Reliability

Certainly not everything in this Monte Carlo model is an accurate description of the Radphi calorimeter. The spectral attenuation of lead glass depends on its history and is almost certainly different for inner and outer blocks. The wrapping probably has air gaps in some places and none in others, which means that neither the black paper nor the aluminum foil models are really exact. Thus some checks are needed that the simulation gets the overall detector response right in regions where we already know the answer, before we can rely on it for guidance in the large-angle region where the response is unknown.

In the forward region $\theta<10^0$ we already know how the shower depth $z$ and yield $p_e$ should depend on photon energy $E$. Up to an overall shift, $z$ should be proportional to the radiation length times $\log{E}$. The simulation data are shown in Fig. 3 where only showers at $7^0\pm 2^0$ have been included. Superimposed on the data are two curves, Eq. 5 in blue and the 2d parameterization in red. The two curves are difficult to distinguish at the resolution of this figure. This just reflects the fact that the depth calculation under the new parameterization is just the same as Eq. 5 in the forward region. Fig. 2 shows that $Z(\theta,E)$ continues smoothly from this region to smaller values at higher angles, partly from the geometric factor $\cos{\theta}$ and partly due to shower leakage.

Figure 3: Same as Fig. 2, but plotted vs shower energy for fixed polar angle, instead of vice versa. The blue curve is the function in Eq. 5 and the red curve is the 2d parameterization described in the text. The two curves cannot be distinguished at the resolution of this plot.

Figure 4: Same as Fig. 1, but plotted vs shower energy for fixed polar angle, instead of vice versa. The blue curve is the function in Eq. 1 with $\epsilon =0.055$ and the red curve is the 2d parameterization described in the text.

Figure 5: Same as Fig. 4, but for showers at high polar angle $\theta \approx 25^0$ where significant leakage is present. The blue curve is the function in Eq. 1 with $\epsilon =-0.062$ and the red curve is the 2d parameterization described in the text.

In the forward region we also already know that the curve $p_e$ vs $E$ should look like Eq. 1. The simulation data are shown in Fig. 4 where only showers at $7^0\pm 2^0$ have been included. Superimposed on the data are two curves, a free fit to Eq. 1 in blue and the 2d parameterization in red. Note that the free fit gives a value of $\epsilon =0.055$, to be compared with a value of 0.06 that was obtained by D. Armstrong directly from Radphi data. The agreement between the red and the blue curves shows that the two embody essentially the same nonlinearity correction. Note that this was not put into the simulation by hand. It emerged from the interplay of internal reflection and attenuation effects on the one hand, and leakage effects on the other. At large angles one would expect leakage effects to win out over attenuation and cause the sign of $\epsilon$ to reverse. That is exactly what the simulation shows. In Fig. 5 is shown the same thing as Fig. 4 except that showers are selected in the angular range $25^0\pm 1^0$. Here the free power-law fit gives an exponent less than 1. Again good agreement is obtained between the fit to Eq. 1 and the 2d parameterization.

In conclusion, the new parameterization will reproduce the present known corrections to shower depth and energy for showers at low polar angle. In addition, it will give us reliable guidance for obtaining corrected values of these variables for large-angle showers. In the remaining section of this report a procedure is presented for how Eq. 2-4 may be solved in analysis code.