\alpha_S from $F_\pi$ and Renormalization Group Optimized Perturbation

TL;DR

This paper introduces a renormalization group optimized perturbation method to accurately compute the pion decay constant ratio and QCD scale, yielding stable results and new precise determinations of the strong coupling constant.

Contribution

It develops a modified variational perturbation approach incorporating renormalization group properties, improving stability and convergence in calculating QCD parameters.

Findings

Determined mbda^{n_f=2} mbda^{n_f=3} with estimated uncertainties.

Provided new values for lpha_S(m_Z) and lpha_S(m_ au) with competitive accuracy.

Demonstrated good stability and convergence of the modified perturbative series.

Abstract

A variant of variationally optimized perturbation, incorporating renormalization group properties in a straightforward way, uniquely fixes the variational mass interpolation in terms of the anomalous mass dimension. It is used at three successive orders to calculate the nonperturbative ratio $F_\pi/\Lambda$ of the pion decay constant and the basic QCD scale in the MSbar scheme. We demonstrate the good stability and (empirical) convergence properties of this modified perturbative series for this quantity, and provide simple and generic cures to previous problems of the method, principally the generally non-unique and non-real optimal solutions beyond lowest order. Using the experimental $F_\pi$ input value we determine \Lambda^{n_f=2}\simeq 359^{+38}_{-25} \pm 5 MeV and \Lambda^{n_f=3}=317^{+14}_{-7} \pm 13 MeV, where the first quoted errors are our estimate of theoretical uncertainties of the method, which we consider conservative. The second uncertainties come from the present uncertainties in F_\pi/F and F_\pi/F_0, where F (F_0) is $F_\pi$ in the exact chiral SU(2) (SU(3)) limits. Combining the \Lambda^{n_f=3} results with a standard perturbative evolution provides a new independent determination of the strong coupling constant at various relevant scales, in particular \alpha_S (m_Z) =0.1174 ^{+.0010}_{-.0005} \pm .001 \pm .0005_{evol} and \alpha_S^{n_f=3}(m_\tau)= 0.308 ^{+.007}_{-.004} \pm .007 \pm .002_{evol}. A less conservative interpretation of our prescriptions favors central values closer to the upper limits of the first uncertainties. The theoretical accuracy is well comparable to the most precise recent {\em single} determinations of \alpha_S, including some very recent lattice simulation determinations with fully dynamical quarks.