The last few weeks have seen a rain of amplitudes papers on arXiv, including quite a few interesting ones.
Over the last year Nima Arkani-Hamed has been talking up four or five really interesting results, and not actually publishing any of them. This has understandably frustrated pretty much everybody. In the last week he published two of them, Cosmological Polytopes and the Wavefunction of the Universe with Paolo Benincasa and Alexander Postnikov and Scattering Amplitudes For All Masses and Spins with Tzu-Chen Huang and Yu-tin Huang. So while I’ll have to wait on the others (I’m particularly looking forward to seeing what he’s been working on with Ellis Yuan) this can at least tide me over.
Cosmological Polytopes and the Wavefunction of the Universe is Nima & co.’s attempt to get a geometrical picture for cosmological correlators, analogous to the Ampituhedron. Cosmological correlators ask questions about the overall behavior of the visible universe: how likely is one clump of matter to be some distance from another? What sorts of patterns might we see in the Cosmic Microwave Background? This is the sort of thing that can be used for “cosmological collider physics”, an idea I mention briefly here.
Paolo Benincasa was visiting Perimeter near the end of my time there, so I got a few chances to chat with him about this. One thing he mentioned, but that didn’t register fully at the time, was Postnikov’s involvement. I had expected that even if Nima and Paolo found something interesting that it wouldn’t lead to particularly deep mathematics. Unlike the N=4 super Yang-Mills theory that generates the Amplituhedron, the theories involved in these cosmological correlators aren’t particularly unique, they’re just a particular class of models cosmologists use that happen to work well with Nima’s methods. Given that, it’s really surprising that they found something mathematically interesting enough to interest Postnikov, a mathematician who was involved in the early days of the Amplituhedron’s predecessor, the Positive Grassmannian. If there’s something that mathematically worthwhile in such a seemingly arbitrary theory then perhaps some of the beauty of the Amplithedron are much more general than I had thought.
Scattering Amplitudes For All Masses and Spins is on some level a byproduct of Nima and Yu-tin’s investigations of whether string theory is unique. Still, it’s a useful byproduct. Many of the tricks we use in scattering amplitudes are at their best for theories with massless particles. Once the particles have masses our notation gets a lot messier, and we often have to rely on older methods. What Nima, Yu-tin, and Tzu-Chen have done here is to build a notation similar to what we use for massless particle, but for massive ones.
The advantage of doing this isn’t just clean-looking papers: using this notation makes it a lot easier to see what kinds of theories make sense. There are a variety of old theorems that restrict what sorts of theories you can write down: photons can’t interact directly with each other, there can only be one “gravitational force”, particles with spins greater than two shouldn’t be massless, etc. The original theorems were often fairly involved, but for massless particles there were usually nice ways to prove them in modern amplitudes notation. Yu-tin in particular has a lot of experience finding these kinds of proofs. What the new notation does is make these nice simple proofs possible for massive particles as well. For example, you can try to use the new notation to write down an interaction between a massive particle with spin greater than two and gravity, and what you find is that any expression you write breaks down: it works fine at low energies, but once you’re looking at particles with energies much higher than their mass you start predicting probabilities greater than one. This suggests that particles with higher spins shouldn’t be “fundamental”, they should be explained in terms of other particles at higher energies. The only way around this turns out to be an infinite series of particles to cancel problems from the previous ones, the sort of structure that higher vibrations have in string theory. I often don’t appreciate papers that others claim are a pleasure to read, but this one really was a pleasure to read: there’s something viscerally satisfying about seeing so many important constraints manifest so cleanly.
I’ve talked before about the difference between planar and non-planar theories. Planar theories end up being simpler, and in the case of N=4 super Yang-Mills this results in powerful symmetries that let us do much more complicated calculations. Non-planar theories are more complicated, but necessary for understanding gravity. Dual Conformal Symmetry, Integration-by-Parts Reduction, Differential Equations and the Nonplanar Sector, a new paper by Zvi Bern, Michael Enciso, Harald Ita, and Mao Zeng, works on bridging the gap between these two worlds.
Most of the paper is concerned with using some of the symmetries of N=4 super Yang-Mills in other, more realistic (but still planar) theories. The idea is that even if those symmetries don’t hold one can still use techniques that respect those symmetries, and those techniques can often be a lot cleaner than techniques that don’t. This is probably the most practically useful part of the paper, but the part I was most curious about is in the last few sections, where they discuss non-planar theories. For a while now I’ve been interested in ways to treat a non-planar theory as if it were planar, to try to leverage the powerful symmetries we have in planar N=4 super Yang-Mills elsewhere. Their trick is surprisingly simple: they just cut the diagram open! Oddly enough, they really do end up with similar symmetries using this method. I still need to read this in more detail to understand its limitations, since deep down it feels like something this simple couldn’t possibly work. Still, if anything like the symmetries of planar N=4 holds in the non-planar case there’s a lot we could do with it.
There are a bunch of other interesting recent papers that I haven’t had time to read. Some look like they might relate to weird properties of N=4 super Yang-Mills, others say interesting things about the interconnected web of theories tied together by their behavior when a particle becomes “soft”. Another presents a method for dealing with elliptic functions, one of the main obstructions to applying my hexagon function technique to more situations. And of course I shouldn’t fail to mention a paper by my colleague Carlos Cardona, applying amplitudes techniques to AdS/CFT. Overall, a lot of interesting stuff in a short span of time. I should probably get back to reading it!
Interesting times indeed. I would be interested to see a discussion somewhere of the hurdles one must overcome to bridge the gap between calculating amplitudes in “toy” theories and the real world. Off the top of my head:
1) include Non-planer diagrams
2) Include massive particles (without choosing a frame)
3) Integrands beyond tree level keeping super symmetry
4) Integrands beyond tree level without super symmetry
5) Doing the integrals efficiently (Put V. Smirnov out of business:)
It sounds like #2 is now solved. It is really starting to look like all the initial work on “toy” models is now thoroughly vindicated. What else would you add to the list? Where do you think we stand on each hurdle?
Plenty of methods already work for 1), just not all. I’d say 1) and 2) are on similar footing now with Nima&Yu-tin’s paper, where we have nice variables and good way to do cuts, but more elaborate stuff like the Amplituhedron, polylog bootstraps, and integrability are still either inapplicable or not there yet. 3) is essentially solved if by supersymmetry you mean N=4, there are methods to calculate them to very high orders and they’re all there in principle in the Amplituhedron. With 4) there’s progress to varying degrees in specific cases and some very powerful methods but I wouldn’t say it’s a solved problem. 5) is probably the trickiest, with Smirnov still the go-to, though Henn’s differential equation method is very very nice.
The main thing I would add to the list (though it’s sort of already part of 5)) is handling elliptic functions. Currently the most powerful methods we have are polylog-only, and elliptics are very generic in realistic theories, so I think that’s the next big hurdle.
Thanks for the update. Yes I meant Maximal super-symmetry (N=4 for spin 1 an N=8 for spin 2). Regardingf #5: Does that mean elliptics do not show up in N=4 SYM ? Can you tell just from the topology of a diagram if the integral will be eliiptic or polylog?
Are you aware of anyone working software to help visualize (by projecting onto 2 or 3D) the (an) amplitehedron?
Elliptics aren’t supposed to show up in some subset of planar N=4 SYM (for the experts, MHV and NMHV amplitudes). They can show up once more particles get involved, so the first case where people are pretty confident they’ll show up is for ten particle scattering. There are some algorithms that you can apply to a graph to see if it could possibly be elliptic, but it’s still the sort of thing where you don’t really know for sure until you’ve worked it out in some detail.
I don’t know of anyone working on amplituhedron visualizations. It’s a bit of an odd thing to do since the space it’s in is a bit abstract: not just space with some number of dimensions but the space of matrices of some dimensions with some properties..not the sort of thing where visualization is usually very helpful.