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

1. Sep 5, 2016

### Urs Schreiber

Which is precisely why people became interested in the heterotic $E_8$ GUT model. (And since it's anomaly free.)

2. Sep 5, 2016

### arivero

Yep but somehow it seems overkilll, or at least unrelated to the low-energy theory.

3. Sep 5, 2016

### Urs Schreiber

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.

4. Sep 5, 2016

### ChrisVer

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?

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.

Last edited: Sep 5, 2016
5. Sep 5, 2016

### cube137

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.

6. Sep 5, 2016

7. Sep 6, 2016

### ChrisVer

they give you something to look for... or better put, to check its impact on your observables. Something people did and ruled out the minimal SU(5) model.
The idea is to write a group that has the SM in its subgroups (so you can reach there by a sequence of symmetry breakings). One simple choice was the SU(5), and that gave testable predictions.
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)

1 thing I remember is that they can change baryon or lepton numbers.

8. Sep 6, 2016

### cube137

First I have mastered the book Deep Down Things: The Breathtaking Beauty of Particle Physics which focus on the mathematics of Gauge theory (like Lie Groups and stuff) so I can understand the paper shared by Urs Schreiber and others. I was reading this thesis a while ago about GUT SU(5) in https://fenix.tecnico.ulisboa.pt/downloadFile/395143154268/Thesis.pdf It mentioned:

"What about the supersymmetric versions of these models? As we had seen in the previous chapter, extending
the theory through SUSY was elegant in the sense that unification arose naturally and out-of-the-box. On
the other hand, SUSY is an interesting theory by itself and has clear predictions at the TeV scale and so we
get a theory with many phenomenological interesting predictions.
But as we have seen unification can be accomplished by adding new fields that are not the superpartners of
the minimal SU(5) model. Fields that can either solve the wrong Yukawa relations between the down-quarks
and charged leptons or can induce natural seesaw mechanisms which will generate light neutrino masses.

Also, by avoiding SUSY one gains looser constrains due proton decay upon the masses of the scalars that
may mediate proton decay"

Do you know of other papers that put extra fields that exactly make the couple constant coincide for the strong and electroweak just like nearly done by SUSY?

9. Sep 6, 2016

### Urs Schreiber

The state of the art of GUT model building without SUSY seems to be this here:

Alexander Dueck, Werner Rodejohann,
"Fits to SO(10) Grand Unified Models"
(arXiv:1306.4468)

Chee Sheng Fong, Davide Meloni, Aurora Meroni, Enrico Nardi,
"Leptogenesis in SO(10)"
(arXiv:1412.4776)

10. Sep 6, 2016

### StandardsGuy

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?

Last edited by a moderator: May 8, 2017
11. Sep 6, 2016

### Urs Schreiber

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

12. Sep 7, 2016

### 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?

13. Sep 7, 2016

### ChrisVer

is there a multiverse version of the hierarchy problem?

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.

14. Sep 7, 2016

### ohwilleke

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

Last edited: Sep 7, 2016
15. Sep 7, 2016

### ohwilleke

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.

Last edited: Sep 7, 2016
16. Sep 7, 2016

### 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)

17. Sep 7, 2016

### ohwilleke

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

18. Sep 8, 2016

### Urs Schreiber

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.

19. Sep 8, 2016

### samalkhaiat

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.

Last edited: Sep 8, 2016
20. Sep 14, 2016

### lpetrich

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.