Coulomb Corrections to the Fermi Nuclear Matrix Element

The Fermi nuclear matrix elements for positron decay between two $T=1\mathrm{}$ states can be evaluated exactly for charge-independent nuclear forces in the limit of nonrelativistic nucleon velocities to give $M_{\mathrm{F}}=\sqrt{2}$. Sherr and Gerhart have used this fact to place limits on the Fierz interference terms for the scalar and vector $\beta$ interactions by comparing the $\mathrm{ft}$ values for the ${\mathrm{O}}^{14}$, ${\mathrm{Al}}^{26}$, and ${\mathrm{Cl}}^{34}$ $0^{+}$→$0^{+}$ $\beta$ decays. The equality of these $\mathrm{ft}$ values to within an experimental uncertainty of ∼10% implies a limit for the contribution of the Fierz interference term to the $\mathrm{ft}$ value of 12%. Such a limit will be modified by the correction terms arising from the Coulomb interaction and from $\frac{v^2}{c^2}$ terms in the nucleon velocities. These Coulomb correction terms are calculated on the coupling model with harmonic-oscillator wave functions and are shown to affect the ratio of the $\mathrm{ft}$ values for ${\mathrm{Cl}}^{34}$ and ${\mathrm{O}}^{14}$ by less than 1.5%. The relativistic corrections to this ratio have been shown to be even smaller. Therefore the uncertainty in the theoretical value for the square of the Fermi nuclear matrix elements is much smaller than the present experimental uncertainty in the $\mathrm{ft}$ values.