When talking about the spinor-helicity formalism in his new textbook on quantum field theory, Matthew D. Schwartz claims as a highly nontrivial example, it is quite easy to use the Parke-Taylor formula to calculate the $gg\to ggg$ scattering cross section at tree level by hand, which is also one of the problems in the book. (Problem is found on page 560, problem 27.6)

Can anyone tell me how to do this exactly? It looks like we cannot avoid summing $24^2$ terms, and in the paper, http://www.sciencedirect.com/science/article/pii/0550321386902300, the cross section of $gg\rightarrow gggg$ spans several pages! This makes me expect the cross section for $gg\rightarrow ggg$ is also complicated. Is there any smart way to do it?

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    $\begingroup$ Could you give more detail about the reaction (for those of us without the book)? $\endgroup$ – Your Majesty Mar 24 '14 at 22:07
  • $\begingroup$ The reaction has initial state with two gluons and final state with three gluons, the theory is SU(N) non-Abelian gauge theory $\endgroup$ – Mr. Gentleman Mar 24 '14 at 23:19
  • $\begingroup$ @JamalS are you sure about the edit? He never asked for $gg\rightarrow gggg$. $\endgroup$ – Your Majesty Mar 25 '14 at 15:53
  • $\begingroup$ @LoveLearning: The paper he referenced says "two-gluon to four-gluon," so I think it's just a typo. If you check his very first post, the OP says three gluon, but then in the next paragraph says four gluon. $\endgroup$ – JamalS Mar 25 '14 at 16:02
  • $\begingroup$ Yes but to my knowledge the problem in the book says $ggg$ (3). $\endgroup$ – Your Majesty Mar 25 '14 at 16:14

I will try to avoid too many details. In case you want to learn more about these things I could suggest this reference: http://arxiv.org/pdf/1308.1697.pdf and from it you can find more references. First of all, the Parke-Taylor formulas is about the so called MHV (Maximal Helicity Violating) Amplitudes and with the use of the spinor-helicity formalism they can be written in a very compact form. MHV are the amplitudes where all gluons except for two have positive helicities (amplitudes with all helicities positive or all except one vanish) They were conjectured in the Parke-Taylor paper and later proven by recursive formulas called Berends-Giele ( http://www.sciencedirect.com/science/article/pii/0550321388904427?via=ihub).
Now, if you want to calculate amplitudes with different polarisations, the situation becomes more complicated. However, it was found (see references in the first paper I gave you) that one can use as building blocks three point on-shell MHV amplitudes and construct them recursively. The most commonly used method is the so called BCFW recursion. You work at the amplitude level and from lower amplitudes you can build higher ones (and since the 3 point MHV amplitudes have very simple form the calculations are relatively simple, especially if you master the use of spinor-helicity formalism). In that case, you can gain a lot since you avoid working with Feynman diagrams which are too many as the number of external particles grows (the number of Feynman diagrams grow usually like $n!$. There are a lot of computations in this subject and that is the reason I avoided giving formulas and examples. But you will find them in the review. Good luck!


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