  An upper limit to distillable entanglement between two disconnected regions
of massless non-interacting scalar field theory has an exponential decay
defined by a geometric decay constant. When regulated at short distances with a
spatial lattice, this entanglement abruptly vanishes beyond a dimensionless
separation, defining a negativity sphere. In two spatial dimensions, we
determine this geometric decay constant between a pair of disks and the growth
of the negativity sphere toward the continuum through a series of lattice
calculations. Making the connection to quantum field theories in three-spatial
dimensions, assuming such quantum information scales appear also in quantum
chromodynamics (QCD), a new relative scale may be present in effective field
theories describing the low-energy dynamics of nucleons and nuclei. We
highlight potential impacts of the distillable entanglement structure on
effective field theories, lattice QCD calculations and future quantum
simulations.
