We evidence the existence of a universal correlation between the binding energies of successive four-boson bound states (tetramers), for large two-body scattering lengths (a), related to an additional scale not constrained by three-body Efimov physics. Relevant to ultracold atom experiments, the atom-trimer relaxation peaks for |a|→∞ when the ratio between the tetramer and trimer energies is ≃4.6 and a new tetramer is formed. The new scale is also revealed for a < 0 by the prediction of a correlation between the positions of two successive peaks in the four-atom recombination process.

A spin-isospin dependent Three-Dimensional approach based on momentum vectors for formulation of the three-nucleon bound state is presented in this paper. The three-nucleon Faddeev equations with two-nucleon interactions are formulated as a function of vector Jacobi momenta, specifically the magnitudes of the momenta and the angle between them with the inclusion of the spin-isospin quantum numbers, without employing a partial wave decomposition. As an application the spin-isospin dependent Faddeev integral equations are solved with Bonn-B potential. Our result for the Triton binding energy with the value of −8.152 MeV is in good agreement with the achievements of the other partial wave based methods.

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The four-body bound state with two-body interactions is formulated in Three-Dimensional approach, a recently developed momentum space representation which greatly simplifies the numerical calculations of few-body systems without performing the partial wave decomposition.The obtained three-dimensional Faddeev-Yakubovsky integral equations are solved with two-body spin-independent and spin-averaged potentials. This is the first step toward the calculations of four-nucleon bound state problem in Three-Dimensional approach. Results for four-body binding energies are in good agreement with achievements of the other methods.

The discrete Efimov scaling behavior, well-known in the low-energy spectrum of three-body bound systems for large scattering lengths (unitary limit), is identified in the energy dependence of atommolecule elastic cross-section in mass imbalanced systems. That happens in the collision of a heavy atom with mass mH with a weakly-bound dimer formed by the heavy atom and a lighter one with mass mL mH . Approaching the heavy-light unitary limit the s−wave elastic cross-section σ will present a sequence of zeros/minima at collision energies following closely the Efimov geometrical law. Our results open a new perspective to detect the discrete scaling behavior from low-energy scattering data, which is timely in view of the ongoing experiments with ultra-cold binary mixtures having strong mass asymmetries, such as Lithium and Caesium or Lithium and Ytterbium.The Efimov effect [1] refers to a discrete scaling symmetry, which emerges in the quantum three-body system at the unitary limit (when the two-body scattering lengths diverge). The optimal condition to observe this discrete scaling symmetry in cold atomic laboratories is now found for heteronuclear three-atom systems with large mass asymmetry and large interspecies scattering lengths. In the Efimov (unitary) limit, the shallow three-body levels are geometrically spaced, namely the ratio between the binding energies of the n and n + 1 levels is given by B = exp (2π/s 0 ), where s 0 is a universal constant, which depends only on the mass ratio and not on the details of the interaction. The energy ratio for three identical bosons is exp (2π/s 0 ) ≈ 515, decreasing for the case of two heavy particles and light one. When m L /m H = 0.01, for example, the value of this energy ratio goes to exp (2π/s 0 ) = 4.698 [2].The Efimov geometric scaling factor has been measured in a cold-atom experiment with mass-imbalance mixtures of Caesium ( 133 Cs) and Lithium ( 6 Li) gases by different groups [3,4]. The ratio between the positions of two successive peaks in the three-body recombination rate, obtained by varying the large negative scattering lengths (a HL ), was found in close agreement with the theory. Complementary to this finding, a fingerprint of the Efimov scaling can be found in the s−wave ultracold atom-molecule cross-section by varying the incident momentum energy k instead of the scattering lengths. Natural, but not yet evidenced experimentally or theoretically. What we expect is beyond the trimer crossing the corresponding continuum, which creates the resonant enhancement of the inelastic collisions of Caesium atoms with Caesium dimers, as observed by Knoop et al. [5].Furthermore, there is an evident strong interest in ultra-cold heteronuclear atom-molecule collisions by experimental groups [6][7][8]. Trap setups with ultra-cold degenerated mixtures of alkali-metal-rare-earth molecules with strong mass-imbalanced systems as Ytterbium and Lithium ( 174,173 Yb− 6 Li) have also been reported in Refs. [9,10]. We should mention that on the theory side [11], reactions at ...

The momentum-space structure of the Faddeev-Yakubovsky (FY) components of weakly bound tetramers is investigated at the unitary limit using a renormalized zero-range two-body interaction. The results, obtained by considering a given trimer level with binding energy B 3 , provide further support to a universal scaling function relating the binding energies of two successive tetramer states. The correlated scaling between the tetramer energies comes from the sensitivity of the four-boson system to a short-range four-body scale. Each excited N th tetramer energy B 4.6, which does not depend on N . We show that both channels of the FY decomposition [atom-trimer (K type) and dimer-dimer (H type)] present high-momentum tails that reflect the short-range four-body scale. We also found that the H channel is favored over the K channel at low momentum, when the four-body momentum scale largely overcomes the three-body scale.

Background:The relativistic three-body problem has a long tradition in few-nucleon physics. Calculations of the triton binding energy based on the solution of the relativistic Faddeev equation, in general, lead to a weaker binding than the corresponding nonrelativistic calculation. Purpose: In this work we solve for the three-body binding energy as well as the wave function and its momentum distribution. The effect of the different relativistic ingredients is studied in detail. Method: Relativistic invariance is incorporated within the framework of Poincaré invariant quantum mechanics. The relativistic momentum-space Faddeev equation is formulated and directly solved in terms of momentum vectors without employing a partial-wave decomposition. Results: The relativistic calculation gives a three-body binding energy which is about 3% smaller than its nonrelativistic counterpart. In the wave function, relativistic effects are manifested in the Fermi motion of the spectator particle. Conclusions: Our calculations show that though the overall relativistic effects in the three-body bound state are small, individual effects by themselves are not necessarily small and must be taken into account consistently.

The homogeneous Lippmann-Schwinger integral equation is solved in momentum space to calculate the masses of heavy tetraquarks with hidden charm and bottom. The tetraquark bound states are studied in the diquark-antidiquark picture as a two-body problem. A regularized form of the diquark-antidiquark potential is used to overcome the singularity of the confining potential at large distances or small momenta. Our numerical results indicate that the relativistic effect leads to a small reduction in the mass of heavy tetraquarks, which is less than 2 % for charm and less than 0.2 % for bottom tetraquarks. The calculated masses of heavy tetraquarks for 1s, 1p, 2s, 1d and 2p states are in good agreement with other theoretical calculations and experimental data. Our numerical analysis predict the masses of heavy tetraquarks for 3s, 2d and 3p states for the first time, and we are not aware of any other theoretical results or experimental data for these states.