Cusped SYM Wilson loop at two loops and beyond

TL;DR

This paper computes the two-loop anomalous dimension of cusped Wilson loops in N=4 SYM, revealing an anomaly term and analyzing ladder diagram summations, with implications for AdS/CFT correspondence.

Contribution

It provides a detailed two-loop calculation of the cusp anomalous dimension, identifies an anomaly term, verifies the loop equation, and analyzes ladder diagram summations in N=4 SYM.

Findings

Anomaly term persists for cusped loops, affecting the cusp anomalous dimension.

Loop equation verification reproduces the cusp anomalous dimension at two loops.

Ladder diagrams sum to a Bessel function but do not match the expected strong coupling behavior.

Abstract

We calculate the anomalous dimension of the cusped Wilson loop in ${\cal N}=4$ supersymmetric Yang-Mills theory to order $\lambda^2$ ($\lambda=g^2_{YM}N$). We show that the cancellation between the diagrams with the three-point vertex and the self-energy insertion to the propagator which occurs for smooth Wilson loops is not complete for cusped loops, so that an anomaly term remains. This term contributes to the cusp anomalous dimension. The result agrees with the anomalous dimensions of twist-two conformal operators with large spin. We verify the loop equation for cusped loops to order $\lambda^2$, reproducing the cusp anomalous dimension this way. We also examine the issue of summing ladder diagrams to all orders. We find an exact solution of the Bethe-Salpeter equation, summing light-cone ladder diagrams, and show that for certain values of parameters it reduces to a Bessel function. We find that the ladder diagrams cannot reproduce for large $\lambda$ the $\sqrt{\lambda}$-behavior of the cusp anomalous dimension expected from the AdS/CFT correspondence.