I’ve got a new paper out this week, a continuation of a story that has threaded through my career since grad school. With a growing collaboration (now Simon Caron-Huot, Lance Dixon, Falko Dulat, Andrew McLeod, and Georgios Papathanasiou) I’ve been calculating six-particle scattering amplitudes in my favorite theory-that-does-not-describe-the-real-world, N=4 super Yang-Mills. We’ve been pushing to more and more “loops”: tougher and tougher calculations that approximate the full answer better and better, using the “word jumble” trick I talked about in Scientific American. And each time, we learn something new.

Now we’re up to seven loops for some types of particles, and six loops for the rest. In older blog posts I talked in megabytes: half a megabyte for three loops, 15 MB for four loops, 300 MB for five loops. I don’t have a number like that for six and seven loops: we don’t store the result in that way anymore, it just got too cumbersome. We have to store it in a simplified form, and even that takes 80 MB.

Some of what we learned has to do with the types of mathematical functions that we need: our “guess” for the result at each loop. We’ve honed that guess down a lot, and discovered some new simplifications along the way. I won’t tell that story here (except to hint that it has to do with “cosmic Galois theory”) because we haven’t published it yet. It will be out in a companion paper soon.

This paper focused on the next step, going from our guess to the correct six- and seven-loop answers. Here too there were surprises. For the last several loops, we’d observed a surprisingly nice pattern: different configurations of particles with different numbers of loops were related, in a way we didn’t know how to explain. The pattern stuck around at five loops, so we assumed it was the real deal, and guessed the new answer would obey it too.

Usually when scientists tell this kind of story, the pattern works, it’s a breakthrough, everyone gets a Nobel prize, etc. This time? Nope!

The pattern failed. And it failed in a way that was surprisingly difficult to detect.

The way we calculate these things, we start with a guess and then add what we know. If we know something about how the particles behave at high energies, or when they get close together, we use that to pare down our guess, getting rid of pieces that don’t fit. We kept adding these pieces of information, and each time the pattern seemed ok. It was only when we got far enough into one of these approximations that we noticed a piece that didn’t fit.

That piece was a surprisingly stealthy mathematical function, one that hid from almost every test we could perform. There aren’t any functions like that at lower loops, so we never had to worry about this before. But now, in the rarefied land of six-loop calculations, they finally start to show up.

We have another pattern, like the old one but that isn’t broken yet. But at this point we’re cautious: things get strange as calculations get more complicated, and sometimes the nice simplifications we notice are just accidents. It’s always important to check.

This result was a long time coming. Coordinating a large project with such a widely spread collaboration is difficult, and sometimes frustrating. People get distracted by other projects, they have disagreements about what the paper should say, even scheduling Skype around everyone’s time zones is a challenge. I’m more than a little exhausted, but happy that the paper is out, and that we’re close to finishing the companion paper as well. It’s good to have results that we’ve been hinting at in talks finally out where the community can see them. Maybe they’ll notice something new!

I finished a new paper recently, it’s up on arXiv now.

This time, we’re collaborating with Jaroslav Trnka, of Amplituhedron fame, to investigate connections between the Amplituhedron and our hexagon function approach.

The Amplituhedron is a way to think about scattering amplitudes in our favorite toy model theory, N=4 super Yang-Mills. Specifically, it describes amplitudes as the “volume” of some geometric space.

Here’s something you might expect: if something is a volume, it should be positive, right? You can’t have a negative amount of space. So you’d naturally guess that these scattering amplitudes, if they’re really the “volume” of something, should be positive.

“Volume” is in quotation marks there for a reason, though, because the real story is a bit more complicated. The Amplituhedron isn’t literally the volume of some space, there are a bunch of other mathematical steps between the geometric story of the Amplituhedron on the one end and the final amplitude on the other. If it was literally a volume, calculating it would be quite a bit easier: mathematicians have gotten very talented at calculating volumes. But if it was literally a volume, it would have to be positive.

What our paper demonstrates is that, in the right regions (selected by the structure of the Amplituhedron), the amplitudes we’ve calculated so far are in fact positive. That first, basic requirement for the amplitude to actually literally be a volume is satisfied.

Of course, this doesn’t prove anything. There’s still a lot of work to do to actually find the thing the amplitude is the volume of, and this isn’t even proof that such a thing exists. It’s another, small piece of evidence. But it’s a reassuring one, and it’s nice to begin to link our approach with the Amplituhedron folks.

This week was the 75th birthday of John Schwarz, one of the founders of string theory and a discoverer of N=4 super Yang-Mills. We’ve dedicated the paper to him. His influence on the field, like the amplitudes of N=4 themselves, has been consistently positive.

It’s paper season! I’ve got another paper out this week, this one a continuation of the hexagon function story.

The story so far:

My collaborators and I have been calculating “six-particle” (two particles collide, four come out, or three collide, three come out…) scattering amplitudes (probabilities that particles scatter) in N=4 super Yang-Mills. We calculate them starting with an ansatz (a guess, basically) made up of a type of functions called hexagon functions: “hexagon” because they’re the right functions for six-particle scattering. We then narrow down our guess by bringing in other information: for example, if two particles are close to lining up, our answer needs to match the one calculated with something called the POPE, so we can throw out guesses that don’t match that. In the end, only one guess survives, and we can check that it’s the right answer.

So what’s new this time?

More loops:

In quantum field theory, most of our calculations are approximate, and we measure the precision in something called loops. The more loops, the closer we are to the exact result, and the more complicated the calculation becomes.

This time, we’re at five loops of precision. To give you an idea of how complicated that is: I store these functions in text files. We’ve got a new, more efficient notation for them. With that, the two-loop functions fit into files around 20KB. Three loops, 500KB. Four, 15MB. And five? 300MB.

So if you want to imagine five loops, think about something that needs to be stored in a 300MB text file.

More insight:

We started out having noticed some weird new symmetries of our old results, so we brought in Simon Caron-Huot, expert on weird new symmetries. He couldn’t figure out that one…but he did notice an entirely different symmetry, one that turned out to have been first noticed in the 60’s, called the Steinmann relations.

The core idea of the Steinmann relations goes back to the old method of calculating amplitudes, with Feynman diagrams. In Feynman diagrams, lines represent particles traveling from one part of the diagram to the other. In a simplified form, the Steinmann conditions are telling us that diagrams can’t take two mutually exclusive shapes at the same time. If three particles are going one way, they can’t also be going another way.

With the Steinmann relations, things suddenly became a whole lot easier. Calculations that we had taken months to do, Simon was now doing in a week. Finally we could narrow things down and get the full answer, and we could do it with clear, physics-based rules.

More bootstrap:

In physics, when we call something a “bootstrap” it’s in reference to the phrase “pull yourself up by your own boostraps”. That impossible task, lifting yourself  with no outside support, is essentially what we do when we “bootstrap”: we do a calculation with no external input, simply by applying general rules.

In the past, our hexagon function calculations always had some sort of external data. For the first time, with the Steinmann conditions, we don’t need that. Every constraint, everything we do to narrow down our guess, is either a general rule or comes out of our lower-loop results. We never need detailed information from anywhere else.

This is big, because it might allow us to avoid loops altogether. Normally, each loop is an approximation, narrowed down using similar approximations from others. If we don’t need the approximations from others, though, then we might not need any approximations at all. For this particular theory, for this toy model, we might be able to actually calculate scattering amplitudes exactly, for any strength of forces and any energy. Nobody’s been able to do that for this kind of theory before.

We’re already making progress. We’ve got some test cases, simpler quantities that we can understand with no approximations. We’re starting to understand the tools we need, the pieces of our bootstrap. We’ve got a real chance, now, of doing something really fundamentally new.

So keep watching this blog, keep your eyes on arXiv: big things are coming.

I’ve got a new paper up this week.

It’s a continuation of my previous work, understanding collisions involving six particles in my favorite theory, N=4 super Yang-Mills.

This time, we’re pushing up the complexity, going from three “loops” to four. In the past, I could have impressed you with the number of pages the formulas I’m calculating take up (eight hundred pages for the three-loop formula from that first Hexagon Functions paper). Now, though, I don’t have that number: putting my four-loop formula into a pdf-making program just crashes the program. Instead, I’ll have to impress you with file sizes: 2.6 MB for the three-loop formula, 96 MB for the four-loop one.

Calculating such a formula sounds like a pretty big task, and it was, the first time. But things got a lot simpler after a chat I had at Amplitudes.

We calculate these things using an ansatz, a guess for what the final answer should look like. The more vague our guess, the more parameters we need to fix, and the more work we have in general. If we can guess more precisely, we can start with fewer parameters and things are a lot easier.

Often, more precise guesses come from understanding the symmetries of the problem. If we can know that the final answer must be the same after making some change, we can rule out a lot of possibilities.

Sometimes, these symmetries are known features of the answer, things that someone proved had to be correct. Other times, though, they’re just observations, things that have been true in the past and might be true again.

We started out using an observation from three loops. That got us pretty far, but we still had a lot of work to do: 808 parameters, to be fixed by other means. Fixing them took months of work, and throughout we hoped that there was some deeper reason behind the symmetries we observed.

Finally, at Amplitudes, I ran into fellow amplitudeologist Simon Caron-Huot and asked him if he knew the source of our observed symmetry. In just a few days he was able to link it to supersymmetry, giving us justification for our jury rigged trick. However, we figured out that his explanation went further than any of us expected. In the end, rather than 808 parameters we only really needed to consider 34.

Thirty-four options to consider. Thirty-four possible contributions to a ~100 MB file. That might not sound like a big deal, but compared to eight hundred and eight it’s a huge deal. More symmetry means easier calculations, meaning we can go further. At this point going to the next step in complexity, to five loops rather than four, might be well within reach.

My new paper went up last night.

It’s on a very similar topic to my last paper, actually. That paper dealt with a specific process involving six particles in my favorite theory, N=4 super Yang-Mills. Two particles collide, and after the metaphorical dust settles four particles emerge. That means six “total” particles, if you add the two in with the four out, for a “hexagon” of variables. To understand situations like that, my collaborators and I created “hexagon functions”, formulas that depended on the states of the six particles.

One thing I didn’t emphasize then was that that calculation only applied to one specific choice of particles, one in which all of the particles are Yang-Mills bosons, particles (like photons) created by the fundamental forces. There are lots of other particles in N=4 super Yang-Mills, though. What happens when they collide?

That question is answered by my new paper. Though it may sound surprising, all of the other particles can be taken into account with a single formula. In order to explain why, I have to tell you about something called superspace.

A while back I complained about a blog post by George Musser about the (2,0) theory. One of the things that irked me about that post was his attempt to explain superspace:

Supersymmetry is the idea that spacetime, in addition to its usual dimensions of space and time, has an entirely different type of dimension—a quantum dimension, whose coordinates are not ordinary real numbers but a whole new class of number that can be thought of as the square roots of zero.

This is actually a great way to think about superspace…if you’re already a physicist. If you’re not, it’s not very informative. Here’s a better way to think about it:

As I’ve talked about before, supersymmetry is a relationship between different types of particles. Two particles related by supersymmetry have the same mass, and the same charge. While they can be very different in other ways (specifically, having different spin), supersymmetric particles are described by many of the same equations as each-other. Rather than writing out those equations multiple times, it’s often nicer to write them all in a unified way, and that’s where superspace comes in.

At its simplest, superspace is just a trick used to write equations in a simpler way. Instead of writing down a different equation for each particle we write one equation with an extra variable, representing a “dimension” of supersymmetry. Traveling in that dimension takes you from particle to particle, in the same way that “turning” the theory (as I phrase it here) does, but it does it within the space of a single equation.

That, essentially, is the trick that we use. With four “superspace dimensions”, we can include the four supersymmetries of N=4 super Yang-Mills, showing how the formulas vary when you go beyond the equation from our first paper.

So far, you may be wondering why I’m calling superspace a “dimension”, when it probably sounds like more of a label. I’ve mentioned before that, just because something is a variable, doesn’t mean it counts as a real dimension.

The key difference is that superspace dimensions are related to regular dimensions in a precise way. In a sense, they’re the square roots of regular dimensions. (Though independently, as George Musser described, they’re the square roots of zero: go in the same direction twice in supersymmetry, and you get back where you’re started, going zero distance.) The coexistence of these two seemingly contradictory statements isn’t some sort of quantum mystery, it’s just a consequence of the fact that, mathematically, I’m saying two very different things. I just can’t think of a way to explain them differently without math.

Superspace isn’t a real place…but it can often be useful to think of it that way. In theories with supersymmetry, it can unify the world, putting disparate particles together into a single equation.