Notes on off-shell conformal integrals and correlation functions at five points

TL;DR

This work constructs a uniform transcendental (UT), pure basis of six two-loop, five-point conformal integrals in ${\mathcal{N}=4}$ SYM by diagonalizing leading singularities under conformal invariance and mapping to four-massive master families via conformal frame fixing. Using canonical differential equations and IBP reduction, the authors obtain integrated results for the five-point half-BPS correlators at symbol level, covering both maximal and non-maximal Grassmann sectors, and provide explicit symbol-level expressions with $106$ letters and $31$ square-root structures. The approach clarifies the UT structure of higher-point conformal integrals and establishes a concrete link to known two-loop four-point master integrals, enabling efficient evaluation and future extensions to higher points or elliptic sectors. These results lay groundwork for broader explorations of Correlahedra, hidden higher-dimensional symmetries, and potential generalizations to more complex operator configurations in ${\mathcal{N}=4}$ SYM.

Abstract

We study five-point off-shell conformal integrals and associated half-BPS correlation functions at the two-loop order in the 't Hooft coupling expansion in maximal supersymmetric Yang-Mills theory. We construct a basis of uniform transcendental, pure integrals, comprising six distinct topologies, through the method of diagonalizing leading singularities under the constraints of conformal invariance, which serve as basis integrals for conformally-symmetric observables at five points and two loops. By employing different conformal frame fixing choices, this integral basis can be mapped onto known two-loop four-massive-particle Feynman integral families. Subsequently, their integrated results are computed using the method of canonical differential equations and integration-by-parts reduction. We present for the first time the integrated results for the two-loop five-point half-BPS correlators, encompassing both maximal and non-maximal sectors, at symbol level.

Figures (2)

• Figure 1: Two-loop five-point relevant topologies.
• Figure 2: Excluded two–loop five–point topologies.