Prior work

The total Cerenkov light yield in an electromagnetic shower measures to within a fraction of one percent the total $e^{\pm}$ track length in the shower. However only a small fraction of the total light generated actually reaches the photomultiplier tube, and that fraction depends on the shower position. Variations in the collection efficiency with where the shower enters the front face of a block are simply folded into the energy resolution of the counter because they are uncorrelated with the shower energy. But variations in the shower depth centroid are correlated with shower energy, and produce a nonlinearity in the measured photoelectron yield with shower energy. It is well known that the mean depth of a shower increases as the log of the shower energy [1]. Using a simple exponential attenuation model for the loss of Cerenkov light propagating down a block to the phototube, S. Teige showed [2] that the measured photoelectron yield vs shower energy is a power law.

$$p_e = E^{1+\epsilon}$$ (1)

where $\epsilon$ is of order a few percent. More recently D. Armstrong has shown that without such a correction one cannot obtain consistent values for the masses of the $\pi^0$ and $\eta$ mesons independent of photon multiplicity in the event.

Attenuation is not the only nonlinear effect present. Another major effect that prior studies have not taken into account explicitly is shower leakage from loss of particles out the back of the LGD. This effect is also dependent on shower energy, but it enters with the opposite sign from attenuation: instead of increasing the gain with higher gamma energy it decreases it. S. Teige has pointed out in discussion that the competition between these two effects can be exploited to optimize the energy resolution at a certain block thickness. He indicated that the Radphi LGD blocks are somewhat longer than optimal for Radphi energies. This means that somewhere beyond $20^0$, where showers begin to leak out the sides of the detector, one should observe the energy resolution first improving and then getting worse as shower leakage begins to dominate over attenuation.

In a prior study [3] I proposed a formula to be used in estimating the mean depth of showers inside the LGD, based on a knowledge of the shower energy $E$. This, in turn, feeds back to the calculation of the polar angle $\theta$ of the photon. Unfortunately this can only work reliably in the region $\theta < 20^0$ where shower leakage effects are small. At larger values of $\theta$ a more complicated procedure is required because there the centroid $z$ of the observed shower is a function both of $E$ and $\theta$. Even worse, $E$ itself is not known directly in this region either because it depends on a shower leakage correction that must depend strongly on $\theta$. So we start off with measured values $p_e$ and cluster coordinates $(x,y)$ and we want to end up with corrected values for $(\theta,\phi)$ and $E$. This system can be solved by iteration, by seeking the solution to the following three constraints.

$$\theta = \Theta(x,y,z)$$ (2)

$$z = Z(\theta,E)$$ (3)

$$p_e = L(\theta,E)$$ (4)

where $\Theta(x,y,z) = \arctan{\sqrt{x^2+y^2}/z}$. The functions $Z(\theta,E)$ and $L(\theta,E)$ must be obtained from Monte Carlo simulation. The function $Z(\theta,E)$ should approach the form given in Eq. 3 of Ref. [3] for small values of $\theta$. The function $L(\theta,E)$ should approach Eq. 1 for small $\theta$. If the simulation passes both of these checks without fiddling then one may have confidence in its reliability in the region of large $\theta$.