Tag Archives: supersymmetry

The State of Four Gravitons

This blog is named for a question: does the four-graviton amplitude in N=8 supergravity diverge?

Over the years, Zvi Bern and a growing cast of collaborators have been trying to answer that question. They worked their way up, loop by loop, until they stalled at five loops. Last year, they finally broke the stall, and last week, they published the result of the five-loop calculation. They find that N=8 supergravity does not diverge at five loops in four dimensions, but does diverge in 24/5 dimensions. I thought I’d write a brief FAQ about the status so far.

Q: Wait a minute, 24/5 dimensions? What does that mean? Are you talking about fractals, or…

Nothing so exotic. The number 24/5 comes from a regularization trick. When we’re calculating an amplitude that might be divergent, one way to deal with it is to treat the dimension like a free variable. You can then see what happens as you vary the dimension, and see when the amplitude starts diverging. If the dimension is an integer, then this ends up matching a more physics-based picture, where you start with a theory in eleven dimensions and curl up the extra ones until you get to the dimension you’re looking for. For fractional dimensions, it’s not clear that there’s any physical picture like this: it’s just a way to talk about how close something is to diverging.

Q: I’m really confused. What’s a graviton? What is supergravity? What’s a divergence?

I don’t have enough space to explain these things here, but that’s why I write handbooks. Here are explanations of gravitons, supersymmetry, and (N=8) supergravity, loops, and divergences. Please let me know if anything in those explanations is unclear, or if you have any more questions.

Q: Why do people think that N=8 supergravity will diverge at seven loops?

There’s a useful rule of thumb in quantum field theory: anything that can happen, will happen. In this case, that means if there’s a way for a theory to diverge that’s consistent with the symmetries of the theory, then it almost always does diverge. In the past, that meant that people expected N=8 supergravity to diverge at five loops. However, researchers found a previously unknown symmetry that looked like it would forbid the five-loop divergence, and only allow a divergence at seven loops (in four dimensions). Zvi and co.’s calculation confirms that the five-loop divergence doesn’t show up.

More generally, string theory not only avoids divergences but clears up other phenomena, like black holes. These two things seem tied together: string theory cleans up problems in quantum gravity in a consistent, unified way. There isn’t a clear way for N=8 supergravity on its own to clean up these kinds of problems, which makes some people skeptical that it can match string theory’s advantages. Either way N=8 supergravity, unlike string theory, isn’t a candidate theory of nature by itself: it would need to be modified in order to describe our world, and no-one has suggested a way to do that.

Q: Why do people think that N=8 supergravity won’t diverge at seven loops?

There’s a useful rule of thumb in amplitudes: amplitudes are weird. In studying amplitudes we often notice unexpected simplifications, patterns that uncover new principles that weren’t obvious before.

Gravity in general seems to have a lot of these kinds of simplifications. Even without any loops, its behavior is surprisingly tame: it’s a theory that we can build up piece by piece from the three-particle interaction, even though naively we shouldn’t be able to (for the experts: I’m talking about large-z behavior in BCFW). This behavior seems to have an effect on one-loop amplitudes as well. There are other ways in which gravity seems better-behaved than expected, overall this suggests that we still have a fair ways to go before we understand all of the symmetries of gravity theories.

Supersymmetric gravity in particular also seems unusually well-behaved. N=5 supergravity was expected to diverge at four loops, but doesn’t. N=4 supergravity does diverge at four loops, but that seems to be due to an effect that is specific to that case (for the experts: an anomaly).

For N=8 specifically, a suggestive hint came from varying the dimension. If you checked the dimension in which the theory diverged at each loop, you’d find it matched the divergences of another theory, N=4 super Yang-Mills. At l loops, N=4 super Yang-Mills diverges in dimension 4+6/l. From that formula, you can see that no matter how much you increase l, you’ll never get to four dimensions: in four dimensions, N=4 super Yang-Mills doesn’t diverge.

At five loops, N=4 super Yang-Mills diverges in 26/5 dimensions. Zvi Bern made a bet with supergravity expert Kelly Stelle that the dimension would be the same for N=8 supergravity: a bottle of California wine from Bern versus English wine from Stelle. Now that they’ve found a divergence in 24/5 dimensions instead, Stelle will likely be getting his wine soon.

Q: It sounds like the calculation was pretty tough. Can they still make it to seven loops?

I think so, yes. Doing the five-loop calculation they noticed simplifications, clever tricks uncovered by even more clever grad students. The end result is that if they just want to find out whether the theory diverges then they don’t have to do the “whole calculation”, just part of it. This simplifies things a lot. They’ll probably have to find a few more simplifications to make seven loops viable, but I’m optimistic that they’ll find them, and in the meantime the new tricks should have some applications in other theories.

Q: What do you think? Will the theory diverge?

I’m not sure.

To be honest, I’m a bit less optimistic than I used to be. The agreement of divergence dimensions between N=8 supergravity and N=4 super Yang-Mills wasn’t the strongest argument (there’s a reason why, though Stelle accepted the bet on five loops, string theorist Michael Green is waiting on seven loops for his bet). Fractional dimensions don’t obviously mean anything physically, and many of the simplifications in gravity seem specific to four dimensions. Still, it was suggestive, the kind of “motivation” that gets a conjecture started.

Without that motivation, none of the remaining arguments are specific to N=8. I still think unexpected simplifications are likely, that gravity overall behaves better than we yet appreciate. I still would bet on seven loops being finite. But I’m less confident about what it would mean for the theory overall. That’s going to take more serious analysis, digging in to the anomaly in N=4 supergravity and seeing what generalizes. It does at least seem like Zvi and co. are prepared to undertake that analysis.

Regardless, it’s still worth pushing for seven loops. Having that kind of heavy-duty calculation in our sub-field forces us to improve our mathematical technology, in the same way that space programs and particle colliders drive technology in the wider world. If you think your new amplitudes method is more efficient than the alternatives, the push to seven loops is the ideal stress test. Jacob Bourjaily likes to tell me how his prescriptive unitarity technique is better than what Zvi and co. are doing, this is our chance to find out!

Overall, I still stand by what I say in my blog’s sidebar. I’m interested in N=8 supergravity, I’d love to find out whether the four-graviton amplitude diverges…and now that the calculation is once again making progress, I expect that I will.


Amplitudes Papers I Haven’t Had Time to Read

Interesting amplitudes papers seem to come in groups. Several interesting papers went up this week, and I’ve been too busy to read any of them!

Well, that’s not quite true, I did manage to read this paper, by James Drummond, Jack Foster, and Omer Gurdogan. At six pages long, it wasn’t hard to fit in, and the result could be quite useful. The way my collaborators and I calculate amplitudes involves building up a mathematical object called a symbol, described in terms of a string of “letters”. What James and collaborators have found is a restriction on which “letters” can appear next to each other, based on the properties of a mathematical object called a cluster algebra. Oddly, the restriction seems to have the same effect as a more physics-based condition we’d been using earlier. This suggests that the abstract mathematical restriction and the physics-based restriction are somehow connected, but we don’t yet understand how. It also could be useful for letting us calculate amplitudes with more particles: previously we thought the number of “letters” we’d have to consider there was going to be infinite, but with James’s restriction we’d only need to consider a finite number.

I didn’t get a chance to read David Dunbar, John Godwin, Guy Jehu, and Warren Perkins’s paper. They’re computing amplitudes in QCD (which unlike N=4 super Yang-Mills actually describes the real world!) and doing so for fairly complicated arrangements of particles. They claim to get remarkably simple expressions: since that sort of claim was what jump-started our investigations into N=4, I should probably read this if only to see if there’s something there in the real world amenable to our technique.

I also haven’t read Rutger Boels and Hui Lui’s paper yet. From the abstract, I’m still not clear which parts of what they’re describing is new, or how much it improves on existing methods. It will probably take a more thorough reading to find out.

I really ought to read Burkhard Eden, Yunfeng Jiang, Dennis le Plat, and Alessandro Sfondrini’s paper. They’re working on a method referred to as the Hexagon Operator Product Expansion, or HOPE. It’s related to an older method, the Pentagon Operator Product Expansion (POPE), but applicable to trickier cases. I’ve been keeping an eye on the HOPE in part because my collaborators have found the POPE very useful, and the HOPE might enable something similar. It will be interesting to find out how Eden et al.’s paper modifies the HOPE story.

Finally, I’ll probably find the time to read my former colleague Sebastian Mizera’s paper. He’s found a connection between the string-theory-like CHY picture of scattering amplitudes and some unusual mathematical structures. I’m not sure what to make of it until I get a better idea of what those structures are.

One, Two, Infinity

Physicists and mathematicians count one, two, infinity.

We start with the simplest case, as a proof of principle. We take a stripped down toy model or simple calculation and show that our idea works. We count “one”, and we publish.

Next, we let things get a bit more complicated. In the next toy model, or the next calculation, new interactions can arise. We figure out how to deal with those new interactions, our count goes from “one” to “two”, and once again we publish.

By this point, hopefully, we understand the pattern. We know what happens in the simplest case, and we know what happens when the different pieces start to interact. If all goes well, that’s enough: we can extrapolate our knowledge to understand not just case “three”, but any case: any model, any calculation. We publish the general case, the general method. We’ve counted one, two, infinity.


Once we’ve counted “infinity”, we don’t have to do any more cases. And so “infinity” becomes the new “zero”, and the next type of calculation you don’t know how to do becomes “one”. It’s like going from addition to multiplication, from multiplication to exponentiation, from exponentials up into the wilds of up-arrow notation. Each time, once you understand the general rules you can jump ahead to an entirely new world with new capabilities…and repeat the same process again, on a new scale. You don’t need to count one, two, three, four, on and on and on.

Of course, research doesn’t always work out this way. My last few papers counted three, four, five, with six on the way. (One and two were already known.) Unlike the ideal cases that go one, two, infinity, here “two” doesn’t give all the pieces you need to keep going. You need to go a few numbers more to get novel insights. That said, we are thinking about “infinity” now, so look forward to a future post that says something about that.

A lot of frustration in physics comes from situations when “infinity” remains stubbornly out of reach. When people complain about all the models for supersymmetry, or inflation, in some sense they’re complaining about fields that haven’t taken that “infinity” step. One or two models of inflation are nice, but by the time the count reaches ten you start hoping that someone will describe all possible models of inflation in one paper, and see if they can make any predictions from that.

(In particle physics, there’s an extent to which people can actually do this. There are methods to describe all possible modifications of the Standard Model in terms of what sort of effects they can have on observations of known particles. There’s a group at NBI who work on this sort of thing.)

The gold standard, though, is one, two, infinity. Our ability to step back, stop working case-by-case, and move on to the next level is not just a cute trick: it’s a foundation for exponential progress. If we can count one, two, infinity, then there’s nowhere we can’t reach.

When It Rains It Amplitudes

The last few weeks have seen a rain of amplitudes papers on arXiv, including quite a few interesting ones.


As well as a fair amount of actual rain in Copenhagen

Over the last year Nima Arkani-Hamed has been talking up four or five really interesting results, and not actually publishing any of them. This has understandably frustrated pretty much everybody. In the last week he published two of them, Cosmological Polytopes and the Wavefunction of the Universe with Paolo Benincasa and Alexander Postnikov and Scattering Amplitudes For All Masses and Spins with Tzu-Chen Huang and Yu-tin Huang. So while I’ll have to wait on the others (I’m particularly looking forward to seeing what he’s been working on with Ellis Yuan) this can at least tide me over.

Cosmological Polytopes and the Wavefunction of the Universe is Nima & co.’s attempt to get a geometrical picture for cosmological correlators, analogous to the Ampituhedron. Cosmological correlators ask questions about the overall behavior of the visible universe: how likely is one clump of matter to be some distance from another? What sorts of patterns might we see in the Cosmic Microwave Background? This is the sort of thing that can be used for “cosmological collider physics”, an idea I mention briefly here.

Paolo Benincasa was visiting Perimeter near the end of my time there, so I got a few chances to chat with him about this. One thing he mentioned, but that didn’t register fully at the time, was Postnikov’s involvement. I had expected that even if Nima and Paolo found something interesting that it wouldn’t lead to particularly deep mathematics. Unlike the N=4 super Yang-Mills theory that generates the Amplituhedron, the theories involved in these cosmological correlators aren’t particularly unique, they’re just a particular class of models cosmologists use that happen to work well with Nima’s methods. Given that, it’s really surprising that they found something mathematically interesting enough to interest Postnikov, a mathematician who was involved in the early days of the Amplituhedron’s predecessor, the Positive Grassmannian. If there’s something that mathematically worthwhile in such a seemingly arbitrary theory then perhaps some of the beauty of the Amplithedron are much more general than I had thought.

Scattering Amplitudes For All Masses and Spins is on some level a byproduct of Nima and Yu-tin’s investigations of whether string theory is unique. Still, it’s a useful byproduct. Many of the tricks we use in scattering amplitudes are at their best for theories with massless particles. Once the particles have masses our notation gets a lot messier, and we often have to rely on older methods. What Nima, Yu-tin, and Tzu-Chen have done here is to build a notation similar to what we use for massless particle, but for massive ones.

The advantage of doing this isn’t just clean-looking papers: using this notation makes it a lot easier to see what kinds of theories make sense. There are a variety of old theorems that restrict what sorts of theories you can write down: photons can’t interact directly with each other, there can only be one “gravitational force”, particles with spins greater than two shouldn’t be massless, etc. The original theorems were often fairly involved, but for massless particles there were usually nice ways to prove them in modern amplitudes notation. Yu-tin in particular has a lot of experience finding these kinds of proofs. What the new notation does is make these nice simple proofs possible for massive particles as well. For example, you can try to use the new notation to write down an interaction between a massive particle with spin greater than two and gravity, and what you find is that any expression you write breaks down: it works fine at low energies, but once you’re looking at particles with energies much higher than their mass you start predicting probabilities greater than one. This suggests that particles with higher spins shouldn’t be “fundamental”, they should be explained in terms of other particles at higher energies. The only way around this turns out to be an infinite series of particles to cancel problems from the previous ones, the sort of structure that higher vibrations have in string theory. I often don’t appreciate papers that others claim are a pleasure to read, but this one really was a pleasure to read: there’s something viscerally satisfying about seeing so many important constraints manifest so cleanly.

I’ve talked before about the difference between planar and non-planar theories. Planar theories end up being simpler, and in the case of N=4 super Yang-Mills this results in powerful symmetries that let us do much more complicated calculations. Non-planar theories are more complicated, but necessary for understanding gravity. Dual Conformal Symmetry, Integration-by-Parts Reduction, Differential Equations and the Nonplanar Sector, a new paper by Zvi Bern, Michael Enciso, Harald Ita, and Mao Zeng, works on bridging the gap between these two worlds.

Most of the paper is concerned with using some of the symmetries of N=4 super Yang-Mills in other, more realistic (but still planar) theories. The idea is that even if those symmetries don’t hold one can still use techniques that respect those symmetries, and those techniques can often be a lot cleaner than techniques that don’t. This is probably the most practically useful part of the paper, but the part I was most curious about is in the last few sections, where they discuss non-planar theories. For a while now I’ve been interested in ways to treat a non-planar theory as if it were planar, to try to leverage the powerful symmetries we have in planar N=4 super Yang-Mills elsewhere. Their trick is surprisingly simple: they just cut the diagram open! Oddly enough, they really do end up with similar symmetries using this method. I still need to read this in more detail to understand its limitations, since deep down it feels like something this simple couldn’t possibly work. Still, if anything like the symmetries of planar N=4 holds in the non-planar case there’s a lot we could do with it.

There are a bunch of other interesting recent papers that I haven’t had time to read. Some look like they might relate to weird properties of N=4 super Yang-Mills, others say interesting things about the interconnected web of theories tied together by their behavior when a particle becomes “soft”. Another presents a method for dealing with elliptic functions, one of the main obstructions to applying my hexagon function technique to more situations. And of course I shouldn’t fail to mention a paper by my colleague Carlos Cardona, applying amplitudes techniques to AdS/CFT. Overall, a lot of interesting stuff in a short span of time. I should probably get back to reading it!

An Amplitudes Flurry

Now that we’re finally done with flurries of snow here in Canada, in the last week arXiv has been hit with a flurry of amplitudes papers.


We’re also seeing a flurry of construction, but that’s less welcome.

Andrea Guerrieri, Yu-tin Huang, Zhizhong Li, and Congkao Wen have a paper on what are known as soft theorems. Most famously studied by Weinberg, soft theorems are proofs about what happens when a particle in an amplitude becomes “soft”, or when its momentum becomes very small. Recently, these theorems have gained renewed interest, as new amplitudes techniques have allowed researchers to go beyond Weinberg’s initial results (to “sub-leading” order) in a variety of theories.

Guerrieri, Huang, Li, and Wen’s contribution to the topic looks like it clarifies things quite a bit. Previously, most of the papers I’d seen about this had been isolated examples. This paper ties the various cases together in a very clean way, and does important work in making some older observations more rigorous.


Vittorio Del Duca, Claude Duhr, Robin Marzucca, and Bram Verbeek wrote about transcendental weight in something known as the multi-Regge limit. I’ve talked about transcendental weight before: loosely, it’s counting the power of pi that shows up in formulas. The multi-Regge limit concerns amplitudes with very high energies, in which we have a much better understanding of how the amplitudes should behave. I’ve used this limit before, to calculate amplitudes in N=4 super Yang-Mills.

One slogan I love to repeat is that N=4 super Yang-Mills isn’t just a toy model, it’s the most transcendental part of QCD. I’m usually fairly vague about this, because it’s not always true: while often a calculation in N=4 super Yang-Mills will give the part of the same calculation in QCD with the highest power of pi, this isn’t always the case, and it’s hard to propose a systematic principle for when it should happen. Del Duca, Duhr, Marzucca, and Verbeek’s work is a big step in that direction. While some descriptions of the multi-Regge limit obey this property, others don’t, and in looking at the ones that don’t the authors gain a better understanding of what sorts of theories only have a “maximally transcendental part”. What they find is that even when such theories aren’t restricted to N=4 super Yang-Mills, they have shared properties, like supersymmetry and conformal symmetry. Somehow these properties are tied to the transcendentality of functions in the amplitude, in a way that’s still not fully understood.


My colleagues at Perimeter released two papers over the last week: one, by Freddy Cachazo and Alfredo Guevara, uses amplitudes techniques to look at classical gravity, while the other, by Sebastian Mizera and Guojun Zhang, looks at one of the “pieces” inside string theory amplitudes.

I worked with Freddy and Alfredo on an early version of their result, back at the PSI Winter School. While I was off lazing about in Santa Barbara, they were hard at work trying to understand how the quantum-looking “loops” one can use to make predictions for potential energy in classical gravity are secretly classical. What they ended up finding was a trick to figure out whether a given amplitude was going to have a classical part or be purely quantum. So far, the trick works for amplitudes with one loop, and a few special cases at higher loops. It’s still not clear if it works for the general case, and there’s a lot of work still to do to understand what it means, but it definitely seems like an idea with potential. (Pun mostly not intended.)

I’ve talked before about “Z theory”, the weird thing you get when you isolate the “stringy” part of string theory amplitudes. What Sebastian and Guojun have carved out isn’t quite the same piece, but it’s related. I’m still not sure of the significance of cutting string amplitudes up in this way, I’ll have to read the paper more thoroughly (or chat with the authors) to find out.

Pop Goes the Universe and Other Cosmic Microwave Background Games

(With apologies to whoever came up with this “book”.)

Back in February, Ijjas, Steinhardt, and Loeb wrote an article for Scientific American titled “Pop Goes the Universe” criticizing cosmic inflation, the proposal that the universe underwent a period of rapid expansion early in its life, smoothing it out to achieve the (mostly) uniform universe we see today. Recently, Scientific American published a response by Guth, Kaiser, Linde, Nomura, and 29 co-signers. This was followed by a counterresponse, which is the usual number of steps for this sort of thing before it dissipates harmlessly into the blogosphere.

In general, string theory, supersymmetry, and inflation tend to be criticized in very similar ways. Each gets accused of being unverifiable, able to be tuned to match any possible experimental result. Each has been claimed to be unfairly dominant, its position as “default answer” more due to the bandwagon effect than the idea’s merits. All three tend to get discussed in association with the multiverse, and blamed for dooming physics as a result. And all are frequently defended with one refrain: “If you have a better idea, what is it?”

It’s probably tempting (on both sides) to view this as just another example of that argument. In reality, though, string theory, supersymmetry, and inflation are all in very different situations. The details matter. And I worry that in this case both sides are too ready to assume the other is just making the “standard argument”, and ended up talking past each other.

When people say that string theory makes no predictions, they’re correct in a sense, but off topic: the majority of string theorists aren’t making the sort of claims that require successful predictions. When people say that inflation makes no predictions, if you assume they mean the same thing that people mean when they accuse string theory of making no predictions, then they’re flat-out wrong. Unlike string theorists, most people who work on inflation care a lot about experiment. They write papers filled with predictions, consequences for this or that model if this or that telescope sees something in the near future.

I don’t think Ijjas, Steinhardt, and Loeb were making that kind of argument.

When people say that supersymmetry makes no predictions, there’s some confusion of scope. (Low-energy) supersymmetry isn’t one specific proposal that needs defending on its own. It’s a class of different models, each with its own predictions. Given a specific proposal, one can see if it’s been ruled out by experiment, and predict what future experiments might say about it. Ruling out one model doesn’t rule out supersymmetry as a whole, but it doesn’t need to, because any given researcher isn’t arguing for supersymmetry as a whole: they’re arguing for their particular setup. The right “scope” is between specific supersymmetric models and specific non-supersymmetric models, not both as general principles.

Guth, Kaiser, Linde, and Nomura’s response follows similar lines in defending inflation. They point out that the wide variety of models are subject to being ruled out in the face of observation, and compare to the construction of the Standard Model in particle physics, with many possible parameters under the overall framework of Quantum Field Theory.

Ijjas, Steinhardt, and Loeb’s article certainly looked like it was making this sort of mistake. But as they clarify in the FAQ of their counter-response, they’ve got a more serious objection. They’re arguing that, unlike in the case of supersymmetry or the Standard Model, specific inflation models do not lead to specific predictions. They’re arguing that, because inflation typically leads to a multiverse, any specific model will in fact lead to a wide variety of possible observations. In effect, they’re arguing that the multitude of people busily making predictions based on inflationary models are missing a step in their calculations, underestimating their errors by a huge margin.

This is where I really regret that these arguments usually end after three steps (article, response, counter-response). Here Ijjas, Steinhardt, and Loeb are making what is essentially a technical claim, one that Guth, Kaiser, Linde, and Nomura could presumably respond to with a technical response, after which the rest of us would actually learn something. As-is, I certainly don’t have the background in inflation to know whether or not this point makes sense, and I’d love to hear from someone who does.

One aspect of this exchange that baffled me was the “accusation” that Ijjas, Steinhardt, and Loeb were just promoting their own work on bouncing cosmologies. (I put “accusation” in quotes because while Ijjas, Steinhardt, and Loeb seem to treat it as if it were an accusation, Guth, Kaiser, Linde, and Nomura don’t obviously mean it as one.)

“Bouncing cosmology” is Ijjas, Steinhardt, and Loeb’s answer to the standard “If you have a better idea, what is it?” response. It wasn’t the focus of their article, but while they seem to think this speaks well of them (hence their treatment of “promoting their own work” as if it were an accusation), I don’t. I read a lot of Scientific American growing up, and the best articles focused on explaining a positive vision: some cool new idea, mainstream or not, that could capture the public’s interest. That kind of article could still have included criticism of inflation, you’d want it in there to justify the use of a bouncing cosmology. But by going beyond that, it would have avoided falling into the standard back and forth that these arguments tend to, and maybe we would have actually learned from the exchange.

The Road to Seven-Loop Supergravity

There’s an obvious way to put together a theory of quantum gravity. And it doesn’t work.

Do the same thing you would with any other theory, and you get infinity. You get repeated infinities, an infinity of infinities. And while you could fix one or two infinities, fixing an infinite number requires giving up an infinity of possible predictions, so in the end your theory predicts nothing.

String theory fixes this with its own infinity, the infinite number of ways a string can vibrate. Because this infinity is organized and structured and well-understood, you’re left with a theory that is still at least capable of making predictions.

(Note that this is an independent question from whether string theory can make predictions for experiments in the real world. This is a much more “in-principle” statement: if we knew everything we might want to about physics, all the fields and particles and shapes of the extra dimensions, we could use string theory to make predictions. Even if we knew all of that, we still couldn’t make predictions from naive quantum gravity.)

Are there ways to fix the problem that don’t involve an infinity of vibrations? Or at least, to fix part of the problem?

That’s what Zvi Bern, John Joseph Carrasco, Henrik Johansson, and a growing cast of collaborators have been trying to find out.

They’re investigating N=8 supergravity, a theory that takes gravity and adds on a host of related particles. It’s one of the easiest theories to get from string theory, by curling up extra dimensions in a particularly simple way and ignoring higher-energy vibrations.

Bern, along with Lance Dixon and David Kosower, invented the generalized unitarity technique I talked about last week. Along with Carrasco and Johansson, he figured out another important trick: the idea that you can do calculations in gravity by squaring the appropriate part of calculations in Yang-Mills theory. For N=8 supergravity, the theory you need to square is my favorite theory, N=4 super Yang-Mills.

Using this, they started pushing forward, calculating approximations to greater and greater precision (more and more loops).

What they found, at each step, was that N=8 supergravity behaved better than expected. In fact, it behaved like N=4 super Yang-Mills.

N=4 super Yang-Mills is special, because in four dimensions (three space and one time, the dimensions we’re used to in daily life) there are no infinities to fix. In a world with more dimensions, though, you start getting infinities, and with more and more loops you need fewer and fewer dimensions to see them.

N=8 supergravity, unexpectedly, was giving infinities in the same dimensions that N=4 super Yang-Mills did (and no earlier). If it kept doing that, you might guess that it also had no infinities in four dimensions. You might wonder if, at least loop by loop, N=8 supergravity could be a way to fix quantum gravity without string theory.

Of course, you’d only really know if you could check in four dimensions.

If you want to check in four dimensions, though, you run into a problem. The fewer dimensions you’re looking at, the more loops you need before N=8 supergravity could possibly give infinity. In four dimensions, you need a forbidding seven loops of precision.

(To compare, the highest precision of things we’ve actually tested in the real world is four loops.)

Still, Bern, Carrasco, and Johansson were up to the challenge. Along with Lance Dixon, David Kosower, and Radu Roiban, they looked at three loops, calculating an interaction of four gravitons, and the pattern continued. Four loops, and it was still going strong.

At around this time, I had just started grad school. My first project was a cumbersome numerical calculation. To keep me motivated, my advisor mentioned that the work I was doing would be good preparation for a much grander project: the calculation of whether the four-graviton interaction in N=8 supergravity diverges at seven loops. All I’d have to do was wait for Bern and collaborators to get there.

I named this blog “4 gravitons and a grad student”, and hoped I would get a chance to contribute.

And then something unexpected happened. They got stuck at five loops.

The method they were using, generalized unitarity, is an ansatz-based method. You start with a guess, then refine it. As such, the method is ultimately only as good as your guess.

Their guesses, in general, were pretty good. The trick they were using, squaring N=4 to get N=8, requires a certain type of guess: one in which the pieces they square have similar relationships to the different types of charge in Yang-Mills theory. There’s still an infinite number of guesses that can obey this, so they applied more restrictions, expectations based on other calculations, to get something more manageable. This worked at three loops, and worked (with a little extra thought) at four loops.

But at five loops they were stuck. They couldn’t find anything, with their restrictions, that gave the correct answer when “cut up” by generalized unitarity. And while they could drop some restrictions, if they dropped too many they’d end up with far too general a guess, something that could take months of computer time to solve.

So they stopped.

They did quite a bit of interesting work in the meantime. They found more theories they could square to get gravity theories, of more and more unusual types. They calculated infinities in other theories, and found surprises there too, other cases where infinities didn’t show up when they were “supposed” to. But for some time, the N=8 supergravity calculation was stalled.

And in the meantime, I went off in another direction, which long-time readers of this blog already know about.

Recently, though, they’ve broken the stall.

What they realized is that the condition on their guess, that the parts they square be related like Yang-Mills charges, wasn’t entirely necessary. Instead, they could start with a “bad” guess, and modify it, using the failure of those relations to fill in the missing pieces.

It looks like this is going to work.

We’re all at an amplitudes program right now in Santa Barbara. Walking through the halls of the KITP, I overhear conversations about five loops. They’re paring things down, honing their code, getting rid of the last few bugs, and checking their results.

They’re almost there, and it’s exciting. It looks like finally things are moving again, like the train to seven loops has once again left the station.

Increasingly, they’re beginning to understand the absent infinities, to see that they really are due to something unexpected and new.

N=8 supergravity isn’t going to be the next theory of everything. (For one, you can’t get chiral fermions out of it.) But if it really has no infinities at any loop, that tells us something about what a theory of quantum gravity is allowed to be, about the minimum necessary to at least make sense on a loop-by-loop level.

And that, I think, is worth being excited about.