Shell-model treatment of the $\beta$ decay of $^{99}$Tc

In the present paper we treat the second-forbidden non-unique (2nd-nu) ground-state-to-ground-state $\beta^-$ decay $^{99}\textrm{Tc}(9/2^+)\to\,^{99}\textrm{Ru}(5/2^+)$, with a 100$\%$ branching ratio, within the framework of the nuclear shell model (NSM). The energy spectrum of the electrons emitted in this $\beta$-decay transition ($\beta$-electron spectrum) is sensitive to the wave functions of the involved initial ($9/2^+$) and final ($5/2^+$) nuclear ground states through the many involved nuclear matrix elements (NME). The $\beta$-electron spectrum of this transition is potentially indicative of the effective value, $g_{\rm A}^{\rm eff}$, of the weak axial coupling, $g_{\rm A}$, of crucial importance for extraction of information on beyond-the-standard-model physics from the results of the present and future rare-events experiments. We describe the $\beta$ spectral shape of this decay by using a state-of-the-art $\beta$-decay formalism and compute the many involved NME using the well established NSM Hamiltonians $jj45pnb$ and $glekpn$. We have found a strong dependence of the $\beta$ spectral shape on the value of $g_{\rm A}$ making it a good candidate for determination of the value of $g_{\rm A}^{\rm eff}$ through comparison with the corresponding experimental $\beta$ spectral shape. We have also found an interesting dependence of the $\beta$ spectral shape on the value of the so-called small relativistic vector NME, sNME, used to match the computed half-life with the measured one.