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

[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.

 

[arivero]

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.

Is not Hill doing the exact opposite? He is binding the top (and I am not sure if all the top - light quarks pairs too). While we need to bind all the non-top pairs.

 

[arivero]

Ah no, he uses all the sectors. Interesting. So the sBootstrap is the complementary subset?

 

[mitchell porter]

Well, the idea of the sBootstrap in its essence (please correct me if I misrepresent it) is that by considering superpartners of diquarks and mesons formed from the five light quarks udscb, you get all the fundamental fermions of the standard model. So first you have to add supersymmetry to Hill's scenario. But okay, maybe we can do that.

More vexing is the circularity of the sBootstrap with respect to the light quarks themselves. One way around this is to think in terms of UV and IR. The "fundamental" udscb can be UV degrees of freedom, and the "phenomenological" udscb can be IR degrees of freedom. To me this suggests Seiberg duality, and Strassler's 1995 paper in which he describes deforming N=2 Nc=3 Nf=6 super-Yang-Mills, to get an N=1 theory in the IR which has emergent meson superfields. It's as if we want a version where one of the six flavors has a mass even in the far UV, while the others remain massless, but in the IR we still get back six quark flavors as well as emergent leptons.

If we follow this logic, it means out of Hill's "spectrum of composite states", we only have those formed from quark fields, since in the UV where the binding occurs, only quark fields exist. The leptons will emerge in the IR, as superpartners of Hill's (1,2,1/2) states.

 

[mitchell porter]

I have just run across a 2019 paper from Japan that we seem to have missed, "Dynamical supersymmetry for the strange quark and ud antidiquark in the hadron mass spectrum". As with hadronic supersymmetry, this is not about fundamental supersymmetry, it's about an emergent symmetry that involves a boson and a fermion.

There are some novelties here. The authors get somewhere by treating the strange quark and the ud antidiquark as having about the same mass; this allows them to predict that certain multiplets of baryons (that form representations of the emergent supersymmetry) also have about the same mass. However, they are talking about the constituent mass of the strange quark, not the current mass.

Also, there is no intimation that the masses are similar for any deep reason. Nonetheless, it suggests an interesting refinement of an idea expressed earlier in this thread. Is there an infrared theory derived from the standard model, one that includes leptons, mesons, diquarks, and constituent quarks, that realizes the supersymmetry of the sBootstrap?

 

[mitchell porter]

Some recent explorations:

I was worrying about the baryon number of a "quark-diquark superfield". I don't know how it could be, that the fermionic and bosonic components of a supermultiplet, could differ in quantum numbers other than spin.

In looking around, I discovered that Miyazawa's 1966 paper that introduced hadron supersymmetry, is actually called "Baryon Number Changing Currents"! Some parts are evocative but unfortunately I don't understand the old paradigm of "current algebra".

One approach to baryon number is topological: the winding number of a skyrmion. Usually these are obtained from a 2-flavor chiral model. This is an opportunity to mention a few facts about chiral symmetry. Mathematically you can write down chiral symmetry for four or five flavors, but the heavy quarks break it so badly that it's basically useless. So chiral symmetry in our world really only applies for the three light flavors.

From a sbootstrap perspective there's an interesting twist. As mentioned in earlier comments, since pions, kaons and eta mesons are made of light quarks, they are goldstone bosons of chiral symmetry breaking; and in seeking superpartners for them, one may use the paradigm of quasi goldstone fermions.

Heavy mesons - containing one or more heavy quarks - are not modeled as goldstones. However, heavy quarks very naturally allow for certain forms of hadronic supersymmetry, e.g. heavy quark + light antiquark, and heavy quark + light diquark, have some similarities. Whether this could be unified with the preceding form of supersymmetry, I can't say.

Returning to chiral symmetry, nucleons are usually constructed as 2-flavor skyrmions. They can still be obtained in the 3-flavor chiral model; see the appendix of Witten's "Current algebra, baryons, and quark confinement".

There is a technical pitfall associated with these 3-flavor skyrmions. Skyrmions are often used as models of nucleons, in an approximation where the number N of QCD colors is treated as large. This is 't Hooft's planar limit, in which planar Feynman diagrams dominate. This is OK for two flavors, but when you have three flavors, considering the wrong large-N "baryons" will give you models of the proton in which the valence quarks can be strange quarks, which is wrong. So you have to look at a special subset of the large-N 3-flavor baryons, to obtain valid models of the nucleon, in 3-flavor large-N QCD.

It turns out this situation has an analogue, in another relative of QCD that has already been considered in this thread, "orientifold field theory". This is SU(N) Yang-Mills with a fermion in the "antisymmetric two-index" representation. It provides a model of hadronic supersymmetry, in which the meson is an oriented bosonic string, and the baryon is an unoriented fermionic string, i.e. string with a fermionic charge smeared along it. This field theory can be obtained by "orientifolding" a string theory. For many details, see this big review of the subject by Armoni, Shifman, and Veneziano.

The promised analogy is that the skyrmion in orientifold field theory is something different and more complicated than the simple unoriented fermionic string. This may all seem rather esoteric, but it may end up mattering, e.g. for the right treatment of baryon number in hadronic supersymmetry.

Orientifold field theory, in its simplest form, is related only to one-flavor QCD. However, the big review by Armoni, Shifman, and Veneziano, has something to say about obtaining three-flavor QCD too (from "orienti/2f theory"). Meanwhile, one-flavor baryons have been the subject of recent theoretical progress - see recent comments in this thread about work by Komargodski and by Karasik.

Basically, Skyrme found that multi-flavor baryons could be found as topological solitons in a sigma model of pseudoscalar mesons. Komargodski recently obtained single-flavor baryons as edge excitations of eta-meson membranes. And Karasik unified the two, by showing (?) that single-flavor baryons can be obtained from the sigma models employed by Skyrme and his school, by adding the right vector mesons. It's probably related to the fact that single-flavor diquarks are vector diquarks.

Just to round out this discussion, I'll mention that Fiorenza, Sati and Schreiber had a paper late last year, part of their quest for the proper formulation of M-theory, in which they claim to get a kind of supersymmetric 2-flavor skyrmion, on an M5-brane near an orientifold plane. And they cite hadron supersymmetry and holographic vector mesons as an inspiration.

 

[arivero]

Some recent explorations:
...

In looking around, I discovered that Miyazawa's 1966 paper that introduced hadron supersymmetry, is actually called "Baryon Number Changing Currents"! Some parts are evocative but unfortunately I don't understand the old paradigm of "current algebra".

...
Just to round out this discussion, I'll mention that Fiorenza, Sati and Schreiber had a paper late last year, part of their quest for the proper formulation of M-theory, in which they claim to get a kind of supersymmetric 2-flavor skyrmion, on an M5-brane near an orientifold plane. And they cite hadron supersymmetry and holographic vector mesons as an inspiration.

Funny.

From a sbootstrap perspective there's an interesting twist. As mentioned in earlier comments, since pions, kaons and eta mesons are made of light quarks, they are goldstone bosons of chiral symmetry breaking; and in seeking superpartners for them, one may use the paradigm of quasi goldstone fermions.

Heavy mesons - containing one or more heavy quarks - are not modeled as goldstones.

Ah, but what is a heavy quark anyway?

However, heavy quarks very naturally allow for certain forms of hadronic supersymmetry, e.g. heavy quark + light antiquark, and heavy quark + light diquark, have some similarities. Whether this could be unified with the preceding form of supersymmetry, I can't say.

 

[mitchell porter]

Koide's latest is a five-flavor preon theory! Although he only gets one generation at a time, and needs three further "family preons" to get three generations. And while some composite states are two flavor-preons plus a family preon, others are one flavor-preon plus two family-preons - whereas, in the sbootstrap, everything has two flavor-preons... On the positive side, he's working with the Weyl fermions of the full standard model, rather than just the Dirac fermions of SU(3) x U(1) physics.

This should be compared to his original preon theories, the sbootstrap, our attempts at hyperbootstrap, the "string roadmap" from #239 forward, etc. (Just in case, I'll also mention a recent paper on "SU(5)_L x U(1)_Y electroweak unification".)

 

[ohwilleke]

Koide's preon paper is interesting, although using eight preons to explain the 12 fermion and 3 boson fundamental masses in the SM doesn't seem like that big of an improvement (and you can already get one of those boson masses from SM electroweak theory with ratios of EM and weak force coupling constants, so there are really only 14 free masses, and the original Koide's rule gets it down to 13 free masses).

Yershov's preon papers were IMHO some of the most notable ones that I've seen (although my Wikipedia article on Yershov was stricken for lack of notability (although the late Marni Dee Sheppeard's work also caught my eye). The first paper takes on the SM fermions, the second takes on the SM bosons. Yershov's papers on the subject were:

The First Paper

Fermions as topological objects
Authors: V. N. Yershov
Comments: Latex2e, 20 pages, 12 figures, 3 tables, (V8: formulae compactified)
Subj-class: General Physics

A preon-based composite model of fermions is discussed. The preon is regarded as a topological object with three degrees of freedom in a dual (3+1)-dimensional manifold. It is shown that dualism of this manifold gives rise to a set of preon structures, which resemble three families of fermions. The number of preons in each structure is readily associated with its mass. Although just a sketch, our model predicts masses of fermions to an accuracy of about $10^{-6}$ without using experimental input parameters.

The Second Paper

Date: Thu, 16 Jan 2003 09:54:57 GMT (18kb)
Date (revised v2): Fri, 7 Mar 2003 18:07:30 GMT (18kb)

Neutrino masses and the structure of the weak gauge boson
Authors: V.N.Yershov
Comments: LaTex2e, 4 pages (V2: minor linguistical corrections)
Subj-class: General Physics

It is supposed that the electron neutrino mass is related to the structures and masses of the $W^\\pm$ and $Z^0$ bosons. Using a composite model of fermions (described elsewhere), it is shown that the massless neutrino is not consistent with the high values of the experimental masses of $W^\\pm$ and $Z^0$. Consistency can be achieved on the assumption that the electron-neutrino has a mass of about 4.5 meV. Masses of the muon- and tau-neutrinos are also estimated.

Yershov's is the only preon model that really nails the particle masses (and does so in a quite innovative way). A figure from Yershov's first paper above:

It doesn't really do a great job of explaining why there are only three generations, but there are ways to get there (e.g. too many preons can't hold together, or the W and Z boson widths that facilitate the changes between states don't allow for any preon composites with a width less than the top quark).

There is some wiggle room in the theory to improve the fit, as the first paper notes, as well:

The results presented in Table 2 show that our model agree with experiment to an accuracy better then 0.5%. The discrepancies should be attributed to the simplifications we have assumed here (e. g., neglecting the binding and oscillatory energies, as well as the neutrino residual masses, which contribute to the masses of many structures in our model).

Alas, the fits have not aged very well.

A sort of composite Higgs mass relationship:

Yershov's paper didn't take on the Higgs boson, which wasn't confirmed to exist at the time that his papers were posted. But it isn't too difficult to extend it to include a massive Higgs boson as a composite particle in a manner very different from technicolor theories.

The hypothesis that two times the rest mass of the Higgs boson mass is equal to the sum of the electroweak boson rest masses (W+, W-, Z and the photon) is consistent with the experimental data at better than 2 sigma and would imply a best fit binding energy of 723 MeV. If the W boson has about 2 sigma less rest mass, as global electroweak fits to the W boson mass prefer, the match is even tighter and less binding energy is required.

Since bosons obey Bose statistics, the binding energy wouldn't have to be nearly so high as in a composite particle made up of fermions since they can be in the same place at the same time. So, the binding energy would just need to be slightly more than what is necessary to hold the EM force between the W+ and W- together.

This binding energy is ballpark on the same order of magnitude of the EM contribution to the proton mass. A June 18, 2014 paper estimates that differences in electromagnetic field strength between the proton and neutron account for 1.04 +/- 0.11 MeV, but the W bosons are much more massive than the up and down quarks by a factor of about 16,271. After adjusting for 723 MeV of binding energy v. 1.04 MeV of binding energy, and using a greater distance between the W+ and W- to reduce the amount of binding energy to overcome the EM force, this is equivalent to a distance apart 4.83 times as great in a two Higgs boson pair as the average distance between quarks in a proton. This is not an implausible order of magnitude match.

 

[ohwilleke]

I'll also mention a recent paper on "SU(5)_L x U(1)_Y electroweak unification".

The paper notes in the introduction:

The main result - that allows for a plethora of new degrees of freedom beyond those coming from the Standard Model (SM) - regards the mass spectrum of the model.

This is a serious bug and not a feature to brag about.

 

[arivero]

Motl has mentioned/reviewed a recent interview with John Schwarz for the oral histories collection
https://www.aip.org/history-programs/niels-bohr-library/oral-histories/45439
and his reminiscence of earlier theories is short:

So we called the theory the dual pion model. But anyway, that’s just a historical thing which is very forgettable, because the modern interpretation is entirely different.

The general topic is mentioned as dual resonance theory. So I have taken some time to review inspire-hep looking for the alternative names that are the topic of this thread, just as a refresher

1969 K. Bardakci(UC, Berkeley), M.B. Halpern(UC, Berkeley) Possible born term for the hadron bootstrap
1969 M.B. Halpern(UC, Berkeley), J.A. Shapiro(UC, Berkeley), S.A. Klein(Claremont Coll.) Spin and internal symmetry in dual feynman theory
1970 K. Bardakci(UC, Berkeley), M.B. Halpern(UC, Berkeley) New dual quark models (this is the string bit theory, is it? or is it more?) (topcited > 300)
1971 J.H. Schwarz(Princeton U.) Dual quark-gluon model of hadrons "Our proposal is to interpret the Ramond fermions as quarks and the "Dual-pion model" bosons as gluons"
1971 M.B. Halpern(UC, Berkeley), Charles B. Thorn(UC, Berkeley) Two faces of a dual pion - quark model. 2. Fermions and other things
1971 A. Neveu(Princeton U.), J.H. Schwarz(Princeton U.) Quark Model of Dual Pions (topcite > 500) Interacting pseudoscalar pions are incorporated into Ramond's model of free dual fermions. By considering the emission of N−1 pions and factorizing in the quark-antiquark channel, we recover the same N-pion amplitudes as were proposed in a previous paper.
1971 Stephen Dean Ellis(Caltech) A Dual Quark Model with Spin
1971 I. Bars(Yale U.) Degeneracy breaking in a ghost-free dual model with spin and su(3)
1972 P.G.O. Freund(Imperial Coll., London and Chicago U., EFI) Quark spin in a dual-resonance model The foundations are laid for a dual-resonance model with a spectrum characteristic ofU6×U6×O3 symmetry. The model provides an automatic mechanism for the breaking of the collinearU6×O2 symmetry. The states on the leading Regge trajectory with the exception of the lowest (« ground ») state are all parity doubled. It is argued that there may exist « mesonic » strings with a quark at one end and anSU3-singlet spin-zero boson at the other end. These complex hadrons would have all the quantum numbers (half-integer spin, nonvanishing triality, etc.) of quarks, while not being really quarks (e.g., a baryon would not be built of three of them).
1972 Edward Corrigan(Cambridge U., DAMTP and CERN), David I. Olive(Cambridge U., DAMTP and CERN) Fermion meson vertices in dual theories
1972 S.D. Ellis(Fermilab) Regge pole model of pion nucleon scattering with explicit quarks
1973 K. Bardakci(UC, Berkeley), M.B. Halpern(UC, Berkeley) DUAL M - MODELS
1973 John H. Schwarz(Caltech) Dual resonance theory ...A modification of the Veneziano model incorporating SU( N ) symmetry in a dynamical fashion is shown to have critical dimension 26− N
1973 L. Brink(Durham U. and Goteborg, ITP), D.B. Fairlie(Durham U.) Pomeron singularities in the Fermion meson dual model
1974 J.H. Schwarz(Caltech) Dual quark-gluon theory with dynamical color A modification of a previously proposed dual resonance theory of quarks and gluons is presented. It consists of incorporating new oscillator modes carrying color indices. The specific properties of these operators and the way they are included into the theory are completely determined by various consistency requirements. This modification of the theory has two important consequences. First, quark statistics are properly taken into account. Second, the critical dimension of space-time is reduced to d = 10−2 N , where N is the number of colors. Thus, the physically preferred choices N = 3 and d = 4 are compatible.
1974 L. Brink(Goteborg, ITP), Holger Bech Nielsen(Bohr Inst.) Two Mass Relations for Mesons from String - Quark Duality
1975 Joel Scherk(Caltech), John H. Schwarz(Caltech) Dual Field Theory of Quarks and Gluons " The 10-dimensional space-time of the spinor dual model is interpreted as the product of ordinary 4-dimensional space-time and a 6-dimensional compact domain, whose size is so small that it is as yet unobserved. This leads to an SU(4) symmetry group with quarks in both a 4 and a 4 multiplet. " (topcited > 200 )
1976 M. Ademollo(Florence U. and INFN, Florence), L. Brink(Goteborg, ITP), et al, Dual String Models with Nonabelian Color and Flavor Symmetries

It seems that dual quark in the early seventies referred to the idea of adding flavour-spin SU(12) or u(6) or similar beasts in order to produce all the mesons. So it stands to reason that Schwarz considers this denomination a different way from pure string theory. He does not see any relationship with SO(32) strings or the like. So his 1971 paper prefers to use the title "quark model of dual pions" to stress the diference with group theoretical flavour games.

1972 is the year of the basic QCD paper https://arxiv.org/abs/hep-ph/0208010
Current Algebra: Quarks and What Else? Harald Fritzsch, Murray Gell-Mann
and then SU(3) colour was still denominated quark-gluon theory, it seems.

In 1975 paper, the approach does not include pions anymore, it is "gluons", and the conclusions explain that "The approach of this paper departs from the conventional philosophy of trying to use dual models to construct a ·more or less realistic approximstion to the hadron S matrix. Instead, we are suggesting the use of the spinor dual model as an alternative kind of quark-gluon field theory in which the input fields have color and presumably do not correspond to physical particles."

 

[arivero]

About the mesons, while reading this old note of Neumaier https://www.physicsoverflow.org/27965/ I see that some consideration should be given to the difference between charged and neutral mesons, because some of the neutral mesons can decay even in the absence of weak force.

 

[mitchell porter]

One recurring theme in this thread, is the idea that the standard model might arise as the low energy limit of a theory which, at high energy, is a super-QCD (with quark superfields). I have run across a paper (and accompanying video) which studies a regime of SQCD which is promising from this perspective.

The author's objective is actually to prove properties of QCD with various numbers of flavors, as a limit of the corresponding N=1 theory. The question is, what to do with the squarks and gauginos, which are not part of QCD. The answer is to use a special method of susy breaking, anomaly-mediated supersymmetry breaking, which makes the squarks and gauginos massive. An interesting technical detail is the analogy between AMSB, and QCD in curved space. The lagrangian for QCD in curved space is simply the usual lagrangian, multiplied by a universal factor of sqrt(-g), where g is the metric. AMSB has a similar coupling, but it's to a fermionic deformation of a superspace, i.e. a generalized geometry with a fermionic direction. See around 37:00 in the video.

From a sbootstrap perspective, a key moment is on page 3 of the paper. The superpotential has two minima and the author can't work out which is lower apriori. However, one has massive mesinos and the other one has massless mesinos. The author wants to obtain non-supersymmetric QCD and so he opts for the one with massive mesinos (since they can drop out of the effective theory, once they become massive enough). However, from our perspective, we want light mesinos, since that's where the leptons are to come from.

It's QCD with 3 colors and 3 flavors that is being discussed, so these calculations should be compared with the work from the 1980s, mentioned starting at #222 in this thread. (By the way, back at post #49, I actually mentioned AMSB as a promising approach.)

 

[mitchell porter]

Diquarks (and any "diquarkinos") are in the adjoint representation of QCD. QCD with fermions in the adjoint representation is sometimes studied via supersymmetric QCD, since gluons are also in the adjoint, and therefore so are their gluino and sgluon partners. Part 1 of a paper from 2018 reviews some of these relationships. The paper's focus is on SU(2), but Emily Nardoni at Strings 2022 promises a forthcoming paper dealing with all SU(N).

 

[arivero]

Just for the search engines: "biquarks" and "bifermions".
https://arxiv.org/abs/2301.02425 An SU(15) Approach to Bifermion Classification.

This is Frampton's team. They suggest that diquark should be reserved for the SU(3) composite, and use biquark for elementary particles. They had already used the term "bileptons". It is a brief not, sparse, suggesting that SU(15) is an interesting group to work with.

This thread post #254 and post #244 suggest to go down to the 224 and 105 SU(15), but then go special getting the squarks from the (15,3) down the 105, and the sleptons from the (24,1) of the 224 and we are in completely different decompositions. Or more properly, Frampton's team go the discovery way, while our posts were just evaluating Slansky formulae. I can not see the advantage of doing differently, the other way also produces +4/3 particles. They go their way because it is still the idea of having a standard model family all inside a 15 representation, that Frampton defended in the past century.

Another call to the SU(15) door comes from Dobrescu arXiv:2112.15132 arXiv:2211.02211 but it overloads to SU(15) x SO(10) to combine preon and standard model charges so again I fail to relate their decompositions with ours. It would be more interesting if they were able to uplift -or rearrange- it towards say SO(16)x SO(16) or other stringy connections.

 

[Adrian59]

Another call to the SU(15) door comes from Dobrescu arXiv:2112.15132 arXiv:2211.02211 but it overloads to SU(15) x SO(10) to combine preon and standard model charges so again I fail to relate their decompositions with ours. It would be more interesting if they were able to uplift -or rearrange- it towards say SO(16)x SO(16) or other stringy connections.

As you will no doubt surmise from my question, I am not a string theorist, though I have read most of Green, Schwarz and Witten's book. My question is that from an understanding of the standard model of particle physics, the number of generators of a Lie group is equal to the number of force exchange bosons, so by going to ever larger groups like S0(32) for heterotic string theory or SU(15) one gets either 496 or 224 generators, respectively. Do these nearly all represent undiscovered bosons? If so, it makes the non-discovery of supersymmetric particles look trivial.