Real-Time Instanton Dynamics in a Triple-Well to Double-Well Homotopy
We show that the singularities which the analytically continued double-well instanton develops in real (Minkowski) time are not an inherent feature of real-time tunneling solutions. Rather, they can be viewed as arising from a singular limit of a one-parameter family of regular real-time solutions. Using a linear homotopy between a triple-well and a double-well potential, we construct a family of potentials, parametrized by $p\in(0,1)$, whose real-time ($α=0$) instanton solutions are everywhere regular and bounded. For any Wick rotation angle $α\in(0,π/2)$, we prove that regularity is generic up to a measure-zero set. There exists a countably infinite, closed-form family of potential parameters $\{p^{\mathrm{singular}}_{k}(α)\}_{k\geq0}$ at which the instanton becomes singular at exactly two points of the rotated time axis, never more. In the strict double-well limit, $p\to 1$ taken at $α=0$, the entire family collapses onto the single point $(p,α) = (1,0)$ at which the singular pairs of all $k$ occur simultaneously. This phenomenon produces the infinite singular comb found by Cherman and Ünsal. By solving the complexified equations of motion in terms of Weierstrass elliptic functions, we trace this comb to classical imaginary turning points receding to $\pm\mathrm{i}\infty$, giving a unified geometric and algebraic account of when, and exactly how often, real-time instantons become singular. We further show that the action of the continued instanton is independent of $α$ only in its real part: its imaginary part jumps by one real-time period action at each singular angle, the $α$-independence found by Cherman and Ünsal being the degenerate case of the double well, in which no such jump occurs.