The static potential in {\cal N}=4 supersymmetric Yang-Mills at weak coupling

We compute the static potential associated to the locally 1/2 BPS Wilson loop in ${\cal N}$=4 supersymmetric Yang-Mills theory with ${\cal O}(\lambda^2/r)$ accuracy. We also resum the leading logarithms, of ${\cal O}(\lambda^{n+1}\ln^n\lambda/r)$, 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 the use of 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 ${\cal O}(\lambda^{n+3}\ln^n\lambda/r)$, of the static potential associated to an ordinary Wilson loop in the same theory.