What is new with Koide sum rules?

[S.Daedalus]

This whole discussion is way above my level of expertise in high-energy physics, but I have a side question if I may: if I understood correctly, one of the main problems with the Koide approach is that it's a connection between the low-energy masses of the theory, which should have no fundamental significance due to renormalization group flow. But couldn't there be something like supersymmetry nonrenormalization at work, that is, the parameters m that turn up in the low energy theory are actually identical to the high energy masses?

 

[mitchell porter]

Actually those nonrenormalization properties disable the Sumino mechanism for preserving the Koide relation, because it relies on vertex corrections that no longer exist under susy! Koide and Yamagarbagea developed an alternative but it doesn't work as well. Still, perhaps one can hope that susy will simplify the RG equations in some other way.

Some resources.

 

[mitchell porter]

Another useful NCG paper is Kolodrubetz & Marcolli. Also see this lecture, especially slide 10. It seems that one wants to construct a cascade of effective field theories, with a Sumino model at the final stage.

Returning to comment #65... The original Koide triplet relates yukawas from a single mass matrix, but the new triplets for quarks all combine up-type yukawas with down-type yukawas, so the transformation looks unnatural. It's as if we need an extended Higgs mechanism that includes "up-down yukawas". We could suppose they are there and set them to zero... but what would they be? The Standard Model mass matrices tabulate coefficients of Yukawa terms in the Lagrangian. These new "up-down Yukawa terms" would require something new.

Nonetheless:

$$\left( \begin{array}{ccc} y^u_{11} & 0 & y^u_{12} & 0 & y^u_{13} & 0 \\ 0 & y^d_{11} & 0 & y^d_{12} & 0 & y^d_{13} \\ y^u_{21} & 0 & y^u_{22} & 0 & y^u_{23} & 0 \\ 0 & y^d_{21} & 0 & y^d_{22} & 0 & y^d_{23} \\ y^u_{31} & 0 & y^u_{32} & 0 & y^u_{33} & 0 \\ 0 & y^d_{31} & 0 & y^d_{32} & 0 & y^d_{33} \\ \end{array} \right)$$

... if I may be permitted to introduce this interleaving of up and down Yukawa matrices, without exactly saying what it is; and if we suppose that the "up" and "down" parts are each diagonalized as much as possible, with diagonal entries ordered by size; then the Koide waterfall amounts to saying that there is a "Brannen symmetry" for each 3x3 block on the main diagonal.

edit: Whoops, I missed a stage. The Brannen symmetry relates the square roots of the masses. So we would be looking at blocks on the diagonal of a 6x6 matrix whose square is the matrix above.

edit #2: The Brannen transformation for a particular block could look like this:
$$\left( \begin{array}{ccc} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & \sqrt{2} & e^{i\delta} & e^{-i\delta} & 0 & 0 \\ 0 & e^{-i\delta} & \sqrt{2} & e^{i\delta} & 0 & 0 \\ 0 & e^{i\delta} & e^{-i\delta} & \sqrt{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right)$$

 

[mitchell porter]

There's a new paper on Koide triplets today, in which the author experiments with a Brannen parametrization of up-type masses and down-type masses, and comes out with phases of 2/27 and 4/27. The usual Brannen phase for the charged leptons is 2/9, i.e. 6/27. These are numbers which I first saw on Marni Sheppeard's blog, and which I thought she discovered through discussion with Dave Look, so I'll be writing to the author to let him know - as well as to mention the tbcsud "waterfall" of triplets discussed in this thread. I think of the waterfall as real, and tend to dismiss those quark family triplets as spurious. Given the idea that there are unknown "Koide symmetries" responsible for the "authentic triplets", I suppose it's possible that the same symmetries could be present in uct and dsb too, but with a lot more noise.

 

[arivero]

There's a new paper on Koide triplets today,

...
I think of the waterfall as real, and tend to dismiss those quark family triplets as spurious.
...

Of course I am of the same opinion, but still I am sligthly amused that the waterfall uses delta_scb = 3 delta_L, and he gets delta_L= 3 delta_U.

 

[mitchell porter]

Or to put it another way, the scb angle is 2/3, the eμτ angle is 2/9, and the uct angle is 2/27.

We also have that the eμτ mass scale is 313 MeV (one-third the proton mass, i.e. constituent mass of a first-generation quark), and the scb mass scale is three times that.

From Sheppeard's blog (1 2), I get that the mass scale for a uct triplet would be about 20 GeV.

edit: A few months ago I was thinking about what sort of model would produce just these "family phases" - what Zenczykowski calls δ_L, δ_D, δ_U - simply because that's easier to think about. I was interested in an Adler-type 3HDM (three-Higgs-doublet model) with circulant mass matrices. But you could take any model of the charged-lepton sector, that produces a Koide relation, and try to apply it separately to the up-type and down-type quarks - for example, Ernest Ma's supersymmetric model.

 

[arivero]

Or to put it another way, the scb angle is 2/3, the eμτ angle is 2/9, and the uct angle is 2/27.

Of course, it is very problematic to have angles which are not a submultiple of the circumference. Up to now, the main motivation for the factor of three was to consider the case where one of the masses is 0, fixing thus the angle, and then the orthogonality between the triples with 15 degrees and 45 degrees, this is pi/12 and pi/4. The angle of 2/3 I though of it as a perturbation from pi/4, the angle of 2/9 as a perturbation from pi/12.

 

[mitchell porter]

Since we don't know where these quantities come from, I don't think we can say that their form is problematic. Would their origin be easier to understand if they were simple fractions of π? Also, it's hard to think of e.g. 2/9 (the actual phase, for e-μ-τ) as a perturbation of π/12 (the phase for e-μ-τ, in the "modified waterfall" that lands on Harari-Haut-Weyers values for d-u-s masses), because normally a perturbation of a quantity x just gives you "x plus a small mess", it doesn't give you a simple rational number! I have noticed that 2/3 (possible phase for s-c-b) is obtained by the first two terms in the Leibniz formula for π/4, as if it were a truncation. One could start thinking about formulas with Grassmann variables, so all the higher terms vanish...

Another line of investigation would be to look for the significance of the "Brannen angle" in the other frameworks that manage to produce Koide triplets. Sumino, in his paper which tries to explain the exactness of the original Koide formula despite RG running, also presents an original derivation of the triplet itself (from the interactions in the scalar sector of his model). Then there's Ma, mentioned above; then there's Koide's original preon theory. Carl Brannen's formula plays no apparent role in any of these, but I wonder if they still look simple when expressed using his variables?

edit: Some comments on whether m_u=0 is still a live option. (For the general reader of this thread: Alejandro found a "waterfall" of interlocking Koide triplets which works well for the four heaviest quarks and which can be extended to the remaining quarks. The modified waterfall is a version adjusted so that the up quark has exactly zero mass. The heavy quark masses become less accurate but the Brannen angles assume interesting values, and the idea is that the real waterfall is a perturbed version of this modified waterfall, see his paper for details.)

Michael Dine gave a talk as recently as 2009 implying that it was still being considered by theorists like Seiberg and Kaplan. Dine's 1993 review "Topics in string phenomenology" points out two ways to get m_u=0 from string theory, one from anomalous discrete symmetry, the other from a horizontal symmetry as described in a series of papers (1 2 3).

From the other side, 1103.3304 gives in a few sentences (page 83) the reason why workers in lattice QCD might dismiss the m_u=0 option as an explanation for no strong CP violation. This argument needs to be confronted with the ideas in reference 3, listed above.

 

[arivero]

I am curious about how sensible the prediction of the top mass is to the factor 3 in the jump from leptons to quarks. So here is the "bc -l program"

define top(massfactor,anglefactor) {
me=0.000510998910
mmu=0.1056583668
mtau=((sqrt(me)+sqrt(mmu))*(2+sqrt(3)*sqrt(1+2*sqrt(me*mmu)/(sqrt(me)+sqrt(mmu))^2)))^2
m=(me+mmu+mtau)/6
pi=4*a(1); cos=(sqrt(me/m)-1)/sqrt(2); tan=sqrt(1-cos^2)/cos
delta=pi+a(tan)-2*pi/3
mc=massfactor*m*(1+sqrt(2)*c(anglefactor*delta+4*pi/3))^2
ms=massfactor*m*(1+sqrt(2)*c(anglefactor*delta+2*pi/3))^2
mb=massfactor*m*(1+sqrt(2)*c(anglefactor*delta))^2
mtop=((sqrt(mc)+sqrt(mb))*(2+sqrt(3)*sqrt(1+2*sqrt(mc*mb)/(sqrt(mc)+sqrt(mb))^2)))^2
return mtop
}

Newcomers can see that the same formula is used for mtop and mtau. And of course it is also the same formula, in angular form, for the rest of the calculation involving delta and m. Koide everywhere.

so with the factor 3 argued in my paper, we get

top(3,3)
173.2639415940

Which is in the center of the combination of Tevatron (173.18) and LHC (173.34). In fact, the weighed average of Tevatron and CMS (september) should be 173.265 ± .679 GeV, so the prediction is pretty in the center.

Which are the 1 sigma limits for the mass and angle factors, with this average? Well, pretty narrow, but still some place for perturbative corrections:

173.265+0.679
173.944
top(3,3.046)
173.9416301253
top(3.012,3)
173.9569973497
x=3.01;top(x,x)
173.9906886621
173.265-0.679
172.586
x=2.990;top(x,x)
172.5374243669

 

[arivero]

Probably it is a red herring, but some comments from mitchell have indirectly driven me to look at the mass formula for an stack of D-branes. I am not sure in how they are in the superstring case, but already in the bosonic string they look a lot as a generalisation of Koide mixing:

$$M^2 = \big((n + {\theta_i - \theta_j \over 2 \pi}) {R' \over \alpha'}\big)^2 + {N-1 \over \alpha'}$$

THis is f. 174 in arXiv:hep-th/0007170v3

For n=1, N=1, and i,j from 1 to 3 with i different of j, the stack of three D-branes looks Koide's formula. I am not sure of which is the mass formula in this case (nor in the fermionic/superstring case...)
It should be, if M^2 where instead a seesawed product of two masses, $M^2=m_{ij} M_0$


$$m_{ij} = {R'^2 \over M_0 \alpha'^2} (n + {\theta_i - \theta_j \over 2 \pi})^2$$

Note that the basic fact is that the sum of the three differences $\theta_i - \theta_j$ is zero, as in the case of the sum of three cosines in Koide.

EDIT: Michael Rios suggested, last year, to use three coincident branes to emulate Koide.
EDIT2: Today is the birthday of Lubos Motl, this is my birthday gift: string theory becomes predictive

 

[arivero]

Unrelated to the previous comment, except for the fact that strings dof come in groups of 8, it could be worthwhile to rethink again the 12x8 ideas in the light of Koide. The "Koide waterfall" in #58 above, with mass of up quark exactly zero in three Koide steps, provides, if we also use the orthogonality condition, some intriguing pairing of leptons and quarks:

t:174.10 GeV
b:3.64 GeV
tau, c:1.698 GeV
mu, s:121.95 MeV
e, u:0
d:8.75 MeV

On other hand, the most naive way of building a multiplet with 8 degrees of freedom is to use an electroweak pair of Dirac fermions: neutrino, electron for instance, or any up, down combination. This is still possible here, and even it could be convenient if we consider that we are going to broke this pairing of leptons and quarks. But looking at this table, we could take it serioustly and consider that one lepton and the three colours of a quark should be the components of a multiplet. Then the unpaired quarks would correspond to see-sawed neutrinos and the whole table is

$\nu_1,t$: 174.10 GeV
$\nu_2,b$: 3.64 GeV
$\tau,c$: 1.698 GeV
$\mu,s$: 121.95 MeV
$\nu_3,d$: 8.75 MeV
$e,u$: 0

Of course we have sixteen degrees of freedom in each line and it is still to see how they should be managed in groups of 8, either by chirality or by particle/antiparticle.

What is intriguing in any case is the mu,s pairing: a charged lepton with a down type quark. It could point to the need of using a SU(2)xSU(2) L-R symmetry.

 

[mitchell porter]

Pati-Salam as we know it, doesn't allow such a scheme. The orthodox way to embed the waterfall in Pati-Salam would be to use the conventional generation structure (three sets of two "four-color quarks", one of which divides into an up-type quark and a neutrino, the other of which divides into a down-type quark and a charged lepton), a selection of Higgses (there must at least be one to break SU(4)_c to SU(3)_c and another to break U(1)_B-L x SU(2)_R to U(1)_Y), then work out the 3x3 Yukawa matrices for the "four-color quarks", and finally the effective Yukawas for the SM quarks and leptons. And since the waterfall has that intricate structure, probably the best way is via flavons: the Yukawas are VEVs of "flavon" fields. (Koide himself uses flavons in his yukawaon models of recent years.) We can then try to obtain a Pati-Salam waterfall from flavon symmetries.

This doesn't have the simplicity of just directly associating e-mu-tau with u-s-c (or with s-c-b, as might have been suggested by Georgi-Jarlskog), but at least it is a type of theory which it is known can be constructed. If you do it this way, the orthodox way, you do get to preserve the direct association of muon with strange quark. So you might suppose that the second generation is a sort of pivot, where there is approximate equality of masses, connecting waterfall Koide triplets on the quark side, and the usual family Koide triplets on the lepton side.

Intriguingly, if you imagine interleaving the Yukawas for charged and uncharged leptons in the fashion of #73, then the Brannen transformation matrix for family Koide triplets (rather than sequential, waterfall triplets) looks like this:
$$\left( \begin{array}{ccc} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & \sqrt{2} & 0 & e^{i\delta} & 0 & e^{-i\delta} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & e^{-i\delta} & 0 & \sqrt{2} & 0 & e^{i\delta} \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & e^{i\delta} & 0 & e^{-i\delta} & 0 & \sqrt{2} \\ \end{array} \right)$$
(or the obvious counterpart where the two interleaved blocks change places).

As I wrote in #73, an "interleaving of Yukawa matrices" has no physical meaning that I can identify. But what I like about this perspective is that the "family Brannen transformation" and the "sequential Brannen transformation" could both plausibly be part of some larger algebraic structure. In both cases they're based on a 3x3 block within the 6x6 matrix, it's just that the spacing is different. So the idea is that a Pati-Salam embedding of waterfall + original Koide could result from a flavor symmetry containing that "larger algebraic structure", with family symmetries dominating on the lepton side and sequential symmetries dominating on the quark side, and with the second generation providing the bridge.

 

[arivero]

I have scanned a couple of collections which show the history of Koide before Koide. The first set pivotes on Harari-Haut-Weyers and its refutations, the second set is some extra articles of the same age, found while I explored the first selection.

While they will be useful mostly to Carl Brannen, perhaps Mitchell and other crowd can enjoy them too.

https://docs.google.com/open?id=1UflBQIr-r3RsEigmr1ty2kTMpSU_5gM2Z-DFtWfKW7rewRqkMlReePUCCGDj

https://docs.google.com/open?id=1vRfLIl-wvb7BRNlS4z9uukYDFwDLb71OC2x5iB9Z9O4vB4PnIwW634ZOhATP

And yes, I use a monitor which can pivote 90 degrees. But nowadays you can always cancel the gravitational sensor of your iPad, can you?

 

[arivero]

Wow, Zenczykowski paper was accepted for PhysRev D last Thursday (Dec 13, 2012).

There's a new paper on Koide triplets today,

If it is still v2, I am a bit sorry that he has not changed the references (for instance, to include the waterfall) but still a good thing.

 

[arivero]

Just for the record, it is interesting to look to the solutions in the lepton side ascending from the e-mu-tau triple. Remember we are conjecturing a descent where some leptonic object partners with every quark. It could be reasonable to think of Dirac mass terms for neutrinos, for instance.

-, t:174.10 GeV
-, b:3.64 GeV
tau, c:1.698 GeV
mu, s:121.95 MeV
e, u:0
-, d:8.75 MeV

Now, once we have broken the pairing, we can use Koide separately in each sector... just to see if it has some sense.

To ascent from mu, tau to the next two levels, the equation with the above values has discriminant cero, we should look with some care with branch of the answer is it really taken, but anyway here you have both branches. For both of them, the second step is unique, due to negative roots forbidding other solutions.

mtau
1776.96888139816566506171
((sqrt(mtau)-sqrt(mmu))*(2+sqrt(3)*sqrt(1-2*sqrt(mtau*mmu)/(sqrt(mtau)-sqrt(mmu))^2)))^2
7211.73510126774064895083
((sqrt(mtau)+sqrt(m1))*(2+sqrt(3)*sqrt(1+2*sqrt(mtau*m1)/(sqrt(mtau)+sqrt(m1))^2)))^2
268928.53716239525673236427

((sqrt(mtau)-sqrt(mmu))*(2-sqrt(3)*sqrt(1-2*sqrt(mtau*mmu)/(sqrt(mtau)-sqrt(mmu))^2)))^2
1812.91990902666662582893
((sqrt(mtau)+sqrt(m1))*(2+sqrt(3)*sqrt(1+2*sqrt(mtau*m1)/(sqrt(mtau)+sqrt(m1))^2)))^2
121946.96009306194199844666

I like this second branch: the (tau, nu1, nu2) triplet equal to (1.777, 1.813, 121.95). It could be saying that the lepton sequence moves to increase the gap between the two final states. And while we left the 174.1 GeV endpoint, we still are in a nice mass range.

 

[mitchell porter]

Suppose we have six flavors of quark in an SU(4) gauge theory. For the moment, suppose there are no other quantum numbers... Then we will have a 6x6 yukawa matrix.

Next, suppose that these yukawas are flavon vevs, and that the flavon potential has a discrete symmetry generated by the four "sequential" (#73) and two "family" (#82) Brannen transformations, for particular values of δ.

And now, let us augment this "theory", so that the usual electric charges for the quarks arise or are introduced, and so that the usual Pati-Salam higgsing of SU(4) to SU(3) occurs. It seems that the first step should introduce a "checkerboard" texture to the 6x6 yukawa matrix, and then the second step should "double" the yukawa matrix, so there's one 6x6 yukawa checkerboard for three-color quarks and another 6x6 yukawa checkerboard for leptons.

Finally, let us suppose that the sequential symmetries dominate the quark yukawas, and the family symmetries dominate the lepton yukawas (though the residual family symmetries in the quark yukawas may be strong enough to produce recognizable Koide phases of 2/27 for up quarks and 4/27 for down quarks). This can give us the waterfall for the quarks, and the original Koide relation for the charged leptons.

edit: I think the first thing to do, would be to create the theory of the second paragraph. That would be practice at constructing a theory in which a Koide waterfall of masses arose from a 6x6 yukawa matrix.

 

[arivero]

Finally, let us suppose that the sequential symmetries dominate the quark yukawas, and the family symmetries dominate the lepton yukawas (though the residual family symmetries in the quark yukawas may be strong enough to produce recognizable Koide phases of 2/27 for up quarks and 4/27 for down quarks). .

I ack that the publication of Żenczykowski paper has biased me to consider again the family symmetries. It could be that we are seeing different aspects of a larger discrete symmetry group... but how? The real problem is that it does not seem a permutation group because for any pair of masses we exchange, we find that one Koide equation keeps invariant but obviously others, containing only one of the two masses, are not preserved. by the way, the publication of Phys. Rev. D 86, 117303 (2012) officially raises the number of cites of Brannen's and of myself on this topic!. I get a citation to hep-ph/0505220 so that the author can refer indirectly to internet forums with a "Brannen, as cited in...". And Carl gets a second citation, directly to http://brannenworks.com/MASSES2.pdf

 

[mitchell porter]

I think it would be instructive to express all the fermion pole masses as multiples of the Brannen mass parameter for the original Koide triple, M_L, and then use the relationship between the top mass and the Higgs VEV to express the latter in the same units. M_L is presumably the fundamental quantity in the waterfall (because it apparently comes from QCD or SQCD), but I don't think we've thought about how to get the Fermi scale from it. Yet surely this should be playing a role in our thinking about the Higgs.

 

[mitchell porter]

Today I was looking at two new and two old papers. The new papers are "Neutrino Mass and Mixing with Discrete Symmetry" by King and Luhn, and "Top-quark and neutrino composite Higgs bosons" by Adam Smetana. The old papers - well, one was a thesis, Francois Goffinet's thesis, http://cp3.irmp.ucl.ac.be/upload/theses/phd/goffinet.pdf, and the other was the co-authored paper resulting from it, "A New Look at an Old Mass Relation".

Together, they should have something to say about how to extend the waterfall to the neutrinos, to the mixing angles, and to the properties of the Higgs sector. Goffinet's concept of "pseudo-mass" was invented precisely to link the Koide relation to mixing angles. King and Luhn review flavon models with discrete family symmetries, for the neutrino sector. And Smetana tries to get the Fermi scale by having both a top condensate and a neutrino condensate, in a broad class of models featuring a gauged flavor symmetry. To get the numbers right he ends up needing a large number of right-handed neutrinos, so probably he is still missing something essential, but it begins to make the connection I called for in the previous comment.

 

[arivero]

Today I was looking at two new and two old papers. The new papers are "Neutrino Mass and Mixing with Discrete Symmetry" by King and Luhn,

$S_4$ is intriguing. It is equal to $Z_2 \times Z_2 \ltimes S_3$, and I wonder if these, say, four copies of $S_3$ could be the four copies acting in the waterfall. Also, where does it come from? Speculatively, could be a subgroup of SU(5)_flavour -the sBootstrap group-. And more speculatively, what about the permutations of the 4 components of a spinor?

 

[arivero]

Given that a lot of Koide stuff seems related (hat tip to de Vries and Brannen here) to this matrix

$$\begin{pmatrix}-\frac{2\,\mathrm{sin}\left( t\right) }{\sqrt{6}} & \frac{\mathrm{sin}\left( t\right) }{\sqrt{6}}+\frac{\mathrm{cos}\left( t\right) }{\sqrt{2}} & \frac{\mathrm{sin}\left( t\right) }{\sqrt{6}}-\frac{\mathrm{cos}\left( t\right) }{\sqrt{2}}\cr \frac{2\,\mathrm{cos}\left( t\right) }{\sqrt{6}} & \frac{\mathrm{sin}\left( t\right) }{\sqrt{2}}-\frac{\mathrm{cos}\left( t\right) }{\sqrt{6}} & -\frac{\mathrm{sin}\left( t\right) }{\sqrt{2}}-\frac{\mathrm{cos}\left( t\right) }{\sqrt{6}}\cr \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}}\end{pmatrix}$$

I have setup a wxMaxima notebook to play with it. Not that I like Maxima, I used it in a VAX and it was already superseded by REDUCE when Mathematica come. But it comes with Ubuntu and has a graphical interface, which Reduce has not.

 

[mitchell porter]

Koide has a model of leptons with an S4 symmetry embedded in a flavor SU(3).

 

[mitchell porter]

Since this is a Koide thread we have to mention Zenczykowski's latest, though it is about "family triples", and not what I call the "sequential triples" of the waterfall... He's still building on the generalization of the e-mu-tau 2/9 parameter to u-c-t and d-s-b; he proposes that another parameter, which is just "1" for e-mu-tau, is also "1" for the quarks if you use Goffinet's concept of "pseudo-mass". If that's true it's a breakthrough, as well as a headache for the waterfall, because aren't we getting overloaded with too many relationships at once?

He mentions the usual problem, that these relations work best for low-energy masses. We've previously discussed Sumino's efforts to have family gauge bosons cancel out certain QED corrections, so that Koide's relation may be exact; but I was always curious about whether there might be some dual description of physics, in which, rather than thinking of the UV as fundamental, you thought of the physics as "IR + new degrees of freedom at a series of higher energies" - the idea being that the cause of Koide relations might be more transparent in this hypothetical "infrared first" formulation.

Well, I wonder if this paper by Davide Gaiotto (from January 2012) might be relevant: "Domain Walls for Two-Dimensional Renormalization Group Flows". "Renormalization Group domain walls are natural conformal interfaces between two CFTs related by an RG flow. The RG domain wall gives an exact relation between the operators in the UV and IR CFTs." It seems a tiny step towards what I had in mind.

 

[arivero]

If that's true it's a breakthrough, as well as a headache for the waterfall,

The waterfall could happily miss the last triple, d-u-s, in exchange by one of the "standard" ones, but d-u-s does a better prediction of the down mass that d-s-b.

A motivation to follow this track could be, put all the quarks in the faces of a cube, such that all the equations of the waterfall are the faces that meet in some vertex. You will notice that his cube has a property, that opposite faces have opposite weak isospin. You can also notice that we only need three equations to fix the faces.

One of the vertexes of this cube is DSB, and of course is opposite vertex has the faces of the up-type quarks. This is the only axis that does not correspond to a waterfall symmetry, and on other hand the DUS vertex is the only axis which is used in both extremes.

Going to discrete groups, S4 is the group of permutations of the four "Z3 axis" in a cube, while the subgroup S3 is contained in four not-very-different ways, each of them being the permutations that keep one of such axis invariant (you can exchange fully the vertex by the opposite, to implement the Z2 subgroup of S3).

 

[arivero]

The real problem of S4 is to know the physics content, the objects we are permuting. The suggestion of putting quarks (or leptons) in the faces of a cube is rarely seen in the literature.

About PZ and pseudomasses, I think it is not very different of the initial objections to Harari-Haut-Weyers, they also do a similar trick, or a trick that can be interpreted as taking only the diagonal of the undiagonalised mass matrix.

By the way, I note that the abbreviations 2/27 and 4/27 are first used by Sheppeard in her note 342

 

[mitchell porter]

The pseudomass adds up the contributions of all the mass eigenstates to one of the weak eigenstates. So it looks like "sequential" triples, like in the waterfall, apply to mass eigenstates, family triples (as in the original Koide formula) apply to weak eigenstates, and this wasn't noticed until recently because, for charged leptons, the weak eigenstates are the same as the mass eigenstates.

For the quarks, we can then think of the waterfall as the dominant chain of relationships, and then the mixing parameters encode the rotation away from waterfall mass values, required to produce family triples with 2n/27 phases.

For the leptons, perhaps the family triples dominate, and a waterfall is weak or nonexistent. (I'm still not clear on whether right-handed neutrinos could have masses of the order of the quarks, as in #81, and then give rise to the observed small masses via seesaw.)

p.s. Chris Quigg had a paper yesterday - "Beyond Confinement" - in which 2/27 shows up as the exponent in a relation between the top mass and the nucleon mass, in a unified theory!

 

[arivero]

I hated macsyma and I am transferring my hate to its free twin, maxima, but still this is interesting. I got again the waterfall while I was trying to solve for the full S4 symmetric set of Koide equations.

This is, I was trying to find mass values in the six faces of a cube such of for each vertex we have a Koide equation. The group of rotations of the cube is S4; imposing Koide explictly breaks the symmetry as it gives different values to different faces. Not a very convincing motivation, but ok to play a little bit.

Now, in maxima. You define
K(x,y,z):=x^2+y^2+z^2-4*(x*y+y*z+z*x);
so that
expand(K(x,y,z)*K(x,-y,z)*K(x,y,-z)*K(x,-y,-z));
is a degree 8 even polynomial on three variables;
we put all of it

Q(x,y,z):=z^8-28*y^2*z^6-28*x^2*z^6+198*y^4*z^4-1172*x^2*y^2*z^4+198*x^4*z^4-28*y^6*z^2-1172*x^2*y^4*z^2-1172*x^4*y^2*
z^2-28*x^6*z^2+y^8-28*x^2*y^6+198*x^4*y^4-28*x^6*y^2+x^8;

and now we can use maxima "eliminate" to do the equivalent, I guess, of Sylvester matrix.
factor(eliminate([Q(1,x1,x2),Q(1,x2,x3)],[x2])) shows two terms that just validate x^2=x3^2. So in my first step I also canceled these factors:

step1:factor(eliminate([Q(1,x1,x2),Q(1,x2,x3)],[x2]))/(x3-x1)^8/(x3+x1)^8;

In the next step, factor(eliminate([paso1[1],Q(1,x3,x4)],[x3])) happens to have as a factor the polynomial for Q(1,x1,x4) and then it trivially tell us that the whole system of vertexes (1,x1,x2),(1,x2,x3),(1,x3,x4),(1,x4,x1) has solutions. As we want to exhibe actually some solution, I cancel the factor first

step2:factor(eliminate([step1[1],Q(1,x3,x4)],[x3]))/factor(Q(1,x1,x4)^12);

and now I cross each of the factors against the extant equation Q(1,x1,x4)

for i:1 thru 16 do (
pol:part(part(step2[1],i),1),
sl:solve([pol,Q(1,x1,x4)],[x1,x4]),
for k:1 thru length(sl) do (
  s:ev([x1,x4],sl[k]),
  if featurep(s[1],real) and featurep(s[2],real) then 
     (s:abs(s),
     if s[1]>s[2] then s:[s[2],s[1]],
     print(s,float(s)), 
     )
  )
);

The process solves to:
four complete x1,x2,x3,x4,x1 cycles
two sequences x1,x2,x3,x4
two sequences x1,x2,x3

The solutions for the waterfall triplets cbt and bcs appear in the list, numerically as sqrt(t)=10.12, sqrt(b)=1.464, sqrt(s)=0.267, with sqrt(c)=1, in one of the not-closing sequences. I guess I need an expert on discrete groups in order to understand what is going on.

PS. wow, now I notice that Q surely is the trick that Goffinet uses to avoid the square roots somewhere in his thesis.

 

[mitchell porter]

Exciting progress!

wow, now I notice that Q surely is the trick that Goffinet uses to avoid the square roots somewhere in his thesis.

http://cp3.irmp.ucl.ac.be/upload/theses/phd/goffinet.pdf (section 3.2.1).

I guess I need an expert on discrete groups in order to understand what is going on.

A further step would be to look for an S4-symmetric potential where these solutions are the local minima.

 

[arivero]

http://cp3.irmp.ucl.ac.be/upload/theses/phd/goffinet.pdf (section 3.2.1).[/QUOTE]

Indeed it is the same polynomial.

z^8-28*y^2*z^6-28*x^2*z^6+198*y^4*z^4-1172*x^2*y^2*z^4+198*x^4*z^4-28*y^6*z^2-1172*x^2*y^4*z^2-1172*x^4*y^2*
z^2-28*x^6*z^2+y^8-28*x^2*y^6+198*x^4*y^4-28*x^6*y^2+x^8;

  4         3         3        2  2             2        2  2       3
(%o3) z  - 28 y z  - 28 x z  + 198 y  z  - 1172 x y z  + 198 x  z  - 28 y  z
           2           2           3      4         3        2  2       3
 - 1172 x y  z - 1172 x  y z - 28 x  z + y  - 28 x y  + 198 x  y  - 28 x  y
    4
 + x

The minor improvement is that here we are sure that it is an ·"if and only if" relationship; Goffinet, in the text, was worried that the squaring could be introducing spureous solutions. As we have shown that this poly decomposes exactly in the product of the four possible sign combinations of Koide equation, now we are in position to grant that every solution of Goffinet's matrix version (3.30) of the equation is really a Koide solution.

Let me copy here this equation 3.30, setting the determinant of M as a function of the traces in M and M^2:

$$|M| = {2 \over 3 * 32^2} {(7 (Tr M)^2 - 8 Tr M^2)^2 \over Tr M}$$

 

[arivero]

The following Mathematica code will help to find the solutions to the S4 symmetric Koide system.

If this post is the only one you are going to read, remember that we are organising three generations in opposite faces of a cube, and each corner must agree with Koide equation. A way to solve this is to fix one face, say to unity, and then check to four corners of this face.

Please use the code line-by-line; it is listed here without EOF separators! Also, please verify that you select the non common factor in each step, the order could change between versions of Mathematica (this is done with version 9.0 in the free trial period)

K[u_, v_, t_] := u u + v v + t t - 4 (u v + v t + t u)
G[m1_, m2_, m3_] =FullSimplify[K[Sqrt[m1], Sqrt[m2], Sqrt[m3]] K[-Sqrt[m1], Sqrt[m2], Sqrt[m3]] K[ Sqrt[m1], -Sqrt[m2], Sqrt[m3]] K[Sqrt[m1], Sqrt[m2], -Sqrt[m3]]]
Expand[G[1, a, b]]
step1 = FactorList[Resultant[G[1, x, y2], G[1, y2, x2], y2]]
Resultant[step1[[2, 1]], G[1, x2, y], x2]
step2a = FactorList[Resultant[step1[[3, 1]], G[1, x2, y], x2]]
step2b = FactorList[Resultant[step1[[4, 1]], G[1, x2, y], x2]]
step2 = {step2a[[2, 1]], step2a[[4, 1]], step2b[[3, 1]], 
   step2b[[4, 1]]};
step2[[1]]
s1 = N[Solve[{step2[[1]] == 0, G[1, y, x] == 0, x >= 0, y >= 0, 
    x >= y}, {x, y}], 8]
s2 = N[Solve[{step2[[2]] == 0, G[1, y, x] == 0, x >= 0, y >= 0, 
    x >= y}, {x, y}], 8]
s3 = N[Solve[{step2[[3]] == 0, G[1, y, x] == 0, x >= 0, y >= 0, 
    x >= y}, {x, y}], 8]
s4 = N[Solve[{step2[[4]] == 0, G[1, y, x] == 0, x >= 0, y >= 0, 
    x >= y}, {x, y}], 8]
sol = Join[s1, s2, s3, s4];
{1/x*174.1, y/x*174.1} /. sol

You can be intrigued that the solutions are more detailed than a simultaneus Solve[] of the system of four equations G[1, x, y2]==0, G[1, y2, x2]==0, G[1, x2, y]==0, G[1, y, x]==0. I am intrigued too. It seems that some of the particular solutions found by the Resultant method are embedded inside a continuous spectrum of solutions, and then the Solve method avoids listing them twice. I wished to know more on the relationship between resultants and continuous solutions.

It is amusing that the first triplet of the list is Rodejohann-Zhang triplet,
{1, {x -> 102.50258, y -> 2.1435935}}. Scale it times 174.1/102.50 and you get 1.69849, 174.1, 3.64088

My own triplet appears later, as it is generated by the last polynomial... it is
1, {x -> 2.1435935, y -> 0.071796770}. Use the same scale factor than before, and you get 1.69849, 3.64088, 0.12195

Both triplets are of the kind that becomes hidden in the continuous under Solve.

I am not sure about why this resolvent method does not find solutions with a zero, for instance 1.69849, 0.12195, 0. They can be searched by starting from G[0, x, y2]==0, G[0, y2, x2]==0, G[0, x2, y]==0, G[0, y, x]==0

IN:  Solve[{G[0, x, y2] == 0, G[0, y2, x2] == 0, G[0, x2, y] == 0, 
  G[0, y, x] == 0, x == 0.12195, G[r, x, y2] == 0, G[r, y2, x2] == 0, 
  G[r, x2, y] == 0, G[r, y, x] == 0, y >= y2}, {x, y, y2, x2, r}]

OUT: 
{x -> 0.12195, y -> 0.00875562, y2 -> 0.00875562, x2 -> 0.000628625,   r -> 0},
{x -> 0.12195, y -> 0.00875562, y2 -> 0.00875562,  x2 -> 0.12195, r -> 0}, 
{x -> 0.12195, y -> 0.00875562,  y2 -> 0.00875562, x2 -> 0.12195, r -> 0.261411},
{x -> 0.12195, y -> 0.00875562, y2 -> 0.00875562, x2 -> 0.12195,  r -> 3.13693}, 
{x -> 0.12195, y -> 1.69854, y2 -> 0.00875562,  x2 -> 0.12195, r -> 0}, 
{x -> 0.12195, y -> 1.69854, y2 -> 1.69854,  x2 -> 0.12195, r -> 0}, 
{x -> 0.12195, y -> 1.69854, y2 -> 1.69854,  x2 -> 0.12195, r -> 3.64099}, 
{x -> 0.12195, y -> 1.69854,  y2 -> 1.69854, x2 -> 0.12195, r -> 43.6919}, 
{x -> 0.12195,  y -> 1.69854, y2 -> 1.69854, x2 -> 23.6577, r -> 0}}

EDIT: ok, a faster recipe could be

pols:factor(eliminate([G(1,x,a),G(1,a,y)],[a]))/(y-x)^4$
f1:ev(part(pols[1],1),[y=x])$
float(sol1:solve([f1,G(1,a,x)],[x,a]));

the output has the following positive solutions:
[x = 29.85640584694755, a = 0.12453316162267], 
[x = 29.85640584694755, a = 650.4292237442922], 
[x = 29.85640646055102, a = 199.4974226119286], 
[x = 29.85640646055102, a = 13.92820323027551]
[x = 2.143593539448983, a = 102.5025773880714], 
[x = 2.143593539448983, a = 0.071796769724491], 

and thus
(%i58) mc
;
(%o58)                              1.69854
(%i59) mc * 2.143593;
(%o59)                           3.64097845422
(%i60) mc *  0.07179;
(%o60)                           0.1219381866
(%i61) mc * 102.50257;
(%o61)                          174.1047152478

 

[mitchell porter]

$$|M| = {2 \over 3 * 32^2} {(7 (Tr M)^2 - 8 Tr M^2)^2 \over Tr M}$$

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.

 

[arivero]

Let me copy here this equation 3.30, setting the determinant of M as a function of the traces in M and M^2:

$$|M| = {2 \over 3 * 32^2} {(7 (Tr M)^2 - 8 Tr M^2)^2 \over Tr M}$$

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.

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

 

[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(√m_e,√m_μ,√m_τ), 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".

 

[arivero]

A few months ago I sketched a report on the topic of predictions from Koide equation that could be more readable for people used to PhysRev latex format. It is here: http://es.scribd.com/doc/157932274/Koide-equations-for-quark-mass-triplets

 

[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 $Tr \lambda = 0$, $Tr (\lambda^2) = Tr(1)$