Measurement of the beam-helicity asymmetry $I^{\odot}$ in the photoproduction of $\pi^0\pi^{\pm}$-pairs off protons and off neutrons

Beam-helicity asymmetries have been measured at the MAMI accelerator in Mainz for the photoproduction of mixed-charge pion pairs in the reactions $\boldsymbol{\gamma}p\rightarrow n\pi^0\pi^+$ off free protons and $\boldsymbol{\gamma}d\rightarrow (p)p\pi^0\pi^-$ and $\boldsymbol{\gamma}d\rightarrow (n)n\pi^0\pi^+$ off quasi-free nucleons bound in the deuteron for incident photon energies up to 1.4 GeV. Circularly polarized photons were produced from bremsstrahlung of longitudinally polarized electrons and tagged with the Glasgow-Mainz magnetic spectrometer. The charged pions, recoil protons, recoil neutrons, and decay photons from $\pi^0$ mesons were detected in the 4$\pi$ electromagnetic calorimeter composed of the Crystal Ball and TAPS detectors. Using a complete kinematic reconstruction of the final state, excellent agreement was found between the results for free and quasi-free protons, suggesting that the quasi-free neutron results are also a close approximation of the free-neutron asymmetries. A comparison of the results to the predictions of the Two-Pion-MAID reaction model shows that the reaction mechanisms are still not well understood, in particular at low incident photon energies in the second nucleon-resonance region.