Tag Archives: amplitudes

It Was Thirty Years Ago Yesterday, Parke and Taylor Taught the Band to Play…

Just a short post this week. I’m at MHV@30, a conference at Fermilab in honor of Parke and Taylor’s landmark paper from March 17, 1986. I don’t have time to write up an explanation of their work’s importance, but luckily I already have.

It’s my first time visiting Fermilab. They took us on a tour of their neutrino detectors 100m underground. Since we theorists don’t visit experiments very often, it was an unusual experience.

In case you wanted to know what a neutrino beam looks like, look at the target.

The fun thing about these kinds of national labs is the sheer variety of research, from the most abstract theory to the most grounded experiments, that spring from the same core goals. Physics almost always involves a diversity of viewpoints and interests, and that’s nowhere more obvious than here.

Symbology 101

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I work with functions called polylogarithms. There’s a whole field of techniques out there for manipulating these functions, and for better or worse people often refer to them as symbology.

My plan for this post is to give a general feel for how symbology works: what we know how to do, and why. It’s going to be a lot more technical than my usual posts, so the lay reader may want to skip this one. At the same time, I’m not planning to go through anything rigorously. If you want that sort of thing there are plenty of good papers on the subject, here’s one of mine that covers the basics. Rather, I’m going to draw what I hope is an illuminating sketch of what it is we do.

Still here? Let’s start with an easy question.

What’s a log?

Ok, besides one of these.

For our purposes, a log is what happens when you integrate dx/x.

$\log x=\int \frac{dx}{x}$

Schematically, a polylog is then what happens when you iterate these integrations:

$G=\int \frac{dx_1}{x_1} \int \frac{dx_2}{x_2}\ldots$

The simplest thing you can get from this is of course just a product of logs. The next most simple thing is one of the classical polylogarithms. But in general, this is a much wider class of functions, known as multiple, or Goncharov, polylogarithms.

The number of integrations is the transcendental weight. Naively, you’d expect an L-loop Feynman integral in four dimensions to give you something with transcendental weight 4L. In practice, that’s not the case: some of the momentum integrations end up just giving delta functions, so in the end an L-loop amplitude has transcendental weight 2L.

In most theories, you get a mix of functions: some with weight 2L, some with weight 2L-1, etc., all the way down to rational functions. N=4 super Yang-Mills is special: there, everything is at the maximum transcendental weight. In either case, though, being able to manipulate transcendental functions is very useful, and the symbol is one of the simplest ways to do so.

The core idea of the symbol is pretty easy to state, though it takes a bit more technology to state it rigorously. Essentially, we take our schematic polylog from above, and just list the logs:

$\mathcal{S}(G)=\ldots\otimes x_2\otimes x_1$

(Here I have switched the order in order to agree with standard conventions.)

What does that do? Well, it reminds us that these aren’t just some weird functions we don’t understand: they’re collections of logs, and we can treat them like collections of logs.

In particular, we can do this with logs,

$\log (x y)=\log x+\log y$

so we can do it with symbols as well:

$x_1\otimes x y\otimes x_3=x_1\otimes x \otimes x_3+x_1\otimes y\otimes x_3$

Similarly, we can always get rid of unwelcome exponents, like so:

$\log (x^n)=n\log x$

$x_1\otimes x^n\otimes x_3=n( x_1\otimes x \otimes x_3)$

This means that, in general, we can always factorize any polynomial or rational function that appears in a symbol. As such, we often express symbols in terms of some fixed symbol alphabet, a basis of rational functions that can be multiplied to get any symbol entry in the function we’re working with. In general, it’s a lot easier to calculate amplitudes when we know the symbol alphabet beforehand. For six-particle amplitudes in N=4 super Yang-Mills, the symbol alphabet contains just nine “letters”, which makes it particularly easy to work with.

That’s arguably the core of symbol methods. It’s how Spradlin and Volovich managed to get a seventeen-page expression down to two lines. Express a symbol in the right alphabet, and it tends to look a lot more simple. And once you know the right alphabet, it’s pretty straightforward to build an ansatz with it and constrain it until you get a candidate function for whatever you’re interested in.

There’s more technical detail I could give here: how to tell whether a symbol actually corresponds to a function, how to take limits and do series expansions and take derivatives and discontinuities…but I’m not sure whether anyone reading this would be interested.

As-is, I’ll just mention that the symbol is only part of the story. In particular, it’s a special case of something called a coproduct, which breaks up polylogarithms into various chunks. Break them down fully until each chunk is just an individual log, and you get the symbol. Break them into larger chunks, and you get other components of the coproduct, consisting of tensor products of polylogarithms with lower transcendental weight. These larger chunks mean we can capture as much of a function’s behavior as we like, while still taking advantage of these sorts of tricks. While in older papers you might have seen mention of “beyond-the-symbol” terms that the symbol couldn’t capture, this doesn’t tend to be a problem these days.

PSI Winter School

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I’m at the Perimeter Scholars International Winter School this week. Perimeter Scholars International is Perimeter’s one-of-a-kind master’s program in theoretical physics, that jams the basics of theoretical physics into a one-year curriculum. We’ve got students from all over the world, including plenty of places that don’t get any snow at all. As such, it was decided that the students need to spend a week somewhere with even more snow than Waterloo: Musoka, Ontario.

A place that occasionally manages to be this photogenic

This isn’t really a break for them, though, which is where I come in. The students have been organized into groups, and each group is working on a project. My group’s project is related to the work of integrability master Pedro Vieira. He and his collaborators came up with a way to calculate scattering amplitudes in N=4 super Yang-Mills without the usual process of loop-by-loop approximations. However, this method comes at a price: a new approximation, this time to low energy. This approximation is step-by-step, like loops, but in a different direction. It’s called the Pentagon Operator Product Expansion, or POPE for short.

Approach the POPE, and receive a blessing

What we’re trying to do is go back and add up all of the step-by-step terms in the approximation, to see if we can match to the old expansion in loops. One of Pedro’s students recently managed to do this for the first approximation (“tree” diagrams), and the group here at the Winter School is trying to use her (still unpublished) work as a jumping-off point to get to the first loop. Time will tell whether we’ll succeed…but we’re making progress, and the students are learning a lot.

Amplitudes for the New Year

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Ah, the new year, time of new year’s resolutions. While some people resolve to go to the gym or take up online dating, physicists resolve to finally get that paper out.

At least, that’s the impression I get, given the number of papers posted to arXiv in the last month. Since a lot of them were amplitudes-related, I figured I’d go over some highlights.

Everyone once in a while people ask me for the latest news on the amplituhedron. While I don’t know what Nima is working on right now, I can point to what others have been doing. Zvi Bern, Jaroslav Trnka, and collaborators have continued to make progress towards generalizing the amplituhedron to non-planar amplitudes. Meanwhile, a group in Europe has been working on solving an issue I’ve glossed over to some extent. While the amplituhedron is often described as calculating an amplitude as the volume of a geometrical object, in fact there is a somewhat more indirect procedure involved in going from the geometrical object to the amplitude. It would be much simpler if the amplitude was actually the volume of some (different) geometrical object, and that’s what these folks are working towards. Finally, Daniele Galloni has made progress on solving a technical issue: the amplituhedron gives a mathematical recipe for the amplitude, but it doesn’t tell you how to carry out that recipe, and Galloni provides an algorithm for part of this process.

With this new algorithm, is the amplituhedron finally as efficient as older methods? Typically, the way to show that is to do a calculation with the amplituhedron that wasn’t possible before. It doesn’t look like that’s happening soon though, as Jake Bourjaily and collaborators compute an eight-loop integrand using one of the more successful of the older methods. Their paper provides a good answer to the perennial question, “why more loops?” What they find is that some of the assumptions that people made at lower loops fail to hold at this high loop order, and it becomes increasingly important to keep track of exactly how far your symmetries can take you.

Back when I visited Brown, I talked to folks there about some ongoing work. Now that they’ve published, I can talk about it. A while back, Juan Maldacena resurrected an old technique of Landau’s to solve a problem in AdS/CFT. In that paper, he suggested that Landau’s trick might help prove some of the impressive simplifications in N=4 super Yang-Mills that underlie my work and the work of those at Brown. In their new paper, the Brown group finds that, while useful, Landau’s trick doesn’t seem to fully explain the simplicity they’ve discovered. To get a little partisan, I have to say that this was largely the result I expected, and that it felt a bit condescending for Maldacena to assume that an old trick like that from the Feynman diagram era could really be enough to explain one of the big discoveries of amplitudeology.

There was also a paper by Freddy Cachazo and collaborators on an interesting trick to extend their CHY string to one-loop, and one by Bo Feng and collaborators on an intriguing new method called Q-cuts that I will probably say more about in future, but I’ll sign off for now. I’ve got my own new years’ physics resolutions, and I ought to get back to work!

The “Lies to Children” Model of Science Communication, and The “Amplitudes Are Weird” Model of Amplitudes

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Let me tell you a secret.

Scattering amplitudes in N=4 super Yang-Mills don’t actually make sense.

Scattering amplitudes calculate the probability that particles “scatter”: coming in from far away, interacting in some fashion, and producing new particles that travel far away in turn. N=4 super Yang-Mills is my favorite theory to work with: a highly symmetric version of the theory that describes the strong nuclear force. In particular, N=4 super Yang-Mills has conformal symmetry: if you re-scale everything larger or smaller, you should end up with the same predictions.

You might already see the contradiction here: scattering amplitudes talk about particles coming in from very far away…but due to conformal symmetry, “far away” doesn’t mean anything, since we can always re-scale it until it’s not far away anymore!

So when I say that I study scattering amplitudes in N=4 super Yang-Mills, am I lying?

Well…yes. But it’s a useful type of lie.

There’s a concept in science writing called “lies to children”, first popularized in a fantasy novel.

This one.

When you explain science to the public, it’s almost always impossible to explain everything accurately. So much background is needed to really understand most of modern science that conveying even a fraction of it would bore the average audience to tears. Instead, you need to simplify, to skip steps, and even (to be honest) to lie.

The important thing to realize here is that “lies to children” aren’t meant to mislead. Rather, they’re chosen in such a way that they give roughly the right impression, even as they leave important details out. When they told you in school that energy is always conserved, that was a lie: energy is a consequence of symmetry in time, and when that symmetry is broken energy doesn’t have to be conserved. But “energy is conserved” is a useful enough rule that lets you understand most of everyday life.

In this case, the “lie” that we’re calculating scattering amplitudes is fairly close to the truth. We’re using the same methods that people use to calculate scattering amplitudes in theories where they do make sense, like QCD. For a while, people thought these scattering amplitudes would have to be zero, since anything else “wouldn’t make sense”…but in practice, we found they were remarkably similar to scattering amplitudes in other theories. Now, we have more rigorous definitions for what we’re calculating that avoid this problem, involving objects called polygonal Wilson loops.

This illustrates another principle, one that hasn’t (yet) been popularized by a fantasy novel. I’d like to call it the “amplitudes are weird” principle. Time and again we amplitudes-folks will do a calculation that doesn’t really make sense, find unexpected structure, and go back to figure out what that structure actually means. It’s been one of the defining traits of the field, and we’ve got a pretty good track record with it.

A couple of weeks back, Lance Dixon gave an interview for the SLAC website, talking about his work on quantum gravity. This was immediately jumped on by Peter Woit and Lubos Motl as ammo for the long-simmering string wars. To one extent or another, both tried to read scientific arguments into the piece. This is in general a mistake: it is in the nature of a popularization piece to contain some volume of lies-to-children, and reading a piece aimed at a lower audience can be just as confusing as reading one aimed at a higher audience.

In the remainder of this post, I’ll try to explain what Lance was talking about in a slightly higher-level way. There will still be lies-to-children involved, this is a popularization blog after all. But I should be able to clear up a few misunderstandings. Lubos probably still won’t agree with the resulting argument, but it isn’t the self-evidently wrong one he seems to think it is.

Lance Dixon has done a lot of work on quantum gravity. Those of you who’ve read my old posts might remember that quantum gravity is not so difficult in principle: general relativity naturally leads you to particles called gravitons, which can be treated just like other particles. The catch is that the theory that you get by doing this fails to be predictive: one reason why is that you get an infinite number of erroneous infinite results, which have to be papered over with an infinite number of arbitrary constants.

Working with these non-predictive theories, however, can still yield interesting results. In the article, Lance mentions the work of Bern, Carrasco, and Johansson. BCJ (as they are abbreviated) have found that calculating a gravity amplitude often just amounts to calculating a (much easier to find) Yang-Mills amplitude, and then squaring the right parts. This was originally found in the context of string theory by another three-letter group, Kawai, Lewellen, and Tye (or KLT). In string theory, it’s particularly easy to see how this works, as it’s a basic feature of how string theory represents gravity. However, the string theory relations don’t tell the whole story: in particular, they only show that this squaring procedure makes sense on a classical level. Once quantum corrections come in, there’s no known reason why this squaring trick should continue to work in non-string theories, and yet so far it has. It would be great if we had a good argument why this trick should continue to work, a proof based on string theory or otherwise: for one, it would allow us to be much more confident that our hard work trying to apply this trick will pay off! But at the moment, this falls solidly under the “amplitudes are weird” principle.

Using this trick, BCJ and collaborators (frequently including Lance Dixon) have been calculating amplitudes in N=8 supergravity, a highly symmetric version of those naive, non-predictive gravity theories. For this particular, theory, the theory you “square” for the above trick is N=4 super Yang-Mills. N=4 super Yang-Mills is special for a number of reasons, but one is that the sorts of infinite results that lose you predictive power in most other quantum field theories never come up. Remarkably, the same appears to be true of N=8 supergravity. We’re still not sure, the relevant calculation is still a bit beyond what we’re capable of. But in example after example, N=8 supergravity seems to be behaving similarly to N=4 super Yang-Mills, and not like people would have predicted from its gravitational nature. Once again, amplitudes are weird, in a way that string theory helped us discover but by no means conclusively predicted.

If N=8 supergravity doesn’t lose predictive power in this way, does that mean it could describe our world?

In a word, no. I’m not claiming that, and Lance isn’t claiming that. N=8 supergravity simply doesn’t have the right sorts of freedom to give you something like the real world, no matter how you twist it. You need a broader toolset (string theory generally) to get something realistic. The reason why we’re interested in N=8 supergravity is not because it’s a candidate for a real-world theory of quantum gravity. Rather, it’s because it tells us something about where the sorts of dangerous infinities that appear in quantum gravity theories really come from.

That’s what’s going on in the more recent paper that Lance mentioned. There, they’re not working with a supersymmetric theory, but with the naive theory you’d get from just trying to do quantum gravity based on Einstein’s equations. What they found was that the infinity you get is in a certain sense arbitrary. You can’t get rid of it, but you can shift it around (infinity times some adjustable constant 😉 ) by changing the theory in ways that aren’t physically meaningful. What this suggests is that, in a sense that hadn’t been previously appreciated, the infinite results naive gravity theories give you are arbitrary.

The inevitable question, though, is why would anyone muck around with this sort of thing when they could just use string theory? String theory never has any of these extra infinities, that’s one of its most important selling points. If we already have a perfectly good theory of quantum gravity, why mess with wrong ones?

Here, Lance’s answer dips into lies-to-children territory. In particular, Lance brings up the landscape problem: the fact that there are 10^500 configurations of string theory that might loosely resemble our world, and no clear way to sift through them to make predictions about the one we actually live in.

This is a real problem, but I wouldn’t think of it as the primary motivation here. Rather, it gets at a story people have heard before while giving the feeling of a broader issue: that string theory feels excessive.

Why does this have a Wikipedia article?

Think of string theory like an enormous piece of fabric, and quantum gravity like a dress. You can definitely wrap that fabric around, pin it in the right places, and get a dress. You can in fact get any number of dresses, elaborate trains and frilly togas and all sorts of things. You have to do something with the extra material, though, find some tricky but not impossible stitching that keeps it out of the way, and you have a fair number of choices of how to do this.

From this perspective, naive quantum gravity theories are things that don’t qualify as dresses at all, scarves and socks and so forth. You can try stretching them, but it’s going to be pretty obvious you’re not really wearing a dress.

What we amplitudes-folks are looking for is more like a pencil skirt. We’re trying to figure out the minimal theory that covers the divergences, the minimal dress that preserves modesty. It would be a dress that fits the form underneath it, so we need to understand that form: the infinities that quantum gravity “wants” to give rise to, and what it takes to cancel them out. A pencil skirt is still inconvenient, it’s hard to sit down for example, something that can be solved by adding extra material that allows it to bend more. Similarly, fixing these infinities is unlikely to be the full story, there are things called non-perturbative effects that probably won’t be cured. But finding the minimal pencil skirt is still going to tell us something that just pinning a vast stretch of fabric wouldn’t.

This is where “amplitudes are weird” comes in in full force. We’ve observed, repeatedly, that amplitudes in gravity theories have unexpected properties, traits that still aren’t straightforwardly explicable from the perspective of string theory. In our line of work, that’s usually a sign that we’re on the right track. If you’re a fan of the amplituhedron, the project here is along very similar lines: both are taking the results of plodding, not especially deep loop-by-loop calculations, observing novel simplifications, and asking the inevitable question: what does this mean?

That far-term perspective, looking off into the distance at possible insights about space and time, isn’t my style. (It isn’t usually Lance’s either.) But for the times that you want to tell that kind of story…well, this isn’t that outlandish of a story to tell. And unless your primary concern is whether a piece gives succor to the Woits of the world, it shouldn’t be an objectionable one.

Map Your Dead Ends

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I’m at Brown this week, where I’ve been chatting with Mark Spradlin and Anastasia Volovich, two of the founding figures of my particular branch of amplitudeology. Back in 2010 they figured out how to turn this seventeen-page two-loop amplitude:

into a formula that just takes up two lines:This got everyone very excited, it inspired some of my collaborators to do work that would eventually give rise to the Hexagon Functions, my main research project for the past few years.

Unfortunately, when we tried to push this to higher loops, we didn’t get the sort of nice, clean-looking formulas that the Brown team did. Each “loop” is an additional layer of complexity, a series of approximations that get closer to the exact result. And so far, our answers look more like that first image than the second: hundreds of pages with no clear simplifications in sight.

At the time, people wondered whether some simple formula might be enough. As it turns out, you can write down a formula similar to the one found by Spradlin and Volovich, generalized to a higher number of loops. It’s clean, it’s symmetric, it makes sense…and it’s not the right answer.

That happens in science a lot more often than science fans might expect. When you hear about this sort of thing in the news, it always works: someone suggests a nice, simple answer, and it turns out to be correct, and everyone goes home happy. But for every nice simple guess that works, there are dozens that don’t: promising ideas that just lead to dead ends.

One of the postdocs here at Brown worked on this “wrong” formula, and while chatting with him here he asked a very interesting question: why is it wrong? Sure, we know that it’s wrong, we can check that it’s wrong…but what, specifically, is missing? Is it “part” of the right answer in some sense, with some predictable corrections?

As it turns out, this is a very interesting question! We’ve been looking into it, and the “wrong” answer has some interesting relationships with some of our Hexagon Functions. It may have been a “dead end”, but it still could turn out to be a useful one.

A good physics advisor will tell their students to document their work. This doesn’t just mean taking notes: most theoretical physicists will maintain files, in standard journal article format, with partial results. One reason to do this is that, if things work out, you’ll have some of your paper already written. But if something doesn’t work out, you’ll end up with a pdf on your hard drive carefully explaining an idea that didn’t quite work. Physicists often end up with dozens of these files squirreled away on their computers. Put together, they’re a map: a map of dead ends.

There’s a handy thing about having a map: it lets you retrace your steps. Any one of these paths may lead nowhere, but each one will contain some substantive work. And years later, often enough, you end up needing some of it: some piece of the calculation, some old idea. You follow the map, dig it up…and build it into something new.

Hexagon Functions III: Now with More Symmetry

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I’ve got a new paper up this week.

It’s a continuation of my previous work, understanding collisions involving six particles in my favorite theory, N=4 super Yang-Mills.

This time, we’re pushing up the complexity, going from three “loops” to four. In the past, I could have impressed you with the number of pages the formulas I’m calculating take up (eight hundred pages for the three-loop formula from that first Hexagon Functions paper). Now, though, I don’t have that number: putting my four-loop formula into a pdf-making program just crashes the program. Instead, I’ll have to impress you with file sizes: 2.6 MB for the three-loop formula, 96 MB for the four-loop one.

Calculating such a formula sounds like a pretty big task, and it was, the first time. But things got a lot simpler after a chat I had at Amplitudes.

We calculate these things using an ansatz, a guess for what the final answer should look like. The more vague our guess, the more parameters we need to fix, and the more work we have in general. If we can guess more precisely, we can start with fewer parameters and things are a lot easier.

Often, more precise guesses come from understanding the symmetries of the problem. If we can know that the final answer must be the same after making some change, we can rule out a lot of possibilities.

Sometimes, these symmetries are known features of the answer, things that someone proved had to be correct. Other times, though, they’re just observations, things that have been true in the past and might be true again.

We started out using an observation from three loops. That got us pretty far, but we still had a lot of work to do: 808 parameters, to be fixed by other means. Fixing them took months of work, and throughout we hoped that there was some deeper reason behind the symmetries we observed.

Finally, at Amplitudes, I ran into fellow amplitudeologist Simon Caron-Huot and asked him if he knew the source of our observed symmetry. In just a few days he was able to link it to supersymmetry, giving us justification for our jury rigged trick. However, we figured out that his explanation went further than any of us expected. In the end, rather than 808 parameters we only really needed to consider 34.

Thirty-four options to consider. Thirty-four possible contributions to a ~100 MB file. That might not sound like a big deal, but compared to eight hundred and eight it’s a huge deal. More symmetry means easier calculations, meaning we can go further. At this point going to the next step in complexity, to five loops rather than four, might be well within reach.

Scooped Is a Spectrum

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Historic Montreal

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I’m at a conference in Montreal this week, so it’s going to be a short post. The University of Montreal’s Centre de Recherches Mathématiques has been holding a program on the various hidden symmetries of N=4 super Yang-Mills since the beginning of the summer. This week is the amplitudes-focused part of the program, so they’ve brought in a bunch of amplitudes-folks from around the world, myself included.

It’s been great hanging out with fellow members of my sub-field, as always, both at the conference and at dinner afterwards. Over possibly too much wine I heard stories of the heady days of 2007, when James Drummond and Johannes Henn first discovered one of the most powerful symmetries of N=4 super Yang-Mills (a duality called dual conformal invariance) and Andrew Hodges showed off the power of a set of funky variables called twistors. It’s amazing to me how fast the field moves, sometimes: by the time I started doing amplitudes work in 2011 these ideas were the bedrock of the field. History operates on different scales, and in amplitudes a few decades have played host to an enormous amount of progress.

History in the real world can move surprisingly fast too. After seeing cathedrals in Zurich that date back to the medieval era, I was surprised when the majestic basilica overlooking Montreal turned out to be less than a century old.

In retrospect the light-up cross should have made it obvious.

4 gravitons

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