Real-Time Instanton Dynamics in a Triple-Well to Double-Well Homotopy
We show that the real-time singularities found in the analytically continued double-well instanton are not an inherent feature of Minkowski tunneling, but a degenerate limiting phenomenon. 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 but not universal. There exists a countably infinite, closed-form family of potential parameters $\{p_{s,k}(α)\}_{k\geq0}$ at which the instanton develops exactly two real-time singularities, never more. In the strict double-well limit ($p\to1$) taken at $α=0$, these collapse into 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.