Three-body dynamics of the Roper resonance

The enigmatic Roper resonance is notoriously difficult to determine in lattice QCD. Its quantum numbers, JP=1/2⁺, allow for two pions and a nucleon in relative S-waves. This absence of a centrifugal barrier leads to strong three-body $ππN$ effects on top of the two-body $πN$ channel -maybe the reason why the Roper is obscured in lattice QCD close to physical masses. Guided by unitarity, an amplitude is constructed in which three-body states are populated through channels in f₀(500)N(939)1/2⁺, πΔ(1232)3/2⁺, and πN(1440)1/2⁺ configurations. The latter channel is required by unitarity in the presence of nucleon exchange, and it contains the nucleon as a bound state, generating the $πN$ channel automatically. With the sub-channels matching the corresponding two-body amplitudes from experiment and nucleon properties, the P₁₁ amplitude from SAID is fitted, allowing for an accurate determination of the Roper poles close to the complex $πΔ$ threshold. The Roper poles and residues are, for the first time, determined by explicitly including this and other non-analyticities in combination with strict two- and three-body unitarity. Contour deformation allows for a direct pole search. Systematic uncertainties from the parametrization dependence are quantified. The result for the Roper S-matrix pole is √sR/ MeV =1346± 2⁺⁶₋₃ - i(81± 2⁺⁷₋₈). The amplitude developed here will serve as the central input for a quantization condition that allows for future determinations of the Roper resonance from lattice QCD.