The wrong turn of string theory: our world is SUSY at low energies

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"Quasi Nambu-Goldstone Fermions" (Buchmuller et al, 1983) is one of the fundamental papers on this topic. They describe how, in passing from a coset sigma model to its supersymmetric counterpart, the symmetry group is complexified, doubling the real coordinates of the coset space, and adding to the original Goldstone bosons, a set of "quasi Goldstone bosons". Taken together, these are the superpartners of the goldstone fermions.

The Goldstone scalars form the coordinates of a geometric space, a Kahler manifold. For the nonsupersymmetric case, the metric of this Kahler manifold is unique, and it uniquely determines the sigma-model lagrangian. But for the supersymmetric case, the quasi Goldstones double the coordinates of the geometry, and away from the "Goldstone hyperplane" the metric - and consequently the lagrangian - is no longer unique.

These sigma models are effective theories. The parts of the sigma-model lagrangian that are not determined by the coset geometry, are determined by details of the deeper theory that has undergone spontaneous symmetry breaking. For example, suppose we had a brane stack in a compactification, with some strongly coupled supersymmetric theory as its worldvolume theory. The basic properties of the brane stack may imply a particular coset sigma model as effective theory, while the geometric details of the compactification may determine the details of the lagrangian.

In terms of the sbootstrap, one could then proceed as follows. Identify a sbootstrap supersymmetric sigma model, such that the SM fermions are its goldstone fermions; and perhaps a specific potential for the fermion masses. Then find a brane configuration which implements that sigma model, and a compactification geometry which induces the desired potential. This 2016 paper offers a small start by considering possible supersymmetric mass terms for pions.
A small observation:

The sbootstrap has two parts: a version of the traditional quark-diquark "hadronic supersymmetry", and an extension to include a lepton-meson supersymmetry.

These supersymmetric coset models can actually give us the latter, e.g. see the paper in #222. And they can also give the full SM fermion spectrum, e.g. Kugo-Yanagida.

But in Kugo-Yanagida, there is nothing like quark-diquark supersymmetry. Quarks and leptons are on the same level, whereas I would look for quark-diquark supersymmetry to be realized at a deeper level (perhaps as in Brodsky et al).

That would mean that quarks are fundamental, while leptons are super-composites, as in #222 (though the phenomenological quarks may also be super-composites). #222 offers an SQCD implementation of this for Dirac fermions.

But what about the chiral fermions of the full SM? In #239 and #241, @arivero described an N=1 SU(5) theory. Superficially, it's just another GUT; but it's supposed to provide the context for a chiral implementation of the sbootstrap.

My small observation is just that, if we stick with the idea that the quark superfields are fundamental and the lepton superfields are composite... the leptons all come from the 24. So that would mean that, for a chiral sbootstrap theory, the 15 and 15-bar would be what's fundamental.
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By the way, Mitchel, have you already commented on Mourad-Sagnotti here in this thread? Interestingly, the most cited paper on USp(32) seems to be from Sugimoto, the one of the Sakai-Sukimoto model. It would seem that the open-strings guys are into something.


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Branching down from USp(32) the main difference is that the "padding" (the dim 30 irreps that are foreign to our construct but complement the also extra (10,6) irreps of su5 + su3 ) gets itself arranged as a SU(2) doublet instead of oppositely U(1)-charged objects.
  1. usp32 to su2 ⊕ usp30
    • 32 = (2,1) + (1,30)
    • 495 = (2,30)+ (1,434)+(1,1)
    • 528 = (3,1) + (2,30) + (1,465)
  2. And then down to su15 ⊕ u1(R):
    • 434 = (224)(0) ⊕ (105)(2) ⊕ (\bar 105)(−2)
    • 30 = (15)(1) ⊕ (\bar 15)(−1)
And then su15 ⊃ su5 ⊕ su3(S) proceeds in the same way that above

Objection here is that the 495 in the USp(32) string is only for fermions, not for scalars. Also, once USp is in the game, one should consider the ""dual" SO(33), should we? On the other hand, we could have considered to start the game from SO(11) instead of SO(10) or SU(5). It seems that it adds more complexity without further physics.
I don't think that I've ever seen a symplectic algebra in GUT model building. I decided to look at the algebra breakdown in more detail, using my SemisimpleLieAlgebras Mathematica notebook that I'd written. For SU(5) model building, one wants 1, 5, 10, 10*, 5*, 1 for the elementary fermions, 24 for the gauge fields, and 5, 5* for the Standard-Model Higgs particle. We also Higgs-EF interactions H(5).F(1).F(5*), H(5).F(10).F(10), H(5*).F(5*).F(10), and Higgs-Higgs interaction H(5).H(5).

Symplectic-algebra irreps:
Vector: {1,0,0,...}, Antisymmetric 2-tensor: {0,1,0,...}, Symmetric 2-tensor, adjoint: {2,0,0,...}
The antisymmetric one is sort of traceless, with the algebra's antisymmetric form subtracted out.

Unitary-algebra irreps:
Vector: {1,0,0,...}, Vector conjugate: {0,...,0,0,1}, Antisymmetric 2-tensor: {0,1,0,...}, Antisymmetric conjugate 2-tensor: {0,...,0,1,0}, Symmetric 2-tensor: {2,0,0,...}, Symmetric conjugate 2-tensor: {0,...,0,0,2}, Adjoint: {1,0,0,...,0,0,1} ((vector * conjugate vector) - scalar)

The first one is from extension splitting, as I call it: Sp(2(m+n)) -> Sp(2m) * Sp(2n)
Sp(32) -> Sp(2) * Sp(30) -- with Sp(2) ~ SU(2)

The second one is what I call root demotion, with the long root reduced to a U(1) factor: Sp(2n) -> SU(n) * U(1)
Sp(30) -> SU(15) * U(1)

The third one is a Kronecker-product decomposition: SU(m*n) -> SU(m) * SU(n)
This is a sort of outer product with original first indices flattened, and original second indices flattened. It is done on the fundamental rep.
SU(15) -> SU(5) * SU(3)

I'll now decompose the original irreps:
Vector: 32 = (2,1) + (1,30)
= (2,1,0) + (1,15,1/2) + (1,15*,-1/2)
= (2,1,1,0) + (1,5,3,1/2) + (1,5*,3*,-1/2)
Antisymmetric: 495 = (2,30) + (1,434) + (1,1)
= (2,15,1/2) + (2,15*,-1/2) + (1,105,1) + (1,105*,-1) + (1,224,0) + (1,1,0)
= (2,5,3,1/2) + (2,5*,3*,-1/2) + (10,6,1) + (10*,6*,-1) + (15,3*,1) + (15*,3,-1) + (24,8,0) + (24,1,0) + (1,8,0) + (1,1,0)
Symmetric: 528 = (3,1) + (2,30) + (1,465)
= (3,1,0) + (2,15,1/2) + (2,15*,-1/2) + (1,120,1) + (1,120*,-1) + (1,224,0) + (1,1,0)
= (3,1,1,0) + (2,5,3,1/2) + (2,5*,3*,-1/2) + (15,6,1) + (15*,6*,-1) + (10,3*,1) + (10*,3,-1) + (24,8,0) + (24,1,0) + (1,8,0) + (1,1,0)

One can get singlets for both the 5-5* and the 5*-5*-10 SU(5)-model interactions. But it does not seem to be possible for the 5-10-10 one. This model also seems to lack right-handed neutrinos.
Oops, some typos. The (MS)SM Higgs interaction is supposed to be H(5).H(5*). I also omitted some initial 1's in the antisymmetric and symmetric 2-tensor results:
AntiSym = (2,5,3,1/2) + (2,5*,3*,-1/2) + (1,10,6,1) + (1,10*,6*,-1) + (1,15,3*,1) + (1,15*,3,-1) + (1,24,8,0) + (1,24,1,0) + (1,1,8,0) + (1,1,1,0)
Sym = (3,1,1,0) + (2,5,3,1/2) + (2,5*,3*,-1/2) + (15,6,1) + (1,15*,6*,-1) + (1,10,3*,1) + (1,10*,3,-1) + (1,24,8,0) + (1,24,1,0) + (1,1,8,0) + (1,1,1,0)

The tensor ones have SU(5) irreps in addition to what appears in the Standard Model: 15 and 15* (symmetric 2-tensor and its conjugate).
5* -> (1,2,-1/2) and (3*,1,1/3) -- L lepton, R down quark
10 -> (3,2,1/6) and (3*,1,-2/3) and (1,1,1) - L quark, R up quark, R electron
15 -> (3,2,1/6) and (6,1,-2/3) and (1,3,1) - L quark, R up quark with QCD multiplet 6* instead of 3, R electrons with charges 0, -1, and -2
Thus making some elementary fermions that we do not observe.


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Have you uploaded/published your package somewhere? I have always found intriguing that Wolfram has not a standard package for this task.

AntiSym = ... + (1,15,3*,1) + (1,15*,3,-1) + ... + (1,24,1,0)
Thus making some elementary fermions that we do not observe.
To me, this (1,15,3*,1) and the colour singlet (1,24,1,0) are the real meat that should survive at low energy. They contain three (or six, which could be unsurprising if mirrors are required) generations of something. Also, the last U(1) charge is proportional to "baryon number" so it can be used to align leptons and quarks.


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The big thing of the 15 as it goes down to three "pairs of generations" is that while we can not see the electroweak force, we can see that if it appears it is going to be chiral: each "pair of generations" has two -1/3, two +2/3 and one 4/3 quark. If we want to suppress the latter at low energy, we need a combination of SU(2) unable to see it.

Note that during this thread I was more in the side of pretending that this representation was only involving the scalar partners of the 3-Gen MSSM. But it could also allow for this interpretation, as fermions with mirror fermions in the game. It could be argued that they were to be expected, as a GUT starting with real or pseudoreal representations -and this is why we do not usually see USP(2n) in the game- needs them.
Discussion here on obtaining SM fermions from unconventional representations of SU(5). It shows how the 15 branches to produce a left-handed quark SU(2) doublet. But my problem is, what about the right-handed quark singlet? It seems like it has to come from the 5 or 10. Does that mean that 15s and 24s are not enough?

Let me also expand a little on #252. The idea is that 15 x 15* = 1 + 24 + 200. So you could start with an N=1 U(5) gauge theory which only had 15 and 15* matter (chiral superfields), but the 24 would come out of the 15 x 15* meson superfield.

Also, a note for the future, I have discovered that there is an obscure ancient thesis containing a twistor model for quark-diquark systems. It is not online and the author (Aleks Popovich) left physics, but I hope we can track it down later this year.


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Discussion here on obtaining SM fermions from unconventional representations of SU(5).
Well, not so unconventional, as the SU(3) subgroup there is still supposed to be colour, and not family. On the contrary, I was working with the expectation that both subgroups of SU(5) are "family groups", and colour is not seen until upgrading to SU(15).

Of course the SU(5) products can also be seen reflected in SU(15)

15 ⊗ 15* = (200) ⊕ (1) ⊕ (24) versus 15 ⊗ 15* = (224) ⊕ (1)
15* ⊗ 15* = (70′ ) ⊕ (50) ⊕ (105) versus 15* ⊗ 15* = (120) ⊕ (105)

but the amusing/careful point here is that while the 105 of SU(15) branches down to SU(5)xSU(3) producing colour triplets (15,3) plus colour sextets (10,6), this coloured 15 that we got is not the 15 we started from. The 15 of SU(15) also branches down, to (5,3).

On the other hand, the production of the 24 seems -at first glance, I have not checked in detail - to go similarly in both cases, as it is a colour singlet.

Also note that the need of working both with N and N* is the hint that really invites us to climb up to SO(30) or USp(30) at least.
Have you uploaded/published your package somewhere? I have always found intriguing that Wolfram has not a standard package for this task.
It's in this archive: -- I have Mathematica, Python, and C++ versions.

Its user interface is not very fancy. One specifies algebras with it as {family, rank}, where family is 1, for A, 2 for B, up to 7 for G. One specifies irreps as highest-weight vectors, like {1,0,0,0}. Its outputs are lists of lists. For product reps, it outputs a list of {multiplicity, highest-weight vector}. It also does powers of reps (plethysms), breaking them down by symmetry. Each symmetry type has a list like for product reps.

It does compound algebras, like SU(3)(color) * SU(3)(flavor) * SU(2)(spin), and in a file on notable physics results, I obtain the light-quark baryon spectra.

For maximal subalgebras, it supports just about every one that I could work out projection matrices for. This includes root demotions (algebra root -> U(1) factor), extension splitting (add a root then remove another root), SO(even) -> SO(odd) + SO(odd) (the other parities are handled by the previous two types), SU(m*n) -> SU(m)*SU(n) and similar for SO and Sp, and Slansky's list of exceptional-algebra breakdowns, including my favorite, E8 -> G2*F4. However, I don't have ones like SU(6) -> SU(3), because I couldn't work out general formulas for them, though I've worked out (algebra) -> SU(2). One uses the heights of roots (sum of root components) in it.

It specifies irreps as basis sets, something like magnetic-quantum-number values for angular momentum. It has a list of {multiplicity, root, weight}. It also breaks down into Weyl orbits, each one specified with its highest weight. For each orbit, one can find a list of {root, weight} in it.
So for Sp(32), we have
Extension splitting of Sp(32) at 1
Sp(30) * SU(2)
Root demotion of Sp(30) at 15
SU(15) * SU(2) * U(1)
Product splitting of SU(15) into 5*3
SU(5) * SU(3) * SU(2) * U(1)
Root demotion of SU(5) at 3
SU(3)^2 * SU(2)^2 * U(1)^2
more-or-less (Standard Model)^2

One can get all the Standard Model's multiplets out of it, some with multiplicities that can be multiple generations.
I was working with the expectation that both subgroups of SU(5) are "family groups", and colour is not seen until upgrading to SU(15).
This also means that I have been wrong since #240, in my assertions that these latest excursions are meant to implement a chiral sbootstrap. I thought that the SU(2) was supposed to be SU(2)L, but in fact it's something like a (gauged?) flavor symmetry.

OK. So what is actually going on? The original, "non-chiral" sbootstrap, looks at meson and diquark pairings of the five lighter-than-top flavors of quark in the standard model, and obtains electric charge values corresponding to all the elementary fermions of the standard model, so one asks if there is a implicit supersymmetry in the standard model, or if some supersymmetric QCD can unfold into the full standard model when super-composites are considered.

Around #237, the 54 of SO(10) was introduced as a kind of unified heuristic model of these mesons and diquarks, with 15 and 15* standing for (anti)diquarks, and 24 standing for mesons. In the further branching under SU(3)xSU(2)xU(1) described in #239, the only part that is actually SM-relevant is the U(1), which will mix with other U(1)s to imitate SM electric charge.

Then in #242 and #244, SU(3)c is introduced, as part of a scheme to obtain all these groups from certain large groups (SO(32), E8xE8, now USp(32)) appearing as the ten-dimensional gauge group in various string theories. Since SO(32) is the gauge group of the Type I open string, there may have been an intention to recover a stringy structure of the "mesons" and "diquarks", if these branchings could be implemented there.

But SU(2)L has not been introduced, and the U(1)s are combining to imitate electric charge, not weak hypercharge. So even if the scheme can be implemented as intended, it won't give us the standard model, it will give us the non-chiral SU(3)xU(1)em effective theory that follows electroweak symmetry breaking.

I am going to have to rethink where this has gotten us, but meanwhile I want to say something about how SM-like models are actually obtained in the string theories with a ten-dimensional SO(32) gauge group, heterotic SO(32) and the type I string. The key problem is how to obtain chiral fermions.

This paper
tells us that in the heterotic SO(32) theory, "orbifold and Calabi-Yau compactifications" and also "toroidal compactification with magnetic fluxes" can do this, and presents a model of the last type, in which SO(32) is broken to G_SM by the fluxes, and some but not all of the SM fermions are obtained from the 496 (see section 3.1). I believe that the 496 branching described by @arivero could very probably be obtained in one of these flux models, but as I have explained, it would at best give us that non-chiral effective theory and not the full standard model.

As for the type I theory, the SO(32) open string actually derives from 32 space-filling D9-branes. There are type-I models where other D-branes are also introduced, but it seems like the models closest to the philosophy espoused in recent posts, would be those in which the D9-branes are the only ones. From the literature I have gleaned the following: these type I models are often dual to heterotic Z_n orbifolds. "D9-branes only" corresponds to n odd, while n even corresponds to D9s and D5s. D9+D5 is considered more promising phenomenologically, but type-I/heterotic duality can be easier to prove with D9s only, since a D5 maps to an NS5-brane in the heterotic theory, i.e. the M5-brane, whose worldvolume theory is not well understood.


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This also means that I have been wrong since #240, in my assertions that these latest excursions are meant to implement a chiral sbootstrap. I thought that the SU(2) was supposed to be SU(2)L, but in fact it's something like a (gauged?) flavor symmetry.
More than wrong, lets say mainstream :-) But yep, our SU(5) here looks as a flavor symmetry, and so the same when decomposed in SU(3)xSU(2); in the diquark picture, SU(3) moves flavour between d,s,b "subquarks" and SU(2) moves between "u,c" subquarks. The only hint of chirality is the total content of the 15, were we have a pair number of objects of charge -1/3 and of charge +2/3, but only an odd number of objects of charge +4/3. This is a small hint, telling us that perhaps we need to reorganize objects to have _L and _R symmetries, and that during such reorganization the odd object, +4/3, should dissappear (of the low energy spectrum, at least).

Your review is accurate; we see that breaking SO(32)/Usp(32) we get a "infrared standard model", this is, the limit where only colour and electromagnetism survive. And we need other way down where the "standard standard model" (uh, I need a better name) appears but also with three generations, or perhaps with mirror generations if we do not see how to produce different complex representations.

My current speculation is that the heuristic of thinking in terms of the 54 of SO(10) could be complemented with a 27 + 27 from elsewhere (perhaps E6, perhaps some 26 plus a singlet, perhaps a 24+3) and that this view should be the one showing the L and R gauge symmetries. In the ten-dimensional heaven, it would correspond to the connection between SO(32) and E8xE8, or to a connection between USp(32) and some other type 0 theory.

About getting "only" SU(3)xU(1), -by now-, I would not be very disappointed. At least it means that we are not claiming -yet- forbidden miracles such as to get chiral theories from real representations. And a lot of the game in string theories is about "effective theories", i.e, about the content of a theory in an extreme limit. Three generations of colored electromagnetism is the limit of the SM where the yukawas of the fermions are cero but the electroweak vacuum (or at least the mass of W and Z) is infinite.
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Matthew Strassler just posted a memoriam of working with Joe Polchinski, which focuses on their paper introducing an AdS/CFT dual for confinement. It contains an example (page 44, "QCD-like vacua") that is provocative from our point of view. They are starting with Maldacena's original duality: Type IIB string in AdS5 x S^5, which is dual to N=4 super-Yang-Mills.

Polchinski and Strassler add 5-branes in the AdS space and obtain various vacua of "N=1* theory", which is N=4 theory broken into an N=1 vector multiplet and three massive N=1 chiral multiplets. Specifically, each 5-brane shows up as a "shell" in AdS5, a sphere of a certain radius (and the remaining dimensions are compactified on the S^5). The 5-branes also have D3-charge (internal flux), this allows one 5-brane with k units of D3-charge to carry an SU(k) gauge theory.

The example on page 44 is pretty simple - a D5 with n units of D3 charge, an NS5 with N-n units of D3 charge, both lying at about the same radius in AdS5. The D5 carries an SU(n) gauge field, the NS5 carries an SU(N-n) field. "In the field theory this corresponds to a vacuum with a broken SU(n) sector, a U(1) vector multiplet, and a confining SU(N−n) sector."

The D5 charges will be analogous to flavor, the NS5 charges analogous to color, D5-NS5 strings analogous to quarks. A meson is two D5-NS5 strings connected (I think!) within the NS5 by internal flux. These mesons decay to D5-D5 strings, "bypassing the NS5-brane altogether...Indeed this almost happens in nature; charged pions decay through isospin gauge multiplets ... because they couple to light leptonic states — which could also be represented here, if there were a need."

The immediate possibility is that this could provide a model of pion-to-muon decay in which there is naturally a near-degeneracy of pion and muon masses.


Once I had an idea and I called few physicists. some were unknown and replied arrogantly. When I called Polchinski has was such a gentle and humble man, just offered an advice in the most respectable way. A mark of a great character beside being a great physicist.
Since my cleaning up of the thread there doesn't appear any interest, so I think closing it now is probably appropriate. As always if anyone wants to reopen it please contact the mentors either via PM or using the report function and it will be looked at.

I have had correspondence that people would like the thread kept open. There is no rule violation or anything like that so - open again it is.

But please note - threads with no activity for an appreciable amount of time are often closed and anyone that wants to continue requested to start a new thread. It can of course refer to this thread so nothing is lost.

A few times in this thread, we have mentioned N=2 Nc=3 Nf=6 SQCD. It's in the vicinity of the sBootstrap - supersymmetry, three colors, six flavors - but it lacks electric charge or hypercharge. Gerchkovitz and Karasik have written two papers (1 2) on the strings of N=2 SQCD when the quarks also have U(1) charges. In the second paper they study S- and T-dualities of the strings of N=2 Nc=2 Nf=4 SQCD, a theory which has also been studied by Shifman and Yung, who say that at a special self-dual point, its string is equivalent to the Type IIA superstring on a particular background. (I wonder if there is a relation to the self-dual point of the heterotic string, recently studied.) I believe these papers offer technical progress towards assessing the sBootstrap.

Meanwhile, Brodsky et al have extended their supersymmetric light-front holographic QCD, all the way to charm and bottom. It therefore now offers an alternative, and perhaps more realistic, foundation on which to formulate the full sBootstrap.
An and Wise describe a theoretical limit of QCD, in which a diquark made of two heavy quarks, behaves similarly to a single heavy quark. They describe an effective theory in which this diquark couples via gluon to light quarks. This is possible because the doubly heavy diquark is sufficiently small... Some encouragement that "quark-diquark supersymmetry" could be a manifestation of a fundamental supersymmetry, and not just accidental and emergent.

I have been thinking that N=1 U(3) Yang-Mills theory has some significance for the sbootstrap. The sbootstrap requires color charge and electromagnetic charge, and U(3) supplies both of those. But one stumbling block for the sbootstrap has always been, what to do with the gauginos? The sbootstrap combinatorics involve quarks, diquarks, and mesons, and then leptons enter as mesinos. The gauginos don't have a role, and yet in conventional susy, if you have gauge bosons, you also have gauginos.

Meanwhile, if we just look at pure N=1 U(3) theory (i.e. no quark superfields for now), along with the gluons and the photon, we have gluinos and the photino. The gluinos are like quarks in being colored fermions, while the photino is like the neutrino, a colorless neutral fermion. Indeed, in the very early days of supersymmetric phenomenology, there were attempts to obtain the standard model fermions as superpartners of standard model gauge bosons (see Pierre Fayet), but it didn't work. Perhaps we should look at N=2 U(3) strings a la Karasik, and see if there can be a sbootstrap-like sector.
Perhaps we should look at N=2 U(3) strings ... and see if there can be a sbootstrap-like sector.
There's a paper today on N=2 U(N) strings and their N=1 limit - so I should say something more about the prospects and difficulties for this approach to the sbootstrap.

The core results here pertain to strings in N=2 field theories with separate gauge superfields and quark superfields. This goes all the way back to Seiberg & Witten's 1994 model of confinement. Anyway - these are open strings with charged objects at the ends. Progress in understanding the formation of strings in supersymmetric field theory is great. But for the sbootstrap, we want the string itself to have a superpartner. This is why special values for which the field-string becomes a genuine string-theoretic object (see #269) are important - because then we know that the fermionic string exists too.

Another issue could be called "getting to chirality". N=2 theories are non-chiral, but N=1 is chiral, which is why standard susy phenomenology involves N=1 theories. Meanwhile, the sbootstrap combinatorics involve electric charge, but it's hypercharge that is fundamental in the standard model. Electric charge is what you're left with in the non-chiral fermionic world that follows electroweak symmetry breaking. It's a combination of hypercharge and weak isospin, which aren't even well-defined for hadrons.

And yet in the sbootstrap we want e.g. the leptons, which have hypercharge, to be superpartners of mesons, which are hadrons. We can definitely have N=1 field theories in which something like this is true - see the discussion of goldstone fermions (e.g. #222). So we need to keep probing to see how close this kind of model can get to the standard model. But I do wonder if we need some fresh perspective on electroweak symmetry breaking and the accompanying transition between chiral and non-chiral physics. @arivero expressed many thoughts on this over the years, and perhaps there is more of a connection between QCD and EWSB than we know (a clue being the similarity of the Fermi scale and the QCD scale).

Then there's quark-diquark supersymmetry, the original hadronic supersymmetry and the part of the sbootstrap that looks most like a bootstrap, given its self-referential nature. For me, the latest hope here is something called the "Melosh transformation". I have recently read that in the 1970s, this was pursued as a way of "transitioning between current and constituent quarks", but as an idea it "utterly failed" and was "insidiously counterproductive". Well, in this thread we love lost 1970s ideas about the strong interactions - and in fact that's where string theory came from - so let's have a look! And it turns out there was at least one attempt to apply Melosh transformations to hadronic supersymmetry. It doesn't have many citations, but one of them is the original sbootstrap paper...

What I suspect, is that there is some kind of duality or symmetry relating the light quarks to the heavy quarks. We already have a phenomenon in which QCD at high densities recapitulates low-density QCD. This is seen in color-flavor locking (the diquark condensates), and just this week, Ma and Rho had a paper elaborating on this recapitulation at high density (e.g. they propose that a high-density analogue of deconfinement exists, in which skyrmions come apart into instanton-like half-skyrmions). So I will be looking for supersymmetric Melosh transformations in these N=1 and N=2 theories, as the possible basis of quark-diquark supersymmetry.


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Hans has reported in the Koide thread,, of his advance on a construction to put three generations of Weinberg-Salam fermions as a single fermion of 128 components. Lets remember that the number of components of a fermion exponentiates as dimension increases, so the work can be seen also as an -another- attempt of finding the standard model inside a D=10 or D=11 fermion, can it?

On other hand, we could consider a general topic the construction of field theories in higher dimensions, and what kind of anomalies must be addressed.

Hans de Vries

Science Advisor
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Hans has reported in the Koide thread,, of his advance on a construction to put three generations of Weinberg-Salam fermions as a single fermion of 128 components. Lets remember that the number of components of a fermion exponentiates as dimension increases, so the work can be seen also as an -another- attempt of finding the standard model inside a D=10 or D=11 fermion, can it?

On other hand, we could consider a general topic the construction of field theories in higher dimensions, and what kind of anomalies must be addressed.
Hi Alejandro, Good to see you!

Allow me to explain the work: We define the following extension to the Dirac field:

$$\mbox{Dirac field}:~~\psi=
\xi_{_L} \\ \xi_{_R}
\mbox{Unified Fermion field}:~~\psi=
\xi_{_{L}} \\ \pm{\mathbf{\mathsf{i}}}_g\,\xi_{_{L}} \\ \pm~~\,\xi_{_{R}} \\ \pm{\mathbf{\mathsf{i}}}_g\,\xi_{_{R}}

This field has four spinors and thus a total of 16 coefficients (8 complex)

Then we give the explicit representation of all Standard Model fermions (24 in total) in this four spinor field. More particles are possible.

Next we provide the 16x16 matrix product ##\check{\psi}\,\hat{\psi}## which can be seen as a matrix equivalent of ##\psi^*\psi##

The result is the 16x16 bilinear field matrix that contains all relevant information of the fermion in a highly organized manner. The columns correspond to the 16 bilinear field components ##\bar{\psi}\psi,~\bar{\psi}\gamma^\mu\psi,~\bar{\psi}\sigma^{\mu\nu}\psi,~\bar{\psi}\gamma^\mu\gamma^5\psi,~\bar{\psi}\gamma^5\psi##,


The rows determine the coupling: They determine the generation of the fermion and too which electroweak boson they couple. All the couplings correspond to those of the Standard Model for the specific Weinberg angle with ##\sin^2\theta_w=0.25##. This means that the mixing is already included.

All Standard Model fermions are eigenvectors of a single generator with only the e.m. charge as input. All these values have the right Lorentz transform. Neutrinos for instance exhibited parity violation. Quarks have the right electric charge, they have the correct source current contributions to the neutral weak current, and so on and so on.

The electroweak part of the Standard Model often seems a mess. This shows that it's actually extremely elegant.

The image below describes (1) how the group-structure is derived from the field and (2) How the bilinear field matrix is corresponds to the group structure.


Document: The Unified Fermion Field

Additional materials: (mathematica files, matlab application with interactive GUI)



Gold Member
Somehow I though you were up to 128, yesterday while I was seeing the video. Ok, it is less. Still, increasing the number of components of an spinor is a fine signal of extra dimensions.

Well, more than string theory, the work looks closer to Connes-Lott models :-D

Hans de Vries

Science Advisor
Gold Member
There are a lot of components in the bilinear field matrix, the image shown above (256).

If you are looking for a fit with the title of this thread then I would suggest the SUSY part, for the way how one can describe the e.m. field and the four Maxwell equations with gamma matrices and the boost and rotation operators typically associated with fermions:

We obtain the fundamental covariant description of the electromagnetic field:
$$/\!\!\! \partial\mathbf{A} = \mathbf{F}~~~~~~ ~~~/\!\!\!\partial\mathbf{F} = \mathbf{J}
$$\mbox{with}~~~~~\left\{ \begin{array}{lrcl}
\mbox{mass dimension 1:}~~~~ & \mathbf{A} &=& \gamma^\mu A_\mu \\
\mbox{mass dimension 2:}~~~~ & \mathbf{F} &=& \vec{K}\cdot\vec{E}-\vec{J}\cdot\vec{B} \\
\mbox{mass dimension 3:}~~~~ & \mathbf{J}\, &=& \gamma^\mu~j_\mu \\

There's a Mathematica file for this.
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