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

Abstract

Background: We consider the muon capture reaction μ-+H2→νμ+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 (ΓD). The total capture rate for the muon capture μ-+He3→νμ+H3 (Γ0) is also analyzed, although, in this case, the nn system is not so clean anymore. Purpose: We investigate whether ΓD (and Γ0) could be sensitive to the nn S-wave scattering length (ann), and we check on the possibility to extract ann from an accurate measurement of ΓD. 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 Λ=500 MeV is used, but the low-energy constant (LEC) determining ann is varied so as to obtain ann=-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 cD and cE, present in the three-nucleon potential and axial-vector current (cD), are constrained to reproduce the A=3 binding energies and the triton Gamow-Teller matrix element. Results: The capture rate ΓD is found to be 399(3)s-1 for ann=-18.95 and -16.0 fm; and 400(3)s-1 for ann=-22.0 fm. However, in the case of ann=+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 Γ0 for muon capture on He3 is found to be 1494(15), 1491(16), 1488(18), and 1475(16) s-1 for ann=-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 ann with an uncertainty of less than ∼±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 ΓD. Finally, a comparison with the already available experimental data discourages the possibility of a bound di-neutron state (positive ann).