Analog-antianalog isospin mixing in ${}_{47}K {\beta }^{-}$ decay

We have measured the isospin mixing of the $I^{\pi }=1/2^{+}, E_x=2.599$ MeV state in nearly doubly magic ${}_{47}\mathrm{Ca}$ with the isobaric analog $1/2^{+}$ state of ${}_{47}K$. Using the TRIUMF atom trap for $\beta$ decay, we have measured a nonzero asymmetry of the progeny ${}_{47}\mathrm{Ca}$ with respect to the initial ${}_{47}K$ spin polarization, which together with the $\beta$ asymmetry implies a nonzero ratio of Fermi to Gamow-Teller matrix elements $y=0.098\pm 0.037$ for the $1/2^{+}\to 1/2^{+}$ transition. Interpreting $y$ as mixing between this state and the isobaric analog state implies a Coulomb matrix element magnitude $101\pm 37$ keV. This relatively large matrix element supports a model from the literature of analog-antianalog isospin mixing, which predicts large matrix elements in cases involving excess neutrons over protons occupying more than one major shell. The result supports pursuing a search for time-reversal odd, parity-even, isovector interactions using a correlation in ${}_{47}K\beta$ decay.