Possible solution to basic problems regarding the coupling constants $G_V$ and $G_A$

The existing experimental evidence on the vector and axial-vector coupling constants, $G_V$ and $G_A$, for neutron $\beta$-decay exhibits two prominent problems: (i) the unitarity for the first row of the CKM matrix seems violated; (ii) one obtains different values of $G_A/G_V$ according to whether one uses as input the neutron lifetime or neutron-decay correlation observables. We show that the in-medium modification of $G_A$ can influence both the inner and outer radiative corrections used in deducing $G_V$ from the super-allowed Fermi transitions and that this effect may resolve the long-standing problems (i) and (ii) simultaneously.