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.