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.     

Atom-dimer scattering for confined ultracold fermion gases

We solve the three-body problem of a quasi-one-dimensional ultracold Fermi gas with parabolic confinement length a (perpendicular) and 3D scattering length a. On the two-body level, there is a Feshbach-type resonance at a (perpendicular)/a approximately 1.46, and a dimer state for arbitrary a (perpendicular)/a. The three-body problem is shown to be universal, and described by the atom-dimer scattering length a(ad) and a range parameter b(ad). In the dimer limit a (perpendicular)/a>1, we find a repulsive zero-range atom-dimer interaction. For a (perpendicular)/a<-1, however, the potential has long range, with a(ad)>0 and b(ad)>a(ad). There is no trimer state, and despite a(ad)=0 at a( perpendicular)/a approximately 2.6, there is no resonance enhancement of the interaction.     

Structure and dynamics of few-helium clusters using soft-core potentials

In this work we investigate the structure and dynamics of small clusters of Helium atoms. We consider bound states of clusters having A = 2, 3, 4, 5, 6 atoms and continuum states in the three-atom system. Motivated by the fact that the He-He system has a very large scattering length a compared to the range r ₀ of the He-He potential (r ₀/a < 1/10), we propose the use of a soft-core interparticle potential. We use an attractive gaussian potential that reproduces the values of the dimer binding energy and the atomatom scattering length obtained with one of the widely used He-He interactions, the LM2M2 potential. In addition, we include a repulsive three-body force to reproduce the trimer binding energy. With this model, consisting in the sum of a two- and a three-body potential, we show the spectrum of the four, five, and sixparticle systems and phase-shifts and inelasticities in the three-atom system. Comparisons to calculations using realistic He-He potentials are given. In addition some universal relations are explored.     

Atom-dimer and dimer-dimer scattering in fermionic mixtures near a narrow Feshbach resonance

We develop a diagrammatic approach for solving few-body problems in heteronuclear fermionic mixtures near a narrow interspecies Feshbach resonance. We calculate s-, p-, and d-wave phaseshifts for the scattering of an atom by a weakly-bound dimer. The fermionic statistics of atoms and the composite nature of the dimer lead to a strong angular momentum dependence of the atom-dimer interaction, which manifests itself in a peculiar interference of the scattered s- and p-waves. This effect strengthens with the mass ratio and is remarkably pronounced in ⁴⁰K-(⁴⁰K-⁶Li) atom-dimer collisions. We calculate the scattering length for two dimers formed near a narrow interspecies resonance. Finally, we discuss the collisional relaxation of the dimers to deeply bound states and evaluate the corresponding rate constant as a function of the detuning and collision energy.     

Three-Body System with Two-Channel Zero-Range Interaction Model of Feshbach Resonance

We perform three-body calculations of trimers and atom-dimer scattering near a Feshbach resonance using two interaction models. The first model is a one-channel zero-range model, where the scattering length follows the phenomenological dependence on the external magnetic field. The second is a two-channel model capable to describe the Feshbach resonance. The scattering length dependence on magnetic detuning is recovered. We compare the predictions of these two models, and show that near a Feshbach resonance important differences are expected.     

3H at Next-to-Next-to-Next-to Leading Order of the Chiral Expansion

The chiral three-nucleon force (3NF) at next-to-next-to-next-to leading order (N³LO) is used to calculate the triton wave function and the doublet nucleon–deuteron scattering length. This allows us to fix the values of the low-energy constants which are free parameters of the theory. The obtained values of these parameters, the expectation values of the kinetic energy, two- and three-body potentials and individual contributions of different parts of 3NF are given.     

Infinite-Cutoff Renormalization of the Chiral Nucleon–Nucleon Interaction up to N3LO

Naively, the “best” method of renormalization is the one where a momentum cutoff is taken to infinity while maintaining stable results due to a cutoff-dependent adjustment of counterterms. We have applied this renormalization method in the non-perturbative calculation of phase-shifts for nucleon–nucleon (NN) scattering using chiral NN potentials up to next-to-next-to-next-to-leading order (N³LO). For lower partial waves, we find that there is either no convergence with increasing order or, if convergence occurs, the results do not always converge to the empirical values. For higher partial waves, we always observe convergence to the empirical phase shifts (except for the ³G₅ state). Furthermore, no matter what the order is, one can use only one or no counterterm per partial wave, creating a rather erratic scheme of power counting that does not allow for a systematic order-by-order improvement of the predictions. The conclusion is that infinite-cutoff renormalization is inappropriate for chiral NN interactions, which should not come as a surprise, since the chiral effective field theory (chiral EFT), these interactions are based upon, is designed for momenta below the chiral-symmetry breaking scale of about 1 GeV. Therefore, this value for the hard scale should also be perceived as the appropriate upper limit for the momentum cutoff.     

Calculation of the two-body scattering T-matrix in configuration space

In order to carry out a solution of the three-body Faddeev integral equations in configuration space, the calculation of the two-body scattering T-matrices and related integrals are required as an input. The formulation of the three-body Faddeev solution, as well as the computational steps used for the calculation of the T-matrices are presented, and results for the latter are illustrated for the case of the scattering of two helium atoms.     

Application of EFT at thermal energies

We have evaluated some observables of n-d systems by using pionless effective field theory (EFT( ${\not}\pi$ )) and insertion of the three-body force up to next-to-next to leading order (N²LO). The evaluated data has been compared with experiment and a three-nucleon calculation of the total cross section with modern realistic two- and three-nucleon force AV18/UrbIX potential models.     

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.     

Universal dimer in a collisionally opaque medium: experimental observables and Efimov resonances

A universal dimer is subject to secondary collisions with atoms when formed in a cloud of ultracold atoms via three-body recombination. We show that in a collisionally opaque medium, the value of the scattering length that results in the maximum number of secondary collisions may not correspond to the Efimov resonance at the atom-dimer threshold and thus cannot be automatically associated with it. This result explains a number of controversies in recent experimental results on universal three-body states and supports the emerging evidence for the significant finite range corrections to the first excited Efimov energy level.     

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.     

Exact solution of a model for πd induced reactions and its application to pion-nucleus reactions

Reduction techniques are applied to πd elastic scattering and π-absorption in a theory without anti-nucleons. In a one-pion approximation we derive two sets of exact coupled-channel equations for respectively the amplitudes T_πd,NN, T_{Nδ, NN} and T_{πd, πd}, T_{d, NΔ}- Alternatively we express in terms of the absorption amplitude T_{πd, NN} available solutions for a three-body problem restricted to the πd and NΔ channels. We explicitly demonstrate that our model (which comes close to the one of Thomas, Mizutani and Koltun) strictly respects the Pauli principle and avoids double-counting. Using the same technique we determine amplitudes for the (π, 2N) reaction and for π (in)elastic scattering on general nuclei in terms of amplitudes amongst the NN, NNπ channels. Both the elastic amplitude and the πA optical potential are shown to decompose into a multiple scattering part based on an input πN amplitude without the P₁₁ partial wave and calculable absorption corrections.     

Rigorous estimates of the e-(μ-p) scattering amplitude below the μ- excitation threshold

Starting from an operator formulation of Kohn's variational principle, we derive an estimate of the non-relativistic e^-(μ^-p) scattering amplitude for scattering energies below the μ^- excitation threshold. Our investigations proceed via an appropriate several-step Born expansion of this scattering amplitude. In order to construct majorants of these Born expansions, we perform estimates of certain operator norms by means of rearrangement inequalities and an extension of Hilbert's inequality.     

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.     

Not to Normal Order—Notes on the Kinetic Limit for Weakly Interacting Quantum Fluids

The derivation of the Nordheim-Boltzmann transport equation for weakly interacting quantum fluids is a longstanding problem in mathematical physics. Inspired by the method developed to handle classical dilute gases, a conventional approach is the use of the BBGKY hierarchy for the time-dependent reduced density matrices. In contrast, our contribution is motivated by the kinetic theory of the weakly nonlinear Schrödinger equation. The main observation is that the results obtained in the latter context carry over directly to weakly interacting quantum fluids provided one does not insist on normal order in the Duhamel expansion. We discuss the term by term convergence of the expansion and the equilibrium time correlation 〈a(t)^* a(0)〉.     

���Ѧ� Interaction Leading to N* and ��* Resonances

We have performed a calculation for the three-body ????? system by using the fixed center approximation to Faddeev equations, taking the interaction between ?? and ??, ?? and ??, and ?? and ?? from the chiral unitary approach. We find several peaks in the modulus squared of the three-body scattering amplitude, indicating the existence of resonances, which can be associated to known I?=?1/2, 3/2 and J ^P ?=?1/2⁺, 3/2⁺ and 5/2⁺ baryon states.