Logarithmic Singularities in Processes with Two Final-State Interactions

The effects of logarithmic singularities in rescattering processes are investigated. The reaction $\pi N\to \pi \pi N$ is considered, but treated purely as an $S$-wave, spinless model. A particular triangle graph is analyzed in detail; it contains as an intermediate state the (3,3) nucleon isobar $I$, which is described as a spinless particle of complex mass. The graph is calculated from a dispersion relation as a function of the mass $s$ of the two pions in the final state, for low values of the over-all c.m. system energy $W$. The relation is then analytically continued in $W$. For a narrow range in $W$, an enhancement of the square of the amplitude is found near $s=4\mathrm{}$ (the pion mass is unity). The analogous enhancement also appears in the $W$ channel near $W=I+1\mathrm{}$, for a small range of $s$ only, near $s=4\mathrm{}$. The prominence of the effect depends on the width of $I$, being closely connected with the nearness to the physical region of one of the two logarithmic singularities (anomalous thresholds) of the graph: this distance increases sharply with the isobar width. The positions of the singularities are interpreted as the phase-space limits for the simultaneous production of states with mass $s$ and $I$. The conclusion is that such a "double excitation" process leads to an enhancement of the triangle amplitude only if, in general, $s$ and $I$ fall in certain narrow ranges. The implications of this result for models of the higher resonances in the elastic channel ($\pi N\to \pi N$) is briefly discussed.