Absolute $\gamma \gamma$ Acceptance for $\pi ^-p\rightarrow \gamma \gamma n$

The absolute two-photon acceptance for the $\pi ^-p\rightarrow \gamma \gamma n$ reaction was determined from Monte Carlo simulations using Beder's tree-level prediction (7). In Beder's calculation of double radiative capture the main contributions originate from $\pi\pi$ annihilation graphs, $NN$ bremsstrahlung graphs, and their interference. The $\pi\pi$ annihilation alone account for 64% of the total branching ratio and yield a distribution that is peaked at small opening angles. The $NN$ bremsstrahlung graphs alone account for 20% of the total branching ratio and yield a distribution that is peaked at large opening angles. At threshold, the pion bremsstrahlung contributions vanish, and effects of vector meson exchange and delta resonance excitation are calculated to be very small.

Figure 6.5: The RMC GEANT simulated $\pi ^-p\rightarrow \gamma \gamma n$ opening angle (top) and two-photon energy sharing $X = \frac{\vert E_{\gamma 1}-E_{\gamma 2}\vert}{E_{\gamma 1}+E_{\gamma 2}}$ (bottom left) and two-photon sum energy (bottom right) spectra. Solid and broken lines correspond to AHC cut(3,3) and (4,6) respectively.

Figure 6.6: The RMC GEANT simulated two-photon phase space spectra of opening angle (top) and two-photon energy sharing $X = \frac{\vert E_{\gamma 1}-E_{\gamma 2}\vert}{E_{\gamma 1}+E_{\gamma 2}}$ (bottom left) and two-photon sum energy (bottom right). Solid and broken lines correspond to AHC cut(3,3) and (4,6) respectively.

The absolute two-photon acceptance was determined by generating 5 $\times$ 10$^{6}$ $\pi ^-p\rightarrow \gamma \gamma n$ Monte Carlo events incorporating Beder's tree-level calculation. We used the formula for the graphs documented in the appendix of Reference (7). The process of Monte Carlo generation of these rare $\pi ^-p\rightarrow \gamma \gamma n$ (predicted branching ratio $\sim$ 5.1 $\times$ 10$^{-5}$) two-photon events was time consuming, and the process was concluded after generating enough of these events which could be compared statistically to the measured $\pi ^-p\rightarrow \gamma \gamma n$ signal. The set of cuts used to analyze the measured and Monte Carlo data is listed in Table 6.1.3. With the exception of the beam telescope cut and C counter timing cut (all Monte Carlo events were prompt), all other cuts were applied to the Monte Carlo generated $\pi ^-p\rightarrow \gamma \gamma n$ events.

Table 6.2: The final set of cuts used to obtain the signal $\pi ^-p\rightarrow \gamma \gamma n$ two-photon events from measured and Monte Carlo data.

	$\pi ^-p\rightarrow \gamma \gamma n$ Data
Description of The Applied Cuts
	Measured	Monte Carlo
Tracking cut	yes	yes
Photon cut	yes	yes
AHC cut	yes	yes
SSP cut	yes	yes
C counter timing cut	yes	no
Opening angle cut	yes	yes
Low energy cut	yes	yes
Beam telescope cut	yes	no

From analysis of these events, 407 reconstructed photon-pairs were found for data set AHC cut (3,3) and 286 reconstructed photon-pairs were found for data set AHC cut (4,6) resulting in $\pi ^-p\rightarrow \gamma \gamma n$ acceptances of 8.14 $\times$ 10$^{-5}$ and 5.72 $\times$ 10$^{-5}$ for the two data sets respectively. Using the prompt time cut efficiency of 99% (Section 5.4.1), and the multiplicative factor $F$=0.90 $\pm$ 0.09, and the corresponding pion stop weighted average of the two data sets (45% and 55% respectively), the absolute two-photon acceptance of the RMC photon-pair spectrometer for the $\pi ^-p\rightarrow \gamma \gamma n$ measurement was found to be,

$$\begin{eqnarray*} \epsilon \Delta \Omega \cdot F = (0.45\times 8.14 \times 10^{-... ....72 \times 10^{-5} )\times 0.99\times 0.90 = 6.07 \times 10^{-5} \end{eqnarray*}$$