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Register a new userBoson-Faddeev in the Unitary Limit and Efimov States
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Authors
H.S. K{o}hler
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A numerical study of the Faddeev equation for bosons is made with two-body interactions at or close to the Unitary limit. Separable interactions are obtained from phase-shifts defined by scattering length and effective range. In EFT-language this would correspond to NLO. Both ground and Efimov state energies are calculated. For effective ranges $r_0 > 0$ and rank-1 potentials the total energy $E_T$ is found to converge with momentum cut-off $Lambda$ for $Lambda > sim 10/r_0$ . In the Unitary limit ($1/a=r_0= 0$) the energy does however diverge. It is shown (analytically) that in this case $E_T=E_uLambda^2$. Calculations give $E_u=-0.108$ for the ground state and $E_u=-1.times10^{-4}$ for the single Efimov state found. The cut-off divergence is remedied by modifying the off-shell t-matrix by replacing the rank-1 by a rank-2 phase-shift equivalent potential. This is somewhat similar to the counterterm method suggested by Bedaque et al. This investigation is exploratory and does not refer to any specific physical system.
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We review the properties of neutron matter in the low-density regime. In particular, we revise its ground state energy and the superfluid neutron pairing gap, and analyze their evolution from the weak to the strong coupling regime. The calculations of the energy and the pairing gap are performed, respectively, within the Brueckner--Hartree--Fock approach of nuclear matter and the BCS theory using the chiral nucleon-nucleon interaction of Entem and Machleidt at N$^3$LO and the Argonne V18 phenomenological potential. Results for the energy are also shown for a simple Gaussian potential with a strength and range adjusted to reproduce the $^1S_0$ neutron-neutron scattering length and effective range. Our results are compared with those of quantum Monte Carlo calculations for neutron matter and cold atoms. The Tan contact parameter in neutron matter is also calculated finding a reasonable agreement with experimental data with ultra-cold atoms only at very low densities. We find that low-density neutron matter exhibits a behavior close to that of a Fermi gas at the unitary limit, although, this limit is actually never reached. We also review the properties (energy, effective mass and quasiparticle residue) of a spin-down neutron impurity immersed in a low-density free Fermi gas of spin-up neutrons already studied by the author in a recent work where it was shown that these properties are very close to those of an attractive Fermi polaron in the unitary limit.
The three-body system inside the unitary window is studied for three equal bosons and three equal fermions having $1/2$ spin-isospin symmetry. We perform a gaussian characterization of the window using a gaussian potential to define trajectories for low-energy quantities as binding energies and phase shifts. On top of this trajectories experimental values are placed or, when not available, quantities calculated using realistic potentials that are known to reproduce experimental values. The intention is to show that the gaussian characterization of the window, thought as a contact interaction plus range corrections, captures the main low-energy properties of real systems as for example three helium atoms or three nucleons. The mapping of real systems on the gaussian trajectories is taken as indication of universal behavior. The trajectories continuously link the physical points to the unitary limit allowing for the explanation of strong correlations between observables appearing in real systems and which are known to exist in that limit. In the present study we focus on low-energy bound, scattering and virtual states.
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.
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.
In chiral effective field theory the leading order (LO) nucleon-nucleon potential includes two contact terms, in the two spin channels $S=0,1$, and the one-pion-exchange potential. When the pion degrees of freedom are integrated out, as in the pionless effective field theory, the LO potential includes two contact terms only. In the three-nucleon system, the pionless theory includes a three-nucleon contact term interaction at LO whereas the chiral effective theory does not. Accordingly arbitrary differences could be observed in the LO description of three- and four-nucleon binding energies. We analyze the two theories at LO and conclude that a three-nucleon contact term is necessary at this order in both theories. In turn this implies that subleading three-nucleon contact terms should be promoted to lower orders. Furthermore this analysis shows that one single low energy constant might be sufficient to explain the large values of the singlet and triplet scattering lengths.
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