What is new with Koide sum rules?

[arivero]

Time ago the hard printed edition of the pdf had a last page with an illustration of how the measures had evolved across editions. Remembering that, I have setup a similar evolution for the Koide tuples

https://arivero.github.io/pdghist/predictions.html

 

[mitchell porter]

How do these papers fit into a larger research program of either of the authors?

Neither author has prior work on Koide that I can see. Hübner did some QCD and thus he knows QFT topics like renormalization schemes. Shulga worked on qubits from a physical rather than computational perspective (e.g. superconducting qubits), and maybe something there inspired his mechanism. But otherwise it seems a coincidence that two independent authors had their (very different) first Koide papers released on arxiv on the same day.

 

[arivero]

it seems a coincidence that two independent authors had their (very different) first Koide papers released on arxiv on the same day.

Actually three Mine was removed because just resending now causes an autodelete, even if the resend includes doi and journal. It is a sort of peak surely caused by the AI leverage, but still funny.

 

[Roberto Pavani]

I've been looking at the geometric origin of the phase $\delta = \pi/12$ (15°) that appears in the Koide parametrization for charged leptons.

In 4D Euclidean space, the rotation group decomposes as $SO(4) \cong SU(2)+ \times SU(2)-$ (isoclinic decomposition).
If you have a helical structure with $C_3$ symmetry (three-phase, 120° shifts), the closure condition at a nodal surface forces a total twist of $\pi/2$.
The three-phase offset accounts for $\pi/3$ of that, leaving a residual of $\pi/6$, distributed equally between the two isoclinic planes: $\pi/12$ per plane.

This gives $\delta = \pi/12$ as a geometric necessity, not a fit. The factor 3 between leptons ($\pi/12$) and quarks ($\pi/4 = 3 \times \pi/12$) would then correspond to the three color directions in the transverse space.

Has anyone explored the connection between the isoclinic decomposition of $SO(4)$ and the Koide phase? The coincidence $\pi/12 \leftrightarrow 15°$ seems too precise to be accidental, especially given that $SO(4) \cong SU(2) \times SU(2)$ is unique to 4 dimensions.

 

[mitchell porter]

π/12 appears as part of a formula for a Koide phase in the neutral leptons (see Brannen's "Koide Mass Formula for Neutrinos"), not charged leptons. The phase for the latter is 2/9 radians (no appearance of π). The neutrino masses are not known with great precision and so there is no empirical confirmation of this π/12 formula, the way there is for the 2/9 formula, it is more theory-driven. It would be interesting to try to adapt Shulga's mechanism, mentioned in immediately preceding posts, and which introduces and then cancels factors of π, to the Brannen neutrino phase. And it would be miraculous to find an "isoclinic" origin of 2/9!

 

[Roberto Pavani]

Thank you for the correction — you're right that the Brannen phase for charged leptons is 2/9 radians (no π), not π/12. I was imprecise.

Let me reformulate the observation more carefully.
The connection I'm exploring is indirect: in 4D Euclidean space with a three-phase helical structure ($C_3$ symmetry, 120° shifts), the closure of the helix at a nodal surface forces the gradient perpendicular to the axis (total angle $\pi/2$).
The three-phase offset accounts for $\pi/3$ of that rotation. The residual $\pi/6$, distributed equally between the two isoclinic planes of $SO(4) \cong SU(2)_+ \times SU(2)_-$, gives $\pi/12$ per plane.

This $\pi/12$ doesn't claim to be the Brannen phase. Rather, it enters the coherent closure condition as $7\pi/12 = 2\pi/3 - \pi/12$ (the target phase for counter-rotating closure). When this closure is imperfect (small mismatch $\delta$), the forced precession onto one of the three $C_3$ directions introduces a correction of the form $\sqrt{m_i} \propto 1 + A\cos(\theta_0 + 2\pi(i{-}1)/3)$, and the trifase identity $\sum_{\tau=0}^{2}\cos^2(\theta_0 + 2\pi\tau/3) = 3/2$ then gives $K = 3/2$ exactly independently of $A$ and $\theta_0$.

So the question I should have asked is: has anyone explored the isoclinic decomposition of $SO(4)$ as the geometric origin of the closure condition from which a Koide-type parametrization can emerge?
The factor 3 between leptons and quarks would then correspond to the number of transverse directions (color), not to the phase itself.

Regarding Shulga's mechanism, interesting. His compact dimension with antiperiodic boundary conditions and the Casimir-like shift producing 2/9 is structurally reminiscent of a closure condition on a cycle.
I'd be curious whether his "a-field" dressing could be reinterpreted as a frame connection on a helical structure.

 

[arivero]

they are a lot of elegant results that are just generic of the theory of quadrics. Basically it is possible, except for the integrality, to repreoduce all the symmetries of https://en.wikipedia.org/wiki/Markov_number Markow trees of tuples, find the critical coefficient where the atractor of the tree becomes fractal, look for accumulation points. A loot of very pretty geometry, but that really one can do with different kinds of quadrics.

 

[jedishrfu]

Your post seems to be related to your Zenodo papers. I’d like to remind you that these questions are okay as stand-alone questions but not if they bring in your unpublished Zenodo papers. PF explicitly state that we only discuss peer-reviewed papers from reputable journals. Sadly, zenodo does not count as either a published paper nor as a reputable journal.

If the discussion does extend into your Zenodo papers then the PF mentors will be forced to delete your relevant threads and give you a personal speculation warning or personal theory site ban.

Please take some time to read our site global guidelines.

 

[Roberto Pavani]

Noted. My posts in this thread discuss the isoclinic decomposition of SO(4) and its relation to closure conditions, standard differential geometry. No personal papers are referenced or needed to follow the argument.

 

[arivero]

Linking the question of Pavani with mine, the point is that here there is a fine thread for Mathematics that somehow is not fully exposed in online forums: to characterise the solutions of permutation invariant equations Q(x_1,x_2,...x_n)=0. Markov's
$$x^2 + y^2 + z^2 - K x y z = 0$$
have been very studied because it has Diophante solutions when K=3.
Koide generalisations
$$(x^2 + y^2 + z^2 )+ K \times (xy + yz + zx) = 0$$
are fine objects too, but they have no such elegant diophantine solutions and it is not easy to find studies going concrete with this, or at least digitised ones. This history goes as deep as Newton and Girard, and of course linked to the problem of having all the solutions as roots of a single cubic equation. Pure maths, anyway.

EDIT: I am recalled that the fruity meme problem is also in this category of permutation invariant equations, with k=4 in

$$(k+2)(x+y+z)(x^2+y^2+z^2)+2(k+3)xyz - k (x+y+z)^3=0$$

and I think that we already did the joke of writing koide in fruity way.

 

[mitchell porter]

In "Fermion mass relations in one-parameter modular models", page 18, we are given an ansatz

m_e : m_µ : m_τ = 4√2 ϵ^6 : 2 ϵ^2 : 1

Is this like any of the rewritten forms of Koide?

 

[arivero]

IFIC, Irvine, mexico and INFT, quite a collaboration! textured modulad forms look always as a way to hide the Wilzeck-Zee-Nambu perturbative approach to the muon mass.

 

[mitchell porter]

the Wilzeck-Zee-Nambu perturbative approach to the muon mass.

I wondered what this is and found an older comment from you

https://www.physicsforums.com/threa...ncidence-or-significant.1058414/#post-6979218

AI suggests you are combining "“A New Approach to the Electron-Muon Mass Ratio" (and successor papers) by Barr and Zee, with “Horizontal Interaction and Weak Mixing Angles” by Wilczek and Zee...

As for the comparison with modular symmetry... These Zee et al models basically work by loop effects (radiative corrections, usually involving interaction with new fields), and give mass matrices where the entries are powers of some small parameter. My understanding of modular symmetry models is still very fuzzy, but there I think that, although you can get similar-looking mass matrices, the various powers arise from representation theory of the modular group - modular forms have "q expansions" and the modular representation of a field means that only one term in the expansion remains nonzero and physically relevant... There might be a more concrete explanation when you look for realizations of modular symmetry within string theory, but it depends on the particular mechanism (there is more than one way to obtain modular symmetry within string theory), e.g. in one stringy approach, it's the winding number of a virtual string which controls the powers... The best guide for model building is probably to try to obtain circulant mass matrices a la @CarlB.

 

[arivero]

Yes! From AI hallucinated citations to AI correcting hallucinations. Yes, WZ is the one that help to the frenzy of CKM angles, while BZ is about trying to justify "alphaology" mass ratios.

 

[mitchell porter]

Meanwhile, AI has also pointed out to me

"The exact column texture: tree-level Yukawa universality in heterotic Z_3 x Z_3 orbifolds" (Navid Ardakanian)

which claims (section 8.3) that up-quark and down-quark yukawa matrices in this class of models are, in a first approximation, left-circulant. (If they are exactly left-circulant, there is no CKM mixing, so there must be corrections, possible sources of which are also itemized in the paper.)

I do not recall ever before seeing a string paper with circulant yukawas so this grabs my attention. Of course originally we want this for the charged leptons, and in these models the leptons arise differently to the quarks (section 2, second paragraph). Nonetheless, approximately-circulant yukawas for up and down quarks are my preferred explanation for the waterfall (#291 in this thread), and @arivero 's new paper reports a good "inverse-Koide" formula for the downquarks which also implies circulant yukawas (#321, #324). So this surely deserves some kind of study.

However: the author is an independent researcher with his own genAI business and no visible history as a physicist prior to this year (he has four papers on arxiv now), who lists Claude in the acknowledgments. So one might want to be on guard, even more than usual, as to whether AI is telling us what we want to hear, rather than the truth.

 

[mitchell porter]

"The Very Nearly Right Theory of Flavor" (Arkani-Hamed, Figueiredo, Hall, Andrea Manzari)

Even more than #319, this seems compatible with the waterfall since it is complementary - it's about the phases in the quark mass matrices, rather than the mass eigenvalues. I just had a long dialog with "GPT 5.6-Sol high", and after singling out the texture identified on page 18 as number 24 in figure 6 (because the mass of the charm quark has a special origin in this scenario, and the charm quark is arguably a pivot point in the waterfall), "we" (mostly the AI) went on to identify a list of mechanisms that would start with circulant up and down quark yukawa matrices (as arguably suggested by the waterfall), and then perturb them to put the running values of everything within 1% of the standard model error bars. The implied field theory would be full of new fields and interactions of the kind that are popular with mainstream model builders, something which is unappealing to those of us who expect beyond-standard-model physics to be more minimal and elegant, but it's interesting to see (1) that the waterfall can dovetail with a cutting-edge mainstream proposal (2) that AI can already engage in theory construction at this level.

 

[mitchell porter]

"On the Flavor Yukawa coupling for the Lepton Mass Hierarchy and the Koide Relation" (Olivier Rousselle)

This paper builds on #327 by Shulga, using ansatze in a QFT framework. The author is French and cites the oft-overlooked Francis Goffinet; he seems to have a math PhD from Sorbonne and some association with a Louis de Broglie Foundation.

To obtain Higgs-fermion couplings for the charged leptons, he supposes a complex 3x3 field S which he calls a flavor spurion and which another author might have called a flavon. The field coupling schematically is

(left-fermion) (S-transpose) (Higgs) (S) (right-fermion)

S is supposed to initially possess a U(3) flavor symmetry that is broken to U(1) x Z3. This gives S a circulant vev and the charged lepton yukawa matrix is (S-transpose) (S). The Koide constant Q comes out as 2/3 because energies associated with the flavor-singlet and flavor-octet parts of S are assumed equal; the Brannen phase comes out as 2/9 because of a "topological condensation" that obeys a formula due to Shulga (equations 12-13 here).

I am here just paraphrasing the paper's own assertions. Whether the physical definitions and mechanisms actually make sense, I cannot yet testify, but it seems plausible that some version of them should make sense.

The paper also makes use of an alternative formulation of the Koide relation (ultimately due to Koide himself), in terms of quantities z0,...,z3, that @arivero also used in a 2024 paper (see #277 forward, this thread). In Rousselle, these come from the singlet-octet decomposition of the eigenvalues of S; in Rivero, they are charges of preons, raising the question as to whether this Rousselle-Shulga scenario could be realized within Rivero's framework.