I’m at Brown this week, where I’ve been chatting with Mark Spradlin and Anastasia Volovich, two of the founding figures of my particular branch of amplitudeology. Back in 2010 they figured out how to turn this seventeen-page two-loop amplitude:

into a formula that just takes up two lines:This got everyone very excited, it inspired some of my collaborators to do work that would eventually give rise to the Hexagon Functions, my main research project for the past few years.

Unfortunately, when we tried to push this to higher loops, we didn’t get the sort of nice, clean-looking formulas that the Brown team did. Each “loop” is an additional layer of complexity, a series of approximations that get closer to the exact result. And so far, our answers look more like that first image than the second: hundreds of pages with no clear simplifications in sight.

At the time, people wondered whether some simple formula might be enough. As it turns out, you can write down a formula similar to the one found by Spradlin and Volovich, generalized to a higher number of loops. It’s clean, it’s symmetric, it makes sense…and it’s not the right answer.

That happens in science a lot more often than science fans might expect. When you hear about this sort of thing in the news, it always works: someone suggests a nice, simple answer, and it turns out to be correct, and everyone goes home happy. But for every nice simple guess that works, there are dozens that don’t: promising ideas that just lead to dead ends.

One of the postdocs here at Brown worked on this “wrong” formula, and while chatting with him here he asked a very interesting question: why is it wrong? Sure, we know that it’s wrong, we can check that it’s wrong…but what, specifically, is missing? Is it “part” of the right answer in some sense, with some predictable corrections?

As it turns out, this is a very interesting question! We’ve been looking into it, and the “wrong” answer has some interesting relationships with some of our Hexagon Functions. It may have been a “dead end”, but it still could turn out to be a useful one.

A good physics advisor will tell their students to document their work. This doesn’t just mean taking notes: most theoretical physicists will maintain files, in standard journal article format, with partial results. One reason to do this is that, if things work out, you’ll have some of your paper already written. But if something doesn’t work out, you’ll end up with a pdf on your hard drive carefully explaining an idea that didn’t quite work. Physicists often end up with dozens of these files squirreled away on their computers. Put together, they’re a map: a map of dead ends.

There’s a handy thing about having a map: it lets you retrace your steps. Any one of these paths may lead nowhere, but each one will contain some substantive work. And years later, often enough, you end up needing some of it: some piece of the calculation, some old idea. You follow the map, dig it up…and build it into something new.