Amplitudes in N = 4 super Yang-Mills: an exploration of kinematical limits

Abstract

In this thesis we explore aspects of scattering amplitudes in planar N = 4 super Yang-Mills. In particular we shall focus on studying the mathematical structure of scattering amplitudes in different kinematical limits. First we use linear combinations of differential operators and the properties of multiple polylogarithms to solve for a differential equation obeyed by a 2-loop, 5-point dual conformal scalar integral in a coplanar kinematical limit. Next we dedicate the bulk of this thesis to planar amplitudes in multi-Regge kinematics (MRK) and we exploit the simplifications due to this limit to completely classify their mathematical structure.

We show that in MRK, the singularity structure of the amplitude corresponds to finite cluster algebras and thus may be described entirely by single-valued multiple polylogarithms. We then present a factorised form for the amplitude expressed as a Fourier-Mellin dispersion integral and proceed to derive novel results at leading logarithmic accuracy (LLA) for both MHV and non-MHV configurations. Specifically we show that amplitudes at L loops are determined by amplitudes with L + 4 legs and classify their leading singularities in MRK. Next we go beyond LLA by using 2-loop, 7-point data to extract corrections to the BFKL central emission vertex which is the only quantity in the dispersion integral not known to all orders. Finally we utilise the corrections to the central emission vertex to conjecture a finite coupling expression and thus extend the dispersion integral for amplitudes in MRK to all orders as well as all multiplicities and helicity configurations.

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