Hypothesis of Partially Conserved Axial-Vector Current and Meson-Exchange Effects in $p+p\to d+{\pi }^{+}$ and ${\mathrm{H}}^3\to {\mathrm{He}}^3+e^{-}+{\overline{\nu }}_e$

The hypothesis of partially conserved axial-vector current (PCAC) is applied to relate the threshold production amplitude for the process $\alpha \to \beta +{\pi }^{+}$ to the meson-exchange contributions in $\alpha \to \beta +e^{-}+{\overline{\nu }}_e$, where $\alpha$ and $\beta$ are nuclear states. The latter contributions are calculated by assuming that the two-nucleon $\beta$-decay process $N_i+N_j\to N_i^{}{}_{}{}^{\prime }+N_j^{}{}_{}{}^{\prime }+e^{-}+{\overline{\nu }}_e$ is dominated by the one-pion-exchange contribution. Particular attention is given to the radial dependence of the effective two-body operator which is used to describe both $p+p\to d+{\pi }^{+}$ and the meson-exchange effects in ${\mathrm{H}}^3\to {\mathrm{He}}^3+e^{-}+{\overline{\nu }}_e$. It is shown that terms as singular as $x^{-2\mathrm{}}{e}^{-x}$ and $x^{-3\mathrm{}}{e}^{-x}$ can arise naturally in a dynamical (as opposed to a phenomenological) treatment of meson-exchange effects, and that these terms contribute significantly to $p+p\to d+{\pi }^{+}$ but not at all to ${\mathrm{H}}^3\to {\mathrm{He}}^3+e^{-}+{\overline{\nu }}_e$. Some discussion of the present experimental status of these two processes is also given.