Improved calculations of $\beta$ decay backgrounds to new physics in liquid xenon detectors

We present high-precision theoretical predictions for the electron energy spectra for the ground-state to ground-state $\beta$ decays of ${}^{214}\mathrm{Pb}$, ${}^{212}\mathrm{Pb}$, and ${}^{85}\mathrm{Kr}$ 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 ${}^{214}\mathrm{Pb}$ and ${}^{212}\mathrm{Pb}$ spectra below $\approx 100$ 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 ${}^{214}\mathrm{Pb}$ (${}^{212}\mathrm{Pb}$) 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 $g_{\mathrm{A}}=0.7$–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.

Figure 1

Comparison of $\beta$ spectra for the ground-state decay of ${}^{214}\mathrm{Pb}$ shown over the full energy range (left) and at low energies (right). The result of this work is shown as a solid red line calculated using $g_{\mathrm{A}}=0.85$ and applying the atomic exchange correction. The upper and lower bounds of the shaded band show the spectrum obtained with $g_{\mathrm{A}}=0.7$ and $g_{\mathrm{A}}=1.0$, respectively. Spectra are normalized over the full energy range for each value of $g_{\mathrm{A}}$.

Figure 2

Comparison of $\beta$ spectra for the ground-state decay of ${}^{212}\mathrm{Pb}$ shown over the full energy range (left) and at low energies (right). The result of this work is shown as a solid line calculated using $g_{\mathrm{A}}=0.85$ and applying the atomic exchange correction. The upper and lower bounds of the shaded band show the spectrum obtained with $g_{\mathrm{A}}=0.7$ and $g_{\mathrm{A}}=1.0$, respectively. Spectra are normalized over the full energy range for each value of $g_{\mathrm{A}}$.

Figure 3

Comparison of $\beta$ spectra for the ground-state decay of ${}^{85}\mathrm{Kr}$ shown over the full energy range (left) and at low energies (right). The dashed red line shows the spectrum with the exchange effect calculated using the extended formalism for first-forbidden unique transitions. The lower portion of each figure gives the difference between the spectra with the exchange effect calculated as an allowed and a first-forbidden unique transition. Spectra are normalized over the full energy range.