Two Loops, Five Particles

There’s a very long-term view of the amplitudes field that gets a lot of press. We’re supposed to be eliminating space and time, or rebuilding quantum field theory from scratch. We build castles in the clouds, seven-loop calculations and all-loop geometrical quantum jewels.

There’s a shorter-term problem, though, that gets much less press, despite arguably being a bigger part of the field right now. In amplitudes, we take theories and turn them into predictions, order by order and loop by loop. And when we want to compare those predictions to the real world, in most cases the best we can do is two loops and five particles.

Five particles here counts the particles coming in and going out: if two gluons collide and become three gluons, we count that as five particles, two in plus three out. Loops, meanwhile, measure the complexity of the calculation, the number of closed paths you can draw in a Feynman diagram. If you use more loops, you expect more precision: you’re approximating nature step by step.

As a field we’re pretty good at one-loop calculations, enough to do them for pretty much any number of particles. As we try for more loops though, things rapidly get harder. Already for two loops, in many cases, we start struggling. We can do better if we dial down the number of particles: there are three-particle and two-particle calculations that get up to three, four, or even five loops. For more particles though, we can’t do as much. Thus the current state of the art, the field’s short term goal: two loops, five particles.

When you hear people like me talk about crazier calculations, we’ve usually got a trick up our sleeve. Often we’re looking at a much simpler theory, one that doesn’t describe the real world. For example, I like working with a planar theory, with lots of supersymmetry. Remove even one of those simplifications, and suddenly our life becomes a lot harder. Instead of seven loops and six particles, we get genuinely excited about, well, two loops five particles.

Luckily, two loops five particles is also about as good as the experiments can measure. As the Large Hadron Collider gathers more data, it measures physics to higher and higher precision. Currently for five-particle processes, its precision is just starting to be comparable with two-loop calculations. The result has been a flurry of activity, applying everything from powerful numerical techniques to algebraic geometry to the problem, getting results that genuinely apply to the real world.

“Two loops, five particles” isn’t as cool of a slogan as “space-time is doomed”. It doesn’t get much, or any media attention. But, steadily and quietly, it’s become one of the hottest topics in the amplitudes field.

2 thoughts on “Two Loops, Five Particles

    1. 4gravitons Post author

      So, it’s worth being clear that this isn’t solving the “the amplitude isn’t literally the volume of the amplituhedron, it’s a form with singularities on the boundary” issue, it’s addressing a different question. (They don’t claim they are, but I’ve seen a few people see the “long sought-after” in the abstract and get confused. :P)

      Putting the amplituhedron in momentum space (or more specifically spinor-helicity space) isn’t something I was aware people were trying to do, but it does seem like a reasonable outgrowth of the way that Nima, Song He, and collaborators have been able to cast various things in kinematic space rather than an auxiliary space. The paper mentions in the intro that an advantage of this is that it might more naturally generalize to other theories, this does make sense. I am curious whether just did trees and not loops because there’s something more difficult than expected about loops here, or just because they wanted to get a paper out quickly to signpost. Anyway, I guess I’ll hear more about it at Amplitudes in July, and get a clearer idea of the context.

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