Integral representations combining ladders and crossed-ladders

TL;DR

This paper derives integral representations for scalar field theory amplitudes using the worldline formalism, and applies saddle-point methods to estimate the lowest bound state mass in a two-scalar interaction model.

Contribution

It introduces compact integral representations for ladder and crossed-ladder diagrams in scalar field theory using the worldline formalism, enabling semi-analytic bound state estimates.

Findings

Derived integral representations for scalar propagators and ladder diagrams

Provided a semi-analytic approximation for the lowest bound state mass

Applied asymptotic and saddle-point methods to complex diagrams

Abstract

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.