Recoil effects due to electron shake-off following the beta decay of $^6$He

There are currently several experiments in progress to search for new physics beyond the Standard Model by high precision studies of angular correlations in the $\beta$ decay of the helium isotope $^6{\rm He}$ to form $^6{\rm Li} + e^- + \bar{\nu}_e$ [1,2]. After the $\beta$ decay process, the atomic electrons of $^6$Li$^+$ adjust to the sudden change of nuclear charge from 2 to 3. We calculate the probabilities for electron shake-up and shake-off, including recoil effects, by the use of a Stieltjes imaging representation of the final states. A variety of sum rules provides tight consistency checks on the accuracy of the results. Results obtained previously [3] indicate that there is a 7$\sigma$ disagreement between theory and experiment for the additional nuclear recoil induced by the emission of atomic shake-off electrons. This disagreement will be further studied, and the results extended to the $1s2p\;^3P$ and metastable $1s2s\;^3S$ states as initial states of $^6$He before $\beta$-decay.\\[5pt] [1] C. Couratin {\it et al.}, Phys.\ Rev.\ Lett.\ {\bf 108}, 243201 (2012).\\[0pt] [2] O. Naviliat-Cuncic and M. Gonzalez-Alonso, Ann.\ Phys.\ {\bf 525}, SI 600-619 (2013).\\[0pt] [3] Eva Schulhoff and G.W.F. Drake, Phys.\ Rev.\ A {\bf 92}, R050701 (2015).