Microscopic Characterization of Nuclear Collectivity and Branching Ratios from Jost-RPA Pole Residues
Within the continuum Random Phase Approximation formulated via the Jost function method (Jost-RPA), we formulate a microscopic method to (i) evaluate nuclear collectivity through a dimensionless metric normalized to unity for a pure single-channel excitation, and (ii) extract partial decay widths and branching ratios directly from complex channel amplitudes without artificial width broadening. By applying rank-1 Takagi factorization to the complex symmetric scattering matrix residue at resonance poles across multiple Riemann sheets, the transition strength residue is algebraically partitioned into the single-particle radial overlap scale, multi-channel participation coherence, and operator phase alignment. To validate the effectiveness and physical consistency of this framework, benchmark calculations are performed for $^{16}$O across diverse continuum excitation modes: (i) for the isovector giant dipole resonance, the metric identifies a strongly multi-channel collective response decaying concurrently through multiple open channels; (ii) for the isoscalar quadrupole channel, the metric characterizes the strong collectivity of the low-lying resonance concentrating the low-energy transition strength, while higher core-excited states exhibit suppressed direct excitation and destructive continuum interference; and (iii) for the isovector quadrupole channel, the metric clearly reveals that the most prominent strength peak is a non-collective quasi-single-particle resonance trapped behind a centrifugal barrier, whereas collective IVGQR components are pushed up and fragmented across higher-energy Riemann sheets. This work establishes a transparent diagnostic tool for characterizing continuum resonances and decay branching, providing a practical framework for future applications to heavier nuclei where collective phenomena fully develop.