Description

Scattering amplitudes give the probabilities for different outcomes of any particle scattering experiment, for example at the Large Hadron Collider in CERN. They are notoriously diffficult to compute however, so that even many of the scattering amplitudes relevant for analysing results at the LHC and in particular distinguishing new physics from known physics are not known.

Feynman diagrams are a basic method for calculating scattering amplitudes of particles in quantum field theory. However, to compute even quite a simple amplitude can involve summing many thousands of diagrams. However the final expressions can be remarkably simple, resulting from delicate and unexpected cancellations between terms and implying that there should be a better way to compute them.

In recent years a number of new methods have been discovered that radically simplify the computation. They all, in one way or another bypass the need for Feynman diagrams. One method, known as BCFW, replaces the method of Feynman diagrams by a recursion relation. Surprisingly this relies on little more than some general physical principles and Cauchy's theorem in complex analysis. Another recent method, discovered just a few years ago, is known as the amplituhedron. This states that the scattering ampltiudes in a certain theory are given by a direct generalisation of computing areas of polygons. Other methods are unitarity based ones and the recent colour kineatics duality (BCJ).

The project would begin by discussing the basic features of quantum field theory, focussing on understanding Feynman rules. We would then look at the spinor helicity formalism. Then we can go in a few possible directions: derive the BCFW recursion relations; consider maximally supersymmetric Yang-Mills and the amplituhedron; or alternatively consider unitarity methods or BCJ.

Prerequisites

Resources

One will first need a basic familiarity with quantum field theory, in particular Feynman diagrams and gauge theory. The online notes of Elvang and Huang are useful. Introductory texts on quantum field theory such as  Quantum Field Theory in a Nutshell by A.Zee, Princeton University Press, Quantum Field Theory by Itzykson, C. and Zuber, J. B. New York: McGraw-Hill, 1980 or Quantum Field Theory of Point Particles by Brian Hatfield will be useful.
Beyond basic QFT we will base our study of the most recent developments on the recent lecture notes here.