What is new with Koide sum rules?

[fzero]

I am not sure. Consider a decay muon to electron plus a pair neutrino antineutrinos, as usual. As it is possible that the electron is left in the same rest frame that the initial muon, I could say that the energy available for the neutrino pair is the difference of pole masses of muon and electron, not the muon pole mass minus the renormalised electron mass at muon scale. I think I should had put more care when I attended to the undergraduate lectures, twenty years ago.

Of course it is irrelevant for the experimental results, the running of electron fro .511 to 105 is surely negligible.

Thinking a bit more, I think the best way to look at the issue is the most straightforward. If you compute the decay using the bare masses, then a proper treatment of loop corrections automatically takes into account running of the masses and coupling constants. Determining what bare parameters to use amounts to choosing a renormalization scheme and then extracting the pole mass by finding the pole in the full propagator.

I think that any simplification (like just using effective mass parameters) probably leaves too much physics out that is of the same degree of importance.

 

[arivero]

I agree, bare plus corrections seems the best approach, and in fact it is the usual approach to calculate the decay width. But I am intrigued really about the size of phase space, and more particularly about which is the maximum energy that the neutrino pair can carry. In principle is is a measurable quantity. Is it $105.6583668 - 0.510998910$, i.e, $m_\mu(m_\mu) - m_e(m_e)$ (?), or is it $m_\mu(m_\mu) - m_e(m_\mu)$? I think that the solutions to the RG running make the electron mass to _decrease_ when the scale goes up, so the second answer would extract energy from magic. And the first answer is then an example of a physical comparision of lepton mass at different scales. So I'd conclude that the need of comparing masses at the same scale is just a rule of thumb, not a general axiom.

 

[ohwilleke]

The assumption that Lubos makes that pole masses are necessarily more fundamental the the rest masses we know and love isn't necessary right. This is particularly true if decay width, rather than being a truly independent parameter of a particle is actually a function of some other property or properties of that particle according to a function whose form is not currently known.

By analogy, while it is often more helpful to use the expected decay time of a particle adjusted by a Lorentz transformation to reflect its kinetic energy (we could say that this quantity runs with the energy level of the particle), that doesn't necessarily mean that the Lorentz transformed decay time from the perspective of an observer watching the particle wizz by him is really more fundamental than the decay time of the particle from an observer in the particle's rest frame that does not run.

Likewise, until we understand the underlying mechanism by which Koide's formula arises, there is no particularly good reason to conclude that all of the masses in the formula must be computed at the same energy scale as arivero notes in #37.

 

[mitchell porter]

I still haven't fully worked through Sumino's paper, but I want to highlight another curious fact, that the family symmetry group which he proposes is U(3) x SU(2) (later he embeds this in bigger groups). Since that is the SM gauge group, I've been wondering whether his mechanism can be realized by some form of dimensional deconstruction.

 

[mitchell porter]

What do you get if you minimize this expression:

$V = \sum\limits_{1 ≤ i,j,k ≤ 6; \text{ } i,j,k \text{ different}}(\frac{x_i^2 + x_j^2 + x_k^2}{(x_i+x_j+x_k)^2} - \frac{2}{3})^2$

Do you get something like a descending chain of Koide triplets from the squares? (For some ordering of the "x"s.)

 

[arivero]

http://arxiv.org/abs/1205.4068
Neutrino masses from lepton and quark mass relations and neutrino oscillations,
by Fu-Guang Cao,
suggests the use of Koide-like sums for all the six leptons.

 

[arivero]

Hmm, rumours of fermiophobic Higgs!

It is even better than leptophobic; it implies that the Higgs has not role in the mass of the bcsdu quarks. It is agnostic about top, because a 125 GeV Higgs obviously can not decay into top quarks.

 

[hareyvo]

Precisely, it is possible that the fine structure constant has a role in the calculation of the mass.
With α the fine structure constant, e the charge of electron, me its mass, re its length, q the charge of Planck, m its mass, r its length, according to
http://en.wikipedia.org/wiki/Planck_units,
we have
q^2 = 4πc(hbar)ε_0 = 4πmr(c^2)ε_0 = mr.10^7
αq^2 = e^2 = αmr.10^7 ≡ me.re.10^7
Write α = yz and αq^2 = αmr.10^7 = ym.zr.10^7
With ym = me = 9.1093829100.10^-31 kg,
y = me/m = 4.1853163597.10^-23
With zr = re = 2.8179403250.10^-15 m,
z = re/r = 1,7435592744.10^20 = (4.1755948971.10^10)^2
Then y = [(10α)^ 1/3]/(9, 98451148382.10^21)
and z = [(10α)^2/3].(9.9845040300.10^20)
from which
me = ym ≈ m(10^-22)[(10α)^ 1/3] =m(α/10^65)^ 1/3
and
re = zr ≈ 10r(α.10^31)^2/3

 

[arivero]

In order to correct a bit the distorsion introduced by hareyvo (please, guys, do your homework and read the old threads before posting. Ah, and use your blog part if you do not aim for general discussion), let me stress again what the fermiophobic higgs should mean for ALL the low-energy approach to masses: basically that the field becomes open, because we should have experimental evidence of the nullity of the yukawa coupling for particles lighter than the Higgs itself.

Actually, it is a bit of complex, as it also means that Higgs production has smaller rates than the SM. And the current scenario does not tell anything about the top yukawa coupling, as it is negligible as a channel for observation (if the Higgs is at 125 GeV) and surely (can someone confirm?) also as a production channel -we need to produce a top and then collide it again-.

 

[mitchell porter]

A reminder of why the Koide relation, and its generalizations reported at the start of this thread, are challenging: a short paper from India lists the fermion masses at M_Z scale and at GUT scale in various theories (SM, SM + extra higgs doublet, MSSM). Of course, the masses are different at GUT scale, often very different, and yet that is supposed to be where symmetries are more manifest.

The world of QFT (and strings) contains many unexpected equivalencies between different-looking pictures of the same physics. It may be that Koide relations won't really be understood without switching to a "UV/IR-dual" picture in which the IR looks simple and is somehow the starting point for the theory. Since you drop degrees of freedom in the RG flow from high energies to low, that sounds unlikely - the IR just doesn't have the information needed to reconstruct the UV.

But in string theory we already have various constraining relationships between IR and UV properties. So perhaps for the right sort of theory, we can find a new picture, in which the UV can be completely reconstructed from IR + "something else". Somehow, we want the new heavy degrees of freedom to enter at higher energies, yet the way in which they do so is constrained or foreshadowed or otherwise allows deep and nonaccidental relationships between IR quantities.

 

[mitchell porter]

While considering how to produce the "cascade" or "waterfall" of Koide relations for quark masses that Alejandro discovered, I have thought in terms of a range of possible multihiggs models. At one extreme, you have many higgses and they all contribute to the masses of all the fermions (Adler's circulant models, mentioned in comment #14 in this thread), and the cascade structure comes from a complicated inter-scalar potential. At the other extreme, you basically have one distinct higgs per fermion - again, the cascade must come from the potential, but you would also have to have the scalars and fermions appropriately charged under special discrete symmetries, to enforce "one higgs" or "few higgses" per fermion.

Today Bentov and Zee have a paper about the second scenario. I hasten to add that they don't talk about Koide at all, but you can see from their work something of how it would go. In the standard model, there's just one higgs VEV, and the fermion mass spectrum comes from the yukawas - O(1) for the top, much smaller for the other fermions. In these so-called "Private Higgs" models (every fermion has its own private higgs), all the yukawas are O(1), and it's the VEVs which produce the cascade!

My guess is that a model like this can produce the full Koide cascade at tree level, but that loop corrections should spoil the exactness of the Koide relation. But we won't know for sure, until someone tries to make it work...

edit: Other papers to read: Private Higgs, Private Higgs for leptons, and paper by Ernest Ma which explains Koide using the same discrete symmetry group.

 

[arivero]

Today Bentov and Zee have a paper about the second scenario.

Zee is the last of the big phenomenologists.

Let me remember that the first paper which actually brought a Koide equality (albeit with one of the three masses equal a zero), Harari Haut Weyers, was critiquized because its permuting of exchanging left and right quarks across generatations was really a way to present a complicated Higgs structure. Surely the same criticism applies to any other multiple Higgs ideas, but the escape comes if, as it happens in the Koide waterfall, it is always the same kind of step along all the ladder.

By the way, I have noticed that pdg has moved again their evaluation of the mass of the top, now it has the central value at 173.5 ± 0.6 ± 0.8 GeV, so near of the postdiction of the ascending waterfall of vixra:1111.0062v2/arxiv:1111.7232, which is 173.263947(6)

Edit: if we think of an unperturbed Koide, the main problem is that setting electron to zero but keeping the "QCD" mass to 313 GeV gives a slightly higher value for the top, namely 180 GeV. Of course we could scale everything down and set top to the electroweak vacuum, then the unperturbed levels should be 174.10 GeV (top), 3.64 GeV (bottom), 1.70 GeV (charm,tau), 121.9 MeV (strange, muon), 0 eV (up, electron), 8.75 MeV (down). I am not sure that I like it, but it has the merit of using a single input, the Fermi constant - to produce the initial seed of 174.10-. On other hand, Koide triples are a lot of quadratic equations, and surely there are more solutions also producing the 0 eV up quark; this one must impose also the extra condition of being monotonic, always descending, from top to up.

 

[arivero]

Today Bentov and Zee have a paper about the second scenario. I hasten to add that they don't talk about Koide at all,

But their Higgses are proportional to the square root of the mass of each fermion. Perhaps the authors have got the wind.

 

[arivero]

By the way, I have noticed that pdg has moved again their evaluation of the mass of the top, now it has the central value at 173.5 ± 0.6 ± 0.8 GeV, so near of the postdiction of the ascending waterfall of vixra:1111.0062v2/arxiv:1111.7232, which is 173.263947(6)

And now the final evaluation of Tevatron moves it to 173.18 GeV! So since the upload of the paper, the difference has evolved from .36 to .24 and now to .08. The personal combination from Déliot for TeV-LHC is a bit lower, down to 173.1, but let's see how it evolves towards the pdg.

It is also intriguing that the only really wrong mass is the one of the charm quark, where they are finding some stress against the standard model (in CP violating decays).

To be sure, let me quote the table from the preprint, adding the current known (MS scheme) values of quark masses. Reminder, the only inputs are me = 0.510998910 and mu= 105.6583668 and only assumptions are Mq = 3Ml and q = 3l (quasi-orthogonality quarks/leptons). All the rest is to repeat Koide for each triple.

        | prediction          |  (pdg 2012)
========+=====================+===================
tau     | 1776.96894(7) MeV   | 1776.82 ± 0.16 GeV
strange | 92.274758(3) MeV    | 95 ± 5 MeV ev ([URL="http://pdglive.lbl.gov/ideograms/Q123SM.png"]ideogram[/URL] 94.3±1.2)
charm   | 1359.56428(5) MeV   | 1.275 ± 0.025 GeV   
bottom  | 4197.57589(15) MeV  | 4.18 ± 0.03 GeV
top     | 173.263947(6) GeV   | 173.5 ± 0.6 ± 0.8 GeV | 173.18±0.94 GeV (Tevatron arxiv:1207.1069)

 

[arivero]

I have set up a prezi presentation in the Koide ladder (or waterfall)

http://prezi.com/e2hba7tkygvj/koide-waterfall/

feel free to disseminate

 

[arivero]

It is possible to use only the mass of the top, or the electoweak vacuum, and ask for a Koide waterfall chaining solutions until we arrive to a mass of the top equal to zero. There are five such chains, only three of them are actually "falls", and of those only one uses always the same solution of the Koide equation (see my paper, or this thread above). The waterfall is:

t:174.10 GeV--> b:3.64 GeV---> c:1.698 GeV --> s:121.95 MeV ---> u:0 ---> d:8.75 KeV

Note that the last triplet is even older than Koide, from Harari et al.

This descent uses only one input, Fermi scale, and the mases of c and s are even near of tau and muon that in the descent with two inputs. It supports then the idea of an unperturbed spectrum, where charged leptons are degenerated with some quarks, and then a perturbations that somehow commutes with the cause of Koide.

 

[MTd2]

The up quark is massless?

 

[arivero]

The up quark is massless?

Indeed

There are, with some variants, two main arguments here.

- You can consider it pragmatically, that we are just looking for solutions of the Waterfall that happen to produce a small mass for the quark up, say 3 KeV, starting from 175 GeV. That is six orders of magnitude, and it is a convenient device to count the zero mass solutions and then, barring catastrophes, do all the numerical search from it. It should work the same for the down quark in this case.

- Or, you can take seriously the requirement in order to avoid the theta problem of QCD, and claim that the mass of the up is really zero and its measured mass is of a secondary nature, so that really for s,d,u the masses are modified following
$$m'_x=m_x + {m_y m_z \over M}$$ + ...
with M coming from QCD. Note that if the up is really massless at the point where you apply the formula, then s and d are not modifyed in first order.

 

[MTd2]

What do you mean by " its measured mass is of a secondary nature"? I don't understand how you hid its mass.

 

[mitchell porter]

I have been looking into "quark-hadron duality" for an approach to the Koide waterfall that I'm not yet ready to explain. But I have to point out something I just found in the literature - in "The origins of quark-hadron duality" by Close and Isgur: that one manifestation of this duality, is that the same formula can be expressed as the square of a sum or as a sum of squares - see page 4. Doesn't that sound like the Koide formula? - with the "square roots of the masses" as the basic quantities that you sum or that you square.

 

[arivero]

What do you mean by " its measured mass is of a secondary nature"? I don't understand how you hid its mass.

Where is the problem, exactly? I put the mass of the up equal to zero and then I use an expansion to produce a final mass, this is a very usual recipe. The problem is that M is an interaction scale which comes from the chiral scale of QCD, so it is not fundamental in the Koide waterfall, hence the name of "secondary"

BTW, I mean MeV, no KeV, of course.

 

[arivero]

I have been looking into "quark-hadron duality" for an approach to the Koide waterfall that I'm not yet ready to explain. But I have to point out something I just found in the literature - in "The origins of quark-hadron duality" by Close and Isgur: that one manifestation of this duality, is that the same formula can be expressed as the square of a sum or as a sum of squares - see page 4. Doesn't that sound like the Koide formula? - with the "square roots of the masses" as the basic quantities that you sum or that you square.

I am not sure. Koide seems about finite sequences, mostly triples, and duality is about sums over all the states.

 

[arivero]

t:174.10 GeV--> b:3.64 GeV---> c:1.698 GeV --> s:121.95 MeV ---> u:0 ---> d:8.75 MeV

Just for the record, this is the only waterfall with a sensible value of t/b for five steps to zero. If we aim for six steps, there is (only) other solution, rather more peculiar:

t:174.10 GeV-->c:1.859 GeV-->b:3.401 GeV-->s:132.23 MeV-->d:9.49 MeV --->u:0

Really it uses the same triples uds, scb and cbt, but instead of csu to land into the zero, it calls for bsd.

There is not monotonic solution of six steps compatible with t/b "desert" and crossing zero. And this two are really the only six step series with such compatibility. Of these, the former survives better when going to experimental values.

 

[MTd2]

Where is the problem, exactly?

Suppose the meson rho0 or omega0, where both are u anti u, where does it get its mass, from gluons only?

 

[arivero]

Suppose the meson rho0 or omega0, where both are u anti u, where does it get its mass, from gluons only?

Indeed, and this is true for most of the low mass pions. I am not conversant in QCD, but if you open a thread on the topic in the SM subforum, I will try to follow it.

The main use of a massless up quark is to solve the strong CP problem (again, a topic where someone in the SM subforum can be more conversant than me) Here you can see
http://arxiv.org/abs/hep-ph/9403203
to Banks, Nir, and Seiberg telling that they do not believe that the controversy has solved. Since then, data from lattice show up quark with a mass different of zero, but again it is not clear if they are already accounting for some QCD trick (such as the one I told above).

 

[arivero]

If someone wants to explore koide like equations, this tool seems useful.

from __future__ import division
from sympy import *
x, y, z, t = symbols('x y z t')
def koide(a,b,c):
    return 2*(a+b+c)**2-3*(a**2+b**2+c**2)
In [5]: koide (x,y,z)
Out[5]: 
     2      2      2                2
- 3⋅x  - 3⋅y  - 3⋅z  + 2⋅(x + y + z) 
In [7]: solve([koide(0,1,y)],x,y)
Out[7]: 
⎡⎛    ⎽⎽     ⎞  ⎛  ⎽⎽     ⎞⎤
⎣⎝- ╲╱ 3  + 2,⎠, ⎝╲╱ 3  + 2,⎠⎦

In [8]: solve([koide(0,1,y),koide(1,y,z)],x,y,z)
Out[8]: 
⎡⎛    ⎽⎽       ⎞  ⎛    ⎽⎽⎽           ⎽⎽⎽     ⎞  ⎛  ⎽⎽⎽       ⎞  ⎛  ⎽⎽⎽          ⎽⎽⎽     ⎞⎤
⎣⎝- ╲╱ 3  + 2, 0⎠, ⎝- ╲╱ 3  + 2, - 4⋅╲╱ 3  + 12⎠, ⎝╲╱ 3  + 2, 0⎠, ⎝╲╱ 3  + 2, 4⋅╲╱ 3  + 12⎠⎦

 

[arivero]

Sympy produces some errors and I am not sure how exhaustive it is, so I have turned to Mathematica for an exploration of the "Landscape" of koide coupled systems.

This attachment is a notebook to explore the set of solutions for u,s,d having a starting tbc triplet where t>b>c. So the input data is t and b, and of the two possible solutions for c we choose the one lightest than the b (and t) quark.

It takes about three to five hours to solve all the possible sets of coupled equations with all the possible sign combinations for the square roots.

Most of the solutions are either degenerate or with some mass value higher than the mass of the "charm". If we sort the non degenerated solutions according the mass of the answer, we find that the "natural" solution appears already in the second place:

{0.0701502, 0.0540148, 0.00354978} -->  (4.197,x,y) (1.35978,x, z)  (x, y, - z) 
{0.0922984, 0.00818075, 0.0000355252} --> Natural Waterfall (4.197,1.359,-x),(1.359,x,z),(x,z,y)
{0.0922984, 0.0381469, 0.0000355252}
{0.0922984, 0.0381469, 0.00272298}
{0.0922984, 0.0381469, 0.0082687} 
{0.0922984, 0.0530732, 0.00374946}
{0.103209, 0.0922984, 0.0000355252}
{0.12806, 0.0215823, 0.000876337}
{0.16781, 0.0536519, 0.00362577}
{0.191464, 0.0922984, 0.006624}    
{0.195452, 0.00606144, 0.000970879} -------->   (x,y,z) (x,y,4.197) (x,-y,1.359)
{0.195452, 0.0442732, 0.00606144}
{0.195452, 0.08461, 0.00606144}
{0.195452, 0.0922984, 0.00606144}
{0.195452, 0.195452, 0.00606144} 
{0.201489, 0.0922984, 0.00662588}
...

Note that the main evidence for the validity of the natural result is its orthogonality to leptons; any other similar b,c,s triple will work too, for this task. The "Natural" sequence is, as explained in the start of the thread, the one where we always happen to choose the same solution of the equation, in each step.

 

[arivero]

This second attachment is the landscape of all the combinations of Koide equations having boundary conditions at 0 and 174.10 GeV. So there is really one empirical input, the Fermi Scale (equal to top mass because the yukawa of the top is equal to one; you could also say that we are calculating all the possible paths between yukawa =1 and yukawa =0 with six quarks).

Mathematica finds 4140 different solutions, but most of them are degenerated, having at least two quarks with the same mass. Again, let me order them from lower to higher "bottom quark prediction". The table for "b,c,s d" (we assume the zero coupling is the up) nondegenerated starts:

2.56251 degenerated...
2.82403 degenerated...
3.20787, 1.69849 degenerated...
{3.40143, 1.85905, 0.132232, 0.00949385} {t,b,c}, {b,c,-s}, {b,s,d}, {s,0,d}
{3.41535, 1.8493, 0.132774, 0.00953271} {t,b,c}, {b,s,d}, {c,s,0}, {s,0,d}
{3.64088, 1.69849, 1.66928, 0.121946} {t,b,c}, {t,b,c'}, {b,c,-s}, {c',s,0}
{3.64088, 1.69849, 0.121946, 0} {t,b,c}, {b,c,-s}, {c,s,0},{c,s,-d}, 
{3.64088, 1.69849, 0.121946, 0.0151864} {t,b,c} {b,c,-s} {c,s,0} {s,0,d}
{3.64088, 1.69849, 0.261403, 0.121946} 
{3.64088, 1.69849, 1.69849, 0.121946}
{4.51619, 3.64088, 1.69849, 0.121946}
...

So Koide ansatz with the single input of 174.10 GeV predicts the right scale for the other quarks:
3.40..3.64 GeV for the bottom
1.85..1.69 GeV for charm
132...121 MeV for strange
0...9...15 MeV for down

 

[arivero]

Suppose the meson rho0 or omega0, where both are u anti u, where does it get its mass, from gluons only?

Ah, note that now there is a subtopic in the threasBootstrap thread about chiral symmetry breaking, it is somehow related to the question of QCD mass. https://www.physicsforums.com/showthread.php?t=485247&page=10

 

[arivero]

Could Brannen transformation matrix $$N = \left( \begin{array}{ccc} \sqrt{2} & e^{i\delta} & e^{-i\delta} \\ e^{-i\delta} & \sqrt{2} & e^{i\delta} \\ e^{i\delta} & e^{-i\delta} & \sqrt{2} \\ \end{array} \right)$$
be a symmetry of the fermion field, so that the yukawa coupling of the Higgs $$g\phi \bar\psi\psi$$ somehow goes over to each field, say $$\phi (N \bar\psi) (N \psi)$$ and the couplig g is really N square?

Really, I can not find any theory where the square root of masses is subjected to a fundamental symmetry. Nor to any symmetry whatever. Hmm.

 

[mitchell porter]

The Koide triplets overlap. So if each triplet derives from a set of more fundamental quantities - e.g. if a triplet is derived from the eigenvalues of a matrix - then the set of fundamental quantities associated with one triplet ought to overlap with the fundamental quantities behind an overlapping triplet. But then, if a symmetry mixes all the fundamental quantities for one triplet, from the perspective of the overlapping triplet, just a subset of its fundamental quantities is being transformed, and in a way that depends on quantities from "outside" (outside the overlap with the first triplet). This is not apriori impossible, but it's tricky to arrange.

Alternatively, the masses, or the sqrt-mass VEVs from which the masses derive, may be the symmetry-breaking solution of some potential, as in comment #40. But then one would like to know why the ordering of the masses from largest to smallest takes the very specific form "quark with +2/3 charge, followed by its -1/3 charge partner, repeated three times", when (given a potential completely symmetric in the sqrt-masses) any ordering was apriori possible.

 

[arivero]

Well, as for the ordering, I have tried (the mathematica notebooks above) all the possible orderings and most of them produce a next-to-top quark with a mass higher than the measured bottom quark. So the waterfall is one of the few orderings that creates a wide gap between top and bottom, and the best non-degenerated one.

My thinking on the symmetry was that just fixing top=1 then breaks in cascade all the symmetries. But I agree it is tricky.

Perhaps the quark side symmetries are only approximate and it just happens that the approximation in sbc cancels the one in bct and conspires to get an exact top quark value. Not rare thing in history of science, I can imagine a couple of famous examples.

 

[arivero]

Honestly, I am too old, or too tired, to understand how Koide produces naturally the mass from the square root mass here,
http://ccdb5fs.kek.jp/cgi-bin/img/allpdf?198105037
nor how he got the first mass formula from the spurion.

 

[arivero]

Ok I can find two kinds of works where the square root of yukawa coupling has some fundamental role. One is "flavons", as in http://arxiv.org/abs/1203.1489v3 Other is Composite Higgs, and particularly Contino (eg http://arxiv.org/abs/1005.4269v1) works like to make explicit this. My memory fails me, so perhaps Mitchell has already catalogued other cases.

Edith, also the higgs in NCG seems to be a product of two more elementary numbers, from bialgebras or elsewhere

 

[mitchell porter]

also the higgs in NCG seems to be a product of two more elementary numbers, from bialgebras or elsewhere

We had two NCG higgs papers this month, so perhaps we can discuss this.

In Estrada and Marcolli, we have an action with parameters "a ... e" that are functions of the Yukawa matrices, and some other "f" parameters that are more fundamental, and then we have various relations among a ... f that exist at unification energy.

This action is an expansion of a "spectral action functional" which includes variation of a Dirac operator. The Higgs field arises from those "inner fluctuations" of the Dirac operator that are associated with the finite noncommutative part F of the "almost commutative" space M x F that defines the model. (Inner fluctuations associated with M produce the gauge fields.) These fluctuations have the form u[D,u*], where [,] is a commutator and u,u* are I think unitary elements of the algebra A in the spectral triple <A,H,D>. Alejandro, are u and u* what you were talking about?

The new work this month has been about obtaining a 125 GeV Higgs in the noncommutative SM, either by imposing asymptotic safety and reproducing the Shaposhnikov-Wetterich argument (this is what Estrada and Marcolli did), or by including the scalar that gives Majorana mass to the RH neutrino, in the RG equations (this is what Chamseddine and Connes did). The AS argument gives the right value within a GeV; the other model just shifts the range of possible Higgs masses so that it includes the observed value.

Turning to the world of Koide relations, there have been a few studies of how the values of the expression in the Koide formula flow, for different triplets; and we also have the work of Sumino, which imposes boundary conditions on RG flow at the EWSB scale, in order to explain the exactness of the Koide relation for the pole masses.

So the obvious way to explain the Koide formula in the NCG context, would be to use a high-scale (unification-energy) version of the Sumino mechanism, that employs the "a...f constraints" to engineer the necessary low-energy relations. I'm not sure if this is possible, but if it will jumpstart discussion again, I'm willing to think about it...