Higher-Point Correlation Functions in N = 4 Supersymmetric Yang–Mills Theory

Abstract

This thesis investigates loop integrands of correlation functions of scalar half-BPS operators in four-dimensional N = 4 supersymmetric Yang–Mills (SYM) theory. N = 4 SYM provides a central “theoretical laboratory” due to its high degree of symmetry, its role in the AdS/CFT correspondence, and the emergence of integrability in the planar limit. First, we study planar higher-point generating functions, which combine infinitely many fixed-charge correlators into unified expressions using ten-dimensional coordinates that merge spacetime and R-symmetry variables. We compute the five-point generating function at one and two loops, and the six-point function at one loop. The results exhibit a recursive nesting structure in which higher-order ten-dimensional poles are controlled by lower-point correlators. Second, we analyze ten-dimensional null polygons, obtained as a specific kinematic limit of the generating functions, where these factorize into simpler polygon functions with disc topology. We present a twistor-based direct computation of the polygon integrands, and determine the two-loop integrands for any number of external operators. By imposing soft and collinear bootstrap constraints in off-shell kinematics, we uniquely fix the five-point pentagon integrand up to five loops. This provides a useful setting for testing dualities between correlation functions and massive scattering amplitudes. Finally, we go beyond the planar limit and compute the full non-planar five-loop four-point integrand for the lightest scalar operators. This is achieved using a GPU-accelerated algorithm for evaluating Grassmann multiplications in the twistor Feynman rules. The result shows that the integrand can be expressed purely in terms of genus-one contributions. Furthermore, it allows us to extract the non-planar five-loop correction to the Konishi anomalous dimension. These results provide new insight into the structure of higher-point correlation functions in N = 4 SYM and open avenues for further investigations of both planar and non-planar dynamics.