Towards determination of the strong coupling ${\alpha }_s(m_Z)$ from four-flavor lattice QCD using the continuous $\beta$-function method

The precise value of the strong coupling ${\alpha }_s(m_Z)$ at the $Z$-boson mass $m_Z$ is essential for high-energy phenomenology and precision tests of quantum chromodynamics (QCD). We present the status of a program targeting a $\sim 0.3\mathrm{\%}$ determination of ${\alpha }_s(m_Z)$ using the renormalization group $\beta$-function in the infinite volume gradient flow scheme based on lattice QCD simulations of degenerate four-flavor highly improved staggered quark (HISQ) ensembles. In particular, we analyze both tree-level cutoff effects and finite-mass effects. We also outline the next steps of the analysis, including the infinite-volume and continuum extrapolations required for a precise determination of ${\alpha }_s(m_Z)$.