What is new with Koide sum rules?

[arivero]

I think that the drawing is at least the second version from De Rujula, but the only one he found in his archive as it is used in some publication. I saw another version during a talk in my hometown, I was in the first course of the undergraduate physics studies, and someone did a series of talks addressed to secondary school teachers. Around 1985-1986 then.

Yes, all the masses are in MeV. What we are doing here is just apply the mass formula $M(a,b)= k (z_a+z_b)^2$. When one of the charges is constant, say z_q, and the other three sum zero $z_1+z_2+z_3=0$, then we have $k(z_q+ z_i)^2$ is a koide formula if and only if $3 z_q^2 = z_1^2+z^2+ z^3$. The zero sum rule is the tradicional cosine of the most common version here in the thread. I have used alpha instead of delta for the phase because I am doing more jumps than usual. Koide phase is not only periodic 2 pi /3; it also allows reflections on pi/4 and pi/2. So I am not using the usual phase for the leptons but a reflected one.

Besides checking that I am recovering the usual koide formula, I was interested on the singular points and what kind of particles I get. I went with MeV masses because I was tired of exact roots of two, roots of three etc.

The particles (or pairs of preons) are organised according representations of SU(3)xSU(2). When the SU(2) is a singlet, it means the two particles are from the SU(3), ie they are pairs from the ones I call d,s,b. When the SU(2) is a doublet, one of the particles is a "c" or a "u". For these particles, the charge "z_c" does not come from the cosine but from the rule of the average of squares. I mean, the are the "z_0" of usual koide formula.

In the case pi/4 the charge of "s" coincides also with z_0, so what happens is that we get a massless lepton and that the s particle can also act to produce a pretty exotic koide tuple in the quark sector ds,ss,bs

(More properly, the "antiquark" sector, as a pair such as ds is an antiquark)

I am impressed by the "sneutrino" sector, as I had not anticipated the obvious thing that half of the
scalar neutrinos are massless, independently of the koide phase.

 

[arivero]

Mitchel asked me if I can tell something about the preons in each tuple. I do not see any connection beyond the factor three we knew, and the coincidence with the current quark mass, 313MeV.

What we have, with all the masses in GeV and all current values from pdg, is

mt=(√29.67+√59.13)^2
mb=(√29.67-√11.57)^2 =(√0.925+√1.174)^2
mc=(√29.67-√18.65)^2 =(√0.925+√0.028)^2
ms =(√0.925-√1.61)^2

tau=(√0.3139+√0.5972)^2
mu=(√0.3139-√0.05531)^2
e=(√0.3139-√0.289)^2

EDIT: on inspection, there are some near integer multiples:
2 * 29.67 -> 59.13 <- 99*0.5972
2* 0.0277379 = 0.0554759 -> 0.0553085
2* 0.5972 -> 1.174
2* 0.289 -> 0.5972
4* 0.289 -> 1.174
other worse
33*0.028 -> 0.925
40*0.289 -> 11.57
...
and including the original quarks and leptons:
7 * 0.5972 -> mb
3 * 0.5972 -> tau
mt/ms -> mproton/me -> 0.925/me
4*0.3139 -> mc
3*ms -> 0.289

Not sure how relevant; or perhaps I should also check with mc_predicted, ms_predicted instead of measured. Note that the tau, mu, e is precise enough to discard exactitude of all of these two factors. They look just as QCD-like quantities: 289 MeV, 55 MeV, 597 MeV, 313 MeV (the average of all the other three, as required by Koide)...

 

[arivero]

As I mentioned in the sBoostrap thread, I had put in a preprint some calculations of Koide masses using the original composite idea and they have happened to be published as Eur. Phys. J. C 84, 1058 (2024). https://doi.org/10.1140/epjc/s10052-024-13368-3, so as a collateral effect we now have another published paper that mentions:

• the waterfall, in a footnote.
• the tuples (0ds), (scb) and (cbt).
• the relation sum(scb) = 3 sum (leptons)

 

[ohwilleke]

As I mentioned in the sBoostrap thread, I had put in a preprint some calculations of Koide masses using the original composite idea and they have happened to be published as Eur. Phys. J. C 84, 1058 (2024). https://doi.org/10.1140/epjc/s10052-024-13368-3, so as a collateral effect we now have another published paper that mentions:

• the waterfall, in a footnote.
• the tuples (0ds), (scb) and (cbt).
• the relation sum(scb) = 3 sum (leptons)

Congratulations!

 

[mitchell porter]

A few comments on the waterfall, from a broadly orthodox model-building perspective.

The heart of the waterfall is the idea that the quarks "tbcsud" (I'm on my phone and won't attempt sophisticated formats) form a set of four sequential overlapping "Koide triplets" tbc, bcs, csu, sud, each of which satisfies the Koide formula.

Thanks to R. Foot, we know that a Koide triplet of masses can be represented as a certain type of three-dimensional vector. The waterfall as a whole would have some kind of representation in terms of overlapping Foot vectors in a six-dimensional space, a "spiral staircase" shape. One might hope that this shape represents the minimum of a potential, or even an actual arrangement of branes in extra physical dimensions.

However, in the standard model, the quark masses are eigenvalues of the up and down yukawa matrices. Assuming that the yukawas are dynamically determined, one would therefore be looking for a symmetry and/or dynamics in which the eigenvalues of those matrices are coupled appropriately.

Suppose we go further and ask how this looks from the perspective of grand unification, like SU(5) or SO(10). Naive SU(5) comes out wrong since it implies that the masses of d, s, b quarks are the same as the masses of electron, muon, tau. Usually it is hoped that this is ameliorated by the running of the yukawas, sometimes with the influence of some extra new fields included. The Georgi-Jarlskog ansatz is one that has been mentioned a few times here.

With respect to the construction of quark Koide triplets in such a context, the three fermion generations are independent and so one would be free to aim at triplets like tcu or bsd. But the waterfall calls for Koide triplets that have two quarks from the same generation. It's not clear to me yet, if there are extra constraints when trying to construct the waterfall from yukawas in SU(5) unification...

Meanwhile, I will mention another aspect of the waterfall, the relationship between the bcs triplet and the original electron, muon, tau triplet. This contains 5 out of the 6 fermions involved in the SU(5) relationship mentioned earlier, but with charm quark replacing down quark. It also involves a factor of 3, such as shows up in Georgi-Jarlskog (where it's due to the three colors of SU(3)). Could one possibly obtain the waterfall relationship from a twisting or deformation or modification of the SU(5) GUT relationship?

Finally, I'll mention grand unification in the context of "F theory", which is a corner of string theory in which phenomenology involves intersecting branes in the extra dimensions. Gauge fields are associated with individual branes, other particles (fermions and Higgs) at the intersection of two branes, and yukawa interactions (of Higgs, left fermion, right fermion) at the point intersection of three branes.

I mention this because it's a highly geometric framework in which one could try to embed or realise something like the staircase of Foot vectors mentioned above. But to realise the waterfall there, you'd have to deal with the peculiarities already mentioned - the coupling of eigenvalues, and the non-standard but SU(5)-like relationship to the yukawas of the charged leptons.

 

[ohwilleke]

A few comments on the waterfall, from a broadly orthodox model-building perspective.

The heart of the waterfall is the idea that the quarks "tbcsud" (I'm on my phone and won't attempt sophisticated formats) form a set of four sequential overlapping "Koide triplets" tbc, bcs, csu, sud, each of which satisfies the Koide formula.

Thanks to R. Foot, we know that a Koide triplet of masses can be represented as a certain type of three-dimensional vector. The waterfall as a whole would have some kind of representation in terms of overlapping Foot vectors in a six-dimensional space, a "spiral staircase" shape. One might hope that this shape represents the minimum of a potential, or even an actual arrangement of branes in extra physical dimensions.

Improving On The Waterfall and Conceptual Implications Of This Improvement

One notable point is that you can significantly improve the fit of the Koide triple waterfall by adding an adjustment for the third quark to which the middle quark in the waterfall can transform via a W boson interaction with the mass of that third quark and the probably of a W boson transformation to it (from the relevant CKM matrix element squared).

I worked that out here. The waterfall values are:

Inputs
me = 0.510998910 MeV ± 0.000000013 (i.e. one part per 39,307,608)
mμ = 105.6583668 MeV ± 0.0000038 (i.e. one part per 2,780,483)

Outputs
m = 1776.96894(7) MeV (Tau) - PDG 1776.82 +/- 0.16 (i.e. one part per 11,105) (0.93 SD)
mt = 173.263947(6) GeV (top) - PDG 173.070 +/- 0.888 (i.e. one part per 194) (0.22 SD)
mb = 4197.57589(15) MeV (bottom) - PDG 4180 +/- 30 (i.e. one part per 139) (0.58 SD)
mc = 1359.56428(5) MeV (charm) - PDG 1275 +/- 25 (i.e. one part per 51) (3.38 SD)
ms = 92.274758(3) MeV (strange) - PDG 95 +/- 5 (i.e. one part per 19) (0.55 SD)
md = 5.32 MeV (down) - PDG 4.8 +/- 0.4 (i.e. one part per 12) (1.3 SD)
mu = 0.0356 MeV (up) - PDG 2.3 +/- 0.6 (i.e. one part per 4) (2.26 SD)

Koide ratios of PDG mean values of selected triples is as follows:
t-b-c 0.6695
b-c-s 0.4578
c-s-u 0.622
s-c-d 0.60563
s-u-d 0.564

The worst fits (the central c quark and the central u quark) are cases where the probabilities of the central quark transforming to a third quark are the greatest. The best fit t-b-c are cases where the probabilities of the central quark transforming to the third quark are smallest.

The values with the third quark adjustment are:

mt=172.743 GeV PDG 173.070 +/- 0.888 per t-b-c avg adj down with ts (0.16%) and td (7.52*10^-5)
mb=4193 MeV PDG 4180 +/- 30 per b-c-s adjusted down with ub (0.11%)
mc=1293 MeV PDG 1275 +/- 25 per b-c-s adjusted down with cd (4.9%)
ms=92.55 MeV PDG 95 +/- 5 per b-c-s avg adj up with ts (0.16%) and down with us (4.97%)
md= 5.12 MeV PDG 4.8 +/- 0.4 per s-c-d avg adj of up with td (7.52*10^-5) and down with ud (94.9%)
mu=4.60 MeV PDG 2.3 +/- 0.6 per s-u-d avg adj up with ub (0.11%) and up with us (4.97%)

The adjustment bring all of the formula values for quark masses (and the tau lepton) except the up quark to within 0.8 standard deviations of the experimental values and to the right order of magnitude in the case of the up quark - now off by a factor of 2 rather than a factor of 64.6 - much closer to the mark on a percentage basis - without any experimental inputs other than the electron mass, muon mass and several of the four parameter CKM matrix element values! Thus, the formula comes very close to reproducing the Standard Model values despite dispensing with 7 of the experimentally measured parameters of the Standard Model. . . .

Before v. After Adjustments Experimental Standard Deviations Between Theory and Experimental Value
top 0.22 v. 0.368 SD
bottom 0.58 v. 0.433 SD
charm 3.38 v. 0.72 SD
strange 0.55 v. 0.49 SD
down 1.3 v. 0.8 SD
up 2.26 v. 3.83 SD

Conceptually, the idea that this suggests is that the quark masses are the result of dynamical balancing via W boson interaction (real and virtual) of all possible transformations of a particle into a different particle, weighted by the probabilities of each possible transformation, according to a balancing formula of which Koide's Rule is a special case.

Koide's rule is so close to perfect (in contrast with extended Koide's rule for the quarks) because of charged lepton universality, which making the weighting the different transformation possibilities trivial, and because the neutrino masses are so negligible that they don't materially tweak the values reached with the charged leptons alone.

This also suggests that, conceptually, the CKM matrix probabilities are conceptually prior to the quark masses. Likewise, charged lepton universality is conceptually prior to the charged lepton masses.

If I had a bit more mathematical and particle physics chops, I would think that it would be possible to take this basic intuition and craft it into a more rigorous and natural mathematical formulation and to simultaneously solve for all of the quarks at once (as the approach I used is really a tree level adjustment that doesn't take into account "higher loop" effects where the adjustments affect other adjustment and core Koide value calculations).

The respective roles of the Higgs boson and W boson in setting fundamental fermion masses

This view essentially gives the W boson more importance and the Higgs boson and field less importance in setting the particular fundamental fermion masses.

The Higgs vev which is a function of the W boson mass and the weak force coupling constant, sets the mass scale for all of the fundamental Standard Model particles collectively (the sum of the Higgs field Yukawas of the fundamental Standard Model particles is within two sigma of exactly 1, so each Yukawa is basically allocating a percentage of the Higgs vev to that particular particle), under the phenomenological LP & C relationship.

But the value of those Yukawas is dynamically balanced between particles based upon their W boson transformations into each other.

The Quark and Charged Lepton Mass Coincidence

The coincidence that allows the mass scale of the quarks to be derived from the mass scale of the charged leptons is something of a mystery in this model. But there is probably something to it because it works.

Maybe W and Z boson decay probabilities, which have a three x factor for quarks due to their three color charge variants plays a part in this coincidence.

Footnote On Neutrino Masses

It is also notable that the ratio of the lightest neutrino mass to the electron mass is probably on the same order of magnitude as the ratio of the weak coupling constant to the electromagnetic coupling constant, perhaps suggesting perhaps that the first generation lepton masses may be a function of their self-interactions (something that has been suggested before in the case of the electron). Of course, as Brannan correctly discerned, the neutrino masses can't form a good Koide triple without a sign change for one of those masses. Still, I'm optimistic that the neutrino masses could be due to W and Z boson interactions with neutrinos, rather than the Higgs mechanism, which would allow them to have Dirac mass despite not having both left and right handed versions.

Notably, all fundamental SM particles that have non-zero masses have weak force interactions, while all fundamental SM particles that have zero masses do not interact via the weak force. This is also suggestive of a W boson role in establishing those masses.

 

[arivero]

mc = 1359.56428(5) MeV (charm) - PDG 1275 +/- 25 (i.e. one part per 51) (3.38 SD)

yep I did not go very deep on this on the paper; it was supposed to be a letter and I was expecting the referees to ask me to delete some section. But it is amusing that if one just ignores or corrects the charm failure, it goes better.

It seems to me that your idea is to relate the failures to mixing. That would be very in the line of the first paper in the saga, H. Harari H. Haut J. Weyers Phys. Lett. B 78 459–461 (1978), I believe.

 

[ohwilleke]

yep I did not go very deep on this on the paper; it was supposed to be a letter and I was expecting the referees to ask me to delete some section. But it is amusing that if one just ignores or corrects the charm failure, it goes better.

It seems to me that your idea is to relate the failures to mixing. That would be very in the line of the first paper in the saga, H. Harari H. Haut J. Weyers Phys. Lett. B 78 459–461 (1978), I believe.

Thanks for the reference.

 

[mitchell porter]

We are lacking a mechanism to produce a Koide relation among masses in a triple like (uds), as opposed to triples like (uct). The latter masses are eigenvalues of a single mass matrix, and one can express the formula in terms of the properties of a single matrix, e.g. Carl Brannen's work. (uds) mingles eigenvalues from up and down mass matrices. The (uds) mass values imposed by symmetries in Harari et al form a Koide triple, but not in a stable way (i.e. the property is not preserved if you change the parameters).

We should investigate e.g. dependence of matrix eigenvalues on parameters, as a step towards understanding joint dependence of eigenvalues of two matrices on the same parameters, as a step towards finding ways to enforce a waterfall relation.

@ohwilleke suggests a "dynamic balancing" involving the weak interaction, which is at least capable of coupling u-type and d-type quarks. (I would suggest focusing, not just on the W boson, but on its spin-0 component specifically, which after all comes from the Higgs field.)

Now, we already know effective mass and charge can be renormalized by virtual particles. However, this doesn't involve the kind of reciprocity among particle species that Andrew's slogan suggests to me. Reciprocity is more reminiscent of the flavon idea, according to which yukawas aren't just parameters, but are actually vevs of dynamical scalar fields that can interact with each other via some potential.

So I'm inclined to think in terms of some Higgs-flavon interaction potential (and from a stringy perspective, all these vevs might be moduli, i.e. sizes of cycles in the extra dimensions, distances and angles between branes, magnitudes of stringy p-form "fluxes", and so forth).

 

[mitchell porter]

A practice problem would be:

Suppose you have two 2x2 matrices f_ij and g_ij which diagonalized are diag(a,c) and diag(b,d).

What kind of "interaction potential" involving the fs and gs, results in (abc) and (bcd) satisfying the Koide formula?

edit: Also, a further thing to think about. I have thought of a Koide triplet across the fermion generations, like the original (e mu tau), as more natural, because (considered as mass matrix eigenvalues) those quantities all come from the same yukawa matrix, whereas these "sequential" triplets like (scb) mingle parts of different yukawa matrices.

However, eigenvalues always come with eigenvectors. So if we really need a "matrix" perspective on these unnatural sequential triplets, we can assemble a corresponding matrix out of the eigenvectors from the original yukawa matrices.

If that's too abstract: We are dealing with two yukawa matrices, one for up-type quarks, the other for down-type quarks. The quark masses are eigenvalues of the yukawa matrices. As eigenvalues, they will have accompanying eigenvectors. Each triplet in the waterfall consists of two quarks of one type, and one quark of the other type. So for each such triplet, we can assemble an "artificial matrix" out of the corresponding eigenvectors. The question then, is whether a potential expressed in terms of these "artificial matrices" (which would be most natural for obtaining the waterfall), could be re-expressed in terms of the physical matrices?

(This chain of thought inspired by reading about the "eigenvector-eigenvalue identity" that was rediscovered by neutrino physicists.)