formula.

Category:Leptons

Category:Unsolved problems in physics

Category:Empirical laws

Category:1980s in science