Amplitudes Megapost

If you met me on a plane and asked me what I do, I’d probably lead with something like this:

“I come up with mathematical tricks to make particle physics calculations easier.”

People like me, who research these tricks, are sometimes known as Amplitudeologists. We studying scattering amplitudes, mathematical formulas used to calculate the probabilities of different things happening when sub-atomic particles collide.

Why do we want to make calculations easier? Because particle physics is hard!

More specifically, calculations in particle physics can be hard for three broad reasons: lots of loops, lots of legs, or more complicated theories.

Loops measure precision. They’re called loops because more complicated Feynman diagrams contain “loops” of particles, while the simplest, with no loops at all, are called “trees”. The more loops you include, the more precise your calculation becomes, but it also becomes more complicated.

Legs are the number of particles involved. If two particles collide and bounce off each other, then there are a total of four legs: two from the incoming particles, two from the outgoing ones. Calculations with more legs are almost always more complicated than calculations with fewer.

Most of the time, our end-goal is to calculate things that are relevant to the real world. Usually, this means QCD, or Quantum Chromodynamics, the theory of quarks and gluons. QCD is very complicated, though. Often, we work to hone our techniques on simpler theories first. N=4 super Yang-Mills has been called the simplest quantum field theory, particularly the further simplified, planar version. If you want a basic overview of it, check out the Handy Handbooks tab at the top of my blog. Often, progress in amplitudeology involves adapting tricks from planar N=4 super Yang-Mills to more complicated, and more realistic, theories.

I should point out that our goal in amplitudeology isn’t always to do more complicated calculations. Sometimes, it’s about doing a calculation we already know how to do, but in a way that’s more insightful. This lets us learn more about the theories we’re studying, as well as gaining insights about larger problems like the nature of space and time.

So what sorts of tricks do we use to do all this? Well, there are a few broad categories…

Generalized Unitarity

The prizewinning idea that started it all, generalized unitarity came out of the collaboration of Zvi Bern, Lance Dixon, and David Kosower, starting in the 90’s. The core of the idea is difficult to describe in a quick sentence, but it essentially boils down to noticing that, rather than thinking about every single multi-loop Feynman diagram independently, you can think of loop diagrams as what you get when you sew trees together.

This is a very powerful idea. These days, pretty much everyone who studies amplitudeology learns it, and it’s proven pivotal for a wide array of applications.

In planar N=4 super Yang-Mills it’s one of the techniques that can go to exceptionally high loop order, to six or seven loops. If you drop the “planar” condition, it’s still quite powerful. If you do things right, as Zvi Bern, John Joseph Carrasco, and Henrik Johansson found, you can get results in N=8 supergravity “for free”. This raises what has ended up being one of the big questions of our sub-field: does N=8 supergravity behave like most attempts at theories of quantum gravity, with pesky infinite results that we don’t know how to deal with, or does it behave like N=4 super Yang-Mills, which has no pesky infinities at all? Answering this question requires a dizzying seven-loop calculation, the mystique of which got me in to the field in the first place. Unfortunately, despite diligent efforts from Bern and collaborators, they’ve been stuck at four loops for quite some time. In the meantime they’ve been extending things in all the other amplitudes-directions: more legs, more complicated theories (in this case, supergravity with less supersymmetry), and more insight. Recently, it looks like they may have found a way around this hurdle, so the mystery at seven loops may not be so far away after all.

Generalized Unitarity is also one of the most powerful amplitudes tricks for real-world theories, in particular QCD. In this case, it’s main virtue is in legs, not loops, going up to seven-particles at one loop for practical, LHC-relevant calculations. There’s also a major effort to push this to two loops, with some success.

BCFW Recursion

If generalized unitarity was the trick that got experimentalists to sit up and take notice, BCFW is the one that got the attention of the pure theorists. In the mid-2000s Ruth Britto, Freddy Cachazo, and Bo Feng (later joined by theoretical physics superstar Ed Witten) figured out a way to build up tree amplitudes to any number of legs recursively for any theory, starting with three particles and working their way up. Their method was both fairly efficient and extremely insightful, and it’s another trick that’s made its way into every amplitudeologist’s arsenal. Further developments led to a recursive procedure that could work up to any number of loops in planar N=4 super Yang-Mills, which while not especially efficient did lead to…

The Positive Grassmannian, and the Amplituhedron

The work of Nima Arkani-Hamed, Jacob Bourjaily, Freddy Cachazo, Alexander Goncharov, Alexander Postnikov, and Jaroslav Trnka on the Positive Grassmannian (and more recently the Amplituhedron) has pushed the “more insight” direction impressively far. The Amplituhedron in particular captured the public’s imagination, as well as that of mathematicians, by packaging the all-loop amplitude into a particularly clean, mathematically meaningful form. Now they’re working on pushing this deep understanding to non-planar N=4 super Yang-Mills.

Integration Tricks

Generalized unitarity and the Amplituhedron have one thing in common: neither gives the full result. Calculating scattering amplitudes traditionally is a two-step process: first, add up all possible Feynman diagrams, then add up (integrate) all possible momenta. Generalized unitarity and the Amplituhedron let you skip the diagrams, but in both cases you still need to integrate. There’s a whole lore of integration techniques, from breaking things up into a basis of known “master” integrals (an example paper on this theme here), to attacking the integrations numerically via a process known as sector decomposition (one of the better programs that does this here). Higher-loop integrations are typically quite tough, even with these techniques.


These integrals will usually result in a type of mathematical functions called polylogarithms, or transcendental functions. Understanding these functions has led to an enormous amount of progress (and I’m not just saying that because it’s what I work on 😉 ).

It all started when Alexander Goncharov, Mark Spradlin, Cristian Vergu, and Anastasia Volovich figured out how to write a laboriously calculated seventeen-page two-loop six-particle amplitude in just two lines. To do this, they used mathematical properties of polylogarithms that were previously largely unknown to physicists. Their success inspired Lance Dixon, James Drummond, and Johannes Henn to use these methods to guess the correct answer at three loops, work that was completed with my involvement.

Since then, both groups have made a lot of progress. In general, Spradlin, Volovich, and collaborators have been pushing things farther in terms of legs and insight, while Dixon and collaborators have made progress at higher loops. So far we’ve gotten to four loops (here, plus unpublished work), while the others have proposals for any number of particles at two loops and substantial progress for seven particles at three loops.

All of this is still for planar N=4 super Yang-Mills. Using these tricks for more complicated theories is trickier. However, while you usually can’t just guess the answer like you can for N=4, a good understanding of the properties of polylogarithms can still take you quite far.


Why did the polylogarithm folks start with six particles? Wouldn’t four or five have been easier?

As it turns out, four and five particle amplitudes are indeed easier, so much so that for planar N=4 super Yang-Mills they’re known up to any loop order. And while a number of elements went in to that result, one that really filled in the details was integrability.

Integrability is tough to describe in a short sentence, but essentially it involves describing highly symmetric systems all in one go, without having to use the step-by-step approximations of perturbation theory. For our purposes, this means bypassing the loop-by-loop perspective altogether.

Integrability is a substantial field in its own right, probably bigger than amplitudeology. There’s a lot going on, and only some of it touches on amplitudes-related topics. When it does, though, it’s quite impressive, with the flagship example being the work of Benjamin Basso, Amit Sever, and Pedro Vieira. They are able to compute amplitudes in planar N=4 super Yang-Mills for any and all loops, instead making an approximation based on the particle momenta. These days, they’re working on making their method more complete and robust, while building up understanding of other structures that might eventually allow them to say something about the non-planar case.

CHY and the Ambitwistor String

Ed Witten’s involvement in BCFW didn’t come completely out of left field. He had shown interest in N=4 super Yang-Mills earlier, with the invention of the twistor string. The twistor string calculates tree amplitudes in N=4 super Yang-Mills as the result of a string-theory-like framework. The advantage to such a framework is that, while normal quantum field theory involves large numbers of different diagrams, string theory only has one diagram “shape” for each loop.

This advantage has been thrust back into the spotlight recently via the work of Freddy Cachazo, Song He, and Ellis Ye Yuan. Their CHY formula works not just for N=4 super Yang-Mills, but for a wide (and growing) variety of other theories, allowing them to examine those theories’ properties in a particularly powerful way. Meanwhile, Lionel Mason and David Skinner have given the CHY formula a more solid theoretical grounding in the form of their ambitwistor string, which they have recently been able to generalize to a loop-level proposal.

Amplitudeology is a large and growing field, and there are definitely important people I haven’t mentioned. Some, like Henriette Elvang and Yu-tin Huang, have been involved with many different things over the years, so there wasn’t a clear place to put them. Others are part of the European community, where there’s a lot of work on string theory amplitudes and on pushing the boundaries of polylogarithms. Still others were left out simply because I ran out of room. I’ve only covered a small part of the field here, but I hope that small part gives you an idea of the richness of the whole.

2 thoughts on “Amplitudes Megapost

  1. Pingback: An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes | Shadows of Simplicity

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