Exact Semiclassical Dynamics in a Multidimensional Quartic Potential: Multi-Flavor Instantons and Pre-Factor Dominance at Strong Coupling

Abstract

We present an exact semi-classical analysis of a coupled multi-degree-of-freedom quantum system governed by a symmetric quartic potential with four degenerate minima. Starting from the Euclidean path integral, we identify distinct paths that are longitudinal, transverse, and diagonal, each corresponding to distinct instanton configurations that mediate tunneling between vacua. The continuous translational zero mode for each coupled trajectory is extracted rigorously by transforming the fluctuation operator into a comoving rotating frame, explicitly incorporating curvature-induced fictitious forces. By evaluating the exact functional determinants via the Gelfand-Yaglom method and mapping the multi-flavor dilute instanton gas onto a weighted $K_4$ adjacency matrix, we analytically derive the transition amplitudes and ground-state energy splittings. Crucially, our exact continuum treatment reveals that for strong coupling, the usual exponential can be overpowered in certain parameter ranges by the "pre-factor". This quantity, which arises from quadratic level fluctuations in the direction transverse to the free direction, turns out to have an essential singularity that would be hard to spot in standard discretized instanton approximations.

41 pages, 10 figures, This manuscript supersedes and substantially expands upon the earlier, withdrawn preprint arXiv:2504.15607, featuring a fully revised analytic treatment of the functional determinant and multi-flavor dilute gas summation