The (e,e'K$^+$) reaction

Since the momentum transfer of the (e,e'K$^+$) reaction is almost the same as that of the $( \pi ^ + ,{\rm K}^ +)$ reaction, it is expected to preferentially populate high-spin bound hypernuclear states. However, in contrast to reactions with meson beams, the electromagnetic reaction will populate spin-flip hypernuclear states as well as non-spin-flip states, since the transition operator has spin-independent ($f$) and spin-dependent ($g$) terms [21,22]. Although the spin independent term is significantly smaller than the spin-dependent term, the spin-flip and non-spin-flip components in the spin-dependent term have comparable amplitudes.

Also the (e,e'K$^+$), in contrast to the $( \pi ^ + ,{\rm K}^ +)$ and $( {\rm K}^ -,\pi ^ - )$ reactions, converts a proton to a $\Lambda$ hyperon. This results in proton-hole-$\Lambda$-particle states in the configuration [ $(lj)_N^{-1}(lk)^{\Lambda}$]$_J$. When the proton hole state is $j_> = l + 1/2$, the highest spin states of $J = J_{max} = j_> + j_>^{\Lambda} = l_N + l_{\Lambda} + 1$ are favorably excited. These hypernuclear states are of unnatural parity when the original proton orbit is $J_{>}$. On the other hand, if the hole state has spin $j = j_< = l - 1/2$, the highest spin states of the multiplet $J'_{max} = j_< + j_>^{\Lambda} = l_N + l_{\Lambda}$ with natural parity are strongly populated. This selectivity is particularly important as it allows us to directly study the spin-dependent structure of $\Lambda$ hypernuclei.

Experimentally, the most important characteristics of the (e,e'K$^+$) reaction is that it can provide significantly better energy resolution because the reaction is initiated with a primary electron beam of extremely good beam emittance, in contrast to secondary meson beams. With a high performance spectrometer, energy resolution of a few 100 keV can be achieved.

The unique characteristics of the (e,e'K$^+$) reaction are summarized below.

• Extremely good energy resolution when coupled with the high resolution kaon spectrometer (HKS) under construction in Japan.
• The production of natural and non-natural parity states.
• The production of neutron rich hypernuclei as the reaction replaces a proton by a $\Lambda$.
• calibration of the (e,e'K$^+$) reaction can be performed by the elementary p(e,e'K$^+$)$\Lambda$ reaction on a CH$_{2}$ target which provides an exact energy scale.

Although the (e,e'K$^+$) reaction has many advantages for hypernuclear spectroscopy, it has disadvantage that the cross section is much smaller than reactions using hadronic beams. For example, the calculated cross section for the $^{12}$C(e,e'K$^+$) $^{12}_{\Lambda }$B$_{gr}$ is two orders of magnitude smaller than that of the corresponding $^{12}$C($\pi ^+$,K$^+$) $^{12}_{\Lambda }$C$_{gr}$ reaction. With the E89-009 setup, hypernuclear yields of the ground state of $^{12}_{\Lambda }$B are smaller by almost two order of magnitude compared with that of $^{12}_{\Lambda }$C by the SKS experiment. However, this disadvantage can be overcome by employing a new geometry which we propose for this experiment. The new geometry uses a new Kaon spectrometer, HKS, which is described in the next section.