# What is new with Koide sum rules?

• arivero
Lets quote here also the paper Mitchell mentioned elsewhere, https://arxiv.org/abs/1903.00160, as it could be useful to anyone looking for Koide-like relationships in the diquark sector.

ohwilleke
A paper by Goldman and Stephenson today, promotes the idea that the standard model mass matrices can be obtained by "democratic" yukawa couplings that all have the same value, plus small perturbations.

The reason is as follows. Suppose we have a 3x3 matrix in which all matrix entries are the same (e.g. they could all be equal to 1). You can diagonalize this matrix, by multiplying by a "tribimaximal" matrix. The resulting matrix will be diag(m,0,0) for some m. But for quarks and charged leptons, we have that the third generation is much more massive than the first two. So in all cases, the mass matrix can be approximated by a matrix of the form diag(m,0,0).

Goldman and Stephenson perform an inverse tribimaximal transformation on the quark mass matrices in order to show just how close to democratic they are (eqn 6 and 7), and they show that, for a particular parametrization, the deviations from democracy are small (equation 11)... the largest of these perturbations is still just .02, so if a model can be found, it can be analyzed perturbatively. They proposed in a previous paper that these perturbations might arise from interactions with dark-matter sterile neutrinos, but they don't provide a model. The other potentially significant thing they observe, is that some of the perturbation parameters need to be complex, so they propose that this is where CP violation comes from (section IV B).

They call their idea Higgs Universality, since the idea is that to a first approximation, the coupling of all fermions to the Higgs is the same.

They don't present a model. However, I will point out that recent work by Koide and Nishiura (mentioned, e.g., at #141 in this thread) to some extent is such a model. Koide and Nishiura have a universal ansatz for the mass matrices, which involves contributions from the democratic matrix, the unit matrix, and a matrix diag(√e,√μ,√τ). Ironically, however, for the charged leptons, the contribution from the democratic matrix is zero. This is ironic, not only because Goldman and Stephenson assert (calculations promised for a future paper) that the charged lepton masses can also be obtained by a small perturbation of a democratic matrix; but Koide himself obtained them that way, in earlier work!

If I look at the history of Koide's attempts to explain his own formula, I see three kinds of model. First, the preon model where he first obtained it. Second, the democratic model. Third, the perturbed democratic model with Nishiura. It is my understanding that @arivero's sbootstrap was partly inspired by the preon model, perhaps because some of the preons can be paired up in a fashion reminiscent of quark-diquark supersymmetry. (This should be compared with Risto Raitio's approach to supersymmetric preons.) It would be intriguing if one could close the circle of Koide's models, and obtain the "perturbed democratic model" by having democratically interacting preons mix with their own composites - the latter providing the "√e,√μ,√τ" perturbation.

Speaking of supersymmetry, the study of the supermathematics of Grassmann, Berezin, etc, has given me a new perspective on where the problematic phase of 2/9, discovered by @CarlB, could come from (see e.g. #173 in this thread). Phases that are rational multiples of π are much more natural. I had previously noticed that the well-known expansion of π/4 as 1 - 1/3 + 1/5 - ... contains a 2/3 in its first two terms, so if the analogous expansion for π/12 were somehow truncated there, one could obtain 2/9. The only problem was that I couldn't think of a good reason for such a truncation. One just had to construct a model with a π/12 phase and hope, perhaps, that it approximated Carl's ansatz well enough.

However - that expansion can be obtained as a Taylor series in x, for x=1. Meanwhile, for a grassmann number θ, θ^2 (and all higher powers) equals zero, because of anticommutativity: ab=-ba, so θ.θ = -θ.θ = 0. So, what if you took a Taylor series for x=1, and superanalytically continued it to x=θ...? All powers of x equal to x^2 or higher, will drop out. Unfortunately, 1/3 or 1/9 doesn't naturally show up as the coefficient of x, but rather as the coefficient of x^3, and I haven't thought of a sensible way to associate it with x^1.

arivero and ohwilleke
A paper by Goldman and Stephenson today, promotes the idea that the standard model mass matrices can be obtained by "democratic" yukawa couplings that all have the same value, plus small perturbations.

The reason is as follows. Suppose we have a 3x3 matrix in which all matrix entries are the same (e.g. they could all be equal to 1). You can diagonalize this matrix, by multiplying by a "tribimaximal" matrix. The resulting matrix will be diag(m,0,0) for some m. But for quarks and charged leptons, we have that the third generation is much more massive than the first two. So in all cases, the mass matrix can be approximated by a matrix of the form diag(m,0,0).

Goldman and Stephenson perform an inverse tribimaximal transformation on the quark mass matrices in order to show just how close to democratic they are (eqn 6 and 7), and they show that, for a particular parametrization, the deviations from democracy are small (equation 11)... the largest of these perturbations is still just .02, so if a model can be found, it can be analyzed perturbatively. They proposed in a previous paper that these perturbations might arise from interactions with dark-matter sterile neutrinos, but they don't provide a model. The other potentially significant thing they observe, is that some of the perturbation parameters need to be complex, so they propose that this is where CP violation comes from (section IV B).

They call their idea Higgs Universality, since the idea is that to a first approximation, the coupling of all fermions to the Higgs is the same.

They don't present a model. However, I will point out that recent work by Koide and Nishiura (mentioned, e.g., at #141 in this thread) to some extent is such a model. Koide and Nishiura have a universal ansatz for the mass matrices, which involves contributions from the democratic matrix, the unit matrix, and a matrix diag(√e,√μ,√τ). Ironically, however, for the charged leptons, the contribution from the democratic matrix is zero. This is ironic, not only because Goldman and Stephenson assert (calculations promised for a future paper) that the charged lepton masses can also be obtained by a small perturbation of a democratic matrix; but Koide himself obtained them that way, in earlier work!

If I look at the history of Koide's attempts to explain his own formula, I see three kinds of model. First, the preon model where he first obtained it. Second, the democratic model. Third, the perturbed democratic model with Nishiura. It is my understanding that @arivero's sbootstrap was partly inspired by the preon model, perhaps because some of the preons can be paired up in a fashion reminiscent of quark-diquark supersymmetry. (This should be compared with Risto Raitio's approach to supersymmetric preons.) It would be intriguing if one could close the circle of Koide's models, and obtain the "perturbed democratic model" by having democratically interacting preons mix with their own composites - the latter providing the "√e,√μ,√τ" perturbation.

Speaking of supersymmetry, the study of the supermathematics of Grassmann, Berezin, etc, has given me a new perspective on where the problematic phase of 2/9, discovered by @CarlB, could come from (see e.g. #173 in this thread). Phases that are rational multiples of π are much more natural. I had previously noticed that the well-known expansion of π/4 as 1 - 1/3 + 1/5 - ... contains a 2/3 in its first two terms, so if the analogous expansion for π/12 were somehow truncated there, one could obtain 2/9. The only problem was that I couldn't think of a good reason for such a truncation. One just had to construct a model with a π/12 phase and hope, perhaps, that it approximated Carl's ansatz well enough.

However - that expansion can be obtained as a Taylor series in x, for x=1. Meanwhile, for a grassmann number θ, θ^2 (and all higher powers) equals zero, because of anticommutativity: ab=-ba, so θ.θ = -θ.θ = 0. So, what if you took a Taylor series for x=1, and superanalytically continued it to x=θ...? All powers of x equal to x^2 or higher, will drop out. Unfortunately, 1/3 or 1/9 doesn't naturally show up as the coefficient of x, but rather as the coefficient of x^3, and I haven't thought of a sensible way to associate it with x^1.

FWIW, my hypothesis is that Koide-type relationships and the mass hierarchy general arises because (1) the CKM matrix is logically prior to the mass matrix, and (2) the mass matrix represents a dynamic balancing of the mass of each particle of one type, with each of the particles it could transition to via the W boson, adjusted for transition probabilities, in a simultaneous equation that covers and balances all transitions at once.

Charged lepton transitions are approximately democratic, because the neutrino contribution is so small. The CKM matrix is predominantly one factor equal to the likelihood of a first to second generation transition and a second factor equal to a second to third generation transition, with the probability of a first to third generation transition equal to the product of the two probabilities. CP violation and differences between up-type and down-type quarks in transition probabilities are second or lower order effects.

Here is a nifty new little paper:

Phenomenological formula for CKM matrix and physical interpretation
Kohzo Nishida
(Submitted on 3 Aug 2017)
We propose a phenomenological formula relating the Cabibbo--Kobayashi--Masukawa matrix VCKM and quark masses in the form (md‾‾‾√ms‾‾‾√mb‾‾‾√)∝(mu‾‾‾√mc‾‾‾√mt‾‾‾√)VCKM. The results of the proposed formula are in agreement with the experimental data. Under the constraint of the formula, we show that the invariant amplitude of the charged current weak interactions is maximized.
Subjects: High Energy Physics - Phenomenology (hep-ph)
Cite as: arXiv:1708.01110 [hep-ph]
(or arXiv:1708.01110v1 [hep-ph] for this version)

Has anyone checked this with the square root neutrino masses, one of which is negative? If not, I'm inclined to do it myself.

His formula seems very natural; he's proposing that V is a unitary matrix that takes a real unit vector to a real unit vector. Just what you expect a unitary matrix to do.

The next thing to note is that V is taking a real vector to a real vector. The usual definition of the CKM and MNS matrices allows one to multiply any row or column by a complex phase; this doesn't change the matrix. But the restriction that the matrix take this real vector to that real vector defines these arbitrary complex phases. For example, if you multiply any row of V by a complex phase you can see that his formula will be broken as it won't preserve real vectors.

It is a fact that any unitary matrix can be put, (typically in 4 ways for 3x3 unitary matrices, see equation (56) of http://vixra.org/pdf/1511.0083v1.pdf for the CKM matrix) into a form where the complex phases are defined by requiring that all rows and columns sum to 1, called "magic" in the literature in reference to magic squares. This is an additional requirement to the fact that the sum of the absolute squares of a row or column of a unitary matrix are one.

That any 3x3 unitary matrix can be put into magic form was proved by Gibbs: http://vixra.org/abs/0907.0002 When one puts a unitary matrix into that form, the vector (1,1,1) becomes an eigenvector with eigenvalue 1. My intuition suggests that these are related problems. And that also implies to me that there will be four solutions to these sorts of problems (for a typical random unitary matrix, and an infinite number of solutions for special cases).

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arivero and ohwilleke
Has anyone checked this with the square root neutrino masses, one of which is negative? If not, I'm inclined to do it myself.

I just realized a fairly large problem with how I was thinking of this. For the 2x2 case, his formula is providing two complex equations (i.e. real equations which imply that the imaginary part is zero) which is 4 real restrictions. That happens to match the number of real degrees of freedom in a 2x2 unitary matrix so it determines the answer.

But for the 3x3 case he's only providing three complex equations which gives six restrictions but 3x3 unitary matrices have 9 degrees of freedom. Five of those nine are arbitrary complex phases multiplying rows or columns and the other four determine the probabilities. So he's got enough to decide all the complex phases (which look to be restricted indeed by the real nature of the vectors) and one of the 4 restrictions on probabilities. Rereading the article, he notes this saying "Our basic postulate is to interpret one of the unitary matrices,V" where the emphasis could be on "one of".

In calculating V, he uses a standard parameterization for the unitary matrices which I think are quite ugly when compared to mine.

I have updated the comparison of the extended Koide's formula for quarks of @arivero with updated data including the FLAG 19 report quark masses and the PDG 2018 data to see how this looks relative to when this was originally proposed in 2011. I have also updated the comparison of this new data to the LC & P hypothesis.

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arivero and CarlB
On Dr Koide's personal website, there is a July 6 update in Japanese (you may need to switch your browser's encoding to see it properly), in which he expresses regret for not updating the website more often, and says that he has been in poor health for the past two years. Physics research is the "driving force" of his life, he seems to have received a new research grant, and he links to a copy of a recent research report (again in Japanese) summarizing his latest papers.

In this thread we have regularly entertained various generalizations of Koide's formula, and explanations of varying eccentricity. But let me try to say something about it from a "high orthodox" perspective. I compare it to Balmer's formula for the hydrogen spectrum. Today we can understand that formula in terms of quantum mechanics; but it's the particle masses, in particular, which lack explanation.

Despite a growth in pop-science skepticism about it, string theory is still the most promising framework for explaining the unexplained parameters of the standard model (I did say this would be taking an orthodox perspective!), e.g. as arising from the compactification geometry in various ways. Since string phenomenologists are always looking for ways to narrow the range of vacua that they need to consider, Koide's formula could be an excellent clue.

However, there are reasons why it is neglected. Most importantly, it is an exact relation among pole masses, whereas the renormalization group leads us to expect exact relations only among running masses at high scales. This reason for neglect is independent of string theory, it represents the "common sense" of quantum field theorists.

Nonetheless, this is not the final word. An infrared fixed point can impose relations among infrared quantities. Also - this is not as well understood, but may be significant - when quantum gravity is taken into account, there can be unusual relations between UV and IR of a quantum field theory. And in general, QFT still contains many hidden complexities. One might therefore take the attitude that the counterintuitive nature of Koide's formula - counterintuitive with respect to field theorist's common sense - is a further clue, about what needs to be investigated. One should directly investigate what would have to be true, for a theory to exhibit just this kind of unlikely or impossible-seeming infrared relationship.

(And fortunately, some of that necessary work may already have been done by Yukinari Sumino. Koide himself has also written a few papers approaching his formula from the perspective that the exactness for pole masses is just a coincidence, and that its approximate validity for the running masses is the real fact to be explained; so that kind of approach is possible too.)

Another thing which I think may hinder a successful investigation of Koide's formula by orthodoxy - and here I begin to deviate from orthodoxy, but in ways that many others have also begun to do - is the common assumption of naturalness of masses, achieved by TeV-scale supersymmetry. Such an assumption is not logically incompatible with Koide's formula - Koide has written many papers in which a supersymmetric framework is assumed. But usually (though not always) it doesn't add much to the explanation, it is instead there because he is a competent particle physicist and knows that it is a good and useful theoretical framework.

Nonetheless, the LHC results appear to be telling us that the world works in a different way. No partners (super or otherwise) have been found, and the Higgs mass is close to critical. So new paradigms like the relaxion (and, in string theory, nonsupersymmetric phenomenology) are slowly growing in popularity. Combining this with the peculiar infrared exactness of the Koide formula, suggests to me that one should be trying to explain the formula, in the context of some such new paradigm. Of course this makes life harder for the serious theorist; but it is a logical conclusion.

One might therefore take the attitude that the counterintuitive nature of Koide's formula - counterintuitive with respect to field theorist's common sense - is a further clue, about what needs to be investigated. One should directly investigate what would have to be true, for a theory to exhibit just this kind of unlikely or impossible-seeming infrared relationship.

Yes. It's a bit like MOND. It may be a phenomenological relationship not grounded in theory, but any theory has to reproduce it because it compactly describes the evidence.

Nonetheless, the LHC results appear to be telling us that the world works in a different way.

What an extraordinarily delicate way to express that sentiment.

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Koide and Nishiura have posted another installment of their U(3)xU(3)' universal seesaw model (in this thread, see posts #156, #182 for previous papers). But that is a technical refinement in one particular, rigorously defined BSM theory of theirs.

I find much more intriguing, the updated version of a 2018 paper (here is version 1) that Koide posted at the same time. Version 1 is a quick introduction to Koide's relation, the pole mass problem, Sumino's solution, and the amended Sumino models that Koide has considered.

Version 2 is a completely new paper, "Are the Charged Leptons in the Simultaneous Eigenstates of Mass and Family?" This raises an issue also posed by Goffinet in a 2008 thesis that has also been discussed in this thread - more on that in a moment.

But first, see the beginning of this version-2 paper. Koide starts by discussing how the quark, CKM mixing matrix arises as a product of unitary matrices that diagonalize the SM's quark yukawa matrices (see PDG for more details). The (P)MNS mixing matrix for leptons can be similarly decomposed, as a product of U_nu associated with neutrinos and U_e associated with charged leptons; and in a Sumino-type model with a U(3) family symmetry, asking "Are the charged leptons in the simultaneous eigenstates of mass and family?", means asking whether U_e equals the identity matrix. Koide deduces some observable consequences if it does and if it does not.

Meanwhile, Goffinet 2008 introduced a concept of "pseudo-mass" (equation 3.32) - a quantity resulting when a vector of masses is multiplied by one of these diagonalizing unitary matrices - which has given rise to one of the successful generalizations of the original Koide relation. Sheppeard (#74-#76, this thread) proposed Brannen-style Koide relations for the quarks, and Zenczykowski (#93) found that they worked better if one used Goffinet pseudo-masses.

The significant new development: Koide has pointed out that the diagonalizing matrices have a physical meaning in the context of a Sumino model, which introduces family gauge bosons in order to explain how the Koide formula can work for pole masses. So we have here a circle of concepts, connecting the appearance of pole masses in the Koide formula, with the appearance of pseudo-masses in its Goffinet-Sheppeard-Zenczykowski generalization. Unfortunately I haven't yet had the time to properly analyse what happens when you close that circle.

ohwilleke
One benchmark for comparing Koide's rule and other efforts to determine fundamental particle masses on a theoretical basis is the accuracy with which we can determine light hadron masses with QCD which is part of the Standard Model and widely believed to be true. There really aren't even any fundamental alternatives to Standard Model QCD that receive active investigation:

Light Hadron Masses from a Matrix Model for QCD
Mahul Pandey, Sachindeo Vaidya
(Submitted on 6 Dec 2019)
The SU(3) Yang-Mills matrix model coupled to fundamental fermions is an approximation of quantum chromodynamics (QCD) on a 3-sphere of radius R. The spectrum of this matrix model Hamiltonian is estimated using standard variational methods, and is analyzed in the strong coupling limit. By employing a matching prescription to determine the dependence of the Yang-Mills coupling constant g on R, we relate the asymptotic values of the energy eigenvalues in the R→∞ (flat space) limit to the masses of light hadrons. We find that the matrix model estimates the light hadron spectrum fairly accurately, with the light baryon masses falling within 10%, and most light meson masses falling within about 30% of their observed values.
Another approach does a bit better but is still hardly precision physics:
Using a confining, symmetry-preserving regularisation of a vector×vector contact interaction, we compute the spectra of ground-state pseudoscalar and vector (fg¯) mesons, scalar and axial-vector (fg) diquarks, and JP=1/2+,3/2+ (fgh) baryons, where f,g,h∈{u,d,s,c,b}. The diquark correlations are essentially dynamical and play a key role in formulating and solving the three-valence-quark baryon problems. The baryon spectrum obtained from this largely-algebraic approach reproduces the 22 known experimental masses with an accuracy of 2.9(2.4) %. It also possesses the richness of states typical of constituent-quark models, predicting many heavy-quark baryons not yet observed. This study indicates that diquark correlations are an important component of all baryons; and owing to the dynamical character of the diquarks, it is typically the lightest allowed diquark correlation which defines the most important component of a baryon's Faddeev amplitude.
Pei-Lin Yin, Chen Chen, Gastao Krein, Craig D. Roberts, Jorge Segovia, Shu-Sheng Xu "Masses of ground-state mesons and baryons, including those with heavy quarks" (March 1, 2019).

Other work has estimated the mass of the proton from first principles to about a 1% margin of error.

Another is the experimental measurement uncertainty for various fundamental masses of the Standard Model, but while that can rule out theoretical proposals, the theoretical estimates of hadron masses are much less precise than their experimental measurements at the current time.

The electron mass is 0.510 998 946 10 ± 0.0000000031 MeV.

The muon mass is 105.658 374 5 ± 0.0000024 MeV.

The tau lepton mass is 1,776.86 ± 0.12 MeV.

Incidentally, the masses of the electron, muon and tau lepton are related to each other in a manner that is consistent with Koide's rule to within the current margins of error in these measurements, which made a prediction regarding the mass of the tau lepton as a function of the mass of the electron and the muon in 1981, six years after the tau lepton was first discovered. If Koide's rule is correct, the mass of the tau lepton, given the known electron mass and muon mass is 1776.96894(7). The difference between the measured mass of the tau lepton and its predicted value given Koide's rule is 0.1 MeV, which is less than one standard deviation of measurement error from the predicted value.

For the quarks these are:

2018 Quark Masses v. 2019 Quark Masses v. FLAG19 (in MeV)

top quark 173,000 ± 400 v. 173,000 ± 400 v. 172,900 ± 400
bottom quark 4,180 +40-/-30 v. 4,180 + 30/-20 v. 4,198 ± 12
charm quark 1,275 +25/-35 v. 1,270 ± 20 v. 1,282 ± 17
strange quark 95+9/-3 v. 93 +11/-5 v. 93.12 ± 0.69
down quark 4.7 + 0.5/-0.4 v. 4.67 +0.48/-0.17 v. 4.88 ± 0.2
up quark 2.2 +0.5/-0.4 v. 2.16 + 0.49/-0.26 v. 2.5 ± 0.17

Quark Type - Adjusted Extended Koide Mass - FLAG 19 Mass- PDG Mass (all in MeV)

top 172,743 v. 173,000 +/- 400 (-0.64 sigma) v. 173,000 +/- 400 (-0.64 sigma)
bottom 4192.98 v. 4,198 +/- 12 (-0.49 sigma) v. 4,180 +40-/-30 (+ 0.32 sigma)
charm 1293.21 v. 1,282 +/- 17 (+0.66 sigma) v. 1,275 +25/-35 (+ 0.73 sigma)
strange 92.274758 v. 93.12 +/- 0.69 (-1.22 sigma) v. 95+9/-3 (- 0.91 sigma)
down 5.32 v. 4.88 +/- 0.2 (+2.2 sigma) v. 4.7 + 0.5/-0.4 (+1.24 sigma)
up 0.0356 v. 2.5 +/- 0.17 (-14.5 sigma) v. 2.2 +0.5/-0.4 (- 5.4 sigma)

The W boson mass is 80,379 ± 12 MeV. A global electroweak fit suggests that the true value is closer to 80,356 +/- 9 MeV.

The Z boson mass is 91,187.6 ± 0.0021 MeV.

As of 2019, the PDG Higgs boson mass is 125,100 ± 140 MeV. The Higgs vacuum expectation value is 246,227,957.9 ± 1 MeV.
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After all these years I'm still working on the Koide coincidences. My original paper had to do with spin path integrals. In those, a spin keeps changing from pointing in the x to y to z directions in any order. My latest research is in generalizing this to allowing it to change between ± x, ± y and ± z. This implies the full octahedral point group symmetry.

And I realized that I could write things in terms of a generalization of the Dirac / Weyl wave equations. This increases the number of Pauli spin matrices or gamma matrices and correspondingly creates a large number of coupled wave equations. Uncoupling them reveals the standard model symmetry. I'm still writing it but you can see the temporary version on vixra.

Hans de Vries, ohwilleke and arivero
A curious paper from China today, "A translational flavor symmetry in the mass terms of Dirac and Majorana fermions" by Zhi-zhong Xing. The symmetry in the title is a "discrete shift in flavor space" whose definition I don't understand, but it has two consequences of interest. First, it implies that in each triple that respects the symmetry, one mass should equal zero. This fits the "m_e = m_u = 0" version of the waterfall. (As Xing says, small nonzero masses are then to come from a small breaking of the symmetry.) Second, the coefficients of the shift in flavor space, equal certain elements of the matrix that diagonalizes the mass matrix. It sounds very peculiar, but reminds me a little of Goffinet's "pseudo-masses", which play a role in Zenczykowski's refinement of the Brannen-Sheppeard phases. (Incidentally, Xing is cited many times in Zenczykowski's paper.)

By the way, Koide had a new paper in December, but it was not about his sum rule.

ohwilleke
A curious paper from China today, "A translational flavor symmetry in the mass terms of Dirac and Majorana fermions" by Zhi-zhong Xing. The symmetry in the title is a "discrete shift in flavor space" whose definition I don't understand, but it has two consequences of interest. First, it implies that in each triple that respects the symmetry, one mass should equal zero. This fits the "m_e = m_u = 0" version of the waterfall.

Hmm, of course, this is very similar to the content of the original paper on (0,d,s) "koide" triple. Harari Haut and Weyers did a permutation in flavour space that was criticised in later papers because it was not motivated dynamically. So Xing contribution could be perhaps the motivation.

ohwilleke
I have to say I still don't understand the fermionic translational symmetry (Friedberg-Lee symmetry) that Xing's paper is based on. The simplest paper about it that I could find, is this by Jarlskog... And if it is to apply to the waterfall as a whole, something has to change since most of the triples in the waterfall don't have a zero-mass element.

But I did realize that Xing must be one of China's top phenomenologists (insofar as one can identify the top phenomenologists, in an era of scarce experimental clues). Also, he was PhD supervisor of He Zhang, whose work with Rodejohann began this thread ten years ago... I have been browsing a 2019 review paper by Xing, "Flavor structures of charged fermions and massive neutrinos", and like it.

arivero
Perhaps we should mention the B-decay anomalies. I see two reasons to do so:

First, there are anomalies seen not just in b->s but also b->c, and the (b,c,s) triple is at the heart of the Koide "waterfall" discussed in this thread.

Second, one might hope that family gauge bosons like those posited by Sumino and Koide, could produce the anomalies! See equation 1.6 in this 2015 paper from Koide, for what the interactions look like.

My new paper (out for review at Foundations of Physics) has a connection to Koide but it isn't mentioned: https://vixra.org/abs/2105.0146

There are a couple of relations. First, Marni Sheppeard recognized the way I redid the Koide formula as a Discrete Fourier Transform and supposed that what we needed was a Discrete Fourier Transform for a non Abelian (well she said non commutative) symmetry. My new paper is exactly about that.

And the paper generalizes the Dirac / Weyl equation to one with more interesting Pauli spin matrices. But the underlying symmetry is a point group which implies that space is a lattice. For this the paper cites Iwo Bialynicki-Birula's paper on the Weyl / Dirac equation on a lattice: https://arxiv.org/abs/hep-th/9304070

That paper shows that you can get the special-relativity compatible Weyl / Dirac equation on a cubic lattice of quantum cellular automata provided you use a specific formula for updating the cellular automata. That formula is given by his equations (10) thru (12). But if you work out those equations, you'll find that both his paper and my old Koide paper https://arxiv.org/abs/1006.3114 in its equation (11) are about making steps in the +-x, +-y, and +-z directions. Except that while his considers all possible signs, mine is about +x, +y and +z only. The result is that where my paper is dedicated to the (1,1,1) direction where, over the long term, you have equal steps in the +x, +y and +z direction, his paper shows how to generalize it to steps averaging in any direction. And this tells precisely how to interpret the extra group of size three needed in my new paper; just as in my old Koide paper, the group of size 3 corresponds to assigning a factor of exp(2i k pi / 3) to steps in three directions. Such an assignment can be done 3 ways and still have phases cancel over different paths with the same beginning and end.

Carl

ohwilleke and arivero
Carl, about your old approaches to hadrons. Did you consider the Koide tuple of pi,D,B mesons? If not, why? If yes, do you remember where?

ohwilleke and CarlB
Carl, about your old approaches to hadrons. Did you consider the Koide tuple of pi,D,B mesons? If not, why? If yes, do you remember where?
I didn't, but that shouldn't make you hesitate. All avenues need to be explored.

Not sure if I've ever written this down completely. My reasoning, based on the Spin Path Integral paper, is that the three generations form by a transformation on the phases of the path integrals when considered as steps in the +-x, +-y and +-z directions. That is, if we take those parts of the path integral and write them as three separate groups, Px, Py and Pz, then we can assemble the groups to make a propagator for some particle X. Now we multiply these groups individually by 1, w and w* where w^3 = 1 and check if they can still be assembled into a propagator. Recall that a propagator has to satisfy the QQ=Q relation in order to preserve particle identity and quantum numbers.

The underlying idea here is that the three charged leptons all have the same quantum numbers and therefore there are no superselection rules that forbid their mixing by superposition. And in fact, by the weak force, the charged leptons are indeed produced in superposition as indicated by the PMNS matrix (or the CKM matrix for the quarks). Indeed, once you produce a charged lepton by the weak force you can only determine which lepton it is by measuring its mass which may not be that easy. The idea therefore is that the Koide relationship is about quantum states that differ in mass and mass only.

Consider the paths that begin at a point p0 = (x0,y0,z0) and ends at some other point p1 = (x1,y1,z1). Such a path must have a number of +y and -y steps that sums to give y1-y0, that is,
y1-y0 = N(+y) - N(-y)
where "N" is the number of steps in a path that happens to go from p0 to p1. Changing to a new path with one extra +y and one extra -y leaves y1-y0 unchanged so it contributes to the same sum in the path integral sum but the phase rule will be unchanged by such a path change. Uh, let me explain that better. The idea is that whatever change we make to the +x legs we make the negative of that change to the -x legs and same for y and z legs. And the phase changes are cubed roots of unity.

The effect of the complex phases 1, w and w* is to change the phase of a path integral only for its +-y and +-z parts as the +-x parts take the 1. Consider the +-y parts. Such a path will get a phase of 2 pi/3 (N(+y) - N(-y)) = 2pi/3(x1-x0) so this does not depend on the path. Similar for the +-z steps. In particular, if the path happens to have y1-y0 a multiple of 3, and also z1-z0 a multiple of 3, the phase change causes no change in the path integrals and that part of the propagator is unchanged by the phase change. From this you can see that for any particular pair of beginning and ending points p0 and p1, the phase change will only have an effect of multiplying those paths by either 1, w or w* and therefore, the paths themselves will do whatever interference they would have done without the phase change only the final phase is changed. And from that you can see that the phase change preserves the path integral in that if the paths make a rational propagator (in the sense of unitarity and preserving whatever the particle identity is) before the phase change it also is rational afterwards. Only thing that has changed are the relative phases at neighboring points.

And I assume that the energy depends partly on the difference in values between neighboring points but that other quantum numbers are determined by the actual things stepping. Hence the effect of the phase change is to preserve all quantum numbers except for energy = mass.

So for the pi, I would look for Koide triplets among the pi and its excited states. And for me, they have to have the same quantum numbers. A possibility would be (pi, pi[1300], pi[1800]).

Unfortunately, the lower mass mesons are a bit of a mess because their masses are not at all sharp. The lack of sharp masses means that two different mesons (of the same quantum numbers) can often be close enough together that it is impossible to distinguish them (as they have the same quantum numbers and their masses overlap). Consequently, those labels like pi[1300] with mass 1300(100) cannot be trusted to give you a list of all the states, some are undoubtedly shared. And the unsharp masses give unsatisfying coincidences.

Where I would try the Koide rules first is for mesons with sharply defined masses. Conveniently, these will also give sharper equalities. These occur at the higher mass mesons. For example, the Upsilon[1s], Upsilon[2s], Upsilon[3s], Upsilon[4s], Upsilon[10753], Upsilon[10860] and Upsilon[11020] are b b-bar mesons all with the same quantum numbers, i.e. the same I^G(J^PC) = 0-(1--). Their masses are quite accurately known, i.e. the Upsilon[1s] mass is 9460.30(.26) and can be included easily in Koide relationships. My guess is they occur in groups of 3+3 and that one of the Upsilons is misidentified, with one of the triplets corresponding to the charged leptons and the other to the neutrinos. In my papers, the difference between two pairs as in an up quark part and the corresponding down quark part (ie part = left handed etc), is one of a sign change. So they are closely related and maybe the corresponding change for mesons does not correspond to an obvious quantum number so they come in groups of 6. Or maybe they really can be excited so the 1s, 2s, 3s and 4s are parts of four different Koide triplets. I don't know.

If I were continuing to pursue this (and if I had more mes I would do just that), I would try to find meta relations between different sets of Koide triplets by looking for coincidences in the sharp high mass mesons and then see if they can organize the low mass mesons, that is, tell us which low mass mesons are superpositions of different mesons.

Carl

And I assume that the energy depends partly on the difference in values between neighboring points but that other quantum numbers are determined by the actual things stepping. Hence the effect of the phase change is to preserve all quantum numbers except for energy = mass.

So for the pi, I would look for Koide triplets among the pi and its excited states. And for me, they have to have the same quantum numbers. A possibility would be (pi, pi[1300], pi[1800]).

Yeah but if we agree that e mu and tau have the same quantum numbers, so happens with pi, D, B... they differ by flavour charges. So I got surprised you only went for excited states. At some point I checked for ground states and they are very Koidistic:

with the surprising point that neutral mesons did a better Koide fit than charged mesons... but well, symmetry breaking everywhere.

ohwilleke
Let me share some python3 code with only fundamental states of mesons (whole table can be copy pasted of https://pdg.lbl.gov/2021/html/computer_read.html if you want to do more general)

Python:
#!/usr/bin/env python
from math import sqrt
from itertools import permutations, combinations

p="""Particle ID(s)                  Mass  (GeV)       Errors (GeV)       Width (GeV)       Errors (GeV)      Name          Charges

211                          1.3957039E-01     +1.8E-07 -1.8E-07  2.5284E-17        +5.0E-21 -5.0E-21 pi                  +
111                          1.349768E-01      +5.0E-07 -5.0E-07  7.81E-09          +1.2E-10 -1.2E-10 pi                  0
221                          5.47862E-01       +1.7E-05 -1.7E-05  1.31E-06          +5.0E-08 -5.0E-08 eta                 0
331                          9.5778E-01        +6.0E-05 -6.0E-05  1.88E-04          +6.0E-06 -6.0E-06 eta'(958)           0
321                          4.93677E-01       +1.6E-05 -1.6E-05  5.317E-17         +9.0E-20 -9.0E-20 K                   +
311                          4.97611E-01       +1.3E-05 -1.3E-05                                      K                   0
411                          1.86966E+00       +5.0E-05 -5.0E-05  6.33E-13          +4.0E-15 -4.0E-15 D                   +
421                          1.86484E+00       +5.0E-05 -5.0E-05  1.605E-12         +6.0E-15 -6.0E-15 D                   0
431                          1.96835E+00       +7.0E-05 -7.0E-05  1.305E-12         +1.0E-14 -1.0E-14 D(s)                +
521                          5.27934E+00       +1.2E-04 -1.2E-04  4.018E-13         +1.0E-15 -1.0E-15 B                   +
511                          5.27965E+00       +1.2E-04 -1.2E-04  4.333E-13         +1.1E-15 -1.1E-15 B                   0
531                          5.36688E+00       +1.4E-04 -1.4E-04  4.342E-13         +1.7E-15 -1.7E-15 B(s)                0
541                          6.27447E+00       +3.2E-04 -3.2E-04  1.291E-12         +2.3E-14 -2.3E-14 B(c)                +
441                          2.9839E+00        +4.0E-04 -4.0E-04  3.20E-02          +7.0E-04 -7.0E-04 eta(c)(1S)          0
443                          3.096900E+00      +6.0E-06 -6.0E-06  9.26E-05          +1.7E-06 -1.7E-06 J/psi(1S)           0
553                          9.46030E+00       +2.6E-04 -2.6E-04  5.40E-05          +1.3E-06 -1.3E-06 Upsilon(1S)         0
200553                          1.03552E+01       +5.0E-04 -5.0E-04  2.03E-05          +1.9E-06 -1.9E-06 Upsilon(3S)         0
"""
p.split("\n")[3].split()
m=dict()
for line in p.split("\n")[2:-1]:
d=line.split()
print(d)
m[d[-2]+d[-1]]=float(d[1])
result=[]
for triplet in permutations(m,3):
a,b,c = triplet
a,b,c = m[a],m[b],m[c]
if b < c:
k=((a+b+c)/(-sqrt(a)+sqrt(b)+sqrt(c))**2)
result.append([abs(k-2/3),triplet,k,"-++"]) #use Python3 for floats!
for doublet in combinations(m,2):
b,c = doublet
a,b,c = 0,m[b],m[c]
k=((a+b+c)/(-sqrt(a)+sqrt(b)+sqrt(c))**2)
result.append([abs(k-2.00/3),doublet,k,"0++"])
for triplet in combinations(m,3):
a,b,c = triplet
a,b,c = m[a],m[b],m[c]
k=((a+b+c)/(sqrt(a)+sqrt(b)+sqrt(c))**2)
result.append([abs(k-2/3),triplet,k,"+++"]) #use Python3 for floats!
for doublet in permutations(m,2):
b,c = doublet
a,b,c = 0,m[b],m[c]
k=((a+b+c)/(-sqrt(a)-sqrt(b)+sqrt(c))**2)
result.append([abs(k-2.00/3),doublet,k,"0-+"])
result.sort()
for x in result:
print(f'{"|".join(x[1]):<30}',"\t",x[2],x[3])

looking at the output:

Code:
pi0|D(s)+|eta(c)(1S)0               0.6663837057987507 -++
pi0|D+                              0.6661367723217316 0++
pi+|D(s)+|B+                        0.6673593324132077 -++
pi+|D(s)+|B0                        0.6673602750393526 -++
pi0|D0                              0.6658883844469711 0++
pi+|D(s)+|B(s)0                     0.667632095646159 -++
pi0|D(s)+|J/psi(1S)0                0.6654846261441458 -++
pi+|D(s)+                           0.6678662052942218 0++
pi+|pi0|Upsilon(1S)0                0.6682566138237737 +++
pi0|D(s)+|B(c)+                     0.6683538259548514 -++
pi0|D(s)+|B(s)0                     0.6647562935736748 -++
pi0|D(s)+|B0                        0.6644668556345085 -++
pi0|D(s)+|B+                        0.6644658496344038 -++
pi+|D(s)+|J/psi(1S)0                0.6690366259117551 -++
pi0|D+|J/psi(1S)0                   0.6698183238780191 -++
pi0|D+|B+                           0.6698242106935278 -++
pi0|D+|B0                           0.6698253129918151 -++
pi+|D(s)+|eta(c)(1S)0               0.6699885664600504 -++
pi0|D0|J/psi(1S)0                   0.6700415949749136 -++
pi0|D0|B+                           0.670097549041931 -++
pi0|D0|B0                           0.6700986560032072 -++
pi0|D+|B(s)0                        0.6701414512426769 -++
pi0|D0|B(s)0                        0.6704160867174216 -++
pi+|D+                              0.6629167285356727 0++
pi0|D+|eta(c)(1S)0                  0.6706380841323579 -++
pi+|D0                              0.6626683899479252 0++
pi0|D0|eta(c)(1S)0                  0.6708574511027877 -++
pi+|D(s)+|B(c)+                     0.6710682904109853 -++
pi0|D(s)+                           0.6710861088843688 0++
pi+|D+|B+                           0.6727737789483542 -++
pi+|D+|B0                           0.67277481621395 -++
pi+|D0|B+                           0.6730499585060149 -++
pi+|D0|B0                           0.6730510003496702 -++
pi+|D+|B(s)0                        0.6730728758703244 -++
eta0|Upsilon(1S)0|Upsilon(3S)0      0.6602536515686964 -++
pi+|D0|B(s)0                        0.6733503288525908 -++
pi+|D+|J/psi(1S)0                   0.673443593712495 -++
pi+|D0|J/psi(1S)0                   0.6736705861236503 -++
pi0|D+|B(c)+                        0.6739753428121785 -++
pi0|D0|B(c)+                        0.674261376733939 -++
pi+|D+|eta(c)(1S)0                  0.6743176053201272 -++
pi+|D0|eta(c)(1S)0                  0.6745407648041061 -++
K+|B(c)+                            0.6578600967642535 0++
K0|B(c)+                            0.6570972277563282 0++
pi+|D+|B(c)+                        0.6767411895047183 -++
pi+|D0|B(c)+                        0.6770298238881567 -++
...

there are one or two problems: more noticeably, that tuples of charged and neutral seem to fare better. But on the other hand, neutrals, particularly etas, are known to be more mixed and it is not easy to decide which mass values should be use.

Last edited:
ohwilleke
Addendum: python code to check all the known masses in the pdg listing.

Not considering error bands in the sorting. It could be used to generate some histograms, statistics and, adding the error bands, scores of the most common values of Koide formula and get some insight about the likeliness of getting a coincidence.

Python:
#!/usr/bin/env python
# coding: utf-8
from math import sqrt
from itertools import permutations, combinations
p="""* Particle ID(s)                  Mass  (GeV)       Errors (GeV)       Width (GeV)       Errors (GeV)      Name          Charges
24                          8.0379E+01        +1.2E-02 -1.2E-02  2.08E+00          +4.0E-02 -4.0E-02 W                   +
23                          9.11876E+01       +2.1E-03 -2.1E-03  2.4952E+00        +2.3E-03 -2.3E-03 Z                   0
25                          1.2525E+02        +1.7E-01 -1.7E-01  3.2E-03           +2.8E-03 -2.2E-03 H                   0
11                          5.109989461E-04   +3.1E-12 -3.1E-12  0.E+00            +0.0E+00 -0.0E+00 e                   -
13                          1.056583745E-01   +2.4E-09 -2.4E-09  2.9959837E-19     +3.0E-25 -3.0E-25 mu                  -
15                          1.77686E+00       +1.2E-04 -1.2E-04  2.267E-12         +4.0E-15 -4.0E-15 tau                 -
1                          4.67E-03          +0.5E-03 -0.2E-03                                      d                -1/3
2                          2.16E-03          +0.5E-03 -0.3E-03                                      u                +2/3
3                          9.3E-02           +1.1E-02 -5.0E-03                                      s                -1/3
4                          1.27E+00          +2.0E-02 -2.0E-02                                      c                +2/3
5                          4.180E+00         +3.0E-02 -2.0E-02                                      b                -1/3
6                          1.725E+02         +7.0E-01 -7.0E-01  1.42E+00          +1.9E-01 -1.5E-01 t                +2/3
211                          1.3957039E-01     +1.8E-07 -1.8E-07  2.5284E-17        +5.0E-21 -5.0E-21 pi                  +
111                          1.349768E-01      +5.0E-07 -5.0E-07  7.81E-09          +1.2E-10 -1.2E-10 pi                  0
221                          5.47862E-01       +1.7E-05 -1.7E-05  1.31E-06          +5.0E-08 -5.0E-08 eta                 0
9000221                          6.0E-01           +2.0E-01 -2.0E-01  4.5E-01           +3.5E-01 -3.5E-01 f(0)(500)           0
113     213                  7.7526E-01        +2.3E-04 -2.3E-04  1.491E-01         +8.0E-04 -8.0E-04 rho(770)          0,+
223                          7.8266E-01        +1.3E-04 -1.3E-04  8.68E-03          +1.3E-04 -1.3E-04 omega(782)          0
331                          9.5778E-01        +6.0E-05 -6.0E-05  1.88E-04          +6.0E-06 -6.0E-06 eta'(958)           0
9010221                          9.90E-01          +2.0E-02 -2.0E-02  6.E-02            +5.0E-02 -5.0E-02 f(0)(980)           0
9000111 9000211                  9.80E-01          +2.0E-02 -2.0E-02  7.5E-02           +2.5E-02 -2.5E-02 a(0)(980)         0,+
333                          1.019461E+00      +1.6E-05 -1.6E-05  4.249E-03         +1.3E-05 -1.3E-05 phi(1020)           0
10223                          1.166E+00         +6.0E-03 -6.0E-03  3.75E-01          +3.5E-02 -3.5E-02 h(1)(1170)          0
10113   10213                  1.2295E+00        +3.2E-03 -3.2E-03  1.42E-01          +9.0E-03 -9.0E-03 b(1)(1235)        0,+
20113   20213                  1.23E+00          +4.0E-02 -4.0E-02  4.2E-01           +1.8E-01 -1.8E-01 a(1)(1260)        0,+
225                          1.2755E+00        +8.0E-04 -8.0E-04  1.867E-01         +2.2E-03 -2.5E-03 f(2)(1270)          0
20223                          1.2819E+00        +5.0E-04 -5.0E-04  2.27E-02          +1.1E-03 -1.1E-03 f(1)(1285)          0
100221                          1.294E+00         +4.0E-03 -4.0E-03  5.5E-02           +5.0E-03 -5.0E-03 eta(1295)           0
100111  100211                  1.30E+00          +1.0E-01 -1.0E-01  4.0E-01           +2.0E-01 -2.0E-01 pi(1300)          0,+
115     215                  1.3182E+00        +6.0E-04 -6.0E-04  1.07E-01          +5.0E-03 -5.0E-03 a(2)(1320)        0,+
10221                          1.35E+00          +1.5E-01 -1.5E-01  3.5E-01           +1.5E-01 -1.5E-01 f(0)(1370)          0
9000113 9000213                  1.354E+00         +2.5E-02 -2.5E-02  3.30E-01          +3.5E-02 -3.5E-02 pi(1)(1400)       0,+
9020221                          1.4088E+00        +2.0E-03 -2.0E-03  5.01E-02          +2.6E-03 -2.6E-03 eta(1405)           0
10333                          1.416E+00         +8.0E-03 -8.0E-03  9.0E-02           +1.5E-02 -1.5E-02 h(1)(1415)          0
20333                          1.4263E+00        +9.0E-04 -9.0E-04  5.45E-02          +2.6E-03 -2.6E-03 f(1)(1420)          0
1000223                          1.41E+00          +6.0E-02 -6.0E-02  2.9E-01           +1.9E-01 -1.9E-01 omega(1420)         0
10111   10211                  1.474E+00         +1.9E-02 -1.9E-02  2.65E-01          +1.3E-02 -1.3E-02 a(0)(1450)        0,+
100113  100213                  1.465E+00         +2.5E-02 -2.5E-02  4.0E-01           +6.0E-02 -6.0E-02 rho(1450)         0,+
100331                          1.475E+00         +4.0E-03 -4.0E-03  9.0E-02           +9.0E-03 -9.0E-03 eta(1475)           0
9030221                          1.506E+00         +6.0E-03 -6.0E-03  1.12E-01          +9.0E-03 -9.0E-03 f(0)(1500)          0
335                          1.5174E+00        +2.5E-03 -2.5E-03  8.6E-02           +5.0E-03 -5.0E-03 f(2)'(1525)         0
9010113 9010213                  1.661E+00         +1.5E-02 -1.1E-02  2.4E-01           +5.0E-02 -5.0E-02 pi(1)(1600)       0,+
9020113 9020213                  1.655E+00         +1.6E-02 -1.6E-02  2.5E-01           +4.0E-02 -4.0E-02 a(1)(1640)        0,+
10225                          1.617E+00         +5.0E-03 -5.0E-03  1.81E-01          +1.1E-02 -1.1E-02 eta(2)(1645)        0
30223                          1.670001E+00         +3.0E-02 -3.0E-02  3.15E-01          +3.5E-02 -3.5E-02 omega(1650)         0
227                          1.667E+00         +4.0E-03 -4.0E-03  1.68E-01          +1.0E-02 -1.0E-02 omega(3)(1670)      0
10115   10215                  1.6706E+00        +2.9E-03 -1.2E-03  2.58E-01          +8.0E-03 -9.0E-03 pi(2)(1670)       0,+
100333                          1.680E+00         +2.0E-02 -2.0E-02  1.5E-01           +5.0E-02 -5.0E-02 phi(1680)           0
117     217                  1.6888E+00        +2.1E-03 -2.1E-03  1.61E-01          +1.0E-02 -1.0E-02 rho(3)(1690)      0,+
30113   30213                  1.720E+00         +2.0E-02 -2.0E-02  2.5E-01           +1.0E-01 -1.0E-01 rho(1700)         0,+
9000115 9000215                  1.70E+00          +4.0E-02 -4.0E-02  2.7E-01           +6.0E-02 -6.0E-02 a(2)(1700)        0,+
10331                          1.704E+00         +1.2E-02 -1.2E-02  1.23E-01          +1.8E-02 -1.8E-02 f(0)(1710)          0
9010111 9010211                  1.810E+00         +9.0E-03 -1.1E-02  2.15E-01          +7.0E-03 -8.0E-03 pi(1800)          0,+
337                          1.854E+00         +7.0E-03 -7.0E-03  8.7E-02           +2.8E-02 -2.3E-02 phi(3)(1850)        0
9050225                          1.936E+00         +1.2E-02 -1.2E-02  4.64E-01          +2.4E-02 -2.4E-02 f(2)(1950)          0
119     219                  1.967E+00         +1.6E-02 -1.6E-02  3.24E-01          +1.5E-02 -1.8E-02 a(4)(1970)        0,+
9060225                          2.01E+00          +6.0E-02 -8.0E-02  2.0E-01           +6.0E-02 -6.0E-02 f(2)(2010)          0
229                          2.018E+00         +1.1E-02 -1.1E-02  2.37E-01          +1.8E-02 -1.8E-02 f(4)(2050)          0
9080225                          2.297E+00         +2.8E-02 -2.8E-02  1.5E-01           +4.0E-02 -4.0E-02 f(2)(2300)          0
9090225                          2.35E+00          +5.0E-02 -4.0E-02  3.2E-01           +7.0E-02 -6.0E-02 f(2)(2340)          0
321                          4.93677E-01       +1.6E-05 -1.6E-05  5.317E-17         +9.0E-20 -9.0E-20 K                   +
311                          4.97611E-01       +1.3E-05 -1.3E-05                                      K                   0
9000311 9000321                  8.45E-01          +1.7E-02 -1.7E-02  4.68E-01          +3.0E-02 -3.0E-02 K(0)*(700)        0,+
313                          8.9555E-01        +2.0E-04 -2.0E-04  4.73E-02          +5.0E-04 -5.0E-04 K*(892)             0
323                          8.9167E-01        +2.6E-04 -2.6E-04  5.14E-02          +8.0E-04 -8.0E-04 K*(892)             +
323                          8.955E-01         +8.0E-04 -8.0E-04  4.62E-02          +1.3E-03 -1.3E-03 K*(892)             +
10313   10323                  1.253E+00         +7.0E-03 -7.0E-03  9.0E-02           +2.0E-02 -2.0E-02 K(1)(1270)        0,+
20313   20323                  1.403E+00         +7.0E-03 -7.0E-03  1.74E-01          +1.3E-02 -1.3E-02 K(1)(1400)        0,+
100313  100323                  1.414E+00         +1.5E-02 -1.5E-02  2.32E-01          +2.1E-02 -2.1E-02 K*(1410)          0,+
10311   10321                  1.43E+00          +5.0E-02 -5.0E-02  2.7E-01           +8.0E-02 -8.0E-02 K(0)*(1430)       0,+
315                          1.4324E+00        +1.3E-03 -1.3E-03  1.09E-01          +5.0E-03 -5.0E-03 K(2)*(1430)         0
325                          1.4273E+00        +1.5E-03 -1.5E-03  1.000E-01         +2.1E-03 -2.1E-03 K(2)*(1430)         +
9000313 9000323                  1.67E+00          +5.0E-02 -5.0E-02  1.6E-01           +5.0E-02 -5.0E-02 K(1)(1650)        0,+
30313   30323                  1.718E+00         +1.8E-02 -1.8E-02  3.2E-01           +1.1E-01 -1.1E-01 K*(1680)          0,+
10315   10325                  1.773E+00         +8.0E-03 -8.0E-03  1.86E-01          +1.4E-02 -1.4E-02 K(2)(1770)        0,+
317     327                  1.779E+00         +8.0E-03 -8.0E-03  1.61E-01          +1.7E-02 -1.7E-02 K(3)*(1780)       0,+
20315   20325                  1.819E+00         +1.2E-02 -1.2E-02  2.64E-01          +3.4E-02 -3.4E-02 K(2)(1820)        0,+
9010315 9010325                  1.99E+00          +6.0E-02 -5.0E-02  3.49E-01          +5.0E-02 -3.0E-02 K(2)*(1980)       0,+
319     329                  2.048E+00         +8.0E-03 -9.0E-03  1.99E-01          +2.7E-02 -1.9E-02 K(4)*(2045)       0,+
411                          1.86966E+00       +5.0E-05 -5.0E-05  6.33E-13          +4.0E-15 -4.0E-15 D                   +
421                          1.86484E+00       +5.0E-05 -5.0E-05  1.605E-12         +6.0E-15 -6.0E-15 D                   0
423                          2.00685E+00       +5.0E-05 -5.0E-05                                      D*(2007)            0
413                          2.01026E+00       +5.0E-05 -5.0E-05  8.34E-05          +1.8E-06 -1.8E-06 D*(2010)            +
10421   10411                  2.343E+00         +1.0E-02 -1.0E-02  2.29E-01          +1.6E-02 -1.6E-02 D(0)*(2300)       0,+
10423   10413                  2.4221E+00        +6.0E-04 -6.0E-04  3.13E-02          +1.9E-03 -1.9E-03 D(1)(2420)        0,+
20423                          2.412E+00         +9.0E-03 -9.0E-03  3.14E-01          +2.9E-02 -2.9E-02 D(1)(2430)          0
425     415                  2.4611E+00        +7.0E-04 -8.0E-04  4.73E-02          +8.0E-04 -8.0E-04 D(2)*(2460)       0,+
431                          1.96835E+00       +7.0E-05 -7.0E-05  1.305E-12         +1.0E-14 -1.0E-14 D(s)                +
433                          2.1122E+00        +4.0E-04 -4.0E-04                                      D(s)*               +
10431                          2.3178E+00        +5.0E-04 -5.0E-04                                      D(s0)*(2317)        +
20433                          2.4595E+00        +6.0E-04 -6.0E-04                                      D(s1)(2460)         +
10433                          2.53511E+00       +6.0E-05 -6.0E-05  9.2E-04           +5.0E-05 -5.0E-05 D(s1)(2536)         +
435                          2.5691E+00        +8.0E-04 -8.0E-04  1.69E-02          +7.0E-04 -7.0E-04 D(s2)*(2573)        +
521                          5.27934E+00       +1.2E-04 -1.2E-04  4.018E-13         +1.0E-15 -1.0E-15 B                   +
511                          5.27965E+00       +1.2E-04 -1.2E-04  4.333E-13         +1.1E-15 -1.1E-15 B                   0
513     523                  5.32470E+00       +2.1E-04 -2.1E-04                                      B*                0,+
515                          5.7395E+00        +7.0E-04 -7.0E-04  2.42E-02          +1.7E-03 -1.7E-03 B(2)*(5747)         0
525                          5.7372E+00        +7.0E-04 -7.0E-04  2.0E-02           +5.0E-03 -5.0E-03 B(2)*(5747)         +
531                          5.36688E+00       +1.4E-04 -1.4E-04  4.342E-13         +1.7E-15 -1.7E-15 B(s)                0
533                          5.4154E+00        +1.8E-03 -1.5E-03                                      B(s)*               0
535                          5.83986E+00       +1.2E-04 -1.2E-04  1.49E-03          +2.7E-04 -2.7E-04 B(s2)*(5840)        0
541                          6.27447E+00       +3.2E-04 -3.2E-04  1.291E-12         +2.3E-14 -2.3E-14 B(c)                +
441                          2.9839E+00        +4.0E-04 -4.0E-04  3.20E-02          +7.0E-04 -7.0E-04 eta(c)(1S)          0
443                          3.096900E+00      +6.0E-06 -6.0E-06  9.26E-05          +1.7E-06 -1.7E-06 J/psi(1S)           0
10441                          3.41471E+00       +3.0E-04 -3.0E-04  1.08E-02          +6.0E-04 -6.0E-04 chi(c0)(1P)         0
20443                          3.51067E+00       +5.0E-05 -5.0E-05  8.4E-04           +4.0E-05 -4.0E-05 chi(c1)(1P)         0
10443                          3.52538E+00       +1.1E-04 -1.1E-04  7.E-04            +4.0E-04 -4.0E-04 h(c)(1P)            0
445                          3.55617E+00       +7.0E-05 -7.0E-05  1.97E-03          +9.0E-05 -9.0E-05 chi(c2)(1P)         0
100441                          3.6375E+00        +1.1E-03 -1.1E-03  1.13E-02          +3.2E-03 -2.9E-03 eta(c)(2S)          0
100443                          3.68610E+00       +6.0E-05 -6.0E-05  2.94E-04          +8.0E-06 -8.0E-06 psi(2S)             0
30443                          3.7737E+00        +4.0E-04 -4.0E-04  2.72E-02          +1.0E-03 -1.0E-03 psi(3770)           0
100445                          3.9225E+00        +1.0E-03 -1.0E-03  3.52E-02          +2.2E-03 -2.2E-03 chi(c2)(3930)       0
9000443                          4.0390E+00        +1.0E-03 -1.0E-03  8.0E-02           +1.0E-02 -1.0E-02 psi(4040)           0
9010443                          4.191E+00         +5.0E-03 -5.0E-03  7.0E-02           +1.0E-02 -1.0E-02 psi(4160)           0
9020443                          4.421E+00         +4.0E-03 -4.0E-03  6.2E-02           +2.0E-02 -2.0E-02 psi(4415)           0
553                          9.46030E+00       +2.6E-04 -2.6E-04  5.40E-05          +1.3E-06 -1.3E-06 Upsilon(1S)         0
10551                          9.8594E+00        +5.0E-04 -5.0E-04                                      chi(b0)(1P)         0
20553                          9.8928E+00        +4.0E-04 -4.0E-04                                      chi(b1)(1P)         0
10553                          9.8993E+00        +8.0E-04 -8.0E-04                                      h(b)(1P)            0
555                          9.9122E+00        +4.0E-04 -4.0E-04                                      chi(b2)(1P)         0
100553                          1.002326E+01      +3.1E-04 -3.1E-04  3.20E-05          +2.6E-06 -2.6E-06 Upsilon(2S)         0
20555                          1.01637E+01       +1.4E-03 -1.4E-03                                      Upsilon(2)(1D)      0
110551                          1.02325E+01       +6.0E-04 -6.0E-04                                      chi(b0)(2P)         0
120553                          1.02555E+01       +5.0E-04 -5.0E-04                                      chi(b1)(2P)         0
100555                          1.02686E+01       +5.0E-04 -5.0E-04                                      chi(b2)(2P)         0
200553                          1.03552E+01       +5.0E-04 -5.0E-04  2.03E-05          +1.9E-06 -1.9E-06 Upsilon(3S)         0
300553                          1.05794E+01       +1.2E-03 -1.2E-03  2.05E-02          +2.5E-03 -2.5E-03 Upsilon(4S)         0
9000553                          1.08852E+01       +2.6E-03 -1.6E-03  3.7E-02           +4.0E-03 -4.0E-03 Upsilon(10860)      0
9010553                          1.1000E+01        +4.0E-03 -4.0E-03  2.4E-02           +8.0E-03 -6.0E-03 Upsilon(11020)      0
2212                          9.38272081E-01    +6.0E-09 -6.0E-09  0.E+00            +0.0E+00 -0.0E+00 p                   +
2112                          9.39565413E-01    +6.0E-09 -6.0E-09  7.485E-28         +5.0E-31 -5.0E-31 n                   0
12112   12212                  1.440E+00         +3.0E-02 -3.0E-02  3.5E-01           +1.0E-01 -1.0E-01 N(1440)           0,+
1214    2124                  1.515E+00         +5.0E-03 -5.0E-03  1.10E-01          +1.0E-02 -1.0E-02 N(1520)           0,+
22112   22212                  1.530E+00         +1.5E-02 -1.5E-02  1.50E-01          +2.5E-02 -2.5E-02 N(1535)           0,+
32112   32212                  1.650E+00         +1.5E-02 -1.5E-02  1.25E-01          +2.5E-02 -2.5E-02 N(1650)           0,+
2116    2216                  1.675E+00         +5.0E-03 -1.0E-02  1.45E-01          +1.5E-02 -1.5E-02 N(1675)           0,+
12116   12216                  1.685E+00         +5.0E-03 -5.0E-03  1.20E-01          +1.0E-02 -5.0E-03 N(1680)           0,+
21214   22124                  1.72E+00          +8.0E-02 -7.0E-02  2.0E-01           +1.0E-01 -1.0E-01 N(1700)           0,+
42112   42212                  1.710E+00         +3.0E-02 -3.0E-02  1.4E-01           +6.0E-02 -6.0E-02 N(1710)           0,+
31214   32124                  1.720E+00         +3.0E-02 -4.0E-02  2.5E-01           +1.5E-01 -1.0E-01 N(1720)           0,+
1218    2128                  2.18E+00          +4.0E-02 -4.0E-02  4.0E-01           +1.0E-01 -1.0E-01 N(2190)           0,+
1114    2114    2214    2224  1.2320E+00        +2.0E-03 -2.0E-03  1.170E-01         +3.0E-03 -3.0E-03 Delta(1232)  -,0,+,++
31114   32114   32214   32224  1.57E+00          +7.0E-02 -7.0E-02  2.5E-01           +5.0E-02 -5.0E-02 Delta(1600)  -,0,+,++
1112    1212    2122    2222  1.610E+00         +2.0E-02 -2.0E-02  1.30E-01          +2.0E-02 -2.0E-02 Delta(1620)  -,0,+,++
11114   12114   12214   12224  1.710E+00         +2.0E-02 -2.0E-02  3.0E-01           +8.0E-02 -8.0E-02 Delta(1700)  -,0,+,++
11112   11212   12122   12222  1.860E+00         +6.0E-02 -2.0E-02  2.5E-01           +7.0E-02 -7.0E-02 Delta(1900)  -,0,+,++
1116    1216    2126    2226  1.880E+00         +3.0E-02 -2.5E-02  3.3E-01           +7.0E-02 -6.0E-02 Delta(1905)  -,0,+,++
21112   21212   22122   22222  1.90E+00          +5.0E-02 -5.0E-02  3.0E-01           +1.0E-01 -1.0E-01 Delta(1910)  -,0,+,++
21114   22114   22214   22224  1.92E+00          +5.0E-02 -5.0E-02  3.0E-01           +6.0E-02 -6.0E-02 Delta(1920)  -,0,+,++
11116   11216   12126   12226  1.95E+00          +5.0E-02 -5.0E-02  3.0E-01           +1.0E-01 -1.0E-01 Delta(1930)  -,0,+,++
1118    2118    2218    2228  1.930E+00         +2.0E-02 -1.5E-02  2.8E-01           +5.0E-02 -5.0E-02 Delta(1950)  -,0,+,++
3122                          1.115683E+00      +6.0E-06 -6.0E-06  2.501E-15         +1.9E-17 -1.9E-17 Lambda              0
13122                          1.4051E+00        +1.3E-03 -1.0E-03  5.05E-02          +2.0E-03 -2.0E-03 Lambda(1405)        0
3124                          1.5190E+00        +1.0E-03 -1.0E-03  1.60E-02          +1.0E-03 -1.0E-03 Lambda(1520)        0
23122                          1.600E+00         +3.0E-02 -3.0E-02  2.0E-01           +5.0E-02 -5.0E-02 Lambda(1600)        0
33122                          1.674E+00         +4.0E-03 -4.0E-03  3.0E-02           +5.0E-03 -5.0E-03 Lambda(1670)        0
13124                          1.690E+00         +5.0E-03 -5.0E-03  7.0E-02           +1.0E-02 -1.0E-02 Lambda(1690)        0
43122                          1.80E+00          +5.0E-02 -5.0E-02  2.0E-01           +5.0E-02 -5.0E-02 Lambda(1800)        0
53122                          1.79E+00          +5.0E-02 -5.0E-02  1.1E-01           +6.0E-02 -6.0E-02 Lambda(1810)        0
3126                          1.820E+00         +5.0E-03 -5.0E-03  8.0E-02           +1.0E-02 -1.0E-02 Lambda(1820)        0
13126                          1.825E+00         +5.0E-03 -5.0E-03  9.0E-02           +3.0E-02 -3.0E-02 Lambda(1830)        0
23124                          1.890E+00         +2.0E-02 -2.0E-02  1.2E-01           +4.0E-02 -4.0E-02 Lambda(1890)        0
3128                          2.100E+00         +1.0E-02 -1.0E-02  2.0E-01           +5.0E-02 -1.0E-01 Lambda(2100)        0
23126                          2.09E+00          +4.0E-02 -4.0E-02  2.5E-01           +5.0E-02 -5.0E-02 Lambda(2110)        0
3222                          1.18937E+00       +7.0E-05 -7.0E-05  8.209E-15         +2.7E-17 -2.7E-17 Sigma               +
3212                          1.192642E+00      +2.4E-05 -2.4E-05  8.9E-06           +9.0E-07 -8.0E-07 Sigma               0
3112                          1.197449E+00      +3.0E-05 -3.0E-05  4.450E-15         +3.2E-17 -3.2E-17 Sigma               -
3114                          1.3872E+00        +5.0E-04 -5.0E-04  3.94E-02          +2.1E-03 -2.1E-03 Sigma(1385)         -
3214                          1.3837E+00        +1.0E-03 -1.0E-03  3.6E-02           +5.0E-03 -5.0E-03 Sigma(1385)         0
3224                          1.38280E+00       +3.5E-04 -3.5E-04  3.60E-02          +7.0E-04 -7.0E-04 Sigma(1385)         +
13112   13212   13222          1.660E+00         +2.0E-02 -2.0E-02  2.0E-01           +1.0E-01 -1.0E-01 Sigma(1660)     -,0,+
13114   13214   13224          1.675E+00         +1.0E-02 -1.0E-02  7.0E-02           +3.0E-02 -3.0E-02 Sigma(1670)     -,0,+
23112   23212   23222          1.75E+00          +5.0E-02 -5.0E-02  1.5E-01           +5.0E-02 -5.0E-02 Sigma(1750)     -,0,+
3116    3216    3226          1.775E+00         +5.0E-03 -5.0E-03  1.20E-01          +1.5E-02 -1.5E-02 Sigma(1775)     -,0,+
23114   23214   23224          1.91E+00          +4.0E-02 -4.0E-02  2.2E-01           +8.0E-02 -7.0E-02 Sigma(1910)     -,0,+
13116   13216   13226          1.915E+00         +2.0E-02 -1.5E-02  1.2E-01           +4.0E-02 -4.0E-02 Sigma(1915)     -,0,+
3118    3218    3228          2.030E+00         +1.0E-02 -5.0E-03  1.80E-01          +2.0E-02 -3.0E-02 Sigma(2030)     -,0,+
3322                          1.31486E+00       +2.0E-04 -2.0E-04  2.27E-15          +7.0E-17 -7.0E-17 Xi                  0
3312                          1.32171E+00       +7.0E-05 -7.0E-05  4.02E-15          +4.0E-17 -4.0E-17 Xi                  -
3314                          1.5350E+00        +6.0E-04 -6.0E-04  9.9E-03           +1.7E-03 -1.9E-03 Xi(1530)            -
3324                          1.53180E+00       +3.2E-04 -3.2E-04  9.1E-03           +5.0E-04 -5.0E-04 Xi(1530)            0
203312  203322                  1.690E+00         +1.0E-02 -1.0E-02                                      Xi(1690)          -,0
13314   13324                  1.823E+00         +5.0E-03 -5.0E-03  2.4E-02           +1.5E-02 -1.0E-02 Xi(1820)          -,0
103316  103326                  1.950E+00         +1.5E-02 -1.5E-02  6.0E-02           +2.0E-02 -2.0E-02 Xi(1950)          -,0
203316  203326                  2.025E+00         +5.0E-03 -5.0E-03  2.0E-02           +1.5E-02 -5.0E-03 Xi(2030)          -,0
3334                          1.67245E+00       +2.9E-04 -2.9E-04  8.02E-15          +1.1E-16 -1.1E-16 Omega               -
203338                          2.252E+00         +9.0E-03 -9.0E-03  5.5E-02           +1.8E-02 -1.8E-02 Omega(2250)         -
4122                          2.28646E+00       +1.4E-04 -1.4E-04  3.25E-12          +5.0E-14 -5.0E-14 Lambda(c)           +
14122                          2.59225E+00       +2.8E-04 -2.8E-04  2.6E-03           +6.0E-04 -6.0E-04 Lambda(c)(2595)     +
104122                          2.62811E+00       +1.9E-04 -1.9E-04                                      Lambda(c)(2625)     +
204126                          2.88163E+00       +2.4E-04 -2.4E-04  5.6E-03           +8.0E-04 -6.0E-04 Lambda(c)(2880)     +
4112                          2.45375E+00       +1.4E-04 -1.4E-04  1.83E-03          +1.1E-04 -1.9E-04 Sigma(c)(2455)      0
4212                          2.4529E+00        +4.0E-04 -4.0E-04                                      Sigma(c)(2455)      +
4222                          2.45397E+00       +1.4E-04 -1.4E-04  1.89E-03          +9.0E-05 -1.8E-04 Sigma(c)(2455)     ++
4114                          2.51848E+00       +2.0E-04 -2.0E-04  1.53E-02          +4.0E-04 -5.0E-04 Sigma(c)(2520)      0
4214                          2.5175E+00        +2.3E-03 -2.3E-03                                      Sigma(c)(2520)      +
4224                          2.51841E+00       +2.1E-04 -1.9E-04  1.478E-02         +3.0E-04 -4.0E-04 Sigma(c)(2520)     ++
4232                          2.46771E+00       +2.3E-04 -2.3E-04  1.444E-12         +1.7E-14 -1.7E-14 Xi(c)               +
4132                          2.47044E+00       +2.8E-04 -2.8E-04  4.31E-12          +1.8E-13 -1.8E-13 Xi(c)               0
4322                          2.5782E+00        +5.0E-04 -5.0E-04                                      Xi(c)'              +
4312                          2.5787E+00        +5.0E-04 -5.0E-04                                      Xi(c)'              0
4314                          2.64616E+00       +2.5E-04 -2.5E-04  2.35E-03          +2.2E-04 -2.2E-04 Xi(c)(2645)         0
4324                          2.64510E+00       +3.0E-04 -3.0E-04  2.14E-03          +1.9E-04 -1.9E-04 Xi(c)(2645)         +
104314                          2.7939E+00        +5.0E-04 -5.0E-04  1.00E-02          +1.1E-03 -1.1E-03 Xi(c)(2790)         0
104324                          2.7919E+00        +5.0E-04 -5.0E-04  8.9E-03           +1.0E-03 -1.0E-03 Xi(c)(2790)         +
104312                          2.81979E+00       +3.0E-04 -3.0E-04  2.54E-03          +2.5E-04 -2.5E-04 Xi(c)(2815)         0
104322                          2.81651E+00       +2.5E-04 -2.5E-04  2.43E-03          +2.6E-04 -2.6E-04 Xi(c)(2815)         +
4332                          2.6952E+00        +1.7E-03 -1.7E-03  2.46E-12          +2.6E-13 -2.2E-13 Omega(c)            0
4334                          2.7659E+00        +2.0E-03 -2.0E-03                                      Omega(c)(2770)      0
5122                          5.61960E+00       +1.7E-04 -1.7E-04  4.475E-13         +2.7E-15 -2.7E-15 Lambda(b)           0
5112                          5.81564E+00       +2.7E-04 -2.7E-04  5.3E-03           +5.0E-04 -5.0E-04 Sigma(b)            -
5222                          5.81056E+00       +2.5E-04 -2.5E-04  5.0E-03           +5.0E-04 -5.0E-04 Sigma(b)            +
5114                          5.83474E+00       +3.0E-04 -3.0E-04  1.04E-02          +8.0E-04 -8.0E-04 Sigma(b)*           -
5224                          5.83032E+00       +2.7E-04 -2.7E-04  9.4E-03           +5.0E-04 -5.0E-04 Sigma(b)*           +
5132                          5.7970E+00        +6.0E-04 -6.0E-04  4.19E-13          +1.1E-14 -1.1E-14 Xi(b)               -
5232                          5.7919E+00        +5.0E-04 -5.0E-04  4.45E-13          +9.0E-15 -9.0E-15 Xi(b)               0
5332                          6.0461E+00        +1.7E-03 -1.7E-03  4.0E-13           +5.0E-14 -4.0E-14 Omega(b)            -
"""
p.split("\n")[3].split()
m=dict()
for line in p.split("\n")[1:-1]:
d=line[32:].split()
m[d[-2]+d[-1]]=float(d[0])
total=0
class limlist(list):
def append(self,e):
global total
total = total + 1
if e[0]/0.6666666666666666 < 0.015:
return super().append(e)
result=limlist()
for triplet in permutations(m,3):
a,b,c = triplet
a,b,c = m[a],m[b],m[c]
if b < c:
try:
k=((a+b+c)/(-sqrt(a)+sqrt(b)+sqrt(c))**2)
result.append([abs(k-2/3),triplet,k,"-++"]) #use Python3 for floats!
except:
print(triplet)
for doublet in combinations(m,2):
b,c = doublet
a,b,c = 0,m[b],m[c]
try:
k=((a+b+c)/(-sqrt(a)+sqrt(b)+sqrt(c))**2)
result.append([abs(k-2.00/3),doublet,k,"0++"])
except:
print(doublet)
for triplet in combinations(m,3):
a,b,c = triplet
a,b,c = m[a],m[b],m[c]
k=((a+b+c)/(sqrt(a)+sqrt(b)+sqrt(c))**2)
result.append([abs(k-2/3),triplet,k,"+++"]) #use Python3 for floats!
for doublet in permutations(m,2):
b,c = doublet
a,b,c = 0,m[b],m[c]
try:
if b < c:
k=((a+b+c)/(-sqrt(a)-sqrt(b)+sqrt(c))**2)
result.append([abs(k-2.00/3),doublet,k,"0-+"])
except:
print(doublet)

print(len(result),total, len(result)/total)
result.sort()
for x in result:
print(f'{"|".join(x[1]):<40}',"\t",x[2],x[3])

Some interesting tuples (and a lot of mad mixes, selected for illustration)

Code:
Z0|phi(1020)0|K*(1680)0,+                     0.6666643553495888 +++
e-|mu-|tau-                                   0.6666605124107325 +++
u+2/3|pi0|Sigma-                              0.6663665393367705 -++
d-1/3|pi+|p+                                  0.6670577228854672 -++
W+|eta(1475)0|p+                              0.6661180018623803 +++
e-|K+|B(c)+                                   0.6672821110571778 -++
pi+|D(s)+|B+                                  0.6673593324132077 -++
pi+|D(s)+|B0                                  0.6673602750393526 -++
pi0|D0                                        0.6658883844469711 0++
Z0|D0|n0                                      0.6655131325199812 +++
pi+|D(s)+                                     0.6678662052942218 0++
H0|pi+|B(c)+                                  0.665088269066683 +++
u+2/3|s-1/3|D+                                0.6650850283126151 +++
Z0|e-|B(c)+                                   0.6682520792016591 +++
Z0|tau-|n0                                    0.668553309238286 +++
s-1/3|c+2/3                                   0.6647669161994931 0++
pi0|D(s)+|B0                                  0.6644668556345085 -++
pi0|D(s)+|B+                                  0.6644658496344038 -++
t+2/3|B+|p+                                   0.6644561729935445 +++
t+2/3|B0|p+                                   0.6644518594228188 +++
c+2/3|b-1/3|t+2/3                             0.6693241713057847 +++
pi+|D(s)*+|B(c)+                              0.6635394131136364 -++
pi0|D+|B+                                     0.6698242106935278 -++
pi0|D0|B0                                     0.6700986560032072 -++
u+2/3|s-1/3|D(s)+                             0.6704131350152354 +++
pi0|D0|B(s)0                                  0.6704160867174216 -++
pi+|D+                                        0.6629167285356727 0++
s-1/3|c+2/3|b-1/3                             0.6745970625597004 -++
pi+|D+|B+                                     0.6727737789483542 -++

Last edited:
ohwilleke
The Koide formula is a relation among changes in generation. It works for the charged leptons where the triplet is (e,mu,tau), one from each generation. It doesn't work for either of the quark sets (d,s,b) and (u,c,t) so it would be a huge miracle to me if it worked for mesons that differ only in their valence quark content. (That said, for the quark Koide equations see the Piotr ̇Zenczykowski papers i.e. https://arxiv.org/abs/1301.4143 There the coincidence that is applied to the quarks is about the phase 2/9 used for the charged leptons changing into 2/3 and 1/3 that when used for the two quark sets. Accordingly, I'd be more interested in coincidences involving that phase and I don't think anyone has looked for them.)

A triplet of mesons that differ by generation would be a triplet that is different from having d, s or b valence quark(s) or (u,c,t), that is, something like a triplet (d/d, s/s, b/b) if we transform both of the quarks or (d/d, d/s, d/b) if we keep the first constant at d. The whole problem is made more difficult by the fact that the mesons do not have precise quark composition but instead are superpositions. For example, pi0 is partly d/d and partly u/u.

It would help me if you gave the quark composition, with superpositions, of the mesons that the Koide equation relates. Of course the problem with looking for coincidences among a list of states without a restriction of this sort is that the statistics have to be adjusted for the fact that any list of random numbers will have coincidences and the longer the list the more and better coincidences. This contributes to why the particle people like 5 sigma statistics but that's after the theoretical justification for the coincidence search not before they put every known meson into the hopper of the woodchipper.

arivero
It would help me if you gave the quark composition, with superpositions, of the mesons that the Koide equation relates. Of course the problem with looking for coincidences among a list of states without a restriction of this sort is that the statistics have to be adjusted for the fact that any list of random numbers will have coincidences and the longer the list the more and better coincidences.

By the way, this is one of the amazing things of the original Koide Tuple: that even after you put all the random combinations in, it stands out, even without considering error bands of each mass (to consider error bars, one could generate for each tuple one dozen or so of "mass measurements", and consider average and standard deviation when pondering the "quality" of the tuple)

As for quark content, now that I have generated the full list I am a bit amazed that the "Koide tuple" pi+,Ds+,B+ and the asociated "HHW tuple" pi+,Ds+ are better than the neutrals and better than the pi+,D+,B+.

 e- mu- tau- 0.66666 +++ pi+ D(s)+ B+ 0.667359 -++ pi0 D0 0.665888 0++ pi+ D(s)+ 0.667866 0++

So the best quark content seems to be (ud), (cs), (ub). I am not surprised because the main theme in generations is mixing, so getting isolated generations for the quark sector is not so desirable as it is for charged leptons.

On other hand, charged pi,D,B, quark content (ud) (uc) (ub) gives a Koide tuple with ratio 0.6727. Poor, but perhaps tolerable. With neutrals, pi,D,B, quark content (uu+dd), (cd+dc), (bd+db) gives a ratio 0.6701

For reference, the quark tuples, with pdg masses, are:

 c b t 0.669324 +++ s c b 0.674597 -++ s c 0.664767 0++

Worse that the tuples with mesons, but they -except for the top- are calculated, no measured, masses.

Last edited:
CarlB and ohwilleke
To get anything approaching meaningful matches for quarks you need to use an t-b-c-s-u-d array alternating up type and down type to fit the waterfall of decays. And, to make it really fit well, I think you need to think of a primary set of transitions up and down in mass, and then adjust for other possible transitions.

CarlB
To get anything approaching meaningful matches for quarks you need to use an t-b-c-s-u-d array alternating up type and down type to fit the waterfall of decays. And, to make it really fit well, I think you need to think of a primary set of transitions up and down in mass, and then adjust for other possible transitions.
And I do not forget your mixed lepton-quark tuple for the first generation.

But I am not thinking in term of transitions but of broken symmetries.

CarlB and ohwilleke
As far as proving significance, if you go to the trouble of programming the whole thing, it might be useful to make a chart showing non standard values of the Koide parameter. Then a "hit" would be a value at 2/3 that is significantly higher than the background. From that you ought to be able to get an estimate of the sigma. For a reasonably high sigma it will be a chart where it is totally obvious that 2/3 is a magic number.

This is just an ill-thought out idea. The basic objective is to make a graph where the significance of the relation is obviously above chance. Another way of doing that might be to make a graph with modified particle masses and show that the graph with the real particle masses is a lot better (but again, with this you have to avoid the selection bias so you need to include all particle masses or something like that). What I've done is to try automatic algorithms on sets of 6 masses for excitations of heavy mesons with random numbers instead of actual meson masses. I got some good data but not enough to really bother about.

ohwilleke and arivero
I ran some extra ideas and... well, the only positive thing is that the charged lepton tuple is always in the first positions.

For reference, let me include here the python algo to produce gaussian error statistics for each tuple. The code generates a sample of 1000 random masses distributed gaussian and then it averages the result:

Code:
e=dict()
for line in p.split("\n")[1:-1]:
d=line[32:].split()
e[d[-2]+d[-1]]=[float(d[1]),float(d[2])]

import numpy as np
def rmass(mass,errors):
scale = - errors[1]/errors[0]
base  = np.random.normal(mass, errors[0], 1000)
if scale==1:
calc = base
else:
calc = np.where(base < mass, base*scale, base)
return  np.where(calc < 0, 0, calc)

def rkoide(triplet,signs):
if signs[0]=="0":
a=0
mb,mc = map(m.get,triplet)
eb,ec = map(e.get,triplet)
b,c = rmass(mb,eb), rmass(mc,ec)
sign = 0
else:
ma,mb,mc = map(m.get,triplet)
ea,eb,ec = map(e.get,triplet)
a,b,c = rmass(ma,ea), rmass(mb,eb), rmass(mc,ec)
sign = +1 if signs[0]=="+" else -1
koide=(a+b+c)/np.square(sign*np.sqrt(a)+np.sqrt(b)+np.sqrt(c))
return np.mean(koide), np.std(koide ) #, np.std(koide, ddof=1)

%%time
for x in result:
k,std=rkoide(x[1],x[3])
x[0]=max(abs(k+std-2/3),abs(max(k-std,0)-2/3))
#print(f'{"|".join(x[1]):<40}',"\t{:.8f} +- {:.8f}".format(abs(k-2/3),std))

CarlB
The basic objective is to make a graph where the significance of the relation is obviously above chance. Another way of doing that might be to make a graph with modified particle masses and show that the graph with the real particle masses is a lot better (but again, with this you have to avoid the selection bias so you need to include all particle masses or something like that). What I've done is to try automatic algorithms on sets of 6 masses for excitations of heavy mesons with random numbers instead of actual meson masses. I got some good data but not enough to really bother about.
I like the idea of having a hypothesis testing comparison of Koide predicted values v. other theories advanced in the literature, or one could do Chi-squares for all of the hypotheses as a non-parametric statistic.

CarlB
Update: So I had sent the above paper to arXiv and the idiots put it "on hold" for "moderation". Of course there was no explanation. I would think that the abstract alone was enough to make it an important paper. It was there for 3 weeks and I couldn't do anything with it, just had to wait and get steadily more angry. Finally it got through moderation and was announced to be out in the next release but they had also given me back the "X" button so I deleted it. If my paper is junk it doesn't matter and if it's great they can damn well pay me for it or see people get it off of Vixra.

And at Foundations of Physics, it is still "reviewers assigned" but they updated the date from June 6 to June 17th IIRC. I'm guessing that someone waved off a review and got replaced so I'm taking that as they are working on it. If they managed to make 3 months without accepting or rejecting it I will surprise them by suddenly pulling it out and publishing it at JMP which has, like Vixra, always treated me nicely and quickly.

Finally, I'm writing a guest post for a popular blog. I'm going to blame the whole thing on Steven Weinberg's observation that density matrices have more general symmetries than state vectors. Since the Standard Model depends on temperature (that is, the SU(2) is a high temperature limit thing and is broken by electric charge at low temperatures) and since density matrices cover thermodynamics much better than state vectors, it is imperative that the density matrix symmetries be understood enough to see if they can be used to organize the Standard Model. That's basically what my paper does, but the objective of the blog is to define the possibility that there is some treasure to be dug in that region (in the event mine has an error or isn't the right way).

ohwilleke
My May 28 paper at Foundations of Physics moved from "reviewers assigned" to "under review" on August 1. I'm delaying writing the blog post until I find out what FoP is going to do with it. And I'm think that a better argument for why the subject is interesting would go roughly as follows:

(1) Was Steven Weinberg correct in his paper that mixed density matrices can have more interesting symmetries than state vectors?
(2) If yes, does this imply that we should do some research into density matrix symmetry so we can distinguish them as generalizations of state vector symmetry?
(3) Is it true that mixed density matrices are often better at modeling quantum problems that depend on temperature?
(4) If so, since the SU(2) of the Standard Model SU(3)xSU(2)xU(1) is a high temperature approximation (and is broken by electric charge at low temperatures), does this suggest that we should explore using a mixed density matrix symmetry instead of a state vector symmetry for them?

[edit, 9/22/2021] And the paper just got changed to "Reviewers Assigned" again. I'm supposing that, as expected, this is a difficult paper to review. Meanwhile, I'm working on gauge bosons. The basic idea is to first modify the quantum cellular automata to handle a single fermion of Standard Model. That should in fact be able to handle any number of such fermions provided that they are mutually orthogonal, say a spin-down and a spin-up electron, or a neutrino and an electron, etc. Then see if that can be related to a gauge boson created by the annihilation of a fermion with anti-fermion.[/edit]

Last edited:
arivero and ohwilleke
One of the deep issues in Koide type formulas that naively are based on some sort of pole mass of the fundamental particles is the nail down just how to define that concept outside of top quark mass and lepton masses that (at least in principle) can be measured directly, rather than confined with hadrons.

A new preprint examples multiple definitional choices and comes to terms with the fact that the series approximation used to convert MS mass to pole mass is not convergent and that the minimum adjustment form adding additional loop terms takes place at fewer loops as the MS mass of the quark gets smaller. Essentially, the less massive the quark, the less well defined its pole mass is in relation to its MS mass and the less meaningful the concept of a pole mass becomes.

It is https://arxiv.org/abs/2108.04861 and is well worth a lengthy read of the full text that teases out the relevant issues. A key passage in the body text states:

we observe that the top mass series attains its smallest term at the eighth order in perturbation theory, far beyond the four-loop order currently known. On the other hand, the bottom series reaches its minimal term at this order, while the charm series starts to diverge from the two-loop order, which renders the charm pole mass of limited use for phenomenology. From a pragmatic point of view, the minimal term represents the ultimate accuracy beyond which the purely perturbative use of the pole quark mass ceases to be meaningful.

[Submitted on 10 Aug 2021]

### Pole mass renormalon and its ramifications​

Martin Beneke
I review the structure of the leading infrared renormalon divergence of the relation between the pole mass and the MS⎯⎯⎯⎯⎯⎯⎯⎯⎯ mass of a heavy quark, with applications to the top, bottom and charm quark. That the pole quark mass definition must be abandoned in precision computations is a well-known consequence of the rapidly diverging series. The definitions and physics motivations of several leading renormalon-free, short-distance mass definitions suitable for processes involving nearly on-shell heavy quarks are discussed.

The extended Koide's rule does produce light quark masses that are in the right ballpark of the MS mass at 1-2 GeV for the strange, down and up quarks, despite the fact that pole mass is completely meaningless for these quarks which are always confined in hadrons not less massive than the ca. 130 MeV pion, which is orders of magnitude more than the MS masses of these quarks. So, literal pole mass is clearly not what Koide's rule is pointing towards. But, it isn't at all obvious which of the half dozen mass renormalization schemes discussed in this article really comes closest to what the extended Koide's rule is pointing us towards.

The same definitional issues arise when trying to evaluate the LP&C conjecture that the sum of the square of the fundamental particle masses of the Standard Model is equal to the square the Higgs vacuum expectation value, or equivalently, that sum of the Yukawas (or Yukawa equivalents) of the fundamental particles of the Standard Model are equal to exactly 1. Indeed, perhaps the strength of the Higgs field coupling of a fundamental particle, rather than its "pole mass" is really what both LP&C and any extended Koide's rule should actually be chasing.

Last edited:
CarlB
The citation of this paper is particular provocative. https://link.springer.com/article/10.1140/epjc/s10052-016-3990-3 the abstract says:

Two empirical formulas for the lepton and quark masses (i.e. Kartavtsev’s extended Koide formulas), 𝐾𝑙=(∑𝑙𝑚𝑙)/(∑𝑙𝑚𝑙‾‾‾√)2=2/3Kl=(∑lml)/(∑lml)2=2/3 and 𝐾𝑞=(∑𝑞𝑚𝑞)/(∑𝑞𝑚𝑞‾‾‾√)2=2/3Kq=(∑qmq)/(∑qmq)2=2/3, are explored in this paper. For the lepton sector, we show that 𝐾𝑙=2/3Kl=2/3, only if the uncertainty of the tauon mass is relaxed to about 2𝜎2σ confidence level, and the neutrino masses can consequently be extracted with the current experimental data. For the quark sector, the extended Koide formula should only be applied to the running quark masses, and 𝐾𝑞Kq is found to be rather insensitive to the renormalization effects in a large range of energy scales from GeV to 10121012 GeV. We find that 𝐾𝑞Kq is always slightly larger than 2/3, but the discrepancy is merely about 5 %.

See also https://arxiv.org/abs/0812.2103 (mildly interesting by a Koide collaborator) and https://arxiv.org/abs/1809.00425 (koide reflecting)

https://www.amazon.com/gp/product/B08ZCF9T99/?tag=pfamazon01-20 (unimpressive)

Surely as a consequence of Ethan's post, this week I received an email about Koide fórmula, criticising the format of formula

Somehow it seems that the use of sum and product is not an intuitive way to express the solution of koide formula. I find it useful, for instance if one of the masses is zero, it is immediately apparent the factor of Harari et al.

ohwilleke