Four-body Scale Research Articles

Four-Boson Continuous Scale Symmetry Breaking

We show that the homogeneous Faddeev–Yakubovski formalism for the bound state of four identical bosons with zero-range interaction in the unitary limit (infinite two-body scattering length) presents scale invariance in the ultraviolet (UV) region. By resorting to an approximate form of the integral equations in the UV limit (considering exclusively an attractive pairwise zero-range interaction), we demonstrate that a pair of log-periodic solutions, with a cycle distinct from the three-boson one, exists with a four-body scale required to define the phase between them.

Four-Body Scale in Universal Few-Boson Systems.

The role of an intrinsic four-body scale in universal few-boson systems is the subject of active debate. We study these systems within the framework of effective field theory. For systems of up to six bosons we establish that no four-body scale appears at leading order (LO). However, we find that at next-to-leading order (NLO) a four-body force is needed to obtain renormalized results for binding energies. With the associated parameter fixed to the binding energy of the four-boson system, this force is shown to renormalize the five- and six-body systems as well. We present an original ansatz for the short-distance limit of the bosonic A-body wave function from which we conjecture that new A-body scales appear at N^{A-3} LO. As a specific example, calculations are presented for clusters of helium atoms. Our results apply more generally to other few-body systems governed by a large scattering length, such as light nuclei and halo states, the low-energy properties of which are independent of the detailed internal structure of the constituents.

Trimer–Tetramer Interwoven States in the Scaling Limit

We report here some results we have obtained on the scale dependence of tetramer energies at the unitary limit, considering the number of tetramer energy levels appearing between the ground and the excited Efimov trimers. Our numerical investigation is done by solving a renormalized set of Faddeev–Yakubovsky equations for identical bosons with zero-range interaction, requiring a four-body scale, which in principle can be independent of the trimer properties. The ratio between the three- and four-body scales is introduced by considering the two lower trimer states and corresponding associated tetramers. We conclude that at least three tetramers are possible to exist between two Efimov states by varying the relation between such scales, and considering the relation between three-body Efimov states in the unitary limit. The results for the trimer–tetramer interwoven states are shown through a correlation between tetramers attached to consecutive Efimov trimer states.

Range Corrections to Universal Tetramer Properties

The universal properties of weakly-bound tetramers close to the scaling limit are investigated by solving a subtracted set of Faddeev–Yakubovsky (FY) equations for identical bosons with a zero-range interaction. The solution demands a four-body scale independent of the trimer properties. Furthermore, the effect of a finite effective range is introduced in the FY equations, which we show produces results that are distinct from the scale variation. In particular range effects to two universal scaling functions for the tetramers are investigated. The correlation between successive tetramer energies corresponding to states within two Efimov trimer energies, proposed before and studied close to the unitary limit; and the correlation between the position of the four-atom recombination peaks. In this case, we found a shift in the scaling function due to the range, which can be associated to the shift of the data found for caesium atoms, with respect to zero-range calculations, due to a nonvanishing range in the actual experimental setups.

Four-Boson Systems Close to a Universal Regime

Universal properties of weakly-bound four-boson systems near the scaling limit are discussed by considering recent results obtained from the solution of Faddeev-Yakubovsky (FY) equations, which confirm a previous conjecture on a four-body scale dependence. In the present contribution, within a discussion on our numerical results obtained for the binding energies of two consecutive tetramer states, we are analyzing the relative relevance of the two possible configurations of the four-body system.

Universality in Four-Boson Systems

We report recent advances on the study of universal weakly bound four-boson states from the solutions of the Faddeev-Yakubovsky equations with zero-range two-body interactions. In particular, we present the correlation between the energies of successive tetramers between two neighbor Efimov trimers and compare it to recent finite range potential model calculations. We provide further results on the large momentum structure of the tetramer wave function, where the four-body scale, introduced in the regularization procedure of the bound state equations in momentum space, is clearly manifested. The results we are presenting confirm a previous conjecture on a four-body scaling behavior, which is independent of the three-body one. We show that the correlation between the positions of two successive resonant four-boson recombination peaks are consistent with recent data, as well as with recent calculations close to the unitary limit. Systematic deviations suggest the relevance of range corrections.

Tjon Lines and Scaling Limit in Four-Body Systems

The fixed-slope correlation between tetramer and trimer binding energies, observed by Tjon in the context of nuclear physics, is mainly a manifestation of the dominance of the two-nucleon force in the nuclear potential, which makes the four-body scale on the order of the three-body one. In a more general four-boson case, the correlation between tetramer and trimer binding energies has a non-fixed slope, which expresses the dependence on the new scale. The associated scaling function generates a family of Tjon lines. This conclusion relies on a recent study with weakly-bound four identical bosons, within a renormalized zero-range Faddeev-Yakubovsky formalism.

Binding and structure of tetramers in the scaling limit

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 $B_4^{(N)}$ moves as the short-range four-body scale changes, while the trimer properties are kept fixed, with the next excited tetramer $B_4^{(N+1)}$ emerging from the atom-trimer threshold for a universal ratio $B_4^{(N)}/B_3 = B_4^ {(N)}/B_4^{(N+1)} \simeq 4.6$, 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, which reflect the short-range four-body scale. We also found that the $H-$channel is favored over $K-$channel at low momentum when the four-body momentum scale largely overcomes the three-body one.

Scaling Properties of Universal Tetramers

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.

An α-Particle Model for 16O: Is There a New Four-Body Scale?

The 16O nucleus is modelled as a system of four α-particles. It is shown that a Hamiltonian that includes established two- and three-body forces reproducing the α-α phase shifts, the 8Be resonance, and the first and second 0+ states of 12C overbinds the 16O nucleus. To rectify this a novel four-body force is introduced, which then leads to a good agreement with the low-lying experimental spectrum of 16O.