Three-body Binding Research Articles

The Movement of Efimov States in 2n Halo Nucleus: probing the roles of mass asymmetry and short range of interaction

This brief communication presents the results of our work to study the movement of Efimov states in 2-neutron halo nuclei. We discuss the emergence and evolution of Efimov states in to resonances beyond the two-body (n-Core) breakup threshold and also the movement of the Efimov state in a 2-neutron halo nucleus to the region of the so called, Thomas Collapse. We especially, probe the roles of mass asymmetry of the three-body system (n-n-core) and the short range of the n-core interaction in the movement of the Efimov states. The analysis has been carried out within the framework of a three-body model, assuming two-body, non-local, separable potentials for the binary sub-systems, namely, the n-n and n-core systems. The primary motivation has been to scan the parameter space of the range parameter and determine the value(s) for which the system can support the Efimov states. The analysis shows that the value of the range parameter $\beta$ = 5.2$\alpha$ ( $\alpha$ being the energy parameter given in terms of the deuteron binding energy ) produces the realistic three-body binding energy for $^{20}$C and also generates the excited Efimov states. However, as the value of $\beta$ is increased corresponding to decrease in the range of the two-body (n-core) interaction the ground and excited state binding energies increase drastically. In other words, as the range of the two-body interaction tends to zero the system enters the Thomas Collapse region away from the Efimov region.

The charge and mass symmetry breaking in the KK K¯ system

In the framework of the Faddeev equations in configuration space, we investigate the K(1460) meson as a resonant state of the KKK¯ kaonic system. We perform calculations for the particle configurations K 0 K + K − and K0K+K0¯ within two models: the ABC model, in which all three particles are distinguishable, and the AAC model when two particles are identical. The models differ in their treatment of the kaon mass difference and the attractive Coulomb force between the K + K − pair. We found that the Coulomb shift adds over 1 MeV to the three-body binding energy. The expected correction to the binding energy due to mass redistribution from AA to AB is found to be negligible, up to a maximum of 6% of the relative mass correction. At the same time, the symmetry of the wave function is distorted depending on the mass ratio value. We found that the repulsive KK interaction plays an essential role in the binding energy of the KKK¯ system and reports the mass of 1461.8 or 1464.1 MeV for the neutral K 0(1460) and 1466.5 or 1468.8 MeV for the charged K +(1460) resonances, respectively, depending on the parameter sets for KK and KK¯ interactions.

Geometrical properties of ΩNN three-body states by realistic NN and first principles Lattice QCD ΩN potentials

The Faddeev equations in coordinate space are solved to study the ΩNN and ΩΩN three-body systems using the latest ΩNS25 and ΩΩ S01 interactions developed by the HAL QCD Collaboration. We recalculate the binding energy of the ΩNN system by examining three NN potentials, i.e., modern realistic AV18 potential, Yukawa-type Malfliet-Tjon (MT) interaction, and Gogny-Pires-Tourreil (GPT) soft and local potential. We take into account the contribution of the Coulomb potential. Our numerical calculations for Ωd(T=0) in maximum spin 5/2+ confirm ground state binding energy of 20.953, 19.368, and 20.439 MeV and a matter radius of 1.097, 1.373, and 1.309 fm using MT, GPT, and AV18 NN potentials, respectively. In the case of Ωd(0)5/2+ system, our numerical analysis shows that considering higher partial waves than s wave in NN interactions leads to an increase of about 0.2 MeV using GPT and about 0.1 MeV reduction with AV18 potentials. We study the convergence of three-body binding energies in a cluster model using the hyperspherical harmonics method and investigate the geometrical properties of Ωd(0)5/2+ ground states.

Universality of excited three-body bound states in one dimension

We study a heavy–heavy–light three-body system confined to one space dimension in the regime where an excited state in the heavy–light subsystems becomes weakly bound. The associated two-body system is characterized by (i) the structure of the weakly-bound excited heavy–light state and (ii) the presence of deeply-bound heavy–light states. The consequences of these aspects for the behavior of the three-body system are analyzed. We find a strong indication for universal behavior of both three-body binding energies and wave functions for different weakly-bound excited states in the heavy–light subsystems.

Cooperativity as quantification and optimization paradigm for nuclear receptor modulators.

Nuclear Receptors (NRs) are highly relevant drug targets, for which small molecule modulation goes beyond a simple ligand/receptor interaction. NR–ligands modulate Protein–Protein Interactions (PPIs) with coregulator proteins. Here we bring forward a cooperativity mechanism for small molecule modulation of NR PPIs, using the Peroxisome Proliferator Activated Receptor γ (PPARγ), which describes NR–ligands as allosteric molecular glues. The cooperativity framework uses a thermodynamic model based on three-body binding events, to dissect and quantify reciprocal effects of NR–coregulator binding (KID) and NR–ligand binding (KIID), jointly recapitulated in the cooperativity factor (α) for each specific ternary ligand·NR·coregulator complex formation. These fundamental thermodynamic parameters allow for a conceptually new way of thinking about structure–activity-relationships for NR–ligands and can steer NR modulator discovery and optimization via a completely novel approach.

Emergent four-body parameter in universal two-species bosonic systems

The description of unitary few-boson systems is conceptually simple: only one parameter – the three-body binding energy – is required to predict the binding energies of clusters with an arbitrary number of bosons. Whether this correlation between the three- and many-boson systems still holds for two species of bosons for which only the inter-species interaction is resonant depends on how many particles of each species are in the system. For few-body clusters with species A and B and a resonant AB interaction, it is known that the emergent AAB and ABB three-body scales are correlated to the ground-state binding energies of the AAAB and ABBB systems, respectively. We find that this link between three and four bodies is broken for the AABB tetramer whose binding energy is neither constrained by the AAB nor by the ABB trimer. From this de-correlation, we predict the existence of a scale unique to the AABB tetramer. In our explanation of this phenomenon, we understand the AABB and AAAB/ABBB tetramers as representatives of two different universal classes of N-body systems with distinct renormalization-group and discrete-scaling properties.

Investigation of $$\varXi ^- nn$$ ($$S=-2$$) hypernucleus in low-energy pionless halo effective theory

We study the \(\varXi ^- nn\) (\(S=-2,\,I=3/2,\,J^P={1/2}^+\)) three-body system using low-energy effective field theory (EFT). Due to the acute inadequacy of empirical information in this sector, there exists substantial degree of ambiguity in determining various few-body observables, some of which are expected to yield vital clues to resolving longstanding contentious issues in hypernuclear physics. Moreover, in astrophysical studies, a precise determination of neutron star equation of state (EoS) of putative hyperonic cores relies on essential input from the \(S=-2\) sector. In this obscure current scenario, a pionless EFT analysis provides a systematic model-independent framework for assessing the feasibility of light three-particle-stable bound states, utilizing low-energy universality. Here we take recourse to a simplistic speculation of the three-body system by eliminating the repulsive spin-singlet \(\varXi ^- n\) sub-system, while retaining the predominantly attractive (possibly bound) spin-triplet \(\varXi ^- n\) and the virtual bound spin-singlet nn sub-systems. In particular, a qualitative leading order EFT investigation by introducing a sharp momentum ultraviolet cut-off parameter \(\varLambda _{\mathrm{reg}}\) into the coupled integral equations indicates a discrete scaling behavior akin to a renormalization group limit cycle, thereby suggesting the formal existence of Efimov states in the unitary limit, as \(\varLambda _{\mathrm{reg}}\rightarrow \infty \). Our subsequent non-asymptotic analysis indicates that the three-body binding energy \(B_3\) is sensitively dependent on the cut-off without the inclusion of three-body contact interactions. Furthermore, our analysis reproduces several values of the binding energy \(B_3\sim 3{-}4\) MeV, predicted in context of existing potential models, with the regulator \(\varLambda _{\mathrm{reg}}\) in the range \(\sim 350{-}460\) MeV. Finally, based on these model inputs for \(B_3\), a ballpark estimate of the three-body scattering length in the range \(2.6{-}4.9\) fm, is naively constrained by our EFT analysis. Despite approximations, the resulting Phillips line is expected to yield a robust feature of the halo-bound \(\varXi ^- nn\) system. For pedagogical reasons, using a simple toy model interacting three-bosons system, we highlight in the appendices the typical universal features leading to emergence of RG limit cycle and Efimov states which are amenable to a low-energy EFT formalism.

HΛΛ5 and HeΛΛ5 hypernuclei reexamined in halo/cluster effective theory

The J=1/2 iso-doublet double-$\Lambda$-hypernuclei, namely, ${}_{\Lambda\Lambda}^{5}$H and ${}_{\Lambda\Lambda}^{5}$He, are examined as the three-body cluster states, $\Lambda\Lambda t$ ($t\equiv {}^3$H or triton) and $\Lambda\Lambda h$ ($h\equiv {}^3$He or helion), respectively, in a model independent framework utilizing pionless halo effective theory. Both singlet and triplet states of the constituent $\Lambda T$ ($T\equiv t,h$) sub-system are used in the elastic channel for the study of $_\Lambda^4$H$-\Lambda$ and $_\Lambda^4$He$-\Lambda$ scattering processes. A prototypical leading order investigation using a sharp momentum cut-off regulator, irrespective of the type of the elastic channel chosen, yields almost identical cut-off dependence of the three-body binding energy or the double-$\Lambda$-separation energy ($B_{\Lambda\Lambda}$) is obtained for the mirror partners, evidently suggesting good isospin symmetry in these three-body systems. Subsequently, upon normalization of our solutions to the integral equation with respect to a single pair of input data from an {\it ab initio} potential model analysis for each mirror hypernuclei, yields $B_{\Lambda\Lambda}$ which agrees fairly well with various erstwhile regulator independent potential models for our choice of the cut-off regulator, $\sim 200$ MeV. This is either consistent with pionless effective theory or with its slightly augmented version with a hard scale around $2m_\pi$, where low-energy $\Lambda$-$\Lambda$ interactions dominated by $\pi\pi$ or $\sigma$-meson exchange. Finally, to demonstrate the predictability of our effective theory, we present preliminary estimates of the S-wave $\Lambda\Lambda T$ three-body scattering lengths and the $\Lambda$-separation energies.

Properties of dibaryons in nuclear medium

Properties of six-quark dibaryons in nuclear medium are considered by example of $A=6$ nuclei within the three-cluster $\alpha+2N$ model. Dibaryon production in nuclei leads to the appearance of a three-body force between the dibaryon and nuclear core. This non-conventional scalar force is shown to provide an additional attractive contribution to the three-body binding energy. This three-body contribution improves noticeably agreement between theoretical results and experimental data for the majority of observables. The most serious difference between the traditional $NN$-force models and the dibaryon-induced model is found for the nucleon momentum distribution, the latter model providing a strong enrichment of the high-momentum components both for $^6$Li and $^6$He cases.

The Thomas theorem and the Efimov States within a generalized Bohr model

Using a generalized Bohr model and the hyper-spherical formalism for a three-body system, we derive the universal energy functions. We apply our model to 4He atom where only the Coulomb potential is dominant and find that its binding energy is well reproduced (less than 5.5% difference). Later, we focus on the equal mass three-body systems and derive the Thomas theorem assuming a simple interaction depending on the range of the potential. We discuss the conditions for which an unbound two-body system produces a bound three-body system and apply our model to 4He and triton atoms as well as to the triton nucleus. Using their scattering lengths and effective ranges, we are able to reproduce the two-body or the three-body binding energies (less than 5% difference) with only one parameter fitted. Prediction for excited (Efimov) levels are also given and in particular we demonstrate that for some hyper-angles two equal minima appear which indicate a phase (shape) transition similar to the Landau’s theory of phase transition. We suggest that the observed excited levels in two different experiments for the triton nucleus are indeed Efimov levels and there may be more surprises.

A computational framework for a Lyapunov-enabled analysis of biochemical reaction networks.

Complex molecular biological processes such as transcription and translation, signal transduction, post-translational modification cascades, and metabolic pathways can be described in principle by biochemical reactions that explicitly take into account the sophisticated network of chemical interactions regulating cell life. The ability to deduce the possible qualitative behaviors of such networks from a set of reactions is a central objective and an ongoing challenge in the field of systems biology. Unfortunately, the construction of complete mathematical models is often hindered by a pervasive problem: despite the wealth of qualitative graphical knowledge about network interactions, the form of the governing nonlinearities and/or the values of kinetic constants are hard to uncover experimentally. The kinetics can also change with environmental variations. This work addresses the following question: given a set of reactions and without assuming a particular form for the kinetics, what can we say about the asymptotic behavior of the network? Specifically, it introduces a class of networks that are “structurally (mono) attractive” meaning that they are incapable of exhibiting multiple steady states, oscillation, or chaos by virtue of their reaction graphs. These networks are characterized by the existence of a universal energy-like function called a Robust Lyapunov function (RLF). To find such functions, a finite set of rank-one linear systems is introduced, which form the extremals of a linear convex cone. The problem is then reduced to that of finding a common Lyapunov function for this set of extremals. Based on this characterization, a computational package, Lyapunov-Enabled Analysis of Reaction Networks (LEARN), is provided that constructs such functions or rules out their existence. An extensive study of biochemical networks demonstrates that LEARN offers a new unified framework. Basic motifs, three-body binding, and genetic networks are studied first. The work then focuses on cellular signalling networks including various post-translational modification cascades, phosphotransfer and phosphorelay networks, T-cell kinetic proofreading, and ERK signalling. The Ribosome Flow Model is also studied.

A three-dimensional momentum-space calculation of three-body bound state in a relativistic Faddeev scheme

In this paper, we study the relativistic effects in a three-body bound state. For this purpose, the relativistic form of the Faddeev equations is solved in momentum space as a function of the Jacobi momentum vectors without using a partial wave decomposition. The inputs for the three-dimensional Faddeev integral equation are the off-shell boost two-body t–matrices, which are calculated directly from the boost two-body interactions by solving the Lippmann-Schwinger equation. The matrix elements of the boost interactions are obtained from the nonrelativistic interactions by solving a nonlinear integral equation using an iterative scheme. The relativistic effects on three-body binding energy are calculated for the Malfliet-Tjon potential. Our calculations show that the relativistic effects lead to a roughly 2% reduction in the three-body binding energy. The contribution of different Faddeev components in the normalization of the relativistic three-body wave function is studied in detail. The accuracy of our numerical solutions is tested by calculation of the expectation value of the three-body mass operator, which shows an excellent agreement with the relativistic energy eigenvalue.

Efimov effect in a D-dimensional Born–Oppenheimer approach

We study a three-body system, formed by two identical heavy bosons and a light particle, in the Born–Oppenheimer approximation for an arbitrary dimension D. We restrict D to the interval 2 < D < 4, and derive the heavy–heavy D-dimensional effective potential proportional to 1/R2 (R is the relative distance between the heavy particles), which is responsible for the Efimov effect. We found that the Efimov states disappear once the critical strength of the heavy–heavy effective potential 1/R2 approaches the limit . We obtained the scaling function for the 133Cs–133Cs–6Li system as the limit cycle of the correlation between the energies of two consecutive Efimov states as a function of D and the heavy-light binding energy ED2. In addition, we found that the energy of the (N + 1)th excited state reaches the two-body continuum independently of the dimension D when , where is the Nth excited three-body binding energy.

A New Feature of the Efimov-Like Structure in the Hadron System: Long-Range Force as a Recoil Effect

On the three-body kinematics, we investigate the threshold behavior which appears not only at the three-body break-up threshold (3BT), but also at the quasi two-body threshold (Q2T) for the reactions: A+(BC)rightarrow A+B+C, and (ABC)rightarrow A+(BC), respectively. Recently, the author proposed a generalparticle {transfer} (GPT) potential which appears, not only at the 3BT, but also at the Q2T between A and (BC). The new potential indicates a Yukawa-type potential for short range, but a 1/r^n-type potential for long range. The long range part of the GPT potential for n=1 indicates an attractive Coulomb-like or a gravitation-like potential. While, n=2 indicates the Efimov-like potential between A and (BC). The three-body binding energy: E_n=epsilon +zeta _n with the two-body binding energy epsilon , and the separation energy zeta _n for (ABC)rightarrow A+(BC) satisfies E_n/E_{n+1}=zeta _n/zeta _{n+1}=const for epsilon =0 or the two-body scattering length: arightarrow infty (i.e. the two-body unitary limit). At the Q2T, the condensation of the three-body binding energy is given by the GPT-potential in the form of E_n/E_{n+1}=(zeta _n+epsilon )/(zeta _{n+1}+epsilon )rightarrow 1 (const) for nrightarrow infty (with zeta _nrightarrow 0) which implies the existence of Efimov-like states at the Q2T in the hadron systems, thereby the possibility of “ultra low energy nuclear transformation”, where the three-body binding energies degenerate at zero energy. Finally, the origin of such a long range potential will be clarified.

On Mass Polarization Effect in Three-Body Nuclear Systems

We evaluate the mass polarization term of the kinetic-energy operator for different three-body nuclear $AAB$ systems by employing the method of Faddeev equations in configuration space. For a three-boson system this term is determined by the difference of the doubled binding energy of the $AB$ subsystem $2E_{2}$ and the three-body binding energy $E_{3}(V_{AA}=0)$ when the interaction between the identical particles is omitted. In this case: $\left\vert E_{3}(V_{AA}=0)\right\vert >2\left\vert E_{2}\right\vert$. In the case of a system complicated by isospins(spins), such as the kaonic clusters $ K^{-}K^{-}p$ and $ppK^{-}$, the similar evaluation impossible. For these systems it is found that $\left\vert E_{3}(V_{AA}=0)\right\vert <2\left\vert E_{2}\right\vert$. A model with an $AB$ potential averaged over spin(isospin) variables transforms the later case to the first one. The mass polarization effect calculated within this model is essential for the kaonic clusters. Besides we have obtained the relation $|E_3|\le |2E_2|$ for the binding energy of the kaonic clusters.

Emergence of Clusters: Halos, Efimov States, and Experimental Signals.

We investigate the emergence of halos and Efimov states in nuclei by use of a newly designed model that combines self-consistent mean-field and three-body descriptions. Recent interest in neutron heavy calcium isotopes makes ^{72}Ca (^{70}Ca+n+n) an ideal realistic candidate on the neutron dripline, and we use it as a representative example that illustrates our broadly applicable conclusions. By smooth variation of the interactions we simulate the crossover from well-bound systems to structures beyond the threshold of binding, and find that halo configurations emerge from the mean-field structure for three-body binding energy less than ∼100 keV. Strong evidence is provided that Efimov states cannot exist in nuclei. The structure that bears the most resemblance to an Efimov state is a giant halo extending beyond the neutron-core scattering length. We show that the observable large-distance decay properties of the wave function can differ substantially from the bulk part at short distances, and that this evolution can be traced with our combination of few- and many-body formalisms. This connection is vital for interpretation of measurements such as those where an initial state is populated in a reaction or by a beta decay.

Ground-State Properties of Unitary Bosons: From Clusters to Matter.

The properties of cold Bose gases at unitarity have been extensively investigated in the last few years both theoretically and experimentally. In this Letter we use a family of interactions tuned to two-body unitarity and very weak three-body binding to demonstrate the universal properties of both clusters and matter. We determine the universal properties of finite clusters up to 60 particles and, for the first time, explicitly demonstrate the saturation of energy and density with particle number and compare with bulk properties. At saturation in the bulk we determine the energy, density, two- and three-body contacts, and the condensate fraction. We find that uniform matter is more bound than three-body clusters by nearly 2 orders of magnitude, the two-body contact is very large in absolute terms, and yet the condensate fraction is also very large, greater than 90%. Equilibrium properties of these systems may be experimentally accessible through rapid quenching of weakly interacting boson superfluids.

Molecular recognition of ternary complexes: a new dimension in the structure-guided design of chemical degraders.

Molecular glues and bivalent inducers of protein degradation (also known as PROTACs) represent a fascinating new modality in pharmacotherapeutics: the potential to knockdown previously thought ‘undruggable’ targets at sub-stoichiometric concentrations in ways not possible using conventional inhibitors. Mounting evidence suggests these chemical agents, in concert with their target proteins, can be modelled as three-body binding equilibria that can exhibit significant cooperativity as a result of specific ligand-induced molecular recognition. Despite this, many existing drug design and optimization regimens still fixate on binary target engagement, in part due to limited structural data on ternary complexes. Recent crystal structures of protein complexes mediated by degrader molecules, including the first PROTAC ternary complex, underscore the importance of protein–protein interactions and intramolecular contacts to the mode of action of this class of compounds. These discoveries have opened the door to a new paradigm for structure-guided drug design: borrowing surface area and molecular recognition from nature to elicit cellular signalling.

Retardation Effects and Virtual Annihilation Interactions in Relativistic Three-body Systems

The Hamiltonian formalism of quantum field theory and the variational methods in a reformulated model have been used to investigate the influence of retardation effects or virtual annihilation interactions in relativistic three-body wave equations for scalar two particles and one antiparticle interacting via a massive or massless mediating scalar field. The results show that the inclusion of virtual annihilation interactions or retardation effects can have noticeable consequence on the three-body binding energy at strong coupling.

Three-body bound states with zero-range interaction in the Bethe–Salpeter approach

The Bethe–Salpeter equation for three bosons with zero-range interaction is solved for the first time. For comparison the light-front equation is also solved. The input is the two-body scattering length and the outputs are the three-body binding energies, Bethe–Salpeter amplitudes and light-front wave functions. Three different regimes are analyzed: (i) For weak enough two-body interaction the three-body system is unbound. (ii) For stronger two-body interaction a three-body bound state appears. It provides an interesting example of a deeply bound Borromean system. (iii) For even stronger two-body interaction this state becomes unphysical with a negative mass squared. However, another physical (excited) state appears, found previously in light-front calculations. The Bethe–Salpeter approach implicitly incorporates three-body forces of relativistic origin, which are attractive and increase the binding energy.