Amplitude structure of 3→ 3 scattering in a Mandelstam variable representation
Abstract
We construct a dispersive representation of the relativistic 3→ 3 scattering amplitude for three identical spinless particles in the S-wave. The two-particle subenergy, instead of the total energy, is used as the dispersive variable. This choice keeps the physical dispersive contour free of the kinematical cuts that complicate total-energy dispersion relations. By separating discontinuities across the two-particle subenergy cuts from that across the three-body cut, we derive a linear integral equation with one-particle exchange as the driving term. We further show that the solution satisfies three-body unitarity as a consequence of two-body unitarity, analyticity, and crossing symmetry. For pair-wise interactions, this representation can be rewritten into the form used for isobar-spectator scattering. Finally, we give prescriptions for contour deformation and the subtraction of poles in the two-body subsystem amplitudes, which continue the amplitude onto adjacent unphysical Riemann sheets and thus provide direct access to the analytic structure relevant for three-body resonance poles.
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BibTeX
@misc{zhang2026amplitude,
title = {{Amplitude structure of $3\to 3$ scattering in a Mandelstam variable representation}},
author = {Xu Zhang and Feng-Kun Guo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00985},
eprint = {2609.00985},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00985},
}
RIS
TY - UNPB
TI - Amplitude structure of 3→ 3 scattering in a Mandelstam variable representation
AU - Xu Zhang
AU - Feng-Kun Guo
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00985
UR - https://arxiv.org/abs/2609.00985
AB - We construct a dispersive representation of the relativistic 3→ 3 scattering amplitude for three identical spinless particles in the S-wave. The two-particle subenergy, instead of the total energy, is used as the dispersive variable. This choice keeps the physical dispersive contour free of the kinematical cuts that complicate total-energy dispersion relations. By separating discontinuities across the two-particle subenergy cuts from that across the three-body cut, we derive a linear integral equation with one-particle exchange as the driving term. We further show that the solution satisfies three-body unitarity as a consequence of two-body unitarity, analyticity, and crossing symmetry. For pair-wise interactions, this representation can be rewritten into the form used for isobar-spectator scattering. Finally, we give prescriptions for contour deformation and the subtraction of poles in the two-body subsystem amplitudes, which continue the amplitude onto adjacent unphysical Riemann sheets and thus provide direct access to the analytic structure relevant for three-body resonance poles.
ER -