Abstract
We discuss properties of three-particle Dalitz distributions in coupled channel systems in presence of triangle singularities. The single channel case was discussed long ago where it was found that as a consequence of unitarity, effects of a triangle singularity seen in the Dalitz plot are not seen in Dalitz plot projections. In the coupled channel case we find the same is true for the sum of intensities of all interacting channels. Unlike the single channel case, however, triangle singularities do remain visible in Dalitz plot projections of individual channels.
Highlights
Under specific kinematic conditions [2], triangle diagrams [3] have singularities that can mimic resonance poles
The triangle diagram can be expressed through a dispersive integral in which the on-shell amplitude describing t-channel exchange of a particle of mass λ is projected onto the s-channel partial wave and unitarized
For a range of λ2, determined by the Coleman-Norton condition [2], one of these branch points, sT is located infinitesimally below the real s-axis and above the s-channel threshold, sβ. This leads to a logarithmic branch point in the dispersive integral located on the second sheet just below the physical region
Summary
Under specific kinematic conditions [2], triangle diagrams [3] have singularities that can mimic resonance poles. The triangle diagram can be expressed through a dispersive integral in which the on-shell amplitude describing t-channel exchange of a particle of mass λ is projected onto the s-channel partial wave and unitarized. For example in the analysis of the a1(1420) [12] the t-channel exchange of a stable kaon connects the f0(980)π and K∗K , aka KKπ three-particle states In this cases it is necessary to invoke three-body unitary to constrain the triangle amplitude. As discussed below Eq (1), cross channel exchanges are physical in the s-channel and lead to additional (beyond the one determined by s-channel unitarity) complexity of the s-channel partial waves. Instead it is replaced by the relation for the discontinuity given in Eq (6) [22]
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