A Three generations of Fermions from octonions Clifford alegbras

[kodama]

TL;DR Summary

standard model octonions clifford alegbras

Quanta has this article,

The Peculiar Math That Could Underlie the Laws of Nature​

New findings are fueling an old suspicion that fundamental particles and forces spring from strange eight-part numbers called “octonions.”

https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/

John Baez has this

Can We Understand the Standard Model Using Octonions?​

https://math.ucr.edu › home › baez › standard

Octonions and the Standard Model (Part 1) | The n-Category ...​

https://golem.ph.utexas.edu › category › 2020/07 › oct...Jul 17, 2020 — I want to talk about some attempts to connect the Standard Model of particle physics to the octonions. I should start out by saying I don't .

Octonions and the Standard Model - Perimeter Events (Indico)​

https://events.perimeterinstitute.ca › event › overviewFeb 8, 2021 — Over the years, various researchers have suggested connections between the octonions and the standard model of particle physics.

In this talk we review some lessons from grand unified theories and also from recent work using the octonions. The gauge group of the Standard ...

there are a lot of heavily cited research papers on using octonions and clifford alebgras to explain, among other things, that the 3 generations of fermions in the standard model is the result of octonions

and I would like to know where this line of research is and seems independent from string theory

Three fermion generations with two unbroken gauge symmetries from the complex sedenions​

Adam B. Gillard, Niels G. Gresnigt

We show that three generations of leptons and quarks with unbroken Standard Model gauge symmetry SU(3)c×U(1)em can be described using the algebra of complexified sedenions C⊗S. A primitive idempotent is constructed by selecting a special direction, and the action of this projector on the basis of C⊗S can be used to uniquely split the algebra into three complex octonion subalgebras C⊗O. These subalgebras all share a common quaternionic subalgebra. The left adjoint actions of the 8 C-dimensional C⊗O subalgebras on themselves generates three copies of the Clifford algebra Cℓ(6). It was previously shown that the minimal left ideals of Cℓ(6) describe a single generation of fermions with unbroken SU(3)c×U(1)em gauge symmetry. Extending this construction from C⊗O to C⊗S naturally leads to a description of exactly three generations.

Comments:	22 pages, 2 figures
Subjects:	High Energy Physics - Theory (hep-th)
Cite as:	arXiv:1904.03186 [hep-th]

Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra​

N. Furey

A considerable amount of the standard model's three-generation structure can be realized from just the 8C-dimensional algebra of the complex octonions. Indeed, it is a little-known fact that the complex octonions can generate on their own a 64C-dimensional space. Here we identify an su(3)⊕u(1) action which splits this 64C-dimensional space into complexified generators of SU(3), together with 48 states. These 48 states exhibit the behaviour of exactly three generations of quarks and leptons under the standard model's two unbroken gauge symmetries. This article builds on a previous one, [1], by incorporating electric charge.
Finally, we close this discussion by outlining a proposal for how the standard model's full set of states might be identified within the left action maps of R⊗C⊗H⊗O (the Clifford algebra Cl(8)). Our aim is to include not only the standard model's three generations of quarks and leptons, but also its gauge bosons.

Comments:	10 pages, 2 figures
Subjects:	High Energy Physics - Theory (hep-th); High Energy Physics - Phenomenology (hep-ph); Mathematical Physics (math-ph)
Cite as:	arXiv:1910.08395 [hep-th]

High Energy Physics - Theory​

[Submitted on 14 Feb 2017 (v1), last revised 8 May 2018 (this version, v3)]

The Standard Model Algebra - Leptons, Quarks, and Gauge from the Complex Clifford Algebra Cl6​

Ovidiu Cristinel Stoica

A simple geometric algebra is shown to contain automatically the leptons and quarks of a family of the Standard Model, and the electroweak and color gauge symmetries, without predicting extra particles and symmetries. The algebra is already naturally present in the Standard Model, in two instances of the Clifford algebra Cℓ6, one being algebraically generated by the Dirac algebra and the weak symmetry generators, and the other by a complex three-dimensional representation of the color symmetry, which generates a Witt decomposition which leads to the decomposition of the algebra into ideals representing leptons and quarks. The two instances being isomorphic, the minimal approach is to identify them, resulting in the model proposed here. The Dirac and Lorentz algebras appear naturally as subalgebras acting on the ideals representing leptons and quarks. The resulting representations on the ideals are invariant to the electromagnetic and color symmetries, which are generated by the bivectors of the algebra. The electroweak symmetry is also present, and it is already broken by the geometry of the algebra. The model predicts a bare Weinberg angle θW given by sin2θW=0.25. The model shares common ideas with previously known models, particularly with Chisholm and Farwell, 1996, Trayling and Baylis, 2004, and Furey, 2016.

Subjects:	High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Representation Theory (math.RT)
Cite as:	arXiv:1702.04336 [hep-th]

 

[jambaugh]

I believe that invoking "octonians" per se may be a bit of a red herring. Given their non-associative structure it is very hard to give them any kind of practical interpretation in a physical theory which can't be equally applied through the associative algebra of left and right actions by their elements. (The non-associativity becomes non-commutativity of left with right actions.)

The construction of the octonians relies however on a a more key mathematical phenomenon which is triality which is a symmetry between the vector and two spinor representations of spin(8). One uses this symmetry to construct the octonians by cycling between these three interpretations of a generic 8-vector in the product structure: ab=c.

I think that triality and the 8-cycles of Bott periodicity may be the significant mathematical foundation to study. I think triality underpins much of why we have those few odd exceptional simple Lie algebras/groups.

But I'm getting too old to dig deeply into this level of math so take my opinions here with a pinch of salt.

 

[ohwilleke]

FWIW describing “octonions" as "peculiar" is a bit extreme. They are really just a very straightforward generalization of complex amplitudes that are ubiquitous in quantum physics. Complex amplitudes represent far more of a conceptual leap from real number valued scalars and vectors, than octonions do from complex amplitudes.

Likewise, prominently describing these mathematical objects as operating in 8 dimensional or 64 dimensional space obscures more than it explains for an educated lay reader or even a non-specialist physicist or mathematician, making it seem more exotic than it really is.

 

[PeterDonis]

Moderator's note: Thread moved to Beyond the Standard Model forum and thread level changed to "A".

 

[jambaugh]

FWIW describing “octonions" as "peculiar" is a bit extreme. They are really just a very straightforward generalization of complex amplitudes that are ubiquitous in quantum physics. Complex amplitudes represent far more of a conceptual leap from real number valued scalars and vectors, than octonions do from complex amplitudes.

I stand by my "peculiar" qualifier. There is no reason in standard QM to divide by the complex numbers. We do not use them as a division algebra but rather as an ideal of the compact U(1)=SO(2) group. The critical feature is that the phase is a gauge quantity, and the amplitude is sufficiently complex (pardon the pun) to allow continuous non-trivial evolution between states while staying normalized. We thus can manifest interference. In a pure real theory we couldn't, while maintaining normalization evolve from 1 to -1 so that e.g. the amplitude from one slit could destructively interfere with the amplitude from the other in our double slit experiment.

This was apparent when Finkelstein et al studied quaternionic quantum mechanics.
Foundations of Quaternion Quantum Mechanics (1961)
They resolved that it is effectively SU(2)=Sp(1) gauge theory. This is to say, as I understand it, you excise the non-commutativity of the "scalars" in the amplitude into the non-commutativity of the gauge group. It is the group structure (preserving probability as squared norm of the amplitude) that matters, not the fact that you have a division algebra.

It is the exceptional group structures we get due to triality which allows for, among other things, our ability to construct an 8 dimensional division algebra, albeit a division algebra that is peculiar in that it is non-associative. I assert that for the quantum mechanics we don't need the division algebra, we need the deeper group structure. I doubt we can glean any new physical prediction from saying "octonians" as opposed to saying "spin(8) with triality". The latter has the virtue of ignoring the non-associate product structure forced on us to make all non-zero octonians have reciprocals.

Likewise, prominently describing these mathematical objects as operating in 8 dimensional or 64 dimensional space obscures more than it explains for an educated lay reader or even a non-specialist physicist or mathematician, making it seem more exotic than it really is.

I thought that by excising the superfluous components one would obscure less. I assert that the octonians are more exotic and esoteric to the lay reader than say generalizing rotations to 8 or even 64 dimensional space. We don't teach vector calculus using quaternions. Once discovered you find them a cool way to represent rotations in 3-space and 4-space. But they are not less exotic they are more so. We can use quaternions due to the coincidence that the third clifford algebra = fourth even clifford sub-algebra is isomorphic to the quaternions. One is using the quaternions as the corresponding clifford algebra and when one extends to higher dimensional calculus one must follow those clifford algebras, not the short finite chain of division algebras.

 

[mitchell porter]

What's actually fermionic about any of these constructions?

 

[kodama]

What's actually fermionic about any of these constructions?

A simple geometric algebra is shown to contain automatically the leptons and quarks of a family of the Standard Model, and the electroweak and color gauge symmetries, without predicting extra particles and symmetries. The algebra is already naturally present in the Standard Model, in two instances of the Clifford algebra Cℓ6, one being algebraically generated by the Dirac algebra and the weak symmetry generators, and the other by a complex three-dimensional representation of the color symmetry

arXiv:1702.04336

leptons and quarks of a family of the Standard Model,fermionic

 

[kodama]

What's actually fermionic about any of these constructions?

have you had a opportunity to get to read the articles

We show that three generations of leptons and quarks with unbroken Standard Model gauge symmetry SU(3)c×U(1)em can be described using the algebra of complexified sedenions C⊗S.

leptons and quarks are fermionic

 

[mitchell porter]

have you had a opportunity to get to read the articles

We show that three generations of leptons and quarks with unbroken Standard Model gauge symmetry SU(3)c×U(1)em can be described using the algebra of complexified sedenions C⊗S.

leptons and quarks are fermionic

I will give a crude analogy. By one way of counting there are a dozen elementary fermions in the standard model. In some countries, you can also buy eggs in groups of a dozen. So what if I said, "we show that three generations of leptons and quarks can be described using a carton of a dozen eggs", because you can give each egg in the carton the name of a specific quark or lepton. Nonetheless, there are two problems with suggesting that quarks and leptons are really eggs: first, actual eggs have properties that have to be ignored or removed from consideration; second, actual quarks and leptons have properties that the eggs don't have, which have to somehow be added.

@jambaugh has pointed out that certain essential properties of octonions apparently go unused in these "models". That's the first problem in action. I'm asking about the second kind of problem: actual quarks and leptons are fermions. Fermion fields, at least in a path integral, are Grassmann-valued, octonions aren't. How does fermion-ness enter the picture? Is it supposed to arise from anti-commutativity?

 

[kodama]

I will give a crude analogy. By one way of counting there are a dozen elementary fermions in the standard model. In some countries, you can also buy eggs in groups of a dozen. So what if I said, "we show that three generations of leptons and quarks can be described using a carton of a dozen eggs", because you can give each egg in the carton the name of a specific quark or lepton. Nonetheless, there are two problems with suggesting that quarks and leptons are really eggs: first, actual eggs have properties that have to be ignored or removed from consideration; second, actual quarks and leptons have properties that the eggs don't have, which have to somehow be added.

@jambaugh has pointed out that certain essential properties of octonions apparently go unused in these "models". That's the first problem in action. I'm asking about the second kind of problem: actual quarks and leptons are fermions. Fermion fields, at least in a path integral, are Grassmann-valued, octonions aren't. How does fermion-ness enter the picture? Is it supposed to arise from anti-commutativity?

What about taking the standard model as it is, with Fermion fields that are Grassmann-valued, and then explaining 3 generations in terms of octonions, the nature of flavor.

so to use your crude analogy, we have 1 generation of fermions with the right properties, and in order to explain why there is flavor and 3 generations, these papers invoke octonions, as well as connections with SU(3) to 3 generations and 3 colors in QCD

 

[mitchell porter]

What about taking the standard model as it is, with Fermion fields that are Grassmann-valued, and then explaining 3 generations in terms of octonions, the nature of flavor.

so to use your crude analogy, we have 1 generation of fermions with the right properties, and in order to explain why there is flavor and 3 generations, these papers invoke octonions, as well as connections with SU(3) to 3 generations and 3 colors in QCD

Why would that give you three generations, rather than eight?

Besides, that's not what any of these papers claim to do, is it? They are supposed to work in some other way.

Meanwhile, today we have yet another paper, this one claiming to get one generation from an octonionic derivation of E8.

https://arxiv.org/abs/2204.05310
Octions: An E8 description of the Standard Model
Corinne A. Manogue, Tevian Dray, Robert A. Wilson
[Submitted on 11 Apr 2022]

We interpret the elements of the exceptional Lie algebra e₈₍₋₂₄₎ as objects in the Standard Model, including lepton and quark spinors with the usual properties, the Standard Model Lie algebra su(3)+su(2)+u(1), and the Lorentz Lie algebra so(3,1). The resulting model naturally contains GUTs based on SO(10) (Georgi--Glashow), SU(5) (Georgi), and SU(4)×SU(2)×SU(2) (Pati--Salam). We then briefly speculate on the role of the remaining elements of e₈, and propose a mechanism leading to exactly three generations of particles.

 

[kodama]

Why would that give you three generations, rather than eight?

Besides, that's not what any of these papers claim to do, is it? They are supposed to work in some other way.

Meanwhile, today we have yet another paper, this one claiming to get one generation from an octonionic derivation of E8.

https://arxiv.org/abs/2204.05310
Octions: An E8 description of the Standard Model
Corinne A. Manogue, Tevian Dray, Robert A. Wilson
[Submitted on 11 Apr 2022]

Why would that give you three generations, rather than eight?

Besides, that's not what any of these papers claim to do, is it? They are supposed to work in some other way.

Meanwhile, today we have yet another paper, this one claiming to get one generation from an octonionic derivation of E8.

https://arxiv.org/abs/2204.05310
Octions: An E8 description of the Standard Model
Corinne A. Manogue, Tevian Dray, Robert A. Wilson
[Submitted on 11 Apr 2022]

I was referring to this claim by N Fuery

"Here we identify an su(3)⊕u(1) action which splits this 64C-dimensional space into complexified generators of SU(3), together with 48 states. These 48 states exhibit the behaviour of exactly three generations of quarks and leptons under the standard model's two unbroken gauge symmetries."the paper you cite states"We then briefly speculate on the role of the remaining elements of e8, and propose a mechanism leading to exactly three generations of particles."so to use your analogy, instead of eggs, we definitely have quarks and leptons, with the properties of quarks and leptons, and now we want to know why there are 3 generations.Is octonions a viable way to explain why exactly three generations of particles, given if you write down quarks and leptons from the standard model.

these papers on octonions by different researchers claim octonions explains why there are 3 generations - so it seems perhaps quarks and leptons with the right properties are a given.

 

[mitchell porter]

I'm not sure what else to say. None of these papers actually constructs a quantum field theory. They play with algebraic objects and obtain some of the groups and algebras employed in the standard model. They do not explain how they will go further and obtain an actual field theory, and in some cases there are obvious problems with their methods (e.g. when they want to put bosons and fermions in the same multiplet, but using only a purely bosonic symmetry; you need an algebra with fermionic elements, i.e. a superalgebra, in order to turn bosons into fermions or vice versa). The web of relationships among small groups and algebras is as dense as the web of relationships among the numbers from 1 to 10, and just because someone finds a mathematical relationship involving groups from the standard model, that doesn't mean it's physically relevant.

What's really, really missing from these papers, is construction of any kind of quantum theory. For example, a Hilbert space of quantum states, with a Schrodinger equation of motion, and observable properties given by eigenvalues of self-adjoint operators. If the objects they study are actually quarks and leptons, why can't they talk about those objects in a properly quantum way? @jambaugh and I have already given some reasons - some of the mathematically essential properties are physically irrelevant, some of the physically essential properties are mathematically missing.

There have definitely been sophisticated attempts to make quantum theories out of higher division algebras. @jambaugh #5 mentions quaternions. Baez and others have pursued octonionic algebrology in the context of the exceptional Jordan algebra of 3x3 self-adjoint octonionic matrices, which resembles a quantum operator algebra. But the papers above don't do this.

 

[kodama]

so what do you think is the most promising research that explains why there are 3 generations of fermions in the standard model?

 

[kodama]

I'm not sure what else to say. None of these papers actually constructs a quantum field theory. They play with algebraic objects and obtain some of the groups and algebras employed in the standard model. They do not explain how they will go further and obtain an actual field theory, and in some cases there are obvious problems with their methods (e.g. when they want to put bosons and fermions in the same multiplet, but using only a purely bosonic symmetry; you need an algebra with fermionic elements, i.e. a superalgebra, in order to turn bosons into fermions or vice versa). The web of relationships among small groups and algebras is as dense as the web of relationships among the numbers from 1 to 10, and just because someone finds a mathematical relationship involving groups from the standard model, that doesn't mean it's physically relevant.

What's really, really missing from these papers, is construction of any kind of quantum theory. For example, a Hilbert space of quantum states, with a Schrodinger equation of motion, and observable properties given by eigenvalues of self-adjoint operators. If the objects they study are actually quarks and leptons, why can't they talk about those objects in a properly quantum way? @jambaugh and I have already given some reasons - some of the mathematically essential properties are physically irrelevant, some of the physically essential properties are mathematically missing.

There have definitely been sophisticated attempts to make quantum theories out of higher division algebras. @jambaugh #5 mentions quaternions. Baez and others have pursued octonionic algebrology in the context of the exceptional Jordan algebra of 3x3 self-adjoint octonionic matrices, which resembles a quantum operator algebra. But the papers above don't do this.

"What's really, really missing from these papers, is construction of any kind of quantum theory."

Okay I understand what is missing.

Could you combine ideas and constructions of these papers with existing theories that have
a quantum theory, for example, and so what is missing in those papers is combined with papers that do have a quantum field theory

one possible example is,

Euclidean Twistor Unification​

Peter Woit

Taking Euclidean signature space-time with its local Spin(4)=SU(2)xSU(2) group of space-time symmetries as fundamental, one can consistently gauge one SU(2) factor to get a chiral spin connection formulation of general relativity, the other to get part of the Standard Model gauge fields. Reconstructing a Lorentz signature theory requires introducing a degree of freedom specifying the imaginary time direction, which will play the role of the Higgs field.
To make sense of this one needs to work with twistor geometry, which provides tautological spinor degrees of freedom and a framework for relating by analytic continuation spinors in Minkowski and Euclidean space-time. It also provides internal U(1) and SU(3) symmetries as well as a simple construction of the degrees of freedom of a Standard Model generation of matter fields. In this proposal the theory is naturally defined on projective twistor space rather than the usual space-time, so will require further development of a gauge theory and spinor field quantization formalism in that context.

Comments:	48 pages, 1 figure
Subjects:	High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc)
Cite as:	arXiv:2104.05099 [hep-th]

 

[mitchell porter]

Euclidean Twistor Unification
Peter Woit

That proposal doesn't have a quantum theory worked out either. Note the abstract: "... will require further development of a gauge theory and spinor field quantization formalism".

 

[kodama]

That proposal doesn't have a quantum theory worked out either. Note the abstract: "... will require further development of a gauge theory and spinor field quantization formalism".

so...twistor theory has been around all these years (decades?) since Penrose proposed it and including Edward Witten and no one working on twistor theory ever worked a quantum theory worked out for this, including twistor string theory or twisted geometries?

 

[mitchell porter]

so what do you think is the most promising research that explains why there are 3 generations of fermions in the standard model?

I won't say that it is the most promising of all; but given the other ingredients emphasized in this thread - octonions, now twistors - I would suggest Hermann Nicolai's quest to obtain the standard model from a deformation of N=8 supergravity. N=8 supergravity was a theory-of-everything focus among people like Gell-Mann, before the 1984 superstring revolution, and Nicolai is its leading proponent (as a description of the world) these days.

In this case, the three generations are to come from the SU(3) flavor symmetry.

https://arxiv.org/abs/1412.1715
Standard Model Fermions and N=8 supergravity
Krzysztof A. Meissner, Hermann Nicolai
[Submitted on 4 Dec 2014 (v1), last revised 23 Feb 2015 (this version, v2)]
In a scheme originally proposed by M. Gell-Mann, and subsequently shown to be realized at the SU(3)xU(1) stationary point of maximal gauged SO(8) supergravity by N. Warner and one of the present authors, the 48 spin 1/2 fermions of the theory remaining after the removal of eight Goldstinos can be identified with the 48 quarks and leptons (including right-chiral neutrinos) of the Standard Model, provided one identifies the residual SU(3) with the diagonal subgroup of the color group SU(3)_c and a family symmetry SU(3)_f. However, there remained a systematic mismatch in the electric charges by a spurion charge of ±1/6. We here identify the `missing' U(1) that rectifies this mismatch, and that takes a surprisingly simple, though unexpected form.

 

[kodama]

I won't say that it is the most promising of all; but given the other ingredients emphasized in this thread - octonions, now twistors - I would suggest Hermann Nicolai's quest to obtain the standard model from a deformation of N=8 supergravity. N=8 supergravity was a theory-of-everything focus among people like Gell-Mann, before the 1984 superstring revolution, and Nicolai is its leading proponent (as a description of the world) these days.

In this case, the three generations are to come from the SU(3) flavor symmetry.

"In this case, the three generations are to come from the SU(3) flavor symmetry."

could this work without SUSY? i.e. just the standard model and add SU(3) flavor symmetry

 

[mitchell porter]

no one working on twistor theory ever worked [out] a quantum theory ... ?

Plenty of twistor quantum theories exist. It's Woit's specific proposal which lacks a quantum theory.

"In this case, the three generations are to come from the SU(3) flavor symmetry."

could this work without SUSY? i.e. just the standard model and add SU(3) flavor symmetry

There are various ways to do this. The three generations are the same except for masses and mixings, so you can say that the standard model has an SU(3) flavor symmetry broken by the yukawa matrices. You could also posit a new SU(3) flavor gauge field whose bosons are heavy (like W and Z, but a lot heavier).

But this isn't really explaining why three generations rather than some other number. It's just elaborating on the unexplained three-ness that is already there. Whereas if Nicolai's program worked out, the SU(3) flavor symmetry would genuinely be a side effect of something else. The starting axiom would be "let there be an N=8 extended supersymmetry", and the resulting fermions would have the three-generation structure. Incidentally, in this scheme, supersymmetry doesn't consist of adding superpartners to everything in the standard model - instead, the existing particles would be tied to each other by a web of super-relations.

However, so far this hasn't worked out. None of the known vacua of N=8 supergravity actually give you the standard model. Nicolai hopes you can do it by deforming the algebra somehow.

 

[kodama]

Plenty of twistor quantum theories exist. It's Woit's specific proposal which lacks a quantum theory.

Couldn't Woit's specific proposal reuse already existing twistor quantum

 

[kodama]

There are various ways to do this. The three generations are the same except for masses and mixings, so you can say that the standard model has an SU(3) flavor symmetry broken by the yukawa matrices. You could also posit a new SU(3) flavor gauge field whose bosons are heavy (like W and Z, but a lot heavier).

But this isn't really explaining why three generations rather than some other number. It's just elaborating on the unexplained three-ness that is already there. Whereas if Nicolai's program worked out, the SU(3) flavor symmetry would genuinely be a side effect of something else. The starting axiom would be "let there be an N=8 extended supersymmetry", and the resulting fermions would have the three-generation structure. Incidentally, in this scheme, supersymmetry doesn't consist of adding superpartners to everything in the standard model - instead, the existing particles would be tied to each other by a web of super-relations.

However, so far this hasn't worked out. None of the known vacua of N=8 supergravity actually give you the standard model. Nicolai hopes you can do it by deforming the algebra somehow.

you got me wondering, what would it take for a proposal to explain the standard model's 3 generations, flavor physics, to be generally accepted by HEP

"Nicolai hopes you can do it by deforming the algebra somehow."

would using the algebra of octonions help?

btw do you know anything about the entropy of extremel black holes?

 

[mitchell porter]

Couldn't Woit's specific proposal reuse already existing twistor quantum

A recent comment from him.

what would it take for a proposal to explain the standard model's 3 generations, flavor physics, to be generally accepted by HEP

It would help if it made correct new predictions.

 

[kodama]

A recent comment from him.

It would help if it made correct new predictions.

what if a theory doesn't make new predictions but can retroactively explain fermion masses?

I understand that electrons muons and tau differ in flavor, and their masses is determined by Yukawa couplings between them and the Higgs. The Higgs field interactions more strongly with tau than muons, which in term is stronger than electrons.

does Nicolai's program which explain 3 generations also explain why the electron and muon and tau have the masses and thus Yukawa couplings they do?

I also wonder if the octonion proposals by Baez et al also address this.

 

[mitchell porter]

what if a theory doesn't make new predictions but can retroactively explain fermion masses?

It depends on the nature and quality of the explanation. There are all kinds of beyond-standard-model theories which are still full of free parameters, but which try to explain some of the qualitative patterns in the particle masses and mixings, e.g. the endless papers on yukawa "textures" which try to explain patterns in the yukawa mass matrices as arising from simple new symmetries. A theory like that doesn't explain the exact values of the fermion masses, but rather something like, why the third generation is much heavier than the others. Such explanations may not be that compelling, but one of them could turn out to be true.

Then on the other hand we have higher-precision formulas which aim at explaining more of the numerical details of the masses. Most such formulas are dismissed as "numerology" because they are not a part of an actual physical theory (with equations of motion, etc). The Koide formula is one that we keep coming back to on this forum.

None of the papers discussed so far in this thread, explain fermion masses in either of these ways. As I mentioned, most of the papers don't even describe quantum field theories, just vague correspondences between particular algebraic objects and particular sets of particles. N=8 supergravity is an actual field theory, but the attempt to make it look like the standard model has yet to even produce something like a Higgs mechanism, let alone yukawa mass matrices, let alone yukawa mass matrices with the right form.

In the main Koide thread, a paper was mentioned which is trying to obtain fermion masses from an octonionic framework. Despite what the abstract says, it doesn't actually predict the fermion masses. Instead, they are trying to obtain the fermion masses from combinations of a certain limited set of numbers (certain matrix eigenvalues). As such it is a work of physics numerology, but they would hope to turn the numerology into a physical proposition within their framework.

 

[kodama]

Has there been any proposals along the lines that flavor comes in "charges" of discrete quantized units.

first generation fermions have 1 unit of flavor charge, second generation 2, and third generation 3.

so for example, an electron would have 1 1 unit of flavor charge, muon 2 unit of flavor charge, tau 3 unit of flavor charge.

3 is the most unit of flavor charge an elementary particle may hold, and hence 3 generations.

more flavor charges means it binds to the Higgs field more strongly which is why the third generation is heaviest.

perhaps there are additional reasons besides the Higgs field that contribute to particle masses.

 

[mitchell porter]

Has there been any proposals along the lines that flavor comes in "charges" of discrete quantized units.

Froggatt-Nielsen models are a well-known approach with a flavor charge, but they are very complicated.

A yukawa coupling in the standard model, links two chiral fermions via the Higgs field. In a Froggatt-Nielsen model, each chiral fermion has an integer flavor charge. There's also a "flavon" field, which has a nonzero condensate analogous to the Higgs field. There are also extra fermions, called messenger fermions, also carrying flavor charge.

The size of the yukawa coupling in the standard model, is explained in a Froggatt-Nielsen model, as (I think) the ratio of the flavon condensate scale to the mass of the messenger fermions (call it ϵ), raised to the power of the difference in flavor charges between the two chiral fermions (e.g. n_u - n_d). So the 3x3 yukawa matrix is an array of terms of the form ϵ^(n₁-n₂).

For what this looks like as Feynman diagrams, see figure 6, page 33, in this paper. F and G are the messenger fermions, and s is the flavon. So the pointlike standard model yukawa interaction, resolves into a series of interactions with the flavon field, each of which changes the flavor charge by 1. So e.g. if up quark has flavor charge -1 and down quark has flavor charge +1, there should be a series of flavon interactions which go: up quark with flavor charge -1, messenger with flavor charge 0, down quark with flavor charge +1. That is what would be happening "inside" the interaction that the standard model represents as an up quark-down quark yukawa coupling.

If it sounds unattractively complicated, I agree. But perhaps there are simpler ways to make it work.

 

[kodama]

interesting.

so do all seriously considered proposals of flavor and 3 generations require BSM physics including new particles and fields? there's no proposals to explain 3 generations, flavor, and yukuwa couplings solely with just the known standard model physics?

btw what other BSM topics do you like to comment on this forum?

 

[mitchell porter]

so do all seriously considered proposals of flavor and 3 generations require BSM physics including new particles and fields? there's no proposals to explain 3 generations, flavor, and yukuwa couplings solely with just the known standard model physics?

From a standard model perspective, the generation structure is a postulate, and the couplings are free parameters, so you do need "something more" to explain all that; and yes, new fields are usually part of the "something more". I would be very surprised if you could explain all the yukawas e.g. just from fixed points (attractors of RG flow) and flavor symmetries, which are two of the ways in which parametric freedom of couplings in a QFT might be constrained.

Actually, I just remembered that Gerard 't Hooft had the idea that coupling the standard model to conformal gravity would only allow certain values of the standard model parameters. So there would be a discrete landscape of possibilities, similar to string theory, but without all the extra string states... But I don't think 't Hooft had any good evidence for this.

edit: the paper by 't Hooft

 

[kodama]

From a standard model perspective, the generation structure is a postulate, and the couplings are free parameters, so you do need "something more" to explain all that; and yes, new fields are usually part of the "something more". I would be very surprised if you could explain all the yukawas e.g. just from fixed points (attractors of RG flow) and flavor symmetries, which are two of the ways in which parametric freedom of couplings in a QFT might be constrained.

Actually, I just remembered that Gerard 't Hooft had the idea that coupling the standard model to conformal gravity would only allow certain values of the standard model parameters. So there would be a discrete landscape of possibilities, similar to string theory, but without all the extra string states... But I don't think 't Hooft had any good evidence for this.

edit: the paper by 't Hooft

what is the idea of "explain all the yukawas e.g. just from fixed points (attractors of RG flow) and flavor symmetries,"
what are your favorite ideas on this topic?

is there

 

[mitchell porter]

I feel like I should state some mainstream perspectives on these questions first. Not to necessarily endorse them, but just as conceptual orientation... The mainstream perspective is that you have field theory to describe most physics, and field theories have lots of coefficients which are just free parameters; and then string theory as a fundamental theory with a large landscape of ground states, each of which is approximated by some field theory whose coefficients are determined by string geometry (Calabi-Yau etc).

The old paradigm was that the world is in a string theory ground state with weakly broken supersymmetry that keeps the Higgs mass natural. The newer paradigm (e.g. as advocated by Arkani-Hamed) is that supersymmetry still exists, but at higher scales that play no role in making the Higgs natural; instead, the Higgs, along with the cosmological constant, is low for anthropic reasons.

Within this slight adjustment to the old paradigm, the yukawas (and thus masses and mixings) are still to be explained in the same way, as ultimately determined by string geometry, e.g. distances and angles between branes.

I mentioned fixed points and flavor symmetries as something that could affect the yukawas. A fixed point is an attractor for the running of a coupling. Here is a 2018 paper which claims that the masses of all the heavy standard model fermions arise as fixed points in a slight extension of the supersymmetric standard model. This would mean that for a particular family of field theories, the high-energy details (e.g. of string geometry) are not even needed, because the choice of gauge group and field representations alone, creates a renormalization flow tending towards a particular spectrum of masses.

As for flavor symmetries, at the level of field theory, those are extra postulated symmetries that can force yukawas to have certain relations with each other, and which again could have a geometric origin in string theory (e.g. see threads here on modular symmetry).

For our discussion here, one theme of which is the attempt to explain everything just with field theory, and maybe even with only a minimal extension to the standard model... my point is that fixed points and flavor symmetries are meaningful just at the level of field theory, as mechanisms which can constrain the allowed values of all these couplings; but if you work only with the standard model fields, those mechanisms don't seem to be enough to explain everything.

 

[jambaugh]

Pardon the random interjection on an earlier question. I thought I'd bring up the Weinberg (weak mixing) angle as a classic a posteriori prediction of a new theory. It is very closely predicted by certain GUT models (SU(5) models specifically if my memory serves). I would consider this significant retroactive evidence for them.

Unfortunately, the GUT models also predict proton decay which we've now checked with quite strong negative results, enough so that we lean towards ruling out conventional GUT field theories.

Taken together this, to my mind suggests our next refinement must occur at a deeper level (and I have some preliminary ideas but they are too half-baked to mention here.)

The main point I would make is that one may charge forward with speculative explanations for the ad hoc structure of current orthodox theories but it is comparable to tracking a car on a side road after taking a wrong fork. You may feel you're getting closer and that feeling may be valid. But, to actually progress we need to apply the level of methodological rigor which becomes the map to see how one can "get there from here" so to speak. Epiphanies are wonderful except when they are false. The true and false epiphanies feel the same but you have to test them against reality to distinguish them.

 

[ohwilleke]

interesting.

so do all seriously considered proposals of flavor and 3 generations require BSM physics including new particles and fields? there's no proposals to explain 3 generations, flavor, and yukuwa couplings solely with just the known standard model physics?

btw what other BSM topics do you like to comment on this forum?

There is no deeper theory in the Standard Model to explain why its two dozen or so experimentally measured independent degrees of freedom have the values they do (a few parameters of the Standard Model are related to each other in electroweak unification, so there are a few more parameters than there are independent degrees of freedom). These are not derived quantities.

There is also no deeper theory to explain why there are three generations of fundamental fermions (although there are some fairly good arguments from the Standard Model and available experimental data to strongly suggest that there are exactly three generations, rather than more than three), or why there are three color charges (indeed lattice QCD is routinely done using various non-SM numbers of quark flavors and color charges to show trend lines that can be extrapolated to a world with the SM number of quarks and color charges, we can calculate what the world were like it these quantities were different).

The Standard Model's equations do imply that there should be an up-type quark, a down-type quark, a charged lepton, and an electroweak neutrino state in each generation, so if, for example, we discovered a fourth generation active left handed neutrino of 50 GeV of mass, if the Standard Model is correct, this would imply the existence of an entire fourth generation.

The number of gluon color combinations (eight) flows from the fact that there are three kinds of quark color charge and the SU(3) group of the strong force. The existence of antiparticles is likewise predicted by the Standard Model.

 

[kodama]

what would a deeper theory to explain why there are three generations of fundamental fermions candidate resemble ?

 

[ohwilleke]

what would a deeper theory to explain why there are three generations of fundamental fermions candidate resemble ?

I think that it is probably easier to come up with elaborations adding additional content to the Standard Model that make more of its free parameters into derived ones, than it is to explain why there are exactly three generations of fermions, although the latter question might be better defined and more suggestive of an answer if you solved the first one.

The structure of the CKM matrix and PMNS matrix and a few other observations provide some hints at the nature of how the SM could be extended to explain it more deeply and with fewer free parameters. Connecting the dots of these hints is a task that has evaded a legion of geniuses for half a century, but some of the more notable hints can be fairly easily stated.

Some of the hints that I find notable include:

* All of the fundamental particles in the Standard Model that interact via the weak force have rest mass, all of the fundamental particles in the Standard Model that don't interact via the weak force (i.e. photons and gluons) are massless. This would be true in a theory of everything that added a massless graviton as well. This suggests that fundamental particle mass is a weak force mediated process which the Higgs field mechanism of mass generation in the Standard Model basically is, even though it passes the buck on the question of why the fundamental particle masses are what they are by replacing it with the question of why the fundamental particle Yukawas are what they are. The Higgs mechanism is basically a part of the electroweak portion of the Standard Model.

* The formula 2M_W + M_Z = 2M_H would predict a Higgs boson mass of 125.95 GeV (using the global electroweak fit value for the W boson mass) which is inconsistent with the measured value of 125.25 ± 0.17 GeV, a 4.1 sigma difference (the CMS experiment uses a value of 125.38 GeV in its papers to make predictions). But, if one could imagine that the W and Z boson masses were related to the Higgs boson mass via some sort of process with negative binding energy or loop effects or some sort of shielding could tweak this by 0.7 GeV or so, with this equation as a first order tree level term. Thus, the Higgs boson mass might have an origin in the sum of these masses. In some supersymmetry theories this kind of correction to produce a Higgs boson mass from the vector boson masses is present with a correction on this order of magnitude.

* The neutrino oscillation described by the PMNS matrix can be described as a W boson mediated process. It is fair to think of both the CKM matrix and the PMNS matrix as properties of the W boson.

* There is an extension of Koide's rule that might work for the neutrinos although without absolute mass values or ratios of absolute mass values for them (which we lack) it is hard to tell if this is correct and it is different in form from Koide's rule and its simplest extension of quarks which is first order of magnitude correct but not exact.

* The sum of the square of the fundamental Standard Model constants is consistent to within two sigma with the square of the Higgs vacuum expectation value, which is a function of Fermi's constant, which is a function of the W boson mass and the weak force coupling constant. Put another way, the Higgs vacuum expectation value seems to set the overall scale of the fundamental fermion (and fundamental boson) masses. (Uncertainty in the relationship is dominated by uncertainty in the top quark mass and to a lesser extent, the Higgs boson mass.)

* Koide's rule works perfectly to the limits of experimental measurement for the charged leptons (possibly, in part, because the neutrino masses are too small to make a meaningful tweak to their masses), in what as an ex ante prediction at the time since the tau lepton mass was not known very precisely when it was formulated and the electron and muon masses were known less precisely than they are now.

* An extension of Koide's rule is a good first approximation of the quark masses. You can consider t-b-c, b-c-s, c-s-u, and s-u-d, that alternate up and down type quarks as would happen in two sequential W boson transformations.

* One way to think about the extended Koide's rule for quarks is that they are focused on the middle quark in the triple and omit one possible decay. So, for example, the triple triple t-b-c is really encoding the possible W boson mediated transformations of the b quark which are b-t, b-c, and b-u.

* The extended Koide's rule triples for quarks where the CKM matrix element for the omitted W boson transformation is smallest are the closest to the perfect Koide rule value. For example, the most accurate triple is t-b-c in which the omitted W boson interaction is b-u which has a best fit CKM matrix value of 0.00382 corresponding to a probability of a b-u decay (to three significant digits) when all decays are conservation of energy-matter permitted of 0.00146%.

* When the omitted W boson interaction from the triple is greatest, the extended Koide's rule is much less accurate, but adjusting the extended Koide's rule value by the probability of the omitted W boson transformation (i.e. the square of the CKM matrix element omitted) time the mass of the quark in the omitted interactions is much closer to the experimental value.

* This is suggestive of the idea that the relative magnitude of the fundamental fermion Yukawas could arise from a dynamic balancing of the fundamental fermion masses in W boson interactions whose overall magnitude is set by the W boson mass and the weak force interaction coupling. If this idea were developed properly it would make it possible to calculate all of the quark masses from first principles with just a couple of experimentally measured inputs (or less).

* The masses of the first generation fundamental fermions (i.e. the lightest neutrino mass eigenvalue, the electron mass, and the up and down quark masses) are on the order of the self-interactions of those particles via the forces that they interact with. There is also a plausible first principles explanation of electron-neutrino mass in a Majorana mass scenario, although it isn't clear how this generalizes to three generations of neutrinos.

* The CKM matrix elements are not primarily a function of the masses of the particles in the W boson interaction described. Instead, the probability of a first to second generation change is quite similar whether it is a u-s or d-c, the probability of a second to third generation change is quite similar whether it is a c-b or t-s transition, and the probability of a first to third generation change is quite similar to the probability of a first to second generation change times a second to third generation change. This suggests that fundamental fermion generations are in some way more fundamental than fundamental fermion masses.

* The CKM matrix in the Wolfenstein parameterization is described by four parameters, λ, A, ρ, and η which makes the CKM matrix at low order (with the image of the CKM matrix elements and meanings first):

For λ = 0.2257+0.0009 −0.0010, A = 0.814+0.021 −0.022, ρ = 0.135+0.031 −0.016, andη = 0.349+0.015 −0.017.

* The diagonals entries of the CKM matrix differ in sign and complex conjugation, which are nothing to do with the masses of the quarks involved and suggest a deeper structure of the generations.

* Whatever accounts for the generations seems to have not net effect on strong force and electromagnetic force interactions or parity or particle-antiparticle classification, all of which are identical for particles of a particular type of each generation. Particles of the same type are identical at each generation except for mass (their decay properties are derived in the Standard Model and all follow the Standard Model rules for decays).

* Aλ² is equal to (2λ)⁴ at the 0.1 sigma level of precision, and there is no place in the Wolfenstein parameterization of the CKM matrix where this substitution cannot be made. So, except for a CP violating parameter which is a complex number p-iη and its complex conjugate (p+iη), which shows up only in element t-d and b-u up to the third power of λ, the CKM matrix can be described as a function of just one experimentally measured physical constant λ. See, e.g., this paper.

* The relative probability of a second to third generation transition is thus roughly (2λ)⁴, while the probability of a first to second generation transition is roughly λ. So, if you are looking for a theory of the three generations you want one that somehow makes the square root of the second to third generation probability (2λ)⁴ relative to the first to second generation transition probability of λ flow from the structure of the deeper theory.

* There is suggestive evidence that the neutrinos have a normal mass hierarchy, parallel to the quark mass hierarchy, which would be consistent with a single process being in play for the generations of quarks and the generation of leptons.

* There is a hypothesis out there called quark-lepton complementarity, first suggested by Foot and Law in 1990. This flows from the observation that:

This suggests that the two matrixes have a functional relationship with each other due to a symmetry, and that there is in some sense a preferred parameterization of the CKM and PMNS matrixes of the infinite number of possibilities for doing so.

But as measurements of these parameters improve it isn't clear that these approximate relationships hold well, although small discrepancies could be addressed by considering the relevant energy scale at which to evaluate them and by considering higher order loop effects with the simple relationship merely being a tree level one.

If this were to hold true, it would also shed light on the nature of the three generations.

If quark-lepton complementarity of some kind is true, this also means that one could, for example, treat the PMNS matrix parameters in the Standard Model as derived.

* It seems very plausible to me that the source of CP violation in W boson interactions (the only place that they occur in the Standard Model) could have a source independent conceptually from the source of the three generations of the Standard Model, even though they manifest together in the CKM matrix and PMNS matrix. Ideally one could come up with a formula for it in a deeper theory that heavily utilizes know facts and a non-experimentally measured axiom.

If an independent explanation for CP violation in W boson interactions could be worked out, that would simplify the matter of explaining the three generations since the resulting residual CKM and PMNS matrixes with just a single free parameter each (just one single free parameter for both if quark-lepton complementarity exist) would be less complicated mathematically.

* The sum of the square of the masses of the fundamental bosons is larger than the share of sum of the square of the masses of the fundamental fermions (their combined Yukawas add to exactly 1 within margins of error). It isn't implausible to think that the CP violation in W boson interactions could be related to this approximate but not exact fundamental fermion mass to fundamental boson mass symmetry, with their magnitudes conceivably related by some formula.

* Not a single violation of lepton number or baryon number has been observed although the Standard Model predicts it is possible while conserving B-L in ultrahigh energy sphaleron interactions which we have not been able to produce in the labs due to a lack of collider power (it would take a generation of colliders beyond the next post-LHC collider to see it). This is sometimes considered "accidental" but may be more profound and shed light on a deeper theory.

* It is notable that quarks and antiquarks have electromagnetic charges of +/- 1/3 or +/- 2/3, that neutrinos and anti-neutrinos have an electromagnetic charge of 0, and the charged leptons and W bosons have electromagnetic charges +/- 1, is suggestive of a fundamental electromagnetic charge of +/- 1/3 and a composite picture for fundamental particle structure, but this preon intuition is undermined for reasons discussed below. Likewise the fact that the magnitude of color charges of quarks and of gluons is always the same and that it is in a three quark color, eight gluon type scheme consistent with the SU(3) group is notable, again pointing at a preon-like structure, but also possibly described differently, such as a topological description.

* The fact that the width (i.e. mean lifetime) of the top quark is very close to the width of the W boson which it transforms to another kind of quark is very close (the top quark is only slightly more long lived than the W boson), that width is a derived quantity in the Standard Model which would imply a much shorter mean lifetime for a fourth generation quark, and that the Standard Model implies that each generation of fermions must be complete, provides a good Standard Model justification for why there are not mere than three generations, even though it doesn't explain why we don't have only two or only one generation of fermions. This could fit with a composite structure excitation idea, but also, for example, with excitations of a string.

* Hadrons, both bosons (including two valence quark mesons, but also tetraquarks and heaxquarks) with integer spin, and fermions (including ordinary three valence quark baryons, but also pentaquarks) can have excited states (see, e.g. this article, describing the spectrum of excited state protons and neutrons). So, it is tempting to think that the fundamental fermion generations in the Standard Model could be due to excited states of first generation quarks and leptons which are actually composite particles made out of preons, and this avenue of reasoning has been pursued in some detail. But experimental efforts to detect quark or lepton compositeness have not been successful up to some of the highest interaction energies of any of the experimental exclusions of beyond the Standard Model phenomena.

Likewise, if the Standard Model fundamental bosons were composite, we'd expect at least the W and Z bosons to have excited states. But an excited W boson is excluded for masses up to 6 TeV and an excited Z boson is excluded for masses up to 5.1 TeV. By comparison, the mass gap between ground state mesons and their first excited states is much, much smaller these these gaps as a percentage of the ground state mass, again suggesting that the W and Z boson are not composite (even though in the electroweak model we think of the W and Z bosons as a blend of the W₃ and B bosons in spontaneous symmetry breaking quantified by the weak mixing angle.

So, as tempting as it is to think of the three generations of fermions as excited states of a composite particle made of preons, there is good reason not to look for an explanation on that path.

* If these ideas were fleshed out appropriately it is conceivable that you could reduce the number of Standard Model free parameters to six the W boson mass, the three coupling constants, two CKM matrix parameters one real valued and one complex valued to explain CP violation, thus making nineteen more current experimentally measured Standard Model parameters derived: the twelve fundamental fermion masses, the Z boson mass, the Higgs boson mass, four PMNS matrix parameters, and one CKM matrix parameter.

In a gravity theory without a cosmological constant or dark matter or dark energy (which is pretty much necessary in a quantum gravity theory and for which there are several suggested methods by which it could be achieved) you would add one more coupling constant, for a total of seven (eight if complex parameters count as two), down from twenty-nine parameters in the core theory of the Standard Model plus gravity. You would have the W boson mass, four coupling constants, one CKM/PMNS matrix parameter, and one complex valued CP violation parameter.

* Topological explanations, or excitation of strings in string-like theories would seem to make more sense at explaining the existence of exactly three fermion generations than true composite models. The late physicist Marni Shepard had some interesting ideas on this front (see also here) seeing fundamental particles in the Standard Model as ribbons that were braided and twisted in different ways that were topologically distinct.

* FWIW while the Higgs boson is sometimes called the "God particle", the W boson is far more deserving of the name. It takes nine experimentally measured parameters (its mass, four CKM matrix elements, and four PMNS matrix elements) to describe it and is at the heart of a lot of unsolved problems of physics. It is the sole source of CP violation in the SM and GR. It (together with the Z boson) are the only particles that treat left parity and right parity differently in interaction strength making the SM electroweak theory chiral. It is the only means by which SM fundamental fermions can change into other SM fundamental fermions. It is at the core of the question of the existence of three generations. It sets the mass scale of the SM.

 

[kodama]

interesting.

is the flavor of a particle in any way connected to they yukawa couplling of the particle to the Higgs field and three generations? is flavor similar to electric or color charge or mass? is the flavor of a first generation electron the same flavor as a first generation up or down quark, and if so why are their yukawa coupling differ (with the quarks being more massive than an electron)

 

[ohwilleke]

interesting.

is the flavor of a particle in any way connected to they yukawa couplling of the particle to the Higgs field and three generations?

Trivially, yes. Particle generation and type determine the Yukawa coupling of that particle. But there isn't any deeper formula except that higher generation particles of the same type are more massive than lower generation particles of the same type, in a magnitude which Koide's rule predicts exactly to the limits of experimental measurement for charged leptons.

is flavor similar to electric or color charge or mass?

Flavor, as noted above, is related to mass, but discrete (also due to the mass-energy relation E=mc², flavor is not the only source of mass which can also come, for example, from the energy of gluon fields in hadrons).

It is not similar to electric charge or color charge, which are perfectly conserved in the SM. Mass-energy is perfectly conserved (as is CPT), but flavor and mass are not perfectly conserved except incident to conservation of baryon number, lepton number, electromagnetic charge, and mass-energy.

 

[mitchell porter]

All of the fundamental particles in the Standard Model that interact via the weak force have rest mass, all of the fundamental particles in the Standard Model that don't interact via the weak force (i.e. photons and gluons) are massless. This would be true in a theory of everything that added a massless graviton as well. This suggests that fundamental particle mass is a weak force mediated process which the Higgs field mechanism of mass generation in the Standard Model basically is

There is a completely orthodox sense in which you could say that mass generation in the standard model is a "weak isospin mediated process" - which is very close to being weak-force-mediated (though not exactly the same), since weak isospin is the property that is gauged to produce the weak force.

What I'm referring to: the Higgs field has weak isospin and hypercharge - as do all standard model fermions - and the Higgs-yukawa interaction terms in the standard model, are precisely those combinations of Higgs field and fermion fields that are invariant under weak and hypercharge gauge transformations (it is a general principle of gauge theories that the allowed terms are all and only those field combinations which are invariant under the gauge group of the theory).

However, note that these Higgs-yukawa interaction terms, while containing fields charged under the gauge charges, do not contain the gauge fields themselves. The Higgs field "H" and the various fermions "psi", have weak quantum numbers, but the weak gauge field W does not make an appearance in the yukawa terms.

From a conventional field-theoretic perspective, if asked to explain why mass is distributed the way it is in the standard model, one might say: The default state in a gauge theory is for all the elementary fields to be massless, for reasons to do with renormalization. The Higgs mechanism is the one way to add mass at an elementary level in a gauge theory, and in the standard model, this involves particles with electroweak charges because the Higgs field of the standard model happens to carry electroweak charge, but this is a contingent thing; if the Higgs field had carried some other kind of charge, the yukawa couplings would have involved particles with that other charge.

From an unorthodox perspective: several times it has been suggested that the weak gauge field SU(2)_left can be combined with an SU(2)_right from the Ashtekar variables for gravity, into a single gauge field. It was suggested here, and it's also part of Woit's unification proposal. So one might be tempted to suppose that standard model mass always involves an SU(2) charge, because that SU(2) charge is somehow partnered with an SU(2) hidden in the gravitational field. The problem with this concept is that from a conventional perspective, you can couple gravity to any field theory and there's no need to have an SU(2) gauge field included. So probably this particular connection is just a coincidence.

 

[ohwilleke]

Graviweak Unification

Graviweak unification is also discussed in this previous PF thread that cites a number of additional sources. https://www.physicsforums.com/threa...tion-at-ilqgs-online-talk-by-marciano.670299/

Graviweak unification also competes with QCD squared approaches relating gravity in functional form and mathematical properties as a non-Abelian theory to QCD.

This said, unifying gravity and a SM force be it the weak force or the strong force, still doesn't really get you far on the path of explaining why there are exactly three generations of SM fermions.

Other Reasons To Think That There Are More Than Three SM Fundamental Fermion Generations

In addition to the W boson width v. top quark width argument for no more than three generations, there are a couple of strong experimental hints that there indeed aren't more than three generations of SM fermions for whatever the reason.

There is something to the extended Koide's formula which seems to apply quite accurately to the heavier quark masses and the charged leptons. If one extends the formula based upon recent data on the mass of the bottom and top quarks and presumes that there is a b', t, b triple, and uses masses of 173.4 GeV for the top quark and 4.190 GeV for the bottom quark, then the predicted b' mass would be 3.563 TeV (i.e. 3,563 GeV) and the predicted t' mass would be about 83.75 TeV (i.e. 83,750 GeV). These possibilities haven't been experimentally ruled out. But, iof the relationship between decay time for fundamental fermions and mass were extrapolated in any reasonable way to these masses, they would have decay times far shorter than that of the W boson that facilitates this process. Thus, the bar to fourth generation quarks is similar to the physics that prevents top quarks from hadronizing.

Of course, even if Koide's formula is not correct in this domain, it is suggestive of the kinds of masses for fourth generation quarks that one would expect and the estimated masses need not be very precise to give rise to the same conclusion.

Similarly, the extension of a Koide triple for charged leptons (to a muon, tau, tau prime triple) would imply a 43.7 GeV tau prime, which has been excluded at the 95% confidence level for masses of less than 100.8 GeV and with far greater confidence at 45.09 GeV and below, which would otherwise be produced at a significant and easy to measure frequency in Z boson decays and Higgs boson decays and have many other global implications for the SM.

And, since there are no active neutrinos of less than 45,090,000,00 eV, as determined by Z boson decays, and all three of the neutrino masses are very likely all under 0.1 eV (given cosmology limits), the likelihood of a fourth generation of SM neutrino more massive than that seems very low indeed.

So, since fundamental fermion generations, by virtue of symmetries in the equations of the SM, are all or nothing affairs and that one cannot have just three generations of quarks, while having four generations of leptons, for example, the non-detection of fourth generation leptons strongly disfavors the existence of fourth generation SM particles in general.

What are the second and third generations good for? Not much.

A world without second and third generation Standard Model fermions wouldn't be all that different from our own world. A world without third generation Standard Model fermions would be even less different and difficult to distinguish without advanced scientific equipment designed to look for them.

The longest lived of them is the muon at 1O^-6 seconds of mean lifetime. The tau lepton is much shorter lived. Hadrons containing second and third generation fermions are mostly shorter lived, or similar in mean lifetime. Particles, fundamental and composite alike, that last for a microsecond or less are never going to make up a large share of the universe by mass or by number of particles, vis-a-vis particles that are either stable (protons, bound neutrons, electrons, electron neutrinos), or metastable (a free neutron has a mean lifetime of about 15 minutes).

Charm quarks, bottom quarks, top quarks, and tau leptons all have masses of more than 1 GeV which takes lots of collider energy to produce on shell, and there are very few natural processes that give rise to them at any time in the last 13.6 billion years or so.

Strange quarks and muons take less energy to create and are created in nature, but still play a very minor role. Atomic nuclei would decay slightly different because kaons, which contain second generation quarks (strange quarks), are one of the many meson mediators of the residual strong force that binds nucleons into atomic nuclei that is http://www.scholarpedia.org/article/Nuclear_Forces. but to which kaons make a minor contribution (see also here). There are practical applications of muons that were developed in the last fifty years, but not really any natural processes involving muons that would rock our world if they didn't exist.

The differences between a world with three active neutrinos and just one kind of active neutrino would be almost impossible to discern without very advanced scientific equipment and a reason to look for more than one. It would tweak a few cosmology parameters every so slightly, but that isn't anything you could know without state of the art telescopes like Planck and an immense amount of computational power to apply to the data it collects.

Usually physics is reductionist. But, to explain the things that the Standard Model explains it was necessary to create a greatly more complex model of the Universe, which in the case of second and third generation fermions, the hadrons they create, and the CKM and PMNS matrixes that they necessitate, do very little to explain what is going on in Nature when we don't poke and prod it with particle accelerators, and have very little engineering applications (except for the muon).

We could have a perfectly sensible universe with an up quark, down quark, electron, electron neutrino, W boson, Z boson, gluons, photons, and a Higgs boson. Indeed, it is dirty little secret that a lot of BSM theories find a way to do just that while giving no serious thought other than hand waving to how to bring their theories from pre-falsified one generation of fermions theories to real life replicating three generations of fermion theories.

 

[ohwilleke]

A new preprint discusses a flavon theory of the three generations and discusses what would be necessary to detect it. I don't endorse it, but it is one explanation that is out there.

 

[kodama]

Is flavor the sole reason the top quark binds to the higgs field more strongly than a up quark?

does a tau and a top quark have the same flavor or different flavor, as both are same 3rd generation?

 

[ohwilleke]

A very different approach to three generations.

arXiv:2206.02557 (cross-list from physics.gen-ph) [pdf, ps, other]
The Three Faces of U(3)
J. LaChapelle

SU(3)×U(1) as a gauge symmetry is known to be phenomenologically problematic unless the U(1) factor lives in a "dark" or "hidden" sector. By contrast, U(3) is a semi-direct product group that is characterized by non-trivial homomorphisms mapping U(1) into the automorphism group of SU(3). Since the semi-direct product includes mutual coupling among all gauge field mass-energy, U(3) at least has a chance of being phenomenologically sensible as a gauge symmetry outside of any dark or hidden sector. For U(3), there are three different non-trivial homomorphisms that induce three separate defining representations. In a toy model of U(3) Yang-Mills (endowed with a suitable inner product) coupled to massive fermions, this renders three distinct covariant derivatives acting on a single matter field. By employing a mod3 permutation of the vector space carrying the defining representation induced by a "large" gauge transformation, the three covariant derivatives and one matter field can alternatively be expressed as a single covariant derivative acting on three distinct species of matter fields possessing the same U(3) quantum numbers but different renormalized masses. One can interpret this as three generations of matter fields.

Also, from the conclusion, is this little tidbit:

"We did not consider U(2) as a replacement for SU(2) × U(1), but off hand the same mechanism would appear to apply and it should be studied in the context of spontaneous symmetry breaking."

 

[mitchell porter]

the three covariant derivatives and one matter field can alternatively be expressed as a single covariant derivative acting on three distinct species of matter fields

Doesn't make sense to me. Just to be a little clearer about what's going on: instead of considering the usual product of gauge groups SU(3) x U(1) - which is arguably what you have in the standard model after the Higgs breaks the electroweak group - the author wants the gauge group to be the semi-direct product SU(3) ⋉ U(1). Part of making a gauge theory is to construct a covariant derivative based on the gauge group, and the author says there are three ways to do it for SU(3) ⋉ U(1), each of which involves a different representation of the group. (Anyone seriously checking whether this paper makes sense, should also see if the construction of a positive definite inner product in 2.1 also works, but I'll skip that for now.)

The author's idea seems to be that you can start with "one matter field" (as in the quote above) but have it interact with the SU(3) ⋉ U(1) gauge field in three different ways, corresponding to the three different representations. But how is that magically the same as having three matter fields? I don't get it. The way I see it, either it's equal to one matter field interacting with three copies of the gauge field (so the matter field has a different representation under each copy of the gauge group), or it's equal to one matter field interacting with one gauge field in a peculiarly redundant way.

edit: Maybe the idea could be, that the three generation copies of the matter field, would be three different ways that the matter field could be "dressed" with a cloud of virtual particles, according to the three different couplings?? ... More work is required to show that this makes QFT sense.

 

[ohwilleke]

A couple of new Octonion papers:

arXiv:2206.06911 [pdf, other]
An E8⊗E8 unification of the standard model with pre-gravitation, on an octonion-valued twistor space
Priyank Kaushik, Vatsalya Vaibhav, Tejinder P. Singh
Comments: 8 pages, 1 figure, to be presented at the conference 'When ℏ meets G', 27 June-1 July 2022, Institut d'Astrophysique Paris, this http URL
Subjects: High Energy Physics - Phenomenology (hep-ph)
We propose an E8⊗E8 unification of the standard model with pre-gravitation, on an octonionic space (i.e. an octonion-valued twistor space equivalent to a 10D space-time). Each of the E8 has in its branching an SU(3) for space-time and an SU(3) for three fermion generations. The first E8 further branches to the standard model SU(3)c⊗SU(2)L⊗U(1)Y and describes the gauge bosons, Higgs and the left chiral fermions of the standard model. The second E8 further branches into a right-handed counterpart (pre-gravitation) SU(3)grav⊗SU(2)R⊗U(1)g of the standard model, and describes right chiral fermions, a Higgs, and twelve gauge bosons associated with pre-gravitation, from which general relativity is emergent. The extra dimensions are complex and they are not compactified, and have a thickness comparable to the ranges of the strong force and the weak force. Only classical systems live in 4D; quantum systems live in 10D at all energies, including in the presently observed low-energy universe. We account for 208 out of the 496 degrees of freedom of E8⊗E8 and propose an interpretation for the remaining 288, motivated by the trace dynamics Lagrangian of our theory.

AND

[Submitted on 14 Jun 2022]

Octonion Internal Space Algebra for the Standard Model​

Ivan Todorov

Our search for an appropriate notion of internal space for the fundamental particles starts with the Clifford algebra Cℓ10 with gamma matrices expressed as left multiplication by octonion units times a pair of Pauli matrices. Fixing an imaginary octonion unit allows to write 𝕆=ℂ⊕ℂ3 reflecting the lepton-quark symmetry. We identify the preserved unit with the Cℓ6 pseudoscalar, ω6=γ1⋯γ6. It is fixed by the Pati-Salam subgroup of Spin(10), GPS=Spin(4)×Spin(6)/ℤ2, which respects the splitting Cℓ10=Cℓ4⊗̂ Cℓ6, while =12(1−iω6) is the projector on the 16-dimensional particle subspace (annihilating the antiparticles). We express the generators of the subalgebras Cℓ4 and Cℓ6 in terms of fermionic oscillators describing flavour and colour, respectively. The standard model gauge group appears as the subgroup of GPS that preserves the sterile neutrino (identified with the Fock vacuum). The ℤ2-graded internal space algebra  is then included in the projected tensor product: ⊂Cℓ10=Cℓ4⊗Cℓ06. The Higgs field appears as the scalar term of a superconnection, an element of the odd part, Cℓ14, of the first factor. As an application we express the ratio mHmW of the Higgs to the W-boson masses in terms of the cosine of the theoretical Weinberg angle.

Comments:	32 pages, Extended version of a lecture presented at the Workshop Octonions and the Standard Model, Perimeter Institute, Waterloo, Canada, February-May 2021, and at the 14th International Workshop Lie Theory and Its Applications to Physics (LT 14), Sofia, June 2021
Subjects:	High Energy Physics - Theory (hep-th); High Energy Physics - Phenomenology (hep-ph); Mathematical Physics (math-ph)
Report number:	IHES/P/22/01
Cite as:	arXiv:2206.06912 [hep-th]

 

[ohwilleke]

Is flavor the sole reason the top quark binds to the higgs field more strongly than a up quark?

does a tau and a top quark have the same flavor or different flavor, as both are same 3rd generation?

To state the obvious, in the SM, a particle's coupling strength of the Higgs field is proportional to its rest mass, and there are twelve SM rest masses that are derived from Higgs field interactions (six quark masses, three charged lepton masses, the W mass, the Z mass, and the Higgs mass).

For SM fundamental fermions, rest mass is a function of generation, but also of whether a particle is an up-type quark, down-type quark, charged lepton, or neutrino.

By definition, each of the twelve different kinds of SM fermions has a different flavor: "The Standard Model counts six flavours of quarks and six flavours of leptons."

So, a tau and a top quark have different flavors even though they in the same 3rd generation.

Since fermion rest mass in the SM is entirely and uniquely determined by flavor, it is fair to say that flavor is the sole reason that the top quark binds to the Higgs field more strongly than an up quark.

Generation v. Flavor In Quarks v. Charged Leptons

Also, while the tau in the SM is truly just a heavy electron or muon, which is identical in all respects except mass to the electron and muon (a SM rule which is being questioned by some experimental data from the LHC ), this simple description of the differences between generations isn't quite true in the case of the quarks.

Up quarks and charm quarks have the same electromagnetic and strong force properties and couplings as the top quark controlling for mass (and for that matter the same gravitational force properties and couplings as the top quark controlling for mass).

But the CKM matrix elements implicated the up quark, the charm quark, and the top quark, respectively, differ from each other in ways that can only be determined experimentally in the SM. Top quarks almost always transform into bottom quarks. Up quarks very infrequently transform to bottom quarks (even in the absence of conservation of mass-energy limitations), often transform into down quarks, and occasionally transform into strange quarks. Charm quarks most often transform into strange quark, but sometimes transform (mass-energy conservation permitting) into bottom or down quarks.

In contrast, the probability of transformations one one charged lepton into another charged lepton, or of a W or Z boson into a particular generation of charged lepton, are identical (i.e. there is charged lepton universality), even though this doesn't hold for transformations of one kind of neutrino into another generation of neutrino. There is also (as I understand it) a too small to observe SM correction to charged lepton universality predicted transition probabilities, at loop level, to account for indirect transformations of charge lepton flavor via virtual neutrinos, which could oscillate according to the PMNS matrix, mediated by virtual W bosons.

An Aside Re Neutrinos

The SM is agnostic in the source and nature of the neutrino masses. Indeed, there is some pedantic debate over whether neutrino mass is really part of the SM although from a practical perspective, most people would say that it is when they are talking about SM physics predictions.

If neutrinos did couple to the Higg field, they would have such a weak coupling that we would probably never observe Higgs boson decays to neutrinos even if they did, like other SM fundamental particles, have a coupling to the Higg field proportional to their rest mass. I'll illustrate this conclusion with the following back of napkin class calculations.

The branching faction of the charged leptons is for tau-lepton pairs, 6.27% (observed), for muon pairs, 0.021 8% (observed), and for electron-positron pairs, 0.000 000 5% (not yet observed).

Roughly speaking branching fractions ratios are on the same order of magnitude of the square of mass ratios.

Electrons are 511,000 eV v. something on the order of 0.050 eV for the largest neutrino mass eigenstate, a ratio of 10⁷.

This implies that if neutrinos got their mass via the same SM Higgs mechanism that applies to other SM fermions (whether or not that makes sense for other reasons), that a branching fraction from Higgs boson decays for neutrinos on the order of 10¹⁴ smaller than that of the roughly 5*10^-7 for electrons.

This would imply a Higgs boson branching fraction on the order of not more than 10^-21 for any kind of neutrino, when muon pairs with a branching fraction of about 2*10^-2 are at the current experimental detection threshold.

In addition, a smaller proportion of neutrinos passing through a detector are actually seen than a proportion of charged leptons passing through a detector, since neutrinos interact more weakly with other matter, so a Higgs boson decay into neutrinos would not be possible at the same branching fraction that a charge lepton Higgs boson decay can be detected.

Realistically, detector precision would have to improve by a factor of about 10²² or more over current technology to directly observe neutrino decays from Higgs bosons in a statistically significant way.

I don't expect that to happen anytime during the lives of anyone who ever encounters me alive.

 

[kodama]

arXiv:2206.06911 [pdf, other]
An E8⊗E8 unification of the standard model with pre-gravitation, on an octonion-valued twistor space
Priyank Kaushik, Vatsalya Vaibhav, Tejinder P. Singh

I wonder if some variation of this idea could be used with Woit's 4D Euclidean Twistor space.

 

[kodama]

so what determine the flavor of a particle. and how is flavor tied in with generations, since the second and third generation are heavier than first.

 

[mitchell porter]

arXiv:2206.06911 [pdf, other]
An E8⊗E8 unification of the standard model with pre-gravitation, on an octonion-valued twistor space
Priyank Kaushik, Vatsalya Vaibhav, Tejinder P. Singh

I wonder if some variation of this idea could be used with Woit's 4D Euclidean Twistor space.

This paper is almost an ultimate development of trends seen earlier in this thread... One of the contrasts between the mainstream of theory, and the alternatives we are discussing in this thread, is that the mainstream theories may not describe reality, but they are at least mathematically well-defined as full quantum theories; whereas these alternative theories involve wishful thinking, even just at the mathematical level.

Here one should also note a curious superficial similarity with one of the best-known approaches to a theory of everything, the E8xE8 heterotic string in ten dimensions. In this theory, one also has E8xE8 and ten dimensions, but the hoped-for mechanisms are almost completely different.

For the sake of comparison, let me say a bit about how the standard model and gravity were to be obtained from the heterotic string... First of all, in that model, gravity has nothing to do with E8xE8, it comes from a separate gravitational sector of string states. E8xE8 is the gauge sector of the string, all the observed gauge fields must come from within this. But the heterotic string also has N=1 supersymmetry, meaning that there are fermionic superpartners to those E8xE8 gauge bosons, and the observed fermions must come from among those. Finally, an effect associated with compactification reduces one of those E8s to E6, one of the known candidate groups for grand unification. So in the field theory limit, you end up with a GUT with E6 symmetry group and representations (which then must be higgsed down to the standard model), and a second, unbroken E8 sector which might contribute to dark matter. This was the high hope of unification in the late 1980s (e.g. it's the topic of Brian Greene's PhD thesis), before string theorists came up with several other approaches (braneworlds, F-theory), and before the LHC threw the usual conceptions of supersymmetry into doubt.

OK, so what's going on in this paper? It would take quite a few hours of work to chase down everything they are saying, what follows is just what I can make out already.

First, this paper is the work of Tejinder Singh's group in India. I already cited another paper of theirs in comment #25 in this thread. Their work is distinctive and ambitious enough to warrant being singled out. They're trying to do everything from explain quantum mechanics per se (via a version of Adler's "trace dynamics"), to explaining the actual particle masses (see #25).

So, what are they doing with E8xE8? I think at heart it's similar to the chiral graviweak unification mentioned in #38-#39 in this thread. This is the idea that the weak force and gravity can be described together by an SU(2)_L x SU(2)_R gauge theory. The first SU(2) is the weak force, the second SU(2) is gravity in Ashtekar variables... Here, Singh et al want to get SU(3)xSU(2)xU(1) from one E8, and gravity from the other E8. So this is the first significant divergence from string theory's use of E8xE8: in string theory, gravity has a separate origin.

Part II of this paper is called "Branching of E8 x E8", so this is where their plans for the group are described in detail. One E8 is broken to an SU(3) symmetry among generations, times the standard model gauge group. The other one is broken the same way, except that SU(3)xSU(2)xU(1) is here interpreted as a weird "pre-gravitational" gauge group. There's supposed to be an SU(3) of "gravi-gluons" which only matter for gravity on a subatomic scale, and a "U(1)_g" which apparently gives rise to ordinary gravity but I don't see how.

I also don't see where they get their fermions from. In the first paragraph of the paper they say "An important aspect of E8 is that the adjoint representation is the fundamental representation, therefore we are able to write the fermions and the bosons using the same rep". Well, it's true enough that E8's adjoint rep is also its fundamental rep; but a bosonic field is still a different thing from a fermionic field. That's why the heterotic string uses E8 superfields, with fermionic and bosonic components. (This was also a problem of Garrett Lisi's E8 theory: he tried to get bosons and fermions from the same E8 gauge field, but if the field components have different particle statistics, it's no longer an E8 object but instead something simpler.) Singh et al have a section on triality and maybe they think they get fermions that way somehow, but I don't see it, and in general, it seems to be another example (see comment #9) in which alleged "fermions" aren't, and even can't be, true fermions.

There are other peculiar details, e.g. everything is ultimately made of 2-branes in octonionic twistor space, and there's more numerology relating masses to the degrees of gauge groups. I return to my introductory comment, that a theory like this involves a lot of mathematical wishful thinking. Rather than exhibiting a mathematical framework clear enough to be predictive, instead we are given pieces of a jigsaw puzzle, with the hope that the pieces can be assembled into a single well-defined object, while retaining all their desired properties. There's no law against this kind of theorizing, but in this case, the large number of peculiar propositions and leaps of faith make it apriori very unlikely that the whole thing really does hang together.

 

[kodama]

So, what are they doing with E8xE8? I think at heart it's similar to the chiral graviweak unification mentioned in #38-#39 in this thread. This is the idea that the weak force and gravity can be described together by an SU(2)_L x SU(2)_R gauge theory. The first SU(2) is the weak force, the second SU(2) is gravity in Ashtekar variables... Here, Singh et al want to get SU(3)xSU(2)xU(1) from one E8, and gravity from the other E8. So this is the first significant divergence from string theory's use of E8xE8: in string theory, gravity has a separate origin.I also don't see where they get their fermions from. In the first paragraph of the paper they say "An important aspect of E8 is that the adjoint representation is the fundamental representation, therefore we are able to write the fermions and the bosons using the same rep".

There are other peculiar details, e.g. everything is ultimately made of 2-branes in octonionic twistor space, and there's more numerology relating masses to the degrees of gauge groups. I return to my introductory comment, that a theory like this involves a lot of mathematical wishful thinking. Rather than exhibiting a mathematical framework clear enough to be predictive, instead we are given pieces of a jigsaw puzzle, with the hope that the pieces can be assembled into a single well-defined object, while retaining all their desired properties. There's no law against this kind of theorizing, but in this case, the large number of peculiar propositions and leaps of faith make it apriori very unlikely that the whole thing really does hang together.

Do you find Peter Woit's derivation of fermionic fields in his proposal to be satisfactory?

The reason I ask is that this proposal makes use of twistor space and so does Woits, both are Twistor.

Could the twistor theory of Woit theory be some way combined with the use of octonionic twistor space in this paper?

Maybe some combination of Woit's proposal with some form of octonions to get 3 generations, in twistor space.

 

[mitchell porter]

Do you find Peter Woit's derivation of fermionic fields in his proposal to be satisfactory?

The reason I ask is that this proposal makes use of twistor space and so does Woits, both are Twistor.

Could the twistor theory of Woit theory be some way combined with the use of octonionic twistor space in this paper?

Maybe some combination of Woit's proposal with some form of octonions to get 3 generations, in twistor space.

Well, first let's review what's involved in making a fermion field. I can only think of two ways to do it. One is that you take an appropriate classical quantity, treat it as an operator, and then postulate that it is anticommuting. This is standard textbook quantization. The other approach is to have a classically anticommuting quantity and do a path integral over it. This is the approach that requires Grassmann numbers.

Woit wants to start in Euclidean space, and then obtain formulae appropriate for our empirical Minkowski space-time, through analytic continuation of the Euclidean formulae. OK, that's a standard thing, except that in the case of fermion fields, there are technical problems and Woit doesn't like the standard approach to them - see appendices B.2 and B.3 of his 2021 paper. Then on top of this, Woit wants to work in Euclidean twistor space, and you want him to try the octonionic version of this. These seem to be domains where even the basics of quantum field theory haven't been studied much. But at least they are well-defined. The most efficient course of action may be to ask him directly.

I will mention that in mainstream theory, there are objects called supertwistors, which are twistors with some Grassmann components. This must be the standard way to get a fermion field from a twistor path integral. But I don't know if it's compatible with Woit's construction or philosophy.