Conclusion

We analyzed the invariant mass distributions from $\pi^0 \to 2\gamma$ and $\eta \to 2\gamma$ decays to extract the single shower energy and position resolution. To a good approximation, the width of the $\eta$ mass peak is mainly determined by the energy resolution, while the $\pi ^0$ mass is heavily affected by the spatial resolution. This feature is related to the fact that the detector is close to the target and $\pi ^0$ shower pairs from have separations close to the detector acceptance. In the first approximation we neglected the spatial contribution to the $\eta$ mass resolution and measured the squared-mass variance for a set of well-defined shower energies ($E_1,E_2$). From this measurement we a obtained model-independent solution to the energy resolution function. This solution is in good agreement with the standard expression used to describe the energy resolution of lead glass. In order to incorporate the $\pi ^0$ data into the same description, we introduced a standard expression for the shower centroid resolution. The spatial contribution to the mass resolution is scaled by a free parameter $C$. The $\eta$ data showed no sensitivity to the value of the parameter $C$. In order to fix the set of parameters that governs the LGD resolution, we performed a simultaneous fit to 34 $\eta$ and $\pi ^0$ mass resolution measurements. From the fit, we obtained for the energy resolution function

$\frac{\sigma_E}{E} =3.6\% + \frac{7.3\%}{\sqrt{E}}$,

and for the shower position uncertainty

$\sigma_{\rho} = \sqrt{ \left(\frac{0.64}{\sqrt{E}}\right)^2 + \left(X_0\sin{\theta} \right)^2 }$, $\rho \sigma_{\phi} = \frac{0.64}{\sqrt{E}}$,

where ($\rho,\phi$) are the polar coordinates of the shower in the face of the LGD. All lengths in these relations are in mm and all energies in GeV. The mass widths of both the $\pi ^0$ and $\eta$ from $2\gamma$ events are explained very well with this single set of parameters.