A Quantum Treatment of the Landau-Pomeranchuk-Migdal Effect

Abstract

In this letter the high-energy expansion for scattering from extended targets, which the authors previously applied to beamstrahlung radiation and pair production, is applied to the problem of radiation in a medium with multiple scattering. The suppression of the emission of long wave-length photons, the Landau-Pomeranchuk-Migdal effect, is treated and explained in physical terms. This extends previous classical treatments of the problem to the quantum domain and corrects certain approximations made in these earlier works; for example, the effects of finite target thickness can be treated. A model of a random scattering medium is defined that allows a quantum treatment of multiple scattering and the resultant suppression of bremsstrahlung radiation. Submitted to Physical Review Letters ∗Work supported by Department of Energy contract DE–AC03–76SF00515. Perhaps the most ubiquitous process occurring in high-energy physics is the bremsstrahlung of photons by a charged particle in the field of an atom first described by Bethe and Heitler [1]. Following experimental confirmation in 1993 of the Landau–Pomeranchuk–Migdal (LPM) effect [2-6], there is renewed interest in extensions of this process, as well as in its strong interaction analogue, gluon radiation at very high energies in heavy nuclei. We describe here the application of eikonal techniques developed for the beamstrahlung process [7] that lead to a simpler, more straightforward, and physically transparent quantum mechanical derivation of the LPM suppression of soft photon radiation from high-energy electrons in dense matter. This effect was first described by Landau and Pomeranchuk [8] who treated the classical radiation of a high-energy particle in the fluctuating and ‘random’ field inside an infinitely thick medium. The longitudinal momentum transfer q|| of a high-energy electron of momentum p and mass m, radiating a photon of momentum k ≡ (1 − x)p, has a minimum value given by qmin || = m2(1 − x)/2xp. The uncertainty principle can be used to define the formation length lf = (1/q min || ) which at high energies (p >> m) and soft photon emission (1− x) << 1 can grow quite large relative to the interatomic spacing. In their classical derivation, which is appropriate to this kinematics limit, Landau and Pomeranchuk were the first to show that the familiar Bethe-Heitler radiated photon spectrum dN ∼ dk/k is modified by the multiple scattering of the electron as it traverses the rapidly varying electric fields of the medium. When the mean free path of the electron αL is comparable or less than the formation length lf , they found that the spectrum is suppressed, ultimately achieving the form dN ∼ dk/(p √ Lk) . (1)