Relativistic scattering operators for Dirac particles: Structure, symmetries, and reconstruction

A systematic construction of the general forms of relativistic scattering operators for spin-1/2 particles is presented. Lorentz-invariant two-body scattering operators for the spin-1/2-spin-1/2 and spin-1/2-spin-0 systems are explicitly represented in a manner which is the relativistic generalization of the Wolfenstein representation of Galilean-invariant scattering operators. The simpler spin-1/2-spin-0 case serves as a prototype for the much more complicated spin-1/2-spin-1/2 case. For each case a complete set of independent Lorentz invariants is obtained directly in terms of the available four-momenta and the Dirac {γ} matrices. The most general forms of the scattering operators are derived in terms of these.The implications of parity and time-reversal symmetries are obtained and the resultant constraints are imposed upon the scattering operator. The resulting analysis is in terms of independent true scalar amplitudes that are invariant to all the symmetries imposed. In this way a separation of the purely off-mass-shell components of the scattering operators is obtained and the characteristic structure of such components is explicated. Partially and fully on-mass-shell limits are subsequently obtained. The full implications of the Pauli principle are developed for the spin-1/2-spin-1/2 identical particle case and a generalization of the Fierz transformation is developed.Isospin symmetry is also incorporated into the analysis. The connection to the optical potential for the scattering of a spin-1/2 particle from a composite spin-0 object (such as a nucleus) is described, com- pleting a technical basis for the construction of such optical models from spin-1/2-spin-1/2 scattering operators. The systematic development also provides a natural basis for extensions to investigate symmetry violating contributions in a relativistic context for both the identical and nonidentical particle spin-1/2-spin-1/2 cases. A general framework for the reconstruction of invariant scattering operators from (known) matrix elements in the barycentric frame is developed.This framework reveals the relationship between the relativistic invariants and the Pauli spin structure of their matrix elements in each of the {ρ}-spin sectors of the full space. Constraints upon, ambiguities of, and alternatives in the reconstruction procedure are made apparent. A specific reconstruction scheme is developed which directly maps Pauli-space rotational invariant operators to Dirac-space Lorentz invariant operators via a covariant extension technique. This scheme for direct reconstruction on the full Dirac space precludes the inadvertent introduction of kinematic singularities, as well as instabilities in fitting and approximations; it also provides a vehicle for the analysis of alternative schemes. The formalism developed here is compared to methods employed in recent work on the reconstruction of the invariant nucleon-nucleon scattering operator from matrix elements of the solution of a relativistic scattering equation. A prospectus for further work is given.