Sensitivity of the ${}^{3,4}$He($K^{-}$, ${\pi }^0$) production ratio to the $\mathrm{\Lambda }$ binding energy of ${}_{\mathrm{\Lambda }}^3$H

We study the production of ${}_{\mathrm{\Lambda }}^3$H and ${}_{\mathrm{\Lambda }}^4$H in the ${}^{3,4}$He($K^{-}$,${\pi }^0$) reactions at $p_{K^{-}}=1.0$~GeV/$c$ within the distorted-wave impulse approximation, using the optimal Fermi-averaged $K^{-}p\to {\pi }^0\mathrm{\Lambda }$ amplitude. Because the ${}_{\mathrm{\Lambda }}^3$H ground state is extremely weakly bound, the $d$–$\mathrm{\Lambda }$ wave function becomes spatially extended. We calculate the integrated cross sections ${\sigma }_{\mathrm{l}\mathrm{a}\mathrm{b}}$ and their ratio $R_{34}={\sigma }_{\mathrm{l}\mathrm{a}\mathrm{b}}{(}_{\mathrm{\Lambda }}^3\mathrm{H})\mathrm{/}{\sigma }_{\mathrm{l}\mathrm{a}\mathrm{b}}{(}_{\mathrm{\Lambda }}^4\mathrm{H})$ for forward angles ${\theta }_{\mathrm{l}\mathrm{a}\mathrm{b}}=0^{\circ }$–${20}^{\circ }$. The production strength of ${}_{\mathrm{\Lambda }}^3$H and the ratio $R_{34}$ are strongly sensitive to the $\mathrm{\Lambda }$ binding energy $B_{\mathrm{\Lambda }}$, which is constrained to be approximately 0.05–0.15~MeV by comparison with experimental data from the J-PARC E73 experiment. This indicates that the ${}^3$He($K^{-}$,${\pi }^0$) reaction provides a sensitive probe of the weak binding of ${}_{\mathrm{\Lambda }}^3$H.