Photon splitting in strong magnetic fields:S-matrix calculations

The $S$-matrix approach to the treatment of photon splitting in a magnetized vacuum, with the electron propagators expressed in the Landau representation, is discussed critically. Although the analytic results of Mentzel, Berg and Wunner are confirmed, we propose that their available numerical results may be subject to two previously unidentified sources of error associated with the sum over principal quantum number $n$, leading to spurious contributions to the amplitude, and the extremely slow convergence of the sum for weak fields. It is shown how the sums may be rearranged to avoid the spurious contributions. If the Euler-Maclaurin summation formula is used to evaluate the infinite sums over $n$, the $S$-matrix approach then reproduces results derived by the effective Lagrangian and proper-time techniques in the weak-field, low-frequency limit. This method gives reliable results, for $B\gtrsim 0.01$ and $\omega \lesssim 0.1,$ that reproduce those obtained by proper-time techniques. The $S$-matrix approach simplifies in the strong-field limit, $B\gg 1$, where the sum over $n$ converges rapidly. Our results show that the branching ratio for the splittings $\overrightarrow{\perp }\perp \perp$ and $\overrightarrow{\perp }\parallel \parallel$ decreases from its known value $\sim 3.4$ for $B\ll 1$ towards zero for $B\gg 1.$ For weak fields the $S$-matrix approach is unnecessarily cumbersome, and future numerical work should be based on the alternative approaches.