Integral representations combining ladders and crossed-ladders

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.

39 pages, 10 pdf figures

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• Worldline master formulas for the dressed electron propagator, part 2: On-shell amplitudes
• High-Energy Gravitational Scattering and Bose-Einstein Condensates of Gravitons
• Propagator from Nonperturbative Worldline Dynamics
• The Generalised LKF Transformations for Arbitrary $N$-point Fermion Correlators
• Non-perturbative Quantum Propagators in Bounded Spaces