Electron spectral shape of the third-forbidden $\beta$-decay of $^{87}$Rb measured using a Rb$_2$ZrCl$_6$ crystal scintillator

In recent years, interest in experimental studies of $\beta$-decay electron spectra -- often referred to as $\beta$ spectra -- has been growing. This is particularly true for $\beta$ transitions where the electron spectra are sensitive to the effective value of the weak axial coupling, $g_{\rm A}$. Such measurements serve as important benchmarks for nuclear physics calculations and can also be used to characterize background in astroparticle physics experiments. In this work, a dedicated experiment has been carried out to investigate the spectral shape of the third-forbidden $^{87}$Rb $\beta$-decays, with the goal of estimating the effective $g_{\rm A}$ value for this transition and of deriving the T$_{1/2}$ value. This was done by comparing the experimental spectral shape with the estimates from various phenomenological models. The $^{87}$Rb source was embedded directly within the detector material of a new Rb$_2$ZrCl$_6$ crystal scintillator; the data taking was performed deep underground at Gran Sasso National Laboratory. The obtained experimental half-life value for the studied process is T$_{1/2} = 5.08(13) \times$ 10$^{10}$ yr; while a $g_{\rm A}$ value in the range 0.4 to 0.6 is obtained when accounting for uncertainties and depending on the model adopted as discussed in detail in the text.