Four-body System Research Articles

The Kozai-Lidov Mechanism in Binary Star Systems and the Stabilizing Effect of Binary Stars

This study used Python simulations to compare the orbits of planets in single-planet binary systems and two-planet binary systems with equivalent single star systems to examine the differences of the Kozai-Lidov mechanism in three-body single star systems and four-body binary star systems. The Python program simulated the forces between the stars and the planets, and produced inclination graphs and orbital diagrams. In the single-planet scenario, no Kozai-Lidov effect is observed as the additional star did not affect the planet’s inclination, but the orbit is considerably rotated. In the two-planet scenario, the Kozai-Lidov oscillation period decreases from 650 years to 370 years, and the maximum inclination decreases from 0.690 radians to 0.280 radians, indicating that the binary star system acts against the Kozai-Lidov Mechanism. Additional trials with modified initial configurations of the two-planet binary star system confirm the universality of the suppressive effect in varying initial conditions in binary star systems.

Stability of the coplanar planetary four-body system

We consider the coplanar planetary four-body problem, where three planets orbit a large star without the cross of their orbits. The system is stable if there is no exchange or cross of orbits. Starting from the Sundman inequality, the equation of the kinematical boundaries is derived. We discuss a reasonable situation, where two planets with known orbits are more massive than the third one. The boundaries of possible motions are controlled by the parameter c2E. If the actual value of c2E is less than or equal to a critical value (c2E)cr, then the regions of possible motions are bounded and therefore the system is stable. The criteria obtained in special cases are applied to the Solar System and the currently known extrasolar planetary systems. Our results are checked using N-body integrator.

Masses of doubly heavy tetraquarks TQQ′ in a relativized quark model

In the present work, the mass spectra of doubly heavy tetraquarks $T_{QQ^\prime}$ are systematically investigated in a relativized quark model. The four-body systems including the Coulomb potential, confining potential, spin-spin interactions, and relativistic corrections are solved within the variational method. Our results suggest that the $IJ^P=01^+$ $bb \bar u \bar d$ state is 54 MeV below the relevant $\bar B \bar B$ and $\bar B \bar B^*$ thresholds, which indicates that both strong and electromagnetic decays are forbidden, and thus this state can be a stable one. Its large hidden color component and small root mean square radius demonstrate that it is a compact tetraquark rather than a loosely bound molecule or point-like diquark-antidiquark structure. Our predictions of the doubly heavy tetraquarks may provide valuable information for future experimental searches.

Coupling of dual mass-transferring white dwarf binaries as a variable gravitational-wave emitter

ABSTRACT We study evolution of a hierarchical four-body (2 + 2) system composed by a pair of mass-transferring white dwarf binaries. Applying a simplified model around the synchronous state of two inner orbital periods, we newly find that the four-body system could settle down to a limit cycle with a small period gap. The period gap generates an amplitude variation of emitted gravitational waves as a beat effect. Depending on model parameters, the beat period could be 1–10 yr and a large amplitude variation might be observed by space gravitational-wave detectors.

Possible Lightest Ξ Hypernucleus with Modern ΞN Interactions.

Experimental evidence exists that the Ξ-nucleus interaction is attractive. We search for NNΞ and NNNΞ bound systems on the basis of the AV8 NN potential combined with either a phenomenological Nijmegen ΞN potential or a first principles HAL QCD ΞN potential. The binding energies of the three-body and four-body systems (below the d+Ξ and ^{3}H/^{3}He+Ξ thresholds, respectively) are calculated by a high precision variational approach, the Gaussian expansion method. Although the two ΞN potentials have significantly different isospin (T) and spin (S) dependence, the NNNΞ system with quantum numbers (T=0, J^{π}=1^{+}) appears to be bound (one deep for Nijmegen and one shallow for HAL QCD) below the ^{3}H/^{3}He+Ξ threshold. Experimental implications for such a state are discussed.

Adiabatic hyperspherical potentials with localized correlated Gaussians

The adiabatic hyperspherical approach is a natural extension of the well-known three-dimensional polar coordinate method to solving a Schr\odinger equation of a few-body system. To evaluate the matrix element of an adiabatic Hamiltonian at a fixed hyper-radius is crucially important in that approach, but due to the difficulty of its calculation real applications have been limited mostly up to four-body systems. To resolve this limitation I introduce a localized hyper-radial function and show that the matrix element needed for $N$-body system can be obtained using correlated Gaussians with arbitrary angular momentum. I demonstrate its feasibility in the systems of $N=3\text{--}6\phantom{\rule{4pt}{0ex}}\ensuremath{\alpha}$ particles. It is pointed out that an extension into correlated Gaussians with double global vectors is desirable for further realistic descriptions of the hyperangular motion.

Four-body calculation of energy levels of muonic molecule dtμe in muon catalyzed fusion

SynopsisWe investigate dtμe (= d + t + μ + e) four-body system which includes the most important bound states (dtμ)11 in muon catalyzed fusion (μCF). We directly estimate the energy shift more accurately than the previous perturbation calculation. The shift found to be one order of magnitude larger than previous calculation and to be more significant in the calculation of the μCF reaction rate.

On Possible Formation of Matter-Antimatter Exotic Molecular Structures

Possible formation of exotic matter-antimatter molecular structure is considered as one of the most challenging problems at International Laboratories of Particle Physics. In the present work, elaborate computer codes built for investigating four-body systems are employed for calculating the binding energies of exotic molecules composed of electrons, protons, muons, pions and their antiparticles. The results confirm the stability of these molecules against dissociation to their lowest possible channels. Based on these results, it is argued that possible creation of two universes immediately after the Big Bang should be considered. Particularly, it is proposed that an overlapping area might exist between the universe and antiuniverse in which continuous creation and annihilation of simple and complicated particle-antiparticle structures might occur. Antiparticles escaping from this area are considered as the origin of the minimal traces of antiparticles appearing in our universe. Recent interpretations of cosmic-rays and gamma-radiations observed at the edge of our universe could be thought of as evidences for supporting this argument. Furthermore, it is argued that possible formation of matter-antimatter molecular structures could open the gate in front of a new field of chemistry to be referred to as Antimatter Chemistry.

Zero dynamics analysis and adaptive tracking control of underactuated multibody systems with flexible links

This paper studies the adaptive tracking control problem for underactuated multibody systems with flexible links in the presence of unknown parameters. A four-body system with input signals acting on the first and fourth bodies is chosen as a benchmark model, which can be decomposed into a control dynamics subsystem and a zero dynamics subsystem. A new and detailed stability analysis is conducted to show that such a zero dynamic system is Lyapunov stable and is partially input-to-state stable under the condition that the speeds of first and fourth bodies are synchronous. The physical meaning of such partial input-to-state stability is clarified. An adaptive controller is developed to ensure such a needed system stabilisation condition and thus the boundedness of the desired closed-loop system signal and asymptotic speed tracking of the control dynamics subsystem. Detailed closed-loop system stability and tracking performance analysis are given, in which the tracking errors satisfy the performance. Extensions to the other multibody systems with flexible links are derived. The developed adaptive controller is applied to a realistic train dynamic model, and simulation results verify the desired system performance.

Observation of resonant scattering between ultracold heteronuclear Feshbach molecules

We report the observation of a dimer-dimer inelastic collision resonance for ultracold Feshbach molecules made of bosonic sodium and rubidium atoms. This resonance, which we attribute to the crossing of the dimer-dimer threshold with a heteronuclear tetramer state, manifests itself as a pronounced inelastic loss peak of dimers when the interspecies scattering length between the constituent atoms is tuned. Near this resonance, a strong modification of the temperature dependence of the dimer-dimer scattering is observed. Our result provides insight into the heteronuclear four-body system consisting of heavy and light bosons and offers the possibility of investigating ultracold molecules with tunable interactions.

Hill stability of coplanar four-body systems with application to the Solar System and extrasolar systems

The $c^{2}E$ condition, for determining the Hill stability of a coplanar four-body system, is analyzed for the particular situation where the mass of a body is much greater than the other three. In this case, the criterion can be expressed in a closed form, which makes it easier to judge the stability of a system or predict the stable regions described by physical and orbital parameters. The criterion is applied to nine known four-body systems in our Solar System. Besides, the situations where a passing star moving on a parabolic or hyperbolic orbit encounters a triple system are considered. The Hill stability of all extrasolar triple systems are investigated during encounters of a passing star.

The onset of ΛΛ hypernuclear binding

Binding energies of light, A≤6, ΛΛ hypernuclei are calculated using the stochastic variational method in a pionless effective field theory (π̸EFT) approach at leading order with the purpose of assessing critically the onset of binding in the strangeness S=−2 hadronic sector. The π̸EFT input in this sector consists of (i) a ΛΛ contact term constrained by the ΛΛ scattering length aΛΛ, using a range of values compatible with ΛΛ correlations observed in relativistic heavy ion collisions, and (ii) a ΛΛN contact term constrained by the only available A≤6 ΛΛ hypernucler binding energy datum of ΛΛ6He. The recently debated neutral three-body and four-body systems ΛΛ3n and ΛΛ4n are found unbound by a wide margin. A relatively large value of |aΛΛ|≳1.5 fm is needed to bind ΛΛ4H, thereby questioning its particle stability. In contrast, the particle stability of the A=5 ΛΛ hypernuclear isodoublet ΛΛ5H–ΛΛ5He is robust, with Λ separation energy of order 1 MeV.

Efimov universality with Coulomb interaction

The universal properties of charged particles are modified by the presence of a long-range Coulomb interaction. We investigate the modification of Efimov universality as a function of the Coulomb strength using the Gaussian expansion method. The resonant short-range interaction is described by Gaussian potentials to which a Coulomb potential is added. We calculate binding energies and root mean square radii for the three- and four-body systems of charged particles and present our results in a generalised Efimov plot. We find that universal features can still be discerned for weak Coulomb interaction, but break down for strong Coulomb interaction. The root-mean-square radius plateaus at increasingly smaller values for strong Coulomb interaction and the probablity distributions of the states become more concentrated inside the Coulomb barrier. As an example, we apply our universal model to nuclei with an alpha-cluster substructure. Our results point to strong non-universal contributions in that sector.

Universality of size-energy ratio in four-body systems

Universal relationship of scaled size and scaled energy, which was previously established for two- and three-body systems in their ground state, is examined for four-body systems, using Quantum Monte Carlo simulations. We study in detail the halo region, in which systems are extremely weakly bound. Strengthening the interparticle interaction we extend the exploration all the way to classical systems. Universal size-energy law is found for homogeneous tetramers in the case of interaction potentials decaying predominantly as r−6. In the case of mixed tetramers, we also show under which conditions the universal line can approximately describe the size-energy ratio. The universal law can be used to extract ground-state energy from experimentally measurable structural characteristics, as well as for evaluation of theoretical interaction models.

Three- and four-body kaonic nuclear states: Binding energies and widths

Using separable potentials for $\bar{K}N-\pi\Sigma$ interaction, we investigated four-body kaonic nuclear systems such as $K^{-}ppn$ and $K^{-}K^{-}pp$, with the Faddeev AGS method in the momentum representation. The Faddeev calculations are based on the quasi-particle method and the method of the energy dependent pole expansion was used to obtain the separable representation for the integral kernels in the three- and four-body equations. Different types of $\bar{K}N-\pi\Sigma$ potentials based on phenomenological and chiral SU(3) approach are used and it was shown that the kaonic nuclear systems under consideration are tightly bound.

Kinetics of the OH+HCl→H2 O+Cl reaction: Rate determining roles of stereodynamics and roaming and of quantum tunneling.

The OH + HCl → H2 O + Cl reaction is one of the most studied four-body systems, extensively investigated by both experimental and theoretical approaches. Here, as a continuation of our previous work on the OH + HBr and OH + HI reactions, which manifest an anti-Arrhenius behavior that was explained by stereodynamic and roaming effects, we extend the strategy to understand the transition to the sub-Arrhenius behavior occurring for the HCl case. As previously, we perform first-principles on-the-fly Born-Oppenheimer molecular dynamics calculations, thermalized at four temperatures (50, 200, 350, and 500 K), but this time we also apply a high-level transition-state-theory, modified to account for tunneling conditions. We find that the theoretical rate constants calculated with Bell tunneling corrections are in good agreement with extensive experimental data available for this reaction in the ample temperature range: (i) simulations show that the roles of molecular orientation in promoting this reaction and of roaming in finding the favorable path are minor than in the HBr and HI cases, and (ii) dominating is the effect of quantum mechanical penetration through the energy barrier along the reaction path on the potential energy surface. The discussion of these results provides clarification of the origin on different non-Arrhenius mechanisms observed along this series of reactions. © 2018 Wiley Periodicals, Inc.

Four-Body Bound-States Systems in Relativistic Scalar Quantum Field Theory: Variational Basis-State Approach

The variational method within the Hamiltonian formalism of quantum field theory has been used in order to investigate the effect of virtual pairs for four-body scalar systems consisting of two particles and two antiparticles of the same mass. The scalar particles and antiparticles interact via a massive or massless mediating scalar field. The ground state energy solutions of Fock-space variational trial states ( $|N{\bar {N}}N{\bar {N}}{ \rangle }+|N{\bar {N}}N{\bar {N}}N{\bar {N}\rangle }$ ) of the relativistic wave equations have been studied. We have compared these results with the previous work of four-body system (variational trial states of the form $|N{ \bar {N}}N{\bar {N}}{\rangle }$ ) and we have shown that the inclusion of virtual pairs has a noticeable effect at low coupling and at high coupling the energy of the system is changed by an important amount. In other words, the calculations show that the inclusion of virtual pairs augments the binding energy of the four-body system by a substantial amount at strong coupling. This study can pave the way for some new ideas in order to investigate the effect of virtual pairs, for example, for a bound-states quark-antiquark or tetraquark systems in future.

Hill stability of the satellite in the elliptic restricted four-body problem

We consider an elliptic restricted four-body system including three primaries and a massless particle. The orbits of the primaries are elliptic, and the massless particle moves under the mutual gravitational attraction. From the dynamic equations, a quasi-integral is obtained, which is similar to the Jacobi integral in the circular restricted three-body problem (CRTBP). The energy constant $C$ determines the topology of zero velocity surfaces, which bifurcate at the equilibrium point. We define the concept of Hill stability in this problem, and a criterion for stability is deduced. If the actual energy constant $C_{\mathrm{ac}}\ ( {>} 0 ) $ is bigger than or equal to the critical energy constant $C_{\mathrm{cr}}$ , the particle will be Hill stable. The critical energy constant is determined by the mass and orbits of the primaries. The criterion provides a way to capture an asteroid into the Earth–Moon system.

Hypernuclear no-core shell model

We extend the No-Core Shell Model (NCSM) methodology to incorporate strangeness degrees of freedom and apply it to single-$\Lambda$ hypernuclei. After discussing the transformation of the hyperon-nucleon (YN) interaction into Harmonic-Oscillator (HO) basis and the Similarity Renormalization Group transformation applied to it to improve model-space convergence, we present two complementary formulations of the NCSM, one that uses relative Jacobi coordinates and symmetry-adapted basis states to fully exploit the symmetries of the hypernuclear Hamiltonian, and one working in a Slater determinant basis of HO states where antisymmetrization and computation of matrix elements is simple and to which an importance-truncation scheme can be applied. For the Jacobi-coordinate formulation, we give an iterative procedure for the construction of the antisymmetric basis for arbitrary particle number and present the formulae used to embed two- and three-baryon interactions into the many-body space. For the Slater-determinant formulation, we discuss the conversion of the YN interaction matrix elements from relative to single-particle coordinates, the importance-truncation scheme that tailors the model space to the description of the low-lying spectrum, and the role of the redundant center-of-mass degrees of freedom. We conclude with a validation of both formulations in the four-body system, giving converged ground-state energies for a chiral Hamiltonian, and present a short survey of the $A\le7$ hyper-helium isotopes.

Nonadiabatic rotational states of the hydrogen molecule.

We present a new computational method for the determination of energy levels in four-particle systems like H2, HD, and HeH+ using explicitly correlated exponential basis functions and analytic integration formulas. In solving the Schrödinger equation, no adiabatic separation of the nuclear and electronic degrees of freedom is introduced. We provide formulas for the coupling between the rotational and electronic angular momenta, which enable calculations of arbitrary rotationally excited energy levels. To illustrate the high numerical efficiency of the method, we present the results for various states of the hydrogen molecule. The relative accuracy to which we determined the nonrelativistic energy reached the level of 10-12-10-13, which corresponds to an uncertainty of 10-7-10-8 cm-1.