Abstract
We present high-precision theoretical predictions for the electron energy spectra for the ground-state to ground-state $\beta$ decays of $^{214}$Pb, $^{212}$Pb, and $^{85}$Kr most relevant to the background of liquid xenon dark matter detectors. The effects of nuclear structure on the spectral shapes are taken into account using large-scale shell model calculations. Final spectra also include atomic screening and exchange effects. The impact of nuclear structure effects on the $^{214}$Pb and $^{212}$Pb spectra below $\approx100$ keV, pertinent for several searches for new physics, are found to be comparatively larger than those from the atomic effects alone. We find that the full calculation for $^{214}$Pb ($^{212}$Pb) predicts 15.0-23.2% (12.1-19.0%) less event rate in a 1-15 keV energy region of interest compared to the spectrum calculated as an allowed transition when using values of the weak axial vector coupling in the range $g_{\rm A}=0.7-1.0$. The discrepancy highlights the importance of both a proper theoretical treatment and the need for direct measurements of these spectra for a thorough understanding of $\beta$ decay backgrounds in future experiments.
Highlights
The discovery of rare events caused by new physics requires that backgrounds which could mimic the signal be reduced as much as possible
Searches for new physics based on the dual-phase liquid xenon (LXe) time projection chamber (TPC) have exciting potential for the discovery of dark matter and for the discovery of new neutrino properties
The spectra were calculated as allowed in Ref. [9], without any nuclear shape adjustment, unlike the present work. For both Pb isotopes the present calculations result in a lower rate in the low-energy region of interest for several new physics searches compared to the allowed calculation
Summary
The discovery of rare events caused by new physics requires that backgrounds which could mimic the signal be reduced as much as possible. While multiple techniques are used to infer the final level of each isotope realized in an experiment, the modeling of this background depends on the ground-state decay branching ratios assumed and, more crucially, the precise shape of the β energy spectra. For first-forbidden nonunique transitions, such as the ground-state decays of 214Pb and 212Pb, the spectral shape can depend heavily on the details of the nuclear structure. Nonunique β transitions can depend strongly on the effective value of the weak axial vector coupling constant gA This dependence on gA was studied in the nuclear shellmodel framework in Ref. In this work we report on the ground-state β shapes of 214Pb and 212Pb obtained by calculating the necessary NMEs for a firstforbidden nonunique transition and by employing a formalism for atomic exchange corrections that has been extended to include forbidden unique transitions. The same exchange formalism is applied to the ground-state β decay of 85Kr
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