QCD static potential in D<4 dimensions at weak coupling

We study the static potential of a color singlet quark-antiquark pair with (fixed) distance r in D=3 and D=2 space-time dimensions at weak coupling (αr 1 and gr 1, respectively). Using the effective theory potential nonrelativistic QCD (pNRQCD) we determine the ultrasoft contributions, which cannot be computed in conventional perturbative QCD. We show in detail how the ultrasoft renormalization in pNRQCD is carried out. In three dimensions the precision of our results reaches O(α3r2), i.e. next-to-next-to-leading order (NNLO) in the multipole expansion, and next-to-next-to-leading logarithmic order (NNLL) in a α/ΔV expansion, where ΔV∼αln (αr). We even present results up to partly N4LL order and compare them to existing lattice data. Finally we discuss the relevance of the perturbative calculation in two dimensions, where the exact result is known. © 2010 The American Physical Society.