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Neutron-neutron scattering length from $π^+$ photoproduction on the deuteron

We discuss the possibility of extracting the neutron-neutron scattering length $a_{nn}$ and effective range $r_{nn}$ from cross section data ($d^2σ/dM_{nn}/dΩ_π$), as a function of the $nn$ invariant mass $M_{nn}$, for $π^+$ photoproduction on the deuteron ($γd\to π^+nn$). The analysis is based on a $γd\to π^+nn$ reaction model in which realistic elementary amplitudes for $γp\to π^+n$, $NN\to NN$, and $πN\to πN$ are built in. We show that $M_{nn}$ dependence (lineshape) of a ratio $R_{\rm th}$, $d^2σ/dM_{nn}/dΩ_π$ normalized by $dσ/dΩ_π$ for $γp\toπ^+ n$ and the nucleon momentum distribution inside the deuteron, at the kinematics with $θ_π=0^\circ$ and $E_γ\sim 250$ MeV is particularly useful for extracting $a_{nn}$ and $r_{nn}$ from the corresponding data $R_{\rm exp}$. It is found that $R_{\rm exp}$ with 2% error, resolved into the $M_{nn}$ bin width of 0.04 MeV (corresponding to the $p_π$ bin width of 0.05 MeV$/c$), can determine $a_{nn}$ and $r_{nn}$ with uncertainties of $\pm 0.21$ fm and $\pm 0.06$ fm, respectively, for the case of $a_{nn}=-18.9$ fm and $r_{nn}=2.75$ fm. The requirement of such narrow bin widths indicates that the momenta of the incident photon and the emitted $π^+$ have to be measured with high resolutions. This can be achieved by utilizing virtual photons of very small $Q^2$ from electron scattering at Mainz MAMI facility. The proposed method for determining $a_{nn}$ and $r_{nn}$ from $γd\to π^+ nn$ has a great experimental advantage over the previous one utilizing $π^- d\toγnn$ for being free from the formidable task of controlling the neutron detection efficiency and its uncertainty.