Transverse Momentum Dependence of the Landau-Pomeranchuk-Migdal Effect

We study the transverse momentum dependence of the Landau-Pomeranchuk-Migdal effect in QED, starting from the high energy expansion of the solution of the Dirac equation in the presence of an external field. The angular integrated energy loss formula differs from an earlier expression of Zakharov by taking finite kinematical boundaries into account. In an expansion in powers of the opacity of the medium, we derive explicit expressions for the radiation cross section associated with N=1, 2 and 3 scatterings. We verify the Bethe-Heitler and the factorization limit, and we calculate corrections to the factorization limit proportional to the square of the target size. A closed form expression valid to arbitrary orders in the opacity is derived in the dipole approximation. The resulting radiation spectrum is non-analytic in the coupling constant which is traced back to the transverse momentum broadening of a hard parton undergoing multiple small angle Moliere scattering. In extending the results to QCD, we test a previously used dipole prescription by comparing to direct pQCD results for N=1 and 2. For N=1, the QCD dipole prescription reproduces exactly the Bertsch-Gunion radiation spectrum. For N=2, we find a sizeable correction which reduces to a multiplicative factor 17/8 at large separation.