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!


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