# Hexagon Functions III: Now with More Symmetry

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.

## 4 thoughts on “Hexagon Functions III: Now with More Symmetry”

1. Thorsten

Nice paper! Although I dont understand the details I can see that a lot of different aspects went into the making of the final result.

I got a few laymen questions:

1) Is it possible to give an intiuitve explanation for the many 2d cross ratio plots that are shown?

2) How long do you project will it take to make the 5-loop computation given the same constraints? Or do you expect to find new constraints that again simplify that calculation?
What can we learn from it? Do we want to go as high as possible, to discover more and more intricate non-trivial properties?

3) In the end you elude to a connection with the positive grassmanian (to appear). Can you give an estimate when you plan to publish it?

Thanks again for this blog

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1. 4gravitonsandagradstudent Post author

For 1) (with your clarification), my older posts should be a lot more informative.

Essentially, this is all about trying to understand scattering of “six” particles, where “six” is counting both what goes in and what comes out. So two particles colliding and becoming four particles would work, as would three interacting to form three, or one decaying to five. If all the particles are gluons, some of the behavior is known, and some was explored in our prior work. If the particles are anything else in the theory then there’s a correction, and that’s this “ratio function”: the ratio of the amplitude for other particles to the amplitude for gluons. (The reason we can lump all the other particles together like this is due to something called superspace, which I talk more about in one of the linked posts).

Hexagon Functions are the general type of functions that make up V and Vtilde, as well as the older functions we looked at for the pure-gluon setup.

For 2), there are actually a few simplifying constraints that we’re going to try out, so it shouldn’t be too much more difficult than four loops was. It’s pretty hard to say how long these things take: we might well be done with it in a year, but that doesn’t mean the calculation itself will take a year.

We probably won’t be able to go beyond five, as the expressions involved start getting too difficult to manipulate with our software. The goal is less to observe intricate properties at higher loops (in fact, one of our most surprising observations is that there are very few such properties, higher loops actually look much like lower loops) but rather to understand how far these techniques can go, and what their limits are.

For 3), we’re going to be meeting to plan the paper this week. We’ve already done most of the relevant calculations, and while I don’t want to commit to anything here, I suspect that it could be out in a few months if all goes well.

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2. Thorsten

@ 1) sorry I did not mean “cross ratios” but “ratio functions” V or V_bar .. the one that is also in the title of the paper. What are the implications of these functions .. why are they important? Is it just a complex function that governs important properties of the theory? Is it the same as the Hexagon function?

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