Abstract
It is shown that the singularities of the triangle or rescattering diagram cannot produce peaks in the total transition rate. Rescattering bands can be seen on the Dalitz plot, but their projections into invariant-mass plots do not produce peaks, if all events on the Dalitz plot are included. In other words, the Peierls mechanism and its recent variants do not work, even if the singularity is on the physical boundary. The reason is a cancellation of the peak, which occurs when the diagram without rescattering is added coherently to the triangle diagram. It is also shown that interference of overlapping resonance bands does not produce the previously expected peaks in the total transition rate or in the two-body mass plots in three different cases. This result holds to first order in the ratio of the resonance width to the diameter of the Dalitz plot.
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