The precision of correlation measurements in neutron and nuclear $\beta$ decay has now reached the level of about 1% and better. At this level of precision, higher-order corrections such as recoil-order corrections induced by the strong interaction and radiative corrections cannot necessarily be neglected anymore. We provide here an update of the $\mathcal{F}t$ values of the isospin $T=1/2$ mirror $\beta$ decays including the neutron, of interest to determine the $V_{\mathrm{ud}}$ quark-mixing matrix element. We also provide an overview of current experimental and theoretical knowledge of the most important recoil term, weak magnetism, for both these mirror $\beta$ decays and a large set of $\beta$ decays in higher isospin multiplets. The matrix elements determining weak magnetism were calculated in the nuclear shell model and cross-checked against experimental data, showing overall good agreement. We show that the neutron and the mirror nuclei now effectively contribute to the value of $V_{\mathrm{ud}}$, but we also stress the need for further work on the radiative correction ${\mathrm{\Delta }}_R^V$. Our results provide new insight into the size of weak magnetism, extending the available information to nuclei with masses up to $A=$ 75. This provides important guidance for planning and improved sensitivity for interpreting correlation measurements in searches for new physics or to extract $V_{\mathrm{ud}}$ in mirror $\beta$ decays. It can also be of interest for further theoretical work related to the reactor neutrino problem.

• Received 23 September 2021
• Revised 28 July 2022
• Accepted 14 October 2022

DOI:https://doi.org/10.1103/PhysRevC.107.015502

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society