Effective Field Theory Analysis of Three-Boson Systems at Next-To-Next-To-Leading Order

We use an effective field theory for short-range forces (SREFT) to analyze systems of three identical bosons interacting via a two-body potential that generates a scattering length, a, which is large compared to the range of the interaction, ?. The amplitude for the scattering of one boson off a bound state of the other two is computed to next-to-next-to-leading order (N²LO) in the ?/a expansion. At this order, two pieces of three-body data are required as input in order to renormalize the amplitude (for fixed a). We apply our results to a model system of three Helium-4 atoms, which are assumed to interact via the TTY potential. We generate N²LO predictions for atom-dimer scattering below the dimer breakup threshold using the bound-state energy of the shallow Helium-4 trimer and the atom-dimer scattering length as our two pieces of three-body input. Based on the convergence pattern of the SREFT expansion, as well as differences in the predictions of two renormalization schemes, we conclude that our N²LO phase- shift predictions will receive higher-order corrections of < 0.2 %. In contrast, the prediction of SREFT for the binding energy of the “deep” trimer of Helium-4 atoms displays poor convergence.     

The Universality of the Efimov Three-body Parameter

In this paper we discuss the recent discovery of the universality of the three-body parameter (3BP) from Efimov physics. This new result was identified by recent experimental observations in ultracold quantum gases where the value of the s-wave scattering length, a = a _?, at which the first Efimov resonance is created was found to be nearly the same for a range of atomic species — if scaled as a _?/r _vdW, where r _vdW is the van der Waals length. Here, we discuss some of the physical principles related to these observations that emerge from solving the three-body problem with van der Waals interactions in the hyperspherical formalism. We also demonstrate the strong three-body multichannel nature of the problem and the importance of properly accounting for nonadiabatic effects.     

Efimov Physics in Small Bosonic Clusters

We study small clusters of bosons, A = 2, 3, 4, 5, 6, characterized by a resonant interaction. Firstly, we use a soft-gaussian interaction that reproduces the values of the dimer binding energy and the atom-atom scattering length obtained with LM2M2 potential, a widely used ⁴He-⁴He interaction. We change the intensity of the potential to explore the clusters’ spectra in different regions with large positive and large negative values of the two-body scattering length and we report the clusters’ energies on Efimov plot, which makes the scale invariance explicit. Secondly, we repeat our calculation adding a repulsive three-body force to reproduce the trimer binding energy. In all the region explored, we have found that these systems present two states, one deep and one shallow close to the A ? 1 threshold, and scale invariance has been investigated for these states. The calculations are performed by means of Hyperspherical Harmonics basis set.     

Six-Bodies Calculations Using the Hyperspherical Harmonics Method

In this work we show results for light nuclear systems and small clusters of helium atoms using the hyperspherical harmonics basis. We use the basis without previous symmetrization or antisymmetrization of the state. After the diagonalization of the Hamiltonian matrix, the eigenvectors have well defined symmetry under particle permutation and the identification of the physical states is possible. We show results for systems composed up to six particles. As an example of a fermionic system, we consider a nucleon system interacting through the Volkov potential, used many times in the literature. For the case of bosons, we consider helium atoms interacting through a potential model which does not present a strong repulsion at short distances. We have used an attractive gaussian potential to reproduce the values of the dimer binding energy, the atom-atom scattering length, and the effective range obtained with one of the most widely used He–He interaction, the LM2M2 potential. In addition, we include a repulsive hypercentral three-body force to reproduce the trimer binding energy.     

Spectroscopy and decay properties of B and Bs mesons

Lattice simulations of light nuclei necessarily take place in finite volumes, thus affecting their infrared properties. These effects can be addressed in a model-independent manner using Effective Field Theories. We study the model case of three identical bosons (mass m with resonant two-body interactions in a cubic box with periodic boundary conditions, which can also be generalized to the three-nucleon system in a straightforward manner. Our results allow for the removal of finite-volume effects from lattice results as well as the determination of infinite-volume scattering parameters from the volume dependence of the spectrum. We study the volume dependence of several states below the break-up threshold, spanning one order of magnitude in the binding energy in the infinite volume, for box side lengths L between the two-body scattering length a and L = 0.25a. For example, a state with a three-body energy of ?3/(ma ²) in the infinite volume has been shifted to ?10/(ma ²) at L = a. Special emphasis is put on the consequences of the breakdown of spherical symmetry and several ways to perturbatively treat the ensuing partial-wave admixtures. We find their contributions to be on the sub-percent level compared to the strong volume dependence of the S-wave component. For shallow bound states, we find a transition to boson-diboson scattering behavior when decreasing the size of the finite volume.     

An exact treatment for the BEC with three-body recombination and time-dependent scattering length

In this study, Gross-Pitaevskii (GP) equation with three-body recombination and with a time-dependent scattering length is investigated. It is shown that GP equation admits an exact solution for a special choice of the time-dependent scattering length. The effect of Feshbach resonance on the collapse dynamics is investigated for large ∣a∣ for which the three body recombination rate K₃ grows with a⁴.     

Scales, Universality and Finite-Range Correction in Three-body Systems

The scale invariance manifested by the weakly-bound Efimov states implies that all the Efimov spectrum can be merged in a single scaling function. By considering this scaling function, the ratio between two consecutive energy levels, ${E_3^{\rm (N+1)}}$ and ${E_3^{\rm (N)}}$ , can be obtained from a two-body low-energy observable (usually the scattering length a), given in units of the three-body energy level N. The zero-ranged scaling function is improved by incorporating finite range corrections in first order of r ₀/a (r ₀ is the potential effective range). The critical condition for three-identical bosons in s-wave, when the excited ${E_3^{\rm (N+1)}}$ state disappears in the 2 + 1 threshold, is given by ${\sqrt{E_2/E_3^{\rm (N)}} \approx 0.38+0.12 ({r_0}/{a})}$ .     

Scattering length and effective range in closed form for the Coulomb plus Yamaguchi potential

From the known expression for the off-shell T-matrix corresponding to the potential consisting of the sum of the Coulomb potential and the Yamaguchi potential, the physical scattering amplitude can be derived in a satisfactory way. We derive simple exact closed formulae for the scattering length a_cs and the effective range r_cs from this amplitude. These are compared with approximate formulae derived by Harrington. Also a few numerical calculations are reported and compared with results obtained by Harrington and by Ali et al.     

Efimov effect in an analytically solvable model

The Efimov effect is demonstrated in a model consisting of two heavy particles and a light one when the light-heavy interaction leads to a zero-energy two-body bound state. The model is solved in the Born-Oppenheimer approximation with the light-heavy interaction taken to be a separable S-wave potential of Yamaguchi form. It is demonstrated that in the case of a- two-body zero-energy bound state the binding energy of the light particle in the two-center potential exactly yields an effective $\text{1}\text{r}{}^2$ potential for the relative motion of the heavies. If the light-heavy mass ratio is made small enough, infinitely many bound states (the Efimov effect) are obtained. The approach to this limit is studied and the nature of the potential for large scattering length is obtained. An upper bound for the number of bound states is calculated using a result of Bargmann and Calogero and Efimov's $\text{ln}\text{(}\text{a}\text{r}{}_0\text{)}$ result is found. We note that the long-range effect arises from the large extent of the bound state, the pair wave function being essentially $\text{exp}\text{(?}\text{r}\text{a}\text{)}$ when the scattering length a is large.     

α8Be cluster model for the 0 2 + resonance in the 12C nucleus

The 0 ₂ ⁺ resonance in the ¹²C nucleus is treated on the basis of the α+⁸Be two-cluster model. An equation for the function describing the relative motion of the clusters is derived by using the s-wave differential Faddeev equations for the 3α system and by relying on the simplest version of the resonating-group method. A phenomenological potential is taken to simulate the pair αα interaction. A three-body potential binding three alpha particles together gives rise to a resonance in two-cluster α+⁸Be scattering. The calculated resonance features and the calculated parameters of the wave function of the system are compared with the results obtained by other authors.     

Cross sections for scattering of deuterium mesic atoms on deuterium nuclei

Cross sections have been calculated for elastic scattering of deuterium mesic atoms in ortho and para states on deuterium nuclei in the 0–50 eV energy range; cross sections for spin-flip have also been calculated. The calculations have been performed in the adiabatic representation where the initial problem of slow collisions in a three-body system is reduced to the multichannel scattering problem. The temperature dependence of the ortho-para transition rate λ_hf(T) for dμ atoms is obtained for a deuterium target. The calculated rate 37.2×10⁶ s^-1 (at 30 K) is in good agreement with the recent experimental value, λ_exp(30 K) = (37.0±0.74)×10⁶ s^-1 (Vienna-SIN group, 1983).     

Calculation of the elastic scattering properties for cold and ultracold 39K atoms in a triplet state

The elastic scattering properties for collisions between cold and ultracold 39K atoms in a triplet state are investigated. Based on the recent theoretical and experimental results, the improved hybrid potential is presented for a triplet α3∑u^＋ ground state of K2. Our calculated value of the s-wave scattering length a by using the Numerov method for the triplet state is 79.578α0 and found to be in good agreement with the previous ones. The numbers of bound states are supported by the molecular potential. Pronounced shape resonances appear for the l = 3 partial waves for the α3∑u^＋ state. Furthermore, the s-wave scattering cross section, the total cross section and energy positions of shape resonances for the α3∑u^＋ state are calculated.     

Investigation of the reaction 2H(n,np) at θn = 0°

The differential cross section of the ²H(n, np) reaction was measured at θ_n = 0°. Because of the applied special geometry the experimental data extend over a large fraction of the phase space, including several final state interaction regions as well as regions far from the dominance of quasi-two-body processes. The experiment was analysed with an exact three-body calculation using s-wave separable potentials (CEE). The analysis of the n-n and n-p final state interaction peaks gave a_nn = ?16.3 ± 1.6 fm for the n-n ¹S₀ scattering length and r_nn = 3.15 ± 0.7 fm for the effective range. Remarkable deviations from the calculated cross sections-were found in regions where no strong final state interaction was present.     

Small copper clusters: Study of the local atomic and electronic structure by the XANES and DFT methods

The Cu L ₃ edge X-ray absorption spectra of free Cu _n clusters containing 5–15 atoms were obtained for the first time. It is shown that the geometry of small clusters, including the bond length and bond angle, can be studied by analyzing the X-ray absorption spectra. The experimental X-ray absorption spectra of Cu₁₃ clusters were theoretically interpreted using the self-consistent method of full multiple scattering, the finite difference method in the muffin-tin potential approximation, and the full potential method for the icosahedral cluster structure. Good agreement between the theoretical and experimental spectra is achieved for the calculation in the full potential approximation.     

The Ground State Energy of Dilute Bose Gas in Potentials with Positive Scattering Length

The leading term of the ground state energy/particle of a dilute gas of bosons with mass m in the thermodynamic limit is 2p(^h/_2p)² a r/m{2\pi \hbar^2 a \varrho/m} when the density of the gas is r{\varrho}, the interaction potential is non-negative and the scattering length a is positive. In this paper, we generalize the upper bound part of this result to any interaction potential with positive scattering length, i.e, a > 0 and the lower bound part to some interaction potentials with shallow and/or narrow negative parts.     

Three-Body Recombination in Ultracold Systems: Prediction of Weakly-Bound Atomic Trimer Energies

The three-body recombination coefficient of an ultracold atomic system, together with the corresponding two-body scattering length a, allow us to predict the energy E₃ of the shallow trimer bound state, using a universal scaling function. The production of dimers in trapped Bose-Einstein condensates, from three-body recombination processes, in the regime of short magnetic pulses near a Feshbach resonance, is also studied in line with the experimental observation.     

The Three-Boson System at Next-to-Next-to-Leading Order

We discuss effective field-theory treatments of the problem of three particles interacting via short-range forces (range R ≪ a ₂, with a ₂ the two-body scattering length). We show that forming a once-subtracted scattering equation yields a scattering amplitude whose low-momentum part is renormalization-group invariant up to corrections of O(R ³/a ₂ ³). Since corrections of O(R/a ₂) and O(R ²/a ₂ ²) can be straightforwardly included in the integral equation’s kernel, a unique solution for 1 + 2 scattering phase shifts and three-body bound-state energies can be obtained up to this accuracy. We use our equation to calculate the correlation between the binding energies of Helium-4 trimers and the atom-dimer scattering length. Our results are in excellent agreement with recent three-dimensional Faddeev calculations that used phenomenological inter-atomic potentials.     

Binding effects in the optical potential of pionic atoms

We calculate the contribution from nucleon binding to the coefficient Re B₀ in the optical potential of pionic atoms. The relevant Feynman graphs are deduced for nuclear matter. They depend on off-shell values for pion-nucleon scattering lengths a_πN and for the nucleon-binding potential V_NA. Off-shell effects are found to be very important: A pole model for a_πN, extrapolated off-shell, increases binding effects by a factor of three over the use of on-shell values. Two simple models are proposed for the off-shell continuation of V_NA. One leads to attractive, the other to repulsive contributions to Re B₀.     

Three-body bound states in Amado's model

We have calculated the eigenvalue trajectories of the scattering kernel for a system of three, identical particles with Yamaguchi interactions, for both spin half and spin zero particles. For potential strengths corresponding to large values of the two-body scattering length, we find several, weakly-bound, three-body excited states of the type proposed by Efimov, and calculate their binding energies.