Talk:Koide formula

Unpublished developments

 * Please read No original research. You are misunderstanding how Wikipedia works. --mfb (talk) 14:23, 21 March 2015 (UTC)


 * Hmm, the argument is that if it would not ever go to the main page, it does not deserve even a discussion in Talk. I tend to agree, but the problem I see is that the state-of-the-art of this topic will most probably include always some "sourced but unpublished" observations and that visitors of the page are very likely to rediscover them, so perhaps Talk can act as a kind of buffer, citing the -usually web- sources of the unpublished observations. Arivero (talk) 18:39, 21 March 2015 (UTC)

Some idea from symmetry
This is a thing I have mentioned in a thread of physicsforums which collects most of the speculation on the topic and also in some talks, but I am afraid it is not going to be published in journal literature.

Some mechanism should give mass first to an unbroken Pati-Salam or GUT group, with a mass for each of three generations that we can call

M3 = e up bottom M2= tau charm down M1= mu top strange

If the masses of the this unified system agree with Koide equation, K(M1,M2,M3)=0, we have really nine Koide equations: K(tau,mu,e)=0 K(up,charm,top)=0, K(up,charm,strange)=0, K(up,down,top)=0, K(up,down,strange)=0 K(bottom,charm,top)=0, K(bottom,charm,strange)=0, K(bottom,down,top)=0, K(bottom,down,strange)=0

Then something breaks the unification group in a way that not all the Koide equations are broken. The ones in bold somehow survive, and link all the quarks at the cost of recovering the old idea of an initial zero mass for the up quark. Arivero (talk) 19:00, 21 March 2015 (UTC)

Mass Predictions
The formula allows one to estimate the masses of a hypothetical fourth generation of leptons. It gives a mass of 43GeV for a 4th generation lepton (yet this is ruled out otherwise it would be produced in W boson decay which have a mass around 90GeV, unless the fourth generation neutrino is very heavy), and quark masses of 3.5TeV and 84TeV for down and up-type quarks. Continuing in this way one can have a maximum of 9 generations below the Plank Mass scale. The sequence settles into one where each mass is 22.956 $$=(23+5 \sqrt{21} )/2$$ larger than the last. — Preceding unsigned comment added by 87.216.217.210 (talk) 02:22, 21 March 2015 (UTC)

Another Interesting Coincidence?
Consider an xyz Cartesian coordinate system and the vectors (1, 1, 0), (1, 0, 1) and (0, 1, 1), the projections of the vector (1, 1, 1) onto the xy, xz, and yz planes respectively. Modify the projection vectors so that they have the lengths, respectively, that correspond to the masses of the electron, muon, and tau particle in units of eV/c2. Applying the pythagorean theorem to each of the projection vectors, we have the system of equations

x2 + y2 = (0.000511)2, x2 + z2 = (0.1057)2, y2 + z2 = (1.777)2.

The solution to this system is given by	x2 = -1.5732, y2 = 1.57327, z2 = 1.58445.

The input data only has three place accuracy, so the solutions can not be relied upon beyond three significant figures. But it is interesting (and perhaps only coincidental) that these values cluster about the value of pi/2 = 1.57079... . Samdhatte (talk) 00:14, 19 February 2008 (UTC)

Yep, the one you have found is, it seems to me, other way to arrive to the observation by R. Foot about how to reinterpret Koide formula in terms of a vector 45 degrees away from 1,1,1. 83.138.204.146 (talk) 02:24, 23 February 2008 (UTC)

No, what you've done is contrary to the spirit of Koide's formula (in that it deals with squares of masses instead of square roots of masses) and is also not very accurate.

Koide's formula is accurate to all digits in the tau mass. Currently the tau measurement is 1776.84(.17), see the Particle Data Group website for 2008 data. Koide's estimate for the tau was 1776.96, last time I checked www.brannenworks.com/MASSES2.pdf, which was with the previous PDG numbers for electron and muon, but they don't change much. By the way, this is far better than the 3-digit accuracy implied in the other comment. Also, I rewrote Koide's formula in eigenvector form (as explained in the above link), and extended it to the neutrinos. For a published reference to the neutrino mass formula, see Koide's article on my extension to the neutrinos arxiv.org/abs/hep-ph/0605074. The eigenvalue form for Koide's formula (for the electron, muon, and tau) is:

$$\sqrt{m_n/313.85773 MeV} = (1 + \sqrt{2}\cos(2/9 + 2n \pi/3))$$,

where m_n is the electron, muon, and tau, for n=1,2,3. It is accurate to around 5 digits or so. It is not quite as accurate as Koide's formula because the lepton masses require the "2/9" to be made slightly smaller (to instead be about about 0.222222047 if I recall). This formula is in the peer reviewed literature, see www.worldscinet.com/mpla/22/2204/S0217732307022621.html.

Carl Brannen 64.184.170.248 (talk) 10:39, 8 November 2008 (UTC)

Actual Result
With 3 significant digits.

$$Q = \frac{m_e + m_{\mu} + m_{\tau}}{(\sqrt{m_e}+\sqrt{m_{\mu}}+\sqrt{m_{\tau}})^2}$$

$$Q = \frac{0.511\ \rm{MeV}/c^2 + 105.7\ \rm{MeV}/c^2 + 1777\ \rm{MeV}/c^2}{(\sqrt{0.511\ \rm{MeV}/c^2}+\sqrt{105.7\ \rm{MeV}/c^2}+\sqrt{1777\ \rm{MeV}/c^2})^2}$$

$$Q = \frac{1.88 \times 10^3\ \rm{MeV}/c^2}{(7.15 \times 10^1\ \sqrt{\rm{MeV}/c^2} + 1.028 \times 10^2\  \sqrt{\rm{MeV}/c^2} + 4.215 \times 10^2\  \sqrt{\rm{MeV}/c^2})^2}$$

$$Q = \frac{1.88 \times 10^3\ \rm{MeV}/c^2}{(5.32 \times 10^1\ \sqrt{\rm{MeV}/c^2})^2}$$

$$Q = \frac{1.88 \times 10^3\ \rm{MeV}/c^2}{(5.32 \times 10^1\ \sqrt{\rm{MeV}/c^2})^2}$$

$$Q = \frac{1.88 \times 10^3\ \rm{MeV}/c^2}{2.83 \times 10^3\ \rm{MeV}/c^2}$$

$$Q = 6.64 \times 10^-1 $$
 * If you had used up to date figures for the masses, or if you had any idea how to use significant digits correctly, this might be interesting. 99.48.75.121 (talk) 04:47, 13 January 2011 (UTC)

Justo to clarify: this comment from user 99.48.75.121 assumes that the mass of the tau is known only as about 1777(1) instead of 1776.84(17), and similarly for the other masses, and then applies an engineer rule of thumb about translating the error about an order of magnitude when operating, then coming to 0.664(1) instead of 0.666659(10). It just proofs that the engineers rule is good enough for the usual work. — Preceding unsigned comment added by 155.210.136.50 (talk) 18:46, 28 January 2012 (UTC)

other references online
Some interesting discussion has been generated in this period, but it is still too especulative to add to the main entry. Still, the interested reader should pursue the following blog posts:


 * http://www.science20.com/quantum_diaries_survivor/alejandro_rivero_fermion_mass_coincidences_and_other_fun_ideas-85187
 * http://dispatchesfromturtleisland.blogspot.com/2011/12/more-fun-with-koides-formula.html
 * http://motls.blogspot.com/2012/01/could-koide-formula-be-real.html
 * http://es.scribd.com/doc/157932274/Koide-equations-for-quark-mass-triplets
 * https://johncarlosbaez.wordpress.com/2021/04/04/the-koide-formula/ — Preceding unsigned comment added by Arivero (talk • contribs) 00:49, 18 June 2023 (UTC)

Known but unusual triplets
If we accept negative sign for the square root, then also


 * $$Q = \frac{m_C + m_{B} + m_{S}}{(\sqrt{m_C}+\sqrt{m_{B}}-\sqrt{m_{S}})^2} \approx 0.675 \approx \frac{2}{3}.$$

I am the "discoverer", afaik, of this triplet, and there is a reference in the arxiv, but it is not referenced in the published literature, so I am afraid that detailing the triplet in the main page should be self promotion Arivero (talk) 18:51, 21 March 2015 (UTC)

Also, assuming that the source of mass of the up quark is different of the rest, we put the mass of the up quark to zero, then there is a match


 * $$Q = \left. \frac{m_C + m_u + m_{S}}{(\sqrt{m_C}+\sqrt{m_u}+\sqrt{m_{S}})^2} \right|_{m_u\approx 0} \approx 0.663 \approx \frac{2}{3}.$$

Same that the previous case, I found it empirically and then notice that it amounts to set a very small or negligible mass for the up quark. Such mass is suggested in other published studies as an argument to solve the theta-problem of QCD, but even if this is accepted, the equation itself is not in the published literature so I left it here Arivero (talk) 18:51, 21 March 2015 (UTC)

Due to the proximity of pion mass and strange mass, the above two equations happen to work also when replacing quarks (s,c,b) by mesons Pi0, D0 and B0. — Preceding unsigned comment added by 217.165.114.218 (talk) 13:50, 25 April 2015 (UTC)

Also, with the same assumption of up quark to zero, the equation works for strange and down, a bit worse than for strange and charm:


 * $$Q = \left. \frac{m_D + m_{S}}{(\sqrt{m_D}+\sqrt{m_{S}})^2} \right|_{m_u\approx 0} \approx 0.702 \approx \frac{2}{3}.$$

In this case, simple algebraic manipulation reduces it to a form that was mentioned in the literature. I feel that it can be mentioned as a "koide triplet" in the main text but I do not feel that it should be formulated in Koide form, as it should be to put words not in the original work. So I keep mentioning Harari et al but not writing the formula in the main text. Arivero (talk)

Also with up quark equal or near zero, a crossed Koide tuple can be built with electron, up and down. This was noticed by A.W in his blog dispatchesfromturtleisland Arivero (talk) 22:51, 25 April 2015 (UTC)

Also this unsourced formula has been proposed here: 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 $1/2$ < Q < 1, although the masses of these quarks are known less accurately.

Disputed "prediction" of τ mass
I put in a "disputed" template to the incomprehensible statement in the lede about a spurious "prediction" of the τ mass mass in an 1981 paper. The heavy lepton was discovered in the early 70s; and, by 1975, Perl had estimated its mass pretty well. I have no idea what the baffling assertion is meant to convey, nor where it actually came from. It has a strong whiff of fringe. Cuzkatzimhut (talk) 11:12, 11 April 2018 (UTC)
 * According to page 25 of Perl 1992, for many years the best measurement of the tau mass was a 1978 measurement of 1784 +2 -7 MeV. The next improved measurement, made in 1992, was 1776.9 +.5 -.5 MeV. Koide 1982 had predicted 1777 MeV. Mporter (talk) 06:02, 4 June 2018 (UTC)
 * Granted; however, the 28 March 2018 statement on "prediction" was highly misleading. In 1982, Koide had access to the DELCO 1784 measurement you cite. However, a mere quarter-century drift of the observational value (DELCO-BES)/DELCO by less than 0.4% closer to the subsequent Koide value does not quite meet the standards of "prediction" of the field. WP excessive statements of bogus revelation of this type have been systematically abused by crackpots perpetrating unsound numerological myths, all in the name of "hard facts" furnished by WP in its less than proud moments. Cuzkatzimhut (talk) 14:18, 4 June 2018 (UTC)
 * Uh, what? The prediction was 1 standard deviation below the measured value, but with a much smaller uncertainty as the electron and muon mass were known much more accurately. The method used to make this prediction is debatable, but the prediction was excellent and spot-on. --mfb (talk) 20:07, 4 June 2018 (UTC)
 * We seem to dissonate on the word "predicted", as used originally in the article. Koide was within a σ of the available observed value then and has an indisputably better fit now, 36 years thence.  "Predicted" is stretching the term. "Fits the mass of the τ well within error" is indisputable, but that's not what the abusers understand instead of glomming on an imagined history. Cuzkatzimhut (talk) 22:17, 4 June 2018 (UTC)
 * Tons of predictions in particle physics are like that, and they are always called theory predictions, even when they are made after initial measurements exist. They do not depend on the measured value, that is the important point. If you want to use "prediction" differently from everyone else, fine, but do that somewhere else. --mfb (talk) 00:35, 5 June 2018 (UTC)

One does not postdict a number through numerology, not theory, and then claim he predicted it. Koide did not. I and my coauthors stick to mainstream usage in the mainstream literature.Cuzkatzimhut (talk) 02:53, 5 June 2018 (UTC)
 * I don't know where you publish, but apparently not in particle physics, otherwise we wouldn't have this discussion. This is a particle physics topic. --mfb (talk) 05:53, 5 June 2018 (UTC)
 * Perhaps we have it because I've been refereeing and editing DPF papers for four decades? Cuzkatzimhut (talk) 13:44, 5 June 2018 (UTC)
 * Then I'm sure you refereed many papers that made "predictions" for things measured before. Based on the SM, based on SUSY, based on whatever. So what is the problem? --mfb (talk) 00:57, 6 June 2018 (UTC)

Koide formula in even numbers
The Koide formula in even numbers 0, 1, (7 + 4 * square root of 3 )

https://www.wolframalpha.com/input/?i=%28+0+%2B+1+%2B+a%29%2F%28+sqr+0+%2B+sqr+1+%2B+sqr+a%29%5E2+%3D+2%2F3

Maybe this has meaning to someone?

Answer to above unsigned: yes, the 0 mass Koide triplet was discovered before Koide, it is quoted here and in the main page, as it is a published result Arivero (talk) 11:18, 6 June 2023 (UTC)

Anonymous post from Newcastle
In 2013 and 2018 (https://en.wikipedia.org/w/index.php?title=Koide_formula&diff=prev&oldid=827675064) we got a couple suggestions from a contributor in Newcastle that while interesting, are unsourced. The main idea here is eigenvalues of some 3x3 matrix. The equations go as mass square roots, but of course they could be uplifted to 6th degree, surely such uplifting is buried somewhere in goffinet thesis. The recipe is obvious:


 * $$(a x^3 + b x^2 + c x + d)(a x^3 - b x^2 + c x - d) = a^2 x^6 + 2 a c x^4 - b^2 x^4 - 2 b d x^2 + c^2 x^2 - d^2$$

and then the condition $$b^2=6ac$$

Ratio Factor
Each triple of Koide masses is defined by a mass scale and another variable $$\gamma$$ given by:


 * $$\frac{ \sqrt{e}^3+\sqrt{\mu}^3+\sqrt{\tau}^3}{(\sqrt{e}+\sqrt{\mu}+\sqrt{\tau})^3} = \frac{1}{2}\left(1+\gamma\right)$$

which determines along with the Koide relation the mass ratios $$e:\mu:\tau$$. When one of the masses is zero then $$\gamma=0$$. For the charged lepton triple we find $$\gamma\approx \frac{1}{81}$$.

A problem I have with this is that it does not emerge from a simpler integer relationship nor from a simpler formula. It reminds, and can be confused with, the extra correction Koide did in its original paper Arivero (talk) 00:09, 18 June 2023 (UTC)

As Solutions to a Cubic Equation
The following cubic equation's solutions fit the Koide formula for any value of $$n$$ when considered as square roots of masses up to scale:


 * $$2x^3-6x^2n+3xn^2-3=0$$

A value of $$n=3$$ fits the lepton triple, while a value of $$n=2$$ fits the (top, bottom, charm) triple. A value of $$n=2.5$$ fit's the (strange, down, up) triple only if we give the up-quark mass a value of 0.1MeV. Cubic equations usually arise in symmetry breaking when solving for the Higgs vacuum. This involves finding the eigenvalues of a 3x3 matrix.

NOTE that the author of this contribution seems to forget a dependence on n for the last term, that would justify the explanation and surely also the term 81 in the other contributed formula. — Preceding unsigned comment added by Arivero (talk • contribs) 15:35, 18 June 2023 (UTC) Arivero (talk) 11:20, 6 June 2023 (UTC)

Commented out unclear claim "$n = 3$ for charged leptons"
Someone appended the following statement to the end of the section (now at the article end) that derives the Koide relation from a putative characteristic polynomial. The statement is


 * For the charged lepton triple, the value of $$\ n\ $$ is extremely close to the integer $$\ 3 ~.$$

The claim does not check out: I inserted the known charged lepton masses into the first two elementary symmetric polynomials and divided them (no squaring). The result is
 * $$\ n = 100.213\ \mathsf{MeV} / c^2\ ,$$

which I certainly find interesting, but different from 3, even for very large values of 3. Due to the two elementary symmetric polynomials the above came from, is definitely correct (with MeV/c²) regardless of scaling in the polynomial, if the polynomial is correct. I believe that I appropriately carried out the naïve interpretation of what "$n = 3$" should mean, and it doesn't work in the units used throughout particle physics. I don't know of a system of units where 100 MeV/c² would come out as 3, but I didn't look far.

I'm reasonably confident that the units in the characteristic polynomial must have been scaled out in some fashion (demonstrated below). Without a careful description of how the rescaling was done, a statement, the statement $$\ n = 3\ $$ is vacuous: It can't mean anything without more information. Note that because the Koide formula works out to be scale-free, its derivation doesn't depend on units in the polynomial – only that the polynomial is actually valid, whatever the appropriate scale might be.

The rabbit-from-the-hat polynomial is
 * $$\ 4\ m^3 - 24\ n^2\ m^2 + 9\ n\ (n^3 - 4)\ m - 9\ $$

Casual inspection of it shows that if $m$ has units of MeV / c², then $n$ must have units proportional to $$\ \sqrt{\ \mathsf{MeV} / c\ }\ ;$$ the digit 4 that's subtracted from $$\ n^3\ $$ inside the polynomial's only parentheses, must have units proportional to $$\ (\mathsf{MeV})^{3/2} / c^3\ ,$$ and the final digit 9 must have units proportional to (MeV / c²)³. The integers 4 and 9 cannot of course really have units; clearly, if the characteristic polynomial is at all valid, units must have been scaled out.

I would guess that both $m$ and $n$ have been rescaled. So without discussing the units, the statement $$\ n = 3\ $$ is vacuous – it doesn't mean anything – and is a booby-trap for even a moderately informed reader who might plow into the section. For that reason, I commented it out.

Please note again: Because the Koide formula is scale-free, and the elementary polynomials produce appropriate units, the derivation doesn't depend on handling of units in the polynomial, only that the scaling was done properly and that it is a legitimate polynomial for lepton mass eigenvalues, for which there is a separate "citation needed". — 107.122.85.33 (talk) 05:13, 11 February 2024 (UTC)

Some Possibly Related Coincidences ?
I had various pieces of 'numerology' related to particle mass and the fine structure constant published in July 7th 1983 edition of Nature. Since that publication I have simplified the various relationships into the following....which I find quite remarkable even if it is coincidence...

The following is the mathematical ‘operation’ table for positive times negative integers

For the range 0 to 2 for positive numbers and 0 to –2 for negative numbers.

Now take the values in the above table and use them as powers to which pi is raised

( i.e pi ^ -1  is pi to the power of minus 1)

The fine structure constant 137.03604 happens to be very close to (4 * pi ^ 3) + pi ^ 2 + pi, or 137.03599

This happens to be  pi ^ 3 times the sum of the values in columns A and B ( or rows X and Y ) in the table.

If we take the proton mass to be pi ^ 2, then the sum of the values in column B ( or row Y ) gives you the neutral pion mass to extremely close accuracy, and the sum of the values in column C ( or row Z ) gives the Muon mass very closely. The overall accuracy here is remarkable !

Consider this……that a simple operation on a multiplication operator table gives a simple matrix that provides a remarkably accurate relationship between fine structure constant, muon mass, neutral pion mass,

and proton mass. ( The proton mass does not have to be pi ^ 2 but can clearly be scaled up or down as any power of pi along with the matrix itself being scaled up or down by the same power….so all the relationships can be defined as between integral powers of pi .)

Fine structure constant                                   = 137.03600                         Above table gives 137.036303

Ratio of proton to neutral pion mass              = 6.95230                              Above table gives 6.95223189

Ratio of proton to muon mass                         = 8.87678                              Above table gives  8.87883898

If this is a coincidence, then it is a remarkable one. Most pieces of ‘numerology’ simply provide single relationships…..such as that the proton electron mass ratio is close to 6 * pi ^ 5 ( six times pi to the power 5 ), which has been known for years. Other than Bode’s law for the planets, no other simple ‘numerological’ formula gives such a remarkable relationship between four fundamental constants of nature.

I might expect, by pure chance, a rough proximity to a few particle masses or constants. But what convinces me that the above very likely does have significance is its amazing overall accuracy and simplicity. It is the combination of those two factors that is most striking. Most ‘numerology’ in this arena is convoluted, includes a lot of arbitrary factors to ‘tweak’ the results, and contains precious little in the way of any patterns.

This ‘matrix’ was first published in July 1983 in Nature ( not in the above form )…….along with an editorial comment that recognised the remarkable nature of the relationships but regarded the matter as ‘numerology’. Whilst I’m quite happy to accept criticism that there’s no explanation or theory as to why such a relationship occurs, I find it extremely hard to accept that the above can lightly be dismissed as just ‘coincidence’. Science is the search for patterns between data, and here you have a pattern that cries out for an answer. Why should four fundamental constants just happen to have such a close fit to so simple a symmetry ? Peter Stanbury (talk) 16:45, 18 March 2024 (UTC)