Density-Matrix Formalism for Lepton-Hadron Scattering

Abstract

A density-matrix approach is presented for lepton-hadron reactions. We give a unified treatment of electroproduction, neutrino-induced production, decay, and colliding-beam-type processes. In this formulation, the lepton reactions are treated in a manner completely analogous to the hadronic reactions involving decaying resonances. The general content of our result for the differential cross section can be stated as a product of finite-dimensional representations of the $\mathrm{SO}(2,1)\mathrm{}$ group or $\mathrm{SO}\left(3\right)\mathrm{}$ group for the lepton pair and a hadronic density matrix. The group changes from noncompact $SO(2, 1)$ to compact $\mathrm{SO}\left(3\right)\mathrm{}$, depending upon whether the momentum of the lepton pair involved in the interaction is spacelike or timelike. Formulas are given for general lepton-induced reactions. Specific reactions (total inelastic lepton scattering and leptonic pion, $\rho$, and $N^{*}$ production) are discussed in detail. In this formulation, the implications of Regge-pole theory, low-energy theory, partial conservation of axial-vector current, and current algebra can be easily studied.