Abstract
We review one of the most accurate low-energy determinations of $\alpha_s$. Comparing at short distances the QCD static energy at three loops and resummation of the next-to-next-to leading logarithms with its determination in 2+1-flavor lattice QCD, we obtain $\alpha_s(1.5~{\rm GeV})=0.336^{+0.012}_{-0.008}$, which corresponds to $\alpha_s(M_Z)=0.1166^{+0.0012}_{-0.0008}$. We discuss future perspectives.
Highlights
For many years the average of the strong coupling constant, αs, provided by the Particle Data Group (PDG) has shown a rather stable central value and a steady decrease in the associated error, see figure 1. This satisfactory situation has been challenged in the last years: precise determinations pointing often towards a lower value of αs have appeared, traditionally larger and precise determinations from τ decay have become smaller and less precise [6], accurate determinations from lattice QCD have been questioned
The extraction of αs that follows from comparing perturbative QCD with short distance lattice computations of the QCD static energy is one of these determinations
This quantity is known at three loops and next-to-next-to-next-to leading logarithmic (NNNLL) accuracy
Summary
The color-singlet static potential encodes the contributions from the scale 1/r, while the low-energy contributions are in the term proportional to the two chromoelectric dipoles. An advantage of equation (4) is that it allows for an efficient resummation of the logarithmic contributions to the static potentials and eventually to the static energy This is achieved by evaluating the anomalous dimensions of the potentials through the computation of the ultraviolet divergences in the relevant integrals proportional to the two chromoelectric dipoles and by solving the renormalization group equations for the static potentials [15, 17]. Notice that the integration in (6) can be done (numerically) keeping the strong-coupling constant running at a natural scale of the order of the inverse of the distance This scheme does not generate potentially large logarithms of the type ln νr. In the newer analysis of [3], we have adopted this second method
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