Abstract
We investigate the u=1/2left[mathcal{O}left({Lambda}_{mathrm{QCD}}right)right] and u=3/2left[mathcal{O}left({Lambda}_{mathrm{QCD}}^3right)right] renormalons in the static QCD potential in position space and momentum space using the OPE of the potential-NRQCD effective field theory. This is an old problem and we provide a formal formulation to analyze it. In particular we present detailed examinations of the u = 3/2 renormalons. We clarify how the u = 3/2 renormalon is suppressed in the momentum-space potential in relation with the Wilson coefficient VA(r). We also point out that it is not straightforward to subtract the IR renormalon and IR divergences simultaneously in the multipole expansion. Numerical analyses are given, which clarify the current status of our knowledge on the perturbative series. The analysis gives a positive reasoning to the method for subtracting renormalons used in recent αs(MZ ) determination from the QCD potential.
Highlights
By the growth of αs in the infrared (IR) region and is characterized by the singularities in the Borel transform of the perturbative series [35]
We investigate the u = 1/2 [O(ΛQCD)] and u = 3/2 [O(Λ3QCD)] renormalons in the static QCD potential in position space and momentum space using the operator product expansion (OPE) of the potential-NRQCD effective field theory
There exists no rigorous proof on existence of renormalons in QCD observables, there exist standard arguments based on the operator product expansion (OPE) and renormalization group (RG) equations which show that their existence is consistent and plausible theoretically [35]
Summary
Let us first review briefly the structure of renormalons in QCD observables [35]. Consider a general RG-invariant dimensionless observable X(Q) with a typical energy scale Q. Μ denotes the renormalization scale in the MS scheme It satisfies the RG equation μ2 d dμ. Renormalons of XPT refer to the singularities of BX (t) located on the real axis in the complex t-plane. In the context of the OPE in 1/Q, XPT is identified with the Wilson coefficient of the leading identity operator. Let us denote by Ou the lowest dimension (dimension 2u) renormalized operator responsible for cancellation of the leading renormalon in XPT. We assume that the leading ambiguity induced by the renormalon of C1X as given in eq (2.10) is canceled by the second term of the OPE. The Q-dependence of the renormalon uncertainty of C1X should coincide with that of the second term in the OPE, which can be detected as follows.
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