Determination of $\alpha _s(m_Z)$ by the non-perturbative decoupling method

We present the details and first results of a new strategy for the determination of $\alpha _s(m_Z)$ (ALPHA Collaboration et al. in Phys. Lett. B 807:135571, 2020). By simultaneously decoupling 3 fictitious heavy quarks we establish a relation between the $\Lambda $-parameters of three-flavor QCD and pure gauge theory. Very precise recent results in the pure gauge theory (Dalla Brida and Ramos in Eur. Phys. J. C 79(8):720, 2019; Nada and Ramos in Eur Phys J C 81(1):1, 2021) can thus be leveraged to obtain the three-flavour $\Lambda $-parameter in units of a common decoupling scale. Connecting this scale to hadronic physics in 3-flavour QCD leads to our result in physical units, $\Lambda ^{(3)}_{\overline{\textrm{MS}}} = 336(12)\, {\textrm{MeV}}$, which translates to $\alpha _s(m_Z) = 0.11823(84)$. This is compatible with both the FLAG average (Aoki et al. in FLAG review 2021. arXiv:2111.09849 [hep-lat]) and the previous ALPHA result (ALPHA Collaboration et al., Phys. Rev. Lett. 119(10):102001, 2017), with a comparable, yet still statistics dominated, error. This constitutes a highly non-trivial check, as the decoupling strategy is conceptually very different from the 3-flavour QCD step-scaling method, and so are their systematic errors. These include the uncertainties of the combined decoupling and continuum limits, which we discuss in some detail. We also quantify the correlation between both results, due to some common elements, such as the scale determination in physical units and the definition of the energy scale where we apply decoupling.