Theoretical study of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>γ</mml:mi><mml:mi>d</mml:mi><mml:mo>→</mml:mo><mml:msup><mml:mi>π</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mi>η</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:math> reaction

We have done a theoretical study of the $\gamma d \to \pi^0 \eta d$ reaction starting with a realistic model for the $\gamma N \to \pi^0 \eta N$ reaction that reproduces cross sections and polarization observables at low energies and involves the $\gamma N \to \Delta(1700)\to \eta \Delta(1232) \to \eta \pi^0 N$ process. For the coherent reaction in the deuteron we considered the impulse approximation together with the rescattering of the pions and the $\eta$ on a different nucleon than the one where they are produced. We found this second mechanism very important since it helps share between two nucleons the otherwise large momentum transfer of the reaction. Other contributions to the $\gamma d\to\pi^0\eta d$ reaction, involving the $\gamma N\to \pi^\pm\pi^0 N^\prime$ process, followed by the rescattering of the $\pi^\pm$ with another nucleon to give $\eta$ and a nucleon, have also been included. We find a natural explanation, tied to the dynamics of our model, for the shift of the $\eta-d$ mass distribution to lower invariant masses, and of the $\pi^0-d$ mass distribution to larger invariant masses, compared to a phase space calculation. We also study theoretical uncertainties related to the large momenta of the deuteron wave function involved in the process as well as to the couplings present in the model. Striking differences are found with the experimental angular distribution and further theoretical investigations might be necessary.