Coupled Instantons In A Four-Well Potential With Application To The Tunneling Of A Composite Particle

Coupled instantons are introduced by generalizing the double well potential to multiple mutually coupled wells. Physically this corresponds to the simultaneous tunneling of multiple degrees of freedom. A system with four equal minima is examined in detail. It has three instanton types or flavors with distinct actions. For weak coupling and subject to there being a single large (or small) parameter, the interactive system can be handled perturbatively. The zero mode problem arising from time translation symmetry is handled via the Fadeev-Popov procedure. A diagrammatic procedure allows corrections to the fluctuation determinant to be calculated systematically. Independent instanton contributions are summed over by extending the dilute gas approximation to three flavors and energy splittings of the lowest four states is calculated. All tunneling amplitudes are concisely expressed in terms of elementary functions. While the model is possibly useful for a variety of physical systems, an application is made here to the tunneling of a composite particle in one dimension.