Four-Loop Collinear Anomalous Dimension in N = 4 Yang-Mills Theory

TL;DR

The paper computes the four-loop contribution to the collinear anomalous dimension $g^{(4)}$ in planar N=4 Yang-Mills theory, a key component of the universal subleading infrared structure of gluon amplitudes. Using the obstruction method in Mellin space, the authors extract the $1/\epsilon$ pole coefficients from the four-loop iterative relation, enabling the precise determination of $g^{(4)}$. The key result is $g^{(4)} = -1240.9(3)$, confirming the growing connection between weak- and strong-coupling analyses. An interpolation anchored by the Alday-Maldacena strong-coupling result shows $g(\lambda)$ is consistent with the predicted asymptotics, with the estimated $g^{(4)}$ closely matching a strong-coupling extrapolation within about 7%. The paper also constructs Padé approximants to summarize $g(\lambda)$ at finite coupling, offering a practical representation to guide integrability-based approaches.

Abstract

We report a calculation in N = 4 Yang-Mills of the four-loop term g^4 in the collinear anomalous dimension g(lambda) which governs the universal subleading infrared structure of gluon scattering amplitudes. Using the method of obstructions to extract this quantity from the 1/epsilon singularity in the four-gluon iterative relation at four loops, we find g^4 = -1240.9 with an estimated numerical uncertainty of 0.02%. We also analyze the implication of our result for the strong coupling behavior of g(lambda), finding support for the string theory prediction computed recently by Alday and Maldacena using AdS/CFT.

Figures (2)

• Figure 1: The 12 integrals appearing in the $L \le 4$-loop four-particle amplitudes in ${\mathcal{N}} = 4$ Yang-Mills theory. We refer the reader to Bern:2006ew for all necessary details.
• Figure 2: A Padé approximant for the function $g(\lambda)$ versus $\lambda/16 \pi^2$, obtained by fitting the ansatz (\ref{['eq:padeansatz']}) to all available data at weak and strong coupling.