where k is the gamma ray energy in GeV and fk = 8%. The constant term fo (nominally 2%) is neglected in what follows. The following linear approximation for the 2-photon invariant mass is good to 1% out to an opening angle of 40°.
The propagation of errors on the measured quantities k1 , k2 and x,y leads to
where d is the distance from the center of the target to the effective midplane of the showers inside the lead-glass wall. This assumes that Sx = Sy . To go further I must assume a typical value for the lab energy k1 + k2 of the pi0. While some of the pi0's are directly produced in a quasi-two-body reaction and carry away essentially all of the incident photon energy, others appear in association with other mesons as decay products in a continuum below the two-body peak. Our trigger acceptance being what it is, most of the pi0's we record are of the second variety, as can be seen in Fig. 1, which shows the total energy of two-gamma clusters that form a pi0.
This plot was obtained after having made several calibration passes through the data from runs 3013-3014 to obtain approximate calibration constants for the individual blocks, otherwise it is difficult to tell where to place the window around the pi0 mass. The placement of the pi0 window is shown, for this rough calibration, in Fig. 2.
This data sample contains all of the two-cluster events accepted by the trigger in runs 3013-3014, subject to the following two additional cuts. These were applied offline in order to enrich the sample with pi0's.
Note that the width of the pi0 peak in Fig. 2 is NOT an accurate reflection of the experimental resolution of our apparatus under the above conditions because the plot is made using the same data as was used to generate the calibration. It is shown here only to indicate how the pi0 energy spectrum (Fig. 1) was obtained for this data sample. The total energy cut was suspended in making Fig. 1 so that the full spectrum would be visible.
In the following I consider a pi0 with 1.5GeV in the lab. These pairs have a maximum lab opening angle of 10° and appear as clusters separated by about 23cm in the LGD, where I take d = 124cm, the sum of the target-LGD distance of 109cm and average shower penetration depth 15cm. The full distribution is shown in Fig. 3, including all clusters represented in Fig. 1.
From the above equation for Vm the r.m.s. width of the pi0 mass peak is plotted in Fig. 4 versus the fraction x of the total energy carried by one of the clusters (symmetric under exchange of clusters). The mass resolution degrades as the energy of the pi0 increases above 1.5GeV, where the separation between the clusters has decreased to the point where cluster centroid error becomes dominant in determining the mass resolution. This crossover will take place at higher energies for the eta meson. This shows that if we exclude the very asymmetric cluster pairs then we should be able to achieve a width of less than 15MeV for pi0's below 2GeV total energy.
To show the sensitivity of these results to the inputs Sx and fk I evaluated the pi0 peak r.m.s. at x = 0.5 versus these parameters over a large range in both. The results are shown as a contour plot in Fig. 5. Note that a value for fk greater than 8% can be used to estimate the effects of imperfections in the block-to-block calibration of the LGD.
The red star shows what should be achievable under the current configuration of the experiment. Unfortunately the limited statistics we were able to collect during the May run at 4.0GeV may not be sufficient to demonstrate this on a sample independent from the one used for calibration. But this simple calculation indicates what we can expect to achieve.
Richard Jones,
RadPhi collaboration