Optimization of the experimental condition

The following items have been taken into account to optimize the 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 figure 1 and figure 2. Therefore, both the electron and the kaon spectrometer should be positioned at as forward angles as possible. A splitter magnet that bends scattered electrons and kaons in opposite directions makes the measurement of very forward-going particles possible.
2. The energy of the virtual photons is given as $E_{\gamma} = E_{e} - E_{e'}$, where E$_{e}$ and E$_{e'}$ are energies of the beam and the scattered electrons. Since the elementary cross section of the p($\gamma$,K$^+$)$\Lambda$ reaction has relatively weak $E_{\gamma}$ dependence from 1.1 to 1.6 GeV, we can choose the energy in this range. Once the energy of virtual photons is fixed, outcoming K$^+$ momentum is calculated for $\Lambda$ hypernuclei produced in the reaction. P$_{K^+}$ will be about 1.2 GeV/$c$ for E$_{\gamma}$ = 1.8 - 0.3 = 1.5 GeV, assuming the scattered electron energy of 0.3 GeV as an example.
3. Maximum kaon momentum is optimized considering:
   1. Yield of hypernuclei.
   2. Energy resolution and acceptance of the spectrometer. Naturally, the absolute energy resolution deteriorates for the 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 $\Lambda$ hypernuclear cross sections get larger with higher $\gamma$ energy because the recoil momentum becomes smaller but the elementary cross section of p($\gamma$,K$^+$)$\Lambda$ is almost constant in the energy range of real $\gamma$ from 1.1 to 2.0 GeV.
   2. The survival rate of kaons for a given flight path of the spectrometer becomes higher with a higher kaon momentum.
   3. A larger portion of the hypernuclei produced in the reaction will be captured by the kaon spectrometer for kaons with higher momenta since the angular spread becomes smaller.

   In figure 3, relative hypernuclear yield of $^{12}_{\Lambda }$C ground state as a function of electron energy is given for a scattered electron energy of 0.285 GeV. It can be seen that the higher the energy of the electron beam, the larger the yield of the hypernuclear ground states for a given spectrometer configuration, partly because hypernuclear recoil energy becomes smaller and $\Lambda$ bound probability gets higher.

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. Those channels possibly become sources of kaon background, because bremsstrahlung photons up to the beam energy are produced in the targets. Therefore, electron beam energy is better kept as low as possible from the points of background. Moreover, particle identification will be better done and higher energy resolution can be achieved with lower beam energy.
6. Taking into account above conditions, the optimum kaon momentum was chosen to be 1.2 GeV/$c$ aiming at 2 $\times$ 10$^{-4}$ (FWHM) momentum resolution, which corresponds to about 230 keV energy resolution in hypernuclear excitation spectra.
7. The electron spectrometer also should have a momentum resolution of $\leq$ 4 $\times$ 10$^{-4}$, matching that of the kaon spectrometer. Since the momentum of the scattered electrons is lower compared to that of the kaons, the required momentum resolution is relaxed compared with that for the HKS spectrometer.