$\gamma \gamma$ Acceptance from Multi-$\pi$ Accidentals

The two-photon acceptance of the RMC spectrometer was obtained from the random coincidences of two photons (multi-$\pi$ accidentals) that originate from separate $\pi^- p \rightarrow \gamma n$ and/or $\pi^-p\rightarrow\pi^0n$ capture reactions. Such random photon coincidences can occur when two negative pions arrive at the RMC spectrometer in the same beam bucket. Given a stopping pion rate $N_{\pi}/s$ of 0.5 $\sim$ 0.7 MHz, and a cyclotron RF frequency $f$ of 23 MHz, the beam buckets that contain two pions are estimated using Poisson statistics.
The probability of $k$ coincident pions arriving in one beam bucket was obtained using Poisson statistics via,

$$P(k) = \frac{\lambda^k}{k!} e^{-\lambda}, \quad \mbox{where} \quad \lambda = \frac{N_{\pi}/s}{f} \quad$$ (6.1)

Thus,

$$\frac{P(2)}{P(1)} = \frac{1}{2} \lambda = \frac{1}{2}\frac{N_{\pi}/s}{f}$$ (6.2)

And for an incoming single pion rate of $N_{\pi}/s$, rate of two pions in one beam bucket is estimated as:

$$({\mbox N}_{\pi}/s) \frac{P(2)}{P(1)} = \frac{1}{2}\frac{{(N_{\pi}/s})^2}{f}$$ (6.3)

Figure 6.1: The multi-$\pi$ accidental two-photon 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 sum energy (bottom right) spectra. Solid lines correspond to Monte Carlo generated events and the histograms correspond to measured multi-$\pi$ events corresponding to AHC cut(3,3) data. The histograms are arbitrarily normalized in order to compare the measured data with Monte Carlo generated events.

Figure 6.2: The multi-$\pi$ accidental two-photon 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 sum energy (bottom right) spectra. Solid lines correspond to Monte Carlo generated events and the histograms correspond to measured multi-$\pi$ events corresponding to AHC cut(4,6) data.

The contribution of three or more random coincident pions to the multi-$\pi$ stop rate is found to be negligible. This multi-$\pi$ calibration data and the $\pi ^-p\rightarrow \gamma \gamma n$ data were collected simultaneously and passed through the same cuts (with the exception of the beam telescope amplitude cut). Also, the energy angle range for multi-$\pi$ accidentals covered the entire kinematical range for the $\pi ^-p\rightarrow \gamma \gamma n$ events.

Two different Analog Hit Counter (AHC) trigger requirements were used during data taking. Out of a total of 3.1 $\times$ 10$^{11}$ incident pions, events from 1.39 $\times$ 10$^{11}$ (45%) were required to have at least 3 cell hits in layer 2, and 3 cell clusters in layers 3 and 4 combined ( AHC cut(3,3) ). Events from the remaining 1.71 $\times$ 10$^{11}$ (55%) were required to have hits of at least 4 cells in layer 2 and were required to fire at least 6 clusters in layers 3 and 4 combined ( AHC cut(4,6) ).

The two-photon acceptance from random events from the measured data was obtained from the relation:

$$\epsilon\Delta\Omega = \frac{N_{ran 2 \gamma}}{N_{2\pi} \cdot \epsilon_{s}^2 \cdot \epsilon_{c}}$$ (6.4)

Figure 6.3: Variation of the multi-$\pi$ accidental two-photon acceptance corresponding to AHC cut(3,3). Acceptance is obtained from the different subsets of runs denoted as subset 1 through 6 (see Table 6.1.1).

Figure 6.4: Variation of the multi-$\pi$ accidental two-photon acceptance corresponding to AHC cut(4,6). Acceptance is obtained from the different subsets of runs denoted as subset 7 through 11 (see Table 6.1.1).

where $N_{ran 2 \gamma}$ is the number of reconstructed photon pairs corresponding to $N_{2\pi}$ random coincident multi-pions. The multiplicative factor $\epsilon_{s}$ (0.85) is the ratio of the incident to stopped pions in the target (26), and so, the corresponding correction factor for two pion stops in the target appears in Equation 6.4 as $\epsilon_{s}^2$. The factor $\epsilon_{c}$ (0.97) corresponds to the correction factor due to 4 ns maximum time difference, the pion beam pulse width, between two incident pions in one beam bucket. This time difference slightly decreases both the track reconstruction efficiency and the two-photon acceptance. From Monte Carlo simulations, the loss in acceptance was found to be $\pm$3%.

During the four week long data taking, the running conditions were subject to slight variations from run to run affecting the acceptance of the detector. So the over-all two-photon acceptance was calculated from the average of acceptances obtained from different subsets of the running periods starting from the earliest to the final runs (Table 6.1.1). The variation of the acceptance vs. subsets of runs for the two data sets is shown in Figures 6.3 and 6.4. During the later part of AHC cut(4,6) data, one fastbus TDC module readout was found to get progressively worse contributing to a $\sim$10% decline in the measured two-photon acceptance between the early runs and the late runs as seen from Figure 6.4. The measured acceptance varied by $\pm 4$% from run to run within data set with AHC cut (3,3). Within this data set, the first subset of runs (subset 1) were without SSP cut (this cut is described in Section 3.3) corresponding to 8.5% of the pion stops.

Table 6.1: The subsets of runs each consisting of two-photon event triggers ranging between 200$\times 10^{3}$ to 10$\times 10^{3}$ were analyzed to obtain the two-photon acceptance from the multi-$\pi$ accidentals corresponding to AHC cut(3,3) and AHC cut(4,6) data sets.

Subset Number	Runs Analyzed in that Subset
Data set AHC cut (3,3)
1	8439-8442
2	8490-8494
3	8541-8544
4	8597-8600
5	8640-8645
6	8665-8669
Data set AHC cut (4,6)
7	8743-8764
8	8781-8788
9	8798-8807
10	8848-8854
11	8855-8862

With $\cos\theta > -$0.1 and two-photon energy sum $E_{2\gamma} >$ 80 MeV, from AHC cut(3,3) data, 2.9 $\times$ 10$^{10}$ single pion stops were analyzed and 4.4 $\times$ 10$^{4}$ reconstructed photon-pairs were obtained. From AHC cut(4,6) data, 5.9 $\times$ 10$^{10}$ single pion stops were analyzed and 7.9 $\times$ 10$^{4}$ reconstructed photon-pairs were obtained. A two-photon acceptance $\epsilon\Delta\Omega$ of 2.17 $\times$ 10$^{-4}$, and 1.79 $\times$ 10$^{-4}$ was found for data sets with AHC cut(3,3) and (4,6) respectively. From 5.0 $\times$ 10$^{6}$ Monte Carlo events generated reproducing the same running conditions, the two-photon acceptance $\epsilon\Delta\Omega$ was found to be 2.63 $\times$ 10$^{-4}$, and 1.85 $\times$ 10$^{-4}$ for data sets with AHC cut(3,3) and (4,6) respectively.

Thus the experiment and simulation agreed to 82.5% and 96.8% for AHC cut(3,3) and AHC cut(4,6) respectively. The variations between the Monte Carlo and experimental data reflect the difficulty in simulating the AHC cut due to changes in the chamber efficiencies and fluctuations in the chamber noise. However, as seen from Figures (6.1, 6.2), the energy-angle distributions from experiment and simulation were found to be in very good agreement.

Therefore, on an average, the agreement between experiment and simulation, weighted by the proportion of the number of incident pions, was found to be $0.45\times 82.5 + 0.55 \times 96.8\%$ or 90%. This was used as a multiplicative correction factor $F=0.90$ $\pm 0.09$ to the absolute $\pi ^-p\rightarrow \gamma \gamma n$ acceptance determined from Monte Carlo simulations using Beder's tree-level prediction (7). The 10% uncertainty in the determination of the multiplicative $F$ factor is very conservative, and embodies the entire variation of the measured acceptance over the running period.