I study scattering amplitudes, probabilities that particles scatter off each other.
In particular, I’ve studied them using polylogarithmic functions. Polylogarithmic functions can be taken apart into “logs”, which obey identities much like logarithms do. They’re convenient and nice, and for my favorite theory of N=4 super Yang-Mills they’re almost all you need.
Well, until ten particles get involved, anyway.
That’s when you start needing elliptic integrals, and elliptic polylogarithms. These integrals substitute one of the “logs” of a polylogarithm with an integration over an elliptic curve.
And with Jacob Bourjaily, Andrew McLeod, Marcus Spradlin, and Matthias Wilhelm, I’ve now computed one.
The last few weeks have been a frenzy of work, finishing up our calculations and writing the paper. It’s the fastest I’ve ever gotten a paper out, which has been a unique experience.
Computing this integral required new, so far unpublished tricks by Jake Bourjaily, as well as some rather powerful software and Mark Spradlin’s extensive expertise in simplifying polylogarithms. In the end, we got the integral into a “canonical” form, one other papers had proposed as the right way to represent it, with the elliptic curve in a form standardized by Weierstrass.
One of the advantages of fixing a “canonical” form is that it should make identities obvious. If two integrals are actually the same, then writing them according to the same canonical rules should make that clear. This is one of the nice things about polylogarithms, where these identities are really just identities between logs and the right form is comparatively easy to find.
Surprisingly, the form we found doesn’t do this. We can write down an integral in our “canonical” form that looks different, but really is the same as our original integral. The form other papers had suggested, while handy, can’t be the final canonical form.
What the final form should be, we don’t yet know. We have some ideas, but we’re also curious what other groups are thinking. We’re relatively new to elliptic integrals, and there are other groups with much more experience with them, some with papers coming out soon. As far as we know they’re calculating slightly different integrals, ones more relevant for the real world than for N=4 super Yang-Mills. It’s going to be interesting seeing what they come up with. So if you want to follow this topic, don’t just watch for our names on the arXiv: look for Claude Duhr and Falko Dulat, Luise Adams and Stefan Weinzierl. In the elliptic world, big things are coming.