Neutron-neutron scattering length from $\pi^+$ 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 ($d^2\sigma/dM_{nn}/d\Omega_\pi$), as a function of the $nn$ invariant mass $M_{nn}$, for $\pi^+$ photoproduction on the deuteron ($\gamma d\to \pi^+nn$). The analysis is based on a $\gamma d\to \pi^+nn$ reaction model in which realistic elementary amplitudes for $\gamma p\to \pi^+n$, $NN\to NN$, and $\pi N\to \pi N$ are built in. We show that $M_{nn}$ dependence (lineshape) of a ratio $R_{\rm th}$, $d^2\sigma/dM_{nn}/d\Omega_\pi$ normalized by $d\sigma/d\Omega_\pi$ for $\gamma p\to\pi^+ n$ and the nucleon momentum distribution inside the deuteron, at the kinematics with $\theta_\pi=0^\circ$ and $E_\gamma\sim 250$ 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 $p_\pi$ 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 $\pi^+$ 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 $\gamma d\to \pi^+ nn$ has a great experimental advantage over the previous one utilizing $\pi^- d\to\gamma nn$ for being free from the formidable task of controlling the neutron detection efficiency and its uncertainty.