  We use the worldline formalism to derive integral representations for three
classes of amplitudes in scalar field theory: (i) the scalar propagator
exchanging N momenta with a scalar background field (ii) the "half-ladder" with
N rungs in x - space (iii) the four-point ladder with N rungs in x - space as
well as in (off-shell) momentum space. In each case we give a compact
expression combining the N! Feynman diagrams contributing to the amplitude. As
our main application, we reconsider the well-known case of two massive scalars
interacting through the exchange of a massless scalar. Applying asymptotic
estimates and a saddle-point approximation to the N-rung ladder plus crossed
ladder diagrams, we derive a semi-analytic approximation formula for the lowest
bound state mass in this model.
