A comparison of the bremsstrahlung cross section in two frameworks: classical Lorentz–Abraham–Dirac dynamics and quantum field theory

Abstract

In classical electromagnetic theory, the Lorentz–Abraham–Dirac (LAD) equation describes the dynamics of a charged particle, including radiation reaction. Though the LAD equation is derived from Maxwell’s equations, consistent with the conservation of energy and momentum, it admits unphysical solutions for point-like charged particles. By assuming small radiative effects, Landau and Lifshitz (LL) develop an approximation of the LAD equation that has no pathological solutions. Though the LL approximation is dynamically sound, it is the LAD equation that has firm theoretical underpinning. As such, the difficulties encountered in LAD dynamics suggest, in part, that an electrodynamic treatment of point-like particles lies outside the classical domain. Herein, we compute the cross section of a charged particle scattered by an attractive Coulombic potential using classical LAD and LL dynamics. We then compare this with the analogous cross section in quantum field theory (QFT). In particular, we compute the cross section for a charged scalar particle incident upon another extremely massive charged scalar with a single photon in the final state. We include the following -radiative corrections: scalar-photon vertex correction, infrared photon emission, vacuum polarization, and two-photon exchange. We find that the classical and quantum frameworks do not produce similar cross sections. Relative to the elastic cross section, the classical bremsstrahlung cross section vastly overestimates the relative QFT cross section; this is, in part, due to the fact that the additional radiative corrections in QFT do not have a classical analog in either the LAD or LL framework.