Tag Archives: amplitudes

Amplitudes Megapost

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

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

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

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

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

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

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

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

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

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

Generalized Unitarity

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

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

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

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

BCFW Recursion

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

The Positive Grassmannian, and the Amplituhedron

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

Integration Tricks

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

Polylogarithms

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

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

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

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

Integrability

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

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

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

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

CHY and the Ambitwistor String

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

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

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

What Do You Get When You Put 136 Amplitudeologists into One Room? Amplitudes 2015!

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I’m at Amplitudes this week, my subfield’s big yearly conference, located this year in sweltering but otherwise lovely Zurich.

A typical inhabitant of Zurich.

A typical inhabitant of Zurich.

I gave a talk on Tuesday. They’ve posted the slides online, and I think they’re going to post the talk itself at some point.

This is the first year I’ve been to Amplitudes, and it’s remarkable seeing the breadth of the field. We’ve got everything from people focused heavily on the needs of experimentalists, trying to perfect calculations that will reduce the error on measurements coming out of the LHC, to people primarily interested in some of the more esoteric aspects of string theory. Putting everyone into the same room definitely helps emphasize just how many different approaches there are under the amplitudes umbrella. It’s the first time I’ve really appreciated just how “big” the field is, how much it’s grown to encompass.

The Near and the Far: Motivations for Physics

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When I introduce myself, I often describe my job like this:

“I develop mathematical tools to make calculations in particle physics easier and more efficient.”

However, I could equally well describe my job like this:

“I’m looking for a radical new way to reformulate particle physics in order to solve fundamental problems in space and time.”

These may sound very different, but they’re both correct. That’s because in theoretical physics, like in many branches of science, we have two types of goals: near-term and far-term.

In the near-term, I develop mathematical tools and tricks, which let me calculate things I (and others) couldn’t calculate before. Pushing the tricks to their limits gives me more proficiency, making the tools I develop more robust. In the future, I can imagine applying the tools to more types of calculations, and specifically to more “important” calculations.

All of that still involves relatively near-term goals, though. Develop a new trick, and you can already envision what it might be used for. The far-term goals are generally deeper.

End of the road, not just the next tree.

In the far term, the new techniques that I and others develop might lead to fundamentally new ways to understand particle physics. That’s because a central feature of most of the tricks we develop is that they rephrase the calculation in a way that leaves out something that used to be thought of as fundamental. They’re “revolutions”, overthrowing some basic principle of how we do things. The hope is that the right “revolution” will help us solve problems that our current understanding of physics seems incapable of solving.

Most scientists have both sorts of goals. Someone who studies quantum mechanics might talk about developing a quantum computer, but in the near-term be interested in perfecting some algorithm. A biologist might study how information is stored in a cell, but introduce themself as someone trying to cure cancer.

For some people, the far-term goals are a big component of how they view themselves. Nima Arkani-Hamed, for example, has joked that believing that “spacetime is doomed” is what allows him to get out of bed in the morning. (For a transcript of the relevant parts, see here.) There are plenty of others with similar perspectives, people who need a “big” goal to feel motivated.

Myself, I find it harder to identify with these kinds of goals, because the payoff is so uncertain. Rephrasing particle physics in a new way might be the solution to a fundamental problem…but it could also just be another way to say the same thing. There’s no guarantee that any one project will be that one magical solution. In contrast, for me, near term goals are something I can feel confident I’m making real progress on. I can envision each step along the way, and see the part my work plays in a larger picture, led along by the satisfaction of solving each puzzle as it comes.

Neither way is better than the other, and both are important parts of science. Some people do better with one, some do better with the other, and in the end, everyone can view themselves as accomplishing something they care about.

What’s an Amplitude? Just about everything.

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I am an Amplitudeologist. In other words, I study scattering amplitudes. I’ve explained bits and pieces of what scattering amplitudes are in other posts, but I ought to give a short definition here so everyone’s on the same page:

A scattering amplitude is the formula used to calculate the probability that some collection of particles will “scatter”, emerging as some (possibly different) collection of particles.

Note that I’m using some weasel words here. The scattering amplitude is not a probability itself, but “the formula used to calculate the probability”. For those familiar with the mathematics of waves, the scattering amplitude gives the amplitude of a “probability wave” that must be squared to get the probability. (Those familiar with waves might also ask: “If this is the amplitude, what about the period?” The truth is that because scattering amplitudes are calculated using complex numbers, what we call the “amplitude” also contains information about the wave’s “period”. It may seem like an inconsistent way to name things from the perspective of a beginning student, but it is actually consistent with the terminology in a large chunk of physics.)

In some of the simplest scattering amplitudes particles literally “scatter”, with two particles “colliding” and emerging traveling in different directions.

A scattering amplitude can also describe a more complicated situation, though. At particle colliders like the Large Hadron Collider, two particles (a pair of protons for the LHC) are accelerated fast enough that when they collide they release a whole slew of new particles. Since it still fits the “some particles go in, some particles go out” template, this is still described by a scattering amplitude.

It goes even further than that, though, because “some particles” could also just be “one particle”. If you’re dealing with something unstable (the particle equivalent of radioactive, essentially) then one particle can decay into two or more particles. There’s a whole slew of questions that require that sort of calculation. For example, if unstable particles were produced in the early universe, how many of them would be left around today? If dark matter is unstable (and some possible candidates are), when it decays it might release particles we could detect. In general, this sort of scattering amplitude is often of interest to astrophysicists when they happen to get involved in particle physics.

You can even use scattering amplitudes to describe situations that, at first glance, don’t sound like collisions of particles at all. If you want to find the effect of a magnetic field on an electron to high accuracy, the calculation also involves a scattering amplitude. A magnetic field can be thought of in terms of photons, particles of light, because light is a vibration in the electro-magnetic field. This means that the effect of a magnetic field on an electron can be calculated by “scattering” an electron and a photon.

4gravanom

If this looks familiar, check the handbook section.

In fact, doing the calculation in this way leads to what is possibly the most accurately predicted number in all of science.

Scattering amplitudes show up all over the place, from particle physics at the Large Hadron Collider to astrophysics to delicate experiments on electrons in magnetic fields. That said, there are plenty of things people calculate in theoretical physics that don’t use scattering amplitudes, either because they involve questions that are difficult to answer from the scattering amplitude point of view, or because they invoke different formulas altogether. Still, scattering amplitudes are central to the work of a large number of physicists. They really do cover just about everything.

Hexagon Functions II: Lost in (super)Space

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My new paper went up last night.

It’s on a very similar topic to my last paper, actually. That paper dealt with a specific process involving six particles in my favorite theory, N=4 super Yang-Mills. Two particles collide, and after the metaphorical dust settles four particles emerge. That means six “total” particles, if you add the two in with the four out, for a “hexagon” of variables. To understand situations like that, my collaborators and I created “hexagon functions”, formulas that depended on the states of the six particles.

One thing I didn’t emphasize then was that that calculation only applied to one specific choice of particles, one in which all of the particles are Yang-Mills bosons, particles (like photons) created by the fundamental forces. There are lots of other particles in N=4 super Yang-Mills, though. What happens when they collide?

That question is answered by my new paper. Though it may sound surprising, all of the other particles can be taken into account with a single formula. In order to explain why, I have to tell you about something called superspace.

A while back I complained about a blog post by George Musser about the (2,0) theory. One of the things that irked me about that post was his attempt to explain superspace:

Supersymmetry is the idea that spacetime, in addition to its usual dimensions of space and time, has an entirely different type of dimension—a quantum dimension, whose coordinates are not ordinary real numbers but a whole new class of number that can be thought of as the square roots of zero.

This is actually a great way to think about superspace…if you’re already a physicist. If you’re not, it’s not very informative. Here’s a better way to think about it:

As I’ve talked about before, supersymmetry is a relationship between different types of particles. Two particles related by supersymmetry have the same mass, and the same charge. While they can be very different in other ways (specifically, having different spin), supersymmetric particles are described by many of the same equations as each-other. Rather than writing out those equations multiple times, it’s often nicer to write them all in a unified way, and that’s where superspace comes in.

At its simplest, superspace is just a trick used to write equations in a simpler way. Instead of writing down a different equation for each particle we write one equation with an extra variable, representing a “dimension” of supersymmetry. Traveling in that dimension takes you from particle to particle, in the same way that “turning” the theory (as I phrase it here) does, but it does it within the space of a single equation.

That, essentially, is the trick that we use. With four “superspace dimensions”, we can include the four supersymmetries of N=4 super Yang-Mills, showing how the formulas vary when you go beyond the equation from our first paper.

So far, you may be wondering why I’m calling superspace a “dimension”, when it probably sounds like more of a label. I’ve mentioned before that, just because something is a variable, doesn’t mean it counts as a real dimension.

The key difference is that superspace dimensions are related to regular dimensions in a precise way. In a sense, they’re the square roots of regular dimensions. (Though independently, as George Musser described, they’re the square roots of zero: go in the same direction twice in supersymmetry, and you get back where you’re started, going zero distance.) The coexistence of these two seemingly contradictory statements isn’t some sort of quantum mystery, it’s just a consequence of the fact that, mathematically, I’m saying two very different things. I just can’t think of a way to explain them differently without math.

Superspace isn’t a real place…but it can often be useful to think of it that way. In theories with supersymmetry, it can unify the world, putting disparate particles together into a single equation.

Gravity is Yang-Mills Squared

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There’s a concept that I’ve wanted to present for quite some time. It’s one of the coolest accomplishments in my subfield, but I thought that explaining it would involve too much technical detail. However, the recent BICEP2 results have brought one aspect of it to the public eye, so I’ve decided that people are ready.

If you’ve been following the recent announcements by the BICEP2 telescope of their indirect observation of primordial gravitational waves, you’ve probably seen the phrases “E-mode polarization” and “B-mode polarization” thrown around. You may even have seen pictures, showing that light in the cosmic microwave background is polarized differently by quantum fluctuations in the inflaton field and by quantum fluctuations in gravity.

But why is there a difference? What’s unique about gravitational waves that makes them different from the other waves in nature?

As it turns out, the difference all boils down to one statement:

Gravity is Yang-Mills squared.

This is both a very simple claim and a very subtle one, and it comes up in many many places in physics.

Yang-Mills, for those who haven’t read my older posts, is a general category that contains most of the fundamental forces. Electromagnetism, the strong nuclear force, and the weak nuclear force are all variants of Yang-Mills forces.

Yang-Mills forces have “spin 1”. Another way to say this is that Yang-Mills forces are vector forces. If you remember vectors from math class, you might remember that a vector has a direction and a strength. This hopefully makes sense: forces point in a direction, and have a strength. You may also remember that vectors can also be described in terms of components. A vector in four space-time dimensions has four components: x, y, z, and time, like so:

\left( \begin{array}{c} x \\ y \\ z \\ t \end{array} \right)

Gravity has “spin 2”.

As I’ve talked about before, gravity bends space and time, which means that it modifies the way you calculate distances. In practice, that means it needs to be something that can couple two vectors together: a matrix, or more precisely, a tensor, like so:

\left( \begin{array}{cccc} xx & xy & xz & xt\\ yx & yy & yz & yt\\ zx & zy & zz & zt\\ tx & ty & tz & tt\end{array} \right)

So while a Yang-Mills force has four components, gravity has sixteen. Gravity is Yang-Mills squared.

(Technical note: gravity actually doesn’t use all sixteen components, because it’s traceless and symmetric. However, often when studying gravity’s quantum properties theorists often add on extra fields to “complete the square” and fill in the remaining components.)

There’s much more to the connection than that, though. For one, it appears in the kinds of waves the two types of forces can create.

In order to create an electromagnetic wave you need a dipole, a negative charge and a positive charge at opposite ends of a line, and you need that dipole to change over time.

Change over time, of course, is a property of Gifs.

Gravity doesn’t have negative and positive charges, it just has one type of charge. Thus, to create gravitational waves you need not a dipole, but a quadrupole: instead of a line between two opposite charges, you have four gravitational charges (masses) arranged in a square. This creates a “breathing” sort of motion, instead of the back-and-forth motion of electromagnetic waves.

This is your brain on gravitational waves.

This is why gravitational waves have a different shape than electromagnetic waves, and why they have a unique effect on the cosmic microwave background, allowing them to be spotted by BICEP2. Gravity, once again, is Yang-Mills squared.

But wait there’s more!

So far, I’ve shown you that gravity is the square of Yang-Mills, but not in a very literal way. Yes, there are lots of similarities, but it’s not like you can just square a calculation in Yang-Mills and get a calculation in gravity, right?

Well actually…

In quantum field theory, calculations are traditionally done using tools called Feynman diagrams, organized by how many loops the diagram contains. The simplest diagrams have no loops, and are called tree diagrams.

Fascinatingly, for tree diagrams the message of this post is as literal as it can be. Using something called the Kawai-Lewellen-Tye relations, the result of a tree diagram calculation in gravity can be found just by taking a similar calculation in Yang-Mills and squaring it.

(Interestingly enough, these relations were originally discovered using string theory, but they don’t require string theory to work. It’s yet another example of how string theory functions as a laboratory to make discoveries about quantum field theory.)

Does this hold beyond tree diagrams? As it turns out, the answer is again yes!
The calculation involved is a little more complicated, but as discovered by Zvi Bern, John Joseph Carrasco, and Henrik Johansson, if you can get your calculation in Yang-Mills into the right format then all you need to do is square the right thing at the right step to get gravity, even for diagrams with loops!

zvi-bern-350

carrasco

This trick, called BCJ duality after its discoverers, has allowed calculations in quantum gravity that far outpace what would be possible without it. In N=8 supergravity, the gravity analogue of N=4 super Yang-Mills, calculations have progressed up to four loops, and have revealed tantalizing hints that the uncontrolled infinities that usually plague gravity theories are absent in N=8 supergravity, even without adding in string theory. Results like these are why BCJ duality is viewed as one of the “foundational miracles” of the field for those of us who study scattering amplitudes.

Gravity is Yang-Mills squared, in more ways than one. And because gravity is Yang-Mills squared, gravity may just be tame-able after all.

Caltech Amplitudes Workshop, and Valentines Poem 2014

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This week’s post will be a short one. I’m at a small workshop for young amplitudes-folks at Caltech, so I’m somewhat busy.

(What we call a workshop is a small conference focused on fostering discussion and collaboration. While there are a few talks to give the workshop structure, most of the time is spent in more informal discussions between the participants.)

There have been a lot of great talks, and a lot of great opportunities to bond with fellow young amplitudeologists. Also, great workshop swag!

Yes, that is a Hot Wheels Mars Rover

Yes, that is a Hot Wheels Mars Rover

Unrelatedly, to continue a tradition from last year, and since it’s Valentine’s Day, allow me to present a short physics-themed poem I wrote a long time ago, this one about the sometimes counter-intuitive laws of thermodynamics:

Thermodynamic Hypothesis

A cold object, like a hot one, must be insulated

Cut off from interaction

Immerse the subject in a bath of warmth

And I reach equilibrium

Update on the Amplituhedron

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Awhile back I wrote a post on the Amplituhedron, a type of mathematical object  found by Nima Arkani-Hamed and Jaroslav Trnka that can be used to do calculations of scattering amplitudes in planar N=4 super Yang-Mills theory. (Scattering amplitudes are formulas used to calculate probabilities in particle physics, from the probability that an unstable particle will decay to the probability that a new particle could be produced by a collider.) Since then, they published two papers on the topic, the most recent of which came out the day before New Year’s Eve. These papers laid out the amplituhedron concept in some detail, and answered a few lingering questions. The latest paper focused on one particular formula, the probability that two particles bounce off each other. In discussing this case, the paper serves two purposes:

1. Demonstrating that Arkani-Hamed and Trnka did their homework.

2. Showing some advantages of the amplituhedron setup.

Let’s talk about them one at a time.

Doing their homework

There’s already a lot known about N=4 super Yang-Mills theory. In order to propose a new framework like the amplituhedron, Arkani-Hamed and Trnka need to show that the new framework can reproduce the old knowledge. Most of the paper is dedicated to doing just that. In several sections Arkani-Hamed and Trnka show that the amplituhedron reproduces known properties of the amplitude, like the behavior of its logarithm, its collinear limit (the situation when two momenta in the calculation become parallel), and, of course, unitarity.

What, you heard the amplituhedron “removes” unitarity? How did unitarity get back in here?

This is something that has confused several commenters, both here and on Ars Technica, so it bears some explanation.

Unitarity is the principle that enforces the laws of probability. In its simplest form, unitarity requires that all probabilities for all possible events add up to one. If this seems like a pretty basic and essential principle, it is! However, it and locality (the idea that there is no true “action at a distance”, that particles must meet to interact) can be problematic, causing paradoxes for some approaches to quantum gravity. Paradoxes like these inspired Arkani-Hamed to look for ways to calculate scattering amplitudes that don’t rely on locality and unitarity, and with the amplituhedron he succeeded.

However, just because the amplituhedron doesn’t rely on unitarity and locality, doesn’t mean it violates them. The amplituhedron, for all its novelty, still calculates quantities in N=4 super Yang-Mills. N=4 super Yang-Mills is well understood, it’s well-behaved and cuddly, and it obeys locality and unitarity.

This is why the amplituhedron is not nearly as exciting as a non-physicist might think. The amplituhedron, unlike most older methods, isn’t based on unitarity and locality. However, the final product still has to obey unitarity and locality, because it’s the same final product that others calculate through other means. So it’s not as if we’ve completely given up on basic principles of physics.

Not relying on unitarity and locality is valuable. For those who research scattering amplitudes, it has often been useful to try to “eliminate” one principle or another from our calculations. 20 years ago, avoiding Feynman diagrams was the key to finding dramatic simplifications. Now, many different approaches try to sidestep different principles. (For example, while the amplituhedron calculates an integrand and leaves a final integral to be done, I’m working on approaches that never employ an integrand.)

If we can avoid relying on some “basic” principle, that’s usually good evidence that the principle might be a consequence of something even more basic. By showing how unitarity can arise from the amplituhedron, Arkani-Hamed and Trnka have shown that a seemingly basic principle can come out of a theory that doesn’t impose it.

Advantages of the Amplituhedron

Not all of the paper compares to old results and principles, though. A few sections instead investigate novel territory, and in doing so show some of the advantages and disadvantages of the amplituhedron.

Last time I wrote on this topic, I was unclear on whether the amplituhedron was more efficient than existing methods. At this point, it appears that it is not. While the formula that the amplituhedron computes has been found by other methods up to seven loops, the amplituhedron itself can only get up to three loops or so in practical cases. (Loops are a way that calculations are classified in particle physics. More loops means a more complex calculation, and a more precise final result.)

The amplituhedron’s primary advantage is not in efficiency, but rather in the fact that its mathematical setup makes it straightforward to derive interesting properties for any number of loops desired. As Trnka occasionally puts it, the central accomplishment of the amplituhedron is to find “the question to which the amplitude is the answer”. By being able to phrase this “question” mathematically, one can be very general, which allows them to discover several properties that should hold no matter how complex the rest of the calculation becomes. It also has another implication: if this mathematical question has a complete mathematical answer, that answer could calculate the amplitude for any number of loops. So while the amplituhedron is not more efficient than other methods now, it has the potential to be dramatically more efficient if it can be fully understood.

All that said, it’s important to remember that the amplituhedron is still limited in scope. Currently, it applies to a particular theory, one that doesn’t (and isn’t meant to) describe the real world. It’s still too early to tell whether similar concepts can be defined for more realistic theories. If they can, though, it won’t depend on supersymmetry or string theory. One of the most powerful techniques for making predictions for the Large Hadron Collider, the technique of generalized unitarity, was first applied to N=4 super Yang-Mills. While the amplituhedron is limited now, I would not be surprised if it (and its competitors) give rise to practical techniques ten or twenty years down the line. It’s happened before, after all.

The Amplitudes Revolution Will Not Be Televised (But It Will Be Streamed)

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I’ve been at the Simons Center’s workshop on the Geometry and Physics of Scattering Amplitudes all week, so I don’t have time for a long post. There have been a lot of great talks from a lot of great amplitudes-folks (including one on Tuesday by Lance Dixon discussing this work, and one on the same day explaining the much-hyped amplituhedron). Curious folks can follow the conference link above to find videos and slides for each of the talks, arranged by the talk schedule.

I’ve made some great contacts, picked up a couple running jokes (check out Rutger Boels’s talk on Monday and Lance’s talk on Tuesday), heard the phrase “only seven loops” stated in relative seriousness, and heard the story of why the conference ended up choosing an artist’s conception of the amplituhedron for the workshop poster, which I can relate if folks are especially curious.

The Parke-Taylor Amplitudes: Why Quantum Field Theory Might Not Be So Hard, After All

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If you’ve been following my blog for a while, you know that Quantum Field Theory is hard work. To calculate anything, you have to draw an ever-increasing number of diagrams, translate them into formulas involving the momentum and energy of your particles, and add all those formulas up to get your final result, the amplitude of the process you’re interested in.

As I said in that post, my area of research involves trying to find patterns in the results of these calculations, patterns that make doing the calculation simpler. With that in mind, you might wonder why we expect to find any patterns in the first place. If Quantum Field Theory is so complicated, what insurance do we have that it can be made simpler? Where does the motivation come from?

Our motivation comes from a series of discoveries that show that things really do simplify, often in unexpected ways. I won’t go through all of these discoveries here, but I want to tell you about one of the first discoveries that showed amplitudes researchers that they were on the right track.

Let’s try to calculate a comparatively simple process. Say that we’ve got two gluons (force carrying bosons for the strong force, an example of a Yang-Mills field). Suppose the two gluons collide, and some number of gluons emerge. It could be two again, or it could be three, or more.

For now, let’s just think about diagrams at tree level, that is, diagrams with no loops. The particles can travel from place to place in the diagram, but they can’t form closed loops on the inside.

Gluons have two types of interactions, places where particle lines can come together. You can either have three lines meeting at one point, or four.

If two gluons come in and two come out, we have four possible diagrams:

4ptMHV

Note that while the last diagram looks like it has a loop in it (in the form of the triangle in the middle), actually that triangle just represents that two particles are passing each other without colliding, so that their lines cross.

The number of diagrams increases substantially as you increase the number of outgoing particles. With two particles going to three particles, you get fifteen diagrams. Here are three examples:

5ptMHV

Since the number of diagrams just keeps increasing, you’d expect the final amplitude to become more and more complicated as well. However, Steven Parke and Tomasz Taylor found in 1986 that for a particular arrangement of the spins of the particles (for technical people: this is the Maximally Helicity Violating configuration, or two particles with negative helicity and all the rest with positive helicity) the answer simplifies dramatically. In the sort of variables we use these days, the result can be expressed in an incredibly simple form:

\frac{\langle 1 | 2 \rangle^4}{ \langle 1 | 2 \rangle\langle 2 | 3 \rangle\langle 3 | 4 \rangle \ldots \langle n-1 | n \rangle\langle n | 1 \rangle}

Here the angle brackets represent momenta of the incoming (for 1 and 2) and outgoing (all the other numbers) particles, with n being the total number of particles (two going in, and however many going out). (Technically, these are spinor-helicity variables, and those interested in the technical details should check out chapter 3 of this or chapter 2 of this.)

Nowadays, we know why this amplitude looks so simple, in terms of something called BCFW recursion. At the time though, it was quite extraordinary.

This is the sort of simplification we keep running into when studying amplitudes. Almost always, it means that there is some deeper principle that we don’t yet understand, something that would let us do our calculations much faster and more efficiently. It indicates that Quantum Field Theory might not be so hard after all.