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

[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:

[QUOTE[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.[/QUOTE]

 

[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.