Energy resolution

In the proposed experiment, the hypernuclear mass resolution is governed by the following items. The contributions to the energy resolution are summarized in table 2. The proposed spectrometer system is expected to achieve an energy resolution of 300-400 keV (FWHM), depending on the choice of targets.

1. HKS momentum resolution
   With a Monte Carlo simulation, the momentum resolution of the HKS was estimated to be 110 keV/$c$ (rms) for a spatial resolution of 200$\mu$m of the wire chambers. This corresponds to a missing mass resolution of 230 keV. A detailed discussion is given in section 3.2.4.

2. Beam momentum resolution
   Beam momentum resolution is required to be better than 1 $\times 10^{-4}$. This corresponds to 180 keV/$c$ (FWHM) for 1.8 GeV/$c$ beam. It is also necessary to monitor long term fluctuation of the beam central momentum. If these fluctuation should turn out to be significant, they will be corrected in the off-line analysis.

3. Enge momentum resolution
   Assuming a spatial resolution of 60$\mu$m (rms) and an angular resolution of 2 mr (rms), the momentum resolution of the Enge spectrometer is estimated to be 120 keV/$c$ (FWHM). (see section 3.3.2).
   

4. Kinematical broadening due to uncertainty of the K$^+$ scattering angle
   The uncertainty of the K$^+$ emission angle is dominated by the multiple scattering through the materials between the target and the drift chambers (uncertainty due to the spatial resolution of the chambers are less than 0.2 mrad). Contribution of the following materials are taken into account: the target (100 mg/cm$^2$), vacuum windows (Kevlar 216$\mu$m, Mylar 127$\mu$m), helium bag (100 cm) and chambers (mylar 0.0045'', argon 5.08 cm @ STP).

   The total uncertainty of the K$^+$ angle was estimated to be 2.9 mrad (rms) for the carbon target. This angular uncertainty corresponds to 134 keV (FWHM) ambiguity to the $^{12}_\Lambda$B mass. Similarly, this effect to $^{28}_\Lambda$Al (Si target) will be 56 keV (FWHM).

5. Momentum loss in the target
   Assuming a Vavilov distribution, the average momentum loss and spread of 1.2 GeV/$c$ K$^+$ in 100 mg/cm$^2$ carbon were calculated.The corresponding contribution to the energy resolution is 180 keV (FWHM).

表 2: The energy resolution of the HKS system

Item	Contribution to the mass resolution (keV, FWHM)
Target	C	Si	V	Y
HKS momentum( $\frac{\Delta p}{p} = 2 \times 10^{-4}$)	230
Beam momentum( $\frac{\Delta p}{p} = 2 \times 10^{-4}$ at 1.8 GeV/c)	$\le$ 180
Enge momentum	120
K$^+$ angle( $\Delta \theta < 2.5 degrees$)	134	56	32	18
Target (100 mg/cm$^2$)	$\le$ 180	$\le$ 171	$\le$ 148	$\le$ 138
Overall	$\le$ 390	$\le$ 360	$\le$ 350	$\le$ 345