Analysis of $^{115}$In $\beta$ decay through the spectral moment method

We analyze the $^{115}$In $\beta$-decay energy spectrum through the spectral moment method (SMM), previously introduced in the context of $^{113}$Cd $\beta$ decay. The spectral moments $\mu_n$ are defined as averaged $n^{\rm th}$ powers of the $\beta$ particle energy, characterizing the spectrum normalization ($n=0$) and shape ($n\geq 1$) above a given threshold. For $^{115}$In, we consider three independent datasets characterized by different thresholds. We also consider three nuclear model calculations with two free parameters: the ratio of axial-vector to vector couplings, $r=g_{\rm A}/g_{\rm V}$, and the small vector-like relativistic nuclear matrix element (NME), $s=s$-NME. By using the most recent of the three datasets, we show that the first few spectral moments can determine $(r,\, s)$ values in good agreement with those obtained by full-fledged experimental fits. We then work out the SMM results for the other datasets. We find that, although $g_{\rm A}$ quenching is generally favored, the preferred quenching factors may differ considerably depending on the chosen experimental data and nuclear models. We discuss various issues affecting both the overall normalization and the low-energy behaviour of the measured and computed spectra, and their joint effects on the experimentally quoted half-life values. Further $^{115}$In $\beta$-decay data at the lowest possible energy threshold appear to be crucial to clarify these issues.