Perturbative Correlation Functions and Scattering Amplitudes in Planar N=4 Supersymmetric Yang-Mills

Abstract

In this thesis, we study the integrands of a special four-point correlation function formed of protected half-BPS operators and scattering amplitudes in planar supersymmetric N=4 Yang-Mills.

We use the "soft-collinear bootstrap" method to construct integrands of the aforementioned correlator and four-point scattering amplitudes to eight loops. Both have a unique representation in terms of (dual) conformal integrands with specified coefficients. The result is then extended to ten loops, by introducing two graphical relations, called the "triangle" and "pentagon" rules. These relations provide consistency conditions on the coefficients, and when combined with the "square" rule, prove sufficient to fix the answer to ten loops. We provide derivations for the graphical relations and illustrate their application with examples. The result exposes novel features seen for the first time at eight loops, that continue to be present through to ten loops. For example, the integrand includes terms that are finite even on-shell and terms that are divergent even off-shell (so-called "pseudoconformal" integrals).

We then proceed to study the correlator/amplitude duality by taking six and seven adjacent points of the four-point correlator to be light-like separated. A conformal basis (with rational coefficients) is used to extract amplitude integrands for both six and seven particles up to two loops - more precisely, the complete one-loop amplitude and parity-even two-loop amplitude (at two loops, we use a refined prescriptive basis). We also construct an alternative six-point one-loop basis involving integrands with conformal cross-ratio coefficients, and reverse the duality to algebraically extract integrands from an ansatz, by introducing the Gram determinant. We expect the former approach to be applicable to n-points at arbitrary loop-order l, by going to one extra order of perturbation in the correlator (to determine all parity-odd l-loop ambiguities).