Optimization of the experimental condition

The following items have been taken into account to optimize E01-011 experimental conditions for high-resolution high-efficiency $\Lambda$ hypernuclear spectroscopy.

1. Angular distributions of virtual photons and kaons in the (e,e'K$^+$) reaction are forward peaked as shown in Fig. 1 and Fig. 2. Therefore, both the electron and the kaon spectrometer should be positioned at as forward angles as possible. A splitter magnet which bends scattered electrons and kaons to the opposite directions makes the measurement of very forward-going particles possible.
2. The energy of the virtual photons at 0 degrees is given as $E_{\gamma} = E_{e} - E_{e'}$, 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 of about 900 MeV. Once the energy of virtual photons is fixed, outcoming K$^+$ momentum can be calculated for $\Lambda$ hypernuclei produced in the reaction. Assuming the scattered electron energy of 0.3 GeV as an example, P$_{K^+}$ will be about 1.2 GeV/$c$ for E$_{\gamma}$ = 1.8 - 0.3 = 1.5 GeV.

3. Maximum kaon momentum has been optimized considering:
   1. Yield of hypernuclei.
   2. Energy resolution and acceptance of the spectrometer. Naturally, the energy resolution deteriorates for higher momentum.
   3. Particle identification, particularly between pions and kaons.
   4. Size of the kaon spectrometer and consequently construction cost.

4. 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 to 2.0 GeV. Corresponding kaon momentum is from 0.7 to 1.6 GeV/$c$. However, the hypernuclear cross sections get larger 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 a given flight path of the spectrometer.
   3. With higher kaon momentum, the angular spread of the scattered kaons is smaller. Therefore, a larger portion of the hypernuclei produced in the reaction will be captured for a given solid angle when the spectrometer is positioned at or close to 0 degree.

   In Figure 3, the figure of merit as a function of electron energy is given, in the case the scattered electron energy is 0.285 GeV. It is shown that the higher the energy of the electron beam, the larger yield of the hypernuclear ground states is obtained for a given spectrometer configuration.

5. Although the hypernuclear yield is expected to increase with the electron beam energy, reaction channels for strangeness production other than a $\Lambda$ hyperon open at higher energy and will become sources of kaon background, because that bremstrahlung photons up to the beam energy are produced in the targets. The electron beam energy is, therefore, better kept as low as possible from the points of background. Moreover, particle identification will be easier and higher energy resolution can be achieved with lower beam energy.
6. Taking into account above conditions, the optimum kaon momentum is set to be 1.2 GeV/$c$ aiming 2 $\times$ 10$^{-4}$ (FWHM) momentum resolution. The momentum resolution corresponds to about 170 keV energy resolution in hypernuclear excitation spectra.
7. The electron spectrometer should have momentum resolution of $\leq$ 4 $\times$ 10$^{-4}$, matching that of the kaon spectrometer. Since the momentum of scattered electron is lower compared to that of kaons, the required momentum resolution is modest compared with that for the HKS spectrometer.