Analysis Sensitivity of the $\pi ^-p\rightarrow \gamma \gamma n$ Branching Ratio

In this section we present a test of the sensitivity of our $\pi ^-p\rightarrow \gamma \gamma n$ branching ratio to the AHC configuration, beam ADC pulseheight, $C$ counter timing and the energy-angle cuts.

Table 6.6: $\pi ^-p\rightarrow \gamma \gamma n$ branching ratio for data sets AHC cut (3,3) and AHC cut (4,6).

AHC cut	$\cos\theta$	$\epsilon\Omega\cdot$ F	raw	$\pi ^o$ bkg.	random	signal	$\Gamma$
		acc.	$\gamma \gamma n$		$\pi\pi$ bkg.	$\gamma \gamma n$	BR
		$\times$ 10$^{-5}$	events	events	events	events	$\times$ 10$^{-5}$
(3,3)	$-0.1$	6.65	284	24	28	232	2.97 $\pm$ 0.42
(4,6)	$-0.1$	5.48	351	29	72	250	3.18 $\pm$ 0.49

In the quoted branching ratio, we have taken an average of the two data sets, AHC cut (3,3) and AHC cut (4,6), weighted by their respective pion stops of 45% and 55%. We calculated the branching ratio separately from these individual data sets as well. The result is shown in Table 6.6. By applying the AHC cut (4,6) requirement, trigger rates were reduced due to lower acceptance. Hence we were able to increase the incident pion rate by opening the slits in M9A.

However, for AHC cut (4,6) the random multi-$\pi$ background increased, and as seen from Table 6.6, the multi-$\pi$ random background is estimated to be considerably larger for data set AHC cut (4,6). After subtracting these background events, the branching ratio corresponding to the two data sets were found to be in reasonable agreement ($\sim$7 %). The total uncertainty in both the individual measurements were higher than the uncertainty in our quoted branching ratio due to lower statistics in the individual data sets.

Table 6.7: The sensitivity of the $\pi ^-p\rightarrow \gamma \gamma n$ branching ratio to the beam ADC cut.

beam	$\epsilon\Omega\cdot$ F	raw	$\pi ^o$ bkg.	random	signal	Cut	$\Gamma$
ADC	acc.	$\gamma \gamma n$		$\pi\pi$ bkg.	$\gamma \gamma n$	efficiency	BR
cut	$\times$ 10$^{-5}$	events	events	events	events		$\times$ 10$^{-5}$
1069	6.07	635	53	100	482	0.990	3.05 $\pm$ 0.405
1059	6.07	627	52	97	478	0.989	3.02 $\pm$ 0.403
1049	6.07	618	52	92	474	0.988	3.00 $\pm$ 0.402
1039	6.07	609	51	87	471	0.986	2.99 $\pm$ 0.396

Table 6.7 shows the sensitivity of the extracted branching ratio to the beam ADC cut. As the cut threshold is lowered, efficiency of passing the signal is also lowered. As seen from Table 6.7, the corresponding two-photon events obtained after background subtraction also decreased roughly proportional to the efficiency of passing the $\pi ^-p\rightarrow \gamma \gamma n$ signal. The resulting branching ratios are found to be well within the combined statistical and systematic errors of the measurement.

The $\pm~4~\mbox{ns}~C$ counter timing cut was applied to select coincident two-photon events arising from single pion stops. When relaxed to $\pm~6~\mbox{ns}$, the efficiency of passing the $\pi ^-p\rightarrow \gamma \gamma n$ signal for this cut increased from 99% to 99.4% at the cost of increasing background two-photon events arising from random multi-$\pi$ accidentals coincident in the same beam bucket.

Table 6.8: Variation of $\pi ^-p\rightarrow \gamma \gamma n$ branching ratio vs opening angle cut.

$\cos\theta$	$\epsilon\Omega\cdot$ F	raw	$\pi ^o$ bkg.	random	signal	$\Gamma$	%
cut	acc.	$\gamma \gamma n$		$\pi\pi$ bkg.	$\gamma \gamma n$	BR	change
	$\times$ 10$^{-5}$	events	events	events	events	$\times$ 10$^{-5}$
$-0.2$	6.67	772	118	114	540	3.10 $\pm$ 0.44	+1.6
$-0.1$	6.07	635	53	100	482	3.05 $\pm$ 0.41	0.0
0.0	5.46	552	53	89	410	2.85 $\pm$ 0.38	$-5.9$

In Table 6.8 the tests of the sensitivity of the extracted branching ratio to the opening angle cuts are shown. The percentage variations for the different opening angle cuts are found to be well within the combined statistical and systematic errors of our measurement.

Table 6.9: The sensitivity of the $\pi ^-p\rightarrow \gamma \gamma n$ branching ratio to the threshold energy ($E$) of normalization of the measured multi-$\pi$ spectrum to measured two-photon signal.

$E$	$\epsilon\Omega\cdot$ F	raw	$\pi ^o$ bkg.	random	signal	$\Gamma$	%
cut	acc.	$\gamma \gamma n$		$\pi\pi$ bkg.	$\gamma \gamma n$	BR	change
	$\times$ 10$^{-5}$	events	events	events	events	$\times$ 10$^{-5}$
150	6.07	575	53	55	467	2.95 $\pm$ 0.405	$-3.0$
160	6.07	587	53	59	475	3.00 $\pm$ 0.407	$-2.0$
170	6.07	598	53	63	482	3.05 $\pm$ 0.408	0.0
180	6.07	607	53	72	482	3.05 $\pm$ 0.413	0.0

To subtract the multi-$\pi$ background we normalized the measured multi-$\pi$ spectrum to measured two-photon signal for a threshold energy $E >$ 170 MeV. The sensitivity of our result for the branching ratio to $E$ is given in Table 6.9. As Table 6.9 indicates, the branching ratio changes by only 2-3% when $E$ changes from 150-180 MeV.

Finally, we also compared the theoretical energy-angle distributions from Beder (7) with a Monte Carlo generated phase space energy-angle distribution (Figure 6.6).

The extracted branching ratio is found to be (30):

$$\frac{\Gamma_{\rm Beder}}{\Gamma_{\rm phase \ space}} = 0.93 \pm 0.01$$ (6.11)

ie., a change of only $-7$% indicating that our measured $\pi ^-p\rightarrow \gamma \gamma n$ branching ratio is fairly model independent. A fuller understanding of any model dependency of our result will require new theoretical work.