Photon splitting in strong magnetic fields : asymptotic approximation formulae vs . accurate numerical results

We present the results of a numerical calculation of the photon splitting rate below the electron-pair creation threshold (ω ≤ 2m) in magnetic fields B > ∼ B cr = m 2 /e = 4.414 × 10 9 T. Our results confirm asymptotic approximations derived in the low-field (B < B cr) and high-field (B ≫ B cr) limit, and allow interpolating between the two asymptotic regions. Our expression for the photon splitting rate is a simplified version of a formula given by Mentzel et al. We also point out that, although the analytical formula is correct, the splitting rates calculated there are wrong due to an error in the numerical calculations. 1 The exotic process of magnetic photon splitting, i.e. the decay of a photon into two photons in the presence of a very strong magnetic field, has recently attracted renewed attention, mainly because of the great importance this process may have in the interpretation of the spectra of cosmic γ-ray burst sources. The basic formulae for magnetic photon splitting had already been derived in the seventies [1–5]. In the first approach the photon splitting effect was analyzed using the Heisenberg-Euler effective Lagrangian. This method is justified under the condition ω ≪ m. The result was the (ω/m) 5 (B/B cr) 6 dependency of the attenuation coefficient in the weak field regime (B ≪ B cr) [1,2]. Adler [3] was the first to solve the problem for arbitrary magnetic field strengths and photon energies below the pair creation threshold. He used the gauge invariant proper-time method and presented numerical results up to B = B cr for the two cases ω = m and ω ≪ m. Other gauge-invariant versions of the splitting-amplitude were found later by Stoneham [5], Baier et al. [6], and recently, using a path-integral approach, by Adler and Schubert [7]. Nevertheless, numerical results, apart from those obtained by Adler, and in particular for magnetic fields exceeding B cr , were not available. Mentzel et al. [8] therefore undertook a rederivation of the photon splitting rate using a configuration space representation for the electron propagator in a strong magnetic field with the aim to find an exact analytical expression that would be suitable for numerical evaluation for magnetic fields above B cr. The representation chosen for the electron propagator, though not gauge-invariant, had the advantage that the final expression for the photon splitting rate did not …