  The electronic dispersions of the quasi-one-dimensional organic conductor
TTF-TCNQ are studied by angle-resolved photoelectron spectroscopy (ARPES) with
higher angular resolution and accordingly smaller step width than in previous
studies. Our experimental results suggest that a refinement of the single-band
1D Hubbard model that includes finite-range interactions is needed to explain
these photoemission data. To account for the effects of these finite-range
interactions we employ a mobile quantum impurity scheme that describes the
scattering of fractionalized particles at energies above the standard
Tomonaga-Luttinger liquid limit. Our theoretical predictions agree
quantitatively with the location in the $(k,\omega)$ plane of the
experimentally observed ARPES structures at these higher energies. The
nonperturbative microscopic mechanisms that control the spectral properties are
found to simplify in terms of the exotic scattering of the charge
fractionalized particles. We find that the scattering occurs in the unitary
limit of (minus) infinite scattering length, which limit occurs within
neutron-neutron interactions in shells of neutron stars and in the scattering
of ultracold atoms but not in perturbative electronic condensed-matter systems.
Our results provide important physical information on the exotic processes
involved in the finite-range electron interactions that control the high-energy
spectral properties of TTF-TCNQ. Our results also apply to a wider class of 1D
and quasi-1D materials and systems that are of theoretical and potential
technological interest.
