Analytical investigations of high-energy electromagnetic showers in strong magnetic fields

Abstract

Abstract The diffusion equation of high-energy electromagnetic showers in strong magnetic fields is solved analytically, under the condition that the product of particle energy and field strength considerably exceeds $mc^2H_{\rm c}\simeq 2.3 \times 10^{7}$ TeV G, by applying Mellin and Laplace transforms. Differential and integral energy spectra of shower electrons/positrons and photons are evaluated by applying the saddle point method. Both spectra expressed by asymptotic expansions are also derived based on singularities of the Laplace–Mellin transform of the spectrum. The results are compared with those derived by a Monte Carlo method and numerical integration methods. Energy flows, peak positions, and peak values of transition curves, as well as track lengths of shower particles, are predicted and discussed, together with other characteristic properties of showers in strong magnetic fields, wherein good agreement between the low-energy limit of the power-law index for our differential energy spectra and the low-energy photon index of $\Gamma$ observed in Fermi LAT is pointed out and discussed.