Cusped SYM Wilson loop at two loops and beyond

TL;DR

<3-5 sentence high-level summary>We study the cusped Wilson loop in ${\cal N}=4$ SYM and compute its cusp anomalous dimension $\gamma_{\rm cusp}$ to ${\cal O}(\lambda^2)$, uncovering an anomalous surface term from two-loop internal-vertex diagrams that persists at the cusp and contributes to $\gamma_{\rm cusp}$; we also analyze ladder diagram sums via the Bethe-Salpeter equation, including an exact light-cone solution, and show that ladders alone cannot reproduce the AdS/CFT-predicted $\sqrt{\lambda}$ strong-coupling behavior. The two-loop result agrees with the large-spin twist-two anomalous dimension, confirming the consistency with gauge/string duality and the loop equation in ${\cal N}=4$ SYM. The work highlights the essential role of vertex-induced contributions near the cusp and suggests that universal aspects of the cusp may extend to non-supersymmetric theories, while clarifying the limitations of ladder resummations for capturing strong-coupling scaling.

Abstract

We calculate the anomalous dimension of the cusped Wilson loop in ${\cal N}=4$ supersymmetric Yang-Mills theory to order $λ^2$ ($λ=g^2_{YM}N$). 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 $λ^2$, 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 $λ$ the $\sqrtλ$-behavior of the cusp anomalous dimension expected from the AdS/CFT correspondence.

Figures (6)

• Figure 1: Cusped Wilson loop analytically given by Eq. (\ref{['uv']}).
• Figure 2: Two-loop diagrams relevant for the calculation of the anomalous dimension of the cusped Wilson loop. The dashed lines represent either the Yang-Mills or scalar propagators.
• Figure 3: Two-loop anomaly diagram ($a$) which emerges as a surface term in the sum of the diagrams ($b$), ($c$) and ($d$). An analytic expression is given by the sum of $i)$ and $iv)$ in Eq. (\ref{['twosurf']}).
• Figure 4: Typical path $z(t)$ (represented by the bold line) connecting $x=z(0)$ and $y=z(\tau)$ in the regularization of the delta-function on the right-hand side of the loop equation by Eq. (\ref{['pidreg']}). A typical length of the path is $\sim a$.
• Figure 5: Diagrams of order $\lambda$ for closed cusped Wilson loop. The diagram $(a)$ is the usual one. For the diagrams $(b)$ and $(c)$ one end of the propagator line ends at the regularizing path.