  We construct two-electron scattering states and verify their tensor product
structure in the infrared-regular massless Nelson model. The proof follows the
lines of Haag-Ruelle scattering theory: Scattering state approximants are
defined with the help of two time-dependent renormalized creation operators of
the electrons acting on the vacuum. They depend on ground state wave functions
of the (single-electron) fiber Hamiltonians with infrared cut-off. Convergence
of these approximants as $t\to \infty$ is shown with the help of Cook's method
combined with a non-stationary phase argument. Removal of the infrared cut-off
in the limit $t\to \infty$ requires sharp estimates on the derivatives of these
ground state wave functions w.r.t. electron and photon momenta, with mild
dependence on the infrared cut-off. These key estimates, which carry
information about the localization of electrons in space, are obtained in a
companion paper with the help of iterative analytic perturbation theory. Our
results hold in the weak coupling regime.
