Clustering and the Three-Point Function

TL;DR

This paper develops analytical methods for computing three-point functions of heavy operators in a strongly coupled quantum field theory, using a hexagon approach reformulated as contour integrals and cluster sums, with tests at various couplings.

Contribution

It introduces a new reformulation of the hexagon expansion as contour integrals and cluster sums, enabling explicit calculations of structure constants at strong coupling.

Findings

Matching semiclassical results with string theory at strong coupling

Sum of wrapping corrections expressed as a super-determinant

Analytical framework applicable to heavy operators in integrable models

Abstract

We develop analytical methods for computing the structure constant for three heavy operators, starting from the recently proposed hexagon approach. Such a structure constant is a semiclassical object, with the scale set by the inverse length of the operators playing the role of the Planck constant. We reformulate the hexagon expansion in terms of multiple contour integrals and recast it as a sum over clusters generated by the residues of the measure of integration. We test the method on two examples. First, we compute the asymptotic three-point function of heavy fields at any coupling and show the result in the semiclassical limit matches both the string theory computation at strong coupling and the tree-level results obtained before. Second, in the case of one non-BPS and two BPS operators at strong coupling we sum up all wrapping corrections associated with the opposite bridge to the non-trivial operator, or the "bottom" mirror channel. We also give an alternative interpretation of the results in terms of a gas of fermions and show that they can be expressed compactly as an operator-valued super-determinant.