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

Abstract

We study the production of $^3_\Lambda$H and $^4_\Lambda$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 $K^-p\to\pi^0\Lambda$ amplitude. Because the $^3_\Lambda$H ground state is extremely weakly bound, the $d$--$\Lambda$ wave function becomes spatially extended. We calculate the integrated cross sections $\sigma_{\rm lab}$ and their ratio $R_{34}=\sigma_{\rm lab}(^3_\Lambda{\rm H})/\sigma_{\rm lab}(^4_\Lambda{\rm H})$ for forward angles $\theta_{\rm lab}=0^\circ$--$20^\circ$. The production strength of $^3_\Lambda$H and the ratio $R_{34}$ are strongly sensitive to the $\Lambda$ binding energy $B_\Lambda$, 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 $^3_\Lambda$H.