Fermi matrix element with isospin breaking

Prompted by the level of accuracy now being achieved in tests of the unitarity of the CKM matrix, we consider the possible modification of the Fermi matrix element for the $\beta$-decay of a neutron, including possible in-medium and isospin violating corrections. While the nuclear modifications lead to very small corrections once the Behrends-Sirlin-Ademollo-Gatto theorem is respected, the effect of the $u-d$ mass difference on the conclusion concerning $V_{ud}$ is no longer insignificant. Indeed, we suggest that the correction to the value of $|V_{ud}|^2 \, + \, |V_{us}|^2 \, + \, |V_{ub}|^2$ is at the level of $10^{-4}$.