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!