Precise ${}^{113}$Cd $\beta$ decay spectral shape measurement and interpretation in terms of possible $g_A$ quenching

Highly forbidden $\beta$ decays provide a sensitive test to nuclear models in a regime in which the decay goes through high spin-multipole states, similar to the neutrinoless double-$\beta$ decay process. There are only 3 nuclei (${}^{50}$V, ${}^{113}$Cd, ${}^{115}$In) which undergo a $4^{\mathrm{t}\mathrm{h}}$ forbidden non-unique $\beta$ decay. In this work, we compare the experimental ${}^{113}$Cd spectrum to theoretical spectral shapes in the framework of the spectrum-shape method. We measured with high precision, with the lowest energy threshold and the best energy resolution ever, the $\beta$ spectrum of ${}^{113}$Cd embedded in a 0.43 kg CdWO${}_4$ crystal, operated over 26 days as a bolometer at low temperature in the Canfranc underground laboratory (Spain). We performed a Bayesian fit of the experimental data to three nuclear models (IBFM-2, MQPM and NSM) allowing the reconstruction of the spectral shape as well as the half-life. The fit has two free parameters, one of which is the effective weak axial-vector coupling constant, $g_A^{\text{eff}}$, which resulted in $g_A^{\text{eff}}$ between 1.0 and 1.2, compatible with a possible quenching. Based on the fit, we measured the half-life of the ${}^{113}$Cd $\beta$ decay including systematic uncertainties as $7.73_{-0.57}^{+0.60}\times {10}^{15}$ yr, in agreement with the previous experiments. These results represent a significant step towards a better understanding of low-energy nuclear processes.