Four-body System Research Articles

Secular evolution of circumbinary 2-planet systems with isotropically varying masses

ABSTRACT We investigate the secular evolution of a four-body planetary system, where two planets move around a binary star configuration on quasi-elliptic orbits. It is assumed that the masses of all bodies can change isotropically at different rates and the bodies attract each other according to Newton’s law of universal gravitation. The purpose of this work is to investigate an influence of variability of the masses of the binary stars and their planets on the dynamical evolution of the planetary system. We consider the case of small eccentricities and inclinations of the bodies orbits and assume that their orbits do not intersect during evolution. Differential equations of the perturbed motion in the osculating analogues of canonical Poincaré elements were obtained in a general case of the many-body problem with variable masses in our previous work. Here we solve these equations numerically and investigate the secular evolution of some fictitious circumbinary 2-planet system under assumption that the two stars of the binary system lose their masses independently at different rates. In order to demonstrate the dynamical features of the equations we use the known parameters of the TOI-1338 system. Comparing the results of calculations in the cases of constant and variable masses, we show that the isotropic variability of the masses of bodies can influence substantially the secular perturbation of the orbital elements.

The tetraquark system in a chiral quark model* *Partially supported by the National Natural Science Foundation of China (12305093, 11535005, 11775118), the Natural Science Foundation of Zhejiang Province, China(LQ22A050004), the Ministerio Español de Ciencia e Innovación (PID2019-107844GB-C22, PID2022-140440NB-C22), the Junta de Andalucía under contract Nos. Operativo FEDER Andalucía (2014-2020 UHU-1264517, P18-FR-5057 and also PAIDI FQM-370)

The S-wave tetraquarks, with spin-parities , , and , in both isoscalar and isovector sectors, are systematically studied using a chiral quark model. The meson-meson, diquark-antidiquark, and K-type arrangements of quarks and all possible color wave functions are comprehensively considered. The four-body system is solved using the Gaussian expansion method, a highly efficient computational approach. Additonally, a complex-scaling formulation of the problem is established to disentangle bound, resonance, and scattering states. This theoretical framework has already been successfully applied in various tetra- and penta-quark systems. For the complete coupled channel and within the complex-range formulation, several narrow resonances of and systems are obtained, in each allowed -channel, within the energy regions of GeV and GeV, respectively. The predicted exotic states, which indicate a richer color structure when going towards multiquark systems beyond mesons and baryons, are expected to be confirmed in future high-energy particle and nuclear experiments.

Four-body systems at large cutoffs in effective field theory

Four-body systems at large cutoffs in effective field theory

Sliding Mode-Based Distributed Trajectory Tracking Control of Four-body Train Systems

This paper considers the speed tracking of a four-body train system modelled mathematically based on Newton’s second law, which is described by a large-scale interconnected system with four subsystems. Uncertainties are included in the systems to represent the potential impacts on system performance caused by mechanical wear and external environmental changes. An adaptive sliding mode technique is employed to design a distributed control scheme to guarantee tracking accuracy. Coordinate transformations are introduced to transfer the model of train systems to a system in regular form to facilitate the design of the hyperplane and controllers. The Barbashin-Krasovskii theorem is employed to show the reachability of the hyperplane. In simulations, the Gaussian function is chosen as the desired signal, representing time-varying characteristics relevant to real-world situations, and the result demonstrates the feasibility of the proposed control strategy.

Long-term perturbations in four-body systems with mutual highly inclined orbits

Hierarchical Three Body systems with mutual highly inclined orbits have been well-known since the 1960s. Lidov-Kozai cycles arise within them where the inner orbit eccentricity acquires extreme values. In particular, we focus our research on the motion of exoplanets and exomoons on different Three Body stellar scenarios. Our goal is to study how the LK cycles are perturbed by a fourth body (which we called perturbed LK). We analyze the evolution of the eccentricity and inclination of the inner orbit in two cases: the first involves an exoplanet and the second involves an exomoon. Due to the possible stable configurations of a four-body system, we treat two subcases as well: the Totally Hierarchical Configuration and the 2 + 2 configuration. According to that derived from the particular scenarios of study discussed in the present research, the LK perturbed in exomoon orbits around exoplanets seem to exhibit, in general, way less alterations than the exoplanet orbit around its star.

Light meson emissions of selected charmonium-like states within compact tetraquark configurations* *Supported by the National Natural Science Foundation of China (11705056, 12175037, 11947224, 11475192, 11975245, U1832173), the Key Project of Hunan Provincial Education Department under (21A0039), the State Scholarship Fund of China Scholarship Council (22006725011), the Sino-German CRC 110 “Symmetries and the Emergence of Structure in QCD” project by NSFC

We adopt the quark pair creation model to investigate the light meson emissions of several charmonium-like states. The quark pair creation model is applied to four-body systems, and we calculate the pion/kaon emissions of , , , , , and within compact tetraquark configurations. We find that the pion/kaon decay widths of and are relatively small, whereas the partial decay widths of the resonances , , , and are significant. We expect that our exploration of these decay behaviors will provide useful information for future experimental searches and theoretical interpretations.

Mathematical modelling of elastic vibrations arising from four-body impact interaction

This paper presents a mathematical model of elastic vibrations arising from shock interaction in a four-body system. The resulting model has two applications: the study of linear elastic interaction and the study of nonlinear elastic interaction. The linear elastic body model is presented in analytical form. In the transition to nonlinear interaction, it is difficult to represent the model in analytical approximation. The application of numerical algorithms has made it possible to obtain solution of a nonlinear problem with a specified accuracy sufficient for the simulated process. The model is intended for the description of the behavior of the device for measuring the modulus of elasticity of soil by the stamp method. An algorithm is developed to obtain the displacement and acceleration of the falling load and the loading die. Under laboratory conditions, an experiment with a test rig capable of measuring the acceleration of a loading die during shock-elastic interaction with the falling load is conducted. The measuring part is a three-axis accelerometer sensor connected to a four-channel analog-digital converter, which transmits the data to a mobile device with the developed specialized software. The sampling frequency of the measuring system is 12 kHz for each of the four channels. Polyurethane was used as a nonlinear element of the system, the approximate stiffness of which was determined experimentally by measuring deformations under a given load. When comparing the experimental results with the simulation results, the numerical model parameters were optimized by an algorithm based on gradient descent, which was based on maximizing the correlation coefficient between the theoretical and experimental data.

Wave-packet continuum discretization approach to He2+−He collisions

The two-center wave-packet convergent close-coupling method is generalized to multiply charged ion collisions with helium and applied to the ${\mathrm{He}}^{2+}\text{\ensuremath{-}}\mathrm{He}$ scattering problem. The approach is applicable in a wide range of collision energies including low and intermediate energies, where coupling between various reaction channels and electron exchange between the fragments in the rearrangement channel are important. The target structure is treated using the configuration-interaction method within the frozen-core approximation, where one of the electrons of the target is assumed to stay in the ground state throughout the collision. This accounts for the indistinguishability and the correlations between the electrons of the target. We also use a simpler alternative method that is based on an effective one-electron target description that neglects the electron-electron correlations. In both methods, the continuum of the target atom and the hydrogenlike atom formed after electron capture by the projectile is discretized using the wave-packet approach. We present cross sections for total and state-selective electron capture, excitation, and single ionization of the target. The results are provided for the incident energies from 10 keV/u to 5 MeV/u, where one-electron processes are expected to dominate. Particular attention is focused on the intermediate-energy region where substantial deviations between various theoretical results are found. However, overall, the present results are in good agreement with experimental data and other calculations, where available. We demonstrate that both effective single-electron and two-electron methods can provide a realistic picture of all single-electron scattering processes taking place in ${\mathrm{He}}^{2+}\text{\ensuremath{-}}\mathrm{He}$ collisions in terms of the total cross sections. This creates a reliable platform for modeling differential scattering and ionization in this four-body system.

Saturated absorption spectroscopy of HD at 76 K

Precision spectroscopy of molecular hydrogen can be used to test the quantum electrodynamics theory and determine the proton-to-electron mass ratio in a four-body system. Saturated absorption spectroscopy of the R(1) line in the (2-0) band of HD is measured at 76 K. An asymmetric line profile is observed and analyzed, and the transition frequency is predicted to be $217\phantom{\rule{0.16em}{0ex}}105\phantom{\rule{0.16em}{0ex}}182\phantom{\rule{0.16em}{0ex}}284\phantom{\rule{0.16em}{0ex}}{(11)}_{\mathrm{stat}}{(27)}_{\mathrm{sys}}$ kHz. The value is nine times more accurate than our previous result obtained at room temperature and differs by $6.6\ensuremath{\sigma}$ from the recent result derived with a different line-shape model.

S -wave fully charmed tetraquark resonant states

We calculate the mass spectrum of the $S$-wave fully-charmed tetraquark resonant states $cc\bar c\bar c $ in the nonrelativistic quark model, which successfully describes the charmonium spectrum. The four-body system is solved with the Gaussian expansion method. The complex scaling technique is used to identify the genuine resonances. With the nonrelativistic quark model, our results show the existence of two $cc\bar c\bar c$ resonances in each of the $J^{PC}=$ $0^{++}$, $1^{+-}$ and $2^{++}$ sectors, respectively. In the $S$-wave sector, no resonance is found at the energy region of the $X(6200)$ and $X(6600)$ states. The lower $0^{++}$ and $2^{++}$ resonances are located around $100$ MeV higher than the $X(6900)$ state observed in experiments but have the decay widths consistent with the experiment. The higher $0^{++}$ and $2^{++}$ resonances are found at around $7.2$ GeV with the widths of $60.6$ MeV and $91.2$ MeV, respectively, and they may be good candidates for the $X(7200)$ state.

Cavity-enhanced double resonance spectroscopy of HD

Precision spectroscopy of molecular hydrogen can be used to test the quantum electrodynamics theory and determine the proton-to-electron mass ratio in a four-body system. Cavity-enhanced double-resonance spectroscopy is proposed to determine highly-excited rotation-vibration energies of HD with high precision. As a demonstration, the V-type comb-locked cavity-assisted double resonance (COCA-DR) spectroscopy of HD is measured by pumping the P(1) (2-0) line and probing the R(1) line in the same overtone band. The sum value of the two transition frequencies is determined to be 426 889 423 917 (26) kHz, representing a new test of theoretical calculations.

Three and four identical fermions near the unitary limit

This work analyzes the three and four equal-mass fermionic systems near and at the $s$- and $p$-wave unitary limits using hyperspherical methods. The unitary regime addressed here is where the two-body dimer energy is at zero energy. For fermionic systems near the $s$-wave unitary limit, the hyperradial potentials in the $N$-body continuum exhibit a universal long-range ${R}^{\ensuremath{-}3}$ behavior governed by the $s$-wave scattering length alone. The implications of this behavior on the low-energy phase shift are discussed. At the $p$-wave unitary limit, the four-body system is studied through a qualitative look at the structure of the hyperradial potentials at unitarity for the ${L}^{\ensuremath{\pi}}={0}^{+}$ symmetry. A quantitative analysis shows that there are tetramer states in the lowest hyperradial potentials for these systems. Correlations are made between these tetramers and the corresponding trimers in the two-body fragmentation channels. Universal properties related to the four-body recombination process $A+A+A+A\ensuremath{\leftrightarrow}{A}_{3}+A$ are discussed.

The existence of the quasi-bound state in $$K{\bar{K}}NN$$ four-body system

The existence of the quasi-bound state in $$K{\bar{K}}NN$$ four-body system

Time dependent nature of the Neptune’s arc configuration

By distinguishing the main arc Fraternite from the minor arcs Egalite (2,1), Liberte, Courage, the conservative restricted three-body system is extended to a non-conservative restricted four-body system with the central body Neptune S, the primary body Galatea X, a minor body Fraternite F, and a test body s. Through the equations of motion, it is shown that the locations where the force is null (null points) along the orbit of s correspond to the locations of Egalite (2,1), Liberte, and Courage. Even if all the arcs were captured by three-body CER sites initially, the orbits over the CER potential maxima would be unstable for the minor arcs due to the disturbing force of Fraternite with finite mass, allowing them to be relocated to the four-body null points. On the other hand, the minor arcs do not have the mass to destabilize Fraternite from its three-body CER site, which is enlarged from 8.37 degrees to 9.7 degrees thus accounting for the Fraternite’s span. In this non-conservative restricted four-body system, s is under the effects of the potentials of X and F, therefore the null points are not absolutely stable as the Lagrangian points. The potential of X drives a long period harmonic pendulum oscillation through Φ s x = [ ( n + 1 ) θ s − n θ x − ϕ s ] centered over a four-body null point with T = 1 , 000 d , while the potential of F drives a much longer period singular pendulum oscillation through Δ θ s f centered over Fraternite with T f = 40 , 000 d . In this four-body system, the dynamics of these two oscillations generates a time alternating symmetric arc configuration about Fraternite with a century long time scale. The non-conservative nature of this system could account for the time varying intensity, configuration, and disappearance of the minor arcs.

Four-body variational calculation of a hydrogen-like atom involving an excited muonic molecule

We report a four-body variational calculation of a hydrogen-like atom consisting of an excited muonic molecule consisting of d, t, and μ, and a ground state electron. Due to the compact size of the muonic molecule dtμ, it behaves as a quasi-nucleus for the electron; however, the system is actually a resonance state because the de-excitation energy of dtμ is sufficient to ionize the electron. We calculate resonance energy levels of the four-body system dtμe using a Gaussian expansion method and a stabilization method. Our best calculation results in a four-body energy of –100.159 88 a.u. which is in good agreement with the value estimated by the first order perturbation theory [Harston et al. Zeitschrift für Physik D 22, 635 (1992)]. We indicate that implementation of the electron-induced polarization of dtμ will be indispensable for the further precise determination of the resonance energy. The present result is the first step towards a full four-body calculation of the muonic molecule under the presence of the electron, which is of importance for development of a muon catalyzed fusion kinetics model.

Bound-state properties, positron annihilation and hyperfine structure of the four-body positronium hydrides

Bound-state properties of the ground (bound) -state(s) in the four-body positronium hydrides HPs, HPs (DPs), HPs (TPs) and MuPs are determined and investigated. By using numerical data from our computations of these four-body systems, we have determined a number of different annihilation rates for each of these positronium hydrides and evaluated the hyperfine structure splitting. The properties of the ground (bound) states in the four-body exitonic complexes, where and , have also been evaluated numerically. The neutral four-body systems with are similar to the HPs hydrides. In particular, each of these states has only one bound state. We also discuss applications of the exponential and semi-exponential variational expansions for accurate, bound state computations of the four-body positronium hydrides.

Masses of fully heavy tetraquark states from a four-quark static potential model

In this article, we study the ground-state mass of full-heavy tetraquarks $cc\overline{c}\overline{c}$ and $bb\overline{b}\overline{b}$ with solving the nonrelativistic four-body systems. The flux-tube configurations, the tetraquark four-body potential butterfly, and the dimeson potential flip-flop of SU(3) lattice quantum chromodynamics have been applied to describe the tetraquark interaction. Our numerical analysis indicates that the disconnected and connected static potentials can predict the mass of tetraquark very close to experimental data.

Charge transfer in collisions of H<sup>+</sup>, Li<sup>3+</sup>, Be<sup>4+</sup> and O<sup>7+</sup> ions with He atom based on 4-classical trajectory Monte Carlo method

The classical trajectory Monte Carlo (CTMC) method is a common method to study the charge-transfer and impact-ionization cross sections for the collisions between ions and atoms, and the heavy particle collision in astrophysics and laboratory plasma environment. Here in this work, we use the 4-CTMC method to study a four-body collision process including two bound electrons, and the Hamiltonian equation of the four-body dynamic system is solved numerically. The single/double electron ionization and capture cross sections are calculated for collisions of high charge state ions (Li<sup>3+</sup>, Be<sup>4+</sup> and O<sup>7+</sup>) with helium atom in a wide range of projectile energy. The calculation results show that the results from the 4-CTMC method and the experimental measurements are in better agreement in a projectile energy range of 50－200 keV/amu for proton-helium collision system. In addition, for incident ions with high charge state, the results calculated by the 4-CTMC method are in better agreement with the experimental measurements or other theoretical values in a projectile energy range of 100－500 keV/amu. Though the double ionization and capture cross sections calculated by 4-CTMC or 3-CTMC method are higher than the experimental results due to ignoring the electron correlation, the results from the 4-CTMC method are in better agreement with the experimental results.

Capturing the potential energy landscape of large size molecular clusters from atomic interactions up to a 4-body system using deep learning.

Exploring the structure and properties of molecular clusters with accuracy using the ab initio methods is a resource intensive task due to the increasing cost of the ab initio methods and the number of distinct conformers as the size increases. The energy landscape of methanol clusters has been previously explored using computationally efficient empirical models to collect a database of structurally distinct minima, followed by re-optimization using ab initio methods. In this work, we propose a new method that utilizes the database of stable conformers and borrow the fragmentation concept of many-body-expansion (MBE) methods in ab initio methods to train a deep-learning machine learning (ML) model using SchNet. Picking 684 local minima of (CH3OH)5 to (CH3OH)8 from the existing database, we can generate ∼51 000 data points of one-body, two-body, three-body and four-body molecular systems to train an ML model to reach a mean absolute error (MAE) of 3.19 kJ mol-1 (in energy) and 2.48 kJ mol-1 Å-1 (in forces) tested against ab initio calculations up to (CH3OH)14. This ML model is then used to create a database of low energy isomers of (CH3OH)n (n = 15-20). The proposed scheme can be applied to other hydrogen bonded molecular clusters with an accuracy of first-principles methods and computational speed of empirical force-fields.

Four-body (an)harmonic oscillator in d-dimensional space: S-states, (quasi)-exact-solvability, hidden algebra sl (7)

As a generalization and extension of our previous paper [Turbiner et al., J. Phys. A: Math. Theor. 53, 055302 (2020)], in this work, we study a quantum four-body system in Rd (d ≥ 3) with quadratic and sextic pairwise potentials in the relative distances, rij ≡ |ri − rj|, between particles. Our study is restricted to solutions in the space of relative motion with zero total angular momentum (S-states). In variables ρij≡rij2, the corresponding reduced Hamiltonian of the system possesses a hidden sl(7; R) Lie algebra structure. In the ρ-representation, it is shown that the four-body harmonic oscillator with arbitrary masses and unequal spring constants is exactly solvable. We pay special attention to the case of four equal masses and to atomic-like (where one mass is infinite and three others are equal), molecular two-center (two masses are infinite and two others are equal), and molecular three-center (three infinite masses) cases. In particular, exact results in the molecular case are compared with those obtained within the Born–Oppenheimer approximation. The first and second order symmetries of non-interacting system are searched. In addition, the reduction to the lower dimensional cases d = 1, 2 is discussed. It is shown that for the four-body harmonic oscillator case, there exists an infinite family of eigenfunctions that depend on the single variable, which is the moment of inertia of the system.