Abstract
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.
Highlights
The static potential energy of quantum chromodynamics (QCD) is the non-Abelian analogue of the well-known Coulomb potential energy of quantum electrodynamics
The static energy between a heavy quark and antiquark is an important quantity in QCD as it can be calculated in both perturbation theory and in lattice QCD (LQCD) simulations
We have extended the results for static energy to the four-loop order using the renormalization group and Padeapproximant
Summary
The static potential energy of quantum chromodynamics (QCD) is the non-Abelian analogue of the well-known Coulomb potential energy of quantum electrodynamics. The short distance part of this quantity is calculated in the nonrelativistic QCD (NRQCD)[1,2] framework and involves the evaluation of Feynman diagrams It has been studied extensively in recent years and analytical results are known to three-loop. The renormalization group (RG) improvement of the ultrasoft terms at three-loop order was first discussed in Ref. (iv) We use the RG-summed and unsummed forms of the static energy in momentum space to extract ΛMQCSD to four-loop from the LQCD inputs from Ref. Appendix E contains the final result of uncontrolled contribution to the static energy in position space to the four-loop order
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