B What Are the Key Questions Surrounding SU(3)xSU(2)xU(1) Symmetry Groups?

[Urs Schreiber]

I'm not familiar with the symbol that looks like a C in your formulas. Could you explain it and tell where it comes from?

This is the symbol for inclusion of sets. Here: inclusion of subgroups.

 

[cube137]

Studying GUT SU(5) and SM SU(3)xSU(2)xU(1). I came across neutrino mass and the seesaw mechanism and I have a question. First some background for those not familiar with it (are there?) http://www.quantumfieldtheory.info/TheSeesawMechanism.htm

"1. Background
It may seem unusual to have such low values for masses of neutrinos, when all other particles like electrons, quarks, etc are much heavier, with their masses relatively closely grouped. Given that particles get mass via the Higgs mechanism, why, for example, should the electron neutrino be 105 times or more lighter than the electron, up and down quarks. That is, why would the coupling to the Higgs field be so many orders of magnitude less?

One might not be too surprised if the Higgs coupling were zero, giving rise to zero mass. One might likewise not be too surprised if the coupling resulted in masses on the order of the Higgs, or even the GUT, symmetry breaking scale.

Consider the quite reasonable possibility that after symmetry breaking, two types of neutrino exist, with one having zero mass (no Higgs coupling) and the other having (large) mass of the symmetry breaking scale. As we will see, it turns out that reasonable superpositions of these fields can result in light neutrinos (like those observed) and a very heavy neutrino (of symmetry breaking scale, and unobserved)."

Guys. What I'd like to know is what if there is really no GUT or SU(5) and the SU(3)xSU(2)xU(1) is all there is (maybe programmed). Can we still have Seesaw mechanism between the one neutrino with zero mass (No Higgs coupling) and the other having the mass of the Higgs symmetry breaking scale? If not. When what is the substitute for the small mass of the neutrino? Is there also a Multiverse version just like in the Hierarchy program?

 

[ChrisVer]

Is there also a Multiverse version just like in the Hierarchy program?

is there a multiverse version of the hierarchy problem?

Can we still have Seesaw mechanism between the one neutrino with zero mass (No Higgs coupling) and the other having the mass of the Higgs symmetry breaking scale?

The seesaw mechanism naturally gives you tiny masses. In general you can still have tiny Yukawa couplings. Any rank 5 operator from EFT can give you tiny masses for the neutrinos, but that is not an elegant way to do things.

 

[ohwilleke]

As Urs notes above, the very first step in most GUT models is:

the idea that it is natural to expect that fundamentally the gauge group is a simple Lie group which is spontaneously broken at low energy to the non-simple gauge group we observe.

But, there isn't a whole lot of analysis that goes into why one uses a simple Lie group as opposed to some alternative (well, the assumption that it is "simple" follows almost definitionally from the object of "unification", but that doesn't articulate very well why a "Lie group" in particular should be used).

My question then, is this: Does anyone know of any GUT theorists who work with groups that are "almost" but not quite Lie groups in some very subtle respect?

For example, a Lie group is a group that is also a differentiable manifold. But, what about a group that is also a differentiable manifold, except that (unlike a Lie group) it is subject to a finite number of exception cases (or a minimally infinite number of exception cases such as the zeros of an appropriately defined Reimann Zeta function or a set corresponding to the whole numbers in one dimensional space that becomes increasingly sparse as space-time expands in four or more dimensions) in which the manifold is not differentiable, which probably give rise to singularities (perhaps corresponding physically to the Big Bang and perhaps also to black holes)?

Or, for example, what about a group comparable to a Lie group, but involving some discrete rather than continuous symmetries (a bit along the lines of LQG)?

Or, for example, what if some axiom necessary for it to be a "group", as that is defined in abstract algebra, fails in a few special cases that are outside the domain of applicability of existing physical theories that use Lie groups?

Of course, the point is not that either of these specific examples have any validity (I don't want to go down the long rabbit hole of discussing their validity which is really besides the point). I am merely pointing them out just to provide a flavor of the kind of concept that I am talking about (call them demonstrations of the concept), as I am certainly not claiming to be qualified to be a GUT theorist who can figure out precisely which axioms can be relaxed without screwing up the desirable properties of the group.

The point is that while Nature must be described by something very close to Lie groups (for example, because the SU(3)x(SU2)XU(1) Lie group works for the Standard Model which is so exquisitely confirmed experimentally, and the Lorentz and Poincare Lie groups are likewise important to fundamental physics), perhaps one or two of the several axioms and assumptions necessary to fit the definition of the Lie group is not perfectly true in all cases in Nature in some respect and needs to be relaxed slightly to fit what we observe in Nature. And perhaps, a slightly defective Lie group could resolve the pathologies revealed with previous GUT approaches.

For example, perhaps one of the defects in the otherwise differentiable manifold screws up some aspect of the process that makes proton decay possible in ordinary GUT theories. Or, for example, perhaps this defect that manifests right at the energy scale of gauge unification is what caused the gauge group to be spontaneously broken at energies below the gauge unification GUT scale.

Without some real thought, we wouldn't even know where to look in terms of phenomenology to distinguish between a true Lie group, and an "almost" Lie group with an axiom or two that is slightly relaxed in some respect.

A "close but no cigar" similarity between the fundamental laws of Nature and more naive Lie group GUTs could also explain why predictions (for example, showing similarities between the GUT scale and the Planck scale, regarding estimates of the neutrino mass and regarding the weak mixing angle) arising from naive GUT based reasoning of the kind made by Witten in his 2002 talk linked earlier in the thread at post number 2, end up being close to correct even though most straightforward GUTs don't actually work to reproduce Nature.

Anyway, maybe someone has explored that line of inquiry, and I'd be interested to know if anyone is familiar with any explorations of deviations from that assumption of typical GUT theories. I wouldn't even know what to put into a search engine looking for such a thing as I wouldn't be able to figure out the correct technical buzzwords for it.

 

[ohwilleke]

I have a Q though, since the coupling constants don't seem to simultaneously meet within the SM (something that MSSM was able to achieve), how can we talk for unification of the SM interactions? (if SUSY is not there)

Arguably, the fact that the coupling constants don't unify in the SM is a feature and not a flaw of the SM that recommends it relative to relatively simple SUSY models (SUGRA overcomes the issues I discuss below).

One of the key insights of the asymptotic gravity paper that was used to predict the Higgs boson mass (keeping in mind that it is perfectly possible that this was just a fluke), is that introducing gravity in addition to the other three SM forces to your model slightly tweaks the beta functions by which the coupling constants change with energy scale.

Thus, if your model perfectly unifies the three SM forces in the absence of gravity, it is probably wrong because it is probably also true in that case that the unification will fall apart if you add in gravity to your model and change the beta functions of the coupling constants accordingly. (N.B. it is worth noting that the beta functions of the SM and similar BSM theories can be determined and calculated exactly and do not add any experimentally measured constants to the model. Any discretionary constants in a beta function related to the renormalization scheme appear only at intermediate steps in the process and don't affect the observables that you calculate with the model.)

And, while the SM coupling constants don't perfectly unify, it doesn't take much of a tweak to the beta functions of one or more of them to cause this to happen. A mere 1% tweak in just one of the beta functions, and a smaller tweak if it is distributed among all three beta functions can cause the SM coupling constants to unify at high enough energies, and high energies are just where you would reasonably expect that gravitational contributions would be greatest.

Furthermore, if it turns out that a BSM theory incorporates not just gravity but some other new force such as one relevant primarily to inflation or the self-interactions of dark matter or B/L number violations, all of which would have phenomenology impacts outside the domain of applicability of the SM and beyond our capacity to notice experimentally (but, god forbid, don't let it be the Be8 decay force with a 17 MeV force carrying boson that kodama has been citing in half a dozen threads here lately), and not just the ever elusive quantum gravity, the prospects for coupling constant unification arising at the Planck scale due to tweaks in the various beta functions become positively rosy.

 

[ChrisVer]

Do you have the Higgs-mass-predicting paper that deals with gravity?
I don't know, have people (sucessfully) tried to gauge group-ize gravity?
Then I think any GUT introduces new "forces" let's say... especially if the breakings happen subsequently (you remain with other groups + SM)

 

[ohwilleke]

Do you have the Higgs-mass-predicting paper that deals with gravity?
I don't know, have people (sucessfully) tried to gauge group-ize gravity?
Then I think any GUT introduces new "forces" let's say... especially if the breakings happen subsequently (you remain with other groups + SM)

The paper is http://www.sciencedirect.com/science/article/pii/S0370269309014579

 

[Urs Schreiber]

have people (sucessfully) tried to gauge group-ize gravity?

Gravity has a natural formulation in terms of Cartan geometry over the spacetime symmetry group. This is colloquially called the result of "locally gauging" the spacetime symmetry (see here), but even if one may call the result a kind of gauge theory, it is not a Yang-Mills-type gauge theory like that which governs the other forces. In particular the coupling constants of the Yang-Mills forces are on a different footing than those of gravity.

 

[samalkhaiat]

In particular the coupling constants of the Yang-Mills forces are on a different footing than those of gravity.

This has been long over due problem. May be it is about time to formulate a new gravity theory and get rid of Newton constant which appears in the Einstein action \mathcal{L}^{g}_{EH} \sim \frac{1}{G_{N}} R \ .
This can be done in Yang-Mills type gravity. Getting rid of dimension-full parameters (i.e., scale) has, “recently”, led many people to reconsider Yang-Mills gravity model based on the non-compact group of conformal transformations SO(2,4) \mathcal{L}^{YMg}_{SO(2,4)} \sim \frac{1}{\alpha^{2}_{YM}} F^{2} .
It is still early to tell, but such model comes with immediate bonuses: 1) since the coupling \alpha_{YM} is dimensionless, renormalization may come with less embarrassment compared to \mathcal{L}^{g}_{EH}. 2) observers will no longer need measuring sticks and clocks, all information are obtained by light rays.

We had similar situation in the past: Notice the similarity (in form and problems) between Einstein gravity theory \mathcal{L}^{g}_{EH} and Fermi theory of weak interaction \mathcal{L}^{w}_{F} \sim \frac{1}{G_{F}} J_{\mu}J^{\mu} \ , and recall that Yang-Mills theory of weak interaction \mathcal{L}^{YMw}_{SU(2)} \sim \frac{1}{g^{2}_{YM}} F^{2} \ , proved to be the needed cure for renormalizibility and unitarity.

 

[lpetrich]

Witten observed that another popular GUT group, SU(4)xSU(2)xSU(2) was similar to SO(6)xSO(4) and then to the group of isometries of S5 x S3. ...

I must note that the algebra SU(4) is isomorphic to SO(6) and SU(2)*SU(2) to SO(4).

Some authors distinguish the groups and the algebras by typography: group SO(n) vs. algebra so(n), etc.

I note that the various proposed GUT gauge groups also have numerous subgroups. So if our Universe has an underlying GUT gauge symmetry, the problem is why did it break down to the Standard Model's symmetry group and not some other one. String theory does not seem to have helped. Though one can get at least the spectrum of the Standard Model from string theory, it is far from unique, and I do not know of any progress in resolving that extreme non-uniqueness.

 

[gdixon]

Re:
"Alas, the group SU(3) is not the unit octonions. The unit octonions do not form a group since they aren't associative. SU(3) is related to the octonions more indirectly. The group of symmetries (or technically, "automorphisms") of the octonions is the exceptional group G2, which contains SU(3). To get SU(3), we can take the subgroup of G2 that preserves a given unit imaginary octonion... say e1. This is how Dixon relates SU(3) to the octonions. However, why should one unit imaginary octonion be different from the rest? Some sort of "symmetry breaking", presumably? It seems a bit ad hoc."
=======
The complexified octonions (which I label S = C⊗O) is not a division algebra. Its identity can be resolved into a pair of orthogonal idempotents, and it is from this resolution of the identity that much of what's interesting and beautiful about S arises. This resolution requires a direction in imaginary O space be chosen, and the subgroup of G2 leaving this direction invariant is SU(3). With respect to this resolution S splits into 4 SU(3) multiplets: singlet; antisinglet; triplet; antitriplet. This notion was used by Gürsey at Yale in the 1970s. There's nothing ad hoc about it. It's an ineluctable and beautiful part of the maths. There's nothing vague about it, as two books and numerous papers have attempted to demonstrate ad nauseam. Sigh.

 

[arivero]

This resolution requires a direction in imaginary O space be chosen, and the subgroup of G2 leaving this direction invariant is SU(3)

I wonder if this is related to the branched covering of S4 by CP2. (see atiyah an also google https://www.google.es/search?sourceid=chrome-psyapi2&ion=1&espv=2&ie=UTF-8&client=ubuntu&q=S4 branched covering CP2&oq=S4 branched covering CP2&aqs=chrome..69i57.5235j0j7 ). CP2 fibered with S3 should have an isometry group very as the standard model group.

 

[gdixon]

Well, if you have a house, and inside that house there is a very nice and perfectly functional WC, why go outside and dig a hole in the ground to serve the same purpose, but not as well?

I haven't figured out how to include a quote from a previous comment, but this is in reply to arivero's comment above.

 

[mitchell porter]

gdixon, what do you do for dynamics?

 

[gdixon]

Dynamics, hmm. A possibly poor metaphor: it's like asking Mendeleev, after he presents you with the periodic table, where's the chemistry? A perfectly valid question, and Mendeleev certainly would have been able to answer that. My attitude has always been this:

Adding O to P=C⊗H, yielding T=C⊗H⊗O, results ultimately in an expansion of the Dirac algebra and its associated spinor space, which is where the particle fields reside. As I understand it, one starts building dynamics into Dirac maths by building a Lagrangian density. All the pieces needed to do that are present in T-maths, they're just bigger. So, if one wishes one can construct a Lagrangian density for this more complicated spinor space, and its associated 1,9-spacetime Clifford algebra (the expanded Dirac algebra). I did this in my first book 22 years ago, guaging the result. Very nicely one can read from this all viable particle interactions, for it is not hard to pick out from T^2 the bits that represent individual quarks, leptons, anti-quarks, and anti-leptons.

And then there is the matter of the quarks themselves, and the extra 6 space dimensions, neither of which are seeable in any conventional sense, and both of which are associated with the 6 octonions units that do not occur in the resolution of the identity. As I showed in my last published paper, taking this unseeableness to its logical mathematical conclusion implies that from our 1,3-spacetime we can also not see the anti-matter part of the full 1,9-spacetime, so we appear to live in a universe dominantly matter.

Anyway, I'm retired now. I'll present my last paper at a conference this summer, then I'm dropping the mike.

 

[gdixon]

Keep in mind: what I did was a proof of concept. My hope is that reality is more subtle than this.

 

[kneemo]

"Why SU(3)xSU(2)xU(1)? A truly fundamental theory should explain where this precise set of symmetry groups is coming from.

Greetings

One idea to keep in mind is phase transitions. For example, water in a steam state or liquid state is much more symmetric than an ice state with flaws and cracks. If you rotate the ice with streaks and cracks, it's easy to see you lost rotational symmetry.

Similarly, imagine spacetime itself, as we perceive it now, in an "ice-like" state, with temperatures far cooler than the big bang. The big bang temperature would be akin to "boiling" temperature for spacetime. And this is why one studies quantum gravity, in hopes of finding a deeper theory that can describe the extreme phase transitions expected to occur at black hole event horizons and the big bang itself. Essentially, anywhere one would expect singularities in Einstein's general relativity (akin to an effective spacetime hydrodynamics), this would be the realm of quantum gravity and unified field theory.

Hence, from the perspective of phase transitions, SU(3)xSU(2)xU(1) would be the result of a broken higher symmetry, due to the Universe entering an "ice-like" state after 13.82 billion years. This is reasonable, as my living room is nothing like a black hole event horizon.

The whole point of building huge particle accelerators is to reproduce extreme energies, forcing a phase transition to a more symmetric state for a brief time, and take a snap shot of the resulting symmetry breaking that occurs after a high energy collision.

The appeal of E8, for example, is that it is a unique Lie algebra (mathematically) which gives elegant rules for scattering a robust set of particles (bosonic & fermionic), in a closed manner, where the observed inelegant symmetries of the standard model can be seen as part of a larger symmetric whole.

The quest continues...

 

[gdixon]

https://www.quantamagazine.org/20161215-proton-decay-grand-unification/

 

[gdixon]

PS

"Keep in mind: what I did was a proof of concept. My hope is that reality is more subtle than this."

By this I do not intend to cast doubt on the necessity of C⊗H⊗O as a basis
for any viable TOE. I have no doubt. None. The algebra is necessary
because parallelizable spheres are necessary. And once you have it,
the mathematics gives you everything else.

http://7stones.com/7_new/7_Why.html

 

[arivero]

By the way, what is the status on looking for family structure emerging from octonions and/or trialities? I see Dixon's 2004 paper http://7stones.com/Homepage/123cho.pdf, also some idea in Dray-Manogue https://arxiv.org/pdf/hep-th/9910010.pdf, and then some newcomers, namely Furey and Dubois-Violette. Not sure if Farnsworth-Boyle try generations in some publication; it is mentioned section 5.3.3 of Fansworth's thesis, in https://inspirehep.net/record/1419192/files/Farnsworth_Shane.pdf

I am particularly fascinated by the point that Dubois-Violette produces an extra chiral fermion in each generation, coming from the diagonal of the exceptional jordan algebra, and I wonder if other approaches also derive similar extra "dark" particles.

 

[lpetrich]

Hence, from the perspective of phase transitions, SU(3)xSU(2)xU(1) would be the result of a broken higher symmetry, due to the Universe entering an "ice-like" state after 13.82 billion years.

After that amount of time? It would be early in the Universe's history. Extrapolation of the Standard Model's coupling constants with increasing interaction energy reveals that they meet at around 2*10¹⁶ GeV.

But the overall principle is correct, I think, and a variety of higher-symmetry Grand Unified Theories have been proposed.

 

[arivero]

Others think that U(1) is because of the Complex numbers, SU(2) is due to the quaternions, and SU(3) is due to the octonians. Although, I don't think this is completely worked out yet. And more effort needs to be done to resolve it.

Is this the kind of iteration you were thinking about in the thread about Furey's models?

 

[friend]

Is this the kind of iteration you were thinking about in the thread about Furey's models?

That's classified. Actually, it's a little too involved to state here. I don't know how to summarize without sounding speculative as defined in these forums. If you're interested PM me.