Final-state interactions in muon capture by deuterons

Abstract

Prompted in part by the disparate theoretical estimates in the literature, we present a systematic study of the influence of the $^{1}S_{0}n\ensuremath{-}n$ final-state interaction on rates for the reaction ${\ensuremath{\mu}}^{\ensuremath{-}}+d\ensuremath{\rightarrow}n+n+{\ensuremath{\nu}}_{\ensuremath{\mu}}$, treated in impulse approximation with the standard Primakoff effective weak-interaction Hamiltonian. Instead of beginning with a particular $n\ensuremath{-}n$ potential, we use as our basic input a set of four-parameter Bargmann models of the $^{1}S_{0}n\ensuremath{-}n$ phase shift. By solving a "restricted inverse problem" we generate the necessary radial functions from these phase shifts. This procedure enables us to examine in a consistent way the sensitivity of the total capture rates and neutron energy spectrum to uncertainties in the phase shifts. We find that changes in the phase shifts at either high or low energies do not affect the rates noticeably. Hence, we conclude that this reaction is not a good tool for determining the $n\ensuremath{-}n$ scattering length. In addition, we investigate the effects of varying the off-shell behavior of the $n\ensuremath{-}n$ interaction by means of the method of short-range unitary transforms of Coester et al. We then find that the rates change dramatically. The doublet capture rate, for example, can be shifted from the 375 ${\mathrm{sec}}^{\ensuremath{-}1}$ typical of the local potentials of the restricted inverse problem down to 236 ${\mathrm{sec}}^{\ensuremath{-}1}$. More accurate measurements of this rate could thus serve as constraints on the off-shell extrapolation of the $n\ensuremath{-}n$ interaction. Our results are in general agreement with those of Sotona and Truhl\'{\i}k.NUCLEAR REACTIONS ${\ensuremath{\mu}}^{\ensuremath{-}}+d\ensuremath{\rightarrow}n+n+{\ensuremath{\nu}}_{\ensuremath{\mu}}$; effect of $^{1}S_{0}n\ensuremath{-}n$ final-state interactions on rates, neutron spectra. Varied $n\ensuremath{-}n$ phase shifts and off-shell behavior.