We compute the static potential associated with the locally $1/2$ BPS Wilson loop in $\mathcal{N}=4$ supersymmetric Yang-Mills theory with $\mathcal{O}({\lambda }^2/r)$ accuracy. We also resum the leading logarithms, of $\mathcal{O}({\lambda }^{n+1}{ln}^n\lambda /r)$, and show the structure of the renormalization group equation at next-to-leading order in the multipole expansion. In order to obtain these results it is crucial to use an effective theory for the ultrasoft degrees of freedom. We develop this theory up to next-to-leading order in the multipole expansion. Using the same formalism we also compute the leading logarithms, of $\mathcal{O}({\lambda }^{n+1}{ln}^n\lambda /r)$, of the static potential associated with an ordinary Wilson loop in the same theory.

• Received 18 September 2007

DOI:https://doi.org/10.1103/PhysRevD.77.021701