Limiting Geometries of Two Circular Maldacena-Wilson Loop Operators

TL;DR

The paper investigates two-loop perturbative correlators of two circular Maldacena-Wilson loops in ${\cal N}=4$ SYM, focusing on equal orientation and the operator content revealed by shrinking one loop. It shows that extending the opposite-orientation results to equal orientation is achieved by a sign flip of the second radius and preserves finiteness via a simple $R$-rule. In the small-circle limit, the local operator expansion reveals unprotected operators (notably the Konishi scalar) with one-loop anomalous dimensions arising from interacting graphs, while protected operators remain non-renormalized, with explicit values ${\Delta_K^{(1)}}={\frac{3\lambda}{4\pi^2}}$ and ${\Delta_-^{(1)}}={\frac{\lambda}{2\pi^2}}$. For coincident equal circles, the two-loop test supports the Gaussian matrix-model description by showing cancellation of interacting graphs and ladder contributions agreeing with the matrix model up to ${\cal O}(\lambda^3)$, though a counterexample from a single-circle two-loop non-ladder graph indicates the need for caution and a full ${\cal O}(g^6)$ single-circle calculation.

Abstract

We further analyze a recent perturbative two-loop calculation of the expectation value of two axi-symmetric circular Maldacena-Wilson loops in N=4 gauge theory. Firstly, it is demonstrated how to adapt the previous calculation of anti-symmetrically oriented circles to the symmetric case. By shrinking one of the circles to zero size we then explicitly work out the first few terms of the local operator expansion of the loop. Our calculations explicitly demonstrate that circular Maldacena-Wilson loops are non-BPS observables precisely due to the appearance of unprotected local operators. The latter receive anomalous scaling dimensions from non-ladder diagrams. Finally, we present new insights into a recent conjecture claiming that coincident circular Maldacena-Wilson loops are described by a Gaussian matrix model. We report on a novel, supporting two-loop test, but also explain and illustrate why the existing arguments in favor of the conjecture are flawed.