Three-Point Functions of Twist-Two Operators in N=4 SYM at One Loop

TL;DR

The paper addresses the problem of determining three-point functions with two protected operators and a twist-two operator of even spin $j$ in ${\mathcal N}=4$ SYM at one loop. The authors implement a light-cone projection and a soft-limit where the spin-$j$ operator carries vanishing momentum, transforming the perturbative calculation into a tractable set of two-point-like integrals and eliminating dependence on the one-loop mixing matrix. They derive the one-loop correction to the structure constant in terms of harmonic sums, show that the result matches the Dolan–Osborn structure constants obtained from the OPE of half-BPS four-point functions, and provide a normalisation-invariant formulation of the structure constant. This work provides a direct field-theoretic check of OPE-derived constants and demonstrates a calculational route that leverages symmetry and limiting procedures to simplify loop computations in ${\mathcal N}=4$ SYM.

Abstract

We calculate three-point functions of two protected operators and one twist-two operator with arbitrary even spin j in N=4 SYM theory to one-loop order. In order to carry out the calculations we project the indices of the spin j operator to the light-cone and evaluate the correlator in a soft-limit where the momentum coming in at the spin j operator becomes zero. This limit largely simplifies the perturbative calculation, since all three-point diagrams effectively reduce to two-point diagrams and the dependence on the one-loop mixing matrix drops out completely. The results of our direct calculation are in agreement with the structure constants obtained by F.A. Dolan and H. Osborn from the operator product expansion of four-point functions of half-BPS operators.

Figures (3)

• Figure 1: Feynman diagrams contributing to the three-point function at one loop. For $j$ even the diagrams in the second and third row are identical.
• Figure 2: Due to the momenta from the gluon vertices, the integrand decomposes into simpler integrals, which cancel with the self-energy and four-scalar interaction terms.
• Figure 3: One-loop corrections to the two-point function of $\langle \hat{{\mathcal{O}}}_j \hat{{\mathcal{O}}}_j \rangle$.