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

[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.