Calculating the Hard Way, for Science!

I had a new paper out last week, with Jacob Bourjaily and Matthias Volk. We’re calculating the probability that particles bounce off each other in our favorite toy model, N=4 super Yang-Mills. And this time, we’re doing it the hard way.

The “easy way” we didn’t take is one I have a lot of experience with. Almost as long as I’ve been writing this blog, I’ve been calculating these particle probabilities by “guesswork”: starting with a plausible answer, then honing it down until I can be confident it’s right. This might sound reckless, but it works remarkably well, letting us calculate things we could never have hoped for with other methods. The catch is that “guessing” is much easier when we know what we’re looking for: in particular, it works much better in toy models than in the real world.

Over the last few years, though, I’ve been using a much more “normal” method, one that so far has a better track record in the real world. This method, too, works better than you would expect, and we’ve managed some quite complicated calculations.

So we have an “easy way”, and a “hard way”. Which one is better? Is the hard way actually harder?

To test that, you need to do the same calculation both ways, and see which is easier. You want it to be a fair test: if “guessing” only works in the toy model, then you should do the “hard” version in the toy model as well. And you don’t want to give “guessing” any unfair advantages. In particular, the “guess” method works best when we know a lot about the result we’re looking for: what it’s made of, what symmetries it has. In order to do a fair test, we must use that knowledge to its fullest to improve the “hard way” as well.

We picked an example in the middle: not too easy, and not too hard, a calculation that was done a few years back “the easy way” but not yet done “the hard way”. We plugged in all the modern tricks we could, trying to use as much of what we knew as possible. We trained a grad student: Matthias Volk, who did the lion’s share of the calculation and learned a lot in the process. We worked through the calculation, and did it properly the hard way.

Which method won?

In the end, the hard way was indeed harder…but not by that much! Most of the calculation went quite smoothly, with only a few difficulties at the end. Just five years ago, when the calculation was done “the easy way”, I doubt anyone would have expected the hard way to be viable. But with modern tricks it wasn’t actually that hard.

This is encouraging. It tells us that the “hard way” has potential, that it’s almost good enough to compete at this kind of calculation. It tells us that the “easy way” is still quite powerful. And it reminds us that the more we know, and the more we apply our knowledge, the more we can do.

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