Three-body Case Research Articles

Emergent structural correlations in dense liquids.

The complete quantitative description of the structure of dense and supercooled liquids remains a notoriously difficult problem in statistical physics. Most studies to date focus solely on two-body structural correlations, and only a handful of papers have sought to consider additional three-body correlations. Here, we go beyond the state of the art by extracting many-body static structure factors from molecular dynamics simulations and by deriving accurate approximations up to the six-body structure factor via density functional theory. We find that supercooling manifestly increases four-body correlations, akin to the two- and three-body case. However, at small wave numbers, we observe that the four-point structure of a liquid drastically changes upon supercooling, both qualitatively and quantitatively, which is not the case in two-point structural correlations. This indicates that theories of the structure or dynamics of dense liquids should incorporate many-body correlations beyond the two-particle level to fully capture their intricate behavior.

Corrigendum: Classical n-body system in geometrical and volume variables. I. Three-body case

International Journal of Modern Physics AVol. 36, No. 24, 2192004 (2021) CorrigendumFree AccessCorrigendum: Classical n-body system in geometrical and volume variables. I. Three-body caseis erratum ofClassical n-body system in geometrical and volume variables: I. Three-body caseA. M. Escobar-Ruiz, R. Linares, Alexander V. Turbiner and Willard Miller, Jr.A. M. Escobar-RuizCorresponding author.Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, México, CDMX 09340, Mexico, R. LinaresDepartamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, México, CDMX 09340, Mexico, Alexander V. TurbinerInstituto de Ciencias Nucleares, UNAM, México, CDMX 04510, Mexico and Willard Miller, Jr.School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USAhttps://doi.org/10.1142/S0217751X21920044Cited by:0 Previous AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail [Int. J. Mod. Phys. A, Vol. 36, No. 18, 2150140 (2021), DOI: 10.1142/S0217751X21501402]Published: 2 September 2021 You currently do not have access to the full text article. Recommend the journal to your library today! FiguresReferencesRelatedDetailsRelated articlesClassical n-body system in geometrical and volume variables: I. Three-body case25 Jun 2021International Journal of Modern Physics A Recommended Vol. 36, No. 24 Metrics Downloaded 14 times History Received 19 July 2021 Accepted 21 July 2021 PDF download

Classical n-body system in geometrical and volume variables: I. Three-body case

We consider the classical three-body system with [Formula: see text] degrees of freedom [Formula: see text] at zero total angular momentum. The study is restricted to potentials [Formula: see text] that depend solely on relative (mutual) distances [Formula: see text] between bodies. Following the proposal by J. L. Lagrange, in the center-of-mass frame we introduce the relative distances (complemented by angles) as generalized coordinates and show that the kinetic energy does not depend on [Formula: see text], confirming results by Murnaghan (1936) at [Formula: see text] and van Kampen–Wintner (1937) at [Formula: see text], where it corresponds to a 3D solid body. Realizing [Formula: see text]-symmetry [Formula: see text], we introduce new variables [Formula: see text], which allows us to make the tensor of inertia nonsingular for binary collisions. In these variables the kinetic energy is a polynomial function in the [Formula: see text]-phase space. The three-body positions form a triangle (of interaction) and the kinetic energy is [Formula: see text]-permutationally invariant with respect to interchange of body positions and masses (as well as with respect to interchange of edges of the triangle and masses). For equal masses, we use lowest order symmetric polynomial invariants of [Formula: see text] to define new generalized coordinates, they are called the geometrical variables. Two of them of the lowest order (sum of squares of sides of triangle and square of the area) are called volume variables. It is shown that for potentials which depend on geometrical variables only (i) and those which depend on mass-dependent volume variables alone (ii), the Hamilton’s equations of motion can be considered as being relatively simple. We study three examples in some detail: (I) three-body Newton gravity in [Formula: see text], (II) three-body choreography in [Formula: see text] on the algebraic lemniscate by Fujiwara et al., where the problem becomes one-dimensional in the geometrical variables and (III) the (an)harmonic oscillator.

Three-body non-leptonic heavy-to-heavy B decays at NNLO in QCD

Exclusive non-leptonic two-body decays of B mesons have been studied extensively in the past two decades within the framework of factorization. However, the exploration of the corresponding three-body case has only started recently, in part motivated by new data. We consider here the simplest non-leptonic three-body B decays from the point of view of factorization, namely heavy-to-heavy transitions. We provide a careful derivation of the SCET/QCDF factorized amplitudes to NNLO in αs, and discuss the numerical impact of NLO and NNLO corrections. We then study the narrow-width limit, showing that the three-body amplitude reproduces analytically the known quasi-two-body decay amplitudes, and compute finite-width corrections. Finally, we discuss certain observables that are sensitive to perturbative NLO and NNLO corrections and to higher Gegenbauer moments of the dimeson LCDAs. This is the first study of non-leptonic three-body B decays to NNLO in QCD.

Long-range interaction of Li(2S2)−Li(2S2)−Li+(1S1)

The long-range interactions among two- or three-atom systems are of considerable importance in the cold and ultracold research areas for many-body systems. For an ion and an atom, the long-range interaction potential is dominated by the induction (or polarization) potential resulting from the (classical) effect of the ion's electric field on the atom and the leading term of the induction potential is much stronger than the (quantum mechanical) dispersion (or van der Waals) interaction. The present paper focuses on the long-range interaction of the $\mathrm{Li}(2\phantom{\rule{0.16em}{0ex}}^{2}S)\ensuremath{-}\mathrm{Li}(2\phantom{\rule{0.16em}{0ex}}^{2}S)\ensuremath{-}{\mathrm{Li}}^{+}(1\phantom{\rule{0.16em}{0ex}}^{1}S)$ system, to see what changes this induction effect (originating in the electric field of the ${\mathrm{Li}}^{+}$ ion) yields in the long-range additive and nonadditive interactions of this three-body system. Using perturbation theory for energies, we evaluate the coefficients ${C}_{n}$ in the potential energy for the three well-separated constituents, where $n$ refers to the corresponding order in inverse powers of distance, obtaining the additive interaction coefficients ${C}_{4}, {C}_{6}, {C}_{7}, {C}_{8}, {C}_{9}$ and the nonadditive interaction coefficients ${C}_{7}, {C}_{9}$. The obtained coefficients ${C}_{n}$ are calculated with highly accurate variationally generated nonrelativistic wave functions in Hylleraas coordinates. Our calculations may be of interest for the study of three-body recombination and for constructing precise potential energy surfaces. We also provide precise evaluations of the long-range potentials for the two-body $\mathrm{Li}(2\phantom{\rule{0.16em}{0ex}}^{2}S)\ensuremath{-}{\mathrm{Li}}^{+}(1\phantom{\rule{0.16em}{0ex}}^{1}S)$ system. For both the two-body and three-body cases, we provide results for the like-nuclei cases of $^{6}\mathrm{Li}$ and $^{7}\mathrm{Li}$.

Dissociation dynamics of the water dication following one-photon double ionization. II. Experiment

Author(s): Reedy, D; Williams, JB; Gaire, B; Gatton, A; Weller, M; Menssen, A; Bauer, T; Henrichs, K; Burzynski, P; Berry, B; Streeter, ZL; Sartor, J; Ben-Itzhak, I; Jahnke, T; Dorner, R; Weber, T; Landers, AL | Abstract: We demonstrate the use of cold target recoil ion momentum spectroscopy to perform state-selective measurements of the dissociative channels following single-photon double-ionization of H2O. The two dominant dissociation channels observed lead to two-body (OH++H++2e-) and three-body (2H++O+2e-) ionic fragmentation channels. In the two-body case we observe the presence of an autoionization process with a double-differential cross section that is similar to the single-photon double-ionization of helium well above threshold. In the three-body case, momentum and energy correlation maps in conjunction with new classical trajectory calculations in the companion theory paper by Z. L. Streeter et al. [Phys. Rev. A 98, 053429 (2018)10.1103/PhysRevA.98.053429] lead to the determination of the eight populated dication states and their associated fragmentation geometry. For the latter case, state-specific relative cross sections, median kinetic energy releases, and median angles between asymptotic proton momenta are presented. This benchmark-level experiment demonstrates that, in principle, state-selective fixed-frame triple-differential cross sections can be measured for some dication states of the water molecule.

Four-body decays of B meson with lepton number violation

The existence of heavy meson lepton number violating (LNV) processes shows the Majorana nature of the neutrino. Much of this theoretical and experimental researche focuses on this type of decay. Four-body epton-number violation (LNV) processes of the B meson may have sizable branching ratios as they share the same vertex and mixing parameters with the three-body case. Mixing parameters between the heavy Majorana neutrino and charged leptons extracted from the three-body case can be used to constrain the branching ratios of four-body decays of the B meson. So we can update the upper limits of these mixing parameters with new experimental data of the three-body LNV decays. We also analyze using the updated parameters and estimate some channels’ reconstruction events using current experimental data from Belle.

Post-Newtonian effects on the stability of the triangular solution in the three-body problem for general masses

Continuing work initiated in earlier publications [Ichita, Yamada and Asada, Phys. Rev. D {\bf 83}, 084026 (2011); Yamada and Asada, Phys. Rev. D {\bf 86}, 124029 (2012)], we examine the post-Newtonian (PN) effects on the stability of the triangular solution in the relativistic three-body problem for general masses. For three finite masses, a condition for stability of the triangular solution is obtained at the first post-Newtonian (1PN) order, and it recovers previous results for the PN restricted three-body problem when one mass goes to zero. The stability regions still exist even at the 1PN order, though the PN triangular configuration for general masses is less stable than the PN restricted three-body case as well as the Newtonian one.

Lepton-number violating four-body decays of heavy mesons

Neutrinoless hadron Lepton Number Violating (LNV) decays can be induced by virtual Majorana neutrino, which in turn indubitably show the Majorana nature of neutrinos. Many three-body LNV processes and Lepton Flavour Violating (LFV) processes have been studied extensively in theory and by experiment. As a supplement, we here study 75 four-body LNV (LFV) processes from heavy pseudoscalar $B$ and $D$ decays. Most of these processes have not been studied in theory and searched for in experiment, while they may have sizable decay rates. Since the four-body decay modes have the same vertexes and mixing parameters with three-body cases, so their branching fractions are comparable with the corresponding three-body decays. We calculate their decay widths and branching fractions with current bounds on heavy Majorana neutrino mixing parameters, and estimate some channels' reconstruction events using the current experimental data from Belle.

Avoiding dissipation in a system of three quantum harmonic oscillators

We analyze the symmetries in an open quantum system composed by three coupled and detuned harmonic oscillators in the presence of a common heat bath. It is shown analytically how to engineer the couplings and frequencies of the system so as to have several degrees of freedom unaffected by decoherence, irrespective of the specific spectral density or initial state of the bath. This partial thermalization allows observing asymptotic entanglement at moderate temperatures, even in the nonresonant case. This latter feature cannot be seen in the simpler situation of only two oscillators, highlighting the richer structural variety of the three-body case. When departing from the strict conditions for partial thermalization, a hierarchical structure of dissipation rates for the normal modes is observed, leading to a long transient where quantum correlations such as the quantum discord are largely preserved, as well as to synchronous dynamics of the oscillators quadratures.

LAGRANGE SOLUTIONS TO THE DISCRETE-TIME GENERAL THREE-BODY PROBLEM

There is no known integrator that yields exact orbits for the general three-body problem (G3BP). It is difficult to verify whether a numerical procedure yields the correct solutions to the G3BP because doing so requires knowledge of all 11 conserved quantities, whereas only six are known. Without tracking all of the conserved quantities, it is possible to show that the discrete general three-body problem (d-G3BP) yields the correct orbits corresponding to Lagrange solutions of the G3BP. We show that the d-G3BP yields the correct solutions to the G3BP for two special cases: the equilateral triangle and collinear configurations. For the triangular solution, we use the fact that the solution to the three-body case is a superposition of the solutions to the three two-body cases, and we show that the three bodies maintain the same relative distances at all times. To obtain the collinear solution, we assume a specific permutation of the three bodies arranged along a straight rotating line, and we show that the d-G3BP maintains the same distance ratio between two bodies as in the G3BP. Proving that the d-G3BP solutions for these cases are equivalent to those of the G3BP makes it likely that the d-G3BP and G3BP solutions are equivalent in other cases. To our knowledge, this is the first work that proves the equivalence of the discrete solutions and the Lagrange orbits.

A spectral approach based on generalized Sturmian functions for two- and three-body scattering problems

A methodology based on generalized Sturmian functions is put forward to solve two- and three-body scattering problems. It uses a spectral method which allows for the inclusion of the correct asymptotic behavior when solving the associated driven Schrödinger equation. For the two-body case, we demonstrate the equivalence between the exterior complex scaling (ECS) and the Sturmian approaches and illustrate the latter by using Hulthén Sturmian functions. Contrary to the ECS approach, no artificial cut-off of the potential is required in the Surmian approach. For the three-body scattering problem, the theoretical framework is presented in hyperspherical coordinates and a set of hyperspherical generalized Sturmian functions possessing outgoing asymptotic behavior is introduced. The Sturmian procedure is a direct generalization of the method discussed for the two-body problem; thus, the comparison with the ECS method is similar. For both the two- and three-body cases, Sturmian bases are efficient as they possess the correct outgoing behavior, diagonalize part of the potentials involved and are essentially localized in the region where the unsolved interaction is not negligible. Moreover, with the Sturmian basis, the operator (H − E) is represented by a diagonal matrix whose elements are simply the Sturmian eigenvalues.

Three-body dwell time

The lifetime of an unstable state or resonance formed as an intermediate state in two-body scattering is known to be related to the dwell time or the time spent within a given region of space by the two interacting particles. This concept is extended to the case of three-body systems and a relation connecting the three-body dwell time with the two-body dwell times of the substructures of the three-body system is derived for the case of separable wave functions. The Kapur-Peierls formalism is revisited to discover one of the first definitions of dwell time in the literature. An extension of the Kapur-Peierls formalism to the three-body case shows that the lifetime of a three-body resonance can indeed be given by the three-body dwell time.

N=4 mechanics, WDVV equations and roots

N=4 superconformal multi-particle quantum mechanics on the real line is governed by two prepotentials, U and F, which obey a system of partial differential equations linear in U and generalizing the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation for F. Putting U=0 yields a class of models (with zero central charge) which are encoded by the finite Coxeter root systems. We extend these WDVV solutions F in two ways: the A_n system is deformed n-parametrically to the edge set of a general orthocentric n-simplex, and the BCF-type systems form one-parameter families. A classification strategy is proposed. A nonzero central charge requires turning on U in a given F background, which we show is outside of reach of the standard root-system ansatz for indecomposable systems of more than three particles. In the three-body case, however, this ansatz can be generalized to establish a series of nontrivial models based on the dihedral groups I_2(p), which are permutation symmetric if 3 divides p. We explicitly present their full prepotentials.

CP asymmetries in chargino production and decay: the three-body decay case

We study CP violation in chargino production and decay in the minimal supersymmetric standard model (MSSM) with complex parameters at an e+e- linear collider with longitudinally polarised beams. We investigate CP-sensitive asymmetries by means of triple product correlations and study their dependence on the complex parameters M1 and μ. We give numerical predictions for the asymmetries and their measurability at the future International Linear Collider. Our results show that the CP asymmetries can be measured in a large region of the MSSM parameter space.

Inverse scattering problem for time dependent Hartree-Fock equations in the three-body case

We consider an inverse scattering problem for time dependent Hartree-Fock equations in the three-body case. It will be shown that interaction potentials with simple forms are uniquely determined by the corresponding scattering data. We also give reconstruction formulas.

The Bertlmann-Martin Inequalities in the Three-Body Problem

The Bertlmann-Martin inequalities (BMI) are studied in the three-body case. We consider distinguishable particles and scalar interactions. The systems are described by Schrodinger equations with local potentials. Under these conditions we show that the lower bound character of the BMI is preserved. We discuss the question of the correction factors transforming the inequalities into approximate or exact relationships. As illustrative example, a simple model in the (D = 1)-dimensional space is considered, for which the exact solution is known. We also study what can be learned from the Hartree approximation and the hyperspherical method in the central potential approximation with respect to the BMI inequalities and correction factors.

N-body decomposition of bipartite author networks

In this paper, we present a method to project co-authorship networks, that accounts in detail for the geometrical structure of scientists' collaborations. By restricting the scope to three-body interactions, we focus on the number of triangles in the system, and show the importance of multi-scientist (more than two) collaborations in the social network. This motivates the introduction of generalized networks, where basic connections are not binary, but involve arbitrary number of components. We focus on the three-body case and study numerically the percolation transition.

New many-body problems in the plane with periodic solutions

In this paper we discuss a family of toy models for many-body interactions including velocity-dependent forces. By generalizing a construction due to Calogero, we obtain a class of N-body problems in the plane which have periodic orbits for a large class of initial conditions. The two- and three-body cases (N=2, 3) are exactly solvable, with all solutions being periodic, and we present their explicit solutions. For N⩾4 Painlevé analysis indicates that the system should not be integrable, and some periodic and non-periodic trajectories are calculated numerically. The construction can be generalized to a broad class of systems, and the mechanism which describes the transition to orbits with higher periods, and eventually to aperiodic or even chaotic orbits, could be present in more realistic models with a mixed phase space. This scenario is different from the onset of chaos by a sequence of Hopf bifurcations.

Molecular simulation study of cooperativity in hydrophobic association: clusters of four hydrophobic particles

The multibody contribution to the potential of mean force (PMF) of hydrophobic association of four methane molecules in water was investigated by means of umbrella-sampling molecular dynamics. Two systems were considered: (i) a trigonal pyramid with three methane molecules at contact distance forming a fixed base, the fourth molecule being placed on the top with variable distance from the base; and (ii) a regular uniformly expanding tetrahedron. Methane–methane distances as far as 12.5 Å, i.e. beyond the second solvent-separated minimum of the PMF, were considered to address the baseline problem. In contrast to the small effect in the three-body case studied previously (Protein Sci 9 (2000) 1235), the multibody contribution was found to amount to approximately 0.2 kcal/mol per methane–methane pair, or approximately 25% of the depth of the contact minimum in the PMF. The main effect of the multibody contribution to the PMF is a reduction of the height of the barrier between the contact and solvent separated minima and a narrowing of the region of its maximum, while the region of the contact minimum is affected only weakly. The reduction of the barrier is due to four-body contributions. The cooperative contributions to the PMF agree very well with those computed from the molecular surface of the systems under consideration, which further supports earlier observations that the molecular surface can be used with good accuracy to describe the energetics of hydrophobic association.