Conclusions

We have made the first measurement of double radiative capture on pionic hydrogen by recording $\gamma$-ray coincidences from $\pi ^-$ stops in liquid H$_2$ using the RMC detector at the TRIUMF cyclotron. The total branching ratio for the $\pi ^-p\rightarrow \gamma \gamma n$ process is found to be (32)

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

by assuming the kinematical distributions from Beder (7).

Our data suggest that double radiative capture on pionic hydrogen is dominated by the $\pi\pi$ annihilation mechanism. Our conclusion regarding the dominance of the $\pi\pi$ annihilation mechanism is favored by both the measured branching ratio and the kinematical energy-angle distribution of the two-photon spectra.

While our measured branching ratio is in approximate agreement with the predicted branching ratio for the annihilation graph, it is about ten times higher than the branching ratio as predicted by the $NN$ bremsstrahlung mechanism. Additionally, the single photon energy spectra in Figure 6.9, the two-photon sum energy distribution in Figure 6.10, and the two-photon angular spectra in Figure 6.11 show that our measured data points are in good agreement with the predictions of the annihilation graph.

While the measured energy distributions do not discriminate between the competing mechanisms, the angular distribution clearly favor dominance of $\pi\pi$ annihilation graph. Specifically, as seen in Figure 6.11, in the $-0.1<\cos\theta~<~0.5$ region, the annihilation distribution is approximately constant whereas the bremsstrahlung distribution falls off at smaller angles.

We quote 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 measured branching ratio $BR = 3.05 \times 10^{-5}$ (32) is somewhat smaller than the theoretical branching ratio of $BR = 5.06 \times 10^{-5}$ (7). Since Beder's calculation was performed at tree-level and neglects contributions from pion loops, etc, such higher-order terms might explain the difference between the measured and the predicted branching ratio. A new calculation of double radiative capture on pionic hydrogen using chiral perturbation theory is currently underway (31), and may help in understanding the remaining discrepancy between theory and experiment.

The uncertainty in determining the branching ratio had roughly equal contributions from statistical and systematic errors. The statistical errors originate from both the statistics of the experimental data and the statistics of the $\pi ^o$ Monte Carlo. A longer run of two months could reduce the experimental data statistical error to $\pm$ 3% and additional simulations on faster processors could reduce the Monte Carlo statistical error to $<$ 3%. The systematic uncertainty was completely dominated by our uncertainty in the absolute detection efficiency for the $\gamma \gamma n$ signal. This uncertainty was assumed quite conservatively, and represented the run-to-run variation of the measured acceptance over duration of the entire experiment. These variations were probably caused by noise in the drift chamber, instabilities in the wire chamber, etc. Improved understanding, control, and measurement of the two-photon acceptance of the RMC detector could reduce the systematic uncertainty of our measurement to $<$ 5%.

The opening angle range of the identified $\gamma \gamma n$ events was $-0.1 < \cos\theta <1.0$. To extend the range to smaller $\cos\theta$ (i.e. larger angles) would require reducing the angular resolution tail in the $\pi^o\rightarrow\gamma\gamma$ background. This tail is presumably from energy loss and multiple scattering of the $e^+e^-$, and uncertainties associated with track recognition and photon reconstruction. Extending the opening angle range would be valuable in better separating the role of the $\pi\pi$ annihilation mechanism (forward peaked) and the $NN$ bremsstrahlung mechanism (backward peaked).

The low energy cut-off for photon detection was $\sim$30 MeV due to the $e^+e^-$ energy loss and the $D$ counter trigger requirement. Thus a significant portion of the $\gamma \gamma n$ spectrum was not observable. This region corresponds to energy partitions $\vert X\vert > 0.5$, and is interesting in discriminating the contributions of $NN$ bremsstrahlung ($X$ dependent distribution) and the $\pi\pi$ annihilation ($X$ independent distribution).

Double radiative capture measurements on nuclear $( \pi , 2 \gamma )$ targets have been available for some time; on $^{12}$C, Deutsch et al. (12) obtained a partial branching ratio of $(1.4 \pm 0.2) \times 10^{-5}$, for $E_{\gamma} > 25$ MeV and $\cos{ \theta } < 0.64$, and Mazzucato et al. (13) obtained a partial branching ratio of $(1.2 \pm 0.2) \times 10^{-5}$, for $E_{\gamma} > 17$ MeV and $\cos{ \theta } < 0.64$. Single radiative capture on light nuclei such as $^{12}$C has a very small branching ratio, $\Gamma_{\gamma}$=0.017 $\pm$ 0.001 (33). When expressed as double radiative to single radiative capture partial branching ratios on $^{12}$C, the double radiative capture measurements on $^{12}$C can be expressed as: $\frac{\displaystyle \sigma(\gamma\gamma)}{\displaystyle \sigma(\gamma)} = 7.1\times10^{-4}$ (Deutsch et al.), and $\frac{\displaystyle \sigma(\gamma\gamma)}{\displaystyle \sigma(\gamma)} = 5.9\times10^{-4}$ (Mazzucato et al.), as compared to $\frac{\displaystyle \sigma(\gamma\gamma)}{\displaystyle \sigma(\gamma)} = 0.8\times10^{-4}$ obtained by our measurement on hydrogen.

Thus, the $\frac{\displaystyle \sigma(\gamma\gamma)}{\displaystyle \sigma(\gamma)}$ ratio is much larger for light nuclei than on hydrogen. This difference could be due to the dominance of $p$ wave capture on carbon but $s$ wave capture on hydrogen. As seen from Figures 2.6 and 2.8, and discussions in Section 2.3.3, bremsstrahlung mechanism dominates the $p$ state process whereas the $\pi\pi$ annihilation mechanism dominates the $s$ state process. The nuclear two-photon angular distribution is peaked in the backward angle region (large $\theta$). A comparison of these earlier nuclear measurements with our data on hydrogen is difficult however, since the backward angle region is not observable in our measurement because of the large two-photon background due to $\pi ^o$ decay. This limits a detailed comparison of the nuclear data with our experimental data.

Our finding that the $\pi\pi$ annihilation mechanism dominates in the $\pi ^-p\rightarrow \gamma \gamma n$ reaction is significant since the $\pi^-\pi^+\rightarrow\gamma\gamma$ annihilation graph can be related to the $\pi\gamma\rightarrow\pi\gamma$ Compton scattering graph by crossing symmetry and is potentially sensitive to the charged pion polarizability $\alpha^{\pm}_{\pi}$. Further theoretical studies are encouraged to explore this fascinating possibility.

In conclusion, we hope that our findings will stimulate new experimental work and further theoretical studies on double radiative capture reactions as a novel probe of the $\pi \pi \rightarrow \gamma \gamma$ vertex, the proton's virtual pion cloud and its in-medium modifications.