Poles and Shadow Poles in the Many-ChannelSMatrix

The connection between the partial-wave $S$ matrix on different Riemann sheets is obtained from unitarity and analyticity. Under the assumption that coupling between channels can be varied analytically, it is shown that a resonance pole or bound-state pole may lead also to "shadow poles" on other Riemann sheets. The existence of shadow poles is illustrated by a unitary resonance model based on a sum of Feynman diagrams. In general, the number of shadow poles that can be deduced from an observed resonance depends on the number of channels that still have a particular resonance pole in the absence of coupling between channels. If the pole still appears in all channels, then shadow poles occur on every Riemann sheet; if it appears in only one channel, then shadow poles appear on half the sheets. If the resonance disappears in the absence of channel coupling, our method leads to no conclusions. In connection with the unitary symmetry scheme we note that the existence of shadow poles would permit a simple changeover from the separated poles of a resonance multiplet with broken symmetry to the coincident poles of the multiplet that must occur when the symmetry breaking interaction is switched off.