Hypertriton withS′State and theΛ−NInteraction

The hypertriton ${}_{\Lambda }{}^{}{}_{}{}^{}\mathrm{He}_{}^3{}_{}{}^{}$ is considered with the inclusion of an $S^{\prime }$ state ${\Psi }_{S^{\prime }}$ in the total wave function. For ${\Psi }_{S^{\prime }}$ the nucleons are in a singlet spin state and the space part is correspondingly antisymmetric with respect to exchange of the nucleons. Only a spin dependence of the $\Lambda -N$ interaction can give a nonzero admixture of ${\Psi }_{S^{\prime }}$ to the dominant component ${\Psi }_S$, which is space-symmetric under exchange of the nucleons and which is the only component that has been considered in previous investigations. Central, spin-dependent Yukawa potentials were used. The most flexible trial function used was one with 16 parameters for which ${\Psi }_S$ has the same 6-parameter form as used by Downs and Dalitz and ${\Psi }_{S^{\prime }}$ has a corresponding flexibility. In particular, for an intrinsic range $b=1.5\mathrm{}$ F (corresponding to a Yukawa interaction appropriate to two-pion exchange), the effect of ${\Psi }_{S^{\prime }}$ is quite appreciable; the singlet strength is reduced, the triplet strength slightly increased, and the spin dependence reduced by about a third. For a range corresponding to $K$-meson exchange ($b=0.84\mathrm{}$ F), the effect of ${\Psi }_{S^{\prime }}$ is considerably less. With inclusion of ${\Psi }_{S^{\prime }}$ (for a given $b$), the singlet strength is found to be quite insensitive to the value of the triplet strength and is therefore almost entirely determined by $B_{\Lambda }\left({}_{\Lambda }{}^{}{}_{}{}^{}\mathrm{H}_{}^3{}_{}{}^{}\right)$. The resulting total $\Lambda -N$ cross sections at low energies (≲20 MeV) are compared with the experimental values ${\sigma }_{\exp }$. If it is assumed that the singlet scattering length $a_s$ and effective range $r_s$ are most reliably determined from hypernuclei, then for $b=1.5\mathrm{}$ F (for which the estimated values with a hard core of radius 0.42 F are $a_s\approx -2.3\mathrm{}$ F, $r_s\approx 3.6$ F), acceptable agreement with ${\sigma }_{\exp }$ can be obtained with only a modest increase of $\left|a_t\right|$ ($a_t\approx -1.3\mathrm{}$ F, $r_t\approx 2.9$ F) above the value obtained from hypernuclei. (The maximum value consistent with the hypernuclear results is $a_t\approx -0.9\mathrm{}$ F together with $r_t\approx 3.3$ F.) It is shown that an increase of this order of magnitude could be obtained through suppression of the coupling with the $\Sigma N$ channel in ${}_{\Lambda }{}^{}{}_{}{}^{}\mathrm{He}_{}^5{}_{}{}^{}$. Results are also given for a Yukawa potential with $b=2.07\mathrm{}$ F, which is the intrinsic range for an interaction with a hard core of radius 0.42 F and an attractive Yukawa tail appropriate to an exchanged boson with mass $3m_{\pi }$. Finally, it is argued that there is a tentative indication for the existence of a repulsive core in the $\Lambda -N$ interaction.