What is new with Koide sum rules?

1. May 7, 2014

arivero

http://arxiv.org/abs/1405.1076 gives

now lets wait for pdg updated combination. Recall that Koide prediction is 1776.96894(7) and that currently pdg combines to 1776.82 ±0.16. In principle it should go up, but surely they will discard BES 96, and then it could even go down.

Last edited: May 7, 2014
2. Sep 24, 2014

arivero

To fix the record: R. & Z. removed the observation about the top-bottom-charm triplet in the published version. They substituted it by a generic reference to Goffinet's article, which does not mention the massive quarks. As neither Kartavtsev's article nor mine have been published, the status of the formula for those quarks remains thus as unpublished. :(

3. Oct 3, 2014

arivero

Last edited: Oct 3, 2014
4. Oct 4, 2014

ohwilleke

The slides are a great and comprehensive summary of the literature on the subject, including a few that I had missed previously. Great work.

I do have to admit that I was waiting for the big cymbal crash at the end where you revealed that you had everything figured out now, but hey, that really would have been setting the bar too high.

5. Oct 4, 2014

arivero

Thanks!!! I am sorry that the unique big cymbal sound is my horrible accent. It is not even an accent, it is a different language. I strongly suggest to favour the slides over the video. For the end, I choosed to give three slides that can be printed for reference.

One of them is very general: the standard model particles in a logarithmic scale. If someone wants the TeX code for it, just PM or email me.

Other shows the mesons next to the fermions, following the idea of Koide formula in meson spectrum. And the third is just the plot of variation of Koide masses with the angle delta, so that it is easier to understand how the formula works (partly; it is also important to understand when the formula has either two or four solutions)

6. Oct 21, 2014

Blackforest

Except if I missed something in checking the seven pages, I didn't see the document arxiv:1201.2067v1 [physics.gen-ph], 5 January 2012: "The Koide lepton mass formula and geometry of circles". Do you know if that path has been followed in between? Do you know if someone is working on the S6 group symmetry in relation with that topic? Thanks.

7. Oct 25, 2014

arivero

Nobody as far as I know (which amount to this online circle and some email exchanges here and there). I liked the paper; somebody had commented the similitude of the formula with Descartes's but Kocic did a nice, precise work.

8. Oct 31, 2014

ohwilleke

One of the fairly robust predictions of efforts to generalize Koide's rule to quarks by hypothesizing that there are Koide triples of quarks is that the up quark mass must be very nearly zero to be consistent with the down quark and strange quark masses. This assumption also naturally solves the strong CP problem without resort to axions. (An assumption to that up quark and lightest neutrino mass eigenvalue are both negligible also provides an easy starting point from which to construct extended Koide triples in the Standard Model fermion mass matrix without having to worry about massive interrelatedness of the equations.)

Lattice QCD and experimental evidence tend to favor a non-zero value of the up quark mass of around 2 MeV close to half of the down quark mass, but this calculation is model and QCD methodology dependent. A new preprint argues that it is premature to rule out a negligible up quark mass if one takes another approach that is theoretically legitimate and better cordons off big uncertainties in the calculation. http://arxiv.org/pdf/1410.8505.pdf

If the analysis is right, a more straight forward extension of Koide's rule to quarks than would otherwise be possible can be a decedent fit to the data.

9. Oct 31, 2014

arivero

Nice! Let me to remind the "predictions" of Koide waterfall for the masses of up, strange and down. If we assume that we want the up quark exactly zero, then setting the yukawa of the top equal to one solves all the waterfall, and predicts strange=121.95 MeV and down=8.75MeV.

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

Of course, then the correction $\Delta m_u \propto{ m_s m_d \over \Lambda}$ depends of an extra parameter, but values of about 300 MeV for the denominator look reasonable for the scale the preprint is speaking about.

If instead of fixing the up quark to zero we want to predict it, either from the top and bottom masses, or from the electron and muon plus "orthogonality", the result is similar and it has been discussed before in the thread. The mass of the strange quark is then lower, about 95 MeV, and up quark is still small, on the order of KeVs, so the results are similar and the agreement with measured spectrum is even better. But the prediction with inputs $y_t=1, y_u=0, \Lambda \approx (m_u+m_s+m_c)/6$ is, to me, more impressive.

Last edited: Oct 31, 2014
10. Mar 18, 2015

mitchell porter

Koide and Nishiura have their latest yukawaon model. These are models in which the original Koide relation (for electron, muon, tauon) is assumed to derive from a multiplet of scalars whose VEVs set the yukawas for the charged leptons, and then the yukawas for the other fermions are derived similarly; but only the charged lepton masses are treated as a "Koide triple" obeying a precise relation. The other masses are approached in the fashion usual for flavor physics, as only exhibiting rough relations, e.g. order-of-magnitude hierarchies. So for generalizations of Koide, such as are discussed in this thread, in which new Koide triples are also deemed to be real and needing explanation, these yukawaon papers might seem to be misguided. I mention this one only because, in its appendices, it seems to be claiming a new way to explain the original Koide relation; and that would be of general interest. But I haven't understood it yet.

Koide's relation also made an appearance in a recent, more conventional paper on flavor physics, in which the author tries to cast doubt on its significance by contriving a new relation (his equation 62) which he says works just as well. But it looks like he judges its success against running masses considered at the same scale, whereas the Koide formula is at its most accurate for the pole masses. Sumino's papers remain the only ones known to me, that try to explain this feature.

11. Mar 20, 2015

arivero

I am sorry they are not using the hints from the quark sector. In my view, either the Yukawaon or other similar scalars should give mass first to an unbroken Pati-Salam or GUT group, with a mass for each of three generations that we can call

M3 = e up bottom
M2= tau charm down
M1= mu top strange

If the masses of the this unified system agree with Koide equation, K(M1,M2,M3)=0, we have really nine Koide equations:
K(tau,mu,e)=0
K(up,charm,top)=0, K(up,charm,strange)=0, K(up,down,top)=0, K(up,down,strange)=0
K(bottom,charm,top)=0
, K(bottom,charm,strange)=0, K(bottom,down,top)=0, K(bottom,down,strange)=0

Then something breaks the unification group in a way that not all the Koide equations are broken. The ones in bold somehow survive. The model should explain how and why.

12. Mar 21, 2015

arivero

With more detail, what I think it is happening in the quark sector.

First something divides the GUT model into three families that meet a trivial Koide equation

$(u,b')$ of mass zero
$(t',s)$ of mass $(1- \sqrt 3 /2) m_0$
$(c,d')$ of mass $(1+\sqrt 3 /2) m_0$

Then GUT itself is broken sequentially:

1) the $(u,b')$ generation keeps in Koide relationship with the other two generations, but it moves to fill the two possible solutions, so we are left with

$u$ of mass zero
$(t',s)$ of mass $(1-\sqrt 3/2) m_0$
$(c,d')$ of mass $(1+\sqrt 3/2) m_0$
$b$ of mass $4 m_0$

2) the $(t',s)$ and $(c,d')$ break, again keeping Koide, so that
2.1) $(t,s)$ are two solutions of the triplet "cbx" $((1+\sqrt 3/2) m_0, 4 m_0, x)$
2.2) $(c,d)$ are the two solutions of the triplet "usx" $(0, (1-\sqrt 3/2) m_0, x)$

And then we get the spectrum:

$u$ of mass zero
$d$ as solution of the triplet $(0, (1-\sqrt 3/2) m_0, m_d)$
$s$ of mass $(1-\sqrt 3/2) m_0$
$c$ of mass $(1+\sqrt 3/2) m_0$
$b$ of mass $4 m_0$
$t$ as solution of the triplet $((1+\sqrt 3/2) m_0, 4 m_0, m_t)$

If we assume as input $y_t = 1$, then this spectrum is the still unrealistic "waterfall"
t:174.10 GeV--> b:3.64 GeV---> c:1.698 GeV --> s:121.95 MeV ---> u:0 ---> d:8.75 MeV

Then, or at the same time, some second order effects move slightly the u quark out of its zero mass, choosing a particular branch of solutions of 2.1 and 2.2, and also moving the top quark out of its $y_t$=1 value. The global effect, keeping the four koide equations, would be the realistic waterfall of this thread.

Finally, u quark gets a mass contribution from other mechanism, proportional to $m_s m_d / m_{\Lambda}$

From the point of view of "discrete S4 symmetry", it could be useful to notice that the step 1 uses two intersecting triplets, (u,c,s) (c,s,b) while the step 2 uses the two not intersecting (u,d,s) and (c,b,t) of the four I have boldfaced in the previous post.

Last edited: Mar 21, 2015
13. Apr 21, 2015

arivero

I think that we have never mention this one; Brannen speculated a bit with mesons but I have not found now the explicit mention to the mesonic tuples, namely

$3(B^0+D^0+\pi^0)/(2(\sqrt{B^0 }+\sqrt{D^0}-\sqrt {\pi^0})^2) = 1.005244 \pm 6 \; 10^{-6}$
$3(D^0+\pi^0+0)/(2(\sqrt{D^0 }+\sqrt{\pi^0}+\sqrt {0})^2) = 0.998829 \pm 5 \; 10^{-6})$

compared to lepton formula $e \mu \nu \to 0.999987 \pm 13 \; 10^{-6}$
it is not bad, but while the lepton measurement is still compatible with exactness, the mesons are only in target for the 0.5%.

14. Nov 18, 2015

mitchell porter

Jester at Resonaances informs us that leptoquarks are the BSM flavor of the month. Coincidentally, I recently noticed that there is a "leptoquark" approach to explaining the appearance of the constituent quark mass scale in Carl Brannen's rewrite of the Koide formula (in which the sqrt-masses are eigenvalues of a particular circulant matrix).

It comes from combining two things. First, a simple formula for the constituent mass that I found in Martin Schumacher:

$m_q = g_{\sigma q q} f_\pi$

where $g_{\sigma q q}$ is the sigma-meson-mediated coupling between a bare quark and a pion condensate, and $f_\pi$ is the VEV of the pion condensate, better known as the pion decay constant. $g_{\sigma q q}$ is $2\pi/\sqrt{3}$, and $f_\pi$ is about 90 MeV, leading to $m_q$ ~ 325 MeV.

Second, the interaction term which produces the mass matrix of the charged leptons in Sumino's model (this approach originates with Koide himself):

$\bar{\psi}_{Li} \Phi_{ik} \Phi_{kj} \phi e_{Rj}$

where $\psi_L$ and $e_R$ are left- and right-handed fermions, $\phi$ is the SM Higgs, and $\Phi$ is a 3x3-component scalar whose VEVs squared determine the yukawas.

The idea, then, is to substitute Schumacher's coupling for $\phi$, and to suppose that the VEVs of $\Phi$ are circulant:

$(\bar{\psi}_{Li} \Phi_{ik} \bar{q}) \sigma (q \Phi_{kj} e_{Rj})$

i.e. this $\Phi$ is a leptoquark scalar.

Incidentally, to do something analogous for the quarks, one would apparently want a diquark scalar; and in the MSSM, if you allow R-parity violation, the squarks can have leptoquark and diquark couplings.

Last edited: Nov 18, 2015
15. Dec 21, 2015

mitchell porter

Another month, another anomaly that might be colored scalars - "squarks", even a diquark scalar, according to footnote 2 of that paper, though oddly the other paper cited in that footnote doesn't use the term. I think the diquark counterpart of the interaction term above would be $(\bar{q}DN)\pi(\bar{N}Dq)$, where $D$ is the diquark scalar and $N$ is a nucleon field. (I know it would be odd to have a quark field and a nucleon field in the same effective field theory, though I have seen it done.)

But I also learned something else from this "squark" paper (start of part 3, second point), something that's a problem for the whole concept of colored yukawaons. They need a nonzero vev since by hypothesis, that's where the yukawas come from; but if a colored particle has a nonzero vev, that will break SU(3) symmetry...

16. Jan 9, 2017

mitchell porter

A holographic model of nucleon mass, promising from the perspective of #134, can be found in Gorsky et al, 2013.

17. Jan 11, 2017

ohwilleke

The distinction between the bold and not bold sequences is obvious.

Each of the bold sequences, when arranged in order of mass, involve the most likely W boson transition from the heaviest to the next heaviest, and the most likely W boson transition from the next heaviest to the lightest. They are the "route of least resistance" a.k.a. most probable, decay channel of the heaviest fermion in the triple.

The italic triples are impossible through W boson decay without intermediate steps.

A decay of top->up->down isn't impossible, although it is highly improbable.

A decay of top->bottom->down or of top->down->bottom is impossible without intermediate steps. The decays of bottom-top-down or down-top-bottom require an extremely energy boosted starting point (unlike all of the other decays) and are also highly improbable.

Implications For Quark Triples

This is why I am inclined to think that the relative masses of the fermions arises from a balancing of all available weak force interactions into and out of the fermion in question to other fermions, weighted by their relative probability - basically a function similar in concept to the addition of three vectors for each quark in the quark case, balanced out to an equilibrium state that simultaneous fits values for all six quark masses at once.

This intuition is also supported by the fact that the more dominant a share of the overall probability of decays a triple has relative to all possible decays from the heaviest fermion in the triple, the closer it comes to K(triple)=0, while the lower the share of the overall probability that the triple has, the more it deviated from K(triple)=0.

At first order the adjustment to the mass of the middle member of the triple (when that quark is an up-type quark) is approximately the CKM matrix derived probability of a transition from that up-type quark to the down-type quark that is missing from the triple times the mass of the omitted down-type quark (and visa versa when the middle quark in the triple is a down-type quark).

This is a bit surprising, because the relationship of the mass of the middle mass quark of the triple to the other two quarks in triple (which is approximately a Koide triple relationship) is decidedly non-linear. The correct adjustment for the missing opposite type quark is probably actually non-linear, but I just haven't come up with a clever enough idea to figure out what that is yet.

Definitional Issues For Quarks

Of course, the other dicey piece of making a Koide-like formula work for quarks flows from the definitions of masses that are used.

The charged lepton formula (which is exactly correct to the limits of experimental accuracy), and the top-bottom-charm triple (which is the best fit of the quark triples) exclusively involve pole masses.

The bottom-charm-strange triple, the charm-strange-up triple, and the strange-up-down triple (which are less good fits to the Koide triple rule) all involve a mix of pole masses for the heavy quarks (i.e. evaluated at different energy scales) and MS masses for the light quarks evaluated at a constant energy scale - since light quark pole masses are ill defined.

So, if MS mass at 1 GeV is not the correct generalization of pole masses for light quarks to capture a Koide-like relationship for quarks, then some of the discrepancy between the light quark masses and the masses predicted by a Koide waterfall method could be (in whole or in part) due to using the wrong definitions for the light quark masses.

But, you clearly can't just extrapolate the formula for the running for quark masses at higher energy scales to light quark masses either. This gives you light quark masses in which the mass from the quark content of a pion or kaon would far exceed the mass of the particle itself (as demonstrated in a 1994 paper by Koide).

In contrast, QCD calculations using MS masses for the light quarks that add gluon field mass contributions as for other quarks get you in the right, much lighter, ballpark of what a Koide waterfall calculation would suggest.

Anyway, until you generalize the concept of pole mass for light quarks in a more appropriate way, it is not just experimentally difficult, but theoretically impossible to confirm or reject a generalized Koide's formula for quarks involving pole masses.

The Lepton Case

In the charged lepton case, you get a perfect to the limits of experimental measurement fit, because "vectors" involving neutrinos basically make zero contribution since their masses are so tiny, so only the three charged leptons make any contributions to each others masses and have to be balanced out.

I suspect that, in principle, non-zero neutrino masses probably cause Koide's formula to be not quite perfect, since neutrinos do have W boson interactions with charged leptons, but only at something on the order of the ratio of the charged lepton masses to the corresponding neutrino masses (e.g. 1,776,960,000,000 meV for a tau v. about 50-60 meV for the heaviest neutrino mass; 105,000,000,000 meV for a muon v. about 8 meV for the next heaviest neutrino mass, and 511,000,000 meV v. less than 1 meV for the lightest neutrino mass). But, we only know the tau mass to an accuracy of one part per 14,807, so this slight tweak is impossible to measure until our experimental measurements of the tau mass are more precise by a factor on the order of (at least) about 100,000 times what they are now.

Request

One set of number that would be really useful to have at hand, but which I don't have in any easy reference, is a full set of the Standard Model mass parameters evaluated not at M(Z) but at the Higgs vev of about 246.2 GeV.

If anyone could calculate or find out these values for me, it would be greatly appreciated.

I could probably do it. I can relatively easily track down literature with the relevant beta functions. But, doing the actual calculations is not something that I'm in a good position to do at the moment, and even if I could, it would probably take me forever as I don't have the right kind of software to do it by any means other than with a calculator or Xcel worksheet.

Last edited: Jan 11, 2017
18. Jan 12, 2017

arivero

Well, the problem here is that we have a bit of circularity if we assume that the CKM matrix (and then the "most likely" paths) is fixed after or at the same time that the masses. Could it be possible to postulate another "W" with another (dual, orthogonal, reciprocal?) CKM matrix so that the same criteria should select the italic triples and reject the ones in bold?

19. Jan 12, 2017

ohwilleke

The italic triples are impossible because they have the same charge and the only mechanism for flavor changing in the Standard Model requires that you alternate quark electric charges at each step. by one full unit of electric charge. This isn't circular reasoning, it just a fundamental feature of how the W boson works in the Standard Model. And, it is, in general, possible to order any combination of masses from heaviest to lightest without loss of generality.

The structure of the CKM matrix also does seem more fundamental than the quark masses. Indeed, in the exercise that follows in the rest of this post, one can sketch out a toy model of how one could cut the number of non-neutrino parameters of the Standard Model from 19 to 6 with a slight hypothetically possible tweak to the extended Koide rule model arivero has suggested to make it more accurate, a discovery of a relationship of the aggregate fundamental fermion masses and fundamental boson masses to the Higgs vev suggested by C & LP, a possible special relationship of the two electroweak constants to each other at the Higg vev energy scale, and a tweak to the definition of the Cabibbo angle to reflect the discovery of a third generation of fundamental fermions after it was initially defined.

To be clear, I'm not actually arguing that this toy model is the key to the relationship between the fundamental constants of the Standard Model that could greatly reduce their number. Instead, I'm illustrating what new "within the Standard Model" physics that could do that ought to look like, as a motivational exercise to suggest that we aren't as far from making really major progress in greatly reducing the universe of Standard Model physical constants than it might seem. We aren't that far from the promised land, and we are approaching the point where an Einstein-like genius could, in just a few years, reveal a lot of the connections that had been opaque or purely conjectural until we had accurate enough measurements of the fundamental constants to make provable statements about their relationships.

Conjectures Re Fundamental SM Masses

The fermion masses could quite conceivably emerge dynamically with just a single parameter to set the overall fundamental particle mass scale for both fundamental fermions and fundamental bosons.

On the fundamental boson side, the Weinberg angle is the inverse tangent of the bare electromagnetic force gauge coupling constant g' divided by the bare weak force gauge coupling constant g. The magnitude of the fundamental electric charge "e", in turn, is the bare weak force gauge coupling constant g times the sine of the weak force mixing angle (and thus can be determined solely from g and g'). The mass of the W and Z bosons can be computed from g, g' and the Higgs vacuum expectation value v (246.22 GeV). The measured value of the Higgs boson mass is strongly consistent with the square of the Higgs boson mass is equal to v2/2-(MW)2-(MZ)2. So, all of the fundamental boson masses can be determined from g, g' and the Higgs vev, removing three parameters from the Standard Model.

The sum of the square of the fermion masses, likewise, is very nearly equal to v2/2. As you, arivero, have demonstrated, a couple of extensions of Koide's rule can get you very close to all of the nine fundamental charged fermion masses from the electron mass and the muon mass, if only a quirks in the extension of Koide's rule for quarks can be ironed out properly (most likely by making the appropriate adjustment for the down type quark missing from the triple with a middle mass up type quark, or visa versa). Indeed, with the Higgs vev to set an overall mass scale, the only other parameter you need use that approach to get all nine fundamental fermion masses is the ratio of electron mass to the muon mass.

So, it seems attainable to get all nine of the charged fundamental fermion masses, and all three of the fundamental massive boson masses, from two of the three SM coupling constants, the Higgs vev, the ratio of the electron mass to the muon mass, and the CKM matrix. This would reduce the number of experimentally measured parameters in the Standard Model by 10 out of 26.

Conjectures Re CKM Matrix

The CKM matrix can be expressed quite accurately in a parameterization of just one real parameter (the Cabibbo angle) and one complex parameter associated with CP violation, because in the Wolfenstein parameterization Aλ2 is equal to (2λ)4 at the 0.1 sigma level of precision, and there is no place in the Wolfenstein parameterization of the CKM matrix where this substitution cannot be made. This reduces the number of CKM matrix parameters from 4 to 3.

Combined with the mass conjectures above, that would reduce the number of Standard Model parameters from 26 to 15 (of which 7 are for neutrinos).

The key point about the structure of the CKM matrix as expressed in the Wolfenstein parameterization that makes it seem more fundamental is that, up to adjustments for CP violation, this parameterization suggests that the probability of a first to second generation transition (or second to first generation transition) is λ, that the probability of a second to third generation (or third to second generation) transition is (2λ)4 , and that the probability of a first to third generation (or third to first generation) transition is (2λ)5 (i.e. the product of the probability of making first one of the single generation step transitions and then the second).

The probability of transitioning to a quark of the same generation is the residual probability after the probability of the other two options is subtracted out.

The CKM matrix, so parameterized, suggests an almost atomic energy shell-like sequence of transition probabilities between generations that one can imagine popping out easily from some more fundamental theory that is really more straight forward.

Crazy Talk

Making One Electroweak Coupling Constant Derived

g + g', the sum of the two dimensionless electroweak coupling constants, at the W boson mass, are just a wee bit over 1. But, both of these constants run with energy scale, and it is very tempting to imagine the possibility that at some energy scale, such as the Higgs vev, that g+g' are exactly equal to 1.

If this were the case, we would replace one of the two dimensionless electroweak coupling constants in the set of Standard Model parameters with the Higgs vev, reducing the number of experimentally measured parameters of the Standard Model apart from the neutrino sector, from 8 to 7.

Deriving Wolfenstein CKM Parameter λ From The Electroweak Coupling Constants

It is also tempting to think that the sine of the Cabibbo angle could have a functional relationship of some kind to the Weinberg angle, in some way that could reconcile their 2.48% discrepancy, perhaps by redefining the Cabibbo angle. For example, one could imagine redefining it as the inverse tangent of (the absolute value of CKM matrix element Vus plus the absolute value of CKM matrix element Vub) divided by the absolute value of CKM matrix element Vud which would increase the sine of the Cabibbo angle to about 0.22867, and then multiplying this time one plus the fine structure constant (which is roughly 1/137), which would bring it to 0.23034. This would be within one standard deviation of the square of the sine of the Weinberg angle at the Z boson energy scale given the precision of current experimental measurements (the precision of the Weinberg angle measurement is about six times greater than the precision of the Cabibbo angle measurement).

The extension of the definition of the Cabibbo angle to include the addition of CKM matrix element Vub is very natural. The Cabibbo angle was originally defined before the third generation of Standard Model fermions was discovered. In a two fermion generation Standard Model that Cabibbo angle was simultaneous the probability of a transition to a non-first generation quark and the probability of a transition from a first to a second generation quark. Including the CKM matrix element for a transition to a third generation would generalize it using the latter interpretation of its meaning, rather than the former, which were both identical in the two generation case.

The inclusion of a factor of one plus the fine structure constant is less obvious and somewhat arbitrary. But, given that we are talking about an electroweak process that always involves a W boson with has both a weak force coupling and an electromagnetic coupling, it would hardly be stunning that a formula to derive from first principles a probability of quark generation transitions from one generation to another might involve both the weak mixing angle and the electromagnetic coupling constant.

This would make the Cabibbo angle a function of the two electroweak coupling constants, and they, in turn, could conceivably be a function of either one of those constants and the Higgs vev. You could then work out the Wolfenstein parameter λ, from the redefined Cabibbo angle.

This would mean that the Standard Model experimentally measured parameters (outside the neutrino sector) could be reduced to just six if something along the lines of the kind of toy models I am discussing as conjectures could be worked out:

1. The strong force coupling constant.
2. The value of one of the electroweak coupling constant at the Higgs vev energy scale.
3. The Higgs vev.
4. The ratio of the muon mass to the electron mass.
5.-6. The complex valued CP violating parameter of the CKM matrix in the Wolfenstein parameterization.

We know 2, 3 and 4 to extreme precision. We know 1, 5 and 6 to moderate precision.

This would still leave in the neutrino sector three mass eigenvalues and four PMNS matrix parameters. Discovery of a Koide-like relationship for the neutrino masses might reduce the number of neutrino sector parameters from 7 to 6 that could be confirmed when they could actually be measured more precisely. But, we are too far away from having sufficiently accurate measurements of the PMNS matrix parameters (particularly the Dirac CP violating phase) to be able to speculate about possible relationships between them in more than an idle way at this point.

Really, Really Crazy Talk

For what it's worth, I think we are also at a similar juncture with the dark matter-dark energy problem. I think we could find a modification of gravity that solves both problems in one fell stroke while eliminating the cosmological constant as well, in just a few inspired years of articles from the right scholar (my eyes are on Alexandre Deur or someone follow up on his insights).

Between the SM's 26 constants, GR's 2 constants (Newton's constant and the cosmological constant), Plank's constant and the speed of light, we current have a total of 30 fundamental experimentally measured constants.

I think we could cut the total down to 15 while simultaneously solving the dark matter and dark energy issues. And, given that 6 of those remaining 15 would be for the neutrino sector, some of those could probably be trimmed somehow as well with one or two more breakthroughs in the neutrino sector.

Of course, all of those extra relationships and the reduced number of pieces of the puzzle, might in turn increase the likelihood that someone could find a yet deeper relationship that is even more reductionist and fundamental.

20. Jan 13, 2017

arivero

I was speculating that we could have two kinds of W particles, seeing differently the charge of some of our unconventional "families", so that one of the W grants the Koide equation for the first group of triplets, the boldface ones, and other for the second group. But it is unclear, to see the least. Also, the breaking of (bds) seems small, empirically, so perhaps the unbroken triplets are not really the ones listed above, but bds instead of uds. Or both :-(

btw, Koide has new paper, on Sumino models. https://arxiv.org/abs/1701.01921