Muon capture on the deuteron and the neutron-neutron scattering length

Background: We consider the muon capture reaction mu(-) + H-2 -> nu(mu) + n + n, which presents a "clean" two-neutron (nn) system in the final state. We study here its capture rate in the doublet hyperfine initial state (Gamma(D)). The total capture rate for the muon capture mu(-) + He-3 -> nu(mu) + H-3 (Gamma(0)) is also analyzed, although, in this case, the nn system is not so clean anymore. Purpose: We investigate whether Gamma(D) (and Gamma(0)) could be sensitive to the nn S-wave scattering length (a(nn)), and we check on the possibility to extract a(nn) from an accurate measurement of Gamma(D). Method: The muon capture reactions are studied with nuclear potentials and charge-changing weak currents, derived within chiral effective field theory. The next-to-next-to-next-to-leading-order chiral potential with cutoff parameter Lambda = 500 MeV is used, but the low-energy constant (LEC) determining a(nn) is varied so as to obtain a(nn) = -18.95, -16.0, -22.0, and +18.22 fm. The first value is the present empirical one, while the last one is chosen such as to lead to a di-neutron bound system with a binding energy of 139 keV. The LEC's c(D) and c(E), present in the three-nucleon potential and axial-vector current (c(D)), are constrained to reproduce the A = 3 binding energies and the triton Gamow-Teller matrix element. Results: The capture rate Gamma(D) is found to be 399(3) s(-1) for a(nn) = -18.95 and -16.0 fm; and 400(3) s(-1) for a(nn) = -22.0 fm. However, in the case of a(nn) = +18.22 fm, the result of 275(3) s(-1) [135(3) s(-1)] is obtained, when the di-neutron system in the final state is unbound (bound). The total capture rate Gamma(0) for muon capture on He-3 is found to be 1494(15), 1491(16), 1488(18), and 1475(16) s(-1) for a(nn) = -18.95, -16.0, -22.0, and +18.22 fm, respectively. All the theoretical uncertainties are due to the fitting procedure and radiative corrections. Conclusions: Our results seem to exclude the possibility of constraining a negative a(nn) with an uncertainty of less than similar to +/- 3 fm through an accurate determination of the muon capture rates, but the uncertainty on the present empirical value will not complicate the interpretation of the (forthcoming) experimental results for Gamma(D). Finally, a comparison with the already available experimental data discourages the possibility of a bound di-neutron state (positive a(nn)).