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 ($d^2\sigma \mathrm{/}d{M}_{nn}\mathrm{/}d{\mathrm{\Omega }}_{\pi }$), 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 $\pi N\to \pi N$arebuiltin\mathrm{.}Weshowthat$M_{nn}$ dependence (lineshape) of a ratio $R_{\mathrm{t}\mathrm{h}}$, $d^2\sigma \mathrm{/}d{M}_{nn}\mathrm{/}d{\mathrm{\Omega }}_{\pi }$ normalized by $d\sigma \mathrm{/}d{\mathrm{\Omega }}_{\pi }$ 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 $R_{\mathrm{e}\mathrm{x}\mathrm{p}}$. It is found that $R_{\rm exp}$with2$M_{nn}$ bin width of 0.04 MeV (corresponding to the $p_{\pi }$ bin width of 0.05 MeV$\mathrm{/}c$), 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.