Behavior of partial cross sections and branching ratios in the neighborhood of a resonance

Starting from the treatment of Fano for the behavior of the total cross section in a photoionization (or electron-ion scattering) experiment in the vicinity of a resonance, we present a theoretical formula for the behavior of an individual final-state channel in the neighborhood of a resonance. This result is then used to derive another theoretical formula for the behavior of the ratio of two partial cross sections (i.e., the branching ratio) in the vicinity of a resonance. This branching-ratio formula depends on the profile parameters $q$, $\Gamma$, and ${\rho }^2$ for the resonance, on the branching ratio outside the resonance, and on two new parameters which are explicitly related to scattering-matrix elements and phase shifts.