Using chiral perturbation theory to extract the neutron-neutron scattering length from π^-d→nnγ

The reaction π^-d→nnγ is calculated in chiral perturbation theory so as to facilitate an extraction of the neutron-neutron scattering length (a_nn). We include all diagrams up to O(Q³). This includes loop effects in the elementary π^-p→γn amplitude and two-body diagrams, both of which were ignored in previous calculations. We find that the chiral expansion for the ratio of the quasi-free (QF) to final-state-interaction (FSI) peaks in the final-state neutron spectrum converges well. Our third-order calculation of the full spectrum is already accurate to better than 5%. Extracting a_nn from the shape of the entire π^-d→nnγ spectrum by using our calculation in its present stage would thus be possible at the ±0.8 fm level. A fit to the FSI peak only would allow an extraction of a_nn with a theoretical uncertainty of ±0.2 fm. The effects that contribute to these error bars are investigated. The uncertainty in the nn rescattering wave function dominates. This suggests that the quoted theoretical error of ±0.3 fm for the most recent π^-d→nnγ measurement may be optimistic. The possibility of constraining the nn rescattering wave function used in our calculation more tightly—and thus reducing the error—is briefly discussed.