  Nonperturbative polaron variational methods are applied, within the so-called
particle or worldline representation of relativistic field theory, to study
scattering in the context of the scalar Wick - Cutkosky model. Important
features of the variational calculation are that it is a controlled
approximation scheme valid for arbitrary coupling strengths, the Green
functions have all the cuts and poles expected for the exact result at any
order in perturbation theory and that the variational parameters are
simultaneously sensitive to the infrared as well as the ultraviolet behaviour
of the theory. We generalize the previously used quadratic trial action by
allowing more freedom for off-shell propagation without a change in the
on-shell variational equations and evaluate the scattering amplitude at first
order in the variational scheme. Particular attention is paid to the
$s$-channel scattering near threshold because here non-perturbative effects can
be large. We check the unitarity of a our numerical calculation and find it
greatly improved compared to perturbation theory and to the zeroth order
variational results.

Nuclear Theory;High Energy Physics - Theory