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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 (alphar << 1 and g r << 1, respectively). Using the effective theory pNRQCD wedetermine the ultrasoft contributions, which cannot be computed in conventionalperturbative QCD. We show in detail how the ultrasoft renormalization in pNRQCDis carried out. In three dimensions the precision of our results reachesO(alpha^3 r^2), i.e. NNLO in the multipole expansion, and NNLL in aalpha/DeltaV expansion, where DeltaV ~ alpha ln(alpha r). We even presentresults up to partly N^4LL order and compare them to existing lattice data.Finally we discuss the relevance of the perturbative calculation in twodimensions, where the exact result is known.

The QCD static potential in D<4 dimensions at weak coupling