From correlation functions to scattering amplitudes

TL;DR

The paper uncovers a direct duality between light-like limit correlators of protected half-BPS operators in N=4 SYM and MHV gluon amplitudes, established through a Lagrangian-insertion approach and a novel dual infrared regularization. By mapping to dual coordinates, the authors show that the logarithm of the correlator ratio equals twice the logarithm of the amplitude ratio, with explicit checks at one loop for arbitrary n and at two loops for four- and five-point cases. They provide concrete expressions for the relevant correlator and amplitude integrals, and derive nontrivial integral identities required by the duality, aided by conformal symmetry and soft-limit analyses. The results suggest a deeper connection between correlators and scattering amplitudes, potentially linked to dual conformal symmetry, Wilson loops, and integrability, and pave the way for broader tests at higher points (e.g., six points).

Abstract

We study the correlators of half-BPS protected operators in N=4 super-Yang-Mills theory, in the limit where the positions of the adjacent operators become light-like separated. We compute the loop corrections by means of Lagrangian insertions. The divergences resulting from the light-cone limit are regularized by changing the dimension of the integration measure over the insertion points. Switching from coordinates to dual momenta, we show that the logarithm of the correlator is identical with twice the logarithm of the matching MHV gluon scattering amplitude. We present a number of examples of this new relation, at one and two loops.

Figures (11)

• Figure 1: Feynman diagrams of different types contributing to the correlator (\ref{["defco'"]}) at tree level. Arrowed lines denote free scalar propagators $\langle{\bar{\phi}^{12}(x_i) \phi_{12}(x_j)}\rangle$. In the light-cone limit $x_{i,i+1}^2\to 0$ the leading contribution comes from diagram (a), while that of diagram (b) is suppressed by the factor $x_{34}^2 x_{1n}^2/(x_{3n}^2x_{14}^2)$.
• Figure 2: The $n$-point correlator with one insertion. The solid and wavy lines are hypermultiplet and gauge propagators, respectively. The double dot denotes the insertion of the ${\cal N}=2$ SYM Lagrangian ${\rm Tr}(W^2)$.
• Figure 3: Building blocks for the graphs in Fig. \ref{['2loopgraphs']}. The single dot in (a) denotes a $W$ insertion, the double dot in (b) a ${\rm Tr}(W^2)$ insertion.
• Figure 4: One- and two-loop pseudo-conformal integrals contributing to the correlator $G_4$, Eq. (\ref{['44pt']}), and to the amplitude $A_4$, Eq. (\ref{['MHV4']}). The diagrams with solid lines depict Feynman integrals in $x-$space. The diagrams with dashed lines represent the same integral in the dual momentum space. The straight labels correspond to the points $x_i$, the slanted labels correspond to the momenta $p_i=x_i-x_{i+1}$.
• Figure 5: Two-loop pseudo-conformal integrals of different topologies contributing to the correlator $G_5$, Eq. (\ref{['fin']}), and to the amplitude $A_5$, Eq. (\ref{['amp2']}). The diagrams with solid lines depict Feynman integrals in $x-$space. The diagrams with dashed lines represent the same integral in the (dual) momentum space $p_i=x_i-x_{i+1}$. In the latter case, $(-k)$ stands for the particle with momentum $(-p_k)$. Thin solid lines denote numerators in the $x-$integral. In momentum space, this numerator is given by the squared sum of the momenta flowing through the arrowed dashed lines.