The $2_1^+ \to 3_1^+$ gamma width in $^{22}$Na and second class currents

A previous measurement of the $\beta-\gamma$ directional coefficient in $^{22}$Na $\beta$ decay was used to extract recoil-order form factors. The data indicate the requirement of a significant induced-tensor matrix element for the decay.This conclusion largely relies on a Standard-Model-allowed weak magnetism form factor which was determined using an unpublished value of the analog $2_1^+\to3_1^+$ $\gamma$ branch in $^{22}$Na, with the further assumption that the transition is dominated by its isovector $M1$ component. We obtain for the first time an unambiguous determination of the $2_1^+ \to 3_1^+$ branch in $^{22}$Na to be $0.45(8)\%$. Using the Conserved Vector Current (CVC) hypothesis, our branch determines the weak magnetism form factor for $^{22}$Na $\beta$ decay to be $|b/Ac_1| = 8.7(1.1)$. Together with the $\beta-\gamma$ angular correlation coefficient, we obtain a large induced-tensor form factor for the decay that continues to disagree with theoretical predictions. Two plausible explanations are suggested.