Perturbative Correlation Functions and Scattering Amplitudes in Planar $\mathcal{N}=4$ 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 $\mathcal{N}=4$ Yang-Mills. We use the soft-collinear bootstrap' method to construct integrands of the aforementioned correlator and four-point scattering amplitudes to eight loops. The result is then extended to ten loops, by introducing two graphical relations, called thetriangle' and pentagon' rules. These relations provide consistency conditions on the coefficients, and when combined with thesquare' rule, prove sufficient to fix the answer to ten loops. 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.