Proposed reactions

1. p(e,eK$^+$)$\Lambda$ reaction
   
   
   The reaction is used to calibrate the spectrometer system with a CH$_{2}$ target for the absolute missing mass scale. The procedure has been established in the E89-009 experiment. By the known $\Lambda$ mass, the absolute scale of the binding energy can be determined reliably. Since we aim to determine absolute binding energies of a $\Lambda$ hyperon in the mass region where no emulsion experiments can be applied, the reaction is important for the present experiment. It is in contrast to the $( \pi ^ + ,{\rm K}^ +)$ reaction in which we have to rely on other in direct reactions since a neutron target is not available.

2. $^{12}$C(e,e'K$^+$) $^{12}_{\Lambda }$B reaction
   
   
   

   The reaction is used as a reference reaction for examining overall performance of the spectrometer. Since the excitation function of $^{12}_{\Lambda }$B is expected relatively simple and high statics data can be obtained in relatively short data taking hours, the reaction will allow us to optimize the optics of the entire system thus the overall missing mass resolution. Also, short runs before any beam period will allow us to correct any possible shifts in the system settings.

   In addition, the reaction will provide us with significant physics information. As already mentioned in the preceding section, the $^{12}_{\Lambda }$C hypernucleus studied by the $( \pi ^ + ,{\rm K}^ +)$ reaction  revealed hypernuclear core excited states for the first time. Since then, the excitation energies of the 1$_2^-$ and 1$_3^-$ states which supposed to be generated by coupling of a $\Lambda$ hyperon in the $s$ orbital and the core excited $^{11}$C have been under intensive discussion. Role of intershell mixing of positive-parity states and the relation with the $\Lambda$N spin-spin interaction have been suggested [24,25]. Precision spectrum of the mirror symmetric $\Lambda$ hypernucleus, $^{12}_{\Lambda }$B, with much better resolution will resolve these states unambiguously as demonstrated in Fig. 10 and the excitation energies and cross sections of the states will be determined reliably. We also intend to determine or set the limit of spin-spin splitting of the ground state since the (e,e'K$^+$) reaction populates both states in comparable strengths. Due to the high yield of the new geometry and statistics needed for calibration of system optics, the angular distributions are automatically measured for the major shell states because of the large HKS angular acceptance.

3. $^{28}$Si(e,e'K$^+$) $^{28}_{\Lambda }$Al reaction
   
   
   

   The $^{28}$Si($\pi^{+}$,K$^+$) $^{28}_{\Lambda }$Si reaction was studied using the SKS spectrometer, a spectrum of which is shown in Fig. 11 [2]. In the spectrum, major shell structure corresponding to the $s$ and $p$ orbitals with 2 MeV (FWHM) resolution was seen. At the same time unexpected peak structure was observed between the two peaks, although the origin is not known. Since the mass dependence of $\Lambda$ spin-orbit splitting of the $p$ orbital is expected to be almost maximum at $^{28}_{\Lambda }$Si, it was also aimed to resolve the splitting.

   The excitation spectrum of the $^{28}$Si($\gamma$,K$^+$) $^{28}_{\Lambda }$Al reaction has been calculated at E$_{\gamma}$ = 1.30 GeV and $\theta$ = 3 degrees. A simulated spectrum assuming the spin-orbit strength (V$_{so}$ = 2 MeV) with 300 keV (FWHM) resolution and with expected statistics for the proposed running time is shown in Fig. 12. Peaks corresponding to each major shell orbitals will be distinctively identified and their binding energies will be derived reliably. For a $\Lambda$ hyperon in the $p$ orbital, [ $\pi d_{5/2} ^{-1} \otimes \Lambda p_{3/2}$]4$^-$ and [ $\pi d_{5/2} ^{-1} \otimes \Lambda p_{1/2}$]3$^-$ states are dominantly populated, providing a good opportunity to directly observe the $ls$ splitting. Figure 12 clearly demonstrates possibility of observing the splitting.

4. $^{51}$V(e,e'K$^+$) $^{51}_{\Lambda }$Ti reaction
   
   
   

   In the $^{51}$V target, the neutron $f_{7/2}$ shell is well closed and stable because N=28. The reaction is supposed to convert one of the three protons in the $f$-shell to a $\Lambda$ hyperon. In this hypernuclear mass region, a hyperon is bound up to the $d$-orbital, providing us an opportunity to determine the binding energies up to higher $l$. The hypernucleus, $^{51}_{\Lambda }$V, was studied by the $( \pi ^ + ,{\rm K}^ +)$ reaction at BNL with resolution around 3 MeV (FWHM) and it is shown in Fig.13 [1]. The quality of the spectrum is poor but the major shell structure is seen. For the ($\gamma$,K$^+$) reaction, a model calculation has been carried out similarly as $^{28}$Si($\gamma$,K$^+$) $^{28}_{\Lambda }$Al [26,22]. In Fig.14, the calculated excitation spectrum is shown. The [ $\pi f_{7/2} ^{-1} \otimes \Lambda d_{5/2}$]6$^-$ and [ $\pi f_{7/2} ^{-1} \otimes \Lambda d_{3/2}$]5$^-$ states, which are spin-orbit partners, are expected to be split by more than 1 MeV if V$_{so}$ = 2 MeV. The calculated spectrum suggests that these states will be simultaneously populated and can be observed in the (e,e'K$^+$) reaction.

   Since nuclei in this mass region are rather well described by shell-model wave functions, it is expected that comparison between experimental data and theoretical calculations will have less ambiguities. We will therefore have a good chance to investigate the single-particle nature of a $\Lambda$ hyperon and also investigate the splitting of the single particle states not only in the $s$ and $p$ orbitals but also in the $d$ orbital.
   

5. $^{89}$Y(e,e'K$^+$) $^{89}_{\Lambda}$Sr reaction
   
   
   

   As mentioned already, $^{89}_{\Lambda}$Y is the $\Lambda$ hypernucleus studied with the best statistics in medium-heavy mass region by the $( \pi ^ + ,{\rm K}^ +)$ reaction. As seen in Fig. 2, in addition to the major shell peak structure, splitting of these peaks were observed. If the (e,e'K$^+$) reaction can be applied to this heavier mass region, we will better investigate $\Lambda$ hyperon single-particle nature and also splitting of these states by $\Lambda$N interaction. Therefore, we also propose to conduct an exploratory R&D run with the $^{89}$Y target to examine feasibility of extending the (e,e'K$^+$) hypernuclear spectroscopy to the heavier mass region.