Localized scalar modes of $O(3)$ critical bubbles: partial-wave continuum mergers and wave-function deformation

Abstract

At phase coexistence, a degenerate quartic scalar potential admits an exact planar kink whose normal fluctuation operator is the modified P\"oschl--Teller operator, with translational and positive shape states below a continuum beginning at $\Lambda=4$. We continue the two connected spectral bands through finite supercooling in a smooth one-component quartic benchmark and resolve $\ell=0,1,2$. The $O(3)$ bounce is obtained by singular collocation, while the radial Euclidean Hessian is analyzed by finite-difference diagonalization and independent threshold shooting. The positive $\ell=2$ and $\ell=1$ branches reach the common false-vacuum continuum threshold at $\delta_{{\rm merge},2}=0.0900472$ and $\delta_{{\rm merge},1}=0.1410162$, respectively. At each endpoint, $u_\ell\propto\rho^{-\ell}$ is square integrable and defines a threshold eigenstate; beyond the endpoint, however, no normalizable eigenstate continuation exists for the corresponding branch. The $\ell=0$ shape state remains bound up to the geometric wall crossover. Its planar-mode overlap, radial centroid, and distinct interior and exterior decay lengths reveal asymmetric wave-function deformation. Thus, angular spectral dissolution and the later geometric loss of a true-vacuum-like core are separate phenomena, revealing two distinct finite-supercooling fates of the planar shape mode.