# Why SU(3)xSU(2)xU(1)?

• B

## Main Question or Discussion Point

I have two questions.

But first here's the context of my questions in the following excerpt from Peter Woit book Not Even Wrong:

"Why SU(3)xSU(2)xU(1)? A truly fundamental theory should explain where this precise set of symmetry groups is coming from. In addition, whereas QCD (the SU(3) part of this) has the beautiful property of having no free parameters, introducing the two other groups SU(2) and U(1)) introduces two free parameters and one would like some explanation of why they have the values they do. One of these is the fine structure constant a, and the question of where this number comes from goes back to the earliest days of QED. Another related concern is that the U(1) part of the gauge theory is not asymptotically free, and as a result it may not be completely mathematical consistent."

my questions

1. Please share any arxiv (etc.) papers about why SU(3)xSU(2)xU(1) and anything you have heard about where this precise set of symmetry groups is coming from? It looks like numerology, you know the 3-2-1.

2. Peter Woit is asking why U(1) part of the gauge theory is not asymptotically free. In QCD.. it's asymptotically free. So what would happen if U(1) is also asymptotically free? Please describe the dynamics. And why should it and why an un-asymptotically free U(1) may not be completely mathematical consistent?

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natarajan009

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Urs Schreiber
Gold Member
Of course at this point all speculation as to the fundamental reason for the particular form of the gauge group of the standard model remains just that, speculation. But it's certainly interesting to ask the question and there are some interesting arguments as to what might be going on.

One such 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. This is called GUT. As one works out which simple Lie groups would potentially arise this way, one finds oneself working up the sequence of inclusions

$$\mathrm{SU}(5) \subset \mathrm{Spin}(10) \subset E_6 .$$

This sequence naturally continues as

$$\mathrm{SU}(5) \subset \mathrm{Spin}(10) \subset E_6 \subset E_7 \subset E_8$$

and then it stops.

This makes it natural to speculate that fundamentally the gauge group is the largest exceptional simple Lie group ##E_8##, broken down at low energies to the non-simple gauge group which we observe.

If so, then the original question transmutes into the following: Why ##E_8##? Now that question actually has an answer from first principles.

For a beautiful review and exposition of this line of thought, see

Edward Witten, "Quest For Unification", Heinrich Hertz lecture at SUSY 2002 at DESY, Hamburg arXiv:hep-ph/0207124[/PLAIN] [Broken]

For entertainment, you may also see at universal exceptionalism.

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natarajan009, Spinnor, Ravi Mohan and 4 others
Of course at this point all speculation as to the fundamental reason for the particular form of the gauge group of the standard model remains just that, speculation. But it's certainly interesting to ask the question and there are some interesting arguments as to what might be going on.

One such 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. This is called GUT. As one works out which simple Lie groups would potentially arise this way, one finds oneself working up the sequence of inclusions

$$\mathrm{SU}(5) \subset \mathrm{Spin}(10) \subset E_6 .$$

This sequence naturally continues as

$$\mathrm{SU}(5) \subset \mathrm{Spin}(10) \subset E_6 \subset E_7 \subset E_8$$

and then it stops.

This makes it natural to speculate that fundamentally the gauge group is the largest exceptional simple Lie group ##E_8##, broken down at low energies to the non-simple gauge group which we observe.

If so, then the original question transmutes into the following: Why ##E_8##? Now that question actually has an answer from first principles.

For a beautiful review and exposition of this line of thought, see

Edward Witten, "Quest For Unification", Heinrich Hertz lecture at SUSY 2002 at DESY, Hamburg arXiv:hep-ph/0207124[/PLAIN] [Broken]

For entertainment, you may also see at universal exceptionalism.
Hi, Do you know of any symmetry or dualities that can make matter/forces and information interchange (just think of the Gerald t'Hooft Holographic Principle for the context of what information mean). We attempted for 40 years to make bosons and fermions symmetric (supersymmetry) or make space and time symmetric (successfully in General Relativity). Is there attempt to make matter and information symmetric so they can interchange? This makes a lot of sense. We spent 40 years already since the last success of the Standard Model in 1975 to make symmetry groups of the fundamental particles.. and we already have Nightmare Scenerio with the LHC only possibly detecting the Higgs and nothing more. It may be the SU(3)xSU(2)xU(1) is the last true symmetry of the fundamental particles.. GUT SU(5) requires more symmetry breaking and more parameters. So instead what would happen if information can be united with fundamental particles and forces.. so we will have something like the following...

"The views of matter/forces and information which I wish to lay before you have sprung from the soil of LHC nightmare scenario and null results,, and therein lies their basis. They are radical. Henceforth matter/forces by itself, and information by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."

Mentors. This is not a personal theory. I'm just asking if we can make matter/forces and information become symmetric too (or dualities.. is this the version of symmetry). If there are papers, please share the papers.. If there is none.. then at least let me know there is none. So can focus on purely gauge symmetries without linking it to the holographic principle. Maybe it make more sense if my question is whether there is a connection of gauge symmetries to the holographic principle? Or if the origin of the SU(3)xSU(2)xU(1) is even connected to the holographic principle?

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arivero
Gold Member
The SU(5) and SO(10) GUTs have also an interesting interpretation from the point of view of Kaluza Klein theory in extra dimensions. SO(10) is obviously the group of isometries of the sphere S9 and SU(5) is the group of isometries of CP4, which you can fiber with U(1) to recover S9.

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. Then he noticed that quotienting this space with an U(1) action the resulting 7-dimensional manifold has the isometry group at most of SU(3)xSU(2)xSU(2) and generically of SU(3)xSU(2)xU(1).

So the standard model group could be justified as the Kaluza Klein group of a theory in a space time of 11 dimensions.

nrqed
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.

ChrisVer
Gold Member
Urs Schreiber
Gold Member
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.
This is a vague idea that Geoffrey Dixon once voiced. John Baez recalled it in TWF 104 (1997), where it says the following:
As Corinne Manogue explained to me, this relation between the octonions and Lorentz transformations in 10 dimensions suggests some interesting ways to use octonions in 10-dimensional physics. As we all know, the 10th dimension is where string theorists live. There is also a nice relation to Geoffrey Dixon's theory. This theory relates the electromagnetic force to the complex numbers, the weak force to the quaternions, and the strong force to octonions. How? Well, the gauge group of electromagnetism is U(1), the unit complex numbers. The gauge group of the weak force is SU(2), the unit quaternions. The gauge group of the strong force is SU(3)....

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. However, as Manogue explained, there is a nice way to kill two birds with one stone. If we pick a particular unit imaginary octonion, we get a copy of the complex numbers sitting inside the octonions, so we get a copy of sl(2,C) sitting inside sl(2,O), so we get a copy of so(3,1) sitting inside so(9,1)! In other words, we get a particular copy of the good old 4-dimensional Lorentz group sitting inside the 10-dimensional Lorentz group. So fixing a unit imaginary octonion not only breaks the octonion symmetry group G2 down to the strong force symmetry group SU(3), it might also get us from 10-dimensional physics down to 4-dimensional physics.

Cool, no? There are obviously a lot of major issues involved in turning this into a full-fledged theory, and they might not work out. The whole idea could be completely misguided! But it takes guts to do physics, so it's good that Tevian Dray and Corinne Manogue are bravely pursuing this idea.

fresh_42
Mentor
To me this always sounds like fishing in troubled waters. Doesn't fit? Blow up the dimension. Still not convincing? Change to the exceptionals. Still problems? Vary the scalars. What, still not enough? Grade the entire thing. And if nothing helps, invent some additional universes.

I might be totally wrong and I really like good entertainment. But I can't get the thought out of my head, that many scientists throw their hat in the ring and hope for the best.

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Demystifier, nnunn and kodama
if I understand 2009 Asymptotic safety of gravity and the Higgs boson mass
Mikhail Shaposhnikov, Christof Wetterich

SU(3)xSU(2)xU(1) + Asymptotic safety of gravity is all there is - no new physics from Fermi scale to planck scale.

No GUT's or SUSY or SUGRA. 126 +/- 1GEV. Higgs.

SU(3)xSU(2)xU(1) in this Asymptotic scenario would need to be explained without any reference to higher dimensions, GUT's SUSY or SUGRA producing new particles and new physics above Fermi scale.

Of course at this point all speculation as to the fundamental reason for the particular form of the gauge group of the standard model remains just that, speculation. But it's certainly interesting to ask the question and there are some interesting arguments as to what might be going on.

One such 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. This is called GUT. As one works out which simple Lie groups would potentially arise this way, one finds oneself working up the sequence of inclusions

$$\mathrm{SU}(5) \subset \mathrm{Spin}(10) \subset E_6 .$$

This sequence naturally continues as

$$\mathrm{SU}(5) \subset \mathrm{Spin}(10) \subset E_6 \subset E_7 \subset E_8$$

and then it stops.

This makes it natural to speculate that fundamentally the gauge group is the largest exceptional simple Lie group ##E_8##, broken down at low energies to the non-simple gauge group which we observe.

If so, then the original question transmutes into the following: Why ##E_8##? Now that question actually has an answer from first principles.

For a beautiful review and exposition of this line of thought, see

Edward Witten, "Quest For Unification", Heinrich Hertz lecture at SUSY 2002 at DESY, Hamburg arXiv:hep-ph/0207124[/PLAIN] [Broken]

For entertainment, you may also see at universal exceptionalism.
if you start with E8 would you be able to uniquely single out SU(3)xSU(2)xU(1) to the exclusion of all other possibilities?

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If you guys will notice. The internal generators of the Lie Group or symmetries SU(3)xSU(2)XU(1) are based on real forces and dynamics.
U(1) is gauge for quantum electrodynamics or electromagnetism
U(2) is gauge for weak isospin (or the two doublet interchange of the electroweak bosons)
U(3) is gauge for the 3 quarks and color space

Maybe they are all there is to it. The reason they are 1,2,3 is because nature needs the simplest method and it start with 1,2,3 (and maybe end with it). Why argue for GUT SU(5) and more? The internal generators would be several and do they take a part at all in the dynamics of our world? If none.. maybe there is no GUT SU(5), no E8, or any other higher lie group or gauge symmetries. What is your argument there should be?

ChrisVer
Gold Member
To me this always sounds like fishing in troubled waters. Doesn't fit? Blow up the dimension. Still not convincing? Change to the exceptionals. Still problems? Vary the scalars. What, still not enough? Grade the entire thing. And if nothing helps, invent some additional universes.
That's what you are doing when there is no data- you are "blindly" shooting left and right expecting to hit something good.
"Blindly" is a bad word: most of the time you are following a sensible path, but noone can say whether that path is correct or not... you try to make it look as correct as possible.

if you start with E8 would you be able to uniquely single out SU(3)xSU(2)xU(1) to the exclusion of all other possibilities?
I guess that you cannot uniquely reach the SM group, but you make things so that you will eventually reach there.
eg Spin(10) is not the only subgroup of E6...

Why argue for GUT SU(5) and more?
Well, there is a matter of aesthetics and very large coincidences around the SM... One coincidence is how close the coupling constants get at high scales (they don't intersect but they get very close to each other around the 'GUT' scale).

What is your argument there should be?
Is the SM all there is ? Well a lot of things indicate that it is not. For example the observed density of Dark Matter... we are already moving ahead of the SM in some way, eversince the discovery of massive neutrinos. The known cosmological constant is also something that the standard model cannot predict... Then questions about the naturalness of the SM are still there; although I'm not a "defender" of naturalness arguments since I was never convinced for why should an EFT be natural at all. Another thing is why is the top quark mass what it is...

nnunn
That's what you are doing when there is no data- you are "blindly" shooting left and right expecting to hit something good.
"Blindly" is a bad word: most of the time you are following a sensible path, but noone can say whether that path is correct or not... you try to make it look as correct as possible.

I guess that you cannot uniquely reach the SM group, but you make things so that you will eventually reach there.
eg Spin(10) is not the only subgroup of E6...

Well, there is a matter of aesthetics and very large coincidences around the SM... One coincidence is how close the coupling constants get at high scales (they don't intersect but they get very close to each other around the 'GUT' scale).

Is the SM all there is ? Well a lot of things indicate that it is not. For example the observed density of Dark Matter... we are already moving ahead of the SM in some way, eversince the discovery of massive neutrinos. The known cosmological constant is also something that the standard model cannot predict... Then questions about the naturalness of the SM are still there; although I'm not a "defender" of naturalness arguments since I was never convinced for why should an EFT be natural at all. Another thing is why is the top quark mass what it is...
I meant why should there be more symmetry group than SU(3)xSU(2)XU(1). I'm not arguing there are no new physics like Dark matter. Why.. if there is dark matter or new physics.. is it related to more higher symmetry than SU(3)XSU(2)xU(1). They could be independent, isn't it?

Of course there is more to the SM.. in fact so many things more than physicists are willing to study. They are also doing it tunnel vision.. that is why they will never arrive at any unification... unless new breed of physicists come to the scene.

fresh_42
Mentor
I guess that you cannot uniquely reach the SM group, but you make things so that you will eventually reach there.
eg Spin(10) is not the only subgroup of E6...
The point is: SU(3) x SU(2) x U(1) provides a reliable and minimal structure that explains a lot. As soon as you blow up the group, there are far too many possibilities, in which you can find them as subgroups. Without any sound evidence that points into a certain direction, it's simply still a bit arbitrary.

ChrisVer
Gold Member
The point is: SU(3) x SU(2) x U(1) provides a reliable and minimal structure that explains a lot. As soon as you blow up the group, there are far too many possibilities, in which you can find them as subgroups. Without any sound evidence that points into a certain direction, it's simply still a bit arbitrary.
Arbitrary yes, and that's one problem of theories that give so many possibilities... some argument against them is that if you predict everything it doesn't matter if you also predict this world. But looking from another prespective, what I think one does is that (s)he starts with a group and manipulates the way in order to reach the SM... so what seems arbitrary is in the way of manipulation and not the fact that you end up with a singled out group. There is no evidence appart from the fact that you have to reach the "holy grail".
Otherwise, if you cannot reach the SM, your theory is for trash (as would be GR if it could not somehow give similar predictions to the Newtonian mechanics at the regime where the last was very well tested).

is it related to more higher symmetry than SU(3)XSU(2)xU(1). They could be independent, isn't it?
I guess your comment suggests a group like GxSU(3)XSU(2)XU(1), where G is some other group that contains the new stuff?
Left-Right models are such I guess, since you can have an SU(2) Right as well.
Then if the group G is completely independent from the rest (SM particles' couplings under that group=0), you wouldn't be able to detect DM at any experiment. Also I don't see why would that be the case?
Higher groups allow you to deal with the particles in the same way, and at some point (symmetry breaking point) split them up and change their couplings, creating for example WIMPs. Also you could have a non-vanishing coupling constant that the symmetry would allow you to send at 0 in the end.
Again it's not people's fault that unification cannot be reached. The way they approach things is the best they can do with what we have at hand. You can see that by the fact that we are full of theories but we can't single out one. The lack of experimental indications of new physics is what "sentences" them in the tunnel's darkness.

Arbitrary yes, and that's one problem of theories that give so many possibilities... some argument against them is that if you predict everything it doesn't matter if you also predict this world. But looking from another prespective, what I think one does is that (s)he starts with a group and manipulates the way in order to reach the SM... so what seems arbitrary is in the way of manipulation and not the fact that you end up with a singled out group. There is no evidence appart from the fact that you have to reach the "holy grail".
Otherwise, if you cannot reach the SM, your theory is for trash (as would be GR if it could not somehow give similar predictions to the Newtonian mechanics at the regime where the last was very well tested).

I guess your comment suggests a group like GxSU(3)XSU(2)XU(1), where G is some other group that contains the new stuff?
Left-Right models are such I guess, since you can have an SU(2) Right as well.
Then if the group G is completely independent from the rest (SM particles' couplings under that group=0), you wouldn't be able to detect DM at any experiment. Also I don't see why would that be the case?
But the latest LUX experiment hasn't detected any dark matter. Are you saying that even if dark matter and normal matter are connected only by gravity. G shouldn't be independent from the SM as far as symmetry group is concerned?

ChrisVer
Gold Member
Are you saying that even if dark matter and normal matter are connected only by gravity. G shouldn't be independent from the SM as far as symmetry group is concerned?
Well I may be wrong, but if you had some other group, the SM particles would have to belong in singlet representations of that group... so I guess that you can write down terms that keep the lagrangian invariant with some coupling constant $g_{n}$ that mix the particles that transform non-trivially under this new group with the other particles of the SM... in that case you have to explain why those coupling constants are very close to zero (since you don't observe such decays). Broken symmetries provide such an explanation.

Well I may be wrong, but if you had some other group, the SM particles would have to belong in singlet representations of that group... so I guess that you can write down terms that keep the lagrangian invariant with some coupling constant $g_{n}$ that mix the particles that transform non-trivially under this new group with the other particles of the SM... in that case you have to explain why those coupling constants are very close to zero (since you don't observe such decays). Broken symmetries provide such an explanation.
Supergravity is the field theory that combines matter and gravity. They do this by combining fermions and bosons into supersymmetry and combining this with general relativity. Can supergravity still be true if there is really no supersymmetry?

Does anyone know of any concept elsewhere outside physics where they use similar concept in gauge theories.. for example.. in creating cakes.. the ingredients came from one supersource where symmetry breaking produces the different ingredients? Or do they do this in programming languages.. for example.. to design interactive programs.. does the environment and dynamics of the program need to use some kind of supergravity where environment (spacetime) and dynamics (supersymmetry) is more efficient. My point is.. you can create a universe by using other building blocks. But why is our particular universe designed by gauge symmetries.. is this because we are inside a program or something.. or is it Noether Theorem automatically create gauge symmetries even in a purely solid universe without using any concept of programming? What would happen in a universe not ruled by gauge symmetries? Maybe conservation laws are violated and the universe just implode or not consistent or something?

Anyway. I'll be watching the TTC - Superstring Theory - DNA or Reality tutorial starting tomorrow. Supergravity has natural home in superstrings.

Urs Schreiber
Gold Member
Of course it's speculation at this point, and of course you may have better things to do. But the attitude "What we know for sure right now is all there is, so stop following hints for something deeper." has been wrong before. Of course wild speculation is boring. But educated speculation, putting togther concrete hints, is worthwhile. It will necessarily go down many dead-ends, but by trial and error, it is the only way to eventually make progress.

eloheim
arivero
Gold Member
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.
Yep, most of the interpretations pivot around this, because of the Hopf fibration of the seven-sphere, where you can find all these objects. But Nature has not chosen SO(8), going instead to a group, the SM group, with the half of generators minus one.

Or you could start from the isometries of the Lie Manifold SU(3), which are SU(3)xSU(3) and break one towards the electroweak. Or you can try to consider independently SU(2)xU(1) electroweak and SU(3)xU(1) color+electromagnetism. The touchstone is to get not only the group but also the symmetry breaking mechanism. Extra points if you get chiral fermions before the breaking (and perhaps dirac fermions after the breaking).

Urs Schreiber
Gold Member
Extra points if you get chiral fermions before the breaking
Which is precisely why people became interested in the heterotic ##E_8## GUT model. (And since it's anomaly free.)

arivero
Gold Member
Which is precisely why people became interested in the heterotic ##E_8## GUT model. (And since it's anomaly free.)
Yep but somehow it seems overkilll, or at least unrelated to the low-energy theory.

Urs Schreiber
Gold Member
Yep but somehow it seems overkilll, or at least unrelated to the low-energy theory.
With the Planck scale 14 orders of magnitude away, and 5 times as much dark matter present as visible matter, one may argue that a theory leaving room for undetected effects doesn't decrease its plausibility compared to theories that insist that nothing new is going on. Mankind worked its way upthe energy scale around 12 order of magnitude in the past to incrementally find larger gauge groups and more particles. It seems implausible that this should stop being so across the remaining 14 orders of magnitude.

ChrisVer
Gold Member
Supergravity is the field theory that combines matter and gravity. They do this by combining fermions and bosons into supersymmetry and combining this with general relativity. Can supergravity still be true if there is really no supersymmetry?
no because as you say you need superpartners and a theory that combines the inner symmetries with the spacetime ones. Supersymmetric transformations already do the job for that... but how did SUGRA enter the discussion?

But why is our particular universe designed by gauge symmetries
I don't know if anybody has a more satisfying answer to this... the answer I have for myself is "because it works". When people started building up the SM for example, they didn't have any clue of gauge theories, several alternatives were suggested. The gauge theories seem to have won over the rest and became a norm. So the "it works" is a satisfying argument for me, because if there was a solid reasoning then we wouldn't need those alternatives at the beginnings.
I can't comment on the rest. I don't care or believe in some program or a theorem that is there to help us describe nature.
However, I can question something you mentioned: how does Noether's theorem "predict" or impose gauge theories for the description of Nature? Because as far as I know, we impose those symmetries and then we can apply Noether's theorem.

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mandelu2
SU(3) has 8 generators in terms of the 8 color force
SU(2) has 3 generators in terms of the 3 weak bosons
U(1) has 1 generator in terms of the 1 EM force.
Total generators of SU(3)xSU(2)xU(1) is 12.
SU(5) has 24 generators (what do the excess 12 generators do in our life)?

Notice the generators of SU(3)xSU(2)xU(1) has relevants for our world (because they are the fundemantal forces of nature). How about the 24 generators of SU(5) (or the extra 12)? What are they good for? Only for gauge bookkeeping? I can't decide whether forming higher gauge symmetry group like SU(5) is the way to go for unification.. or needing other procedures.