Magnetic photon splitting: TheS-matrix formulation in the Landau representation

Calculations of reaction rates for the third-order QED process of photon splitting $\overrightarrow{\gamma }\gamma \gamma$ in strong magnetic fields traditionally have employed either the effective Lagrangian method or variants of Schwinger’s proper-time technique. Recently, Mentzel, Berg and Wunner presented an alternative derivation via an S-matrix formulation in the Landau representation. Advantages of such a formulation include the ability to compute rates near pair resonances above pair threshold. This paper presents new developments of the Landau representation formalism as applied to photon splitting, providing significant advances beyond the work of Mentzel, Berg, and Wunner by summing over the spin quantum numbers of the electron propagators, and analytically integrating over the component of momentum of the intermediate states that is parallel to the field. The ensuing tractable expressions for the scattering amplitudes are satisfyingly compact, and of an appearance familiar to S-matrix theory applications. Such developments can facilitate numerical computations of splitting considerably both below and above pair threshold. Specializations to two regimes of interest are obtained, namely the limit of highly supercritical fields and the domain where photon energies are far inferior to that for the threshold of single-photon pair creation. In particular, for the first time the low-frequency amplitudes are simply expressed in terms of the $\Gamma$ function, its integral and its derivatives. In addition, the equivalence of the asymptotic forms in these two domains to extant results from effective Lagrangian or proper-time formulations is demonstrated.