Proposed experimental condition

In the proposed experiment, the followings were taken into account to optimize experimental conditions for high-resolution and high efficiency hypernuclear spectroscopy.

1. As shown in Fig. 6 and Fig. 7, angular distributions of virtual photons and kaons in the (e,e'K$^+$) reaction is forward peaked and thus both the electron and the kaon spectrometer should be positioned at as forward angles as possible.
2. The virtual photons at 0 degrees have the energy, E$_{\gamma}$, is given as,
   
   

   $$E _{\gamma} = E _{e} - E_{e'}$$ (1)

   
   
   where E$_{e}$ and E$_{e'}$ are beam and scattered electron energies. The elementary cross section of the ($\gamma$,K$^+$) reaction has relatively weak $E_{\gamma}$ dependence above the threshold.
3. Once the energy of virtual photon is fixed, outcoming K$^+$ momentum is given assuming hypernuclear mass. P$_{K^+}$ is about 1 GeV/c for E$_{\gamma}$ = 1.8 - 0.3 = 1.5 GeV where the scattered electron energy is assumed to be 0.3 GeV as an example. Photon energy effective for the production of kaons will have a range that corresponds to the energy acceptance of the electron spectrometer. Thus, the momentum acceptance of the kaon spectrometer and the electron spectrometer should match each other.
4. Maximum kaon momentum to be detected should be optimized considering
   
   1. Yield of hypernuclei
   2. Energy resolution and acceptance of the spectrometer. Naturally, the energy resolution becomes worse with higher momentum.
   3. Particle identification, particularly between pions and kaons.
   4. Size of the kaon spectrometer and consequently construction cost.

5. For the yield of $\Lambda$ hypernuclei, three factors contribute,
   
   1. The elementary cross section of p($\gamma$,K$^+$)$\Lambda$ is almost constant for the energy range of real $\gamma$ from 1.1 - 2.0 GeV. Corresponding kaon momentum is from $\sim$ 0.7-1.6 GeV/c. However, the hypernuclear cross sections get greater with the higher $\gamma$ energy because the recoil momentum becomes smaller.
   2. With higher kaon momentum, the survival rate of the kaon becomes higher for the given flight path of the spectrometer.
   3. With higher kaon momentum, the cone of scattered kaons becomes narrower. Thus, larger fraction of the hypernuclei produced in the reaction will be captured for the same solid angle if the spectrometer is positioned at or close to 0 degrees.

   The figure of merit as a function of electron energy assuming the scattered electron energy is 0.285 GeV is shown in Fig. 8. It is shown the higher the energy of the electron beam, the larger the yield of the hypernuclear ground states for a given spectrometer configuration.

6. Although the hypernuclear yield is expected to increase with beam energy, reaction channels strangeness production other than a $\Lambda$ hyperon open at higher energy and will become sources of background, because that bremsstrahlung photons up to the beam energy are produced in the targets. The electron beam energy is better kept as low as possible from the points of background and particle identification.
7. Taking into account above conditions, the optimum kaon momentum is set at 1.2 GeV/c aiming 2 $\times$ 10$^{-4}$ momentum resolution. The momentum resolution corresponds to about 100 keV energy resolution in hypernuclear excitation spectra.
8. The electron spectrometer also should have momentum resolution of $\leq$ 3 $\times$ 10$^{-4}$, matching that of the kaon spectrometer. Since the momentum of scattered electron is low compared to that of kaons, better momentum resolution can be achieved.