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

Background: We consider the muon capture reaction ${\mu }^{-}+{}^2\mathrm{H}\to {\nu }_{\mu }+n+n$, which presents a “clean” two-neutron $\left(nn\right)$ system in the final state. We study here its capture rate in the doublet hyperfine initial state $\left({\Gamma }^D\right)$. The total capture rate for the muon capture ${\mu }^{-}+{}^3\mathrm{He}\to {\nu }_{\mu }+{}^3\mathrm{H}$ $\left({\Gamma }_0\right)$ 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 $\left(a_{nn}\right)$, 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 $\left(c_D\right)$, 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 $399\left(3\right){\mathrm{s}}^{-1}$ for $a_{nn}=-18.95$ and $-16.0$ fm; and $400\left(3\right){\mathrm{s}}^{-1}$ for $a_{nn}=-22.0$ fm. However, in the case of $a_{nn}=+18.22$ fm, the result of $275\left(3\right){\mathrm{s}}^{-1}$ $\mathrm{[}135\left(3\right){\mathrm{s}}^{-1}]$ is obtained, when the di-neutron system in the final state is unbound (bound). The total capture rate ${\Gamma }_0$ for muon capture on ${}^3\mathrm{He}$ is found to be 1494(15), 1491(16), 1488(18), and 1475(16) ${\mathrm{s}}^{-1}$ 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 $\sim \pm 3$ 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 $a_{nn})$.

Figure 1

$c_D\text{−}{c}_E$ trajectories fitted to reproduce the experimental ${}^3\mathrm{H}$ and ${}^3\mathrm{He}$ binding energies for the N3LO18, N3LO16, N3LO22, and $\mathrm{N}3\mathrm{LO}18+$ $\mathit{NN}$ potentials, augmented by the N2LO $\mathit{NNN}$ interaction model. Note that the curves that can be clearly distinguished in this figure are the ${}^3\mathrm{H}$ trajectories of the respective potentials. The corresponding ${}^3\mathrm{He}$ trajectories cannot be distinguished and are essentially identical to the N3LO18 curve. See text for further explanation.

Figure 2

$\mathrm{GT}{}^{\mathrm{TH}}/{\mathrm{GT}}^{\mathrm{EXP}}$ as a function of the LEC $c_D$ for the N3LO18, N3LO16, N3LO22, and $\mathrm{N}3\mathrm{LO}18+$ $\mathit{NN}$ potentials and the N2LO $\mathit{NNN}$ interaction model. The N3LO18 and N3LO22 lines are essentially identical and very close to the N3LO16 one.