The reaction $^2H(\gamma,\pi^0)np$ in the threshold region

Neutral pion photoproduction on the deuteron in the inelastic channel is studied in the threshold region from 142 to 160 MeV. The calculation is based on the use of the diagrammatic approach. Contributions from the pole diagrams, one-loop diagrams both with $n$-$p$ and $\pi$-$N$ rescattering as well as two-loop diagrams with simultaneous inclusion of these rescattering mechanisms are taken into account. We have found a reasonable agreement with a very simple calculation by J.C. Bergstrom et al. [Phys. Rev. C 57 (1998) 3203]. It is shown that the `effective' electric dipole deuteron amplitude $E_d$ is negative in agreement with a prediction of ChPT. We conclude that a deviation of about 20% from a ChPT prediction for $E_d$ discovered in an experiment on coherent $\pi^0$ photoproduction off the deuteron cannot be attributed to inadequate estimates of inelastic channel contributions. We suggest that $\pi$-$N$ rescattering is responsible for the deviation of the free-nucleon $p_1$ threshold amplitude from the value measured in that experiment