# Spin 1-2 Fermions in the Unitary Limit.III - Nuclear Theory

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Abstract: In scattering theory, the unitary limit is defined by an infinitescattering-length and a zero effective range, corresponding to a phase-shift\pi-2, independent of energy. This condition is satisfied by a rank-1 separablepotential Vk,k-=-vkvk- withv^{2}k=4\pi^{2}\Lambda^{2}-k^{2}^{-1-2}, \Lambda being the cut-off inmomentum space.Previous calculations using a Pauli-corrected ladder summationto calculate the energy of a zero temperature many body system of spin 1-2fermions with this interaction gave \xi=0.24 in units of kinetic energyindependent of density and with \Lambda->infinity. This value of \xi isappreciably smaller than the experimental and that obtained from othercalculations, most notably from Monte Carlo, which in principle would be themost reliable. Our previous work did however also show a strong dependence oneffective range r 0 with r 0=0 at unitarity. With an increase to r 0=1.0 theenergy varied from \xi~0.38 at k f=0.6 1-fm to ~0.45 at k f=1.8 1-fm which issomewhat closer to the Monte-Carlo results. These previous calculations arehere extended by including the effect of the previously neglected mean-fieldpropagation, the dispersion correction. This is repulsive and found to increasedrastically with decreasing effective range. It is large enough to suggest arevised value of \xi~0.4 <-> ~0.5 independent of r 0. Off-shell effects arealso investigated by introducing a rank-2 phase-shift equivalent separablepotential. Effects of 10% or more in energy could be demonstrated for r 0>0. Itis pointed out that a computational cut-off in momentum-space brings in anotherscale in the in otherwise scale-less unitary problem.

Author: ** H. S. Kohler**

Source: https://arxiv.org/