Field Theoretic Approach to Interacting Two Body Tunneling
Guo Ye
TL;DR
This work develops a field-theoretic framework for interacting two-body tunneling by coupling a tunneling field to a meson field via a Yukawa interaction and embedding the barrier through a delta-like profile. The authors formulate a Bethe-Salpeter equation from ladder resummations and analyze an instantaneous, positive-energy regime in $1+1$D, obtaining a closed-form solution in Laplace space via Wiener-Hopf methods. They validate the approach by deriving a Born approximation for the tunneling amplitude and showing that in the nonrelativistic limit the formalism reduces to a three-dimensional Lippmann-Schwinger equation with an effective two-body kernel, thereby bridging relativistic quantum field theory and conventional few-body scattering. The results highlight correlated tunneling effects, such as nonzero amplitudes at low momenta and distinct behavior for opposite-sign momenta, and establish a self-consistent framework for exploring nonperturbative two-body tunneling beyond mean-field approximations. This work thus provides a minimal, self-contained relativistic model that captures dynamical correlations, retardation, and virtual excitations in interacting tunneling systems, with potential relevance to cold-atom experiments and mesoscopic devices.
Abstract
Two body tunneling problems are hard to treat analytically due to the incompatibility between tunneling and perturbation theory. The lack of classical solutions of the Euclidean Lagrangian of continuous systems further thwarts semi-classical expansions. To develop an analytic theory which provides insight on interacting two-particle tunneling, we use new results to derive the Bethe-Salpeter equation of a tunneling field theory with Yukawa coupling. We show that in the one plus one dimensional case a closed form solution in the instantaneous positive-energy regime is permitted. We then compute the scattering amplitude by perturbing on interparticle interaction and recover the Lippmann-Schwinger equation to confirm physical consistency and relevancy.