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We compute the purely gluonic contribution to the static QCD potential at three-loop order. This completes the computation of the static potential at this order. PACS numbers: 12.38.Aw,12.38.Bx,14.40.Pq For more than 30 years, the static QCD potential V QCD (r) has been studied extensively for the purpose of elucidating the nature of the interaction between heavy quark and antiquark. Generally, V QCD (r) at short-distances can be computed accurately by perturbative QCD. On the other hand, the potential shape at long-distances should be determined by non-perturbative methods, such as lattice simulations or phenomenological potential-model analyses or computations based on string-inspired models.Computations of V QCD (r) in perturbative QCD has a long history. At tree-level, V QCD (r) is merely a Coulomb potential, −C F α S /r (C F = 4/3 is a color factor), arising from one-gluon-exchange diagram. The 1-loop corrections (with massless and/or massive internal quarks) were computed in [1,2]. The 2-loop correction (with massless internal quarks) was computed in [3]. The 2loop correction due to massive internal quarks was computed in [4] (partly corrected in [5]).[28] The logarithmic correction at 3-loop originating from the ultrasoft scale was first pointed out in [1] and computed in [7]. Renormalization-group (RG) improvement of V QCD (r) at next-to-next-to-leading logarithmic order was performed in [8]. A logarithmic contribution at O(α 5 S ) was computed in [9]. The contributions of the massless quark loops to the 3-loop correction were computed in [10]. The only remaining correction at 3-loop order is the purely gluonic contribution, which we compute in this paper.For a long time, the perturbative QCD predictions of V QCD (r) were not successful in the intermediate distance region, relevant to the bottomonium and charmonium states. In fact, the perturbative series turned out to be poorly convergent at r > ∼ 0.1 fm; uncertainty of the series is so large that one could hardly obtain meaningful prediction in this distance region. Even if one tries to improve the perturbation series by certain resummation prescriptions (such as RG improvement), scheme dependence of the results turns out to be very large; hence, one can neither obtain accurate prediction of the potential in this region. It was later pointed out that the large uncertainty of the perturbative prediction can be understood as caused by the O(Λ QCD ) infrared (IR) renormalon contained in V QCD (r) [11].The situation has changed dramatically since the dis-covery of the cancellation of O(Λ QCD ) renormalons in the total energy of a static quark-antiquark pair E tot (r) ≡ V QCD (r) + 2m pole [12]. Convergence of the perturbative series for E tot (r) improved drastically and much more accurate perturbative predictions for the potential shape became available. It was understood that a large uncertainty originating from the O(Λ QCD ) renormalon in V QCD (r) can be absorbed into twice of the quark pole mass 2m pole . Once this is achieved, perturbative uncertai...

We obtain analytic expressions for the third-order corrections due to the strong interaction Coulomb potential to the S-wave Green function, energy levels and wave functions at the origin for arbitrary principal quantum number n. Together with the known non-Coulomb correction this results in the complete spectrum of S-states up to order α 5 s . The numerical impact of these corrections on the Upsilon spectrum and the top quark pair production cross section near threshold is estimated.

We construct approximate formulas for the O(α 3 s ) QCD contributions to vector, axial-vector, scalar and pseudo-scalar quark current correlators, which are valid for arbitrary values of momenta and masses. The derivation is based on conformal mapping and the Padé approximation procedure and incorporates known expansions in the low energy, threshold and high energy regions. We use our results to estimate additional terms in these expansions.

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