Random background subtraction

The remaining multi-$\pi$ contamination was subtracted using (i) the observed number of 2$\pi$ events with summed energy $> 150$ MeV and (ii) the measured summed energy spectrum for the multi-$\pi$ background.

This multi-$\pi$ accidental background in the final candidates was estimated by counting the number of photon-pairs that remained in the 635 events above the photon-pair energy sum of 150 MeV. The energy sum of the photon-pair is close to the pion rest mass of about 140 MeV, the remaining energy going to the recoil neutron. The spectrometer's energy resolution was 9 MeV (FWHM). Thus, to a good approximation, any event that remained at or above 150 MeV in the raw signal spectra could be attributed to the random multi-$\pi$ accidentals. From the sum energy spectrum (Figure 6.8) of the two-photon events from multi-$\pi$ accidentals, the number of random events above and below 150 MeV was measured, and the total number of random backgrounds remaining in the final signal was estimated from the number of events remaining above 150 MeV, which as a first approximation, were assumed to be solely due to the random background.

Figure 6.8: The energy sharing between two-photon events from multi-$\pi$ accidentals. The fraction of events with $E_{2\gamma} < E$ (n$_{<}$) to $E_{2\gamma} > E$ (n$_{>}$) is determined from this spectra. Solid and broken lines correspond to AHC cut(3,3) and (4,6) respectively.

This however, was complicated by the detector's two-photon energy resolution tail that extended beyond 150 MeV. At an energy when there were no more resolution tail events, the total number of true random two-photon events were expected to be constant. As seen from Table 6.4, estimates obtained from 170 MeV and upward, the total random two-photon background was constant, and was found to be 100 events or 15.7% of the raw signal events. The total number of multi-$\pi$ background events ($N_{\pi\pi}$) is estimated from the observed 37 multi-$\pi$ background events above 170 MeV ($n_{>}$). By calculating the number of multi-$\pi$ background events below 170 MeV ($n_{<}$) from the multi-$\pi$ two-photon energy sharing above and below 170 MeV (0.37:0.63), $N_{\pi\pi}$ was found to be;

$$N_{\pi\pi} = n_{>} + n_{<}$$ (6.5)

$$\Rightarrow N_{\pi\pi} = n_{>} + n_{>} \times \frac{0.63}{0.37}$$ (6.6)

$$\Rightarrow 37 + 63 = 100 \quad \mbox{events}.$$ (6.7)

The uncertainty in the determination of this final 100 random background events comes from the uncertainty associated with the observed 37 events above 170 MeV ($n_{>}$). Therefore we estimate the statistical error as $1/\sqrt{37} \times 100 = 16$ events or 2.5%. Hence a random multi-$\pi$ background of 100 $\pm$ 16 or (15.7 $\pm$ 2.5)% was subtracted from the raw 635 signal events.

Table 6.4: The energy sharing and the total background due to the random multi-$\pi$ accidentals. Contributions from data sets with AHC cut (3,3) and (4,6) are shown for $E$ = 170 MeV. $E$ refers to the upper energy limit, $n_{>}$ ($n_{<}$) is the number of two-photon events above (below) $E$. $n_{>}$ is measured while $n_{<}$ is estimated from the measured fraction $n_{<}:n_{>}$. Starting at 170 MeV, contributions from the detector's energy resolution tail become negligible and the total number of random background events are found.

	N$_{\pi\pi}$		N$_{\pi\pi}$		N$_{\pi\pi}$		Total
$E$	fraction		$E_{2\gamma}$ $>$ $E$		$E_{2\gamma}$ $<$ $E$		N$_{\pi\pi}$
	measured		measured		estimated
MeV	$n_{<}:n_{>}$		$n_{>}$		$n_{<}$		$n_{>}+n_{<}$
150	0.48:0.52		60		55		115 $\pm$ 15
160	0.55:0.45		48		59		107 $\pm$ 16
	0.63:0.37		37		63		100 $\pm$ 16
170	(3,3)	(4,6)	(3,3)	(4,6)	(3,3)	(4,6)	(3,3)	(4,6)
	0.62:0.38	0.64:0.36	11	26	17	46	28	72
180	0.72:0.28		28		72		100 $\pm$ 19

From the two-photon energy distribution of these final 635 events, the background two-photon events from the random coincident multi-$\pi$ accidentals are clearly seen above 150 MeV in Figure 6.10.