Experimental objectives

The proposed experiment is intended to establish high-precision spectroscopy of $\Lambda$ hypernuclei for wide mass range by the (e,e'K$^+$) reaction. Such experimental investigation is possible only by utilizing the high precision and power of the Jlab electron beam, and a new high resolution, large solid-angle spectrometer system under construction.

The experimental objectives of the proposal are summarized below.

1. We propose to obtain high-precision binding energies of $\Lambda$ single particle states from the excitation spectra of $\Lambda$ hypernuclei up to A=50 region.

   The limited resolution of the $( \pi ^ + ,{\rm K}^ +)$ reaction makes it difficult to extract precise $\Lambda$ binding energies, except for the light $\Lambda$ hypernuclei. Energy resolutions of a few 100 keV are needed to extract these binding energies. This information provides the depth of the central potential and possible spin-orbit splittings over a wide mass region. The mass dependence of the single particle levels can be directly compared to calculations using single particle potentials and mean-field theory. Since $p$-shell orbitals are barely bound in $p$-shell hypernuclei, it is essential to extend this measurement to heavier systems. The proposed experiment also provides information on the widths of these single particle orbitals.

   As the effective mass of a $\Lambda$ hyperon appears to be close to that of the free $\Lambda$ hyperon, the potential seems to be local in contrast to ordinary nuclei. Thus the proposed precision measurement of the single particle levels can address the degree of non-locality of the effective $\Lambda$-Nucleus potential. This can be related to the nature of the $\Lambda$N and $\Lambda$NN interactions, and to the $\Lambda$N short range interactions [18].

   In a more exotic way, the binding energies can be discussed in terms of the distinguishability of a $\Lambda$ hyperon in nuclear medium, which will result in different A dependence of the binding energy as suggested by Dover [7].

2. We propose to study the spin-orbit splitting of the $\Lambda$ single particle states in heavier hypernuclei. As previously discussed, the $^{89}$Y spectra taken by the $( \pi ^ + ,{\rm K}^ +)$ reaction show that the higher $l$ states are split by about 1 MeV. These splittings are suggested to be due to the $\Lambda$N $ls$ interaction, although the magnitude of the splitting is much larger than expected from previous measurements in the $p$-shell, and in particular from the recent $\gamma$ ray data of $^9_{\Lambda}$Be and $^{13}_{\Lambda}$C [16,17]. However the splitting could also be due to an interplay of different neutron hole states. With a resolution of about 300 keV, we will be able to entangle closely degenerate hypernuclear states, and clarify the situation. If the origin of the splitting is due to $ls$ interaction it will give us the magnitude of the interaction, and introduce a deeper puzzle about the changes in the $ls$ strength between the p and deeper shell structure. If the splitting is due to hypernuclear structure, it will give us information on the characteristics of medium-heavy hypernuclei. In either case, new information is required to address this issue.

3. We propose to study the $^{12}$C(e,e'K$^+$) $^{12}_{\Lambda }$B reaction as a reference reaction in order to tune the new spectrometer system. As a byproduct of this calibration we will accumulate a high statistics $^{12}_{\Lambda }$B spectrum with a resolution of a few 100 keV. This should clearly separate the controversial core excited states. These states were observed in the $^{12}$C($\pi ^+$,K$^+$) reaction, and a high precision - high statistics study will provide information on inter-shell mixing and perhaps on the splitting of the ground state doublet [14,24,25].