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The first full amplitude analysis of $B^+ \to \psi(2S) K^+ \pi^+ \pi^-$decays is performed using proton-proton collision data corresponding to anintegrated luminosity of $9\,\text{fb}^{-1}$ recorded with the LHCb detector.The rich $K^+ \pi^+ \pi^-$ spectrum is studied and the branching fractions ofthe resonant substructure associated with the prominent $K_1(1270)^+$contribution are measured. The data cannot be described by conventional strangeand charmonium resonances only. An amplitude model with 53 components isdeveloped comprising 11 hidden-charm exotic hadrons. New production mechanismsfor charged charmonium-like states are observed. Significant resonant activitywith spin-parity $J^P = 1^+$ in the $\psi(2S) \pi^+$ system is confirmed and amulti-pole structure is demonstrated. The spectral decomposition of the$\psi(2S) \pi^+ \pi^-$ invariant-mass structure, dominated by $X^0 \to \psi(2S)\rho(770)^0$ decays, broadly resembles the $J/\psi \phi$ spectrum observed in$B^+ \to J/\psi \phi K^+$ decays. Exotic $\psi(2S) K^+ \pi^-$ resonances areobserved for the first time.

Amplitude analysis of $B^+ \to \psi(2S) K^+ \pi^+ \pi^-$ decays