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Precise determination of $α_s$ from relativistic quarkonium sum rules

We determine the strong coupling $α_s(m_Z)$ from dimensionless ratios of roots of moments of the charm- and bottom-quark vector and charm pseudo-scalar correlators, dubbed $R_q^{X,n}\equiv(M_q^{X,n})^\frac{1}{n}/(M_q^{X,n+1})^\frac{1}{n+1}$, with $X=V,P$, as well as from the $0$-th moment of the charm pseudo-scalar correlator, $M_c^{P,0}$. In the quantities we use, the mass dependence is very weak, entering only logarithmically, starting at $\mathcal{O}(α_s^2)$. We carefully study all sources of uncertainties, paying special attention to truncation errors, and making sure that order-by-order convergence is maintained by our choice of renormalization scale variation. In the computation of the experimental uncertainty for the moment ratios, the correlations among individual moments are properly taken into account. Additionally, in the perturbative contributions to experimental vector-current moments, $α_s(m_Z)$ is kept as a free parameter such that our extraction of the strong coupling is unbiased and based only on experimental data. The most precise extraction of $α_s$ from vector correlators comes from the ratio of the charm-quark moments $R_c^{V,2}$ and reads $α_s(m_Z)=0.1168\pm 0.0019$, as we have recently discussed in a companion letter. From bottom moments, using the ratio $R_b^{V,2}$, we find $α_s(m_Z)=0.1186\pm0.0048$. Our results from the lattice pseudo-scalar charm correlator agree with the central values of previous determinations, but have larger uncertainties due to our more conservative study of the perturbative error. Averaging the results obtained from various lattice inputs for the $n=0$ moment we find $α_s(m_Z)=0.1177\pm0.0020$. Combining experimental and lattice information on charm correlators into a single fit we obtain $α_s(m_Z)=0.1170\pm 0.0014$, which is the main result of this article.