Superallowed $0^+ \rightarrow 0^+$ $β$ decay of $T =2$ $^{20}$Mg: $Q_{\textrm{EC}}$ value and $βγ$ branching
\textbf{Background}: Superallowed $0^+ \rightarrow 0^+$ $β$ decays of isospin $T=2$ nuclides can be used to test theoretical isospin symmetry breaking corrections applied to extract the CKM matrix element $V_{ud}$ from $T = 0,1$ decays by measuring precise $ft$ values and also to search for scalar currents using the $β-ν$ angular correlation. Key ingredients include the $Q_{\textrm{EC}}$ value and branching of the superallowed transition and the half life of the parent. \textbf{Purpose}: To determine a precise experimental $Q_{\textrm{EC}}$ value for the superallowed $0^+ \rightarrow 0^+$ $β$ decay of $T=2$ $^{20}$Mg and the intensity of $^{20}$Mg $β$-delayed $γ$ rays through the isobaric analog state in $^{20}$Na. \textbf{Method}: A beam of $^{20}$Mg was produced using the in-flight method and implanted into a plastic scintillator surrounded by an array of high-purity germanium detectors used to detect $β$-delayed $γ$ rays. The high-resolution $γ$-ray spectrum was analyzed to measure the $γ$-ray energies and intensities. \textbf{Results}: The intensity of $^{20}$Mg $β$-delayed $γ$ rays through the isobaric analog state in $^{20}$Na was measured to be $(1.60 \pm 0.04_{\textrm{stat}} \pm 0.15_{\textrm{syst}} \pm 0.15_{\textrm{theo}}) \times 10^{-4}$, where the uncertainties are statistical, systematic, and theoretical, respectively. The $Q_{\textrm{EC}}$ value for the superallowed transition was determined to be $4128.7 \pm 2.2$ keV based on the measured excitation energy of $6498.4 \pm 0.2_{\textrm{stat}} \pm 0.4_{\textrm{syst}}$ keV and literature values for the ground-state masses of $^{20}$Na and $^{20}$Mg. \textbf{Conclusions}: The $β$-delayed $γ$-decay branch and $Q_{\textrm{EC}}$ value are now sufficiently precise to match or exceed the sensitivity required for current low-energy tests of the standard model.