Time-Reversal Symmetry and Geometric Constraints on the Residue Phase of Pion Photoproduction via Δ(1232)

The electromagnetic coupling phase at the complex resonance pole is a fundamental property of nucleon excitations. However, its extraction from pion photoproduction data remains model-dependent, particularly for the Δ(1232) resonance, where modern multichannel analyses report helicity amplitude phases ranging from +3∘ to −18∘. In this Letter, we present a largely model-independent geometric S-matrix formalism that provides a physical constraint for this ambiguity. By imposing Watson’s final-state interaction theorem, a direct consequence of S-matrix unitarity and time-reversal symmetry, on the real energy axis and performing an analytic continuation, we isolate the kinematic threshold barriers of the photoproduction (k·q) and elastic (q3) amplitudes. For the dominant M1+ multipole transition of the Δ(1232), our method yields a kinematically constrained prediction of ϕEM=−8.5∘−1.0∘+0.6∘ for the electromagnetic residue phase. This geometric constraint explains the numerical results of coupled-channel phenomenological fits, validating the extractions by the SAID and Bonn–Gatchina groups, and establishes a theoretical benchmark for evaluating resonance properties.