Opening Angle Cut

Opening angle ( $\theta_{2\gamma}$) is the angle between the two photons. The photons convert into pairs of $e^+e^-$ tracks in the lead sheet wrapped around the $A, A'$ scintillators. These $e^+e^-$ pairs travel forward in circular tracks into the Drift Chamber due to the uniform axial magnetic field surrounding the RMC detector. Energy ($E$) and momenta ( $\vec{p}_{\gamma}$) of each photon are determined from the curvature of the $e^+e^-$ tracks using the Lorentz force equation,

$$\vert\vec{F}\vert = evB\sin\theta$$ (5.5)

$$p = eBr\sin\theta$$ (5.6)

where B is the uniform axial magnetic field (1.2 kG), $e$ is the electronic charge and $r$ is the radius of curvature of the charged track fitted to a circle. The photon momenta are obtained from the vector sum of the $e^+e^-$ momenta,

$$\vec{p}_\gamma = \vec{p}_{e^-} + \vec{p}_{e^+} \quad .$$ (5.7)

The cosine of the opening angle, $\cos\theta_{2\gamma}$ is determined from the dot product of the momentum vectors ( $\vec{p}_{\gamma}$) of the two photons,

$$\vec{p}_{\gamma 1} \cdot \vec{p}_{\gamma 2} = p_{\gamma 1} p_{\gamma 2} \cos\theta_{2\gamma}$$ (5.8)

$$p_{x_1}p_{x_2} + p_{y_1}p_{y_2} + p_{z_1}p_{z_2} = p_{\gamma 1} p_{\gamma 2} \cos\theta_{2\gamma}$$ (5.9)

$$\Rightarrow \quad \cos\theta_{2\gamma} = \frac{\displaystyle p_{x_1}p_{x_2} + p_{y_1}p_{y_2} + p_{z_1}p_{z_2}}{\displaystyle p_{\gamma 1} p_{\gamma 2}}$$ (5.10)

where $\vec{p}_{\gamma i} = p_{x_i}\vec{i}+p_{y_i}\vec{j}+p_{z_i}\vec{k}$, and $p_{\gamma i} = \sqrt{p_{x_i}^2+ p_{y_i}^2+ p_{z_i}^2}$, $i=1,2$ .

The two-photon opening angle tail consisting of $\pi^-p\rightarrow\pi^0n$ events having true opening angles $\cos\theta < -$0.76 yielding measured angles $\cos\theta > -$0.76 was studied using $9\times 10^7$ $\pi ^o$ Monte Carlo events, and the fractional contribution from $\pi ^o$ tail background was found to reach a minimum at $\cos{\theta} = -0.1$ (Figure 5.6). Thus, the two-photon opening angle cut was applied at $\cos{ \theta } > -0.1$.