Theoretical Work

A first theoretical estimate for the rare $\pi ^-p\rightarrow \gamma \gamma n$ process became available in published literature in the appendix of a 1960 paper by Joseph (17). While mainly devoted to calculation of electromagnetic corrections to the tree-level amplitudes for the processes $\pi^- p \rightarrow e^+e^- n$ and $\pi^0 \rightarrow\gamma e^+e^-$, noting that the $\pi ^-p\rightarrow \gamma \gamma n$ process is related to the single-photon $\pi^- p \rightarrow \gamma n$ process by gauge invariance, the two-photon matrix element ($M_{2\gamma}$) was estimated from the single-photon matrix element ($M_{\gamma}$) by,

$$M_{2\gamma} = \bigg[ie\sum_{i}\varepsilon_{2i}\sum_{j}\frac{... ...rtial q_{ji}}\bigg] + [ \epsilon_1 \leftrightarrow \epsilon_2]$$ (2.1)

where $i$= 1, 2; [ $\epsilon_1 \leftrightarrow \epsilon_2$] denotes the expression obtained from interchange of the polarization vectors $\epsilon_1$ and $\epsilon_2$ in the first term of Equation 2.1. Index $j$ distinguishes the various charged particles of momenta $q_j$ involved in the process. The two-photon matrix element as obtained from Equation. 2.1 was based on fairly unrealistic assumptions such as photon energies being vanishingly small compared to the pion rest mass, and so, the validity of the estimate was not clear.

Joseph estimated the relative two-photon branching ratio

$$R_{{2\gamma}/{\gamma}} = \frac{\sigma(2\gamma )}{\sigma(\gamma )}$$ (2.2)

i.e. the ratio of two-photon emission to single-photon emission for pion capture by protons at rest as,

$$\frac{\sigma(2\gamma)}{\sigma(\gamma)} = 1.39 \times 10^{-4} \quad .$$ (2.3)

The single-photon process has a well known measured absolute branching ratio $\Gamma_{\gamma}$= 38.6% (1). Thus the two-photon absolute branching ratio as estimated by Joseph was $\Gamma_{2\gamma}$= 5.4 $\times$ 10$^{-5}$.

Lapidus and Musakhanov (18) considered contributions from various annihilation and bremsstrahlung diagrams to the two-photon matrix element, and calculated the $\pi ^-p\rightarrow \gamma \gamma n$ relative branching ratio ( $R_{{2\gamma}/{\gamma}}$). They also obtained expressions for the two-photon angular distribution and recoil neutron energy spectrum. The relative branching ratio was found to be $R_{{2\gamma}/{\gamma}}$= 1.2 $\times$ 10$^{-4}$. Thus the absolute branching ratio obtained was $\Gamma_{2\gamma}$= 4.6 $\times$ 10$^{-5}$. This estimate had an uncertainty associated with the energy sharing between the two photons.

Finally in 1979, Beder (7) presented a detailed calculation of the $\pi ^-p\rightarrow \gamma \gamma n$ reaction based on a pseudoscalar-coupled pion-nucleon theory. Contribution from $l=0$ and $l=1$ capture were considered at the tree-level. A number of amplitudes were discussed in detail and an estimate of the dominant graphs (Figure 2.1) were obtained.

Figure 2.1: The a) $\pi\pi$ annihilation b) pion bremsstrahlung, and c) & d) $NN$ bremsstrahlung graphs

Following the treatment by Beder, if $T_A($$q)$ represents the invariant amplitude for capture of a pion with momentum $q$, the capture rate is given by:

$$\mbox{Rate}(l=0 \quad \mbox{atomic state}) \propto \vert \psi(0)_{l=0}\vert^2 \times \vert T_A(0)\vert^2$$ (2.4)

$$\mbox{Rate}(l=1 \quad \mbox{atomic state}) \propto \bigg\vert \frac{\partial R}{\partial r}_{r=0}\bigg\vert^2 \times \bigg\vert \nabla_qT_A(q)_{q=0} \bigg\vert^2$$ (2.5)

where $\quad \psi_{l=1,m}=Y_{l m}(\Omega)R(r)$ is the atomic wave function of the pionic atom.
The calculations were performed in a specific gauge where the photon fields are purely transverse in the $\pi\cdot$N center of mass frame. Also, while only the sum of all the Feynman graphs to a given order is gauge invariant, contribution of different diagrams were estimated separately to find the most important contributions to the $\pi ^-p\rightarrow \gamma \gamma n$ process. Noting that the two-photon decay following negative pion capture in hydrogen occurs predominantly from the $s$ state ($l=0$), while the $p$ state ($l=1$) capture occurs mostly in light mass nuclei such as carbon, the differential cross-section obtained for the $\pi ^-p\rightarrow \gamma \gamma n$ capture mode of hydrogen and light nuclei was found to be (Equation 4.3 in reference (7)),

$$\frac{ \mbox{d}\sigma }{ \mbox{d}k_1\mbox{d}k_2 } = [ 256\pi^3 M q_{\mbox{\tiny in}}^{\mbox{\tiny lab}} ]^{-1} (4\pi\alpha \mbox{g}_{\pi^- pn})^2 \cdot \Sigma_l$$ (2.6)

$$\quad {\rm where,} \quad \Sigma_0 = \sum_{\lambda_1\lambda_2 s_1 s_2} \bigg\vert \ T_A(0) \ \bigg\vert^2 (4 \pi\alpha \mbox{g}_{\pi^- pn})^{-2} \quad \quad (l \ = \ 0)$$ (2.7)

$$\Sigma_1 = \sum_{\lambda_1\lambda_2 s_1 s_2} \bigg\vert\nabla_qT_A(q)_{q=0} \bigg\vert^2 (4 \pi\alpha \mbox{g}_{\pi^- pn})^{-2} \quad (l \ = \ 1)$$ (2.8)

and $T_A(q)$ is the invariant transition amplitude, $M$ is the pion mass, $\Sigma$ is the spin, polarization sum over the various $\pi$-$\pi$, $\pi$-N, $n$-$n$ graphs and their interference terms, and $s_1$,$s_2$,$\lambda_1$,$\lambda_2$, are the spin and polarization indices (7) of the nucleons and the photons. The coupling constant $\mbox{g}_{\pi^- pn}$ = $\sqrt{2}{g}_{\pi^o pp}$, where $\mbox{g}^2_{\pi^o\ pp}/4\pi$ = 14.2 and $\alpha$ =1/137. This expression for the two-photon process was multiplied by $\frac{1}{2}$ to avoid photon double counting, and numerically integrated. An expression for the single-photon branching ratio (Equation 4.4 of reference (7)) was obtained in a similar fashion, and the relative branching ratio was found to be $R_{2\gamma/\gamma}=$ 1.31 $\times 10^{-4}$. Thus the absolute branching ratio for the $\pi ^-p\rightarrow \gamma \gamma n$ reaction was found to be $\Gamma_{2\gamma}=$ 5.06 $\times 10^{-5}$.

Beder found that, in the case of $l=0$ atomic capture, the $\pi^-\pi^+$ annihilation graph dominates over the nucleon bremsstrahlung, pion bremsstrahlung and $\Delta$ excitation graphs. The various contributing graphs are shown in Figures 2 and 3 in Reference (7). The annihilation diagram was found to contribute 65% of the total $\gamma \gamma n$ rate and 90% of the $\theta \leq$ 90$^{\circ}$ $\gamma \gamma n$ rate. Nucleon bremsstrahlung graphs contributed 30% to the total $\gamma \gamma n$ rate. Neglecting the small contribution from the interference terms, about 5% of the total contribution came from $\Delta$ excitation diagrams. Unlike the annihilation diagram, the bremsstrahlung diagrams were found to favor large opening angles for the photon-pair, and consequently although the nucleon bremsstrahlung contribution is significant at large two-photon opening angles, it is quite small at small opening angles (see Table 3 and Figure 4 of Reference (7)).

Using Beder's results (Equations A.1-A.10 of Reference (7)) the characteristic angle and energy partition dependences of the various Feynman graphs contributing

Figure 2.2: The photon energy and opening angle dependence of the $\pi ^-p\rightarrow \gamma \gamma n$ reaction transition probability using Beder's equations. The solid curves represent the $\pi^-\pi^+$ annihilation graphs, the dashed curve represents the nucleon bremsstrahlung graphs. In the case of the annihilation graphs the transition probability is plotted for energy partitions $x = (\vert E_{\gamma _1} - E_{\gamma _2}\vert)/(E_{\gamma _1} + E_{\gamma _2})$ from 0.5 to 0.1 in steps of 0.1. In the case of the bremsstrahlung graphs the transition probability is independent of the energy partition.

to the at-rest $\pi ^-p\rightarrow \gamma \gamma n$ reaction are shown in Figure 2.2. The photon energy and opening angle dependences of the transition probability for the $\pi^+\pi^-\rightarrow\gamma \gamma$ annihilation graphs and nucleon bremsstrahlung graphs have notably distinct characteristics. The annihilation graphs are peaked at forward opening angles (small $\theta$) whereas the bremsstrahlung graphs are peaked at backward opening angles (large $\theta$). Also, the annihilation graphs are strongly dependent on, whereas the bremsstrahlung graphs are independent of, the two-photon energy partition. These distinctive opening angle and energy partition dependences of the different graphs allow for the experimental discrimination of the reaction mechanisms. Thus, the two-photon angle and energy partition dependence of the predicted $\pi ^-p\rightarrow \gamma \gamma n$ branching ratio is a critical test of the theoretical calculations.