Category Archives: Amplitudes Methods

Amplitudes 2025 This Week

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Summer is conference season for academics, and this week held my old sub-field’s big yearly conference, called Amplitudes. This year, it was in Seoul at Seoul National University, the first time the conference has been in Asia.

(I wasn’t there, I don’t go to these anymore. But I’ve been skimming slides in my free time, to give you folks the updates you crave. Be forewarned that conference posts like these get technical fast, I’ll be back to my usual accessible self next week.)

There isn’t a huge amplitudes community in Korea, but it’s bigger than it was back when I got started in the field. Of the organizers, Kanghoon Lee of the Asia Pacific Center for Theoretical Physics and Sangmin Lee of Seoul National University have what I think of as “core amplitudes interests”, like recursion relations and the double-copy. The other Korean organizers are from adjacent areas, work that overlaps with amplitudes but doesn’t show up at the conference each year. There was also a sizeable group of organizers from Taiwan, where there has been a significant amplitudes presence for some time now. I do wonder if Korea was chosen as a compromise between a conference hosted in Taiwan or in mainland China, where there is also quite a substantial amplitudes community.

One thing that impresses me every year is how big, and how sophisticated, the gravitational-wave community in amplitudes has grown. Federico Buccioni’s talk began with a plot that illustrates this well (though that wasn’t his goal):

At the conference Amplitudes, dedicated to the topic of scattering amplitudes, there were almost as many talks with the phrase “black hole” in the title as there were with “scattering” or “amplitudes”! This is for a topic that did not even exist in the subfield when I got my PhD eleven years ago.

With that said, gravitational wave astronomy wasn’t quite as dominant at the conference as Buccioni’s bar chart suggests. There were a few talks each day on the topic: I counted seven in total, excluding any short talks on the subject in the gong show. Spinning black holes were a significant focus, central to Jung-Wook Kim’s, Andres Luna’s and Mao Zeng’s talks (the latter two showing some interesting links between the amplitudes story and classic ideas in classical mechanics) and relevant in several others, with Riccardo Gonzo, Miguel Correia, Ira Rothstein, and Enrico Herrmann’s talks showing not just a wide range of approaches, but an increasing depth of research in this area.

Herrmann’s talk in particular dealt with detector event shapes, a framework that lets physicists think more directly about what a specific particle detector or observer can see. He applied the idea not just to gravitational waves but to quantum gravity and collider physics as well. The latter is historically where this idea has been applied the most thoroughly, as highlighted in Hua Xing Zhu’s talk, where he used them to pick out particular phenomena of interest in QCD.

QCD is, of course, always of interest in the amplitudes field. Buccioni’s talk dealt with the theory’s behavior at high-energies, with a nice example of the “maximal transcendentality principle” where some quantities in QCD are identical to quantities in N=4 super Yang-Mills in the “most transcendental” pieces (loosely, those with the highest powers of pi). Andrea Guerreri’s talk also dealt with high-energy behavior in QCD, trying to address an experimental puzzle where QCD results appeared to violate a fundamental bound all sensible theories were expected to obey. By using S-matrix bootstrap techniques, they clarify the nature of the bound, finding that QCD still obeys it once correctly understood, and conjecture a weird theory that should be possible to frame right on the edge of the bound. The S-matrix bootstrap was also used by Alexandre Homrich, who talked about getting the framework to work for multi-particle scattering.

Heribertus Bayu Hartanto is another recent addition to Korea’s amplitudes community. He talked about a concrete calculation, two-loop five-particle scattering including top quarks, a tricky case that includes elliptic curves.

When amplitudes lead to integrals involving elliptic curves, many standard methods fail. Jake Bourjaily’s talk raised a question he has brought up again and again: what does it mean to do an integral for a new type of function? One possible answer is that it depends on what kind of numerics you can do, and since more general numerical methods can be cumbersome one often needs to understand the new type of function in more detail. In light of that, Stephen Jones’ talk was interesting in taking a common problem often cited with generic approaches (that they have trouble with the complex numbers introduced by Minkowski space) and finding a more natural way in a particular generic approach (sector decomposition) to take them into account. Giulio Salvatori talked about a much less conventional numerical method, linked to the latest trend in Nima-ology, surfaceology. One of the big selling points of the surface integral framework promoted by people like Salvatori and Nima Arkani-Hamed is that it’s supposed to give a clear integral to do for each scattering amplitude, one which should be amenable to a numerical treatment recently developed by Michael Borinsky. Salvatori can currently apply the method only to a toy model (up to ten loops!), but he has some ideas for how to generalize it, which will require handling divergences and numerators.

Other approaches to the “problem of integration” included Anna-Laura Sattelberger’s talk that presented a method to find differential equations for the kind of integrals that show up in amplitudes using the mathematical software Macaulay2, including presenting a package. Matthias Wilhelm talked about the work I did with him, using machine learning to find better methods for solving integrals with integration-by-parts, an area where two other groups have now also published. Pierpaolo Mastrolia talked about integration-by-parts’ up-and-coming contender, intersection theory, a method which appears to be delving into more mathematical tools in an effort to catch up with its competitor.

Sometimes, one is more specifically interested in the singularities of integrals than their numerics more generally. Felix Tellander talked about a geometric method to pin these down which largely went over my head, but he did have a very nice short description of the approach: “Describe the singularities of the integrand. Find a map representing integration. Map the singularities of the integrand onto the singularities of the integral.”

While QCD and gravity are the applications of choice, amplitudes methods germinate in N=4 super Yang-Mills. Ruth Britto’s talk opened the conference with an overview of progress along those lines before going into her own recent work with one-loop integrals and interesting implications of ideas from cluster algebras. Cluster algebras made appearances in several other talks, including Anastasia Volovich’s talk which discussed how ideas from that corner called flag cluster algebras may give insights into QCD amplitudes, though some symbol letters still seem to be hard to track down. Matteo Parisi covered another idea, cluster promotion maps, which he thinks may help pin down algebraic symbol letters.

The link between cluster algebras and symbol letters is an ongoing mystery where the field is seeing progress. Another symbol letter mystery is antipodal duality, where flipping an amplitude like a palindrome somehow gives another valid amplitude. Lance Dixon has made progress in understanding where this duality comes from, finding a toy model where it can be understood and proved.

Others pushed the boundaries of methods specific to N=4 super Yang-Mills, looking for novel structures. Song He’s talk pushes an older approach by Bourjaily and collaborators up to twelve loops, finding new patterns and connections to other theories and observables. Qinglin Yang bootstraps Wilson loops with a Lagrangian insertion, adding a side to the polygon used in previous efforts and finding that, much like when you add particles to amplitudes in a bootstrap, the method gets stricter and more powerful. Jaroslav Trnka talked about work he has been doing with “negative geometries”, an odd method descended from the amplituhedron that looks at amplitudes from a totally different perspective, probing a bit of their non-perturbative data. He’s finding more parts of that setup that can be accessed and re-summed, finding interestingly that multiple-zeta-values show up in quantities where we know they ultimately cancel out. Livia Ferro also talked about a descendant of the amplituhedron, this time for cosmology, getting differential equations for cosmological observables in a particular theory from a combinatorial approach.

Outside of everybody’s favorite theories, some speakers talked about more general approaches to understanding the differences between theories. Andreas Helset covered work on the geometry of the space of quantum fields in a theory, applying the method to a general framework for characterizing deviations from the standard model called the SMEFT. Jasper Roosmale Nepveu also talked about a general space of theories, thinking about how positivity (a trait linked to fundamental constraints like causality and unitarity) gets tangled up with loop effects, and the implications this has for renormalization.

Soft theorems, universal behavior of amplitudes when a particle has low energy, continue to be a trendy topic, with Silvia Nagy showing how the story continues to higher orders and Sangmin Choi investigating loop effects. Callum Jones talks about one of the more powerful results from the soft limit, Weinberg’s theorem showing the uniqueness of gravity. Weinberg’s proof was set up in Minkowski space, but we may ultimately live in curved, de Sitter space. Jones showed how the ideas Weinberg explored generalize in de Sitter, using some tools from the soft-theorem-inspired field of dS/CFT. Julio Parra-Martinez, meanwhile, tied soft theorems to another trendy topic, higher symmetries, a more general notion of the usual types of symmetries that physicists have explored in the past. Lucia Cordova reported work that was not particularly connected to soft theorems but was connected to these higher symmetries, showing how they interact with crossing symmetry and the S-matrix bootstrap.

Finally, a surprisingly large number of talks linked to Kevin Costello and Natalie Paquette’s work with self-dual gauge theories, where they found exact solutions from a fairly mathy angle. Paquette gave an update on her work on the topic, while Alfredo Guevara talked about applications to black holes, comparing the power of expanding around a self-dual gauge theory to that of working with supersymmetry. Atul Sharma looked at scattering in self-dual backgrounds in work that merges older twistor space ideas with the new approach, while Roland Bittelson talked about calculating around an instanton background.

Also, I had another piece up this week at FirstPrinciples, based on an interview with the (outgoing) president of the Sloan Foundation. I won’t have a “bonus info” post for this one, as most of what I learned went into the piece. But if you don’t know what the Sloan Foundation does, take a look! I hadn’t known they funded Jupyter notebooks and Hidden Figures, or that they introduced Kahneman and Tversky.

Integration by Parts, Evolved

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I posted what may be my last academic paper today, about a project I’ve been working on with Matthias Wilhelm for most of the last year. The paper is now online here. For me, the project has been a chance to broaden my horizons, learn new skills, and start to step out of my academic comfort zone. For Matthias, I hope it was grant money well spent.

I wanted to work on something related to machine learning, for the usual trendy employability reasons. Matthias was already working with machine learning, but was interested in pursuing a different question.

When is machine learning worthwhile? Machine learning methods are heuristics, unreliable methods that sometimes work well. You don’t use a heuristic if you have a reliable method that runs fast enough. But if all you have are heuristics to begin with, then machine learning can give you a better heuristic.

Matthias noticed a heuristic embedded deep in how we do particle physics, and guessed that we could do better. In particle physics, we use pictures called Feynman diagrams to predict the probabilities for different outcomes of collisions, comparing those predictions to observation to look for evidence of new physics. Each Feynman diagram corresponds to an integral, and for each calculation there are hundreds, thousands, or even millions of those integrals to do.

Luckily, physicists don’t actually have to do all those integrals. It turns out that most of them are related, by a slightly more advanced version of that calculus class mainstay, integration by parts. Using integration by parts you can solve a list of equations, finding out how to write your integrals in terms of a much smaller list.

How big a list of equations do you need, and which ones? Twenty-five years ago, Stefano Laporta proposed a “golden rule” to choose, based on his own experience, and people have been using it (more or less, with their own tweaks) since then.

Laporta’s rule is a heuristic, with no proof that it is the best option, or even that it will always work. So we probably shouldn’t have been surprised when someone came up with a better heuristic. Watching talks at a December 2023 conference, Matthias saw a presentation by Johann Usovitsch on a curious new rule. The rule was surprisingly simple, just one extra condition on top of Laporta’s. But it was enough to reduce the number of equations by a factor of twenty.

That’s great progress, but it’s also a bit frustrating. Over almost twenty-five years, no-one had guessed this one simple change?

Maybe, thought Matthias and I, we need to get better at guessing.

We started out thinking we’d try reinforcement learning, a technique where a machine is trained by playing a game again and again, changing its strategy when that strategy brings it a reward. We thought we could have the machine learn to cut away extra equations, getting rewarded if it could cut more while still getting the right answer. We didn’t end up pursuing this very far before realizing another strategy would be a better fit.

What is a rule, but a program? Laporta’s golden rule and Johann’s new rule could both be expressed as simple programs. So we decided to use a method that could guess programs.

One method stood out for sheer trendiness and audacity: FunSearch. FunSearch is a type of algorithm called a genetic algorithm, which tries to mimic evolution. It makes a population of different programs, “breeds” them with each other to create new programs, and periodically selects out the ones that perform best. That’s not the trendy or audacious part, though, people have been doing that sort of genetic programming for a long time.

The trendy, audacious part is that FunSearch generates these programs with a Large Language Model, or LLM (the type of technology behind ChatGPT). Using an LLM trained to complete code, FunSearch presents the model with two programs labeled v0 and v1 and asks it to complete v2. In general, program v2 will have some traits from v0 and v1, but also a lot of variation due to the unpredictable output of LLMs. The inventors of FunSearch used this to contribute the variation needed for evolution, using it to evolve programs to find better solutions to math problems.

We decided to try FunSearch on our problem, modifying it a bit to fit the case. We asked it to find a shorter list of equations, giving a better score for a shorter list but a penalty if the list wasn’t able to solve the problem fully.

Some tinkering and headaches later, it worked! After a few days and thousands of program guesses, FunSearch was able to find a program that reproduced the new rule Johann had presented. A few hours more, and it even found a rule that was slightly better!

But then we started wondering: do we actually need days of GPU time to do this?

An expert on heuristics we knew had insisted, at the beginning, that we try something simpler. The approach we tried then didn’t work. But after running into some people using genetic programming at a conference last year, we decided to try again, using a Python package they used in their work. This time, it worked like a charm, taking hours rather than days to find good rules.

This was all pretty cool, a great opportunity for me to cut my teeth on Python programming and its various attendant skills. And it’s been inspiring, with Matthias drawing together more people interested in seeing just how much these kinds of heuristic methods can do there. I should be clear though, that so far I don’t think our result is useful. We did better than the state of the art on an example, but only slightly, and in a way that I’d guess doesn’t generalize. And we needed quite a bit of overhead to do it. Ultimately, while I suspect there’s something useful to find in this direction, it’s going to require more collaboration, both with people using the existing methods who know better what the bottlenecks are, and with experts in these, and other, kinds of heuristics.

So I’m curious to see what the future holds. And for the moment, happy that I got to try this out!

Replacing Space-Time With the Space in Your Eyes

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Nima Arkani-Hamed thinks space-time is doomed.

That doesn’t mean he thinks it’s about to be destroyed by a supervillain. Rather, Nima, like many physicists, thinks that space and time are just approximations to a deeper reality. In order to make sense of gravity in a quantum world, seemingly fundamental ideas, like that particles move through particular places at particular times, will probably need to become more flexible.

But while most people who think space-time is doomed research quantum gravity, Nima’s path is different. Nima has been studying scattering amplitudes, formulas used by particle physicists to predict how likely particles are to collide in particular ways. He has been trying to find ways to calculate these scattering amplitudes without referring directly to particles traveling through space and time. In the long run, the hope is that knowing how to do these calculations will help suggest new theories beyond particle physics, theories that can’t be described with space and time at all.

Ten years ago, Nima figured out how to do this in a particular theory, one that doesn’t describe the real world. For that theory he was able to find a new picture of how to calculate scattering amplitudes based on a combinatorical, geometric space with no reference to particles traveling through space-time. He gave this space the catchy name “the amplituhedron“. In the years since, he found a few other “hedra” describing different theories.

Now, he’s got a new approach. The new approach doesn’t have the same kind of catchy name: people sometimes call it surfaceology, or curve integral formalism. Like the amplituhedron, it involves concepts from combinatorics and geometry. It isn’t quite as “pure” as the amplituhedron: it uses a bit more from ordinary particle physics, and while it avoids specific paths in space-time it does care about the shape of those paths. Still, it has one big advantage: unlike the amplituhedron, Nima’s new approach looks like it can work for at least a few of the theories that actually describe the real world.

The amplituhedron was mysterious. Instead of space and time, it described the world in terms of a geometric space whose meaning was unclear. Nima’s new approach also describes the world in terms of a geometric space, but this space’s meaning is a lot more clear.

The space is called “kinematic space”. That probably still sounds mysterious. “Kinematic” in physics refers to motion. In the beginning of a physics class when you study velocity and acceleration before you’ve introduced a single force, you’re studying kinematics. In particle physics, kinematic refers to the motion of the particles you detect. If you see an electron going up and to the right at a tenth the speed of light, those are its kinematics.

Kinematic space, then, is the space of observations. By saying that his approach is based on ideas in kinematic space, what Nima is saying is that it describes colliding particles not based on what they might be doing before they’re detected, but on mathematics that asks questions only about facts about the particles that can be observed.

(For the experts: this isn’t quite true, because he still needs a concept of loop momenta. He’s getting the actual integrands from his approach, rather than the dual definition he got from the amplituhedron. But he does still have to integrate one way or another.)

Quantum mechanics famously has many interpretations. In my experience, Nima’s favorite interpretation is the one known as “shut up and calculate”. Instead of arguing about the nature of an indeterminately philosophical “real world”, Nima thinks quantum physics is a tool to calculate things people can observe in experiments, and that’s the part we should care about.

From a practical perspective, I agree with him. And I think if you have this perspective, then ultimately, kinematic space is where your theories have to live. Kinematic space is nothing more or less than the space of observations, the space defined by where things land in your detectors, or if you’re a human and not a collider, in your eyes. If you want to strip away all the speculation about the nature of reality, this is all that is left over. Any theory, of any reality, will have to be described in this way. So if you think reality might need a totally new weird theory, it makes sense to approach things like Nima does, and start with the one thing that will always remain: observations.

Amplitudes 2024, Continued

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I’ve now had time to look over the rest of the slides from the Amplitudes 2024 conference, so I can say something about Thursday and Friday’s talks.

Thursday was gravity-focused. Zvi Bern’s review talk was actually a review, a tour of the state of the art in using amplitudes techniques to make predictions for gravitational wave physics. Bern emphasized that future experiments will require much more precision: two more orders of magnitude, which in our lingo amounts to two more “loops”. The current state of the art is three loops, but they’ve been hacking away at four, doing things piece by piece in a way that cleverly also yields publications (for example, they can do just the integrals needed for supergravity, which are simpler). Four loops here is the first time that the Feynman diagrams involve Calabi-Yau manifolds, so they will likely need techniques from some of the folks I talked about last week. Once they have four loops, they’ll want to go to five, since that is the level of precision you need to learn something about the material in neutron stars. The talk covered a variety of other developments, some of which were talked about later on Thursday and some of which were only mentioned here.

Of that day’s other speakers, Stefano De Angelis, Lucile Cangemi, Mikhail Ivanov, and Alessandra Buonanno also focused on gravitational waves. De Angelis talked about the subtleties that show up when you try to calculate gravitational waveforms directly with amplitudes methods, showcasing various improvements to the pipeline there. Cangemi talked about a recurring question with its own list of subtleties, namely how the Kerr metric for spinning black holes emerges from the math of amplitudes of spinning particles. Gravitational waves were the focus of only the second half of Ivanov’s talk, where he talked about how amplitudes methods can clear up some of the subtler effects people try to take into account. The first half was about another gravitational application, that of using amplitudes methods to compute the correlations of galaxy structures in the sky, a field where it looks like a lot of progress can be made. Finally, Buonanno gave the kind of talk she’s given a few times at these conferences, a talk that puts these methods in context, explaining how amplitudes results are packaged with other types of calculations into the Effective-One-Body framework which then is more directly used at LIGO. This year’s talk went into more detail about what the predictions are actually used for, which I appreciated. I hadn’t realized that there have been a handful of black hole collisions discovered by other groups from LIGO’s data, a win for open science! Her slides had a nice diagram explaining what data from the gravitational wave is used to infer what black hole properties, quite a bit more organized than the statistical template-matching I was imagining. She explained the logic behind Bern’s statement that gravitational wave telescopes will need two more orders of magnitude, pointing out that that kind of precision is necessary to be sure that something that might appear to be a deviation from Einstein’s theory of gravity is not actually a subtle effect of known physics. Her method typically is adjusted to fit numerical simulations, but she shows that even without that adjustment they now fit the numerics quite well, thanks in part to contributions from amplitudes calculations.

Of the other talks that day, David Kosower’s was the only one that didn’t explicitly involve gravity. Instead, his talk focused on a more general question, namely how to find a well-defined basis of integrals for Feynman diagrams, which turns out to involve some rather subtle mathematics and geometry. This is a topic that my former boss Jake Bourjaily worked on in a different context for some time, and I’m curious whether there is any connection between the two approaches. Oliver Schlotterer gave the day’s second review talk, once again of the “actually a review” kind, covering a variety of recent developments in string theory amplitudes. These include some new pictures of how string theory amplitudes that correspond to Yang-Mills theories “square” to amplitudes involving gravity at higher loops and progress towards going past two loops, the current state of the art for most string amplitude calculations. (For the experts: this does not involve taking the final integral over the moduli space, which is still a big unsolved problem.) He also talked about progress by Sebastian Mizera and collaborators in understanding how the integrals that show up in string theory make sense in the complex plane. This is a problem that people had mostly managed to avoid dealing with because of certain simplifications in the calculations people typically did (no moduli space integration, expansion in the string length), but taking things seriously means confronting it, and Mizera and collaborators found a novel solution to the problem that has already passed a lot of checks. Finally, Tobias Hansen’s talk also related to string theory, specifically in anti-de-Sitter space, where the duality between string theory and N=4 super Yang-Mills lets him and his collaborators do Yang-Mills calculations and see markedly stringy-looking behavior.

Friday began with Kevin Costello, whose not-really-a-review talk dealt with his work with Natalie Paquette showing that one can use an exactly-solvable system to learn something about QCD. This only works for certain rather specific combinations of particles: for example, in order to have three colors of quarks, they need to do the calculation for nine flavors. Still, they managed to do a calculation with this method that had not previously been done with more traditional means, and to me it’s impressive that anything like this works for a theory without supersymmetry. Mina Himwich and Diksha Jain both had talks related to a topic of current interest, “celestial” conformal field theory, a picture that tries to apply ideas from holography in which a theory on the boundary of a space fully describes the interior, to the “boundary” of flat space, infinitely far away. Himwich talked about a symmetry observed in that research program, and how that symmetry can be seen using more normal methods, which also lead to some suggestions of how the idea might be generalized. Jain likewise covered a different approach, one in which one sets artificial boundaries in flat space and sees what happens when those boundaries move.

Yifei He described progress in the modern S-matrix bootstrap approach. Previously, this approach had gotten quite general constraints on amplitudes. She tries to do something more specific, and predict the S-matrix for scattering of pions in the real world. By imposing compatibility with knowledge from low energies and high energies, she was able to find a much more restricted space of consistent S-matrices, and these turn out to actually match pretty well to experimental results. Mathieu Giroux addresses an important question for a variety of parts of amplitudes research, how to predict the singularities of Feynman diagrams. He explored a recursive approach to solving Landau’s equations for these singularities, one which seems impressively powerful, in one case being able to find a solution that in text form is approximately the length of Harry Potter. Finally, Juan Maldacena closed the conference by talking about some progress he’s made towards an old idea, that of defining M theory in terms of a theory involving actual matrices. This is a very challenging thing to do, but he is at least able to tackle the simplest possible case, involving correlations between three observations. This had a known answer, so his work serves mostly as a confirmation that the original idea makes sense at at least this level.

(Not At) Amplitudes 2024 at the IAS

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For over a decade, I studied scattering amplitudes, the formulas particle physicists use to find the probability that particles collide, or scatter, in different ways. I went to Amplitudes, the field’s big yearly conference, every year from 2015 to 2023.

This year is different. I’m on the way out of the field, looking for my next steps. Meanwhile, Amplitudes 2024 is going full speed ahead at the Institute for Advanced Study in Princeton.

With poster art that is, as the kids probably don’t say anymore, “on fleek”

The talks aren’t live-streamed this year, but they are posting slides, and they will be posting recordings. Since a few of my readers are interested in new amplitudes developments, I’ve been paging through the posted slides looking for interesting highlights. So far, I’ve only seen slides from the first few days: I will probably write about the later talks in a future post.

Each day of Amplitudes this year has two 45-minute “review talks”, one first thing in the morning and the other first thing after lunch. I put “review talks” in quotes because they vary a lot, between talks that try to introduce a topic for the rest of the conference to talks that mostly focus on the speaker’s own research. Lorenzo Tancredi’s talk was of the former type, an introduction to the many steps that go into making predictions for the LHC, with a focus on those topics where amplitudeologists have made progress. The talk opens with the type of motivation I’d been writing in grant and job applications over the last few years (we don’t know most of the properties of the Higgs yet! To measure them, we’ll need to calculate amplitudes with massive particles to high precision!), before moving into a review of the challenges and approaches in different steps of these calculations. While Tancredi apologizes in advance that the talk may be biased, I found it surprisingly complete: if you want to get an idea of the current state of the “LHC amplitudes pipeline”, his slides are a good place to start.

Tancredi’s talk serves as introduction for a variety of LHC-focused talks, some later that day and some later in the week. Federica Devoto discussed high-energy quarks while Chiara Signorile-Signorile and George Sterman showed advances in handling of low-energy particles. Xiaofeng Xu has a program that helps predict symbol letters, the building-blocks of scattering amplitudes that can be used to reconstruct or build up the whole thing, while Samuel Abreu talked about a tricky state-of-the-art case where Xu’s program misses part of the answer.

Later Monday morning veered away from the LHC to focus on more toy-model theories. Renata Kallosh’s talk in particular caught my attention. This blog is named after a long-standing question in amplitudes: will the four-graviton amplitude in N=8 supergravity diverge at seven loops in four dimensions? This seemingly arcane question is deep down a question about what is actually required for a successful theory of quantum gravity, and in particular whether some of the virtues of string theory can be captured by a simpler theory instead. Answering the question requires a prodigious calculation, and the more “loops” are involved the more difficult it is. Six years ago, the calculation got to five loops, and it hasn’t passed that mark since then. That five-loop calculation gave some reason for pessimism, a nice pattern at lower loops that stopped applying at five.

Kallosh thinks she has an idea of what to expect. She’s noticed a symmetry in supergravity, one that hadn’t previously been taken into account. She thinks that symmetry should keep N=8 supergravity from diverging on schedule…but only in exactly four dimensions. All of the lower-loop calculations in N=8 supergravity diverged in higher dimensions than four, and it seems like with this new symmetry she understands why. Her suggestion is to focus on other four-dimensional calculations. If seven loops is still too hard, then dialing back the amount of supersymmetry from N=8 to something lower should let her confirm her suspicions. Already a while back N=5 supergravity was found to diverge later than expected in four dimensions. She wants to know whether that pattern continues.

(Her backup slides also have a fun historical point: in dimensions greater than four, you can’t get elliptical planetary orbits. So four dimensions is special for our style of life.)

Other talks on Monday included a talk by Zahra Zahraee on progress towards “solving” the field’s favorite toy model, N=4 super Yang-Mills. Christian Copetti talked about the work I mentioned here, while Meta employee François Charlton’s “review talk” dealt with his work applying machine learning techniques to “translate” between questions in mathematics and their answers. In particular, he reported progress with my current boss Matthias Wilhelm and frequent collaborator and mentor Lance Dixon on using transformers to guess high-loop formulas in N=4 super Yang-Mills. They have an interesting proof of principle now, but it will probably still be a while until they can use the method to predict something beyond the state of the art.

In the meantime at least they have some hilarious AI-generated images

Tuesday’s review by Ian Moult was genuinely a review, but of a topic not otherwise covered at the conference, that of “detector observables”. The idea is that rather than talking about which individual particles are detected, one can ask questions that make more sense in terms of the experimental setup, like asking about the amounts of energy deposited in different detectors. This type of story has gone from an idle observation by theorists to a full research program, with theorists and experimentalists in active dialogue.

Natalia Toro brought up that, while we say each particle has a definite spin, that may not actually be the case. Particles with so-called “continuous spins” can masquerade as particles with a definite integer spin at lower energies. Toro and Schuster promoted this view of particles ten years ago, but now can make a bit more sense of it, including understanding how continuous-spin particles can interact.

The rest of Tuesday continued to be a bit of a grab-bag. Yael Shadmi talked about applying amplitudes techniques to Effective Field Theory calculations, while Franziska Porkert talked about a Feynman diagram involving two different elliptic curves. Interestingly (well, to me at least), the curves never appear “together”, you can represent the diagram as a sum of terms involving one curve and terms involving the other, much simpler than it could have been!

Tuesday afternoon’s review talk by Iain Stewart was one of those “guest from an adjacent field” talks, in this case from an approach called SCET, and at first glance didn’t seem to do much to reach out to the non-SCET people in the audience. Frequent past collaborator of mine Andrew McLeod showed off a new set of relations between singularities of amplitudes, found by digging in to the structure of the equations discovered by Landau that control this behavior. He and his collaborators are proposing a new way to keep track of these things involving “minimal cuts”, a clear pun on the “maximal cuts” that have been of great use to other parts of the community. Whether this has more or less staying power than “negative geometries” remains to be seen.

Closing Tuesday, Shruti Paranjape showed there was more to discover about the simplest amplitudes, called “tree amplitudes”. By asking why these amplitudes are sometimes equal to zero, she was able to draw a connection to the “double-copy” structure that links the theory of the strong force and the theory of gravity. Johannes Henn’s talk noticed an intriguing pattern. A while back, I had looked into under which circumstances amplitudes were positive. Henn found that “positive” is an understatement. In a certain region, the amplitudes we were looking at turn out to not just be positive, but also always decreasing, and also with second derivative always positive. In fact, the derivatives appear to alternate, always with one sign or the other as one takes more derivatives. Henn is calling this unusual property “completely monotonous”, and trying to figure out how widely it holds.

Wednesday had a more mathematical theme. Bernd Sturmfels began with a “review talk” that largely focused on his own work on the space of curves with marked points, including a surprising analogy between amplitudes and the likelihood functions one needs to minimize in machine learning. Lauren Williams was the other “actual mathematician” of the day, and covered her work on various topics related to the amplituhedron.

The remaining talks on Wednesday were not literally by mathematicians, but were “mathematically informed”. Carolina Figueiredo and Hayden Lee talked about work with Nima Arkani-Hamed on different projects. Figueiredo’s talk covered recent developments in the “curve integral formalism”, a recent step in Nima’s quest to geometrize everything in sight, this time in the context of more realistic theories. The talk, which like those Nima gives used tablet-written slides, described new insights one can gain from this picture, including new pictures of how more complicated amplitudes can be built up of simpler ones. If you want to understand the curve integral formalism further, I’d actually suggest instead looking at Mark Spradlin’s slides from later that day. The second part of Spradlin’s talk dealt with an area Figueiredo marked for future research, including fermions in the curve integral picture. I confess I’m still not entirely sure what the curve integral formalism is good for, but Spradlin’s talk gave me a better idea of what it’s doing. (The first part of his talk was on a different topic, exploring the space of string-like amplitudes to figure out which ones are actually consistent.)

Hayden Lee’s talk mentions the emergence of time, but the actual story is a bit more technical. Lee and collaborators are looking at cosmological correlators, observables like scattering amplitudes but for cosmology. Evaluating these is challenging with standard techniques, but can be approached with some novel diagram-based rules which let the results be described in terms of the measurable quantities at the end in a kind of “amplituhedron-esque” way.

Aidan Herderschee and Mariana Carrillo González had talks on Wednesday on ways of dealing with curved space. Herderschee talked about how various amplitudes techniques need to be changed to deal with amplitudes in anti-de-Sitter space, with difference equations replacing differential equations and sum-by-parts relations replacing integration-by-parts relations. Carrillo González looked at curved space through the lens of a special kind of toy model theory called a self-dual theory, which allowed her to do cosmology-related calculations using a double-copy technique.

Finally, Stephen Sharpe had the second review talk on Wednesday. This was another “outside guest” talk, a discussion from someone who does Lattice QCD about how they have been using their methods to calculate scattering amplitudes. They seem to count the number of particles a bit differently than we do, I’m curious whether this came up in the question session.

At Quanta This Week, and Some Bonus Material

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When I moved back to Denmark, I mentioned that I was planning to do more science journalism work. The first fruit of that plan is up this week: I have a piece at Quanta Magazine about a perennially trendy topic in physics, the S-matrix.

It’s been great working with Quanta again. They’ve been thorough, attentive to the science, and patient with my still-uncertain life situation. I’m quite likely to have more pieces there in future, and I’ve got ideas cooking with other outlets as well, so stay tuned!

My piece with Quanta is relatively short, the kind of thing they used to label a “blog” rather than say a “feature”. Since the S-matrix is a pretty broad topic, there were a few things I couldn’t cover there, so I thought it would be nice to discuss them here. You can think of this as a kind of “bonus material” section for the piece. So before reading on, read my piece at Quanta first!

Welcome back!

At Quanta I wrote a kind of cartoon of the S-matrix, asking you to think about it as a matrix of probabilities, with rows for input particles and columns for output particles. There are a couple different simplifications I snuck in there, the pop physicist’s “lies to children“. One, I already flag in the piece: the entries aren’t really probabilities, they’re complex numbers, probability amplitudes.

There’s another simplification that I didn’t have space to flag. The rows and columns aren’t just lists of particles, they’re lists of particles in particular states.

What do I mean by states? A state is a complete description of a particle. A particle’s state includes its energy and momentum, including the direction it’s traveling in. It includes its spin, and the direction of its spin: for example, clockwise or counterclockwise? It also includes any charges, from the familiar electric charge to the color of a quark.

This makes the matrix even bigger than you might have thought. I was already describing an infinite matrix, one where you can have as many columns and rows as you can imagine numbers of colliding particles. But the number of rows and columns isn’t just infinite, but uncountable, as many rows and columns as there are different numbers you can use for energy and momentum.

For some of you, an uncountably infinite matrix doesn’t sound much like a matrix. But for mathematicians familiar with vector spaces, this is totally reasonable. Even if your matrix is infinite, or even uncountably infinite, it can still be useful to think about it as a matrix.

Another subtlety, which I’m sure physicists will be howling at me about: the Higgs boson is not supposed to be in the S-matrix!

In the article, I alluded to the idea that the S-matrix lets you “hide” particles that only exist momentarily inside of a particle collision. The Higgs is precisely that sort of particle, an unstable particle. And normally, the S-matrix is supposed to only describe interactions between stable particles, particles that can survive all the way to infinity.

In my defense, if you want a nice table of probabilities to put in an article, you need an unstable particle: interactions between stable particles depend on their energy and momentum, sometimes in complicated ways, while a single unstable particle will decay into a reliable set of options.

More technically, there are also contexts in which it’s totally fine to think about an S-matrix between unstable particles, even if it’s not usually how we use the idea.

My piece also didn’t have a lot of room to discuss new developments. I thought at minimum I’d say a bit more about the work of the young people I mentioned. You can think of this as an appetizer: there are a lot of people working on different aspects of this subject these days.

Part of the initial inspiration for the piece was when an editor at Quanta noticed a recent paper by Christian Copetti, Lucía Cordova, and Shota Komatsu. The paper shows an interesting case, where one of the “logical” conditions imposed in the original S-matrix bootstrap doesn’t actually apply. It ended up being too technical for the Quanta piece, but I thought I could say a bit about it, and related questions, here.

Some of the conditions imposed by the original bootstrappers seem unavoidable. Quantum mechanics makes no sense if doesn’t compute probabilities, and probabilities can’t be negative, or larger than one, so we’d better have an S-matrix that obeys those rules. Causality is another big one: we probably shouldn’t have an S-matrix that lets us send messages back in time and change the past.

Other conditions came from a mixture of intuition and observation. Crossing is a big one here. Crossing tells you that you can take an S-matrix entry with in-coming particles, and relate it to a different S-matrix entry with out-going anti-particles, using techniques from the calculus of complex numbers.

Crossing may seem quite obscure, but after some experience with S-matrices it feels obvious and intuitive. That’s why for an expert, results like the paper by Copetti, Cordova, and Komatsu seem so surprising. What they found was that a particularly exotic type of symmetry, called a non-invertible symmetry, was incompatible with crossing symmetry. They could find consistent S-matrices for theories with these strange non-invertible symmetries, but only if they threw out one of the basic assumptions of the bootstrap.

This was weird, but upon reflection not too weird. In theories with non-invertible symmetries, the behaviors of different particles are correlated together. One can’t treat far away particles as separate, the way one usually does with the S-matrix. So trying to “cross” a particle from one side of a process to another changes more than it usually would, and you need a more sophisticated approach to keep track of it. When I talked to Cordova and Komatsu, they related this to another concept called soft theorems, aspects of which have been getting a lot of attention and funding of late.

In the meantime, others have been trying to figure out where the crossing rules come from in the first place.

There were attempts in the 1970’s to understand crossing in terms of other fundamental principles. They slowed in part because, as the original S-matrix bootstrap was overtaken by QCD, there was less motivation to do this type of work anymore. But they also ran into a weird puzzle. When they tried to use the rules of crossing more broadly, only some of the things they found looked like S-matrices. Others looked like stranger, meaningless calculations.

A recent paper by Simon Caron-Huot, Mathieu Giroux, Holmfridur Hannesdottir, and Sebastian Mizera revisited these meaningless calculations, and showed that they aren’t so meaningless after all. In particular, some of them match well to the kinds of calculations people wanted to do to predict gravitational waves from colliding black holes.

Imagine a pair of black holes passing close to each other, then scattering away in different directions. Unlike particles in a collider, we have no hope of catching the black holes themselves. They’re big classical objects, and they will continue far away from us. We do catch gravitational waves, emitted from the interaction of the black holes.

This different setup turns out to give the problem a very different character. It ends up meaning that instead of the S-matrix, you want a subtly different mathematical object, one related to the original S-matrix by crossing relations. Using crossing, Caron-Huot, Giroux, Hannesdottir and Mizera found many different quantities one could observe in different situations, linked by the same rules that the original S-matrix bootstrappers used to relate S-matrix entries.

The work of these two groups is just some of the work done in the new S-matrix program, but it’s typical of where the focus is going. People are trying to understand the general rules found in the past. They want to know where they came from, and as a consequence, when they can go wrong. They have a lot to learn from the older papers, and a lot of new insights come from diligent reading. But they also have a lot of new insights to discover, based on the new tools and perspectives of the modern day. For the most part, they don’t expect to find a new unified theory of physics from bootstrapping alone. But by learning how S-matrices work in general, they expect to find valuable knowledge no matter how the future goes.

Amplitudes 2023 Retrospective

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I’m back from CERN this week, with a bit more time to write, so I thought I’d share some thoughts about last week’s Amplitudes conference.

One thing I got wrong in last week’s post: I’ve now been told only 213 people actually showed up in person, as opposed to the 250-ish estimate I had last week. This may seem fewer than Amplitudes in Prague had, but it seems likely they had a few fewer show up than appeared on the website. Overall, the field is at least holding steady from year to year, and definitely has grown since the pandemic (when 2019’s 175 was already a very big attendance).

It was cool having a conference in CERN proper, surrounded by the history of European particle physics. The lecture hall had an abstract particle collision carved into the wood, and the visitor center would in principle have had Standard Model coffee mugs were they not sold out until next May. (There was still enough other particle physics swag, Swiss chocolate, and Swiss chocolate that was also particle physics swag.) I’d planned to stay on-site at the CERN hostel, but I ended up appreciated not doing that: the folks who did seemed to end up a bit cooped up by the end of the conference, even with the conference dinner as a chance to get out.

Past Amplitudes conferences have had associated public lectures. This time we had a not-supposed-to-be-public lecture, a discussion between Nima Arkani-Hamed and Beate Heinemann about the future of particle physics. Nima, prominent as an amplitudeologist, also has a long track record of reasoning about what might lie beyond the Standard Model. Beate Heinemann is an experimentalist, one who has risen through the ranks of a variety of different particle physics experiments, ending up well-positioned to take a broad view of all of them.

It would have been fun if the discussion erupted into an argument, but despite some attempts at provocative questions from the audience that was not going to happen, as Beate and Nima have been friends for a long time. Instead, they exchanged perspectives: on what’s coming up experimentally, and what theories could explain it. Both argued that it was best to have many different directions, a variety of experiments covering a variety of approaches. (There wasn’t any evangelism for particular experiments, besides a joking sotto voce mention of a muon collider.) Nima in particular advocated that, whether theorist or experimentalist, you have to have some belief that what you’re doing could lead to a huge breakthrough. If you think of yourself as just a “foot soldier”, covering one set of checks among many, then you’ll lose motivation. I think Nima would agree that this optimism is irrational, but necessary, sort of like how one hears (maybe inaccurately) that most new businesses fail, but someone still needs to start businesses.

Michelangelo Mangano’s talk on Thursday covered similar ground, but with different emphasis. He agrees that there are still things out there worth discovering: that our current model of the Higgs, for instance, is in some ways just a guess: a simplest-possible answer that doesn’t explain as much as we’d like. But he also emphasized that Standard Model physics can be “new physics” too. Just because we know the model doesn’t mean we can calculate its consequences, and there are a wealth of results from the LHC that improve our models of protons, nuclei, and the types of physical situations they partake in, without changing the Standard Model.

We saw an impressive example of this in Gregory Korchemsky’s talk on Wednesday. He presented an experimental mystery, an odd behavior in the correlation of energies of jets of particles at the LHC. These jets can include a very large number of particles, enough to make it very hard to understand them from first principles. Instead, Korchemsky tried out our field’s favorite toy model, where such calculations are easier. By modeling the situation in the limit of a very large number of particles, he was able to reproduce the behavior of the experiment. The result was a reminder of what particle physics was like before the Standard Model, and what it might become again: partial models to explain odd observations, a quest to use the tools of physics to understand things we can’t just a priori compute.

On the other hand, amplitudes does do a priori computations pretty well as well. Fabrizio Caola’s talk opened the conference by reminding us just how much our precise calculations can do. He pointed out that the LHC has only gathered 5% of its planned data, and already it is able to rule out certain types of new physics to fairly high energies (by ruling out indirect effects, that would show up in high-precision calculations). One of those precise calculations featured in the next talk, by Guilio Gambuti. (A FORM user, his diagrams were the basis for the header image of my Quanta article last winter.) Tiziano Peraro followed up with a technique meant to speed up these kinds of calculations, a trick to simplify one of the more computationally intensive steps in intersection theory.

The rest of Monday was more mathematical, with talks by Zeno Capatti, Jaroslav Trnka, Chia-Kai Kuo, Anastasia Volovich, Francis Brown, Michael Borinsky, and Anna-Laura Sattelberger. Borinksy’s talk felt the most practical, a refinement of his numerical methods complete with some actual claims about computational efficiency. Francis Brown discussed an impressively powerful result, a set of formulas that manages to unite a variety of invariants of Feynman diagrams under a shared explanation.

Tuesday began with what I might call “visitors”: people from adjacent fields with an interest in amplitudes. Alday described how the duality between string theory in AdS space and super Yang-Mills on the boundary can be used to get quite concrete information about string theory, calculating how the theory’s amplitudes are corrected by the curvature of AdS space using a kind of “bootstrap” method that felt nicely familiar. Tim Cohen talked about a kind of geometric picture of theories that extend the Standard Model, including an interesting discussion of whether it’s really “geometric”. Marko Simonovic explained how the integration techniques we develop in scattering amplitudes can also be relevant in cosmology, especially for the next generation of “sky mappers” like the Euclid telescope. This talk was especially interesting to me since this sort of cosmology has a significant presence at CEA Paris-Saclay. Along those lines an interesting paper, “Cosmology meets cohomology”, showed up during the conference. I haven’t had a chance to read it yet!

Just before lunch, we had David Broadhurst give one of his inimitable talks, complete with number theory, extremely precise numerics, and literary and historical references (apparently, Källén died flying his own plane). He also remedied a gap in our whimsically biological diagram naming conventions, renaming the pedestrian “double-box” as a (in this context, Orwellian) lobster. Karol Kampf described unusual structures in a particular Effective Field Theory, while Henriette Elvang’s talk addressed what would become a meaningful subtheme of the conference, where methods from the mathematical field of optimization help amplitudes researchers constrain the space of possible theories. Giulia Isabella covered another topic on this theme later in the day, though one of her group’s selling points is managing to avoid quite so heavy-duty computations.

The other three talks on Tuesday dealt with amplitudes techniques for gravitational wave calculations, as did the first talk on Wednesday. Several of the calculations only dealt with scattering black holes, instead of colliding ones. While some of the results can be used indirectly to understand the colliding case too, a method to directly calculate behavior of colliding black holes came up again and again as an important missing piece.

The talks on Wednesday had to start late, owing to a rather bizarre power outage (the lights in the room worked fine, but not the projector). Since Wednesday was the free afternoon (home of quickly sold-out CERN tours), this meant there were only three talks: Veneziano’s talk on gravitational scattering, Korchemsky’s talk, and Nima’s talk. Nima famously never finishes on time, and this time attempted to control his timing via the surprising method of presenting, rather than one topic, five “abstracts” on recent work that he had not yet published. Even more surprisingly, this almost worked, and he didn’t run too ridiculously over time, while still managing to hint at a variety of ways that the combinatorial lessons behind the amplituhedron are gradually yielding useful perspectives on more general realistic theories.

Thursday, Andrea Puhm began with a survey of celestial amplitudes, a topic that tries to build the same sort of powerful duality used in AdS/CFT but for flat space instead. They’re gradually tackling the weird, sort-of-theory they find on the boundary of flat space. The two next talks, by Lorenz Eberhardt and Hofie Hannesdottir, shared a collaborator in common, namely Sebastian Mizera. They also shared a common theme, taking a problem most people would have assumed was solved and showing that approaching it carefully reveals extensive structure and new insights.

Cristian Vergu, in turn, delved deep into the literature to build up a novel and unusual integration method. We’ve chatted quite a bit about it at the Niels Bohr Institute, so it was nice to see it get some attention on the big stage. We then had an afternoon of trips beyond polylogarithms, with talks by Anne Spiering, Christoph Nega, and Martijn Hidding, each pushing the boundaries of what we can do with our hardest-to-understand integrals. Einan Gardi and Ruth Britto finished the day, with a deeper understanding of the behavior of high-energy particles and a new more mathematically compatible way of thinking about “cut” diagrams, respectively.

On Friday, João Penedones gave us an update on a technique with some links to the effective field theory-optimization ideas that came up earlier, one that “bootstraps” whole non-perturbative amplitudes. Shota Komatsu talked about an intriguing variant of the “planar” limit, one involving large numbers of particles and a slick re-writing of infinite sums of Feynman diagrams. Grant Remmen and Cliff Cheung gave a two-parter on a bewildering variety of things that are both surprisingly like, and surprisingly unlike, string theory: important progress towards answering the question “is string theory unique?”

Friday afternoon brought the last three talks of the conference. James Drummond had more progress trying to understand the symbol letters of supersymmetric Yang-Mills, while Callum Jones showed how Feynman diagrams can apply to yet another unfamiliar field, the study of vortices and their dynamics. Lance Dixon closed the conference without any Greta Thunberg references, but with a result that explains last year’s mystery of antipodal duality. The explanation involves an even more mysterious property called antipodal self-duality, so we’re not out of work yet!

At Amplitudes 2023 at CERN

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I’m at the big yearly conference of my sub-field this week, called Amplitudes. This year, surprisingly for the first time, it’s at the very appropriate location of CERN.

Somewhat overshadowed by the very picturesque Alps

Amplitudes keeps on growing. In 2019, we had 175 participants. We were on Zoom in 2020 and 2021, with many more participants, but that probably shouldn’t count. In Prague last year we had 222. This year, I’ve been told we have even more, something like 250 participants (the list online is bigger, but includes people joining on Zoom). We’ve grown due to new students, but also new collaborations: people from adjacent fields who find the work interesting enough to join along. This year we have mathematicians talking about D-modules, bootstrappers finding new ways to get at amplitudes in string theory, beyond-the-standard-model theorists talking about effective field theories, and cosmologists talking about the large-scale structure of the universe.

The talks have been great, from clear discussions of earlier results to fresh-off-the-presses developments, plenty of work in progress, and even one talk where the speaker’s opinion changed during the coffee break. As we’re at CERN, there’s also a through-line about the future of particle physics, with a chat between Nima Arkani-Hamed and the experimentalist Beate Heinemann on Tuesday and a talk by Michelangelo Mangano about the meaning of “new physics” on Thursday.

I haven’t had a ton of time to write, I keep getting distracted by good discussions! As such, I’ll do my usual thing, and say a bit more about specific talks in next week’s post.

Cabinet of Curiosities: The Deluxe Train Set

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I’ve got a new paper out this week with Andrew McLeod. I’m thinking of it as another entry in this year’s “cabinet of curiosities”, interesting Feynman diagrams with unusual properties. Although this one might be hard to fit into a cabinet.

Over the past few years, I’ve been finding Feynman diagrams with interesting connections to Calabi-Yau manifolds, the spaces originally studied by string theorists to roll up their extra dimensions. With Andrew and other collaborators, I found an interesting family of these diagrams called traintracks, which involve higher-and-higher dimensional manifolds as they get longer and longer.

This time, we started hooking up our traintracks together.

We call diagrams like these traintrack network diagrams, or traintrack networks for short. The original traintracks just went “one way”: one family, going higher in Calabi-Yau dimension the longer they got. These networks branch out, one traintrack leading to another and another.

In principle, these are much more complicated diagrams. But we find we can work with them in almost the same way. We can find the same “starting point” we had for the original traintracks, the set of integrals used to find the Calabi-Yau manifold. We’ve even got more reliable tricks, a method recently honed by some friends of ours that consistently find a Calabi-Yau manifold inside the original traintracks.

Surprisingly, though, this isn’t enough.

It works for one type of traintrack network, a so-called “cross diagram” like this:

But for other diagrams, if the network branches any more, the trick stops working. We still get an answer, but that answer is some more general space, not just a Calabi-Yau manifold.

That doesn’t mean that these general traintrack networks don’t involve Calabi-Yaus at all, mind you: it just means this method doesn’t tell us one way or the other. It’s also possible that simpler versions of these diagrams, involving fewer particles, will once again involve Calabi-Yaus. This is the case for some similar diagrams in two dimensions. But it’s starting to raise a question: how special are the Calabi-Yau related diagrams? How general do we expect them to be?

Another fun thing we noticed has to do with differential equations. There are equations that relate one diagram to another simpler one. We’ve used them in the past to build up “ladders” of diagrams, relating each picture to one with one of its boxes “deleted”. We noticed, playing with these traintrack networks, that these equations do a bit more than we thought. “Deleting” a box can make a traintrack short, but it can also chop a traintrack in half, leaving two “dangling” pieces, one on either side.

This reminded me of an important point, one we occasionally lose track of. The best-studied diagrams related to Calabi-Yaus are called “sunrise” diagrams. If you squish together a loop in one of those diagrams, the whole diagram squishes together, becoming much simpler. Because of that, we’re used to thinking of these as diagrams with a single “geometry”, one that shows up when you don’t “squish” anything.

Traintracks, and traintrack networks, are different. “Squishing” the diagram, or “deleting” a box, gives you a simpler diagram, but not much simpler. In particular, the new diagram will still contain traintracks, and traintrack networks. That means that we really should think of each traintrack network not just as one “top geometry”, but of a collection of geometries, different Calabi-Yaus that break into different combinations of Calabi-Yaus in different ways. It’s something we probably should have anticipated, but the form these networks take is a good reminder, one that points out that we still have a lot to do if we want to understand these diagrams.

Cabinet of Curiosities: The Train-Ladder

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I’ve got a new paper out this week, with Andrew McLeod, Roger Morales, Matthias Wilhelm, and Chi Zhang. It’s yet another entry in this year’s “cabinet of curiosities”, quirky Feynman diagrams with interesting traits.

A while back, I talked about a set of Feynman diagrams I could compute with any number of “loops”, bypassing the approximations we usually need to use in particle physics. That wasn’t the first time someone did that. Back in the 90’s, some folks figured out how to do this for so-called “ladder” diagrams. These diagrams have two legs on one end for two particles coming in, two legs on the other end for two particles going out, and a ladder in between, like so:

There are infinitely many of these diagrams, but they’re all beautifully simple, variations on a theme that can be written down in a precise mathematical way.

Change things a little bit, though, and the situation gets wildly more intractable. Let the rungs of the ladder peek through the sides, and you get something looking more like the tracks for a train:

These traintrack integrals are much more complicated. Describing them requires the mathematics of Calabi-Yau manifolds, involving higher and higher dimensions as the tracks get longer. I don’t think there’s any hope of understanding these things for all loops, at least not any time soon.

What if we aimed somewhere in between? A ladder that just started to turn traintrack?

Add just a single pair of rungs, and it turns out that things remain relatively simple. If we do this, it turns out we don’t need any complicated Calabi-Yau manifolds. We just need the simplest Calabi-Yau manifold, called an elliptic curve. It’s actually the same curve for every version of the diagram. And the situation is simple enough that, with some extra cleverness, it looks like we’ve found a trick to calculate these diagrams to any number of loops we’d like.

(Another group figured out the curve, but not the calculation trick. They’ve solved different problems, though, studying all sorts of different traintrack diagrams. They sorted out some confusion I used to have about one of those diagrams, showing it actually behaves precisely the way we expected it to. All in all, it’s been a fun example of the way different scientists sometimes hone in on the same discovery.)

These developments are exciting, because Feynman diagrams with elliptic curves are still tough to deal with. We still have whole conferences about them. These new elliptic diagrams can be a long list of test cases, things we can experiment with with any number of loops. With time, we might truly understand them as well as the ladder diagrams!