Renormalization-scheme dependence of Padé summation in QCD

We study the renormalization-scheme (RS) dependence of Padé approximants (PA's), and compare them with the principle of minimal sensitivity (PMS) and the effective charge (ECH) approaches. Although the formulas provided by the PA, PMS, and ECH predictions for higher-order terms in a QCD perturbation expansion differ in general, their predictions can be very close numerically for a wide range of renormalization schemes. Using the Bjorken sum rule as a test case, we find that Padé summation (PS) reduces drastically the RS dependence of the Bjorken effective charge. We use these results to estimate the theoretical error due to the choice of RS in the extraction of ${\alpha }_s$ from the Bjorken sum rule, and use the available data at $Q^2=3\mathrm{}$ Ge${\mathrm{V}}^2$ to estimate ${\alpha }_s\left(M_Z\right)={0.117}_{-0.007\mathrm{}}^{+0.004\mathrm{}}\pm 0.002$, where the first error is experimental and the second is theoretical.