High-precision measurements of the $ft$ values for superallowed Fermi beta decays between 0$^+$ isobaric analogue states have provided invaluable probes of the Standard Model description of the electroweak interaction. Theoretical corrections must be applied to the experimentally determined $ft$ values obtained from precise measurements of the half-lives, branching ratios, and $Q$ values of the decays. Of particular interest is the isospin symmetry-breaking correction, $\delta_C$, which is nuclear-structure-model dependent; several theoretical approaches can and have been used to calculate these corrections with varying results. In the most recent survey of superallowed Fermi $\beta$ emitters [1] the selection of a particular $\delta_C$ model depended significantly on four of the least precisely determined corrected-$ft$ values: $^{22}$Mg, $^{38}$Ca, $^{62}$Ga, and $^{74}$Rb for the well-measured cases.

Recently, updated calculations of the universal ``inner'' electroweak radiative correction, $\Delta^V_R$, have been performed [2-4]. This value is used in combination with the corrected superallowed $\mathcal{F}t$ values to extract such quantities as $G_V$, the vector coupling constant, and $\vert V_{ud}\vert$, the most precisely determined element of the CKM quark mixing matrix. With the updated value of $\Delta^V_R$, the first row of the CKM quark mixing matrix now disagrees with unitarity at the 2-4$\sigma$ level, prompting an increased interest in re-investigating the model-dependent nuclear structure corrections, especially those which can be directly constrained experimentally.

We have performed a high-statistics experiment for the superallowed Fermi $\beta^+$ emitter $^{62}$Ga at the Isotope Separator and Accelerator (ISAC) radioactive ion beam facility at TRIUMF using the high-efficiency Gamma-Ray Infrastructure for Fundamental Investigations of Nuclei (GRIFFIN) spectrometer. The high coincidence efficiency of the GRIFFIN spectrometer allowed for a significant expansion of the level scheme, more than doubling the known $\gamma$-ray transitions in the daughter nucleus, $^{62}$Zn. This allowed a new measurement of the superallowed branching ratio with a precision of $\pm$0.0012\%, $\sim$6 times more precise than previously achieved [5]. Gamma-ray intensities were measured down to the 1 ppm level, effectively solving the Pandemonium problem [6] for $^{62}$Ga. For one particularly important cascade, sufficient statistics were obtained to perform a $\gamma-\gamma$ angular correlation measurement. This allowed the previously-conflicting spin-assignments for the 2.34~MeV excited state in $^{62}$Zn [7,8] to be resolved and firmly established this state to have $J^\pi = 0^+$. The assignment of the spin of this state has important implications for the isospin symmetry breaking correction, $\delta_{C1}$. Final results from this analysis will be presented.

[1] J.C. Hardy and I.S. Towner, Phys. Rev. C 91, 025501 (2015).

[2] C. Seng, M. Gorchtein, H.H. Patel, and M.J. Ramsey-Musolf, Phys. Rev. Lett. 121, 241804 (2018).

[3] C. Seng, M. Gorchtein, and M.J. Ramsey-Musolf, Phys. Rev. D 100 , 013001 (2019).

[4] A. Czarnecki, W.J. Marciano, and A. Sirlin, Phys. Rev. D 100, 073008 (2019).

[5] P. Finlay et al., Phys. Rev. C 78, 025502 (2008).

[6] J.C. Hardy, L.C. Carraz, B. Jonson, and P.G. Hansen, Phys. Lett. B 71, 307 (1977).

[7] M. Albers et al., Nucl. Phys. A 847, 180 (2010).

[8] K.G. Leach et al., Phys. Rev. C 88, 031306 (2013).