A Tale of Two Donuts

I’ve got a new paper up this week, with Hjalte Frellesvig, Cristian Vergu, and Matthias Volk, about the elliptic integrals that show up in Feynman diagrams.

You can think of elliptic integrals as integrals over a torus, a curve shaped like the outer crust of a donut.

Do you prefer your integrals glazed, or with powdered sugar?

Integrals like these are showing up more and more in our field, the subject of bigger and bigger conferences. By now, we think we have a pretty good idea of how to handle them, but there are still some outstanding mysteries to solve.

One such mystery came up in a paper in 2017, by Luise Adams and Stefan Weinzierl. They were working with one of the favorite examples of this community, the so-called sunrise diagram (sunrise being a good time to eat donuts). And they noticed something surprising: if they looked at the sunrise diagram in different ways, it was described by different donuts.

What do I mean, different donuts?

The integrals we know best in this field aren’t integrals on a torus, but rather integrals on a sphere. In some sense, all spheres are the same: you can make them bigger or smaller, but they don’t have different shapes, they’re all “sphere-shaped”. In contrast, integrals on a torus are trickier, because toruses can have different shapes. Think about different donuts: some might have a thin ring, others a thicker one, even if the overall donut is the same size. You can’t just scale up one donut and get the other.

This donut even has a marked point

My colleague, Cristian Vergu, was annoyed by this. He’s the kind of person who trusts mathematics like an old friend, one who would never lead him astray. He thought that there must be one answer, one correct donut, one natural way to represent the sunrise diagram mathematically. I was skeptical, I don’t trust mathematics nearly as much as Cristian does. To sort it out, we brought in Hjalte Frellesvig and Matthias Volk, and started trying to write the sunrise diagram every way we possibly could. (Along the way, we threw in another “donut diagram”, the double-box, just to see what would happen.)

Rather than getting a zoo of different donuts, we got a surprise: we kept seeing the same two. And in the end, we stumbled upon the answer Cristian was hoping for: one of these two is, in a meaningful sense, the “correct donut”.

What was wrong with the other donut? It turns out when the original two donuts were found, one of them involved a move that is a bit risky mathematically, namely, combining square roots.

For readers who don’t know what I mean, or why this is risky, let me give a simple example. Everyone else can skip to after the torus gif.

Suppose I am solving a problem, and I find a product of two square roots:

\sqrt{x}\sqrt{x}

I could try combining them under the same square root sign, like so:

\sqrt{x^2}

That works, if x is positive. But now suppose x=-1. Plug in negative one to the first expression, and you get,

\sqrt{-1}\sqrt{-1}=i\times i=-1

while in the second,

\sqrt{(-1)^2}=\sqrt{1}=1

Torus transforming, please stand by

In this case, it wasn’t as obvious that combining roots would change the donut. It might have been perfectly safe. It took some work to show that indeed, this was the root of the problem. If the roots are instead combined more carefully, then one of the donuts goes away, leaving only the one, true donut.

I’m interested in seeing where this goes, how many different donuts we have to understand and how they might be related. But I’ve also been writing about donuts for the last hour or so, so I’m getting hungry. See you next week!

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