Three-body Observables Research Articles

Van der Waals interaction as the starting point for an effective field theory

We consider the system of three ${}^4$He atoms to assess whether a pure van der Waals potential can be used as a starting point for an effective field theory to describe three-body processes in ultracold atomic systems. Using a long-range van der Waals interaction in combination with short-distance two-body counterterms, we analyze the dependence of two- and three-body observables on the short-distance regulator that is required due to the singular nature of the van der Waals interaction. We benchmark our approach with results obtained with the realistic ${}^4$He-${}^4$He LM2M2 interaction and find good agreement. We furthermore show that in this effective field theory approach no three-body force is required at leading order and that universal van der Waals physics leads to a universal correlation between three-body observables in the absence of an Efimov three-body parameter.

Three-particle system in a finite volume: formalism, quantization condition, spectrum and energy shift

Lattice QCD calculations provide an ab initio access to hadronic process. These calculations are usu ally performed in a small cubic volume with periodic boundary conditions. The infinite volume extrapolations for three-body systems are indispensable to understand many systems of high current interest. We derive the three-body quantization condition in a finite volume using an effective field theory in the particle-dimer picture. Our work shows a powerful and transparent method to read off three-body physical observables from lattice simulations. In this paper, we review the formalism, quantization condition, spectrum analysis and energy shifts calculation both for 3-body bound states and scattering states.

Renormalization of a finite-range inverse-cube potential

We study the regularization and renormalization of a finite range inverse cube potential in the two- and three-body sectors. Specifically, we compare and contrast three different regulation schemes frequently used to study few-body systems as well as the associated renormalization group flows. We also calculate bound state and scattering observables over a wide range of cutoffs, demonstrating the sufficiency of a two-body contact interaction to renormalize two- and three-body observables. We supplement these plots with quantified analyses of the observables' residual cutoff dependence.

Finite-range effects in Efimov physics beyond the separable approximation

We study Efimov physics for three identical bosons interacting via a pairwise square-well potential, analyze the validity of the separable approximation as a function of the interaction strength, and investigate what is needed to improve this approximation. We find separable approximations to be accurate for potentials with just one (nearly) bound dimer state. For potentials with more bound or almost bound dimer states, these states need to be included for an accurate determination of the Efimov spectrum and the corresponding three-body observables. We also show that a separable approximation is insufficient to accurately compute the trimer states for energies larger than the finite-range energy even when the two-body T matrix is highly separable in this energy regime. Additionally, we have analyzed three distinct expansion methods for the full potential that give exact results and thus improve on the separable approximation. With these methods, we demonstrate the necessity to include higher partial-wave components of the off-shell two-body T matrix in the three-body calculations. Moreover, we analyze the behavior of the Efimov states near the atom-dimer threshold and observe the formation of non-Efimovian trimer states as the potential depth is increased. Our results can help to elaborate simpler theoretical models that are capable of reproducing the correct three-body physics in atomic systems.

Extracting three-body breakup observables from continuum-discretized coupled-channels calculations with core excitations

Core-excitation effects in the scattering of two-body halo nuclei have been investigated in previous works. In particular, these effects have been found to affect in a significant way the breakup cross sections of neutron-halo nuclei with a deformed core. To account for these effects, appropriate extensions of the continuum-discretized coupled-channels (CDCC) method have been recently proposed. We aim to extend these studies to the case of breakup reactions measured under complete kinematics or semi-inclusive reactions in which only the angular or energy distribution of one of the outgoing fragments is measured. We use the standard CDCC method as well as its extended version with core excitations (XCDCC), assuming a pseudo-state basis for describing the projectile states. Two- and three-body observables are computed by projecting the discrete two-body breakup amplitudes, obtained within these reaction frameworks, onto two-body scattering states with definite relative momentum of the outgoing fragments and a definite state of the core nucleus. The presented method provides a tool to compute double and triple differential cross sections for outgoing fragments following the breakup of a two-body projectile, and might be useful to analyze breakup reactions with other deformed weakly-bound nuclei, for which core excitations are expected to play a role. We have found that, while dynamical core excitations are important for the proton target at intermediate energies, they are very small for the Zn target at energies around the Coulomb barrier.

Effective-field-theory analysis of Efimov physics in heteronuclear mixtures of ultracold atomic gases

We use an effective-field-theory framework to analyze the Efimov effect in heteronuclear three-body systems consisting of two species of atoms with a large interspecies scattering length. In the leading-order description of this theory, various three-body observables in heteronuclear mixtures can be universally parameterized by one three-body parameter. We present the next-to-leading corrections, which include the effects of the finite interspecies effective range and the finite intraspecies scattering length, to various three-body observables. We show that only one additional three-body parameter is required to render the theory predictive at this order. By including the effective range and intraspecies scattering length corrections, we derive a set of universal relations that connect the different Efimov features near the interspecies Feshbach resonance. Furthermore, we show that these relations can be interpreted in terms of the running of the three-body counterterms that naturally emerge from proper renormalization. Finally, we make predictions for recombination observables of a number of atomic systems that are of experimental interest.

Three-body systems in pionless effective field theory

Investigations of three-body nuclear systems using pionless effective field theory ([Formula: see text]) are reviewed. The history of [Formula: see text] in [Formula: see text] and [Formula: see text] scattering is briefly discussed and emphasis put on the use of strict perturbative techniques. In addition renormalization issues appearing in [Formula: see text] scattering are also presented. Bound state calculations are addressed and new perturbative techniques for describing them are highlighted. Three-body breakup observables in [Formula: see text] scattering are also considered and the utility of [Formula: see text] for addressing them.

Three-cluster breakup in deuteron-deuteron collisions: Single-scattering approximation

We present results for the three-cluster breakup in deuteron-deuteron collisions at 130 and 270 MeV deuteron beam energy. The breakup amplitude is calculated using the first term in the Neumann series expansion of the corresponding exact four-nucleon equations. In analogy with nucleon-deuteron breakup where an equivalent approximation is compared with exact calculations, we expect this single-scattering approximation to provide a rough estimation of three-body breakup observables in quasifree configurations. We predict the nucleon-deuteron and deuteron-deuteron three-cluster breakup cross sections to be of a comparable size and thereby question the reliability of the recent experimental data [A. Ramazani-Moghaddam-Arani, Ph.D. thesis, University of Groningen, 2009; A. Ramazani-Moghaddam-Arani et al., EPJ Web Conf. 3, 04012 (2010)], which are smaller by about three orders of magnitude. We also show that an equivalent single-scattering approximation provides a reasonable description of deuteron-deuteron elastic scattering at forward-scattering angles.

Efimov Physics Around the Neutron-RichCa60Isotope

We calculate the neutron-60Ca S-wave scattering phase shifts using state of the art coupled-cluster theory combined with modern ab initio interactions derived from chiral effective theory. Effects of three-nucleon forces are included schematically as density dependent nucleon-nucleon interactions. This information is combined with halo effective field theory in order to investigate the 60Ca-neutron-neutron system. We predict correlations between different three-body observables and the two-neutron separation energy of 62Ca. This provides evidence of Efimov physics along the calcium isotope chain. Experimental key observables that facilitate a test of our findings are discussed.

The three-boson system at next-to-leading order in an effective field theory for systems with a large scattering length

We analyze how corrections linear in the effective range, r0, affect quantities in the three-body sector within an effective field theory for short-range interactions. We demonstrate that observables can be obtained straightforwardly using a perturbative expansion in powers of r0. In particular, we show that two linear-in-r0 counterterms are needed for renormalization at this order if scattering-length-dependent observables are considered. We exemplify the implications of this result using various three-body observables. Analytical results for the running of the next-to-leading-order portion of the three-body force in this effective field theory are provided. Expressions which incorporate O(r0) corrections and relate the positions of features observed in three-atom recombination near a Feshbach resonance are presented.

Relativistic Descriptions of Few-Body Systems

A brief review of relativistic effects in few-body systems, of theoretical approaches, recent developments and applications is given. Manifestations of relativistic effects in the binding energies, in the electromagnetic form factors and in three-body observables are demonstrated. The three-body forces of relativistic origin are also discussed.

Range corrections to three-body observables near a Feshbach resonance

A nonrelativistic system of three identical particles will display a rich set of universal features known as Efimov physics if the scattering length $a$ is much larger than the range $l$ of the underlying two-body interaction. An appropriate effective theory facilitates the derivation of both results in the $\ensuremath{\mid}a\ensuremath{\mid}\ensuremath{\rightarrow}\ensuremath{\infty}$ limit and finite-$l∕a$ corrections to observables of interest. Here we use such an effective-theory treatment to consider the impact of corrections linear in the two-body effective range, ${r}_{s}$, on the three-boson bound-state spectrum and recombination rate for $\ensuremath{\mid}a\ensuremath{\mid}⪢\ensuremath{\mid}{r}_{s}\ensuremath{\mid}$. We do this by first deriving results appropriate to the strict limit $\ensuremath{\mid}a\ensuremath{\mid}\ensuremath{\rightarrow}\ensuremath{\infty}$ in coordinate space. We then extend these results to finite $a$ using once-subtracted momentum-space integral equations. We take the cutoff on these equations to be large compared to $1∕\ensuremath{\mid}{r}_{s}\ensuremath{\mid}$, and find that the first-order effects of $\ensuremath{\mid}{r}_{s}\ensuremath{\mid}$ are independent of the cutoff in this regime. We also discuss the implications of our results for experiments that probe three-body recombination in Bose-Einstein condensates near a Feshbach resonance.

TWO- AND THREE-ALPHA SYSTEMS WITH NONLOCAL POTENTIAL

Two body data alone cannot determine the potential uniquely, one needs three-body data as well. A method is presented here which simultaneously fits local or nonlocal potentials to two-body and three-body observables. The interaction of composite particles, due to the Pauli effect and the indistinguishability of the constituent particles, is genuinely nonlocal. As an example, we use a Pauli-correct nonlocal fish-bone type optical model for the α – α potential and derive the fitting parameters such that it reproduces the two-α and three-α experimental data.

Collisional properties of weakly bound heteronuclear dimers

We consider collisional properties of weakly bound heteronuclear molecules (dimers) formed in a two-species mixture of atoms with a large mass difference. We focus on dimers containing light fermionic atoms as they manifest collisional stability due to an effective dimer-dimer repulsion originating from the exchange of the light atoms. In order to solve the dimer-dimer scattering problem we develop a theoretical approach, which provides a physically transparent and quantitative description of this four-atom system in terms of three- and two-body observables. We calculate the elastic scattering amplitude and the rates of inelastic processes such as the trimer formation and the relaxation of dimers into deeply bound molecular states. Irrespective of whether the heavy atoms are bosons or fermions, the inelastic rate can be significantly lower than the rate of elastic collisions. Moreover, the measurement of the inelastic rate which is a four-body observable, can be an efficient and precise tool for determining three-body observables such as the three-body parameter, positions of Efimov states, and their lifetimes.

Two-pion exchange three-nucleon potential:O(q4)chiral expansion

We present the expansion of the two-pion exchange three-nucleon potential (TPE-3NP) to chiral order ${q}^{4}$, which corresponds to a subset of all possibilities at this order and is based on the $\ensuremath{\pi}N$ amplitude at $\mathcal{O}({q}^{3})$. Results encompass both numerical corrections to strength coefficients of previous $\mathcal{O}({q}^{3})$ terms and new structures in the profile functions. The former are typically smaller than 10% whereas the latter arise from either loop functions or nonlocal gradients acting on the wave function. The influence of the new TPE-3NP over static and scattering three-body observables has been assessed and found to be small, as expected from perturbative corrections.

Three-Boson Problem near a Narrow Feshbach Resonance

We consider a three-boson system with resonant binary interactions and show that for sufficiently narrow resonances three-body observables depend only on the resonance width and the scattering length. The effect of narrow resonances is qualitatively different from that of wide resonances revealing novel physics of three-body collisions. We calculate the rate of three-body recombination to a weakly bound level and the atom-dimer scattering length and discuss implications for experiments on Bose-Einstein condensates and atom-molecule mixtures near Feshbach resonances.

Universality in the three-body problem for4Heatoms

The two-body scattering length a for ${}^{4}\mathrm{He}$ atoms is much larger than their effective range ${r}_{s}.$ As a consequence, low-energy few-body observables have universal characteristics that are independent of the interaction potential. Universality implies that, up to corrections suppressed by ${r}_{s}/a,$ all low-energy three-body observables are determined by a and a three-body parameter ${\ensuremath{\Lambda}}_{*}.$ We give simple expressions in terms of a and ${\ensuremath{\Lambda}}_{*}$ for the trimer binding-energy equation, the atom-dimer scattering phase shifts, and the rate for three-body recombination at threshold. We determine ${\ensuremath{\Lambda}}_{*}$ for several ${}^{4}\mathrm{He}$ potentials from the calculated binding energy of the excited state of the trimer and use it to obtain the universality predictions for the other low-energy observables. We also use the calculated values for one potential to estimate the effective range corrections for the other potentials.

The hypertriton in effective field theory

Doublet Λd scattering and the hypertriton are studied in the framework of an effective field theory for large scattering lengths. As in the triton case, consistent renormalization requires a one-parameter three-body force at leading order whose renormalization group evolution is governed by a limit cycle. Constraining unknown parameters from symmetry considerations and the measured binding energy of the hypertriton, we calculate the low-energy phase shifts for doublet Λd scattering. For the low-energy parameters, we find a Λd =(16.8 +4.4 −2.4) fm and r Λd =(2.3±0.3) fm, where the errors are due to the uncertainty in the hypertriton binding energy. Since the hypertriton is extremely shallow, low-energy three-body observables in this channel are very insensitive to the exact values of the ΛN low-energy parameters.

Calculations of three-body observables in8Bbreakup

We discuss calculations of three-body observables for the breakup of ${}^{8}\mathrm{B}$ on a ${}^{58}\mathrm{Ni}$ target at low energy using the coupled discretized continuum channels approach. Calculations of both the angular distribution of the ${}^{7}\mathrm{Be}$ fragments and their energy distributions are compared with those measured at several laboratory angles. In these observables there is interference between the breakup amplitudes from different spin-parity excitations of the projectile. The resulting angle and the energy distributions reveal the importance of the higher-order continuum state couplings for an understanding of the measurements.

Comparison of relativistic nucleon-nucleon interactions

We investigate the difference between those relativistic models based on interpreting a realistic nucleon-nucleon interaction as a perturbation of the square of a relativistic mass operator and those models that use the method of Kamada and Gl\"ockle to construct an equivalent interaction to add to the relativistic mass operator. Although both models reproduce the phase shifts and binding energy of the corresponding non-relativistic model, they are not scattering equivalent. The example of elastic electron-deuteron scattering in the one-photon-exchange approximation is used to study the sensitivity of three-body observables to these choices. Our conclusion is that the differences in the predictions of the two models can be understood in terms of the different ways in which the relativistic and non-relativistic $S$-matrices are related. We argue that the mass squared method is consistent with conventional procedures used to fit the Lorentz-invariant cross section as a function of the laboratory energy.