Absolute $\pi ^-p\rightarrow \gamma \gamma n$ Branching Ratio

The branching ratio for double radiative capture on pionic hydrogen is obtained via

$$B.R. = \frac{N_{2\gamma}} { N_{\pi^-} \cdot \epsilon\Omega \cdot F \cdot c_{bm} \cdot c_{stop}}$$ (6.10)

where $N_{\pi^-}$ is the number of livetime-corrected pion stops, $N_{\gamma \gamma}$ is the number of background-subtracted $\pi ^-p\rightarrow \gamma \gamma n$ events, and $\epsilon\Omega \cdot F$ is the detector acceptance. The factor $c_{stop} = 0.85 \pm 0.01$ accounts for the fraction of incident pions that stop in hydrogen (see Wright et al. (29) for details) and the factor $c_{bm} = 0.99$ accounts for the efficiency of $\pi ^-p\rightarrow \gamma \gamma n$ events passing the beam telescope cut (Section 5.4.2). The source of errors in our measurement is summarized in Table 6.5. The statistical error of measurement is 8.8%, and the systematic error is $\pm 10$%. The systematic error is completely dominated by the $\pm 10$% uncertainty in the determination of the acceptance $\epsilon\Omega \cdot F$. The uncertainties in $N_{\pi^-}$, $c_{stop}$ and $c_{bm}$ were each $\leq 1$ % and entirely negligible.

Table 6.5: Inventory of uncertainties in the $\pi ^-p\rightarrow \gamma \gamma n$ measurement.

Sources of error in the $\pi ^-p\rightarrow \gamma \gamma n$ measurement	% error
Statistical
Statistical error in N$_{2\gamma}$ (482 from 635 events)	5.2
$\pi ^o$ background subtraction	6.2
Multi-$\pi$ background subtraction	3.3
Total statistical error	8.8
Systematics
Net variation in $\epsilon\Omega \cdot F$	10.0
N$_{\pi^-}$	$<$ 1.0
$c_{stop}$	1.0
C timing cut	$<$ 1.0
$c_{bm}$	$<$ 1.0
Total systematic error	10.0
Total percentage error	13.4

Using Equation 6.10, our measured branching ratio of the double radiative capture mode of pionic hydrogen is found to be,

( 3.05 $\pm$ 0.27 (stat.) $\pm$ 0.31 (syst.) ) $\times$ 10$^{-5}$ = ( 3.05 $\pm$ 0.41) $\times$ 10$^{-5}$.

This result is the total $\pi ^-p\rightarrow \gamma \gamma n$ branching ratio for all photon energies ( $0<~E_{\gamma}<m_{\pi}$) and all opening angles ( $-1.0<\cos{ \theta }<+1.0$). Our quoted branching ratio was extracted assuming the energy-angle distributions of Beder (7), although we actually only observed the region with $\cos{ \theta } > -0.1$ and $E_{\gamma}~>~25$ MeV.