Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N=4 SYM

TL;DR

We present a permutation- and conformal-symmetry–driven, diagram-free construction of the four-point stress-tensor multiplet correlator in ${\cal N}=4$ SYM, valid for both planar and non-planar sectors. Using a graph-theoretic decomposition into $f$- and $P$-graphs, we build a general integrand ansatz and fix its coefficients with Euclidean short-distance and Minkowski light-cone singularities, cross-checked against the planar four-particle amplitude via a duality in the light-cone limit. The planar correlator is completely fixed up to six loops (and conjectured to all loops), while the non-planar sector first shows nontrivial corrections at four loops, with a small finite set of coefficients remaining undetermined by singular limits alone. The work also provides a powerful rung-rule–based recursion that mirrors amplitude structures and yields a nontrivial cross-check against known higher-loop amplitudes, thus strengthening the correlation-amplitude duality in planar ${\cal N}=4$ SYM and offering a scalable framework for high-loop perturbative data.

Abstract

We present a construction of the integrand of the correlation function of four stress-tensor multiplets in N=4 SYM at weak coupling. It does not rely on Feynman diagrams and makes use of the recently discovered symmetry of the integrand under permutations of external and integration points. This symmetry holds for any gauge group, so it can be used to predict the integrand both in the planar and non-planar sectors. We demonstrate the great efficiency of graph-theoretical tools in the systematic study of the possible permutation symmetric integrands. We formulate a general ansatz for the correlation function as a linear combination of all relevant graph topologies, with arbitrary coefficients. Powerful restrictions on the coefficients come from the analysis of the logarithmic divergences of the correlation function in two singular regimes: Euclidean short-distance and Minkowski light-cone limits. We demonstrate that the planar integrand is completely fixed by the procedure up to six loops and probably beyond. In the non-planar sector, we show the absence of non-planar corrections at three loops and we reduce the freedom at four loops to just four constants. Finally, the correlation function/amplitude duality allows us to show the complete agreement of our results with the four-particle planar amplitude in N=4 SYM.

Figures (12)

• Figure 1: $P-$graphs up to three loops. The one-loop graph $P^{(1)}$ shrinks to a point corresponding to the constant numerator in (\ref{['f1']}). The two-loop graph $P^{(2)}$ consists of three disconnected segments. At three loops there are four possible graphs, one connected $P^{(3)}_1$ and three disconnected $P^{(3)}_2$, $P^{(3)}_3$, $P^{(3)}_4$.
• Figure 2: $f-$graphs up to three loops obtained from the $P-$graphs in Fig. \ref{['Pgr']}. The one-loop graph $f^{(1)}$ is non-planar, but it becomes planar after multiplying it with the prefactor in (\ref{['integg']}). The two-loop graph $f^{(2)}$ and one of the tree-loop graphs, $f^{(3)}_2$, are planar. The remaining three-loop graphs $f^{(3)}_1$, $f^{(3)}_3$, $f^{(3)}_4$ are non-planar, even including the prefactor in (\ref{['integg']}).
• Figure 3: Planar $f-$graphs at one, two, three and four loops. One can see how applying the rung rule (i.e. gluing a pyramid across rectangle face shown in blue) to a lower-loop $f-$graph produces higher-loop $f-$graphs.
• Figure 4: Obtaining the graph $f^{(1)}$ from the graph ${\cal F}^{(1)}$. The apex of the pyramid ${\cal F}^{(1)}$ corresponds to the integration point, the four legs stand on the external points. The dashed lines are numerator factors canceling the conformal weight of ${\cal F}^{(1)}$ at the external points.
• Figure 5: Graphical representation of the term $\frac{1}{2} {\left({{\cal F}^{(1)}}\right)}^2$ and of the corresponding $f-$graph.