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Register a new userEfimov physics with $1/2$ spin-isospin fermions
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The structure of few-fermion systems having $1/2$ spin-isospin symmetry is studied using potential models. The strength and range of the two-body potentials are fixed to describe low energy observables in the angular momentum $L=0$ state and spin $S=0,1$ channels of the two-body system. Successively the strength of the potentials are varied in order to explore energy regions in which the two-body scattering lengths are close to the unitary limit. This study is motivated by the fact that in the nuclear system the singlet and triplet scattering lengths are both large with respect to the range of the interaction. Accordingly we expect evidence of universal behavior in the three- and four-nucleon systems that can be observed from the study of correlations between observables. In particular we concentrate in the behavior of the first excited state of the three-nucleon system as the system moves away from the unitary limit. We also analyze the dependence on the range of the three-body force of some low-energy observables in the three- and four-nucleon systems.
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In scattering theory, the unitary limit is defined by an infinite scattering-length and a zero effective range, corresponding to a phase-shift pi/2, independent of energy. This condition is satisfied by a rank-1 separable potential V(k,k)=-v(k)v(k) with v^{2}(k)=(4pi)^{2}(Lambda^{2}-k^{2})^{-1/2}, Lambda being the cut-off in momentum space.Previous calculations using a Pauli-corrected ladder summation to calculate the energy of a zero temperature many body system of spin 1/2 fermions with this interaction gave xi=0.24 (in units of kinetic energy) independent of density and with Lambda---->infinity. This value of xi is appreciably smaller than the experimental and that obtained from other calculations, most notably from Monte Carlo, which in principle would be the most reliable. Our previous work did however also show a strong dependence on effective range r_0 (with r_0=0 at unitarity). With an increase to r_0=1.0 the energy varied from xi~0.38 at k_f=0.6 1/fm to ~0.45 at k_f=1.8 1/fm which is somewhat closer to the Monte-Carlo results. These previous calculations are here extended by including the effect of the previously neglected mean-field propagation, the dispersion correction. This is repulsive and found to increase drastically with decreasing effective range. It is large enough to suggest a revised value of xi~0.4 <--> ~0.5 independent of r_0. Off-shell effects are also investigated by introducing a rank-2 (phase-shift equivalent) separable potential. Effects of 10% or more in energy could be demonstrated for r_0>0. It is pointed out that a computational cut-off in momentum-space brings in another scale in the in otherwise scale-less unitary problem.
Physical systems characterized by a shallow two-body bound or virtual state are governed at large distances by a continuous-scale invariance, which is broken to a discrete one when three or more particles come into play. This symmetry induces a universal behavior for different systems, independent of the details of the underlying interaction, rooted in the smallness of the ratio $ell/a_B ll 1$, where the length $a_B$ is associated to the binding energy of the two-body system $E_2=hbar^2/m a_B^2$ and $ell$ is the natural length given by the interaction range. Efimov physics refers to this universal behavior, which is often hidden by the on-set of system-specific non-universal effects. In this work we identify universal properties by providing an explicit link of physical systems to their unitary limit, in which $a_Brightarrowinfty$, and show that nuclear systems belong to this class of universality.
We discuss weakly bound states of a few-fermion system having spin-isospin symmetry. This corresponds to the nuclear physics case in which the singlet, $a_0$, and triplet, $a_1$, $n-p$ scattering lengths are large with respect to the range of the nuclear interaction. The ratio of the two is about $a_0/a_1approx-4.31$. This value defines a plane in which $a_0$ and $a_1$ can be varied up to the unitary limit, $1/a_0=0$ and $1/a_1=0$, maintaining its ratio fixed. Using a spin dependant potential model we estimate the three-nucleon binding energy along that plane. This analysis can be considered an extension of the Efimov plot for three bosons to the case of three $1/2$-spin-isospin fermions.
We resum the ladder diagrams for the calculation of the energy density $cal{E}$ of a spin 1/2 fermion many-body system in terms of arbitrary vacuum two-body scattering amplitudes. The partial-wave decomposition of the in-medium two-body scattering amplitudes is developed, and the expression for calculating $cal{E}$ in a partial-wave amplitude expansion is also given. The case of contact interactions is completely solved and is shown to provide renormalized results, expressed directly in terms of scattering data parameters, within cutoff regularization in a wide class of schemes. $S$- and $P$-wave interactions are considered up to including the first three-terms in the effective-range expansion, paying special attention to the parametric region around the unitary limit.
Efimov physics relates to 3-body systems with large 2-body scattering lengths a and small effective ranges r. For many systems in nature the assumption of a small effective range is not valid. The present report shows binding energies E of three identical bosons calculated with 2-body potentials that are fitted to scattering data and momentum cut-offs (L) by inverse scattering. Results agree with previous works in the case of r<<a. While energies diverge with momentum cut-off L for r=0, they converge for r>0 when L=~10/r. With 1/a=0 the converged energies are given by E(n) =C(n)/r*r with n labeling the energy-branch and calculated values C(0)=0.77, C(1)=.0028. This gives a ratio ~278 thus differing from the value ~515 in the Efimov case. Efimovs angular dependent function is calculated. Good agreement with previous works is obtained when r<< a. With the increased values of effective range the shallow states still appear Efimov-like. For deeper states the angular dependence differs but is independent of the effective range.
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