Category Archives: Amplitudes

Complex Scattering For Beginners

Quantum field theory is a description of interacting particles. These are the fundamental constituents of our universe. They are real, in the sense that their properties are described by real numbers. For example, a typical particle has a momentum through spacetime described by a vector with four real components.

In my line of work we’re generally interested in finding out what happens when particles scatter. There are various rules that enable you to determine a numerical amplitude from diagrams of the process. These so-called “Feynman rules” combine the real quantities in different ways depending on the structure of the theory.

Trouble is, it can be quite tricky to do the exact calculations from these rules. It’s a bit like trying to put together a complicated piece of Ikea furniture with no idea what the end product is meant to look like! In a sense the task is possible, but you’d be hard pressed not to go wrong. Plus it would take you ages to finish the job.

What we really need is some extra pointers that tell us what we’re trying to build. Turns out that we can get that kind of information by performing a little trick. Instead of keeping all of our particle properties real, we bring in the complex numbers.

The complex numbers are like a souped up version of the real numbers. The extra ingredient is a new quantity i which squares to -1. This might all sound rather contrived at the moment, but in fact mathematically the complex numbers are a lot nicer behaved. By bringing them into play you can extract more information about your original scattering process for free!

Let’s go back to our Ikea analogy. Suppose that you get a mate in to help with the job. Your task is still the same as ever, but now as you construct it you can share tips. This makes everything easier. Moreover you can pool your guesses about what piece of furniture you’re building. The end result is still the same (hopefully!) but the extra input helped you to get there.

The Kallen-Lehmann Representation

Enough waffle, let’s get into some maths. Warning: you might find this hard going if you’re a layman! Consider the propagator of a generic (interacting) quantum field theory

\displaystyle \mathcal{A}(x,y) = \langle 0 | T \phi (x)\phi (y) | 0 \rangle

We’d like to look at it’s analytic properties as a function of the momentum p it carries. The first step is to use the standard completeness relation for the quantum states of the theory

\displaystyle \mathbf{1} = | 0 \rangle \langle 0 | + \sum_{\lambda} \int\frac{d^3p}{2(2\pi)^3E_{\mathbf{p}}(\lambda)} |\lambda_{\mathbf p}\rangle \langle \lambda_{\mathbf p} |

where |\lambda_{\mathbf{p}}\rangle is a general (possibly multiparticle) eigenstate of the Hamiltonian with momentum \mathbf{p}. Inserting this in the middle of the propagator we get

\displaystyle \mathcal{A}(x,y) = \sum_{\lambda} \int\frac{d^3p}{2(2\pi)^3E_{\mathbf{p}}(\lambda)} \langle 0 | \phi(x) |\lambda_{\mathbf p}\rangle \langle \lambda_{\mathbf p} | \phi(y) | 0 \rangle

where we have assumed that the VEV \langle \phi(x) \rangle vanishes, which is equivalent to no interactions at \infty. This is a very reasonable assumption, and in fact is a key assumption for scattering processes. (It’s particularly important in the analysis of spontaneous symmetry breaking, for example).

Now a little bit of manipulation (exercise: use the transformation of the quantities under the full Poincare group) gives us that

\displaystyle \langle 0 | \phi(x) |\lambda_{\mathbf p}\rangle = \langle 0 | \phi(0) | \lambda_{0}\rangle e^{-ip.x}|_{p_0 = E_{\mathbf{p}}}

Now substituting and introducing an integration over p_0 we get

\displaystyle \mathcal{A}(x,y) = \sum_{\lambda} D(x-y, m_{\lambda}^2)Z

where D(x-y,m_{\lambda}^2) is a Feynman propagator and Z = |\langle0|\phi(0)|\lambda_0\rangle|^2 is a renormalization factor. We’ll safely ignore Z for the rest of this post, since it doesn’t contribute to the analytic behaviour we’re interested in.

So why is this useful? One natural way to extract information from this formula might be to distinguish one-particle states. Let’s see how that helps. Recall that our states |\lambda_{\mathbf{p}}\rangle are eigenvalues of the energy-momentum operator (H, \mathbf{P}). Generically we get one-particle states of mass m arranged along a hyperboloid in energy-momentum space, due to special relativity. We also have multiparticle states of mass at least 2m forming a continuum at higher energy and momenta. (This is obvious if you consider possible vector addition of one-particle states).

Now we can use this newfound knowledge to rewrite the sum over \lambda as an isolated one-particle term, plus an integral over the multiparticle continuum as follows

\displaystyle \mathcal{A}(x,y) = D(x-y,m^2)Z + \int_{4m^2}^\infty dM^2 D(x-y, M^2)

This is starting to look promising. Transforming to momentum space is the last step we need to extract something useful. We find

\displaystyle \mathcal{A}(p^2) = \frac{iZ}{p^2-m^2} + \int_{4m^2}^{\infty}dM^2 \frac{iZ}{p^2 - M^2}

Considering the amplitude as an analytic function of the “Mandelstam variable” p^2 we find an isolated simple pole from an on-shell single particle state, plus a branch cut from multiparticle states.

It’s easy to generalize this to all Feynman diagrams. The key point is that all the analytic structure of an amplitude is encoded by the propagators. Indeed, the vertices and external legs merely contribute polarization vectors, internal symmetry factors and possibly positive factors of momentum. Singularities and branch cuts can only arise from propagators.

So what’s the big deal?

We’ve done a lot of work to extract some seemingly abstract information. But now it’s time for a substantial payoff! The analytic structure of Feynman diagrams can help us to determine their values. I won’t go into details here, but I will briefly mention one important application.

Remember that the scattering matrix in any sensible theory must conserve probabilities, and so be unitary. This requirement, coupled with our observations about Feynman diagrams tells us a lot about perturbative results. The result is usually known as the optical theorem and allows you to extract information about the discontinuities of higher loop diagrams from those at lower loops.

Still this seems rather esoteric, until you turn the whole procedure on it’s head. Suppose you are trying to guess a 1-loop amplitude. You know it’s general form perhaps, but need to fix some constants. Well from the S-matrix unitarity we know it has a branch cut and that the discontinuity is encoded by some tree level diagrams. These diagrams are essentially given by “cutting” the loop diagram.

So go ahead and compare the discontinuity you have with the product of the relevant tree diagrams. This will give you constraints on the constants you need to fix. Do this enough times, for different “cuts” and you will have fixed your 1-loop amplitude. Simple!

This method is known as generalized unitarity. It’s a vital tool in the modern amplitudes box, and has been used successfully to attack many difficult loop calculations. I’ll return to it more rigorously later, and promise to show you a genuine calculation too.

Until then, enjoy your weekend!

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Why I Like Supersymmetry

Supersymmetry can be variously described as beautiful, convenient, unphysical and contrived. The truth is that nobody really knows whether we’re likely to find it in our universe. Like most theoretical physicists I hope we do, but even if we don’t it can still be a useful mathematical tool.

There are tons of reasons to like supersymmetry, as well as a good many arguments against it. I can’t cover all of these in a brief post, so I’m just going to talk about one tiny yet pretty application I glanced at today.

Let’s talk about scattering processes again, my favourite topic of (physics) conversation. These are described by quantum field theory, which is itself based on very general principles of symmetry. In the standard formulation (imaginatively called the Standard Model) these symmetries involve physical motions in spacetime, as well as more abstract transformations internal to the theory. The spacetime symmetries are responsible for giving particles mass, spin and momentum, while the internal ones endow particles with various charges.

At the quantum level these symmetries actually provide some bonus information, in the form of certain identities that scattering processes have to satisfy. These go by the name of Ward identities. For example QED has a both a gauge and a global U(1) symmetry. The Ward identity for the global symmetry tells you that charge must be conserved. The Ward identity for the gauge symmetry tells you that longitudinally polarized photons are unphysical.

If you’re a layman and got lost above then don’t worry. All you need to know is that Ward identities are cool because they tell you extra things about a theory. The more information you have, the more constrained the answer must be, so the less work you have to do yourself! And this is where supersymmetry comes into the picture.

Supersymmetry is another (very special) type of symmetry that pairs up fermions (matter) and bosons (forces). Because it’s a symmetry it has associated Ward identities. These relate different scattering amplitudes. The upshot is that once you compute one amplitude you get more for free. The more supersymmetry you have, the more relations there are, so the easier your job becomes.

So what’s the use if supersymmetry isn’t true then? Well, in general terms it’s still useful to look at these simplified situations because it might help us discover tools that would be hard to uncover otherwise. Take the analogy of learning a language, for example. One way to do it is just to plunge headlong in and try to pick things up as you go along. This way you tend to get lots of everyday phrases quickly, but won’t necessary understand the structure of the language.

Alternatively you can go to classes that break things down into simpler building blocks. Okay spending one hour studying the subjunctive alone might not seem very useful at first, but when you go back to having a real conversation you’ll pick up new subtleties you never noticed before.

If you’re still unconvinced here’s a (somewhat trivial) concrete example. Recall that you can show that purely positive helicity gluon amplitudes must vanish at tree level in QCD. The proof is easy, but requires some algebraic fiddling. The SUSY Ward identity tells us immediately than in a Super-Yang-Mills (SYM) theory this amplitude must vanish to all orders in the loop expansion. So how do we connect back to QCD?

Well the gluon superpartners (gluinos) have quadratic coupling to the gluon, so an all gluon scattering amplitude in SYM can’t include any gluinos at tree level. (Just think about trying to draw the diagram if you’re confused)! In other words, at tree level the SYM amplitude is exactly the QCD amplitude, which proves our result.

Not sure what will be on the menu tomorrow – I’m guessing that either color-ordering or unitarity methods will feature. Drop me a comment if you have a preference.

Spinor Helicity Formalism – Twistors to the Rescue

Dang! Didn’t get my teeth into enough supersymmetry today. I’m standing at the gateway though, so I’ll be able to tell you much more tomorrow. For now, let’s backtrack a bit and take a look at spinor helicity formalism.

First things first, I need to remind you that on-shell matter particles (specifically spin \frac{1}{2} fermions) are represented as spinors in quantum field theory. For a massless fermion, the spinor encodes the momentum and helicity of the particle. We introduce the so called spinor helicity notation

\displaystyle [p| = +\textrm{ve helicity particle with momentum }p \\ \langle p| = -\textrm{ve helicity particle with momentum }p

Their hermitian conjugate spinors give the corresponding antiparticles, as you’d expect if you’re familiar with QFT. One can thus naturally define the inner product of a particle and antiparticle state by contracting their corresponding spinors. We see these contractions a lot in the Parke-Taylor formula, for example.

Now it turns out that every null future pointing vector can be represented in terms of it’s corresponding spinor helicity as according to the identification

\displaystyle p \leftrightarrow |p]\langle p|\qquad (A)

This can be made formal easily using the Weyl equation that the spinor states must satisfy. But what exactly is the use of this?

Well we saw that writing down scattering amplitudes in the spinor helicity formalism was particularly easy, since we could keep the amplitudes manifestly “on-shell” throughout the process. However I did sweep under the carpet a little algebraic manipulation. I can be more explicit about that now. The only difficult steps in the simplification I omitted are due to momentum conservation.

Usually momentum conservation for an n particle process takes the form

\displaystyle \sum_{i=1}^n p_i = 0

This is easy to implement in the usual Feynman diagram formalism, because it is linear! But in the spinor helicity world, we see that this formula becomes quadratic on account of the identification (A) above. This quadratic relation is somewhat troublesome to deal with, and requires annoying identity manipulation to impose.

But what if we want to have our cake and eat it? Imagine a world where we could have the spinor helicity simplifications and yet keep the simplicity of linear momentum conservation. Fortunately for us, such a world exists. We can get there by means of an abstract tool called the dual momentum twistor. There’s not enough time to tell you about that now, but watch out for its appearance in a later post.

So tomorrow will be some supersymmetry then, and maybe a short aside on calculating graviton scattering amplitudes easily. Easily, you say? Well it’s a doddle… at tree level… with a small number of particles…

My thanks to Andi Brandhuber for an enlightening discussion on this point.

Recursion Relations for Amplitudes

So yesterday’s discussion was all about how simple some gluon scattering processes (or amplitudes) look. In particular the maximal helicity violating (MHV) ones are special, because hundreds of terms cancel down to give a single neat result. Today I’ve been looking at how to prove this result, so I can now sketch the main ingredients for you.

If you’re from a mathsy background you won’t be surprised to learn that the n-gluon Parke-Taylor identity is proved using induction. For the uninitiated there’s a simple analogy with climbing stairs. If you can get up the first one, and you can get from every one to the next one then you can get to the top!

With this in mind, our first task is to prove the simplest case, which turns out to be n=3. Why exactly? Well the Feynman rules for QCD have 3 and 4 point vertices at tree level, so there’s no tree level 2 point amplitudes! Turns out that the n = 3 case is neatly dealt with using spinor helicity formalism. Roughly speaking this takes into account the special helicity structure of the MHV amplitudes to lock in simplifications right from the start of the calculation! Add in momentum conservation and hey presto the n=3 Parke-Taylor identity drops right out.

So now we need to climb from one stair to the next. This is where recursion relations come in handy. Nine years ago, a group of theoreticians spotted a cunning way to break apart tree level gluon amplitudes into smaller, more manageable pieces. Mathematically they spotted that the n-gluon scattering amplitude factorized into the product of two distinct on-shell amplitudes, each with a complementary subset of the original external legs plus an extra leg with momentum \hat{P}. The only added ingredient needed was a factor of 1/P^2 corresponding to a propagator between the two diagrams.

Woah – hold up there! What’s all this terminology all of a sudden. For the uninitiated I’m guessing that on-shell sounds a bit confusing. But it’s no cause for alarm. In general an on-shell quantity is one which obeys the equations of motion of the system involved. Here the relevant equation is the Weyl equation, which implies that \hat{P}^2 = 0.

Why are these recursion relations so useful? Well, they give us exactly the ingredient we needed for the induction step. And we’re done – the Parke-Taylor identity is proved, with a little bit of algebra I’ve shoved under the rug.

There’s one more point I’ve neglected to mention. How do you go about finding these mythical on-shell recursion relations? The answer comes from doing some subtle complex analysis, transforming momenta into the complex plane. It might not sound very physical to do that, but in fact the method opens up oodles of new possibilities. One reason is that complex integration is both more powerful and easier than its real counterpart, so it can be used to extract valuable identities from the world of scattering processes.

I’ll leave you with this great basic article clarifying the subtleties of the on-shell/off-shell distinction. It goes a bit deeper than that too, so is worth a read even if you’re more of an expert!

Tomorrow I hope I’ll bring you something more supersymmetrical in nature. \mathcal{N}=4 SUSY is a favourite playground for scattering enthusiasts because it is finite (no renormalization needed) and very simple (no free parameters). We’ll encounter its stark beauty in due course.

The Parke-Taylor Formula

Unfortunately I’m not going to have time today to give you a full post, mostly due to an abortive mission to Barking! The completion of that mission tomorrow may impact on post length again, so stay tuned for the first full PhD installment.

Nonetheless, here’s a brief tidbit from my first day. Let’s think about the theory of the strong force, which binds quarks and nuclei together. Mathematically it’s governed by quantum chromodynamics (QCD). At it’s simplest we can study QCD with no matter, so just consider the scattering interactions of the force carrying gluon particles.

It turns out that even this is pretty complicated! At tree level in Feynman diagram calculations, the simplest possible approximation, there are about 12000 terms for a four gluon scattering event. Thankfully these all cancel to give a single, closed form expression for the scattering amplitude. But why?

There’s a simpler way that makes use of some clever tricks to prove the more general Parke-Taylor formula that the maximal helicity violating n gluon amplitude is simply

\frac{\langle 12 \rangle^4}{\langle 12 \rangle \langle 23 \rangle \langle 34 \rangle \dots \langle n1 \rangle }

What does this all mean?

Qualitatively, that there is a formalism in which these calculations come out very simply and naturally. This will be the starting point for my exploration of modern day amplitudology – a subject that ranges through twistor theory, complex analysis and high dimensional geometry!

For the real mathematics behind the formula above, I’m afraid you’ll have to wait until tomorrow or Wednesday!