A Three generations of Fermions from octonions Clifford alegbras

[kodama]

TL;DR

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