Experimental tests of isospin symmetry breaking in superallowed beta decay

In the search for physics beyond the standard model, the unitarity test of the Cabibbo-Kobayashi-Maskawa (CKM) matrix is one of the most demanding. Superallowed $\beta $ transitions between $J^{\pi }=0^{+}$, $T=1$ analog states currently provide the most precise value for $V_{\mathrm{ud}}$, the up-down quark mixing element of the CKM-matrix. Since no axial current can contribute in first order to these transitions, they give a direct access to the vector coupling constant $G_{\mathrm{V}}$ of the weak interaction. The current value of $V_{\mathrm{ud}}$ is $\pm 0.03\% $ accurate [1] and is obtained from fifteen $ft$-values for superallowed $\beta $ decays, all measured with a precision of $0.3\% $ or better. There are four small theoretical corrections (all of the order of $1\% )$ required in the $V_{\mathrm{ud}}$ extraction. The current result's error is dominated by these theoretical corrections. From the experimentalist's perspective, precision can be further improved by testing the reliability of the predicted corrections. A powerful experimental test comes from measurements of mirror pair superallowed transitions [1,2] In these transitions the predicted corrections are relatively large and the ratio of their $ft$-values is very sensitive to the model calculation of the isospin-symmetry-breaking corrections $\delta_{NS} $ and $\delta_{C} $. The talk will focus on the experimental effort required, exemplifying with the mirror pair of superallowed $0^{+}\to 0^{+}\beta $ transitions ${ }^{34}\mbox{Ar}\to { }^{34}\mbox{Cl}$ and ${ }^{34}\mbox{Cl}\to { }^{34}\mbox{S.}$ [1] J. J. C. Hardy and I. S. Towner, Phys. Rev. C \textbf{102}, 045501 (2020) [2] V.E. Iacob \textit{et al.} Phys. Rev. C \textbf{101}, 045501 (2020)