A renormalized equation for the three-body system with short-range interactions

We study the three-body system with short-range interactions characterized by an unnaturally large two-body scattering length. We show that the off-shell scattering amplitude is cutoff independent up to power corrections. This allows us to derive an exact renormalization group equation for the three-body force. We also obtain a renormalized equation for the off-shell scattering amplitude. This equation is invariant under discrete scale transformations. The periodicity of the spectrum of bound states originally observed by Efimov is a consequence of this symmetry. The functional dependence of the three-body scattering length on the two-body scattering length can be obtained analytically using the asymptotic solution to the integral equation. An analogous formula for the three-body recombination coefficient is also obtained.     

Dissociation of Hadrons in Quark Matter Within a Finite-Temperature Field-Theory Approach on the Light Front

We present a relativistic three-body equation to investigate the properties of nucleons in hot and dense nuclear/quark matter. Within the light-front approach we utilize a zero-range interaction to study the three-body dynamics. The relativistic in-medium equation is derived within a systematic Dyson equation approach that includes the dominant medium effects due to Pauli blocking and self-energy corrections. We present the in-medium nucleon mass and calculate the dissociation of the three-body system.     

Three-Body Dynamics in One Dimension: A Test Model for the Three-Nucleon System with Irreducible Pionic Diagrams

 We formulate the three-body problem in one dimension in terms of the (Faddeev-type) integral equation approach. As an application, we develop a spinless, one-dimensional (1-D) model that mimics three-nucleon dynamics in one dimension. Using simple two-body potentials that reproduce the deuteron binding, we obtain that the three-body system binds at about 7.5 MeV. We then consider two types of residual pionic corrections in the dynamical equation; one related to the 2π-exchange three-body diagram, the other to the 1π-exchange three-body diagram. We find that the first contribution can produce an additional binding effect of about 0.9 MeV. The second term produces smaller binding effects, which are, however, dependent on the uncertainty in the off-shell extrapolation of the two-body t-matrix. This presents interesting analogies with what occurs in three dimensions. The paper also discusses the general three-particle quantum scattering problem, for motion restricted to the full line. Received March 5, 2002; accepted July 19, 2002     

Effects of three-body interactions in the parametric and modulational instabilities of Bose-Einstein condensates

The parametric modulational instability for a discrete nonlinear Schrödinger equation with a cubic-quintic nonlinearity is analyzed. This model describes the dynamics of BECs, with both two- and three-body interatomic interactions trapped in an optical lattice. We identify and discuss the salient features of the three-body interaction in the parametric modulational instability. It is shown that the three-body interaction term can both, shift as well as narrow the window of parametric instability, and also change the behavior of a modulationally stable and parametrically unstable BEC with attractive two-body interaction. We explore this instability through the multiple-scale analysis and identify it numerically. The effect of the three body losses have also been investigated.     

Path of Momentum Integral in the Skorniakov-Ter-Martirosian Equation

The Skorniakov-Ter-Martirosian (STM) integral equation is widely used for the quantum three-body problems of low-energy particles (e.g., ultracold atom gases). With this equation these three-body problems can be efficiently solved in the momentum space. In this approach the boundary condition for the case that all the three particles are gathered together is described by the upper limit of the momentum integral, i.e., the momentum cutoff. On the other hand, in realistic systems, the three-body recombination (TBR) process can occur when all these three particles are close to each other. In this process two particles form a deep dimer and the other particle can gain high kinetic energy and then escape from the low-energy system. In the presence of the TBR process, the momentum-cutoff in the STM equation would include a non-zero imaginary part. As a result, the momentum integral in the STM equation should be done in the complex-momentum plane. In this case the result of the integral depends on the choice of the integral path. Obviously, only one integral path can lead to the correct result. In this paper we consider how to correctly choose the integral path for the STM equation. We take the atom-dimer scattering problem in a specific ultracold atom gas as an example, and show the results given by different integral paths. Based on the result for this case we explore the reasonable integral paths for general case.     

Quadratic and Coulomb Terms for the Spectrum of a Three-Electron Quantum Dot

We consider the Hamiltonian of a three-electron quantum dot composed of quadratic plus Coulomb terms and calculate the system’s spectra. We next apply the hyperradius to reduce the three-body Schrödinger equation into a one-variable differential equation that is solvable. To avoid the complexity, the Taylor expansion of the effective potential is enters the problem and thereby a solution is found for the eigenvalues of the corresponding three-body Schrödinger equation in terms of the Wigner parameter.     

The Stability of the Relativistic Three-Body System and In-Medium Equations

We present a relativistic three-body equation to study the stability of the isolated three-body system and the correlations in a medium of finite temperatures and densities. Relativity is implemented utilizing the light-front form. Using a zero-range force we find the relativistic analog of the Thomas collapse and investigate the possibility that the nucleon exists as a Borromean system. Within a systematic Dyson equation approach we calculate the three-body Mott transition and the critical temperature of the color-superconducting phase.     

Bound States for Hypercentral Singular and Exponential Potentials

With a view to obtaining an exact closed form solution to the Schrodinger equation for a variety of hypercentral potentials, we investigate further application of an ansatz. This method is good enough for many kinds of potentials, but in this article it applies to a type of the hypercentral singular potentials V(x) = ax2 bx-4 cx-6 and exponential hypercentral Morse potential U(x) = U0 ( e-2ax -2 e-ax) for three interacting particles. The Morse potential is used for diatomic molecule while this method will be successfully used for many atomic molecules. The three-body potentials are more easily introduced and treated within the hyperspherical harmonic formalism. The internal particle motion is usually described by means of Jacobi relative coordinates ρ, λ, and R, in terms of three particle positions r1,r2, and r3. We discuss some results obtained by using harmonic and anharmonic oscillators, however the hypercentral potential can be easily generalized in order to allow a systematic analysis, which admits an exact solution of the wave equation. This method is also applied to some other types of three-body, four-body, ..., interacting potentials.     

Model Study of Three-Body Forces in the Three-Body Bound State

The Faddeev equation for the three-body bound state with two- and three-body forces is solved directly as three-dimensional integral equation. The numerical feasibility and stability of the algorithm, which does not employ partial wave decomposition is demonstrated. The three-body binding energy and the full wave function are calculated with Malfliet-Tjon-type two-body potentials and scalar two-meson exchange three-body forces. For two- and three- body forces of ranges and strengths typical of nuclear forces the single-particle momentum distribution and the two-body correlation function are similar to the ones found for realistic nuclear forces.     

新生中子星的性质与三体核力效应

在Brueckner-Hartree-Fock理论框架内, 研究了新生中子星的状态方程和性质, 计算了新生中子星的最大质量和新生中子星中质子占总核子数的丰度, 特别是讨论了三体核力和中微子束缚效应的影响以及三体核力和中微子束缚效应的相互影响. 结果表明, 无论是否考虑三体核力, 中微子束缚对新生中子星的状态方程和质子丰度均有明显影响. 中微子束缚导致新生中子星物质中的质子丰度显著增大. 三体核力的贡献是使新生中子星的状态方程变硬并导致新生中子星中质子丰度明显增大. 束缚在中子星物质中的中微子显著减弱了三体核力对于中子星物质中质子丰度的影响.     

Wave Trains Generation in a Delayed Nonlinear Response of Bose-Einstein Condensates with Three-Body Interactions

We investigate the modulational instability of matter-wave condensates in a modified Gross-Pitaevskii equation which takes into account effects of the three-body interaction. This three-body interaction consists of a quintic term and an additional one representing the delayed nonlinear response of condensates which are trapped both in an attractive and a repulsive harmonic potentials. Our theoretical study uses a modified lens-type transformation and we obtain a modulational instability criterion, and an explicit growth rate. We show that the presence of the three-body interaction destabilizes the condensate, and enhances the appearance of instability in the condensate. Numerical experiments agree well with analytical predictions. Furthermore, our numerical simulations show that the three-body interaction modifies the symmetry of the trail of soliton chains created. The expulsive potential enhances the instability, while the attractive potential appears to soften the instability.     

Modulational Instability of Dipolar Bose-Einstein Condensates in Optical Lattices with Three-Body Interactions

Motivated by the recent experiment [Nature 530(2016) 194] in which a stable droplet in a dipolar quantum gas has been created by the interaction-induced instability, we focus on the modulation instability of an optically-trapped dipolar Bose-Einstein condensate with three-body interaction. Within the mean-field level, we analytically solve the discrete cubic-quintic Gross-Pitaevskii equation with dipole-dipole interaction loaded into a deep optical lattice and show how combined effects of the three-body interaction and dipole-dipole interaction on the condition of modulational instability. Our results show that the interplay of the three-body interaction and dipole-dipole interaction can dramatically change the modulation instability condition compared with the ordinary Gross-Pitaevskii equation. We believe that the predicted results in this work can be useful for the future possible experiment of loading a Bose-Einstein condensate of ~(164)Dy atoms with strong magnetic dipole-dipole interaction into an optical lattice.     

Uniqueness of First Order Post-Newtonian Collinear Solutions for Three-Body Problem under a Scalar-Tensor Theory

As a continuing investigation of an earlier work that establishes the collinear solutions to the three-body problem with general masses under a scalar-tensor theory, we study these solutions and prove their uniqueness up to the first order post-Newtonian approximation. With the help of observed bounds on the scalar field in the Solar System,we show that the seventh-order polynomial equation determining the distance ratio among the three masses has either one or three positive roots. However, in the case with three positive roots, it is found that two positive roots break down the slow-motion condition for the post-Newtonian approximation so that only one positive root is physically valid.The resulting uniqueness suggests that the locations of the three masses are very close to their Newtonian positions with post-Newtonian corrections of general relativity and the scalar field. We also prove that, in the framework of the scalar-tensor theory, the angular velocity of the collinear configuration is always less than the Newtonian one when all other parameters are fixed. These results are valid only for three-body systems where upper-bounds on the scalar field are compatible with those of the Solar System.     

A unitary model for three-body collisions

A connected 3 → 3 formalism for three-body collision processes is reduced to a hierarchy of three on-energy-shell integral equations and one off-energy-shell integral equation. Only the on-energy-shell equations, which involve only on-energy-shell three-body and two-body amplitudes, need be solved exactly in order to obtain elastic and break-up amplitudes satisfying the unitarity constraints exactly. Applied to n-d break-up, the on-energy-shell equations ensure that the n-d initial-state interaction, the nucleon-nucleon final-state interactions, and more complicated 3 → 3 processes are correctly described. After angular momentum analysis the on-energy-shell equations are one-dimensional integral equations, even in the case of local two-body potentials. This unitary model provides a practical scheme for calculating approximate three-body elastic and break-up amplitudes when two-body local potentials are used to describe the two-body subsystems.     

Connecting neutron star observations to three-body forces in neutron matter and to the nuclear symmetry energy

Using a phenomenological form of the equation of state of neutron matter near the saturation density which has been previously demonstrated to be a good characterization of quantum Monte Carlo simulations, we show that currently available neutron star mass and radius measurements provide a significant constraint on the equation of state of neutron matter. At higher densities we model the equation of state by using polytropes and a quark matter model. We show that observations offer an important constraint on the strength of the three-body force in neutron matter, and thus some theoretical models of the three-body force may be ruled out by currently available astrophysical data. In addition, we obtain an estimate of the symmetry energy of nuclear matter and its slope that can be directly compared to the experiment and other theoretical calculations.     

Approximate Treatment of 3-Body Coulomb Systems: Discrete Spectrum

We present a method for treatment of three charged particles. The proposed method has universal character and is applicable both for bound and continuum states. A finite-rank approximation is used for Coulomb potential in the Lippman?CSchwinger equation, that results in a system of one-dimensional coupled integral equations. Preliminary numerical results for three-body atomic and molecular systems like H ^?, He, pp?? and other are presented.     

Universality of the three-body parameter for Efimov states in ultracold cesium

We report on the observation of triatomic Efimov resonances in an ultracold gas of cesium atoms. Exploiting the wide tunability of interactions resulting from three broad Feshbach resonances in the same spin channel, we measure magnetic-field dependent three-body recombination loss. The positions of the loss resonances yield corresponding values for the three-body parameter, which in universal few-body physics is required to describe three-body phenomena and, in particular, to fix the spectrum of Efimov states. Our observations show a robust universal behavior with a three-body parameter that stays essentially constant.     

Nonlocal properties of dynamical three-body Casimir-Polder forces

We consider the three-body Casimir-Polder interaction between three atoms during their dynamical self-dressing. We show that the time-dependent three-body Casimir-Polder interaction energy displays nonlocal features related to quantum properties of the electromagnetic field and to the nonlocality of spatial field correlations. We discuss the measurability of this intriguing phenomenon and its relation with the usual concept of stationary three-body forces.     

Correlations in Hot and Dense Quark Matter

 We present a relativistic three-body equation to investigate three-quark clusters in hot and dense quark matter. To derive such an equation we use the Dyson equation approach. The equation systematically includes the Pauli blocking factors as well as the self-energy corrections of quarks. Special relativity is realized through the light front form. Presently we use a zero-range force and investigate the Mott transition. Received October 29, 2001; accepted for publication November 12, 2001