# GUTs vs ToEs: What Are We Unifying Here?

“Grand Unified Theory” and “Theory of Everything” may sound like meaningless grandiose titles, but they mean very different things.

In particular, Grand Unified Theory, or GUT, is a technical term, referring to a specific way to unify three of the fundamental interactions: electromagnetism, the weak force, and the strong force.

In contrast, guts unify the two fundamental intestines.

Those three forces are called Yang-Mills forces, and they can all be described in the same basic way. In particular, each has a strength (the coupling constant) and a mathematical structure that determines how it interacts with itself, called a group.

The core idea of a GUT, then, is pretty simple: to unite the three Yang-Mills forces, they need to have the same strength (the same coupling constant) and be part of the same group.

But wait! (You say, still annoyed at the pun in the above caption.) These forces don’t have the same strength at all! One of them’s strong, one of them’s weak, and one of them is electromagnetic!

As it turns out, this isn’t as much of a problem as it seems. While the three Yang-Mills forces seem to have very different strengths on an everyday scale, that’s not true at very high energies. Let’s steal a plot from Sweden’s Royal Institute of Technology:

Why Sweden? Why not!

What’s going on in this plot?

Here, each $\alpha$ represents the strength of a fundamental force. As the force gets stronger, $\alpha$ gets bigger (and so $\alpha^{-1}$ gets smaller). The variable on the x-axis is the energy scale. The grey lines represent a world without supersymmetry, while the black lines show the world in a supersymmetric model.

So based on this plot, it looks like the strengths of the fundamental forces change based on the energy scale. That’s true, but if you find that confusing there’s another, mathematically equivalent way to think about it.

You can think about each force as having some sort of ultimate strength, the strength it would have if the world weren’t quantum. Without quantum mechanics, each force would interact with particles in only the simplest of ways, corresponding to the simplest diagram here.

However, our world is quantum mechanical. Because of that, when we try to measure the strength of a force, we’re not really measuring its “ultimate strength”. Rather, we’re measuring it alongside a whole mess of other interactions, corresponding to the other diagrams in that post. These extra contributions mean that what looks like the strength of the force gets stronger or weaker depending on the energy of the particles involved.

(I’m sweeping several things under the rug here, including a few infinities and electroweak unification. But if you just want a general understanding of what’s going on, this should be a good starting point.)

If you look at the plot, you’ll see the forces meet up somewhere around 10^16 GeV. They miss each-other for the faint, non-supersymmetric lines, but they meet fairly cleanly for the supersymmetric ones.

So (at least if supersymmetry is true), making the Yang-Mills forces have the same strength is not so hard. Putting them in the same mathematical group is where things get trickier. This is because any group that contains the groups of the fundamental forces will be “bigger” than just the sum of those forces: it will contain “extra forces” that we haven’t observed yet, and these forces can do unexpected things.

In particular, the “extra forces” predicted by GUTs usually make protons unstable. As far as we can tell, protons are very long-lasting: if protons decayed too fast, we wouldn’t have stars. So if protons decay, they must do it only very rarely, detectable only with very precise experiments. These experiments are powerful enough to rule out most of the simplest GUTs. The more complicated GUTs still haven’t been ruled out, but it’s enough to make fewer people interested in GUTs as a research topic.

What about Theories of Everything, or ToEs?

While GUT is a technical term, ToE is very much not. Instead, it’s a phrase that journalists have latched onto because it sounds cool. As such, it doesn’t really have a clear definition. Usually it means uniting gravity with the other fundamental forces, but occasionally people use it to refer to a theory that also unifies the various Standard Model particles into some sort of “final theory”.

Gravity is very different from the other fundamental forces, different enough that it’s kind of silly to group them as “fundamental forces” in the first place. Thus, while GUT models are the kind of thing one can cook up and tinker with, any ToE has to be based on some novel insight, one that lets you express gravity and Yang-Mills forces as part of the same structure.

So far, string theory is the only such insight we have access to. This isn’t just me being arrogant: while there are other attempts at theories of quantum gravity, aside from some rather dubious claims none of them are even interested in unifying gravity with other forces.

This doesn’t mean that string theory is necessarily right. But it does mean that if you want a different “theory of everything”, telling physicists to go out and find a new one isn’t going to be very productive. “Find a theory of everything” is a hope, not a research program, especially if you want people to throw out the one structure we have that even looks like it can do the job.

## 12 thoughts on “GUTs vs ToEs: What Are We Unifying Here?”

1. pete1187

Great post.

I did want to ask one quick two quick things about supersymmetry and a recent Not Even Wrong post that I hope isn’t too off topic. As far as justifications for why supersymmetry may be a part of reality, I thought I had heard that it is the only available symmetry left for the spacetime with the dimensions that we inhabit? I could have sworn it was a post on Stackexchange but I’ve been searching with little luck. This is the closest thing I can come up with, though it still might not quite be what I was originally talking about (the original post I read mentioned Minkowski space and its Poincare group):

In addition, and I know this is right up your alley, I read a post on a speech by Nima Arkani-Hamed over at Not Even Wrong (http://www.math.columbia.edu/~woit/wordpress/?p=8377) and Woit mentions Nima saying: “String theory killed QFT, then QFT killed string theory back, now QFT is king. We’re in a situation where most people think QFT is king and string theory a derivative thing in some limits.”

I never thought String Theory even conflicted with QFT. I mean wouldn’t that make this promising avenue a non-starter from the get-go? I’m just not quite buying it but perhaps Nima is trying to say something else, maybe even related to the Amplituhedron.

Any help is greatly appreciated. Keep up the good work.

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For the SUSY question, this is broadly what you’re looking for. Basically, it’s provably that supersymmetry is the only nontrivial “extra symmetry” available, for certain technical meanings of all of the words in that sentence.

String theory doesn’t conflict with QFT. What Nima is talking about is more of a question of methodology, or of which theory/framework is really a “deeper” description of reality. On one hand, string theory does things that QFT can’t. On the other, through things like AdS/CFT it looks like string theory in certain spaces (and maybe more broadly) is “really” a QFT. This ends up influencing what people find interesting to study, to the extent that these days a lot of string theory people are essentially QFT people.

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2. Wyrd Smythe

So, given that we continue to fail to find evidence of supersymmetry, does that mean not GUT is possible? The three forces don’t, in fact, ever unify?

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Whatever happens at higher energies, if it interacts with one of the three forces, it will modify those curves. Supersymmetry happens to modify them in a particularly nice way, but it’s not impossible for some other modification to also lead to the curves unifying. That said, there will probably be fewer people interested in GUTs as the parameter space for supersymmetry gets smaller, yes.

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1. Wyrd Smythe

Funny thing is, all the years I’ve been reading physics books (since before quarks were real!), I’ve seen that diagram with the converging lines several times, but no one has ever mentioned that the merger depends (or seems to) on supersymmetry.

So I learned something here today! Thanks! 🙂

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3. lucassxs

Great post! String theory provides a few new tools for the study of dark matter and dark energy, but those questions are usually best addressed by older methods, from quantum field theory.

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4. ohwilleke

The plot from Sweden’s Royal Institute of Technology is a bit misleading. We’ve empirically verified the running of the three Standard Model coupling constants up to about 1 TeV and this means that if you are going to have a Supersymmetric GUT, you need to have a SM running of the coupling constants up to some supersymmetric energy scale at which you have a kink in the slope of the running of the coupling constants. It looks a lot less magical and beautiful with the kinks and figuring out how to get the right beta functions starts looking pretty ugly.

The LHC has the potential to move that kink up to considerably higher energy scales, which without a discovery of new physics, makes the kink more pronounced and less natural looking. (It also conceals some weirdness in the shape of the strong force coupling strength at low energies which looks more or less like a lopsided Bell Curve with a peak in the general vicinity of about 1 GeV, falling in strength towards both the IR and UV with either a trivial “zero” strength in the limit as it approaches 0 GeV, or an IR fixed point that is non-trivial (that data and calculations aren’t good enough to distinguish them.)

Also, the particular SUSY Model that almost all of these plots illustrate is the MSSM (minimal supersymmetric standard model) which has pretty much been experimentally ruled out at this point, because its parameters space is over constrained. Even if there is beyond the Standard Model physics, the experimental anomalies that we have seen aren’t great fits for SUSY theories.

The basic GUT/TOE story you tell in your post was a pretty reasonable one in the 1980s and 1990s, but I’m inclined to think that as of 2016, it needs a big fat disclaimer that emphasizes just how unfruitful this line of inquiry has been for the last decade or two. Efforts to preserve string theory and SUSY as viable theories increasingly look like epicycles. And increasingly people are asking if the motivations for SUSY like the hierarchy problem and naturalness ever made much sense to start with (in the alternative they were highly presumptuous assumptions about what Nature looks like that have not been in any way predictive or useful in pointing research in the right direction).

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Unless there’s a complete desert up to the GUT scale, the plots will have kinks regardless, it’s just a matter of whether they’re convenient or inconvenient ones.

And yeah, it’s worth pointing out that these plots are typically for the MSSM. I haven’t seen new plots for the NMSSM or similar, and I’d expect they don’t look quite so nice.

Overall, as I mentioned, GUT research isn’t all that popular right now even among the people still working on SUSY.

I wouldn’t really toss string theory in with low-energy SUSY, though, simply because it typically involves a very different domain. When your goal is to deal with mathematical problems at the Planck length, you don’t really expect to have much interaction with experiment one way or another. (For example, while there are plenty of epicycle-looking things for SUSY, I’m curious what you think are string theory’s epicycles: KKLT maybe?)

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1. ohwilleke

Fair point on SUSY epicycles v. string theory epicycles.

You are right that in string theory, the issue is different. We started out thinking that string theory was like one of those lego sets designed to make something very particular like an Imperial Battle Cruiser and expected to be able to use it to see what particular shape nature looked like. Instead, we discovered that instead we discovered that string theory was more like a general purpose 3D printer, that is infinitely flexible up to certain basic limits, but not very predictive about the shape that Nature looks like and it turns out to be damn hard to reproduce the Imperial Battle Cruiser that we think is reality with it.

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5. Antonio (AKA "Un físico")

Hi, I discovered your blog days ago. Going backwards I only found this interesting post.
I don’t know if you, as a string theorist, are aware of a GUT plot that is similar to the swedish-plot; but instead of joining the three lines in around 0.04 unified gauge coupling value, they climb a little and unify at approximatelly the point (10^17, 0.016). They call this point (String mass, and String unified gauge coupling).