Hidden symmetry of four-point correlation functions and amplitudes in N=4 SYM

TL;DR

The study identifies a hidden permutation symmetry in the integrand of four-point stress-tensor multiplet correlators in $\mathcal{N}=4$ SYM, enabling strong constraints on multi-loop structures via Lagrangian insertions. By combining this symmetry with the amplitude/correlation duality, the authors deterministicallyfix the three-loop planar integrand up to four constants and check consistency against the OPE, recovering the Konishi anomalous dimensions. They extend the framework to four loops and outline a systematic path for higher-loop generalizations, illustrating how lower-loop information can constrain higher-loop amplitudes in the planar limit. The results provide nontrivial cross-checks between correlation functions and scattering amplitudes and suggest a broader utility of permutation symmetries in organizing perturbative data in $\mathcal{N}=4$ SYM.

Abstract

We study the four-point correlation function of stress-tensor supermultiplets in N=4 SYM using the method of Lagrangian insertions. We argue that, as a corollary of N=4 superconformal symmetry, the resulting all-loop integrand possesses an unexpected complete symmetry under the exchange of the four external and all the internal (integration) points. This alone allows us to predict the integrand of the three-loop correlation function up to four undetermined constants. Further, exploiting the conjectured amplitude/correlation function duality, we are able to fully determine the three-loop integrand in the planar limit. We perform an independent check of this result by verifying that it is consistent with the operator product expansion, in particular that it correctly reproduces the three-loop anomalous dimension of the Konishi operator. As a byproduct of our study, we also obtain the three-point function of two half-BPS operators and one Konishi operator at three-loop level. We use the same technique to work out a compact form for the four-loop four-point integrand and discuss the generalisation to higher loops.

Figures (10)

• Figure 1: Diagrammatic representation of the function $f^{(\ell)}(x)$ for $\ell=1$ and $\ell=2$, Eq. (\ref{['f12']}). Each line with labels $i$ and $j$ at the end points denotes a scalar propagator $1/x_{ij}^2$.
• Figure 2: Diagrammatic representation of the different $S_7$ symmetric polynomials $P^{(3)}(x_i)$ defined in (\ref{['eq:11']}). The indices $(\sigma_1,\ldots,\sigma_7)$ correspond to all permutations of the external points $(1,\ldots,7)$. The dashed lines with indices $\sigma_i$ and $\sigma_j$ at the end points denote the factors $x_{\sigma_i\sigma_j}^2$.
• Figure 3: Diagrammatic representation of the four classes of functions $f^{(3)}(x)$ corresponding to the polynomials shown in Fig. \ref{['fig:1']}. Solid lines denote scalar propagators $1/x_{\sigma_i\sigma_j}^2$ while dashed lines stand for numerator factors $x_{\sigma_i\sigma_j}^2$.
• Figure 4: Dual conformal $x-$integrals (solid lines) and momentum $p-$integrals (doted lines) in the three-loop planar four-gluon amplitude: (a) one-loop ladder, (b) two-loop ladder, (c) three-loop ladder and (d) tennis court.
• Figure 5: Diagrammatic representation of the one- and two-loop integrals (\ref{['eq:g+h']}). The solid lines denote scalar propagators and the dashed line denotes a factor of $x^2_{24}$ in the numerator. Each external point has conformal weight $(+1)$.