Three-body Sector Research Articles

Loosely bound few-body states in a spin-1 gas with near-degenerate continua

A distinguishing feature of ultracold collisions of bosonic lithium atoms is the presence of two near-degenerate two-body continua. The influence of such a near degeneracy on the few-body physics in the vicinity of a narrow Feshbach resonance is investigated within the framework of a minimal model with two atomic continua and one closed molecular channel. The model allows analysis of the spin composition of loosely bound dimers and trimers. In the two-body sector the well-established coupled-channel calculation phenomenology of lithium is qualitatively reproduced, and its particularities are emphasized and clarified. In the three-body sector we find that the Efimov trimer energy levels follow a different functional form as compared to a single continuum scenario while the thresholds remain untouched. This three-channel model with two atomic continua complements our earlier developed three-channel model with two molecular channels [Yudkin and Khaykovich, Phys. Rev. A 103, 063303 (2021)] and suggests that the experimentally observed exotic behavior of the first excited Efimov energy level (Yudkin, Elbaz, and Khaykovich, arXiv:2004.02723) is most probably caused by the short-range details of the interaction potential.

Heavy hadron molecules in effective field theory: the emergence of exotic nuclear landscapes

Heavy hadron molecules were first theorized from a crude analogy with the deuteron and the nuclear forces binding it, a conjecture which was proven to be on the right track after the discovery of the $X(3872)$. However, this analogy with nuclear physics has not been seriously exploited beyond a few calculations in the two- and three-body sectors, leaving a great number of possible theoretical consequences unexplored. Here we show that nuclear and heavy hadron effective field theories are formally identical: using a suitable notation, there is no formal difference between these two effective field theories. For this, instead of using the standard heavy superfield notation, we have written the heavy hadron interactions directly in terms of the light quark degrees of freedom. We give a few examples of how to exploit this analogy, e.g. the calculation of the two-pion exchange diagrams. Yet the most relevant application of the present idea is the conjecture of exotic nuclear landscapes, i.e. the possibility of few heavy hadron bound states with characteristics similar to those of the standard nuclei.

Renormalization of a finite-range inverse-cube potential

We study the regularization and renormalization of a finite range inverse cube potential in the two- and three-body sectors. Specifically, we compare and contrast three different regulation schemes frequently used to study few-body systems as well as the associated renormalization group flows. We also calculate bound state and scattering observables over a wide range of cutoffs, demonstrating the sufficiency of a two-body contact interaction to renormalize two- and three-body observables. We supplement these plots with quantified analyses of the observables' residual cutoff dependence.

Conformality lost in Efimov physics

A general mechanism for the loss of conformal invariance is the merger and disappearance of an infrared fixed point and an ultraviolet fixed point of a renormalization group flow. We show explicitly how this mechanism works in the case of identical bosons at unitarity as the spatial dimension $d$ is varied. For $d$ between the critical dimensions $d_{\rm 1}=2.30$ and $d_{\rm 2}=3.76$, there is loss of conformality as evidenced by the Efimov effect in the three-body sector. The beta function for an appropriate three-body coupling is a quadratic polynomial in that coupling. For $d<d_{\rm 1}$ and for $d>d_{\rm 2}$, the beta function has two real roots that correspond to infrared and ultraviolet fixed points. As $d$ approaches $d_{\rm 1}$ from below and as $d$ approaches $d_{\rm 2}$ from above, the fixed points merge and disappear into the complex plane. For $d_{\rm 1}<d<d_{\rm 2}$, the beta function has complex roots and the renormalization group flow for the three-body coupling is a limit cycle.

He3andpdscattering to next-to-leading order in pionless effective field theory

We study the three-body systems of ${}^{3}\mathrm{He}$ and $pd$ scattering and demonstrate, both analytically and numerically, that a new $pd$ three-body force is needed at next-to-leading order in pionless effective field theory. We also show that at leading order these observables require no new three-body force beyond what is necessary to describe $nd$ scattering. We include electromagnetic effects by iterating only diagrams that involve a single photon exchange in the three-body sector.

Wilsonian renormalization group analysis of nonrelativistic three-body systems without introducing dimerons

Low-energy effective field theory describing a nonrelativistic three-body system is analyzed in the Wilsonian renormalization group method. No effective auxiliary field (dimeron) that corresponds to two-body propagation is introduced. The Efimov effect is expected in the case of an infinite two-body scattering length and is believed to be related to the limit cycle behavior in the three-body renormalization group equations (RGEs). If the one-loop property of the RGEs for the nonrelativistic system without the dimeron field, which is essential in deriving RGEs in the two-body sector, persists in the three-body sector, it appears to prevent the emergence of limit cycle behavior. We explain how the multiloop diagrams contribute in the three-body sector without contradicting the one-loop property of the RGEs and derive the correct RGEs, which lead to the limit cycle behavior. The Efimov parameter, ${s}_{0}$, is obtained within a few percent error in the leading orders. We also remark on the correct use of the dimeron formulation. We find rich renormalization group flow structure in the three-body sector. In particular, a novel nontrivial fixed point of the three-body couplings is found when the two-body interactions are absent. We also find, on the two-body nontrivial fixed point, the limit cycle is realized as a loop of finite size in the space of three-body coupling constants when terms with derivatives are included.

Nuclear pairing from chiral pion-nucleon dynamics: Applications to finite nuclei

The 1S0 pairing gap in isospin-symmetric nuclear matter and finite nuclei is investigated using the chiral nucleon-nucleon potential at the N3LO order in the two-body sector, and the N2LO order in the three-body sector. To include realistic nuclear forces in RHB (Relativistic Hartree Bolgoliubov) calculations we rely on a separable form of the pairing interaction adjusted to the bare nuclear force. The separable pairing force is applied to the analysis of pairing properties for several isotopic and isotonic chains of spherical nuclei.

Pauli-blocking effects and Cooper triples in three-component Fermi gases

We investigate the effect of Pauli blocking on universal two- and three-body states in the medium. Their corresponding energies are extracted from the poles of two- and three-body in-medium scattering amplitudes. Compared to the vacuum, the binding of dimer and trimer states is reduced by the medium effects. In two-body scattering, the well-known physics of Cooper pairs is recovered. In the three-body sector, we find a class of positive energy poles which can be interpreted as Cooper triples.

The three-boson system at next-to-leading order in an effective field theory for systems with a large scattering length

We analyze how corrections linear in the effective range, r0, affect quantities in the three-body sector within an effective field theory for short-range interactions. We demonstrate that observables can be obtained straightforwardly using a perturbative expansion in powers of r0. In particular, we show that two linear-in-r0 counterterms are needed for renormalization at this order if scattering-length-dependent observables are considered. We exemplify the implications of this result using various three-body observables. Analytical results for the running of the next-to-leading-order portion of the three-body force in this effective field theory are provided. Expressions which incorporate O(r0) corrections and relate the positions of features observed in three-atom recombination near a Feshbach resonance are presented.

Nuclear Pairing from Chiral Pion-Nucleon Dynamics: Latest Results and Relevant Issues

In a recent paper [P. Finelli, T. Niǩsic and D. Vretenar, arXiv:1111.4946] the S0 pairing gap in isospin-symmetric nuclear matter and finite nuclei has been investigated starting from the chiral nucleon-nucleon potential at the NLO order in the two-body sector, and the NLO order in the three-body sector. To include realistic nuclear forces in RHB (Relativistic Hartree Bolgoliubov) calculations we relied on a separable representation of the pairing interaction. In this paper we would like to show recent results and discuss some issues related to these calculations, in particular how pairing gaps in finite nuclei are sensitive to the choice of ph effective interactions.

The similarity renormalization group for three-body interactions in one dimension

We report on recent progress of the implementation of the similarity renormalization group (SRG) for three-body interactions in a one-dimensional, bosonic model system using the plane wave basis. We discuss our implementation of the flow equations and show results that confirm that results in the three-body sector remain unchanged by the transformation of the Hamiltonian. We also show how the SRG transformation decouples low- from high-momentum nodes in the three-body sector and therefore simplifies the numerical calculation of observables.

Precision calculation of threshold scattering, πN scattering lengths, and the GMO sum rule

We use chiral perturbation theory (ChPT) to calculate the $\pi^- d$ scattering length with an accuracy of a few percent, including isospin-violating corrections both in the two- and three-body sector. In particular, we provide the technical details of a recent letter, where we used data on pionic deuterium and pionic hydrogen atoms to extract the isoscalar and isovector pion-nucleon scattering lengths $a^+$ and $a^-$. We study isospin-breaking contributions to the three-body part of $a_{\pi^-d}$ due to mass differences, isospin violation in the $\pi N$ scattering lengths, and virtual photons. This last class of effects is ostensibly infrared enhanced due to the smallness of the deuteron binding energy. However, we show that the leading virtual-photon effects that might undergo such enhancement cancel, and hence the standard ChPT counting provides a reliable estimate of isospin violation in $a_{\pi^- d}$ due to virtual photons. Finally, we discuss the validity of the Goldberger-Miyazawa-Oehme sum rule in the presence of isospin violation, and use it to determine the charged-pion-nucleon coupling constant.

Nonperturbative calculation of the anomalous magnetic moment in the Yukawa model within truncated Fock space

Within the covariant formulation of light-front dynamics, we calculate the state vector of a physical fermion in the Yukawa model. The state vector is decomposed in Fock sectors and we consider the first three ones: the single constituent fermion, the constituent fermion coupled to one scalar boson, and the constituent fermion coupled to two scalar bosons. This last three-body sector generates nontrivial and nonperturbative contributions to the state vector, which are calculated numerically. Field-theoretical divergences are regularized using Pauli-Villars fermion and boson fields. Physical observables can be unambiguously deduced using a systematic renormalization scheme we have developed previously. As a first application, we consider the anomalous magnetic moment of the physical fermion.

Higher Fock Sectors in the Wick-Cutkosky Model

In the Wick-Cutkosky model we analyze nonperturbatively, in light-front dynamics, the contributions of two-body and higher Fock sectors to the total norm and electromagnetic form factor. It turns out that two- and three-body sectors always dominate. For maximal value of coupling constant $\alpha=2\pi$, corresponding to zero bound state mass M=0, they contribute 90% to the norm. With decrease of $\alpha$ the two-body contribution increases up to 100%. The form factor asymptotic is always determined by two-body sector.

Structure of nonlocalities in meson exchange NN interactions and their role in the NN and 3N system.

The structure of nonlocalities implied by the meson exchange dynamics in the nucleon-nucleon interaction is investigated. It is shown that they have a considerable impact on $\mathrm{NN}$ as well as $3N$ observables. It is argued that, for a precise determination of their effects, a sufficiently complete calculation, consistent in the two- and three-body sector, is required.

Soluble model involving four identical particles

Using a nonrelativistic field theoretic formalism, a soluble model of scattering involving four identical spinless particles is developed and solved numerically. In addition to the elementary "nucleon" (the $n$), two composite particles meant to approximate the deuteron and triton are introduced with the couplings $d\ensuremath{\leftrightarrow}n+n$ and $t\ensuremath{\leftrightarrow}d+n$. By consistently excluding all particle-exchange contributions to the three-body sector, four-body integral equations are obtained for the two-to-two processes: $nt\ensuremath{\rightarrow}nt$, $nt\ensuremath{\rightarrow}dd$ as well as for $dd\ensuremath{\rightarrow}dd$ and $dd\ensuremath{\rightarrow}nt$. Numerical solutions of the equations are found to satisfy unitarity constraints above the two-, three-, and four-body thresholds. The positions of the four-body bound states are obtained and a complete phase shift calculation is performed. The sum of the total three- and four-body breakup cross sections predicted by the model are displayed as a function of energy and the angular distributions for all 2 \ensuremath{\rightarrow} 2 reactions are compared with the four-nucleon data.NUCLEAR REACTIONS Four-body problem. Spinless model of the four-nucleon system.

Many-body potentials in the lee model — II

The energy eigenvalue problem for the bound states of the V-nN sectors(n ≥ 0) of the Lee model is reduced to a single equation involving the energy, particle separations, parity and the renormalized coupling constant. The only functions of these quantities that arise in this n+1-body equation are those characteristic of the one (V) and two (V-N) particle sectors for the case where there are two real solutions in the one-particle system. A detailed investigation of the n=3 case shows that it is possible to analyse the corresponding equation, for special geometric configurations, along the lines developed in the two and three (V-2N) particle sectors. If the particle separations form a regular tetrahedron, the eigenvalue equation factors into four possibilities, three of which are identical with the even parity V-N equations with the same separation, and the fourth is similar to the odd parity V-N equation. The equilateral n=2 case and the foregoing n=3 case admit a multiplicity of the even parity V-N equation equal to the number of N-particles present. It is shown for these symmetrical configurations that two, three and four-body potential energies can coincide for even and odd parity states, and the critical constant for the appearance of complex energies in the four-body case can be smaller or larger than the corresponding quantities in the two and three-body sectors if the separations involved in the comparisons are chosen appropriately.