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Beta decay of the $^6$He halo nucleus to form $^6$Li$^+$ may provide evidence for new physics beyond the standard model [1]. After the decay occurs, the two atomic electrons become redistributed over all possible states of the daughter $^6$Li nucleus, including single- and double-electron emission (shake-off). The present study focuses on the probability for double electron emission to form Li$^{3+}$, where there is a substantial disagreement between theory [2] and experiment [1]. We use pseudospectral representations together with Stieltjes imaging to separate the Li$^{3+} + 2$e$^-$ channel from the energetically overlapping Li$^{2+} + $e$^-$ single ionization channel. We find that the formation of Li$^{3+}$ is strongly suppressed near threshold relative to Li$^{2+}$, thereby accounting for part of the disagreement with experiment. However, there still remains a substantial disagreement in the total probability.
[1] R. Hong et al., Phys. Rev. A 96, 053411 (2017).
[2] E. E. Schulhoff and G. W. F. Drake, Phys. Rev. A 92, R050701 (2015).