# What is new with Koide sum rules?

1. Feb 15, 2013

### mitchell porter

You could also: start with a 6x6 mass matrix including fictitious "up-down yukawas" as I have suggested, impose Goffinet's property on each of the four 3x3 blocks on the diagonal, and on the two larger blocks as in #82, and then finally impose a "checkerboard texture" in which all the "up-down yukawas" are set to zero, as in #73... and then see if the two larger blocks ever resemble the actual yukawa matrices.

Two problems: first, the SM yukawas are complex-valued and underdetermined by the experimental data (PDG). One would need to decide if the elements of the matrix M are the SM yukawas or secondary quantities derived from them. Second, the larger blocks are there in order to produce family Koide triplets, as in Zenczykowski; but Z's Koide triplets are made of Goffinet's pseudomasses, which are obtained by applying the CKM matrix to a vector of masses. It's not clear to me whether or not the larger blocks should be transformed somehow, before the Goffinet property is imposed.

2. Feb 20, 2013

### arivero

Just a thinking... Koide masses are fixed by an angle theta and a mass M_0, which is proportional to the trace. So if we suplement the above equation with the already know Tr M = 6 M_0, it takes a look very much as a the terms one usually sees in generalised Higgs potentials.

$$0 = {1 \over 1536} (252 m_0^2 - 8 Tr M^2)^2 - 6 m_0 Det(M)$$

3. Jun 3, 2013

### mitchell porter

There is a very phenomenological paper from Koide (and colleague Ishida) today. It seems to be the first paper that talks about adapting Sumino's mechanism to the quarks.

But let's take a step back. Koide found his formula 30 years ago. Koide has proposed a number of field-theoretic models to explain it; so have a few other people (actually, who else has made a proper field-theoretic model, apart from Ernest Ma?). All QFT models of the relation have the problem that there should be deviations from the formula, because of quantum corrections, but empirically it is exact within error.

Yukinari Sumino was the first person to develop a model in which the corrections are cancelled. It's a little complicated, but it involves a family symmetry that is gauged and then spontaneously broken. The heavy family gauge bosons do the cancelling of the corrections coming from QED.

Koide and Yamagarbagea adapted Sumino's mechanism to supersymmetry. The present paper does not mention supersymmetry, but it does assume the modified version of Sumino's mechanism (in which the mass hierarchy of the family gauge bosons is inverted, compared to Sumino's original version).

Koide and Ishida's inspiration is a tiny aberration in the data for B meson decays. I still haven't digested the paper, but they seem to say at the end that, naively, even a Sumino meson shouldn't be able to produce the dip (that may be there, or which may go away with more data). But there could be some enhancement, and, importantly for them, if the dip is due to their family bosons, then a corresponding dip will not appear in another particular measurement.

From my perspective, this paper runs ahead of theory, because we still have no field-theoretic model of any of the generalized Koide relations for quarks, let alone adaptations of the Sumino mechanism for such models. Koide's own recent BSM work generally assumes that there's a nonet of scalars whose VEVs are diag(√me,√mμ,√mτ), and then he builds mass matrices for all the SM fermions out of couplings to these. It is from within this theoretical context that he will have guessed at the quark couplings with the Sumino mesons.

Since the quarks have their own Koide relations, it seems very unlikely that their masses are produced in the manner of Koide's recent models. Still, it's always useful to have papers that go "too far ahead" - in this case, trying to interpret a known anomaly as a signal of quark-sector Sumino mesons! Thinking about how the ideas in the paper work, may help those of us still struggling to find an approach to "Koide for quarks".

4. Aug 3, 2013

5. Aug 4, 2013

### arivero

Also, perhaps all the thing about sqrt(M) is a red herring. We could just contemplate a correction "susy-like" going only up to order two,

$$M_i = (1 + \lambda_i + \lambda_i^2) M$$

and then Koide eq is the system $Tr \lambda = 0$, $Tr (\lambda^2) = Tr(1)$

6. Aug 6, 2013

### mitchell porter

Maybe I'm stupid but I don't understand any of those equations. What matrices are $M$, $M_i$, $\lambda_i$? What is the $\lambda$ in the final equations?

edit: Let me guess... The first quantities are all scalars. $M$ is a Koide-Brannen mass scale, $M_i$ is the mass of the $i$th member of the corresponding Koide triple, and $\lambda$ is a matrix with the $\lambda_i$s as eigenvalues??

Last edited: Aug 6, 2013
7. Aug 6, 2013

### arivero

Ok, Trace was a bit of pedantry. Instead, say
$$\lambda_1+ \lambda_2 +\lambda_3 =0$$
$$\lambda_1^2 +\lambda_2^2 +\lambda_3^2 =3$$
And I have forgot a factor 2, have I? It should be
$$M_i=M (1 + 2\lambda_i + \lambda_i^2) = M (1 + \lambda_i)^2$$

Well, perhaps the importance of sqrt(M) is not a redherring, at all.

Last edited: Aug 6, 2013
8. Aug 29, 2013

### mitchell porter

There have been two new "yukawaon" papers.

Koide and Nishiura have made a substantial technical change, in order to make the family-symmetry interactions of the SM fermions anomaly-free (previously, new fields had to be introduced just to cancel the anomalies).

Aulakh and Khosa produced "Grand Yukawonification", one of the few papers not by Koide that even mentions the yukawaon models. Actually their philosophy is rather different. If I am reading it correctly, this is a susy SO(10) model, in which GUT symmetry breaking is achieved by some very high-dimensional representations (e.g. a Higgs with 126 components), and then some of these Higgs components are gauged under an SO(3) family symmetry, and the yukawas come from their VEVs.

It would be edifying to compare and contrast what they do, and what Koide does. They call theirs a top-down approach, as opposed to Koide's bottom-up approach. Koide introduced new yukawaon fields and a new scale for family symmetry breaking; they just put to work some of the components of the GUT Higgs, and the GUT scale is also the family scale.

Also, it seems to me that their approach has something in common with the 1990 paper by Koide which was the first step towards yukawaons (for a very brief history, see this talk). In subsequent work, the SM yukawa terms are produced by operators coupling SM fermions, the usual SM Higgs, and the yukawaon VEVs, but in this paper from 1990, the masses come from direct couplings between SM fermions and yukawaon VEVs (I think). And this seems to be what Aulakh and Khosa are doing. The downside is that they are not explaining the Koide formula (or any of its generalizations)...

9. Dec 17, 2013

### mitchell porter

10. Dec 18, 2013

### ftr

I don't have the time to elaborate, but the paper should be looked upon as seminal. The logic and overall idea is correct, however the detail and the ansatz are probably somewhat wrong at short range.

P.S. and since you are from the same area, can you invite him to participate in the thread!

Last edited: Dec 18, 2013
11. Dec 18, 2013

### MTd2

I think you should trust the Mercator projection to infer distances!

12. Dec 18, 2013

### ftr

what's two hours flight. Take midpoint between their mass centers, then draw a circle with 4000 km radius, you will only see two countries. That is how close they are! Even their flags look the same.

13. Dec 18, 2013

### phyzguy

Why would you think this is seminal? There is a model for an electron, with arbitrary constants adjusted so that the mass comes out right. It is stated that excited states can represent the muon and tau, but no attempt is made to calculate the muon and tau masses. At a very minimum, I would expect it to show the correct ratios of the lepton masses, but it doesn't do that. There is also no attempt to explain why there are only three solutions to the eigenvalue equations. Also, there are statements made about how the scalar field (Theta) corresponds to the Higgs field, but no backing for those statements that I can see. Please explain why you think this is important work. What am I missing?

14. Dec 18, 2013

### ftr

I agree with all your objections(I have already stated some of it), actually these were my own questions, and that is why I asked if the author can participate. However, the concept Is familiar to me because I have thought about it independently and I was surprised to find that the concept was thought about by Dirac(references) and even Lorentz long time ago.

http://web.archive.org/web/20041224...i.nl/physis/HistoricPaper/Dirac/Dirac1962.pdf

But to discuss in detail I do need the time.

Last edited: Dec 18, 2013
15. Dec 22, 2013

### mitchell porter

phyzguy is right about this paper, it doesn't derive the Koide relation at all, and offers no evidence that the proposed model works.

It could be regarded as a soliton model, in which there is a spherically symmetric spinor wave coupled to a similar wave in a scalar field, with a peak of charge and mass density in some central region, which then drops away with distance. The excited states presumably have many hills and valleys surrounding the central peak, like excited states of the simple harmonic oscillator. The author doesn't even try to calculate the energies of the excited states, but just says (page 5, end of part IV) that the first two excited states will correspond to the muon and tauon.

Because these soliton-like objects are spherically symmetric, the author may hope for a convergence with the ideas of Gerald Rosen, who has written a number of papers trying to obtain various modified versions of Brannen's formula from models of particles as dynamical two-dimensional surfaces like Dirac's membrane. (Kruglov cites two of these papers, more can be found in Rosen's publication list.) In this soliton model, the two-dimensional surface might be the surface of maximum charge density or maximum mass density.

There's lots missing from Kruglov's paper - not just calculations for the "muon" and "tauon" states (it may actually be possible to falsify the model as presented, by doing those calculations) - but also the interaction of these "electrons" - e.g. do they scatter like real electrons? Kruglov's theory seems quite simple - QED with a mass term for the electron (i.e. not one obtained via Higgs mechanism), plus the extra scalar - and it may be a QFT equation that has already been thoroughly analyzed...

Then there are all the experimental results and theoretical arguments against the idea of the electron having internal structure and against muon and tauon as excited states of that internal structure - perhaps someone could dig up the details of this, which I admit I only know as a talking point.

In M-theory phenomenological models (like the G2-MSSM of Kane et al), the particles are two-dimensional membranes (M2-branes), but the membranes are Planck-scale in size, and the particle generations don't correspond to excitations of the membranes, but have some other origin, e.g. each generation comes from branes "stuck" to a different singular point in the Kaluza-Klein space.

16. Dec 28, 2013

### arivero

A cross reference to the thread on 14 dimensions:

Not sure if MTd2 is going to ellaborate on it, or if we should expect some more detailed paper. Meanwhile, I return to my tomb and my silent rest.

17. Dec 28, 2013

### MTd2

I am waiting for Kneemo to say something. I am like shooting in the dark... Well, not completely. I am doing some numerology and showing some mathematical stuff and see if there is a useful correlation.

Last edited: Dec 28, 2013
18. Dec 29, 2013

### mitchell porter

We have discussed the possibility of a massless up quark several times in this thread (#51, #60, #78). There are two papers which touch on this idea today.

First, Dvali et al, "On How Neutrino Protects the Axion". This cites an older paper of Dvali's, "Three-Form Gauging of axion Symmetries and Gravity", in which it is said (page 12) that "at low energies, the QCD Lagrangian contains a massless three-form field", and that both axion and massless-up-quark solutions to the strong CP problem can be understood as a Higgsing of this three-form.

According to remarks on page 13, in the case of the massless up quark, it's the eta-prime meson which Higgses the three-form.

The other paper today is "Charge Quantization and the Standard Model from the CP2 and CP3 Nonlinear σ-Models" by Hellerman et al. This paper is part of a research program aiming to get charge quantization without grand unification. Instead of embedding the whole SM gauge group in a larger simple group, as in a GUT, part of the SM gauge group is identified as the locally gauged part of a CPn global symmetry in a "nonlinear sigma model". So in each case, SU(3)c x SU(2)L x U(1)Y is split into two parts, which we could call the unembedded and the embedded part. In the CP1 model, SU(3)c x SU(2)L is unembedded, and U(1)Y is embedded into CP1. In the CP2 model, SU(3)c is unembedded, and SU(2)L x U(1)Y is embedded into CP2. In the CP3 model, SU(2)L is unembedded, and SU(3)c x U(1)Y is embedded into CP3. Models employing CP4 and higher are mathematically possible, but their phenomenological viability is not discussed.

On page 4 of today's paper in this series, we read that in the CP3 model, there is a Goldstone boson with the quantum numbers of an up squark, and that if the supersymmetric CP3 model were considered, there could be a massless up quark. It's also stated that in the CP2 model, it might be possible to get the SM Higgs from the corresponding Goldstone boson, and that it could dovetail with the proposal to explain the Higgs mass as arising from ultra-high-energy boundary conditions (e.g. as in Shaposhnikov-Wetterich, though the present proposal has nothing to do with asymptotic safety).

In my opinion, these extra phenomenological twists should be regarded as a bit untested and opportunistic. The key idea in the papers of Hellerman et al is that these NLSMs provide an alternative to the GUT explanation of charge quantization. They develop that idea, and then they note that there might be a way to incorporate these older ideas (massless up as solution to strong CP, high-scale boundary conditions as reason for Higgs boson mass) into their scheme, but this latter part of their work is still sketchy.

19. Dec 30, 2013

### kneemo

What's curious is the Yukawaon model that Koide uses to derive the Koide relation. He starts with some superpotential and assumes some SUSY vacuum conditions (∂W=0) to get a cubic equation that leads to the Koide relation for the leptons.

In supergravity, critical points of a superpotential ∂W=0 correspond to attractor points of the scalar field trajectories that localize on a black hole horizon. In D=4, the attractor mechanism is obeyed for black hole solutions with non-vanishing quartic invariant I≠0. These are rank four solutions that come in three canonical forms under E7:

a) k(1,(-1,-1,-1))
b) k(1,(1,1,-1))
c) k(1,(1,1,1))

where k > 0 and (1,1,-1), for example, corresponds to a diagonalized 3x3 Hermitian matrix with eigenvalues (1,1,-1) that the reduced structure group E6 can in general act on.

Families b) and c) resemble the forms of the eigenvalues found in the neutrino and lepton Koide relations.

If the Yukawaon model derivation of the Koide relation is really the result of an attractor mechanism, this would explain what the Yukawaon really is and why the situation isn't as messy as it could be. In essence, the Koide relation would just be the result of moduli (complex scalar fields) being stabilized on a microscopic black hole event horizon.

Last edited: Dec 30, 2013
20. Dec 30, 2013