And then I was surprised by the comment of Tom, asking how the pairing was done. Well, I thought that I had discussed it in some thread in BSM, but after looking at it, it seems that I did only a few sparse remarks here and there. On other hand, people was not liking to interrupt the flow of the thread and I have been either contacted privately or suggested to open a new thread. So here it is. The development can be traced in some draft papers:

Next post is my answer to Tom question. Keep in mind that we produce squarks, while the quarks are just the ingredient to terminate the extremes of the string.

It is about taking seriously the ideas of http://dx.doi.org/10.1016/0370-2693(71)90028-1" [Broken] ): the fermion in the dual model is susy to gluonic strings. So now all you need is to terminate the gluonic string. Regretly in 1971 there were only three states available to terminate the string: u, d, and s. Now we have the full history, and the experimental data tell us that we can terminate the gluonic string with five and only five different states: u, d, s, c, b.

So just count, please, just do the SU(5) global flavour game, and count. How many states do you get of charge +1? six, by terminating with particle and antiparticle. How many of charge +2/3? six of each colour, by terminating with an antiparticle at each end of the string. How many of -1/3? six. How many +1/3, -1, -2/3? Same: six, six, six. And how many neutrals? of course, twelve: the other half of the 24 of SU(5).

BONUS: Does it means that string theory, given as input the 3-2-1 gauge theory of the SM, predicts three generations? No exactly; only if we require that the neutral leptons must be produced too. If we only look at the quark sector, then any pairing of [itex](2^{p})[/itex] "up quarks" with [itex](2^{p+1} -1)[/itex] "down quarks" will produce equal number, [itex]2^p (2^{p+1} -1)[/itex] of up and down combinations, and p=1 is just the simplest case. Numerically minded people will notice that p=4 amounts to 496, but a theory with 16 light "down" quarks, 31 light "up" quarks and a total of 248 generations seems not to be the object that Nature has offered us.

EDIT: Allow me a correction to this remark: Of course, the quark sector condition works for any integers [itex]q[/itex] and [itex]2 q -1[/itex], with [itex]q[/itex] an even number, not necessarily a power of two. But that the powers of two are an interesing subset was noted by Peter Crawley in other thread time ago and I am kind of obsessed with this, because it could constitute the way to reconnect with usual string models, via the above p=4 case.

EDIT: other references using "fermion-meson": L. Brink and D. B. Fairlie http://dx.doi.org/10.1016/0550-3213(74)90529-X [Broken] Nuclear Physics B Volume 74, Issue 2, 25 May 1974, Pages 321-342 ; Edward Corrigan and David I. Olive http://www.slac.stanford.edu/spires/find/hep/www?j=NUCIA,A11,749 [Broken] Nuovo Cim.A11:749-773, 1972. Modernly, there are some papers, in the framework of SQCD and also in Holography, that work with "mesinos", generic susy partners of mesons. But note that phenomenologists restrict the name "mesino" to the composite combination of squark and quark

Can you present a table (or a ereference) where the pairing is shown explicitly?
Why do you use SU(5) instead of SU(6)?
How do you count different charges like color, flavor, weak isospin and hypercharge and all that?

Because the top quark doesn't bind into mesons (nor diquarks, for the same token).

This is, I am promoting an experimental peculiarity (that the mass of the top is higher than both the QCD scale and the W mass) to a main role.

But you can look also at it from a pure theoretical side. Take SU(3)xSU(2)xU(1) as given, and assume that SU(3) is the force that, at some scale, has this role of building the open string, this is, of binding pairs of particles. Then:

- First, you ask if there is some number of generations such that the possible pairs of terminations are in the same number that the squarks for these generations. The answer is none, so:

- Second, you go for the lesser goal: ask if there is some number of generations so that a subset of the quarks are the same number that the squarks you should have. The answer is yes, that q quarks of type down and 2 q - 1 quarks of type up, when q>1, combine to from the squarks needed for q (2q-1)/2 generations of particles.

- Third, make features out of bugs: postulate that the quarks that do not participate in the binding must have a high mass. On first approach, you can think that the subset of binding quarks should be massless, and the other of infinite mass. And we know from Nature that it is enough for them to have a mass equal or higher than the electroweak scale.

- Fourth, if you wish, add leptons to the mix. Of course leptons doesn't bind, they are SU(3) neutrals. But you want to produce sleptons. It happens that any solution of the second step also produces the needed number of charged sleptons (check combinations, now with quark/antiquark). And only for the simplest case, q=2, we get the expected number of neutral sleptons.

So, the full answer is, we do not use SU(6), because Nature hints us that we can relax to use another smaller number of flavours. Inspection of the quark sector tell us that we can use SU(q+2q-1) with q>1, and of all of these SU(3q-1), only SU(5) produces also the neutral sleptons.

s in squark and slepton stands, as usual, for "scalar". It refers to the spin zero partners of the elementary fermions, so that the electron has two slepton partners of charge -1, the positron has two slepton partners of charge +1 and so on.

I think you are misinterpreting the paper. The "duality" in dual models (Dolen-Horn-Schmid duality) is that you get the amplitude by summing over s-channel diagrams or by summing over t-channel diagrams, not by summing over both at the same time as in ordinary field theory. The string theory explanation is that the dual-model s-channel sum and the dual-model t-channel sum are just different representations of the same sum over string histories, but with the world-sheet cut in different ways (in order to define a path integral). This is mentioned very briefly in Appendix A of Polchinski (page 332, "Relation to the Hilbert space formalism"), where he calls it "world-sheet duality". But this is not supersymmetry.

Hey, no, I never told that the duality of dual models is a supersymmetry; what we know is that the fermions in the dual model are known to be supersymmetric to the bosons in the dual model. Of course this discovery is going to happen after 1971. First they are going to discover wordsheet susy, then years later they are going to discover that it also implies Space Time susy. But never, neither by them -at least in the mainstream- nor by me, a dual model setup between fermions and bosons has been claimed. I am sorry that my wording could be misconstrued in this sense.

The reason to quote this paper is to show that, in the years after the discovery of the fermionic states of the string, by Ramond and then by Neveau-Schwarz, there was no problem to see the string as a holder both for quark and gluonic states, which is the thing I am using: elementary fermions (quarks and leptons) in the fundamental level of one side, gluons (mesons, diquarks) in the fundamental level of the other.

Some years later, with the standard model already established, it could seem strange to have elementary entities on one side and composites in the other, so Schwarz took the bold step of promoting the bosonic sector also to the status of elementary, moving all the game to Planck scale. My claim is that this was the wrong turn, and that the initial view of fermions plus gluons was the right one, when gluons are terminated with light quark states.

Still, it is amusing that if I forget about the neutral sector, then there is also a solution with a total of 248 generations. This joins to other "2-sigma signals" of a link between the solution with five flavours and the solutions that appear in critical superstring theory. Marcus and Sagnotti found an interpretation of SO(32) as an open string with five different terminations. Usually this is guessed to be related to the tadpole count, related to the space time dimension. And if we had some reason to look only for the particular "Mersenne" solutions (with the number of up quarks being a Mersenne prime instead of a generic 2 q -1), we could invoke hep-th/9904212 to claim that our solution is the result of going from D=10 to D=4

I can find at least one example of supersymmetry between elementary excitations and composite excitations [edit: http://arxiv.org/abs/hep-th/0207232" [Broken] but maybe I got it wrong, will need to read later], so that approach to "hadron supersymmetry" really might work. But in Schwarz 1971, the mesons aren't superpartners of the fermions, the mesons are "DHS-dual" to the fermions.

More precisely, you mean that in a dual theory with fermions and bosons it is possible to build diagrams (the most typical, fermion-boson scattering) where the s channel particle is a fermion and the t channel particle is a boson, or reciprocally. I do not deny this. Both of them are elementary states of the string, are not they?

What I was pointing out, by referring to these old papers, was that in a first impression people has not problem to consider the Ramond fermion as a quark and the bosonic states as mesons or gluons. Then people puzzled about it and preferred to consider that speaking of fermions as quarks was just "customary speak" (example, in Scherk 1975 review) and then finally the whole theory was promoted to the status of GUT-Planck scale entities, so that nobody had to worry about the material interpretation of a open bosonic string as a terminated string. And yes, 20 years later we see D-branes coming as a revenge :-D but we are too far away from the original situation.

But a thing that we have learnt along the way (one learns things, even during a long wrong turn) is that we need to produce the same number of bosonic and fermionic states. And that it must be so for each charge, because strings have susy, and susy commutes with the charge generators.

And amazingly, if one checks the original situation, the standard model with the gluonic string, one finds that it agrees with this requirement: their strings can be terminated, charge by charge, in a way that the number of boson and fermion states matches.

So the "wrong turn" was to consider the bosons in the dual model as fundamental rather than as composite, because what we need is a model exhibiting supersymmetry between fundamental fermions and composite bosons?

For a SUSY theory, not just the spectrum must be supersymmetric, but also the interactions between the particles. Otherwise the supercharges are not conserved. How would that come about here?

So far, I have not found a way to produce the gauginos with the same mechanism, the termination of open strings produces exactly all the needed scalars, but only them.

My personal expectation is that the LHC could find the gauginos but not the scalars, because the scalars are already there as QCD strings. It could be different if we were able to build the gauge sector as a kind of closed strings.

There are two puzzling lateral issues, related to the W and Z. On one side, a sort of "duality": that the sum of all the decays of Z seems to have the same rate that the decay of a pion having the same mass. On other, that the scalars that give mass to the Z and W are, in susy, partners of a chiral fermion, and that then we need six extra scalars (for Z, W, and Z0) for any mass mechanism, and three of them are eaten into the 0 helicities of Z and W. My guess is that these scalars are the ones we produce from uu terminations, which have no role in the reproduction of squarks and sleptons.

This is all quite interesting, but also rather hard to grasp at first glance. So for my own reference, and perhaps the edification of confused onlookers, let me present a two-paragraph idiot's guide to what's going on here.

1. There is an obscure research program or line of thought called hadronic supersymmetry. It proposes that quarks and http://en.wikipedia.org/wiki/Diquark" [Broken]. The second paper, in particular, starts with a nice review of history and motivations, and also contains the most mathematically sophisticated approach that I've seen. I'm not saying it's correct, just that it gives a theorist more to work with.

2. Alejandro Rivero, in his papers listed in #1, proposes to extend hadronic supersymmetry to the leptons.

If anyone wants to understand what this discussion is about, I suggest that those are the two ideas to cling to. Alejandro is trying to motivate or implement his idea by digging up these "dual models" from the dawn of string theory, but it's very unclear to me whether his "path not taken" actually exists. Would different dual models or different string theories have been discovered, ones that we don't know about today? Or would the formal theory have developed in the same way, but with different ideas about phenomenology? For my part, I intend to read Catto 2003 next, understand his model of hadronic supersymmetry, and then see what Alejandro's proposed extension looks like when Catto's approach is used as a base.

Indeed Lichtenberg and Catto are relevant references, but I was disappointed that they were only using it as a way to calculate baryon masses. As they are not interested on the fundamental level, they fail to appreciate the miracle that happens when three generations and five light quarks are considered. On the contrary I think that this miracle, and its uniqueness, is important and tell us something about our expectations to find the partners of the standard model particles.

(still, I am going to re-read them... Thanks for the reminder!)

At least, the scalars. I can not tell anything yet about the other bosons, nor the gauginos. As for the "because", I would not say that "we need". It is just that susy happened as a prediction of the evolved R-NS dual model, and that string theory (and dual models) does not need to take strong positions on the issue of compositeness vs fundamental. Of course, the "right turn" had been to postulate that the endings of a bosonic open string were forcefully light fermions of the same theory, and then in 1975 they had predicted three generations and a non-light top quark.

Currently I am fantasising that at the end of the path we had investigated the case with 16 up quarks and 31 down quarks, producing 16*31 = 496 ud combinations and (31*32)/2 = 496 dd combinations and so 248 generations (each generation, of course, needs two scalars to pair each fermion of a given charge), and then the scalars from the extant (16*17)/2= 136 uu combinations had been used to give mass to some objects, breaking some underlying symmetry group from 248 elements to something with 248 - 136 = 112 elements. Or something so

Spoiler

the joke was about the decomposition of E8 as a sum of representations of SU(2)xE7, this is (3,1)+(1,133)+(2,56): the whole representation is a 248, while the last subrrep is a 112. If you were guessing other family of objects, please tell me!

If you refer to #15, it is only a (half-)joke. Mitchell wondered if the formal theory had developed in the same way, and I answered by pointing out that it was possible to arrive to write SO(32) or E8-like scheme even when starting from this empirical approach.

Or do you refer to the whole idea? I am very surprised that you all are not impressed. OK, I could understand that people only worried by gravity can fail to be impresed by any particle-related juggling. But if someone is into particles, to notice that the exact number of scalars of the SSM can be produced from this simple combination game -and with the right charges for almost all: charge only fails for the six scalar partners of the non-Dirac fermions who marry the W an Z-, well, it should deserve at least some weeks of attention. My opinion, of course.

Hadrons for the first n flavors have an approximate SU(2n) symmetry called "spin-flavor symmetry". http://prd.aps.org/abstract/PRD/v12/i1/p147_1" [Broken] you may see De Rújula, Georgi, and Glashow employing first SU(6) spin-flavor symmetry (u,d,s) and later SU(8) (u,d,s,c) to explain hadronic properties.

As mentioned in http://arxiv.org/abs/hep-th/0302101" [Broken] proposed that they might be placed into a single symmetry multiplet, but to do this he had to anticipate supersymmetry, since mesons are bosons and baryons are fermions. He extended SU(6) to SU(6|21); this was the real beginning of "hadronic supersymmetry".

The most commonly believed explanation of this, within QCD, appears to be that a meson is a gluonic string connecting a quark and an antiquark; and that inside a baryon, you end up with two quarks on top of each other at one end of a gluonic string, and with the third remaining quark at the other end; and that this structural similarity accounts for the shared Regge slope. This is the picture that Lichtenberg and Catto employ; and Nobel laureate Frank Wilczek is http://arxiv.org/abs/hep-ph/0409168" [Broken].

However, there is an alternative way to get http://arxiv.org/abs/0901.4508" [Broken], and it rests on a different, equally simple picture. Instead of a baryon being a string connecting a quark and a diquark, it's a string with a quark at either end and a third quark smeared along the string. In other words, the string itself is a fermionic string.

I think this, and the holographic approach to QCD, and Type II string models where all the standard model particles are open strings stretched between branes, together provide a context where the viability of Alejandro's idea can be explored. http://arxiv.org/abs/0910.5955" [Broken], so it may not even be necessary to regard the two approaches to hadronic supersymmetry as mutually exclusive. The extension to leptons is a lot more problematic, but I think we have here a set of tools flexible enough to explore many variations on the idea, but rigorous enough to ensure that questions do have unequivocal answers.

The problem is that up now it's only algebra w/o any fundamental dynamics. It looks like a bottom-up approach, but I can't see if this will produce something like a dynamical theory - or perhaps it may - but this will then be some sort of string theory again.

Regarding gravity: w/o gravity there is no need for string theory as far as I can see; string theory requires SUSY + additional dimensions - which we do not see in nature. String theory seems to be kind of framework to "produce theories as something sitting on top of vacuum states". OK, this is nice but afaik there's no additional benefit. w/o gravity it seems that string theory is nothing else but a very complicated "dual reformulation" of a huge class of (SUSY) gauge theories.

That's the reason why I am not very much impressed.

Also, leaving the top out is like letting the fat kid not play ball, which conjures up unpleasant childhood memories for me. :tongue2:

(More seriously, though, this is interesting speculation that I unfortunately probably won't have the time to fully familiarize myself with. There seems to be just enough fuzziness such that things might end up being merely a coincidence after all, if a suggestive one. Also, I'm unclear about the mass scales -- do the proposed superpartners have the same masses?)