Confirmation of $g_{\rm A}$ quenching using the revised spectrum-shape method for the analysis of the $^{113}$Cd $\beta$-decay as measured with the COBRA demonstrator

In this article we present an updated spectrum-shape analysis of the $^{113}$Cd fourfold forbidden non-unique $\beta$-decay transition in order to address a quenching of the weak axial-vector coupling $g_{\rm A}$ in low-momentum exchange nuclear processes. The experimental data were collected in a dedicated low-threshold run with the COBRA demonstrator at the LNGS and resulted in 44 individual $^{113}$Cd spectra. These data are evaluated in the context of three nuclear model frameworks based on a revised version of the so-called spectrum-shape method under the conserved vector current hypothesis. The approach leads to consistent mean values of the axial-vector coupling of $\overline{g}_{\rm A}(\text{ISM}) = 0.893 \pm 0.054$, $\overline{g}_{\rm A}(\text{MQPM}) = 0.968 \pm 0.056$ and $\overline{g}_{\rm A}(\text{IBFM-2}) = 0.809 \pm 0.122$. While these values agree well with our previous results, additionally the improved nuclear models are now able to reproduce the precisely known experimental half-life of the $^{113}$Cd $\beta$-decay. This achievement resolves one of the main shortcomings of the spectrum-shape method and indicates a milestone in the description of strongly forbidden $\beta$-decays. Moreover, the consistency of the reported results solidifies the indications for the existence of a quenching of $g_{\rm A}$ in low-momentum exchange nuclear processes.