Theoretical Work

Ericson and Wilkin
The suggestion that the mechanism involved in the rare two-photon capture mode of pionic nuclei involves annihilation of incident negative pions ($\pi ^-$) with the soft positive virtual pions ($\pi^+$) in nuclei, which is then subject to medium modifications by the pion field of the nucleus, was first put forward by Ericson and Wilkin (4). The authors explored various virtual processes such as virtual $\pi^o\rightarrow\gamma\gamma$ decay, $\pi^-\pi^+\rightarrow\gamma\gamma$ $or$ $e^+e^-$ on virtual pions in nuclei, and argued that these processes have observable branching ratios. The Feynman graph corresponding to the annihilation process that contributed to the double radiative pion capture on hydrogen was calculated and the two-photon branching ratio was estimated to be $\Gamma_{2\gamma}~=~5\times~10^{-6}$, an estimate that was an order of magnitude smaller than the ones obtained by other authors (17,18).

Nyman and Rho
Nyman and Rho (8) studied this process in the context of the induced pseudoscalar form factor $g_p(q^2)$ in muon capture in nuclei. The authors explored various possibilities such as pion condensation occurring at or near nuclear densities, nuclear ``renormalization'' of weak and electromagnetic vertices, etc., assuming that muon absorption and doubly radiative pion absorption in nuclei complement each other. The authors pointed out that the dominant underlying process of nuclear double radiative pion capture is the $\pi^-\pi^+$ annihilation in which $\pi^+$ is virtual. The authors further noted that this annihilation process is dominant in the Coulomb gauge for the elementary reaction $\pi ^-p\rightarrow \gamma \gamma n$, while the extent of the contribution of this annihilation process in the corresponding nuclear reaction was subject to further studies.

Christillin and Ericson
Christillin and Ericson (5) developed a theory for the nuclear $\gamma \gamma n$ process in 1979 assuming a normal nucleus with no anomaly due to the pion field. The authors constructed an effective Hamiltonian from the individual gauge invariant nucleonic process. They assumed that the reaction was confined to a single nucleon, and non-relativistic limits applied. They calculated the two-photon branching ratio and angular distribution using the closure approximation and an assumed simple shape of the nuclear excitation spectrum. The emitted photons were above the resonance frequencies (when the photon frequencies match nuclear frequencies), i.e., photon energy $E_\gamma \geq 15$ MeV. The amplitudes for the nucleonic process obtained by the authors were in agreement with the calculations of Beder (7). The authors found that the atomic 1$S$ capture is particularly sensitive to pion effects given by the $\pi^-\pi^+$ annihilation graph, while the 2$P$ atomic capture is dominated by momentum-dependent bremsstrahlung processes. The authors considered the $^{12}{\mbox C}(\pi^-,\gamma\gamma)$ process, which is dominated by capture from the 2$P$ state, and calculated the branching ratio using experimental thresholds. The first estimate of the branching ratio based on a unique nuclear excitation energy $\bar{E}=$ 23 MeV was found to be $\Gamma_{2\gamma}=$ 1.1 $\times 10^{-5}$ for $E_{\gamma_{1}}$, $E_{\gamma_{2}} \geq 25$ MeV, and $\theta \geq 42^\circ$. This method was applied to calculate the single radiative capture rate and the obtained result, $\Gamma_{\gamma}=2.15\%$, was found to be in reasonable agreement with the experimental measurement of $\Gamma_{\gamma}=1.92\pm0.2\%$. Using a spectrum of nuclear excitation energies from 23 Mev to 35 MeV, and without the closure approximation, this partial branching ratio was improved upon, and the final result as quoted by the authors was $\Gamma_{2\gamma}=$ 0.9 $\times 10^{-5}$ for $E_{\gamma_{1}}$, $E_{\gamma_{2}} \geq 25$ MeV, and $\theta \geq 42^\circ$. As we shall discuss in the following section, while the branching ratio agreed with experiments, a qualitative discrepancy was found in the shape of the two-photon angular distribution in the backward angle region (large $\theta$).

Gil and Oset
In the more recent years, using improved many body methods, Gil and Oset (6) calculated the two-photon angular distribution for the $\pi ^-$C and $\pi ^-$Be double radiative processes. Exploring the renormalization effects of virtual pions in nuclear medium, and applying the medium modifications as corrections, the authors calculated the pion two-photon decay width using the local density approximation. They considered an explicit sum over all the occupied states of the nucleons with proper accounting for energy of the nucleonic states. Finite density corrections, Fermi motion and Pauli blocking were accounted for. The authors thus avoided the closure sum and the dependence of their predictions on the average excitation energy of the nucleon. The authors worked in the Coulomb gauge where contributions from the pion annihilation graph dominated. Contributions from the pion and nucleon bremsstrahlung graphs were found to be negligible. To compare their results with experimental measurements, the authors took a weighted average of the capture occurring from the different pionic orbits. The authors however, did not quote a branching ratio for double radiative pion capture in nuclei. As will be discussed in the following section, Gil and Oset's calculations reproduced the strong peak at large angles but disagreed with experiments in the small opening angle region.