The Non-Planar Four-Point Integrand and Konishi Dimension in N=4 Super Yang-Mills Theory at Five Loops

TL;DR

The paper computes the complete non-planar five-loop integrand for the four-point correlator of the $20'$ operators in $\mathcal{N}=4$ SYM by combining a manifestly invariant $f$-graph ansatz with light-cone constraints and a twistor-space reformulation. A GPU-accelerated Grassmann-contraction algorithm enables numerical matching to fix the 21 remaining coefficients, yielding a genus-one non-planar correction with no genus-two contributions at this order. From the resulting integrand, the non-planar five-loop Konishi anomalous dimension is extracted as $\gamma_{\mathcal{K}}^{(1,5)} = 135 \zeta_5 - \frac{234}{4} \zeta_3^2 + \frac{11907}{32} \zeta_7$, consistent with pole cancellation and transcendentality expectations. The work provides a detailed non-planar OPE data set and demonstrates the role of twistorial methods and Gram identities in constraining high-loop amplitudes, with implications for convergence properties at finite $N_c$.

Abstract

We compute the complete non-planar integrand for the correlation function of four lightest scalar operators in N=4 super Yang-Mills theory at five-loop order. This is equivalent to the super-correlator of nine stress-tensor multiplets in the self-dual theory. Starting with an ansatz of f-graphs, we impose constraints from light-cone limits, and fix the remaining freedom by using the reformulation of the theory in twistor space. We develop an efficient GPU-based algorithm for the numerical evaluation of the twistor rules. As an application, we extract the five-loop non-planar anomalous dimension of the Konishi operator. Our code and result are provided in ancillary files.

Figures (5)

• Figure 1: The Konishi Dimension $\Delta_{\mathcal{K}}$ as a function of the coupling $a$ for $N_{\text{c}}=\infty$ (red) and $N_{\text{c}}=3$ (blue) at four loops (dashed) and five loops (solid). The radius of convergence of the planar theory is at $a=1/4$.
• Figure 2: Estimate of the convergence radius of the Konishi dimension weak-coupling expansion, for $N_{\text{c}}=\infty$ (squares, red) and for $N_{\text{c}}=3$ (circles, blue). Shown are the ratios $\abs{ \gamma_\mathcal{K}^{(\ell)}/ \gamma_\mathcal{K}^{(\ell-1)}}$ as in \ref{['eq:RadiusEstimate']}, as a function of $1/\ell$, as well as linear fits whose intersections with $1/\ell=0$ yield the numbers in \ref{['eq:RadiusEstimate']}.
• Figure 3: Heat map showing the time (in seconds) to perform 10,000 five-loop contractions on an Nvidia A100 GPU. The y-axis represents the batch size, and the x-axis corresponds to the threads-per-block configuration.
• Figure 4: Histogram of absolute values in the particular genus-one five-loop integrand solution provided in the ancillary files, excluding $582$ zero coefficients.
• Figure 5: Diagrammatic representation of the two non-planar four-loop master integrals that contribute to the Konishi anomalous dimension. The edges connecting vertices $i$ and $j$ stand for factors $1/x_{ij}^2$.