Magnetic photon splitting: The S-matrix formulation in the Landau representation - art. no. 016003

Calculations of reaction rates for the third-order QED process of photon sp litting 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, Beg and Wunner presented an alter native 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, pro viding significant advances beyond the work of Mentzel, Berg, and Wunner by summing over the spin quantum numbers of the electron propagators, and ana lytically integrating over the component of momentum of the intermediate st ates that is parallel to the held. The ensuing tractable expressions for th e scattering amplitudes are satisfyingly compact, and of an appearance fami liar to S-matrix theory applications. Such developments can facilitate nume rical computations of splitting considerably both below and above pair thre shold. Specializations to two regimes of interest are obtained, namely the limit of highly supercritical fields and the domain where photon energies a re far inferior to that for the threshold of single-photon pair creation. I n particular, for the first time the low-frequency amplitudes are simply ex pressed 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 dem onstrated.