Alejandro Rivero: Fermion Mass Coincidences And Other Fun Ideas
    By Tommaso Dorigo | December 4th 2011 07:59 AM | 42 comments | Print | E-mail | Track Comments
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    The text below was graciously written for this blog by Alejandro Rivero (below),  a friend who has contributed to this blog other times in the past. His theoretical ideas are off the mainstream, but in a way which makes them interesting to me. I hope some of you will appreciate reading about the whole thing in summary here - Alejandro has a few papers out which you may want to read if you are specifically interested in the matter.

    During the last year there has been some news on theoretical hints that I have usually discussed online, in some forums and even in old incarnations of this blog. A couple weeks ago I prepared some very short letters to report on these news and Tommaso has suggested to write
    a guest post about them. An invitation that I gladly accept and for which I am deeply grateful.

    The two topics could perhaps be connected, or perhaps not. One of them is the symmetry of the scalars in the supersymmetric standard model (or beyond), the other is the set of Koide relationships, which has expanded in a surprising way. I will start with this one.

    Koide formula is an equation between charged leptons postulated in the early eighties. As I explain in  arxiv:1111.7232  (also in vixra version if you want to leave comments or bibliography), it does not appear magically in middle air; it is the result of a research in models with two or three generations, trying to predict the Cabibbo angle.

    This research was fueled by an old observation, that the tangent of the angle was close to the square root of the quotient between pions and kaons, or between down and strange quarks. An explanation was searched by people as Wilczek, Zee, Weinberg and a lot of big names, and intermixed (pun here) with other ideas  on "flavour textures", special forms of the mass matrix. It was a rage for some years but by 1983 it has been inconclusive.

    Most of this research was done in the context of discrete symmetry groups and some Unification Theories called "Left-Right models". Then Koide went some steps beyond and considered quarks and leptons with substructure, so that lepton mass quotients could predict the Cabibbo angle too, even if this is a mixing between quarks. But he also found that his models implied a relationship between lepton masses themselves:

    2 (sqrt(M_e)+sqrt(M_mu)+sqrt(M_tau))^2 = 3 ( M_e + M_mu +M_tau).

    In 2005, some of us learnt of Koide equation during an online quest for relationships between masses of the elementary particles. You must understand that this is a fool quest from modern standards, because masses are expected to be predicted at very high energies, unification scale, near Planck scale, and then they run down to get the values we see at low energy.

    So at low energy we should expect not to find any link. There is some escape to this. We could find a relationship that is preserved during the running down; a lot of groups looked at this for Koide and they concluded that the preservation was at most approximate, and it is surprising that the most exact fitting happens at low energy. Or we could also propose that at GUT scale there is just a very singular "mass texture", with a lot of zeros, depending on very specific versions of the Higgs field. In this aspect, Koide equation is a hint for model building, but a very weak one, and only some enthusiastic groups have pursued it.

    Early this year Rodejohann and Zhang, from the MPI in Heidelberg, found  that also the most massive quarks were able to fit the Koide equation. You can wonder how we did not notice it before. Two excuses here. One is experimental, that the error bars for some quarks are very wide. You can fit almost anything to three points with error bars. The second is theoretical prejudice: we like to see the quarks grouped by equal charges, so we look at (u,c,t) with charge +2/3 or to (d,s,b) with charge -1/3. We never thought that it was a good idea to look to charm, bottom and top. If someone of the audience did, this is the opportunity to mention it!

    My letter, last November, helps with the first problem. Once the scales fall from our eyes, we can look to other triplets, and then charm, strange and bottom also meet Koide relationship, a nice surprise!

    But you can say, how did Rodejohann and Zhang missed this one? The answer is the square root. You must take the square root of the strange quark with its negative sign. Still, you could have found it simply by solving the second degree equation implied by Koide for any X,Y,Z masses, putting X and Y to the masses of bottom and charm, and solving for Z. Perhaps they did
    it, but still the error bar for the strange quark is huge, so it is not still a very good argument.

    Here came the second insight: the new triple, with the mass of s got from c and b, and the mass of c itself got from b and t, was a peculiar triplet. It was almost orthogonal to the original lepton triple. I justify this orthogonality by looking at the limit case, when the mass of electron is zero and orthogonality is exact.

    And the funny thing is that by assuming this quasi-orthogonality we can reverse the process: take as input the mass of electron and muon, to predict the tau as Koide did, then build the quasi orthogonal triplet using the rule we discovered in the orthogonal case, and then from strange charm and bottom produce the mass of top quark. We have two experimental inputs and one empirical input, but we get five results: tau, strange, charm, bottom
    and top.

    The "prediction" for the mass of top quark, from Koide ladder, is 173.26 GeV. The masses
    of bottom, charm and strange are calculated to be 4.197 GeV, 1.359 GeV and 92.3 MeV. The mass of the tau, calculated by Koide, is 1.777 MeV. The only inputs, besides the use of the equation of Koide, are the masses of electron and muon and a factor 3 that we use to cross from the lepton to the quark sector, and we justify from the orthogonality.

    I put quotes on "prediction" because it is still empirical. No model, from Koide or elsewhere, has found the equation in the quark sector. So you should consider coincidences; in fact Tommaso already discussed in some blog posts. See here and there, and also Lubos.

    But I think that five outputs from two inputs are worthwhile enough to review old models and see if they have some say for quark masses too. Time will tell.

    The second paper is about composites, flavour and susy.  For five years I have been confident that the scalars of the susy standard model (SSM) have a flavour symmetry based in representations of the SU(5) group. The sBoostrap, as I called it, was a bizarre idea but it had the virtue of providing an explanation for the absence of sfermions and squarks, and for the coincidence between tau and muon masses on one side and QCD objects in the other: the scalars of the SSM did not exist; they were just the mesons and diquarks of the QCD string.

    Other rare hints that I could expect to explain if QCD was involved were the scaling of Z0 decay (see A mistery behind the Z width) and the point that Koide basic mass scales are very QCDish, 313 MeV for the original lepton tuple and 940 MeV for its quasi-orthogonal.

    There was only an issue about the superpartners of diquarks with charge +4/3, because they are not seen anywhere in the standard model and this SU(5) flavour ask for them. I had done, perhaps, some progress to solve the issue precisely in "left-right" models, because in these models part of the charge comes from other symmetry, called "baryon minus lepton number".

    But by now some groups have suggested, to explain asymmetries in top quark production, that we should use scalars with charge +4/3!

    If they exist, it means two things: one, that the flavour symmetry I was building, and that is very peculiar of only three generations of particles, could actually happen in the SSM extended with these scalars. Two, that they were actual objects and no QCD diquarks and so my original idea gets a shoot in the leg at the same time that it is vindicated. Which is almost a revenge, because I devised the theory to counter the modern gravity-scale strings, promoting the meson string instead.

    So the paper is just this, to report on the existence of this SU(5) symmetry that seems very much as a composite of five quarks. It could help if the LHC had some hints of technicolor or some  other way to suspect of composites in the electroweak scale.


    So I see you are also now posting on the arxiv, selling your soul to the devil gleefully while the rest of us die.

    Hi Kea, :-) There is a copy in vixra too! And the arxiv paper includes a reference to your vixra 1102.0010, not a frequent thing for arxiv papers. Actually both papers were frozen during the Thanksgiving week until some moderator got time to authorize them, note the submission dates, I was surprised because I had not been frozen since six or seven years ago.

    There are merits in both archives. Vixra has the comments version and it is actually more faster for papers with exotic content. Arxiv is more widely reached, and it is well integrated with SPIRES. And without the "encoraugement" of the frozing, I had not rewrote one of the preprints for publication. Well, we have discussed recently a lot of this in


    Thanks for your interesting contribution. Regarding Koide's formula for SM fermions, I've suggested in several articles that it arises naturally from universal transition to chaos of nonlinear quantum fields (so called Feigenbaum scenario). See for instance:


    Ervin, your 3-level bifurcation rule, as I understand it, is nothing but the levels of n-ordinal trees (ie. the category theory indexes), which increase the categorical dimension. If that is a reasonable interpretation, then it fits the modern M theory point of view.


    As far as I know, my papers have nothing to do with category theory and/or the levels of n-ordinal trees. I am not talking about any 3-level bifurcation rule and I am not basing my arguments on anything remotely connected to modern M-theory.

    My work targets the Renormalization Group (RG) approach to chaos and its relevance to HEP. Specifically I show that, if field equations or RG flow equations are treated as nonlinear dynamical systems, universality of period doubling bifurcations recovers in the most natural way the entire family structure of SM, including fermions, gauge bosons and coupling charges. It is a purely dynamical view on family replication that does not require additional assumptions or concepts.


    Yes, I know that Ervin. My comment is for potentially interested readers.

    Thanks Ervin, your scenario -independently of the validity or not as a physics model, which is not the point- is a good example of how terrible, or terrific, is to have found two extra triplets that seem to agree with Koide scheme, because it shows that really a single parameter does not do the work. It means back to the blackboards.

    Hmm, the first comment from Kea makes me to remember that I had left some acknowledgements in the ink well. This second round of Koide equations have been very encouraged by Kea herself (Marni Sheppeard) in her blog and by Mitchell Porter around physicsforums, but I should also mention the role in the 2005 quest by Carl Brannen and Hans de Vries. A good view of the state of things two years later, in 2007, can be seen in Carl blog (bonus: a picture of Koide and Carl).

    One more note: the original charged quarks triplets ((u,c,t) and (d,s,b)) do almost work, with parameters that are also closely linked to the lepton ones. I don't believe this is a coincidence.

    What worries me is that if we keep iterating Koide solutions for u (from s an c) and then for d (from u and s) we get a sensible value for down, but a very small one for up, and then I am not sure if we should use these values or the experimental ones for the equal charge triplets. I have not mentioned all the tries on these triples, from Dave Lock and yourself online, and also from some other groups in offline publications, but people can rely on SPIRES to find them. In any case, the use of the original equarl-charge quark triplets is no far from the mainstream research on textures.

    Surely the Fourier series interpretation suggests 'cyclic' iterations rather than linear ones. Note how your list exactly interweaves the two charged triplets, whose phases are 1/3 and 2/3 of the charged lepton value (with the remaining parameter given by Dave's 1.76, derived from the sqrt 2). If you are willing to consider rotated 'approximate' triplets, then these must surely be considered interesting.

    It could be interesting to use mathematica or other tool (I tried unsucesfully with gnuplot tools) to fix the mass of top, say to 1, to move the mass of bottom, say from 1 to 0, and given that all the other masses can be woven down once fixed b and t, then to plot then the r parameter of a charged triplet as a funtion on b mass ans see how far it goes from sqrt(2), or if it moves from sqrt(2) to 1 and back or so. The plotting involves some different functions for each decision of sqrt() signs, or equivalently for each assignment from the triplets to dave triangles, but could be informative as a hint to further analytical work.

    Funny, now google does plots. Here is the bottom-charm plane, for mtop=173

    In my opinion, the discovery of this s-c-b relation is FAR more significant even than Alejandro implies. As he describes in the preprint, the original Koide formula for e-mu-tau can be expressed in terms of a mass scale and a phase. The corresponding formula for s-c-b has exactly three times the mass scale and three times the phase! And the original e-mu-tau mass scale is the same as the mass scale for constituent quarks! The facts are shouting some truth at us here; we just don't know what it is.

    As I mentioned above, to me it is quite clear that the hierarchical structure of fermion families and their mass scaling are a natural outcome of "replication through bifurcation", a universal feature of all nonlinear dynamical systems. Interestingly enough, it can be shown that stability of Renormalization Group equations determines the number of SM generations.
    I also believe that chaotic diffusion and mixing on the Feigenbaum attractor has a lot in common with quantum mixing in both quark and lepton sectors (work in progress).

    Ervin, this is all very well, but if you cannot relate it to a sophisticated approach that promises to recover both the full SM and GR, why should anyone care?

    Extra note 1: the RHS of your delta polynomial looks a lot like the Jones polynomial for a trefoil knot ...

    Kea, this work is in progress for me and others developing fractional dynamics and fractional field theory. Using the concept of fractal measure (in the sense of Lebesgue-Stieltjes), a very intriguing consequence of this approach is that fractional field theory in Minkowski space-time appears to be equivalent to field theory in curved space-time. There is strange duality here that looks like a generalization of the AdS/CFT conjecture to manifolds having non-integer dimensionality.

    Mitchell, I agree, the facts are shouting at us. I had refrained of putting explicit calculations in the post, but really we could show here in a comment, borrowing from the PF thread. First we call bc -l and set up some variables and the input:


    Then we solve Koide equation for tau, using electron and muon as input. This is the old stuff from 1981-1983.


    Then we extract the parameters m and delta of the lepton triple and multiply times 3 both parameters to build the quark triple. This is the new stuff in my letter: the existence of the c s b triple and its quasi orthogonality to the charged leptons.


    And then finally we solve Koide again from charm and bottom to get top. This is the more surprising step, to me, and it is the observation of Rodejohann and Zhang.


    And it is done! Print out the results and compare with PDG:


    Alejandro, although it's obvious nobody has asked it: couldn't the factor of 3 needed to go from leptons to quarks be just the number of colors? After all, s, c and b are not three quarks but actually nine when one takes into account the three possible colors.

    Indeed that was the first idea, back in physicsforums, to explain the factor 3.. But without knowing the exact model, we can not tell, and the point of the orthogonality when m_e=0 was more evident, so at the end I have preferred this argument. It could still be part of the answer, via some sort of flavour/colour locking.

    On support of the role of colour we have that not only the basic mass for s-c-b quarks is 3 times the one of leptons, but also that the value of this basic mass is very QCDish, 313.x MeV for the lepton side, 940 MeV for the quark side, sort of eta' or nucleon mass (In fact a Jay R. Yablon found a hint of the regularities of mesons by dividing all the masses by 940 MeV and taking the arc tan of it).

    Let me clarify what we mean by this basic mass. The point is that every Koide triplet (m1,m2,m3) can be produced from three angles separated 2pi/3 each. The formula of this parametrisation is

    m_k= M ( 1 + sqrt(2) Cos( 2 k pi /3 + delta))^2

    and M and delta characterise the triplet. This M is what we are calling "basic mass". Of course if we had put some extra factor in the front, say pi, sqrt(2) or 234243/4364578 the value should be different, but it was amusing from the start that in this simple form the M for the leptons were 313 MeV, just as a QCD "constituient" quark.

    By the way, I guess you mean that this post was Graciously written, rather than Gracefully, unless you mean us to envisage a guy dipping his nib into the ink bottle between pirouettes.

    Hah thanks Kea!
    Nice summary Alejandro!

    There's a minor typo. The Koide relation for leptons is actually:

    2 (sqrt(M_e)+sqrt(M_mu)+sqrt(M_tau))^2 = 3 ( M_e + M_mu +M_tau).

    The Koide relation suggested by Carl Brannen is also interesting, taking the form:

    2(-sqrt(M_1)+sqrt(M_2)+sqrt(M_3))^2 = 3 (M_1 + M_2 + M_3)

    where the mass M_1 corresponds to the lightest neutrino, based on constraints from oscillation data. Notice the negative sign in the sum of squared masses, which differs from the lepton formula.

    Brannen has argued both relations are eigenvalue relations of the form:

    (a + b + c )^2 = 3/2 (a^2 + b^2 + c^3)

    which in mass matrix form is the equivalence:

    tr(A)^2 = 3/2 tr(A^2).

    As the circulant mass matrix A is Hermitian, the Koide relation can be re-cast as a statement about its Frobenius norm, i.e.:

    ||A||^2=2/3 tr(A)^2

    Automorphisms of the algebra of 3x3 Hermitian operators (i.e., gauge symmetries such as SO(3), SU(3), F4), which are isometries of the corresponding projective plane KP^2, will preserve the norm, hence the Koide relation for the mass matrix.

    In a D-brane context, the Koide relation takes on an interesting interpretation. Let's take the lepton mass matrix Koide relation for example, which can be seen as a statement about three coincident branes with gauge symmetry group U(3). There are three scalars coming from the modes with coordinates transverse to the world-volume of three branes. These are the positions of the three branes in the transverse space. The process of giving some of the scalars expectation values corresponds to moving the branes away from each other in transverse space, breaking U(3) to a subgroup as the brane-brane strings are stretched giving rise to excitations with mass proportional to their length times their tension. This is the Higgs mechanism. The Koide relation in this context is a relation about the positions of the three branes in the transverse space.

    Hence, the Koide relation might be further evidence that quarks and leptons descend from higher-dimensional objects with extended symmetry groups. The generations would then result from the geometrical structure of these higher-dimensional objects.

    Upps you are right, I exchanged to 2 and the 3 in the only formula I wrote... It is OK in the preprint, fortunately. I am asking to Tommaso to correct it, as well as the missing link for Z width.

    Brannen equation is mentioned in the preprint but I could strees more the point of the negative sign there too, you are right.

    For M-theory I can not tell, but in the context of getting Koide equations from GUT or Beyond Standard Model groups plus a symmetry breaking mechanism plus a discrete symmetry, it is interesting to read not only Harari et al, but also an early criticism by J M Frere, which was about the need of an ad-hoc Higgs vacuum.

    Thanks for the references.

    Indeed, an M-theory origin for the Koide relation is what I was hinting at, while also considering a noncommutative geometrical intepretation. Connes used the C*-algebra M(1,C)+M(2,C)+M(3,C) = C+H+M(3,C) for his noncommutative standard model, but at higher energies there could be a single C*-algebra that does the job. For an E6 model, the most natural choice would be the Jordan C*-algebra J(3,CxO), which has the complexified Cayley plane (CxO)P^2 as its space of pure states. E6 is the group of isometries of this exotic projective plane. In 2002 Ohwashi considered an E6 matrix model in this context.

    Note that the link from Connes models to string and M-theory is subtle, the point is the dimensionality of the spectral triple, this was discovered in 2006, and in fact it is a strange puzzle: How has the spectral triple got to be in dimension 6 (mod 8) and still the model from Connes-Chamseddine-Marcolli has not hints of supersymmetry? The work of Evans, or more recently revisions of Baez and Huerta, should apply here. By the way, note that Alain moved some work to C + H + H + M(3,C) too.
    One could wonder if the quantum groups, that were very trendy in the nineties and have a lot of play with roots of the imaginary unit (so cubic roots of 1 in our case) have some role for Koide symmetries. In an appendix to "NCG and Reality", the search for a quantum group symmetry of the spectral triple was suggested.
    Regretly, NCG has an small audience and then it progresses at a very slow pace. Being more enthusiastic than the average person about Connes' aproach, I have attended to his lectures in 1995 and then after in 2006, plus a couple conferences, and probably at some point I had met all the researchers of the field, a claim not easy to do in other fields...

    To kneemo's comment about branes, I would add this remark. If you're going to explain the masses in terms of VEVs of brane displacements, there are two things to explain. First, the form of the Koide relations: what sort of potential is it that leads to such relationships? Second, the exactness of the Koide relations at low energies: why aren't they completely obscured by low-energy corrections? Although they are just about field theory, not about branes, Yoshio Koide's own recent "yukawaon" papers offer a prototype for both explanations: the VEV relations come from the vacuum conditions of an appropriate superpotential, and they are protected at low energies by family symmetries.

    Nice remarks Mitchell. Indeed that's where the crux of the matter lies (pun intended).

    Koide invokes SUSY vacuum conditions, yet has remarked that SUSY must be eventually broken. In looking at the problem my best guess is that the attractor mechanism might be playing a role. From an M-theory perspective, we'd be studying brane configurations, which in lower dimensions resemble non-SUSY extremal black holes. The superpotential for these black holes is determined by the brane charges. If we extremize the entropy function for these black holes we obtain attractor values for the moduli and geometry, where the scalars take fixed values at the horizon, independent of asymptotic values. These fixed values of the moduli near the black hole horizon correspond to the minima of the superpotential. Koide's yukawaons might be scalars of this type in an M-theory framework.

    Hi Alejandro,

    Great post. Very interesting:

    "Then Koide went some steps beyond and considered quarks and leptons with substructure, so that lepton mass quotients could predict the Cabibbo angle too, even if this is a mixing between quarks."

    {(sqrt(M_e)+sqrt(M_mu)+sqrt(M_tau))^2} /( M_e + M_mu +M_tau) = 2/3

    The key factor of 2/3 in the Koide relationship is the fractional electric charge of the up/charm/truth quarks, which arises from a mixing effect. It's the 2/3 electric charge of up/charm/truth quarks that's so interesting. The -1/3 charge of the down/strange/bottom quarks is very easily predicted by analysis of vacuum polarization for the case of the omega minus baryon (Fig. 31 in ). It appears that the square root of the product of two very different masses gives rise to an intermediate mass (see for the maths) that the Koide relationship implies a bootstrap model of fundamental particles (akin to the bootstrap concept Geoffrey Chew was trying to develop to explain the S-matrix in the 1960s before quarks were discovered). The square root of the product of the masses of a neutrino and a massive weak boson may give an electron mass, for instance. This seems to be the deeper significance of the Koide formula, from my perspective for what it's worth. All fundamental particles are connected by various offshell field quanta exchanges, so their "charges" are dependent on other charges around them. This means that the ordinary approach of analysis fails, because of the reductionist fallacy. If your mathematical model of rope is the same for 100 one-foot lengths as for a single 100 foot length, it leads to customer complaints when you automatically send a sailor the former, not the latter. It's no good patiently explaining to the sailor that mathematically they are identical, and the universe is mathematical. If the Koide formula is correct, then it points to an extension of the square root nature of the Dirac equation. Dirac made the error of ignoring Maxwell's 1861 paper on magnetic force mechanisms: the chiral handedness of magnetism (the magnetic field curls left-handed around the direction of propagation of an electron) is explained in Maxwell's theory by the spin of "field quanta" (Maxwell had gear cogs, but in QFT it's just the spin angular momentum of field quanta). Maxwell's theory makes EM an SU(2) Yang-Mills theory, throwing a different light on the Dirac's spinor. It just so happens that the Yang-Mills equations automatically reduce to Maxwell's if the field quanta are massless, because of the infinite self-inductance of electrically charged field quanta, so SU(2) Maxwellian electromagnetism in practice looks indistinguishable from Abelian U(1), explaining the delusions in modern physics.

    Well, Nige, I really do not know how to address your ideas. But a big point of having more Koide triplets is that now we should check if our theory for the lepton triplet also matches with the new ones, I suppose you are thinking on this too, are you?

    I do not put a great significance to fractional charge 2/3 -in fact, you can read in a previous comment that I did a typo in the blog post (not in the paper nor in the calculations, of course) exchanging the 2 and the 3 and I didn't notice-. I think that Left-Right models have a say, if only because they appear in 7-dimensional manifolds :-D... No, seriously, I prefer to see that a part of the charge is B-L, and then in que case of the up quark we have a charge 1/6 coming from baryon number and other 1/2 coming from Left-Right SU(2) groups. Same with the down quark, a charge 1/6 from baryin number and a charge -1/2 from the Left-Rights. The funny part is that while SU(3)xSU(2)xSU(2) is the group of isometries of the seven dimensional manifold CP2xS3, we need to add an extra dimension to put this charge 1/6 apart, say the isometries of S1xCP2xS3, or to disguise it as a fourth colour, say the isometries of S5xS3. So I velieve that this extra dimension, and this extra charge, have a different role.

    Thanks! The very interesting results you give are from equation 4 on page 3, where you solve the Koide formula by writing one mass in terms of the two lepton other generation masses. Koide's formula also implies (my 2009 post):

    Me + Mm + Mt = 4 * [(Me * Mm)^1/2 + (Me * Mt)^1/2 + (Mm * Mt)^1/2]

    where Me = electron mass, Mm = muon mass, Mt = tauon mass. I.e., the simple sum of lepton masses equals four times the sum of square roots of the products of all combinations of the masses, making it seem that if Koide's formula is physically meaningful, then Geoffrey Chew's bootstrap theory of particle democracy must apply to masses (gravitational charge) in 4-d. At high energy, early in the universe, tauons, muons and electrons were all represented and we only see an excess of electrons today because the other generations have decayed, although some of the other masses may actually exist as dark matter, and thus still undergoes the interaction of graviton exchange, which determines the Koide mass spectrum today (this dark matter is analogous to right-handed neutrinos). The basic physics of the Koide formula seems to be the Chew bootstrap applied to gravitation (Chew applied it to the strong force, pre-QCD):

    "By the end of the 1950s, [Geoffrey] Chew was calling this [analytic development of Heisenberg’s empirical scattering or S-matrix] the bootstrap philosophy. Because of analyticity, each particle’s interactions with all others would somehow determine its own basic properties and ... the whole theory would somehow ‘pull itself up by its own bootstraps’.” - Peter Woit, Not Even Wrong, Jonathan Cape, London, 2006, p148. (Emphasis added.)

    The S-matrix went out when the SM was developed (although S-matrix results were used to help determine the Feynman rules), but at some stage a Chew-type bootstrap mechanism for Koide's mass formula may be needed to further develop a physical understanding for the underlying theory of mass mixing, leading to a full theory of mixing angles for both gravitation (mass) and weak SU(2) interactions of leptons and quarks.

    Nige, I can not deny that I agree with you on the importance of Chew ideas: my second preprint is actually an spin-off of the superBoostrap series I started in 2005 in arxiv letters hep-ph/0512065, 0710.1526 and 0910.4793. A problem I saw with democratic mixings is that they, I believed, were developed mostly for bosons -I have been proven wrong, after finding Harari et al-, and then I knew that no self-consistent solution of the bootstrap have been found in the sixties; so I tried a new variant: allow the scalars to be composites of the fermions, get the mixing in the scalar sector, and go from fermions to scalars via susy. When one solves this conditions for the standard model group, one finds that three generations is the simplest working set, and the nicests, even unique in some sense. The main problem with this approach is that susy is only midly broken and then you do not find models in the literature. An interesting hint I want to pursue is to use the permutation symmetry (that basically labels a SM down quark as a mixing of a left down and a right bottom) to explain why the electron has not a superpartner of the same mass.

    I like the hint of mass = gravity charge. It is the second time I heard of it, this week. Of course, one must implement it. Let me remember also the Kaluza (or was it Klein?) idea of mass as speed projection in the extra dimensions. In any case, democratic mixing can be a simpler thing, just some tunneling with all the coefficients the same. Put three quantum mechanic wells in a plane with S3 symmetry, ie in the vertex of an equilateral triangle, and see what happens.

    As some of you are actually reading the paper (thanks!), you could have noticed that in both tables, either starting from top and bottom or starting from muon and electron, are included predictions for the mass of up and down, but that I do not include further analysis. In fact the mass of the down quark is in the ballpark, but the mass of up is near zero. Obviusly it is forzed to zero for the orthogonal case.

    The reason because I have not discussed it is because I am ignorant of the current status of the idea that the measured mass of the up is a combination of instanton mass plus bare mass. Remember that this idea was proposed in a preprint from Georgi and McArthur, where it was told that it was feasible (instanton contribution was proportional to the product of down and strange masses divided by the chiral scale). If this estimate were still valid, we could account for it and consider that the real mass to be used in Koide triples with the up quark is the one "for the Lagrangian", substracting the instanton part.

    A note of caution:

    The ultimate goal of these modeling efforts is to clarify the physics of mass generation and mixing. It seems to me that adding too many inputs to the theory is ill-advised. Invoking concepts such as fitting parameters, VEV’s, Yukawaon fields, superpotentials, instantons, branes, SUSY, non-commutative algebra, higher order symmetry groups (and so on) runs the risk of obscuring the theory and defeating its original purpose. Occam's razor is still alive and well...

    Good warning! It is not exactly about Occam razor , but an Anchor Principle: do not introduce more objects if you don't have really a guess of how to manage them. High level mathematicians explain that, to learn maths, it is important to know really well a particular area, say for example some nice branch of algebra. That is you anchor. From there, you explore other places of the math universe and learn about new objects, theorems, etc by referring to your core branch, but without reducing them to it... the art is to keep the safety anchor and, at the same time, to avoid the man-with-a-hammer syndrome (you know, when, everything looks as a nail... the technical name is, it seems, "Maslow's hammer"). Once you have grasped the new concept, you put it in your toolset to use.

    I wonder if there is some tecnical name also for collecting a lot of tools without really mastering the use of any of them.

    I wonder if there is some tecnical name also for collecting a lot of tools without really mastering the use of any of them.

    "The Ideal Candidate"

    I wonder if there is some tecnical name also for collecting a lot of tools without really mastering the use of any of them.

    In English/French one has the options (i) dilettente (though this is not really appropraite to science) (ii) 'jack of all trades' (and master of none)

    It's more than this, as the model you're developing becomes more abstract, you have to be very careful that you're not modeling how you think it works, and not how it really works, and most people are blind to this bias. That's why the space shuttle has at least 2 different sets of navigational code, to try and eliminate these biases(the 2 systems vote on the answer).
    Never is a long time.
    The instanton calculation is also more detailed in Phys.Rev.Lett.61:794,1988 and Nucl.Phys.B383:58-72,1992. After that, nothing. I can not see if the modern lattice calculations really address the question of which part of the mass comes from the instanton and which part comes from the bare lagrangian; most of them refer to JHEP 9911:027,1999 instead, and some of them leave the issue open.

    Hello T & co. I have set up a prezi presentation for the topic of Koide ladders and waterfalls. You can see it here: