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Register a new userGaussian characterization of the unitary window for $N=3$: bound, scattering and virtual states
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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.
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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.
Solution of the scattering problem turns to be very difficult task both from the formal as well as from the computational point of view. If the last two decades have witnessed decisive progress in ab initio bound state calculations, rigorous solution of the scattering problem remains limited to A$leq$4 case. Therefore there is a rising interest to apply bound-state-like methods to handle non-relativistic scattering problems. In this article the latest theoretical developments in this field are reviewed. Five fully rigorous methods will be discussed, which address the problem of nuclear collisions in full extent (including the break-up problem) at the same time avoiding treatment of the complicate boundary conditions or integral kernel singularities. These new developments allows to use modern bound-state techniques to advance significantly rigorous solution of the scattering problem.
Recent ab initio lattice studies have found that the interactions between alpha particles (4He nuclei) are sensitive to seemingly minor details of the nucleon-nucleon force such as interaction locality. In order to uncover the essential physics of this puzzling phenomenon without unnecessary complications, we study a simple model involving two-component fermions in one spatial dimension. We probe the interaction between two bound dimers for several different particle-particle interactions and measure an effective potential between the dimers using external point potentials which act as numerical tweezers. We find that the strength and range of the local part of the particle-particle interactions play a dominant role in shaping the interactions between the dimers and can even determine the overall sign of the effective potential.
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 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.
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