Category Archives: Amplitudes Methods

Calabi-Yaus in Feynman Diagrams: Harder and Easier Than Expected

I’ve got a new paper up, about the weird geometrical spaces we keep finding in Feynman diagrams.

With Jacob Bourjaily, Andrew McLeod, and Matthias Wilhelm, and most recently Cristian Vergu and Matthias Volk, I’ve been digging up odd mathematics in particle physics calculations. In several calculations, we’ve found that we need a type of space called a Calabi-Yau manifold. These spaces are often studied by string theorists, who hope they represent how “extra” dimensions of space are curled up. String theorists have found an absurdly large number of Calabi-Yau manifolds, so many that some are trying to sift through them with machine learning. We wanted to know if our situation was quite that ridiculous: how many Calabi-Yaus do we really need?

So we started asking around, trying to figure out how to classify our catch of Calabi-Yaus. And mostly, we just got confused.

It turns out there are a lot of different tools out there for understanding Calabi-Yaus, and most of them aren’t all that useful for what we’re doing. We went in circles for a while trying to understand how to desingularize toric varieties, and other things that will sound like gibberish to most of you. In the end, though, we noticed one small thing that made our lives a whole lot simpler.

It turns out that all of the Calabi-Yaus we’ve found are, in some sense, the same. While the details of the physics varies, the overall “space” is the same in each case. It’s a space we kept finding for our “Calabi-Yau bestiary”, but it turns out one of the “traintrack” diagrams we found earlier can be written in the same way. We found another example too, a “wheel” that seems to be the same type of Calabi-Yau.

And that actually has a sensible name

We also found many examples that we don’t understand. Add another rung to our “traintrack” and we suddenly can’t write it in the same space. (Personally, I’m quite confused about this one.) Add another spoke to our wheel and we confuse ourselves in a different way.

So while our calculation turned out simpler than expected, we don’t think this is the full story. Our Calabi-Yaus might live in “the same space”, but there are also physics-related differences between them, and these we still don’t understand.

At some point, our abstract included the phrase “this paper raises more questions than it answers”. It doesn’t say that now, but it’s still true. We wrote this paper because, after getting very confused, we ended up able to say a few new things that hadn’t been said before. But the questions we raise are if anything more important. We want to inspire new interest in this field, toss out new examples, and get people thinking harder about the geometry of Feynman integrals.

Amplitudes 2019 Retrospective

I’m back from Amplitudes 2019, and since I have more time I figured I’d write down a few more impressions.

Amplitudes runs all the way from practical LHC calculations to almost pure mathematics, and this conference had plenty of both as well as everything in between. On the more practical side a standard “pipeline” has developed: get a large number of integrals from generalized unitarity, reduce them to a more manageable number with integration-by-parts, and then compute them with differential equations. Vladimir Smirnov and Johannes Henn presented the state of the art in this pipeline, challenging QCD calculations that required powerful methods. Others aimed to replace various parts of the pipeline. Integration-by-parts could be avoided in the numerical unitarity approach discussed by Ben Page, or alternatively with the intersection theory techniques showcased by Pierpaolo Mastrolia. More radical departures included Stefan Weinzierl’s refinement of loop-tree duality, and Jacob Bourjaily’s advocacy of prescriptive unitarity. Robert Schabinger even brought up direct integration, though I mostly viewed his talk as an independent confirmation of the usefulness of Erik Panzer’s thesis. It also showcased an interesting integral that had previously been represented by Lorenzo Tancredi and collaborators as elliptic, but turned out to be writable in terms of more familiar functions. It’s going to be interesting to see whether other such integrals arise, and whether they can be spotted in advance.

On the other end of the scale, Francis Brown was the only speaker deep enough in the culture of mathematics to insist on doing a blackboard talk. Since the conference hall didn’t actually have a blackboard, this was accomplished by projecting video of a piece of paper that he wrote on as the talk progressed. Despite the awkward setup, the talk was impressively clear, though there were enough questions that he ran out of time at the end and had to “cheat” by just projecting his notes instead. He presented a few theorems about the sort of integrals that show up in string theory. Federico Zerbini and Eduardo Casali’s talks covered similar topics, with the latter also involving intersection theory. Intersection theory also appeared in a poster from grad student Andrzej Pokraka, which overall is a pretty impressively broad showing for a part of mathematics that Sebastian Mizera first introduced to the amplitudes community less than two years ago.

Nima Arkani-Hamed’s talk on Wednesday fell somewhere in between. A series of airline mishaps brought him there only a few hours before his talk, and his own busy schedule sent him back to the airport right after the last question. The talk itself covered several topics, tied together a bit better than usual by a nice account in the beginning of what might motivate a “polytope picture” of quantum field theory. One particularly interesting aspect was a suggestion of a space, smaller than the amplituhedron, that might more accuractly the describe the “alphabet” that appears in N=4 super Yang-Mills amplitudes. If his proposal works, it may be that the infinite alphabet we were worried about for eight-particle amplitudes is actually finite. Ömer Gürdoğan’s talk mentioned this, and drew out some implications. Overall, I’m still unclear as to what this story says about whether the alphabet contains square roots, but that’s a topic for another day. My talk was right after Nima’s, and while he went over-time as always I compensated by accidentally going under-time. Overall, I think folks had fun regardless.

Though I don’t know how many people recognized this guy

Amplitudes 2019

It’s that time of year again, and I’m at Amplitudes, my field’s big yearly conference. This year we’re in Dublin, hosted by Trinity.

Which also hosts the Book of Kells, and the occasional conference reception just down the hall from the Book of Kells

Increasingly, the organizers of Amplitudes have been setting aside a few slots for talks from people in other fields. This year the “closest” such speaker was Kirill Melnikov, who pointed out some of the hurdles that make it difficult to have useful calculations to compare to the LHC. Many of these hurdles aren’t things that amplitudes-people have traditionally worked on, but are still things that might benefit from our particular expertise. Another such speaker, Maxwell Hansen, is from a field called Lattice QCD. While amplitudeologists typically compute with approximations, order by order in more and more complicated diagrams, Lattice QCD instead simulates particle physics on supercomputers, chopping up their calculations on a grid. This allows them to study much stronger forces, including the messy interactions of quarks inside protons, but they have a harder time with the situations we’re best at, where two particles collide from far away. Apparently, though, they are making progress on that kind of calculation, with some clever tricks to connect it to calculations they know how to do. While I was a bit worried that this would let them fire all the amplitudeologists and replace us with supercomputers, they’re not quite there yet, nonetheless they are doing better than I would have expected. Other speakers from other fields included Leron Borsten, who has been applying the amplitudes concept of the “double copy” to M theory and Andrew Tolley, who uses the kind of “positivity” properties that amplitudeologists find interesting to restrict the kinds of theories used in cosmology.

The biggest set of “non-traditional-amplitudes” talks focused on using amplitudes techniques to calculate the behavior not of particles but of black holes, to predict the gravitational wave patterns detected by LIGO. This year featured a record six talks on the topic, a sixth of the conference. Last year I commented that the research ideas from amplitudeologists on gravitational waves had gotten more robust, with clearer proposals for how to move forward. This year things have developed even further, with several initial results. Even more encouragingly, while there are several groups doing different things they appear to be genuinely listening to each other: there were plenty of references in the talks both to other amplitudes groups and to work by more traditional gravitational physicists. There’s definitely still plenty of lingering confusion that needs to be cleared up, but it looks like the community is robust enough to work through it.

I’m still busy with the conference, but I’ll say more when I’m back next week. Stay tuned for square roots, clusters, and Nima’s travel schedule. And if you’re a regular reader, please fill out last week’s poll if you haven’t already!

Hexagon Functions VI: The Power Cosmic

I have a new paper out this week. It’s the long-awaited companion to a paper I blogged about a few months back, itself the latest step in a program that has made up a major chunk of my research.

The title is a bit of a mouthful, but I’ll walk you through it:

The Cosmic Galois Group and Extended Steinmann Relations for Planar N = 4 SYM Amplitudes

I calculate scattering amplitudes (roughly, probabilities that elementary particles bounce off each other) in a (not realistic, and not meant to be) theory called planar N=4 super-Yang-Mills (SYM for short). I can’t summarize everything we’ve been doing here, but if you read the blog posts I linked above and some of the Handy Handbooks linked at the top of the page you’ll hopefully get a clearer picture.

We started using the Steinmann Relations a few years ago. Discovered in the 60’s, the Steinmann relations restrict the kind of equations we can use to describe particle physics. Essentially, they mean that particles can’t travel two ways at once. In this paper, we extend the Steinmann relations beyond Steinmann’s original idea. We don’t yet know if we can prove this extension works, but it seems to be true for the amplitudes we’re calculating. While we’ve presented this in talks before, this is the first time we’ve published it, and it’s one of the big results of this paper.

The other, more exotic-sounding result, has to do with something called the Cosmic Galois Group.

Évariste Galois, the famously duel-prone mathematician, figured out relations between algebraic numbers (that is, numbers you can get out of algebraic equations) in terms of a mathematical structure called a group. Today, mathematicians are interested not just in algebraic numbers, but in relations between transcendental numbers as well, specifically a kind of transcendental number called a period. These numbers show up a lot in physics, so mathematicians have been thinking about a Galois group for transcendental numbers that show up in physics, a so-called Cosmic Galois Group.

(Cosmic here doesn’t mean it has to do with cosmology. As far as I can tell, mathematicians just thought it sounded cool and physics-y. They also started out with rather ambitious ideas about it, if you want a laugh check out the last few paragraphs of this talk by Cartier.)

For us, Cosmic Galois Theory lets us study the unusual numbers that show up in our calculations. Doing this, we’ve noticed that certain numbers simply don’t show up. For example, the Riemann zeta function shows up often in our results, evaluated at many different numbers…but never evaluated at the number three. Nor does any number related to that one through the Cosmic Galois Group show up. It’s as if the theory only likes some numbers, and not others.

This weird behavior has been observed before. Mathematicians can prove it happens for some simple theories, but it even applies to the theories that describe the real world, for example to calculations of the way an electron’s path is bent by a magnetic field. Each theory seems to have its own preferred family of numbers.

For us, this has been enormously useful. We calculate our amplitudes by guesswork, starting with the right “alphabet” and then filling in different combinations, as if we’re trying all possible answers to a word jumble. Cosmic Galois Theory and Extended Steinmann have enabled us to narrow down our guess dramatically, making it much easier and faster to get to the right answer.

More generally though, we hope to contribute to mathematicians’ investigations of Cosmic Galois Theory. Our examples are more complicated than the simple theories where they currently prove things, and contain more data than the more limited results from electrons. Hopefully together we can figure out why certain numbers show up and others don’t, and find interesting mathematical principles behind the theories that govern fundamental physics.

For now, I’ll leave you with a preview of a talk I’m giving in a couple weeks’ time:

The font, of course, is Cosmic Sans

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.

Hexagon Functions V: Seventh Heaven

I’ve got a new paper out this week, a continuation of a story that has threaded through my career since grad school. With a growing collaboration (now Simon Caron-Huot, Lance Dixon, Falko Dulat, Andrew McLeod, and Georgios Papathanasiou) I’ve been calculating six-particle scattering amplitudes in my favorite theory-that-does-not-describe-the-real-world, N=4 super Yang-Mills. We’ve been pushing to more and more “loops”: tougher and tougher calculations that approximate the full answer better and better, using the “word jumble” trick I talked about in Scientific American. And each time, we learn something new.

Now we’re up to seven loops for some types of particles, and six loops for the rest. In older blog posts I talked in megabytes: half a megabyte for three loops, 15 MB for four loops, 300 MB for five loops. I don’t have a number like that for six and seven loops: we don’t store the result in that way anymore, it just got too cumbersome. We have to store it in a simplified form, and even that takes 80 MB.

Some of what we learned has to do with the types of mathematical functions that we need: our “guess” for the result at each loop. We’ve honed that guess down a lot, and discovered some new simplifications along the way. I won’t tell that story here (except to hint that it has to do with “cosmic Galois theory”) because we haven’t published it yet. It will be out in a companion paper soon.

This paper focused on the next step, going from our guess to the correct six- and seven-loop answers. Here too there were surprises. For the last several loops, we’d observed a surprisingly nice pattern: different configurations of particles with different numbers of loops were related, in a way we didn’t know how to explain. The pattern stuck around at five loops, so we assumed it was the real deal, and guessed the new answer would obey it too.

Yes, in our field this counts as surprisingly nice

Usually when scientists tell this kind of story, the pattern works, it’s a breakthrough, everyone gets a Nobel prize, etc. This time? Nope!

The pattern failed. And it failed in a way that was surprisingly difficult to detect.

The way we calculate these things, we start with a guess and then add what we know. If we know something about how the particles behave at high energies, or when they get close together, we use that to pare down our guess, getting rid of pieces that don’t fit. We kept adding these pieces of information, and each time the pattern seemed ok. It was only when we got far enough into one of these approximations that we noticed a piece that didn’t fit.

That piece was a surprisingly stealthy mathematical function, one that hid from almost every test we could perform. There aren’t any functions like that at lower loops, so we never had to worry about this before. But now, in the rarefied land of six-loop calculations, they finally start to show up.

We have another pattern, like the old one but that isn’t broken yet. But at this point we’re cautious: things get strange as calculations get more complicated, and sometimes the nice simplifications we notice are just accidents. It’s always important to check.

Deep physics or six-loop accident? You decide!

This result was a long time coming. Coordinating a large project with such a widely spread collaboration is difficult, and sometimes frustrating. People get distracted by other projects, they have disagreements about what the paper should say, even scheduling Skype around everyone’s time zones is a challenge. I’m more than a little exhausted, but happy that the paper is out, and that we’re close to finishing the companion paper as well. It’s good to have results that we’ve been hinting at in talks finally out where the community can see them. Maybe they’ll notice something new!


Amplitudes in String and Field Theory at NBI

There’s a conference at the Niels Bohr Institute this week, on Amplitudes in String and Field Theory. Like the conference a few weeks back, this one was funded by the Simons Foundation, as part of Michael Green’s visit here.

The first day featured a two-part talk by Michael Green and Congkao Wen. They are looking at the corrections that string theory adds on top of theories of supergravity. These corrections are difficult to calculate directly from string theory, but one can figure out a lot about them from the kinds of symmetry and duality properties they need to have, using the mathematics of modular forms. While Michael’s talk introduced the topic with a discussion of older work, Congkao talked about their recent progress looking at this from an amplitudes perspective.

Francesca Ferrari’s talk on Tuesday also related to modular forms, while Oliver Schlotterer and Pierre Vanhove talked about a different corner of mathematics, single-valued polylogarithms. These single-valued polylogarithms are of interest to string theorists because they seem to connect two parts of string theory: the open strings that describe Yang-Mills forces and the closed strings that describe gravity. In particular, it looks like you can take a calculation in open string theory and just replace numbers and polylogarithms with their “single-valued counterparts” to get the same calculation in closed string theory. Interestingly, there is more than one way that mathematicians can define “single-valued counterparts”, but only one such definition, the one due to Francis Brown, seems to make this trick work. When I asked Pierre about this he quipped it was because “Francis Brown has good taste…either that, or String Theory has good taste.”

Wednesday saw several talks exploring interesting features of string theory. Nathan Berkovitz discussed his new paper, which makes a certain context of AdS/CFT (a duality between string theory in certain curved spaces and field theory on the boundary of those spaces) manifest particularly nicely. By writing string theory in five-dimensional AdS space in the right way, he can show that if the AdS space is small it will generate the same Feynman diagrams that one would use to do calculations in N=4 super Yang-Mills. In the afternoon, Sameer Murthy showed how localization techniques can be used in gravity theories, including to calculate the entropy of black holes in string theory, while Yvonne Geyer talked about how to combine the string theory-like CHY method for calculating amplitudes with supersymmetry, especially in higher dimensions where the relevant mathematics gets tricky.

Thursday ended up focused on field theory. Carlos Mafra was originally going to speak but he wasn’t feeling well, so instead I gave a talk about the “tardigrade” integrals I’ve been looking at. Zvi Bern talked about his work applying amplitudes techniques to make predictions for LIGO. This subject has advanced a lot in the last few years, and now Zvi and collaborators have finally done a calculation beyond what others had been able to do with older methods. They still have a way to go before they beat the traditional methods overall, but they’re off to a great start. Lance Dixon talked about two-loop five-particle non-planar amplitudes in N=4 super Yang-Mills and N=8 supergravity. These are quite a bit trickier than the planar amplitudes I’ve worked on with him in the past, in particular it’s not yet possible to do this just by guessing the answer without considering Feynman diagrams.

Today was the last day of the conference, and the emphasis was on number theory. David Broadhurst described some interesting contributions from physics to mathematics, in particular emphasizing information that the Weierstrass formulation of elliptic curves omits. Eric D’Hoker discussed how the concept of transcendentality, previously used in field theory, could be applied to string theory. A few of his speculations seemed a bit farfetched (in particular, his setup needs to treat certain rational numbers as if they were transcendental), but after his talk I’m a bit more optimistic that there could be something useful there.