Magnetic photon splitting: Computations of proper-time rates and spectra

The splitting of photons gamma --> gamma gamma in the presence of an intense magnetic field has recently found astrophysical applications in polar cap models of gamma-ray pulsars and in ''magnetar'' (i.e., neutron stars with extremely high fields) scenarios for soft gamma repeaters. Numerical computation of the polarization dependent rates of this third-order QED process for arbitrary held strengths and energies below pair creation threshold is difficult; thus, early analyses focused on analytic developments and simpler asymptotic forms. The recent astrophysical interest spurred the use of the S-matrix approach by Mentzel, Berg, and Wunner to determine splitting rates. In this paper, we present numerical computations of a full proper-time expression for the rate of splitting that was obtained by Stoneham and is exact up to the pair creation threshold. While the numerical results derived here are in accord with the earlier asymptotic forms that are due to Adler, our computed rates still differ by as much as a factor of 3 from the S-matrix reevaluation of Wilke and Wunner, reflecting the extreme difficulty of generating accurate S-matrix numerics for fields below about 4.4 x 10(13) G. We find that our proper-time rates appear to be very accurate and exceed Adler's asymptotic specializations significantly only for photon energies just below pair threshold and for supercritical fields, but always by less than a factor of similar to 2.6. We also provide a useful analytic series expansion for the scattering amplitude valid at low energies.