Hypertriton with S ′ State and the Λ − N Interaction

The hypertriton $_{\ensuremath{\Lambda}}\mathrm{He}^{3}$ is considered with the inclusion of an ${S}^{\ensuremath{'}}$ state ${\ensuremath{\Psi}}_{{S}^{\ensuremath{'}}}$ in the total wave function. For ${\ensuremath{\Psi}}_{{S}^{\ensuremath{'}}}$ 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 $\ensuremath{\Lambda}\ensuremath{-}N$ interaction can give a nonzero admixture of ${\ensuremath{\Psi}}_{{S}^{\ensuremath{'}}}$ to the dominant component ${\ensuremath{\Psi}}_{S}$, 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 ${\ensuremath{\Psi}}_{S}$ has the same 6-parameter form as used by Downs and Dalitz and ${\ensuremath{\Psi}}_{{S}^{\ensuremath{'}}}$ has a corresponding flexibility. In particular, for an intrinsic range $b=1.5$ F (corresponding to a Yukawa interaction appropriate to two-pion exchange), the effect of ${\ensuremath{\Psi}}_{{S}^{\ensuremath{'}}}$ 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 ($b=0.84$ F), the effect of ${\ensuremath{\Psi}}_{{S}^{\ensuremath{'}}}$ is considerably less. With inclusion of ${\ensuremath{\Psi}}_{{S}^{\ensuremath{'}}}$ (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 ${B}_{\ensuremath{\Lambda}}(_{\ensuremath{\Lambda}}\mathrm{H}^{3})$. The resulting total $\ensuremath{\Lambda}\ensuremath{-}N$ cross sections at low energies (\ensuremath{\lesssim}20 MeV) are compared with the experimental values ${\ensuremath{\sigma}}_{\mathrm{exp}}$. If it is assumed that the singlet scattering length ${a}_{s}$ and effective range ${r}_{s}$ are most reliably determined from hypernuclei, then for $b=1.5$ F (for which the estimated values with a hard core of radius 0.42 F are ${a}_{s}\ensuremath{\approx}\ensuremath{-}2.3$ F, ${r}_{s}\ensuremath{\approx}3.6$ F), acceptable agreement with ${\ensuremath{\sigma}}_{\mathrm{exp}}$ can be obtained with only a modest increase of $|{a}_{t}|$ (${a}_{t}\ensuremath{\approx}\ensuremath{-}1.3$ F, ${r}_{t}\ensuremath{\approx}2.9$ F) above the value obtained from hypernuclei. (The maximum value consistent with the hypernuclear results is ${a}_{t}\ensuremath{\approx}\ensuremath{-}0.9$ F together with ${r}_{t}\ensuremath{\approx}3.3$ F.) It is shown that an increase of this order of magnitude could be obtained through suppression of the coupling with the $\ensuremath{\Sigma}N$ channel in $_{\ensuremath{\Lambda}}\mathrm{He}^{5}$. Results are also given for a Yukawa potential with $b=2.07$ 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 $3{m}_{\ensuremath{\pi}}$. Finally, it is argued that there is a tentative indication for the existence of a repulsive core in the $\ensuremath{\Lambda}\ensuremath{-}N$ interaction.