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.