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Tessellating cushions: four-point functions in N=4 SYM

Burkhard Eden, Alessandro Sfondrini

TL;DR

This work proposes an integrability-based tessellation of planar tree-level four-point functions in N=4 SYM using modified hexagon form factors in a restricted Drukker-Plefka kinematic regime where one BMN operator and three half-BPS operators lie on a line. By tessellating the four-point sphere into hexagon tiles and embedding the space-time dependence into the hexagon vertices, the authors reproduce tree-level correlators efficiently and show agreement with multiple BMN-BPS cases up to length seven. The approach extends the hexagon framework beyond three-point functions and suggests a path toward an independent, integrability-driven computation of higher-point functions, potentially extendable to loop orders with finite-size corrections. Overall, the tessellated hexagon method provides a promising non-OPE-based route to four-point functions, aligning with integrability structures in AdS/CFT and offering practical computational advantages for a broad class of mixed correlators.

Abstract

We consider a class of planar tree-level four-point functions in N=4 SYM in a special kinematic regime: one BMN operator with two scalar excitations and three half-BPS operators are put onto a line in configuration space; additionally, for the half-BPS operators a co-moving frame is chosen in flavour space. In configuration space, the four-punctured sphere is naturally triangulated by tree-level planar diagrams. We demonstrate on a number of examples that each tile can be associated with a modified hexagon form-factor in such a way as to efficiently reproduce the tree-level four-point function. Our tessellation is not of the OPE type, fostering the hope of finding an independent, integrability-based approach to the computation of planar four-point functions.

Tessellating cushions: four-point functions in N=4 SYM

TL;DR

This work proposes an integrability-based tessellation of planar tree-level four-point functions in N=4 SYM using modified hexagon form factors in a restricted Drukker-Plefka kinematic regime where one BMN operator and three half-BPS operators lie on a line. By tessellating the four-point sphere into hexagon tiles and embedding the space-time dependence into the hexagon vertices, the authors reproduce tree-level correlators efficiently and show agreement with multiple BMN-BPS cases up to length seven. The approach extends the hexagon framework beyond three-point functions and suggests a path toward an independent, integrability-driven computation of higher-point functions, potentially extendable to loop orders with finite-size corrections. Overall, the tessellated hexagon method provides a promising non-OPE-based route to four-point functions, aligning with integrability structures in AdS/CFT and offering practical computational advantages for a broad class of mixed correlators.

Abstract

We consider a class of planar tree-level four-point functions in N=4 SYM in a special kinematic regime: one BMN operator with two scalar excitations and three half-BPS operators are put onto a line in configuration space; additionally, for the half-BPS operators a co-moving frame is chosen in flavour space. In configuration space, the four-punctured sphere is naturally triangulated by tree-level planar diagrams. We demonstrate on a number of examples that each tile can be associated with a modified hexagon form-factor in such a way as to efficiently reproduce the tree-level four-point function. Our tessellation is not of the OPE type, fostering the hope of finding an independent, integrability-based approach to the computation of planar four-point functions.

Paper Structure

This paper contains 9 sections, 26 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. BMN operators and tree-level four point functions
  3. BMN operators at one loop
  4. Four-point functions
  5. The hexagon approach and four-point functions
  6. The hexagon approach
  7. Four-point function by hexagons
  8. Outlook
  9. Some explicit examples

Figures (2)

  • Figure 1: Planar four-point tree-level diagrams involving single-trace operators. Operators are depicted by a black ring, with crimson lines between operators $i,j$ representing the product of $l_{ij}$ propagators.
  • Figure 2: A planar three-point function can be split into an inner and an outer hexagon (green and blue dotted lines, respectively). This splitting partitions the Wick contractions (crimson solid lines) among the two hexagons. In the BKV proposal, we should sum over all these partitions with appropriate weights.