As a scientist, you’re surrounded by puzzles. For every test and every answer, ten new questions pop up. You can spend a lifetime on question after question, never getting bored.
But which questions matter? If you want to change the world, if you want to discover something deep, which questions should you focus on? And which should you ignore?
Last year, my collaborators and I completed a long, complicated project. We were calculating the chance fundamental particles bounce off each other in a toy model of nuclear forces, pushing to very high levels of precision. We managed to figure out a lot, but as always, there were many questions left unanswered in the end.
The deepest of these questions came from number theory. We had noticed surprising patterns in the numbers that showed up in our calculation, reminiscent of the fancifully-named Cosmic Galois Theory. Certain kinds of numbers never showed up, while others appeared again and again. In order to see these patterns, though, we needed an unusual fudge factor: an unexplained number multiplying our result. It was clear that there was some principle at work, a part of the physics intimately tied to particular types of numbers.
There were also questions that seemed less deep. In order to compute our result, we compared to predictions from other methods: specific situations where the question becomes simpler and there are other ways of calculating the answer. As we finished writing the paper, we realized we could do more with some of these predictions. There were situations we didn’t use that nonetheless simplified things, and more predictions that it looked like we could make. By the time we saw these, we were quite close to publishing, so most of us didn’t have the patience to follow these new leads. We just wanted to get the paper out.
At the time, I expected the new predictions would lead, at best, to more efficiency. Maybe we could have gotten our result faster, or cleaned it up a bit. They didn’t seem essential, and they didn’t seem deep.
Fast forward to this year, and some of my collaborators (specifically, Lance Dixon and Georgios Papathanasiou, along with Benjamin Basso) have a new paper up: “The Origin of the Six-Gluon Amplitude in Planar N=4 SYM”. The “origin” in their title refers to one of those situations: when the variables in the problem are small, and you’re close to the “origin” of a plot in those variables. But the paper also sheds light on the origin of our calculation’s mysterious “Cosmic Galois” behavior.
It turns out that the origin (of the plot) can be related to another situation, when the paths of two particles in our calculation almost line up. There, the calculation can be done with another method, called the Pentagon Operator Product Expansion, or POPE. By relating the two, Basso, Dixon, and Papathanasiou were able to predict not only how our result should have behaved near the origin, but how more complicated as-yet un-calculated results should behave.
The biggest surprise, though, lurked in the details. Building their predictions from the POPE method, they found their calculation separated into two pieces: one which described the physics of the particles involved, and a “normalization”. This normalization, predicted by the POPE method, involved some rather special numbers…the same as the “fudge factor” we had introduced earlier! Somehow, the POPE’s physics-based setup “knows” about Cosmic Galois Theory!
It seems that, by studying predictions in this specific situation, Basso, Dixon, and Papathanasiou have accomplished something much deeper: a strong hint of where our mysterious number patterns come from. It’s rather humbling to realize that, were I in their place, I never would have found this: I had assumed “the origin” was just a leftover detail, perhaps useful but not deep.
I’m still digesting their result. For now, it’s a reminder that I shouldn’t try to pre-judge questions. If you want to learn something deep, it isn’t enough to sit thinking about it, just focusing on that one problem. You have to follow every lead you have, work on every problem you can, do solid calculation after solid calculation. Sometimes, you’ll just make incremental progress, just fill in the details. But occasionally, you’ll have a breakthrough, something that justifies the whole adventure and opens your door to something strange and new. And I’m starting to think that when it comes to breakthroughs, that’s always been the only way there.