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

[mitchell porter]

"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.

 

[mitchell porter]

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.

 

[arivero]

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.

 

[arivero]

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.

 

[lpetrich]

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.

 

[lpetrich]

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.

 

[arivero]

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.

 

[arivero]

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.

 

[mitchell porter]

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.

 

[arivero]

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.

 

[lpetrich]

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: SemisimpleLieAlgebras.zip -- 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.

 

[lpetrich]

So for Sp(32), we have
Sp(32)
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.

 

[mitchell porter]

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.

 

[arivero]

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, let's 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.

 

[mitchell porter]

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.

 

[ftr]

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.

 

[bhobba]

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.

Thanks
Bill

 

[bhobba]

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.

Thanks
Bill

 

[mitchell porter]

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.

 

[mitchell porter]

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.

 

[mitchell porter]

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.

 

[arivero]

Hans has reported in the Koide thread, https://www.physicsforums.com/threads/what-is-new-with-koide-sum-rules.551549/page-11#post-6083893, of his advance on a construction to put three generations of Weinberg-Salam fermions as a single fermion of 128 components. Let's 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]

Hans has reported in the Koide thread, https://www.physicsforums.com/threads/what-is-new-with-koide-sum-rules.551549/page-11#post-6083893, of his advance on a construction to put three generations of Weinberg-Salam fermions as a single fermion of 128 components. Let's 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:

$$\mbox{Dirac field}:~~\psi=
\left(\!
\begin{array}{c}
\xi_{_L} \\ \xi_{_R}
\end{array}
\!\right)
~~~~~~\Longrightarrow~~~~~~
\mbox{Unified Fermion field}:~~\psi=
\left(\!\!\!
\begin{array}{rc}
\xi_{_{L}} \\ \pm{\mathbf{\mathsf{i}}}_g\,\xi_{_{L}} \\ \pm~~\,\xi_{_{R}} \\ \pm{\mathbf{\mathsf{i}}}_g\,\xi_{_{R}}
\end{array}
\!\!\right)
$$

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.

Video:

Document: The Unified Fermion Field

Additional materials: https://thephysicsquest.blogspot.com/ (mathematica files, MATLAB application with interactive GUI)

 

[arivero]

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]

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:
$$/\!\!\! \partial\mathbf{A} = \mathbf{F}~~~~~~ ~~~/\!\!\!\partial\mathbf{F} = \mathbf{J}
$$
$$\mbox{with}~~~~~\left\{ \begin{array}{lrcl}
\mbox{mass dimension 1:}~~~~ & \mathbf{A} &=& \gamma^\mu A_\mu \\
\mbox{mass dimension 2:}~~~~ & \mathbf{F} &=& \vec{K}\cdot\vec{E}-\vec{J}\cdot\vec{B} \\
\mbox{mass dimension 3:}~~~~ & \mathbf{J}\, &=& \gamma^\mu~j_\mu \\
\end{array}
\right.$$

There's a Mathematica file for this.

 

[arivero]

I am slightly intrigued by the claim of equivalence between composite higgs models and extra-dimensional models where one of the extra dimensions is supposed to represent renormalization scale and stretches across two branes, IR and UV. This is described for example in the last lectures of Csaki https://www.physicsforums.com/threads/is-the-composite-higgs-still-a-thing.942719/

One of my conjectures about Kaluza Klein on Witten spaces is that the equivalent to electroweak symmetry breaking is an interpolation between D=11, where the gauge symmetry group is the standard model unbroken, and D=9, where the gauge symmetry group is color times electromagnetism. I wonder if it could fit in the above framework.

 

[mitchell porter]

This might be a good time to elaborate on the Polchinski-Strassler paradigm for completing the sbootstrap, described in #265. There one has two adjacent brane-stacks, D5s for flavor and NS5s for color, a pion is a D5 string that passes through the NS5, and a muon is a D5 string that does not. If one had a standard model along these lines, strong physics would be associated with the NS5s and electroweak physics with the D5s.

At a deeper level, the NS5 and the D5 are actually part of the same M5-brane. There is a whole literature on "5-brane webs" made of D5s and NS5s, which actually correspond to a single curved M5 in M-theory, but which resolves into a web of branes when one works in string theory. Briefly, the concept is that one should seek to obtain lepton (mesino) mass and mixing relations, as a fermionic counterpart of meson mass and mixing relations (e.g. Gell-Mann-Okubo), with similarities arising from the fact that leptons are D-strings, mesons are "NS5-strings", the relations originate in the geometry of D- or NS-branes, and those geometries are similar because they ultimately come from the one underlying M5 configuration.

Furthermore, I think two papers by Hung and Seco (1 2), on realizing "almost pure phase" mass matrices via branes in six dimensions, offer a concrete starting point. One can try to realize lepton and perhaps quark mass matrices with a Hung-Seco brane configuration, using a Brannen-like circulant form (at this stage it might be better to use a phase of π/12 rather than the accurate but perplexing 2/9); and then compare it to holographic realizations of GMO, GMOR, etc.

 

[mitchell porter]

Several times in this thread (e.g. #221, #238, #270), I have nominated some specific supersymmetric field theory as worth investigating. I have found yet another formalism that may allow for concrete and relevant investigations - except that I'm not sure whether it's completely legitimate. It's in the completely obscure 1994 Russian paper "Extended Chiral Transformations Including Diquark Fields as Parameters", by Novozhilov et al. It defines an "extended chiral symmetry" that includes diquarks along with the mesons. A related paper by the same authors (published in PhysLettB, but not on the arxiv) was already cited in another obscure Russian paper by Kiyanov-Charsky, which attempted to implement hadronic supersymmetry using superfield formalism.

But that was just placing the scalar diquarks of this extended chiral symmetry alongside the quarks. What I want to do, is to construct the supersymmetric counterpart of extended chiral symmetry. Much is already known, about constructing the supersymmetric counterpart of ordinary chiral symmetry. It is an example of supersymmetrizing a coset model, as reviewed e.g. in Nitta and Sasaki 2014. But supersymmetrizing extended chiral symmetry is likely to introduce extra difficulties. As Novozhilov et al state, the diquark part of their symmetry is anomalous. In their non-arxiv paper, this leads to interactions between pions and diquark currents; I have no idea what happens if you try to supersymmetrize that construction.

A curious side note: @arivero pointed me to one of the few papers by string critic Peter Woit, "Supersymmetric Quantum Mechanics, Spinors And The Standard Model". The argument in this paper is that if you start with supersymmetric quantum mechanics (not yet QFT) on a Euclidean 4-manifold, a little hocus-pocus will give you one standard-model generation, complete with all the necessary quantum numbers. He gets there by looking at auxiliary structures like tangent space, complex structure, spin bundles... that are needed to define the theory. At one point he also resorts to twistor space. Anyway, late in the paper he's now looking at CP^3, which it is appropriate to consider as the coset space U(4)/(U(3)xU(1)). Meanwhile, in Novozhilov et al's 1994 arxiv paper, they consider the case where the diquark coset is also CP^3, but here as SU(4)/(SU(3)xU(1)). For that matter, Nitta and Sasaki consider the overtly supersymmetric CP^(N+1) coset model.

I haven't yet tried to disentangle all these proposals, but it seems like at least one of them will offer hints on how to supersymmetrize extended chiral symmetry, hopefully even the extended chiral symmetry of the sbootstrap.

 

[arivero]

This is even more complicated than the sbootstrap!
https://inspirehep.net/record/1720919?ln=es

In the present paper we propose that every fermion pair binds to form a complex scalar boson, due to a universal attractive interaction at a very high scale, Λ. Amongst many new states, including lepto-quarks, colored isodoublets and singlets, etc., this hypothesis implies the existence of a large number of Higgs bosons.
...
We call this system “Scalar Democracy” as it harkens back to the “Nuclear Democracy” of the late 1960’s.

 

[mitchell porter]

It has occurred to me that one could combine Kiyanov-Charsky, who models quark-diquark supersymmetry with genuine superfields (and not with just the supersymmetric QM of Brodsky et al), with Masiero & Veneziano (introduced in #222), who describe an SQCD with an emergent, genuinely field-theoretic, lepton-meson supersymmetry, by embedding both within the MSSM as the sbootstrap conceived it - namely, with the "squarks" and "sleptons" representing diquarks and mesons.

In other words, one would be using the MSSM to represent a kind of unfolded standard model, in which diquarks and mesons have their own fields, in addition to the usual elementary fields of the SM. One only needs the higgsino and gauginos to be heavy.

This is not yet the full sbootstrap, for reasons I will explain in a moment, but it's a big part of it; and it would be remarkable to demonstrate that the MSSM has even this much utility in the real world. In the absence of conventional superpartners showing up, one is used to thinking that the real world can only be described by a "supersplit" MSSM, in which all the superpartners are superheavy.

If we accept the usual estimate (cited e.g. in Stephen Martin's primer, end of section 6.3) that the MSSM has 105 susy-breaking parameters, then it would be progress just to understand what those parameters should be, in an MSSM used in this way. It's a part of MSSM parameter space never usually considered in phenomenology, since e.g. one normally supposes that there is no scalar superpartner of the muon with about the same mass... Susy will be broken even more mildly than is usually considered (hence the name of this thread). And then having decided to explore this novel part of parameter space, possibly we could then use some of the analytical methods already employed by phenomenologists, e.g. seeking much simpler parametrizations, and motivations for them.

In my opinion, or in my usual way of thinking about these things, the full sbootstrap involves still more than this. By itself, the above would just be a serendipitous applicability of the MSSM to the SM. But the sbootstrap implies that the quarks and leptons should be regarded as composite, or at least that such a perspective exists, and in a paaradoxical way whereby the quarks have to be somehow made of each other.

My best hope for realizing this is still that, in the UV (not necessarily the ultimate UV) there is a six-flavor N=1 SQCD with one flavor heavy; that when run down into the IR it turns into a six-flavor theory with an emergent electroweak sector (the SM described by the MSSM, as above); and that the IR quark superfields are not just the UV quark superfields unchanged, but rather that a nontrivial change of variables has occurred, like the change from electric to magnetic variables in an exact Seiberg duality. Also that this similar form for UV and IR variables would be e.g. a manifestation of a duality, and not just an accident.

 

[mitchell porter]

A timely paper today offers masses for almost all scalar mesons and diquarks made from the first five flavors. These masses are calculated, using just four input parameters - essentially, a "light quark" parameter for u and d, and then one parameter for each of s, c, b. The diquark masses are used to calculate baryon masses too, and for this two further interaction parameters are introduced.

This paper is the latest in Craig Roberts' program to study diquarks and mesons using Dyson-Schwinger equations, mentioned briefly in post #249 in this thread. There is nothing about supersymmetry here, this is just a contribution to QCD, and quite a substantial one if its results are any indication. The authors emphasize that their diquarks are dynamical entities that emerge in the context of the three-body problem for quarks. For example, within a baryon, the lightest possible diquark is usually the one that matters.

The meson masses are on page 4, Table II; the diquark masses on page 6, Table III. Diquarks in square brackets are spin 0, in curly brackets are spin 1. u and d are treated as the same mass, so e.g. the mass of [dc] is presumably the same as the mass of [uc]. Diquarks in which both quarks have the same flavor appear only as spin 1, because spin 0 requires flavor antisymmetry.

If one wishes to embed this kind of calculation in a bigger bootstrap that also determines the masses of the elementary fermions of the SM, one faces the problem that the latter are supposed to come only from couplings to the Higgs. Here the perspective of "Scalar Democracy" (#279) might come in handy.

 

[arivero]

Diquarks in which both quarks have the same flavor appear only as spin 1, because spin 0 requires flavor antisymmetry.

Yep, that is a problem because if on one hand getting rid of uu cc is welcome, it does not get rid of cu, and kills the needed bb,ss,dd :-(

 

[mitchell porter]

that is a problem

It sounds messy, but you could have a spin-1/2, spin-1 multiplet for vector diquarks, and a spin-0, spin-1/2 multiplet for scalar diquarks. It would be neater if this were in the context of an N=2 structure, where you had spin-0, spin-1/2, spin-1 in every multiplet. The dd vector diquark (for example) could be the one from QCD, its spin-1/2 partner can be u-type quarks, and the spin-0 'dd squark' would need to be heavy.

One intriguing aspect pertains to isospin. There is a similarity between W+,W-,Z0 and pi+,pi-,pi0. The spin-1 bosons act on Weyl fermions, the spin-0 pions on Dirac fermions. It already looks a little like N=2 susy. (Fayet suggested that the Higgs is the N=2 superpartner of the Z.) And then one could compare e.g. ways that uu becomes ud, in both contexts.

Then there's Komargodski's recent paper on baryons in one-flavor QCD. If he's right, he has turned up an entirely new aspect of QCD, a kind of "eta-prime membrane model" of one-flavor baryons, comparable to the skyrmion model of multi-flavor baryons. But uuu is in the same multiplet as spin 3/2 uud; how can we understand the skyrmion and the eta membrane as variations on a common theme?

Anyway, normally one says that the spin-1 counterparts of the pions - in the sense of being excited states rather than superpartners - are the rho mesons. Komargodski has an older paper in which he uses SQCD to argue that the rho mesons are actually an example of Seiberg duality. But in Sakai and Sugimoto's holographic QCD, the rho mesons are an echo of higher-dimensional flavor gauge bosons. Meanwhile, the electroweak bosons do actually gauge a small part of the standard model's flavor symmetry. It's as if one should think of baryons and mesons as infrared duals of chiral quarks and electroweak gauge bosons.

 

[ohwilleke]

A timely paper today offers masses for almost all scalar mesons and diquarks made from the first five flavors. These masses are calculated, using just four input parameters - essentially, a "light quark" parameter for u and d, and then one parameter for each of s, c, b. The diquark masses are used to calculate baryon masses too, and for this two further interaction parameters are introduced.

This paper is the latest in Craig Roberts' program to study diquarks and mesons using Dyson-Schwinger equations, mentioned briefly in post #249 in this thread. There is nothing about supersymmetry here, this is just a contribution to QCD, and quite a substantial one if its results are any indication. The authors emphasize that their diquarks are dynamical entities that emerge in the context of the three-body problem for quarks. For example, within a baryon, the lightest possible diquark is usually the one that matters.

The meson masses are on page 4, Table II; the diquark masses on page 6, Table III. Diquarks in square brackets are spin 0, in curly brackets are spin 1. u and d are treated as the same mass, so e.g. the mass of [dc] is presumably the same as the mass of [uc]. Diquarks in which both quarks have the same flavor appear only as spin 1, because spin 0 requires flavor antisymmetry.

If one wishes to embed this kind of calculation in a bigger bootstrap that also determines the masses of the elementary fermions of the SM, one faces the problem that the latter are supposed to come only from couplings to the Higgs. Here the perspective of "Scalar Democracy" (#279) might come in handy.

Flagged the paper for latter reading since it looked interesting. Maybe even more interesting than it appeared.

 

[mitchell porter]

Last week, Shifman and Yung (mentioned in #269), came out with "Quantizing a solitonic string", another chapter in their study of strings in SQCD. Specifically, they say that strings in N=2 U(3) SQCD with 3 flavors, correspond to Type II superstrings on M4 x "O(-3) line bundle over CP2". I do not understand the "O(-3)" notation, but the Calabi-Yau in question has been studied previously by Neitzke and Vafa, who in turn say ("example 2.9") that "it describes the geometry of a Calabi-Yau space containing a CP2, in the limit where we focus on the immediate neighborhood of the CP2".

Meanwhile, at the field-theoretic level I have focused on the prospects for obtaining a "pion-muon superfield", in which the muon is a goldstone fermion, and in which the similarity of pion and muon masses is actually due to supersymmetry. In the MSSM there are sum rules relating fermion and sfermion masses. More precisely, there is a supersymmetric contribution to sfermion mass that comes from the yukawa coupling between (s)fermion superfield and Higgs superfield.

In the SM, muon and pion masses appear to have completely different origins. However, the pion mass is related to the vev of the chiral condensate, which can behave like a Higgs condensate in certain respects (e.g. giving masses to electroweak bosons, see Quigg's work on the higgsless standard model). Another consideration is how chiral symmetry interacts with supersymmetry. The phase structure of SQCD can be vary a lot, depending on number of colors and number of flavors. Here it seems we want a vacuum in which chiral symmetry is spontaneously broken (so that pions exist), and in which supersymmetry is softly broken.

Ultimately, we might want an SQCD in which the square root of mass matters for charged leptons, "for the same reason" that square root of mass matters for mesons. In other words, both the Koide mass formula and the GMOR mass formula would have the same underlying cause, but manifested through fermions and bosons respectively. Masiero and Veneziano (mentioned most recently in #280) is still the best starting point I have for that, and the new possibility to watch for, is that lepton-meson part of the sbootstrap could somehow arise by perturbing Neitzke and Vafa's "local CP2", so as to reduce N=2 susy to N=1.

 

[mitchell porter]

Some recent papers...

June: Sonnenschein et al develop Sonnenschein's HISH model (holography inspired stringy hadrons). "Unlike in the usual string theory, in which the modes of open strings correspond to fields of the standard model or other QFTs, here we associate them with the states of hadrons." These are open strings, with charges at the endpoints. "In the present paper we analyze the neutral string case [i.e. oppositely charged endpoints] and the charged string will be discussed in a sequel paper." Supersymmetric behavior (whether as in Brodsky et al, or otherwise) is not considered, nor is any fermionic string.

July: "Light composite fermions from holography". A brane construction with mesons and mesinos of the same mass. "... we view the fermionic mesinos as potential realizations of composite fermions or top partners." Their model has N=2 supersymmetry but they aim for something more realistic in future.

August: A technically new perspective on the type I string, arising from the recent concept of "symmetry protected topological phases". The SPT classification was devised for the study of low-dimensional condensed-matter systems, but here it is applied to the worldsheet theory of the string, the string having some resemblance to a one-dimensional spin chain. The Type I string has turned up several times in this thread.

 

[arivero]

I saw Urs did some comments on twitter about holography and string theory for QCD.

 

[mitchell porter]

Two September papers:

An attempt to realize Brodsky et al's "light-front holographic QCD", mentioned many times in this thread, within a proper string theory! But the paper will require closer study (than I have had time to give it), in order to see what's really going on. LF hQCD is based on a superconformal mechanics. This author, Harun Omer, speaks of embedding it within a superconformal field theory, which is the kind of theory that defines the string worldsheet on a given background. There is some technical novelty (compared to ordinary string theory) in how a scale arises, so that (page 10) "the tower of eigenstates no longer have energies on the order of the Planck scale and the lowest state is not necessarily of zero energy". Elsewhere (page 4) he says LF hQCD here might be obtained as theory of open strings ending on three branes, which sounds orthodox enough; yet he also says this is "a radical departure from what has been done in the field in the last decades and in a sense a return to the beginning". So it's mysterious but of obvious interest.

There is also a new paper from Craig Roberts, a kind of meditation on the origin of mass scales in QCD. Roberts is mentioned here in #281 for his diquark models of baryons... In this paper he mentions the role of the QCD trace anomaly in generating mass, which is a standard observation; but he seems to be presenting a heterodox interpretation of the vanishing of the pion mass in the "chiral limit" of massless quarks. Apparently one normally supposes that this is because the trace anomaly vanishes in this limit; but for Roberts (see discussion after equation 7), "it is easier to imagine that [this] owes to cancellations between different operator-component contributions. Of course, such precise cancellation should not be an accident. It could only arise naturally because of some symmetry and/or symmetry-breaking pattern." (And he may be presenting his answer, around equation 11.)

It is clearly of interest to know whether Roberts' different perspective on QCD scales, is consistent with Omer's different perspective on scale in string theory! And even better if Roberts' quantitative diquark models of mass, could be realized within that framework.

 

[mitchell porter]

"U(3)xU(3) Supersymmetry with a Twist" by Scott Chapman of Chapman University (the university is named for one of his ancestors) proposes to get "two families of Standard Model left-handed quarks" from the gauginos of N=1 supersymmetric U(3)xU(3) gauge theory, in a way that resembles (page 4) the SU(5) GUT.

These are different supermultiplets from the kind we normally consider in this thread. In the sbootstrap, we wish to treat scalar diquarks and mesons as superpartners of SM fermions - thus, chiral supermultiplets - whereas Chapman wants to obtain (some) SM fermions as superpartners of U(3)xU(3) gauge bosons - thus, vector supermultiplets. There may therefore be no connection, unless both schemes can be embedded in an extended, N>1 supersymmetry.

But Chapman's gauge group may be notable. Groups of the form U(3)^n or SU(3)^n show up in various contexts. Koide and Nishiura are using a U(3)xU(3) family symmetry to implement Sumino's mechanism. Product groups like this can appear through "deconstruction" of extra dimensions. In the absence of yukawa couplings, the SM has a U(3)^5 flavor symmetry. In #270, I proposed that pure N=1 U(3) theory might be a good preparatory study for the sbootstrap; perhaps one should consider a "deconstructed" higher-dimensional version of that theory, with the sbootstrap scalars descending from the spin-0 components of higher-dimensional vectors.

 

[MathematicalPhysicist]

"U(3)xU(3) Supersymmetry with a Twist" by Scott Chapman of Chapman University (the university is named for one of his ancestors) proposes to get "two families of Standard Model left-handed quarks" from the gauginos of N=1 supersymmetric U(3)xU(3) gauge theory, in a way that resembles (page 4) the SU(5) GUT.

These are different supermultiplets from the kind we normally consider in this thread. In the sbootstrap, we wish to treat scalar diquarks and mesons as superpartners of SM fermions - thus, chiral supermultiplets - whereas Chapman wants to obtain (some) SM fermions as superpartners of U(3)xU(3) gauge bosons - thus, vector supermultiplets. There may therefore be no connection, unless both schemes can be embedded in an extended, N>1 supersymmetry.

But Chapman's gauge group may be notable. Groups of the form U(3)^n or SU(3)^n show up in various contexts. Koide and Nishiura are using a U(3)xU(3) family symmetry to implement Sumino's mechanism. Product groups like this can appear through "deconstruction" of extra dimensions. In the absence of yukawa couplings, the SM has a U(3)^5 flavor symmetry. In #270, I proposed that pure N=1 U(3) theory might be a good preparatory study for the sbootstrap; perhaps one should consider a "deconstructed" higher-dimensional version of that theory, with the sbootstrap scalars descending from the spin-0 components of higher-dimensional vectors.

Sons of tenured tracked scientists can get tenure much easily...

 

[mitchell porter]

"Hadronic Strings -- A Revisit in the Shade of Moonshine" by Lars Brink takes us back to the beginnings of string theory as well as the beginnings of this thread. He takes us through the attempt to develop a "dual model" (as string theories were originally known) for mesons made from the light quarks. There is a self-consistency relation (equation 16) which the partition function of the string must satisfy, there is a simple ansatz for the light meson masses (equations 21), and then one can look for modular functions that will construct the partition function while giving those masses.

Brink didn't find such modular functions, and says string theories of mesons were made obsolete by QCD, while string theory went on to become a theory of everything; but this is exactly what @arivero dubbed the "wrong turn" when he created this thread. He wanted the string theorists to go back to 1972, and implement the combinatorics of the sBootstrap in a dual model. Meanwhile in many recent posts, we have documented Brodsky et al's phenomenological supersymmetric models of hadrons, Sonnenschein et al's phenomenological string models of hadrons, and a number of situations from orthodox string theory in which the strings correspond directly to the meson strings of some strongly coupled field theory (Sakai and Sugimoto's holographic QCD being the most advanced example of this).

With respect to our recurring interests in this thread, it would be of great interest to see if Brink's method could be applied to a fermionic dual model of the charged leptons, only now one would be seeking modular functions that implement Koide's mass formula.

 

[mitchell porter]

I have no time to discuss or analyze, but MSSM guru Stephen Martin has written a paper, "Mixed gluinos and sgluons from a new SU(3) gauge group", which looks like a more orthodox analysis of some of the U(3)^n / SU(3)^n possibilities I mentioned in #289.

 

[mitchell porter]

Two more papers:

Stanley Brodsky provides another review of his light-front holographic QCD. LF hQCD is meant to be a new paradigm for a great many aspects of QCD - e.g. in his section 2, he says it offers a distinctive perspective on the origins of confinement and the QCD mass scale - but our interest has been that it also provides a new and modern form of "hadronic supersymmetry".

"Supersymmetric nonlinear sigma models as anomalous gauge theories", by Kondo and Takahashi, addresses the other part of the sbootstrap - fermionic partners for Nambu-Goldstone bosons like the pion. It addresses the supersymmetric CP^N coset model, mentioned in #278 as studied by Nitta and Sasaki. This seems to be a distinctive Japanese approach to the subject, potentially complementary to the 1980s European work of Buchmüller et al on "quasi Goldstone fermions".

 

[ohwilleke]

Two more papers:

Stanley Brodsky provides another review of his light-front holographic QCD. LF hQCD is meant to be a new paradigm for a great many aspects of QCD - e.g. in his section 2, he says it offers a distinctive perspective on the origins of confinement and the QCD mass scale - but our interest has been that it also provides a new and modern form of "hadronic supersymmetry".

From the abstract:

The combined approach of light-front holography and superconformal algebra also provides insight into the origin of the QCD mass scale and color confinement. A key tool is the dAFF principle which shows how a mass scale can appear in the Hamiltonian and the equations of motion while retaining the conformal symmetry of the action. When one applies the dAFF procedure to chiral QCD, a mass scale κ appears which determines the hadron masses in the absence of the Higgs coupling. The result is an extended conformal symmetry which has a conformally invariant action even though an underlying mass scale appears in the Hamiltonian. Although conformal symmetry is strongly broken by the heavy quark mass, the supersymmetric mechanism, which transforms mesons to baryons (and baryons to tetraquarks), still holds and gives remarkable mass degeneracies across the spectrum of light, heavy-light and double-heavy hadrons.

 

[arivero]

"hadronic supersymmetry"

Have we found some paper/work/thesis addressing the same thing with sQCD? Sort of superhadronic supersymmetry.

Still, my thinking is that in theories as sQCD, where fermions are allowed to live both in the adjoint representation and in the fundamental, should allow for bound states where the binding "force" is a fermion. Of course, when a fermion in the fundamental emits or absorb one "adjoint fermion", a violation of angular momentum happens, and it needs interpretation. When a baryon emits a pion the violation of energy preservation can happen during a time h/E, because E and t are conjugates. But angular momentum is conjugate to angle, and it is not easy to understand such uncertainty.

It would be very nice if it could be translated to the requisite of zero distance, because then the "composite" of two fundamental fermions joined by an adjoint fermion would be a point-like particle. Intuitively, as more short a segment becomes, more complicated a measurement of its orientation is.

 

[arivero]

At the same time, I think of Christopher Hill's recent papers (1 2, it's basically the same paper twice), in which

Note that recently Hill has started to use the expression "scalar democracy" for an idea of composite scalar sector very in the spirit of the sBootstrap, but at Planck scale. See section III A of https://arxiv.org/abs/2002.11547 for an instance.

 

[mitchell porter]

A year ago, while we were puzzling over what to do with single-flavor diquarks, I wrote

there's Komargodski's recent paper on baryons in one-flavor QCD. If he's right, he has turned up an entirely new aspect of QCD, a kind of "eta-prime membrane model" of one-flavor baryons, comparable to the skyrmion model of multi-flavor baryons. But uuu is in the same multiplet as spin 3/2 uud; how can we understand the skyrmion and the eta membrane as variations on a common theme? ... Komargodski has an older paper in which he uses SQCD to argue that the rho mesons are actually an example of Seiberg duality...

Now Avner Karasik, mentioned in this thread at #269-270, has obtained the one-flavor eta membrane as a limit of a two-flavor skyrmion, by slightly amending the usual baryon current. He remarks (just after his equation 1.1) that the fields appearing in the current are the vector mesons of flavor (i.e. the rho mesons) and a field ξ that "is roughly the square root of the unitary pion+η' matrix". Sbootstrap aficionados should certainly be interested in the "square root of a pion matrix"! If one were to supersymmetrize Karasik's construction, so it features goldstone fermions as well as goldstone bosons, could we get a Koide-like "square root of a fermion mass matrix"? Also, the eta membrane is the isospin partner of an excited state of the nucleon... There are several other obscurely interesting details, such as the role of the omega meson field, which is implicated in the mass difference between neutron and proton, to be seen on pages 16-17.

 

[mitchell porter]

Last month [URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] tweeted about hadronic supersymmetry, eventually asking whether the WZW term from chiral perturbation theory has ever appeared in a hadronic-susy model. I think not. The WZW term is a phenomenon of field theory; the model of Brodsky et al employs supersymmetric quantum mechanics, not supersymmetric field theory. Hu and Mehen (mentioned at #220 in this thread) described heavy-quark hadronic supersymmetry with a form of "heavy hadron chiral perturbation theory", but I'm not sure whether HHchPT ever got as far as concerning itself with WZW. Kiyanov-Charsky (#278) also uses superfields but only concerns himself with the mass matrix. #220 also mentions a supersymmetrization of chiral perturbation theory by Barnes et al but again, WZW not mentioned.

However, this is something we should remain alert for. Chiral perturbation theory is a kind of coset theory, we have discussed cosets (#251) in the context of goldstone fermions, and supersymmetric coset models can come from gauging supersymmetric WZW models (e.g.).

It should be noted that Karasik's amended baryon current, mentioned in #297, is also motivated by the WZW term (which is just called the WZ term in this context). Mannque Rho's latest paper on hadron-quark duality says "This current comes from the homogeneous Wess-Zumino (hWZ) term in hidden local symmetry Lagrangian". I'll also mention that in Sakai-Sugimoto holographic QCD, the baryonic WZ term comes from a Chern-Simons term in the higher-dimensional gauged flavor theory... again, another hint for the sBootstrap.

Finally, at a more down to Earth level: I noticed that from tables 4, 5, 6 in Nielsen and Brodsky, one may read off the specific pairings of quark and diquark employed in their version of hadronic supersymmetry. This is certainly of interest if one wishes to implement the sBootstrap on their work. The principle seems to be that c-bar and b-bar map to cq and bq respectively, where q is a light quark (u or d). But I believe I spotted an inconsistency regarding superpartner of s quark, at the bottom of table 4: in most mesons, s-bar maps to sq, but at the bottom it maps to ss. (Meanwhile, q-bar maps to ud.)

It would also be interesting to seek consistency between the diquark masses of Brodsky et al, and the diquark masses of Roberts et al (#281).

 

[mitchell porter]

An inspiring paper mentioned early in this thread (#48) is Shifman and Vainshtein 2005 on diquarks. They argue that the color SU(3) of real-world QCD, should contain an echo of "SU(2) color", in which diquarks would be gauge-invariant objects on a par with pions. They posit an intermediate "diquark scale" in real-world QCD that can explain "two old puzzles of the 't Hooft 1/N expansion".

Now L. Glozman proposes to explain some other features of QCD, with the idea that deconfinement proceeds in stages - first of an SU(2) subgroup of color SU(3), and then full deconfinement at a temperature three times higher. The idea seems to be that there is SU(2)-color / isospin locking in the intermediate regime. The word "diquark" doesn't appear in the paper, but the concept is reminiscent of Shifman and Vainshtein's intermediate scale.

 

[mitchell porter]

Recent papers:

"Fermions and baryons as open-string states from brane junctions". Studies mesino superpartners of mesons, in the context of brane intersections.

"Supersymmetric Proximity" by Mikhail Shifman. More on resemblances between certain supersymmetric and non-supersymmetric theories.

"Dichotomy of Baryons as Quantum Hall Droplets and Skyrmions In Compact-Star Matter" by Yong-Liang Ma and Mannque Rho. More on Karasik's current, Seiberg dual of QCD, etc.

"The Data Driven Flavour Model". Described as a refinement of Minimal Flavor Violation. Interested me because the flavor symmetries are made of SM-like groups like U(2) and U(3), something also true of several models mentioned in this thread.