The QCD Static Energy using Optimal Renormalization and Asymptotic Pad\'e-approximant Methods

The perturbative QCD static potential and ultrasoft contributions, which together give the static energy, have been calculated to three- and four-loop order respectively, by several authors. Using the renormalization group, and Pad\'e approximants, we estimate the four-loop corrections to the static energy. We also employ the optimal renormalization method and resum the logarithms of the perturbative series in order to reduce sensitivity to the renormalization scale in momentum space. This is the first application of the method to results at these orders. The convergence behaviour of the perturbative series is also improved in position space using the Restricted Fourier Transform scheme. Using optimal renormalization, we have extracted the value of $\Lambda^{\overline{\textrm{MS}}}_{\textrm{QCD}}$ at different scales for two active flavours by matching to the static energy from lattice QCD simulations.