The invariant mass resolution

The square of the invariant mass of two photons is given by

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

where $E_1$ and $E_2$ are corresponding photon energies, while $\theta_{12}$ is their angular separation. The expression for 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. Taking all derivatives of Eq. (1) one can re-write Eq. (2) in the form

$$\frac{V(M^2)}{M^4} = \frac{V_E(E_1)}{E_1^2} + \frac{V_E(E_2)}{E_2^2} + \frac{V_S(M^2)}{M^4},$$ (3)

where spatial derivatives and variances are grouped into the term $V_S(M^2)$. The usual expression for the energy resolution function is given by [1]

$$\frac{\sigma_E}{E} = \sqrt{\frac{V_E}{E^2}} = A + \frac{B}{\sqrt{E}},$$ (4)

where parameters $A$ and $B$ have to be determined. Spatial derivatives can be expressed in terms of measured photon momenta utilizing the expression

$$\cos(\theta_{12}) = \frac{\vec{P_1} \cdot \vec{P_2}}{\left\vert\v... ...frac{x_1x_2+y_1y_2+z_1z_2}{\sqrt{x_1^2+y_1^2+z_1^2} \sqrt{x_2^2+y_2^2+z_2^2}}.$$ (5)

For example

$$\frac{\partial M^2}{\partial x_1} = -2E_1E_2\frac{\partial \cos(\... ...(P_{y_1}^2 + P_{z_1}^2) \ - P_{x_1}(P_{y_1}P_{y_2} + P_{z_1}P_{z_2} ) \right].$$ (6)

The other spatial derivatives have a similar form and they can be obtained by the proper variable substitution. The $z$ position of the shower maximum is not directly measured. However, it is correlated with the energy and this has been taken into account within the shower-depth and nonlinearity corrections [2]. For the purpose of this analysis a common $z$-plane is fixed at $z_0 = 120$ $cm$. The fluctuations in the shower position that determine $V_X$, $V_Y$ and $V_{XY}$ are also affected by the energy. The energy dependence is proportional to $1/\sqrt{E}$ with a proportionality constant that depends on the size of the LGD block [3]. This term alone would make spatial errors cylindrical and uniform across the face of the LGD. However, one can expect that the uncertainty in the shower position increases with polar angle $\theta$, as $X_0\cos{\theta}$, where $X_0$ is the radiation length of the lead glass. This makes shower centroid errors elliptical. The orientation of the ellipse depends on the shower azimuthal angle $\phi$, and is connected with the correlation term $V_{XY}$. Combining all together,

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

$$V_Y = \sigma_y^2 = \frac{C_E^2}{E} + \left(X_0\sin{\theta}\sin{\phi}\right)^2,$$ (7)

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

with $C_E = 7.1$ mm$\cdot$GeV $^{\frac{1}{2}}$, and $X_0 = 31.6$ mm[4]. In this study, we analyze events with two reconstructed showers. The $\pi ^0$ sample, consisting of $15M$ events, is selected by limiting invariant mass to $M < 0.5$ $GeV$. The corresponding invariant mass distribution is shown on the left side of Fig. 1. The $\eta$ sample is obtained by selecting pairs with shower separation $D_{\gamma\gamma} \geq 20$ $cm$. This distance is approximately twice the average shower size in the LGD plane. The corresponding $\eta$ mass distribution, shown on the right side of Fig. 1, contains $8M$ events. Small peak around 0.2 $GeV$ represents the $\pi ^0$ remnants that have large shower separation. The $\pi ^0$ and $\eta$ peaks are fitted with a Gaussian over a polynomial background. In order to check the influence of spatial resolution on the overall mass resolution, the distribution of $V_s(M^2)/M^4$ is plotted in Fig. 2 (solid line), calculated with the help of Eqs. (6) and (7). The distribution is obtained from the inclusive $2\gamma$ sample without restrictions on $M$ or $D_{\gamma\gamma}$. The dotted and dashed lines represent pairs with invariant mass within a $2\sigma$ window around the $\pi ^0$ and $\eta$ peaks, respectively. Table 1 shows the Gaussian mean and r.m.s. of the two fitted meson peaks from Fig. 1. The third column represents the $V(M^2)/M^4$ obtained from the fit. (see Eq. (8)). The last column represents the mean value of $V_s(M^2)/M^4$ from the dashed and dotted distributions in Fig 2. One can see that the spatial resolution contributes about $10\%$ to the $\eta$ variance, while the $\pi ^0$ width is affected at the level of $50\%$. When two showers have the same energies and the spatial term is small, Eq. (3) is reduced to

$$\frac{V(M^2)}{M^4} =4\frac{V(M)}{M^2} = 2\frac{V_E}{E^2}.$$ (8)

By measuring the mass or squared mass resolution in narrow energy bins one can extract the energy resolution function Eq. (4) and the corresponding parameters. The energy bins should be narrower than the detector energy resolution.

Figure 1: Invariant mass distribution of $\pi ^0(2\gamma )$ (left) and $\eta (2\gamma )$ (right).

Table 1: The mean, $r.m.s.$ and corresponding $V(M^2)/M^4$ obtained from the fit of the two meson peaks in Fig. 1. The last column is the mean of dashed and dotted distributions from Fig 2, respectively.

Data	$\mu$ $\left[GeV\right]$	$\sigma$ $\left[GeV\right]$	$V(M^2)/M^4$	$V_s(M^2)/M^4$
$\pi ^0$	0.1352	0.0183	0.0731	0.0406
$\eta$	0.5539	0.0408	0.0217	0.0030

Figure 2: The $V_S(M^2)/M^2$ distribution in $2\gamma$ events calculated using Eqs. (6) and (7). The dashed and dotted lines represent the distribution for pairs with mass within the $\pi ^0$ and $\eta$ windows, respectively.