  The neutron-rich $^{11}$Li halo nucleus is unique among nuclei with known
separation energies by its ability to emit a proton and a neutron in a $\beta$
decay process. The branching ratio towards this rare decay mode is evaluated
within a three-body model for the initial bound state and with Coulomb
three-body final scattering states. The branching ratio should be comprised
between two extreme cases, i.e.\ a lower bound $6 \times 10^{-12}$ obtained
with a pure Coulomb wave and an upper bound $5 \times 10^{-10}$ obtained with a
plane wave. A simple model with modified Coulomb waves provides plausible
values between between $0.8 \times 10^{-10}$ and $2.2 \times 10^{-10}$ with
most probable total energies of the proton and neutron between 0.15 and 0.3
MeV.

Nuclear Theory;Nuclear Experiment