The cusp anomalous dimension at three loops and beyond

TL;DR

The paper addresses the computation of the cusp anomalous dimension Gamma_cusp for N=4 SYM at three loops by exploiting a dual conformal Regge-limit relation between massive Coulomb-branch amplitudes and infrared divergences of cusped Wilson loops. The authors evaluate the four-point amplitude to three loops, using a Regge-limit expansion of massive integrals to extract Gamma_cusp and present analytic expressions, highlighting the xi-dependent ladder structure. They also isolate a ladder-diagram limit that maps to a one-dimensional Schrödinger problem, enabling exact and perturbative sums that agree with strong-coupling results. The results advance the understanding of cusp dynamics, provide substantial parts of the three-loop QCD cusp, and offer insights into ladder-resummation regimes and their relation to string-theoretic predictions.

Abstract

We derive an analytic formula at three loops for the cusp anomalous dimension Gamma_cusp(phi) in N=4 super Yang-Mills. This is done by exploiting the relation of the latter to the Regge limit of massive amplitudes. We comment on the corresponding three loops quark anti-quark potential. Our result also determines a considerable part of the three-loop cusp anomalous dimension in QCD. Finally, we consider a limit in which only ladder diagrams contribute to physical observables. In that limit, a precise agreement with strong coupling is observed.

Figures (6)

• Figure 1: ( a) A Wilson line that makes a turn by an angle $\phi$ in Euclidean space. The vectors $\vec{n}$, $\vec{n}'$ denote the direction that sets the coupling to the scalar. ( b) Under the plane to cylinder map, the same line is mapped to a quark anti-quark configuration. The quark and antiquark are sitting at two points on $S^3$ at a relative angle of $\delta = \pi - \phi$. Of course, they are extended along the time direction.
• Figure 2: Sample two-loop diagram contributing to the four-particle amplitude. Solid and wavy lines denote massive and (almost) massless particles, respectively. The precise masses are given by the labels. Dual conformal symmetry implies that the same function ${\cal M}(u,v)$ describes two different physical situations: The Regge limit $s\to \infty$ of (a) is equivalent to the Bhabha-type scattering (b), where the outer wavy lines have a small mass that regulates the soft divergences.
• Figure 3: Integrals contributing to the four-particle amplitude to three loops. Solid and wavy lines denote massive and massless propagators, respectively. Overall normalizations of $s$ and $t$, as well as a loop-momentum dependent numerator factor in $I_{3b}$ are not displayed.
• Figure 4: Analytic structure of $\Gamma_{\rm cusp}$. The Euclidean region is $x>0$. Below threshold, $x$ is a phase, $x=e^{i \phi}$. Above threshold, we have that $-1<x<0$, with $x$ having an infinitesimal positive imaginary part. The zigzag line denotes branch cuts along $[-\infty,0].$
• Figure 5: Crossed ladder diagrams that appear in the computation of $\Gamma_{\rm cusp}$ at two and three loops.