A case to remeasure the $\beta^+$ decay spectra of $^{22}$Na

$^{22}$Na, with a half-life of $2.6018(22)~$y, undergoes an allowed $\beta^+$ decay 99.944(14)\% of the time to the $2_1^+$ state in $^{22}$Ne with a $Q$ value of $1568.79(13)~$keV. This is one of the lowest $Q_{\beta^+}$ values known. We are looking into the feasibility for an experiment where the overarching goals in re-measuring the $\beta^+$ decay spectra of $^{22}$Na are threefold: (i) to set a new limit on the $\nu_e$ mass by direct measurement; (ii) to remeasure the $\beta$-decay Fierz interference term, $b$, to higher precision; and (iii) improve on one input for the test of CKM unitarity. The $\nu_e$ mass is poorly constrained, $m_{\nu_e}<225~$eV (95\% C.L.). The masses of $^{22}$Na and $^{22}$Ne are already well known. An adequately precise measurement near the end point of the $\beta^+$ decay for $^{22}$Na will allow a constraint on the $\nu_e$ mass. By examining the shape for the $\beta^+$ decay spectra of $^{22}$Na, the Fierz interference term was previously measured, but is almost 6 decades old, underestimates the uncertainties, and needs to be updated. Lastly, the precise determination of the $^{22}$Na mass also contributes to the determination of the CKM-matrix element $V_{ud}$, which is in turn used to test CKM unitarity. Simulations are being performed to determine the best method to make the proposed measurement. In particular, a novel technique for measuring the $\beta^+$ decay spectra of $^{22}$Na by embedding it in a semiconductor detector will be compared against the sensitivity of a similar measurement using a magnetic spectrometer.