The LGD resolution

The energy and spatial resolution in the LGD are measured using 2-shower invariant mass distributions from $\eta \to 2\gamma$ and $\pi^0 \to 2\gamma$ decays, exploring the formula for the square of the invariant mass

$$M^2 = 4E_1E_2\sin^2{\frac{\theta_{12}}{2}} = 2E_1E_2(1-\cos{\theta_{12}}),$$ (1)

where $E_1$ and $E_2$ represent photon energies, while $\theta_{12}$ is their angular separation. The variance of the squared invariant mass in terms of the variances of energy and shower position in the LGD is given by

$$V(M^2) = \sum_{i=1}^{2} \left[ \ \left(\frac{\partial M^2}{\part... ...rtial x_i}\right)\left(\frac{\partial M^2}{\partial y_i}\right) V_{XY} \right],$$ (2)

where index $i$ counts photons. The expression for the energy resolution $\sigma_E = \sqrt{V_E}$ is given by Eq. 2 (NIM: $A E + Bsqrt{E}$) where parameters $A$ and $B$ have to be determined. Spatial variances are taken to be

$$V_X = \sigma_x^2 = \frac{C_E^2}{E} + \left(X_0\sin{\theta}\cos{\phi}\right)^2,$$

$$V_Y = \sigma_y^2 = \frac{C_E^2}{E} + \left(X_0\sin{\theta}\sin{\phi}\right)^2,$$ (3)

$$V_{XY} = X_0^2\sin^2{\theta}\cos{\phi}\sin{\phi},$$

where $C_E$ depends on the size of the LGD block, and $X_0$ is radiation length [1]. For this LGD $C_E = 7.1$ mm$\cdot$GeV $^{\frac{1}{2}}$, and $X_0 = 31.6$ mm [2]. Finding energy derivatives from Eq. 2 is straight forward while all spatial derivatives can be expressed in terms of measured photon momenta [3]. Taking all this into account, Eq. (2) can be simplified as

$$\frac{V(M^2)}{M^4} = \frac{V_E(E_1)}{E_1^2} + \frac{V_E(E_2)}{E_2^2} + \frac{V_S(M^2)}{M^4},$$ (4)

where spatial derivatives and variances are grouped into the term $V_S(M^2)$. From events with two reconstructed showers the $\pi ^0$ sample was selected by limiting invariant mass to $M < 0.5$ $GeV$ and the $\eta$ sample was obtained by selecting pairs with shower separation $D_{\gamma\gamma} \geq 20$ $cm$. The mass resolution was measured by forming the invariant mass distributions for a set of well-defined shower energies $(E_1,E_2)_k$, with $k=1\dots18$ for the $\eta$ and $k=1\dots16$ for the $\pi ^0$ sample. The $\pi ^0$ and $\eta$ peaks from these distributions were fitted with a Gaussian over a polynomial background. From the Gaussian fit the value $V(M^2)/M^4$ has been calculated. The errors are predominantly systematic, governed by the choice of background function, and estimated to be $5\%$ and $10\%$ for the $\eta$ and $\pi ^0$ data respectively. Examining the $V_S(M^2)$ distributions from $\eta$ and $\pi ^0$ decays it was seen that spatial part in the $\eta$ mass resolution is at $10\%$ level, while it gives significant $50\%$ contribution to the $\pi ^0$ mass width [3]. Consequently, in the first approximation the spatial contribution to the $\eta$ mass resolution has been neglected. Following Eq. (4) the set of $k=18$ equations

$$\left(\frac{V(M^2)}{M^4}\right)_k = F_i + F_j,$$ (5)

was formed, where indexes $i,j$ map different photon energies being used. The least-square solution for the $F_i$ values, $i = 1,\dots,11$, was found without assuming the functional form of $F(E)$. This model-independent solution is free of any restrictions on the $F$ values except that $F_i = F_j$ for $i=j$. The corresponding $\sigma_{E}(E_i) = E_i \sqrt{F_i}$ from the free solution is plotted in Fig. 1. These points agree very well with the standard expression for the energy resolution (Eq. 2), with $A=0.037$ and $B=0.080$ GeV $^{\frac{1}{2}}$. In the next step, the spatial corrections to the mass resolution has been included by recording the mean of the $V_s(M^2)/M^4$ distributions for each ($E_1,E_2$) choice, for both the $\eta$ and $\pi ^0$ samples. The expression for the variance function (Eq. (4)) was modified, using the standard expression for the energy resolution,

$$\frac{V(M^2)}{M^4} = \left(A + \frac{B}{\sqrt{E_1}}\right)^2 + \left(A + \frac{B}{\sqrt{E_2}}\right)^2 + C \frac{V_s(M^2)}{M^4}.$$ (6)

In addition to the parameters $A$ and $B$, parameter $C$ that scales the spatial resolution contribution has been introduced. Within this simple model the $\eta$ data fit with $C$ set to zero is shown as solid line in Fig. 1, while the fit parameters are shown in the first row of Table 1. The $\eta$ data fit when $C$ was let to vary is represented by the dashed line in Fig. 1 and the second row in Table 1. The fit confirms that the $\eta$ mass resolution is not very sensitive to spatial corrections. In the next step all measured points from the $\eta$ and $\pi ^0$ samples were fitted together. The fit is shown as the dotted line in Fig. 1, and the resulting parameters are shown in the last row of Table 1. Both the $\pi ^0$ and $\eta$ mass distributions are described well with this single set of parameters. The scale parameter $C$ can be viewed as a correction to the nominal value of $C_E = 7.1$ mm$\cdot$GeV $^{\frac{1}{2}}$ for the LGD, since the energy-dependent term in the spatial errors is dominant at small and moderate angles $\theta < 15\deg$, while the angular part in Eq. (3) becomes important only at large angles $\theta> 15\deg$. According to the fit results the energy resolution function is

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

while the shower position uncertainty can be expressed as

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

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.

Table 1: The parameter values from different fits to the $\eta$ and $\pi ^0$ mass resolutions, obtained using Eq. (6). The corresponding fits are shown in Fig. 1

Fit	$A$	$B$ $\left[GeV^{-1/2}\right]$	$C$	$\chi ^2$
$\eta$ ($C=0$)	$0.0371\pm0.0006$	$0.0797\pm0.0008$	0	1.05
$\eta$ (full)	$0.036\pm0.008$	$0.080\pm0.003$	$0.26\pm1.76$	1.04
$\eta+\pi^0$ (full)	$0.035\pm0.004$	$0.073\pm0.006$	$0.90\pm0.02$	1.50

Figure 1: Energy resolution of showers in the LGD obtained from analysis of the $2\gamma$ sample. Points represent the free solution to the $\eta$ mass resolution measurements when the contribution from the spatial resolution has been neglected. The solid line represents the fit to the $\eta$ data with the standard energy resolution model (Eq. (6)with $C=0$). The dashed line represents the fit to the $\eta$ data when the spatial contribution is taken into account by Eq. (6). The dotted line corresponds to the simultaneous fit to the $\eta$ and $\pi ^0$ data with the same function (Eq. (6)). Corresponding fit parameters and $\chi ^2$ are shown in Table 1.