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

Gold Member
Mitchell, why are you trying to figure out the proton, some people already got the noble prize for it, right.
Hmm, it has been some years, so ok lets go for a recap :-)

Topic here was generically if String Theory had gone wrong turn at some moment, and particularly if it could be possible that it still does better for the pionic string. The observation for this particular is that one could consider a scenary where supersymmetry is not broken:

or a midly broken scenary

Both of them have the peculiar characteristic that the scalars are composites of the light quarks, or perhaps open strings terminated in such quarks. From here a lot of the discussion in the thread has become about accummulating research that could be related to quark - diquark or lepton-meson symmetries.

Please tell me. Is the supersymmetric particle corresponding to dark matter stable? If it's not stable and can only be produced at laboratory.. how can they exist as dark matter that is always present?

Is the supersymmetric particle corresponding to dark matter stable?
That is a standard idea, yes. But the focus in this thread is on the unusual possibility that there are supersymmetric relations among the already known particles. Dark matter is a separate question, and could be anything.

A new paper compels me to correct some terminology from #221. I wrote as if the goldstone fermions I am interested in (as a realization of the sbootstrap) are goldstinos, but this is not so.

According to the Goldstone theorem, the spontaneous breaking of a continuous symmetry leads to a new particle, a Goldstone particle. Chiral symmetry breaking leads to pions, electroweak symmetry breaking leads to the Higgs boson and the spin-0 components of the W and Z.

All those examples are bosons. However, when supersymmetry is spontaneously broken, the emergent particle is a fermion. This is the goldstino.

The fermions I am talking about are not goldstinos and do not arise from the breaking of supersymmetry. Instead, they are the superpartners of Goldstone bosons like pions, that arise from the breaking of other symmetries.

(The new paper really does try to obtain SM fermions from goldstinos. But because you only get one goldstino per susy generator, to explain many or all of the SM fermions this way, it has to contemplate extended supersymmetries far beyond the usual N=8 limit.)

I will also note the existence of some papers trying to embed one of the classic Goldstone-fermion models of the SM (Kugo-Yanagida model) into string theory. The methods employed may be useful if we do find a sigma model that convincingly implements a sbootstrap-like relation.

Gold Member
32 pages on how Thomson might have discovered supersymmetry

Gold Member
Revisiting the question of connection between the sBootstrap and the waterfall, let me put the waterfall mass levels into the sBootstrap-inspired arrangement of fermions, mesons and diquarks. The first column is the fit to zero and 174.2 GeV, and the second colum is the "rotated" calculation where the sum of the three main levels is fit by asking it to be exactly three times the sum of the original koide triple for leptons (so electron + mu + tau).

$$\begin{array}{|rr||l|l|llll||} \hline 174200 & 173260 & \\ 3640 &4197.57 & \stackrel{\bar c\bar c}{cc}& \nu_2, b_{rgb}, e, u_{rgb}& B^\pm,B_c^\pm & \stackrel{\bar b\bar u}{bu}, \stackrel{\bar b\bar c}{bc} & \stackrel{\bar b \bar s}{bs}, \stackrel{\bar b\bar s}{bd} & B^0, B^0_c, \bar B^0, \bar B^0_c \\ 1698&1359.56 & \stackrel{\bar c\bar u}{cu}& \tau, c_{rgb} , \nu_3, d_{rgb}& D^\pm, D_s^\pm& \stackrel{\bar s\bar c}{sc},\stackrel{\bar d\bar c}{dc} & \stackrel{\bar b\bar b}{bb},\stackrel{\bar d\bar d}{dd} & \eta_b, \eta_c, D^0, \bar {D^0}\\ 121.95 &92.274 & \stackrel{\bar u\bar u}{uu}& \mu, s_{rgb} , \nu_1, t_{rgb}& \pi^\pm, K^\pm& \stackrel{\bar s\bar u}{su}, \stackrel{\bar d\bar u}{du}& \stackrel{\bar s\bar s}{ss}, \stackrel{\bar s\bar d}{sd}& \eta_8, \pi^0, K^0, \bar K^0 \\ 0 & 0.0356 & \\ 8.75 & 5.32 & \\ \hline \end{array}$$

What one could look here is for mass sum rules of the kind expected in supersymmetry breaking, say
$$\sum_{bosons} M_i^2 = 2 \sum_{fermions} M_ i^2$$

I am unable to find any, perhaps some other can try. The best thing I see is for the six charged mesons, but using mass instead of its square, and and extra factor sqrt(2) too:

$$\sum_{bosons} M_i = 2 \sqrt 2 \sum_{fermions} M_ i$$

Where the mass levels of "fermions" are the three "before breaking" from the waterfall, and the masses of "bosons" are the experimental ones. A bit disappointing, but I name it just in case that such mass rule happened to be in the literature.

Code:
(6274.9+5279.32+1968.28+1869.59+493.677+139.57061)/(2*(3640+1698+121.95))
1.4675351981

(6274.9+5279.32+1968.28+1869.59+493.677+139.57061)/(2*(4197.57589+1359.56428+92.274758))
1.4183183404

sqrt(2)
1.41421356237

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Gold Member
Well, here's the problem. You can observe the higher states on the regge trajectories for some hadrons. And orthodox string theory does imply that there are regge trajectories for the SM fermions - but the higher states start unobservably high, at the string scale. So the two string regimes shouldn't have much to do with each other. And on the other hand, if we suppose that SM fermions are somehow peers of QCD hadrons, in a new version of Chew's nuclear democracy, why haven't we seen their excited states too?
Thinking about this. Do they have just the pion without parter, or does it happens that every meson or diquark does not have a partner for its lower energy state?

The absence of partner is a very general result of SUSY QM, coming from the witten index of the superpotential, but I wonder if it could be bypassed somehow, perhaps with imaginative use of dirac delta functions, different backgrounds, etc.

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I have not run across any fundamental barrier to the existence of superpartners for pions in supersymmetric field theory. It would appear to be just an example of the Goldstone fermion construction for a supersymmetric nonlinear sigma model (like Kugo-Yanagida), for the specific case of the supersymmetrization of chiral symmetry breaking. The paper in #222 may provide an explicit example of this. The main complication in constructing such models seems to be anomaly cancellation, which may require spectator fields.

On the other hand, Brodsky et al are employing, not supersymmetric field theory, but supersymmetric quantum mechanics. The archetype of supersymmetric quantum mechanics is an oscillator with one bosonic and one fermionic degree of freedom. It's nonrelativistic; supersymmetry as the square root of spatial translation comes about when you introduce relativity. Supersymmetric quantum mechanics can be obtained from a finite-volume limit of supersymmetric field theory - this is how Witten applies his index to field theories.

But I don't know how to obtain Brodsky et al's superconformal quantum-mechanical model of hadrons from a field theory. Their model only works because of the diquark-antiquark mapping that is specific to SU(3), and they don't worry about where that comes from. "Orientifold planar equivalence" does explain it in terms of a mapping from SQCD, to QCD with a quark in the antisymmetric two-index representation (thus, like a diquark), so maybe Brodsky et al can be obtained as a limit of an "orientifold field theory" (perhaps by way of a holographic interpretation of one of its stringy implementations).

Gold Member
Their model only works because of the diquark-antiquark mapping that is specific to SU(3), and they don't worry about where that comes from.
Well, but we know that we need a trick specific to SU(3) too, so any work searching under this lampost is welcome.

Of course if would be preferable if even SU(3) can be inferred. For instance, from compactification down from 8 or 9 dimensions, guessing that SU(3)xU(1) lives in extra CP2 x S1.

Recently I was thinking if it can be inferred from some need of consistency, even simply from contemplating the need of having all the scattering diagrams. I mean, consider the four pions scattering

I should be able to consider also the diagrams with quark lines going in reverse direction, and this is diquark-pion scattering, with a diquark in one channel and still one pion in the other (not sure which one is s and which t).

Could it be that this is actually posible only, if and only if, the colour group is SU(3)? Even the trick that hides this diagram into a Nucleon-Pion scattering, namely to add another quark line

seems a very SU(3)-ish trick. For lange N, we find ourselves adding N-1 parallel lines to build the barion, while here it still looks elegant.

EDIT: there is a small comment from Mandelstam about G-parity at the end of his short letter https://inspirehep.net/record/83802?ln=es telling that "We could also construct a model with $qq$ intermediate states where we do identify the two g-parities". This is, as opposite to the interpretation discussed in the paper, with only $q \bar q$ mesons and where "we cannot identify Neveu-Schwarz g-parity with physical g-parity".

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Gold Member
Today, some history.

"QUARK OR BOOTSTRAP: TRIUMPH OR FRUSTRATION FOR HADRON PHYSICS? "

I have been reading this discourse, which Chew sent to the Physics Today in 1970; preprint is available here: https://pubarchive.lbl.gov/islandora/object/ir:144169/datastream/PDF/view

Most of it pivots about the different feeling that Chew students and collaborators have about having free parameters. Later this year, Veneziano (Phys Lett 24B, p59) opens an article referring confrontation ally to this one "One of the most common arguments [Phys Today 1970] against considering duality a genuine bootstrap scheme, is that..." And the same year, one of the students of Chew publishes the foundation articles of D=10 superstring theory, at the same time that, citing the previous two, claims to be not worried about the impossibility of fixing all the parameters; in NucPhysB31, p85, Schwarz and Neveu tell that "One possible attitude, closely akin to the bootstrap philosophy, is that of all the many dual model that might be constructed, only the right one is free of ghost and tachyons and gives a pomeron pole. Veneziano has further suggested that even the coupling constant can be determinable. However, this is an attitude we do not share".

So it seems that more than evolving from the bootstrap, string theory was born under the crossed fire between "fundamentalists" and "bootstrapers". This could explain why the wrong turn... under fire, it is more important to move anywhere than to know where to go. The first models try to have some quark content, but the failure to include strangeness drives the stringers to abandon this side too. At the end, they land in the undisputed land of quantum gravity.

Relating to the topic of this thread, this paragraph sounds encouraging, even if the answer of the superBootstrap is not the one that Chew is wishing here:
Chew1970 said:
The unfriendly question raised most often by sharp-witted fundamentalists is how self-consistency can possibly be expected to generate "Internal quantum numbers" like hypercharge and baryon number. It is conceded that mass ratios and coupling constants might all be bootstrappable, but how can you bootstrap a symmetry? A" conceivable response Is that symmetries (or the associated quantum numbers) are related to particle multiplicities, and the nonlinear unitarity condition responds to the number of different particles. Models that incorporate unitarity in some serious fashion (not through a formal but meaningless infinite series) thus have a chance of shedding light on the internal quantum number puzzle. If some future bootstrap-motivated, model succeeds in "explaining." baryon number and hypercharge, the most skeptical of fundamentalists ought to be impressed.

Gold Member
Hmm, fast question: In the R-NS Dual Model, are the diquarks forbidden? If so, how?

(I mean, with the original interpretation where the Ramond sector are quarks, and the NS sector has a "quark-antiquark" sector and a "zero quarks" sector.

Gold Member
Reviewing the thread I find this "joke", and while I can not find any relation to string nor group theory, it could we convenient to list the whole construction.

at the end of the path we had investigated the case with 16 up quarks and 31 down quarks, producing 16*31 = 496 ud combinations and (31*32)/2 = 496 dd combinations and so 248 generations, and then the scalars from the extant (16*17)/2= 136 uu combinations had been used to give mass to some objects
Point is, we consider the general case of r "isospin up" quarks and and s "isospin down" quarks, inserted inside a complete set of n generations. We can try arbitrary values of r and s and see how many scalars the system produces and so how

$$\begin{array}{l|c|r|r} & \text{formula} & r=2, s=3 & r = 16, s = 31 \\ \hline \text{down sector} & r s & 6 & 496 \\ \text{up sector: }&s (s +1 )/2 & 6 & 496 \\ \text{4/3 sector: }& r (r +1 )/ 2 & 3 & 136 \\ \text{muon sector: }&r s & 6 &496 \\ \text{neutral sector:} &(r^2 + s^2 - 1 ) / 2 & 6 & 608 \\ \text{of which, "extra neutrals" } & {(r^2 + s^2 - 1 ) / 2 - r s \\ = ((r - s) ^2 -1) / 2 }& 0& 112 \\ \text{so that 4/3 + extra neutrals:} & & 3 &248 \\ \hline \text{total quark sector} & (r + s) (r+s+1) / 2 & 15 & 1128 \\ \text{total lepton sector} & (r+s)^2 -1 & 24 & 2208 \\ \end{array}$$
The topic in this thread was that it is pretty natural to force the r=2 s=3 solution, e.g. asking for the four standard model charges to give the same number of bosons, or asking for zero neutrals and some other equality. And one wonders if there is some argument that makes logical the r=16 s=31 option, which could have some hope to be matched with superstring-originated groups.

Note that once it has been fixed that s = 2 r - 1 (by asking equal number of up and down bosons) then the formulae are
$$\begin{array}{l|c} & \text{formula} \\ \hline \text{down sector} & r (2 r -1) & \\ \text{up sector: }& r (2 r -1 ) \\ \text{4/3 sector: }& r (r +1 )/ 2 \\ \text{muon sector: }&r (2 r - 1) \\ \text{neutral sector:} &( 5 r^2 - 4 r ) / 2 \\ \text{of which, "extra neutrals" } & r (r - 2 ) / 2 \\ \text{so that 4/3 + extra neutrals:} & r ( 2 r -1) /2 \\ \end{array}$$

So that it seems that this condition also fixes that the sum of extra particles amounts to one half of the other sectors.

Note also that, being based in SU(5) flavour, the r=2 s=3 solution can be also seen as related to SO(10): the sum of quark, antiquark and leptons is 15 + 15 + 24 = 54, and indeed the 54 of SO(10) decomposes down to SU(5) irreps following this sum. On the other hand, the similar game in the big solution should involve SU(47) and SO(94), too big a game.

Here's a thought. Suppose we have six quark superfields with electromagnetic charges as in the SM, and a gluon superfield; and then suppose we make one of the up-type quark superfields very heavy. (I'm also going to ignore, for now, the photon superfield implied by the electromagnetic charges.)

What kind of objects can form at scales below the "top quark mass", in this scenario? We have five flavors of quark and squark, and we have gluons and gluinos. Let's suppose we can have gluon-strings and gluino-strings, terminated by quarks and squarks.

A quark-antiquark gluon-string is just an ordinary meson. A quark-antiquark gluino-string could be a lepton, as in the sBootstrap. And the other interesting twist is that a quark-squark gluon-string is reminiscent of a QCD baryon, if you think of a squark as like a diquark. Also, a squark-antisquark gluon-string is then analogous to a tetraquark; so it's like Brodsky.

We also have gluino-strings containing squarks - a new type of extra state. But what I find interesting, is that this scenario begins to incorporate the whole sBootstrap. Lepton-meson supersymmetry is there overtly, and quark-diquark supersymmetry, somewhat covertly.

arivero
Gold Member
A quark-antiquark gluon-string is just an ordinary meson. A quark-antiquark gluino-string could be a lepton, as in the sBootstrap. .
Yep, but one would need to find that a gluino string, because of some unknown, shows in low evergy as a point-particle, while a gluon string shows as an extended one. Hard to swallow, particularly because the strings have never presented a structure function similar to the experimental ones. Not the same partons, it seems :-(

I am thinking of a roadmap for the sBootstrap that could be palatable to stringers.

First, look at bosonic oriented strings and note that the SO(10) Chan Paton symmetry on it implements three generations of scalars in the tachion.
Separate mesons from diquarks by finding some oriented mesons inside this unoriented string. This could be problematic as usually the quotient goes in the reverse direction. But it could be doable.
Then argue that SO(10) is justified by a supersymmetric bootstrap: that light Dirac superpartners of these scalars must generate again the same sector. This for itself is already an argument to introduce fermions.
Now go down from D=26 to D=10 and try to keep this symmetry alive. Perhaps in the D-instanton, which has SO(10), or perhaps with some creative use of Marcus-Sagnotti fermion labels.
Now, our strings are still in some sense decolored, or SU(N-->infinity), and need to have chiral electroweak charge, instead of only electromagnetic. We should solve this when going down to D=4; the process of going from D=10 to D=9 would assign the broken electroweak components, and D=9 down to D=4 would paint the string with flying SU(3) colours plus B-L numbers.

EDIT: let me add the tables of the SO(10). The 54 down to $SU(5) \times U_1(1)$
$$54 = {15}(4) + \bar{15}(−4) + {24} (0)$$

Then each representation goes down to $SU(3) \times SU(2) \times U_2(1)$
$$\begin{eqnarray*} 15 =& (1, 3)(−6) + (3, 2)(−1) + (6, 1)(4) \\ 24 =& (1, 1)(0) + (1, 3)(0) + (3, 2)(5) + (\bar 3, 2)(−5) + (8, 1)(0) \end{eqnarray*}$$
And from the two hypercharges, we can produce a charge Q

$$\begin{array}{l|c |r|r|c} % a/b = 1/6 or = 2/3???? flav & N& Y_1 & Y_2& Q= \frac 1{30}Y_1 - \frac15 Y_2 \\ % Q= -\frac 2{15}Y_1 - \frac15 Y_2 \hline (6,1) & 6 & 4 & 4 & -2/3 \\ (3,2) & 6 & 4 & -1 & +1/3 \\ (1,3) & 3 & 4 & -6 & +4/3 \\ (\bar 6,1) & 6 &-4 & -4 & +2/3\\ (\bar 3,2) & 6 & -4 & 1 & -1/3\\ (1,\bar 3) & 3 & -4 & 6 & -4/3\\ (\bar 3,2)& 6 & 0 & -5 & +1 \\ (3,2) &6 &0 & 5 & -1 \\ (1,1) & &&& \\ (1,3)&12 &0& 0& 0 \\ (8,1)& &&& \\ \end{array}$$
On the other hand, it is tempting to try some hypercharge that puts away the chiral (1,3) squark

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There is no shortage of theoretical options to investigate. But I now prefer to think of quark-diquark supersymmetry as something which manifests only at the end of an open string, as in Brodsky. In #225 I mentioned that in Seiberg-Witten theory there are N=2 theories in which a single-particle state in one vacuum, can correspond to a multi-particle state in another vacuum. There is also the interesting "diquark monopole" of Vachaspati's dual standard model, which can decay into two "quark monopoles".

So what I would envisage, is a UV theory which is a perturbation of some self-dual supersymmetric theory - N=2 Nf=6 Nc=3 SQCD is still attractive as a candidate for that self-dual theory - which in the IR gives rise to the standard model. From the UV perspective, the IR fields of the standard model would be composite, as in a Seiberg duality, but there is some resemblance between the IR spectrum and the UV spectrum, because of the self-duality.

I would also still think in terms of getting both leptons and hadrons from a single IR theory of open strings, in which all strings are terminated by "quarks" (I'm not sure whether they would be IR quarks or UV quarks). Mesons and baryons are bosonic strings terminated by "quarks", leptons are fermionic strings terminated by "quarks".

Quark-diquark supersymmetry, as the part of the sBootstrap that most resembles a self-duality, would be an echo in the IR of that true self-duality in the UV, and would only pertain to the "quarks" that terminate the strings of the IR string theory. Lepton-meson supersymmetry, on the other hand, would be a supersymmetry of the whole string.

There are various challenges for such an approach, but I consider the main problem to be, getting to chiral fermions. The sBootstrap as currently formulated seems to apply to a non-chiral, SU(3) x U(1)em limit of the standard model, that would be expressed in terms of Dirac fermions with electric charge.

I thought further progress might require a careful review of all the phenomena and interactions of that non-chiral limit, and how they are explained by the true standard model in terms of Weyl fermions with weak hypercharge. Then one could take a non-chiral framework like that in #238, and try to elaborate it in an analogous way.

But it looks like you have just posted a concrete proposal for a hyperBootstrap. If it does make sense, it could immediately serve as the basis for a chiral N=1 field theory as in #222, which could then be studied with a view to IR behavior, formation of nonabelian strings, and so on - whether or not we also had a way to realize it in string theory.

Gold Member
The 54 down to $SU(5) \times U_1(1)$
$$54 = {15}(4) + \bar{15}(−4) + {24} (0)$$Then each representation goes down to $SU(3) \times SU(2) \times U_2(1)$
$$\begin{eqnarray*} 15 =& (1, 3)(−6) + (3, 2)(−1) + (6, 1)(4) \\ 24 =& (1, 1)(0) + (1, 3)(0) + (3, 2)(5) + (\bar 3, 2)(−5) + (8, 1)(0) \end{eqnarray*}$$
There is no shortage of theoretical options to investigate. But I now prefer to think of quark-diquark supersymmetry as something which manifests only at the end of an open string,
I am doing a fast review of the bibliography; I'd say we have accumulated a lot. The main problem with the open string formulation is that there are two simultaneous bootstraps putting charges at the ends of the string: the one of the generation group, via supersymmetry, and the one of su(3), via (3 x 3)_anti = 3. They can not be independent because the 15 of SU(5) goes with the 3 of SU(3), whule the 24 goes with the singlet. Which amazingly could be compatible with a claim (hep-ph/9606467) that 54 and higher representations of SO(10) are always in the singlet of any other factors.

Half a 54, which we can do because it contains both particles and antiparticles, is a 27, and then the search scope becomes too wide. A traditional mention is (3,3,3) + (/3,/3,/3) of SU(3)^3, falling from E6 (but it is more typical to smash it into 16+10+1 of SO(10). In both cases, further breaking is needed if we want to get something close to the above decomposition)

Other 54 pathway, which can appear from branes too, is from the 55 of Sp(10), with only the nuissance of the extra singlet. The appendix of hep-th/0305069 mentions that this 55 could be obtained from orientifolds, but it doesn't give a reference. 1206.0819v2 suggest a realization with D7-branes. hep-th/0204023 Uses D7, D3 and O3 for generic Sp(2N+2M)xSp(2N), but does not evaluate our particular N=3 M=2. Neither do Luty et al hep-th/9603034v2 when looking at Sp(2N) susy. Similarly Witten 83 go for generic Sp(2N). This is an interesting paper even if if it focuses on its use as coloration.

A recent work arXiv:1603.05774v2 considers the "hidden pions" in the 15 both comming from SO(10) and Sp(10), and it proposes mass formulae! I do not get how it presents them as pions and not diquarks.

Also recently arXiv:1608.01675v1, Arkani-Hamed et al, mentions the decomposition from 15 into SU(5) with quantum numbers from the standard model. Comparing with this one, and with Vachaspati, it seems that the innovation here in this thread is the use of the most internal SU(3) for flavour instead of colour; the logic being that the string will take care of colour.

Googling for group decompositions, even with site:arxiv.org flag, is very inneficient, so sure I am missing important references.

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Gold Member
it seems that the innovation here in this thread is the use of the most internal SU(3) for flavour instead of colour; the logic being that the string will take care of colour.
I would like to find some strong -pun intended- argument to use SU(3) instead of any other SU(N), big N etc... Lacking this, it could be worthwhile to note that having a 54 and pasting an U(3) upon it is pretty interesting. It could be nice to have Lisi or Toni Smith or some other big numerologists here in the thread. On my side, lets at least notice that
$$54*9 = 496-1-9$$

Or,
$$\bf (15,3)+(15,6)+(15,3)+(15,6)+(24,8)+(24,1)+(1,8)+(1,1) \\ =45+90+45+90+192+24+8+1\\ =495$$

If this is a branching rule of something into something, I do not know. It looks so, but I am not conversant with large representations.

EDIT: from wikipedia https://en.wikipedia.org/wiki/Green–Schwarz_mechanism
Green describes finding 496 on each side of the equals sign during a stormy night filled with lightning, and fondly recalls joking that "the gods are trying to prevent us from completing this calculation".
A more modern analysis is quoted by Lubos here http://motls.blogspot.com/2010/07/string-universality-theres-no-u1496.html but no clue of where our representations comes from.

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Gold Member
Actually I was thinking other two different venues to approach 496:
1. somehow expand SO(32) as a sum "SO(1+5+10+10+5+1) " and consider only the 10 for particle content. This way could be useful if colour is not invited to the party
2. promote each preon with a colour label, so the matrix is promoted from dim 10 to dim 30. Still, we need to add manually an extra preon/antipreon pair, uncolored, to get up so SO(32). This looks more natural that the previous post where we only have dim 30 and it is the 3x3 submatrices that overfill across the diagonal fo fill up to 495 "states".
Then, of course, both cases -and the previous one- need some argument to extract only the sm-like representations.

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Gold Member
I am starting to suspect that special embeddings where not fully explored in the old age. Witten 84 does SO(32) to SU(5) via the most trivial way, times SO(22) and then not looking for generations nor colour (as it is already considered in the GUT group). Fortunately Gell-Mann, Ramond and Slansky look for colour too... but, uh, do they forgot the $(1, n^2_3 -1, 8^c)$ here?

Formula 2.18 for our case should decompose SO(32) as $$(n \times n)_A= [1,1,1^c] + {\bf (1,24,1^c) }+ (1,1,1^c) +(1,1,8^c) + (2,5,3^c) + (2,\bar 5, \bar 3^c) +{\bf [1,15,\bar3^c]} + {\bf [1, \bar {15}, 3^c]} +[1,10,6^c] + [1, \bar 10, \bar 6^c] + 1,24,8^c$$

For verification, it is possible to branch down to this same result using the new tables of
arxiv.org:1511.08771 via regular branching to $so30$ and $su15$, and then special branching down to $su15 \times su3$.
1. For so30 ⊕ u1(R):
• 496 = (435)(0) ⊕ (30)(2) ⊕ (30)(−2) ⊕ (1)(0)
2. Then for su15 ⊕ u1(R):
• 435 = (224)(0) ⊕ (105)(4) ⊕ (105)(−4) ⊕ (1)(0)
• 30 = (15)(2) ⊕ (15)(−2)
3. and then ⊃ su5 ⊕ su3(S):
• 224 = (24, 8) ⊕ (24, 1) ⊕ (1, 8)
• 105 = (15, 3) ⊕ (10, 6)
• 15 = (5, 3)
• (and none from 120 = (15, 6) ⊕ (10, 3) )
Perhaps some way down via so12⊕ su3(S), and so12 is so(2)xso(10) and then so(2)xsu(5)? Also, perhaps O(n) and U(n) instead of su, so?

I am not sure if this is the right branching, or it is the former one, or
some other, but I find a bit disappointing that the final conclusion
of this thread is going to be to identify the lost scalar partners as bosons of the SO(32) string. Such
idea should already be in the literature somewhere.

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I have been worried all week that you are too optimistically jumping between flavor symmetries and gauge symmetries. For example, that scalar 54 which is supposed to come from the mesons and scalar diquarks of five quark flavors, assumes an SO(10) flavor symmetry, which is rather unusual. But in most of this week's rampage through representation theory, you've been looking at gauge groups, not flavor groups.

As you say, Gell-Mann et al is good because they are looking at flavor and color together. So we have at least one clear example of how to do that. But there are further twists. When the theory is strongly coupled, there may be chiral symmetry breaking that reduces the flavor group, and determining that is an art in itself. (It's very very likely that this is related to the sBootstrap, since the pions are precisely the Goldstone bosons of chiral symmetry breaking.)

Also, "there are no global symmetries in string theory". There is a worldsheet theorem that if a global symmetry exists, there must be a corresponding gauge-boson state of the string, turning it into a local symmetry. In Sakai and Sugimoto's holographic QCD, flavor is gauged. But it's also possible for the global symmetry to just be approximate. There is some discussion here.

arivero
I have just run across two highly relevant papers by Armoni - 1310.2027 and 1310.3653. They came out near the start of this thread's long dormant period, from late 2013 through all of 2014... The first one, in particular, is remarkable for how many of our themes it contains.

I'll set the scene with a remark from that first paper (page 7). We are dealing with a field theory which is realized in string theory by a "Hanany-Witten" brane configuration "identical to the brane configuration that realizes SO(2N) SQCD, except that the D4-branes are replaced by anti D4-branes".

I'm emphasizing this because, if we do have to study this one in detail, we know that the place to begin is with the configuration that realizes SO(2N) SQCD. Armoni is interested in a similar but non-supersymmetric theory; but it may be that we will want to go back to the supersymmetric prototype.

Another thing to note is that these Hanany-Witten configurations can be lifted to M-theory. In Type II theories, they appear as a web of D-branes and NS-branes (and in this case, an orientifold plane), but in M-theory, they can be realized as a single M5-brane, on the right geometric background.

Armoni is concerned with two field theories, an electric theory and a magnetic theory. He is proposing a Seiberg duality. Inter-brane forces which cause the branes to rearrange themselves are also a part of it.

What I want to note here, are the symmetries and some of the particle content. The flavor symmetry is SO(2Nf). There are particles in the non-supersymmetric theory (but which is, remember, descended from a supersymmetric theory) which he calls quarks, squarks, a gluino, a meson, and a mesino. The gluino transforms in an antisymmetric two-index representation of the gauge group, so it might be a toss-up as to whether the gluino or the squarks are more like diquarks.

On page 18, the breaking chain SU(2Nf) -> SO(2Nf) -> U(Nf) is referenced. And the companion paper talks about chiral symmetry breaking.

arivero
Gold Member
The next temptation is to try to classify under chirality those bosons we have got in a 496 (or 495 or whatever), or the original 54 ones. If we assume that they were produced, in SO(10), via a set of five "quark preons" and "five antiquarks", then the next step is again obvious: consider sum and difference of particle-antiparticle, as such is the way to build chirality invariant states, and see what happens with the group? Does it decompose to a product of two groups, one for left, other for right chiral? And when we scale up, what does happen? Does 496 divides in 248+248, or 495 in 1+247+247 ?

(amusingly #247 is the number of this post in the thread... yes, numerology is always a running joke here :-)

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Today I found some old research which might help with the bottom-up issues. In the 1980s, Masaki Yasue wrote papers on obtaining SM fermions as goldstone fermions in SQCD. With Terazawa, he found some mass formulae based on Dashen's formula. Then in the 1990s, Craig D. Roberts studied a common Dyson-Schwinger approach to meson and diquark propagators, form factors, etc. He wrote specifically about Goldstone's theorem and also the Pauli-Gursey symmetry. He doesn't mention supersymmetry, but Dyson-Schwinger equations can be extended to superfields.

Something that bothered me about the recent group-theoretic explorations, is that they didn't take the goldstone nature of pions into account. They were trying to get the SM from open strings, but pions aren't just strings, they are strings that emerge from chiral symmetry breaking.

Meanwhile, Mizoguchi and Yata explicitly talk about deriving the SM from goldstone fermions in string theory, from the spontaneous breaking of some geometric symmetry. They are implementing "coset sigma models", and they say, "The advantage of the coset sigma model approach is that the associated quasi-Nambu-Goldstone fermions are typically chiral."

It also turns out that QCD's chiral symmetry breaking can be understood in these terms (i.e. as a coset sigma model). This encourages me to think that all of this can come together, that the SM really might be obtained from the goldstone fermions of some stringy SQCD.

I even wonder if we could find a stringy construction which has E8L x E8R chiral symmetry, and in which a "496" decomposition is implemented by chiral symmetry breaking.

arivero