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arXiv:2403.10244 · Physical review. D/Physical review. D.

Exploring the Efimov effect in the D * D * D * system

Pablo G. Ortega

Published Aug 13, 2024• 11 citations
arXivHTMLPDF

Abstract

The emergence of the Efimov effect in the D * D * D * system is explored under the assumption that the heavy partner of the T c c + exists as a D * D * molecule with ( I ) J P = ( 0 ) 1 + . The three-to-three relativistic scattering amplitude is obtained from the ladder amplitude formalism, built from an energy-dependent contact two-body potential where the molecular component of the T c c * state can be varied. We find that ( I ) J P = ( 1 2 ) 0 − three-body bound states can be formed, with properties that suggest that the Efimov effect can be realized for reasonable values of the molecular probability and binding energy of the T c c * . Published by the American Physical Society 2024

DOI
10.1103/physrevd.110.034015

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The emergence of the Efimov effect in the D * D * D * system is explored under the assumption that the heavy partner of the T c c + exists as a D * D * molecule with ( I ) J P = ( 0 ) 1 + . The three-to-three relativistic scattering amplitude is obtained from the ladder amplitude formalism, built from an energy-dependent contact two-body potential where the molecular component of the T c c * state can be varied. We find that ( I ) J P = ( 1 2 ) 0 − three-body bound states can be formed, with properties that suggest that the Efimov effect can be realized for reasonable values of the molecular probability and binding energy of the T c c * . Published by the American Physical Society 2024
The possibility of a zero-energy Efimov effect is investigated in a model consisting of three identical heavy particles and a lighter one, when the light-heavy interaction leads to a zero-energy bound state for the two-heavy---one-light subsystem. The model is solved in the Born-Oppenheimer approximation with the light-heavy interaction taken to be a separable sss-wave potential of Yamaguchi form. The heavy-heavy interaction is short range and, if attractive, is to be taken weak enough to support no two- or three-heavy-particle bound states. The relevant parameter is the potential strength $\ensuremath{\lambda}$ of the light-heavy interaction which for $\ensuremath{\lambda}={\ensuremath{\lambda}}_{c}$ supports a single zero-energy light-heavy bound state. If the first zero-energy bound state of the two-heavy---one-light subsystem occurs for $\ensuremath{\lambda}={\ensuremath{\lambda}}^{\ensuremath{'}}$ such that $|{\ensuremath{\lambda}}^{\ensuremath{'}}|<|{\ensuremath{\lambda}}_{c}|$, there are no Efimov four-body bound states. On the contrary, if the chosen heavy-heavy potential is repulsive enough to prevent the existence of three-body bound states for $|\ensuremath{\lambda}|<|{\ensuremath{\lambda}}_{c}|$, then as $\ensuremath{\lambda}\ensuremath{\rightarrow}{\ensuremath{\lambda}}_{c}$ a few four-body Efimov states may emerge but their number remains finite. These states disappear for $|\ensuremath{\lambda}|\ensuremath{\gtrsim}|{\ensuremath{\lambda}}_{c}|$ as the three-body cut overrides them.
Abstract 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/ R 2 ( 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/ R 2 approaches the limit − ( D − 2 ) 2 / 4 . We obtained the scaling function for the 133 Cs– 133 Cs– 6 Li 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 E D 2 . 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 E 2 D / E 3 ( N ) = 0.89 , where E 3 ( N ) is the N th excited three-body binding energy.
Elementary demonstrations of the three-body Efimov effect are provided for several models involving a light particle and two heavy ones, and differing by the nature of the two-body (local or nonlocal) potentials which bind at zero energy the light particle to each of the heavy ones. The proofs rely on Born-Oppenheimer approximations, which are actually projections of the Hamiltonian into suitable subspaces, and they use the correct reduced mass. A numerical estimate of the long-range potential responsible for the accumulation of Efimov bound states is obtained through Pad\'e approximants.
In the past years various new hadronic states were discovered in the charm and bottom meson sector by experiments like Belle, BES and Babar. Many of these states fit into the conventional charmonium or bottomonium spectrum, but a growing number of them does not. Especially, the charged ones which can definitely be no conventional hadrons motivate the term exotic hadrons to classify particles which are neither conventional baryons nor conventional mesons. Their substructure is not yet clear due to the often poor experimental knowledge and there are many different interpretations for these exotic particles. One very popular idea is the hadronic molecule interpretation which states that two ordinary hadrons can form a loosely bound pair similar to the deuteron made of a proton and a neutron. However, in the latter system it is known that after adding a third nucleon there appears a three particle bound state, the triton. The emergence of such a three-body state can be explained by the Efimov effect as a low-energy universality phenomenon. The question which will be answered in this work is if there exists also an Efimov effect in systems of hadronic molecules which scatter off a third meson or baryon. Since the number of candidate systems is large (and still growing due to ongoing experiments) it is useful to find a general expression that tells us if the Efimov effect is present. Moreover, this expression should only depend on the basic particle properties like mass, spin and isospin. Such a relation will be derived in this work for a generic three particle system that can be described in a pionless effective field theory. It will then be applied to a set of possible hadronic molecule systems in order to check if the Efimov effect occurs. The thesis is organized as follows: after an introduction of quantum chromodynamics and a review of the classification of hadrons we continue with a summary, firstly, of experimental observations concerning exotic hadrons, secondly, of the idea of effective field theories and thirdly, of basic scattering theory. In chapter 2 the three identical boson system will be discussed in order to explain the Efimov effect more quantitatively. With this knowledge it is possible to derive the mentioned general expression regarding the existence of Efimov physics in a generic three particle system. After the detailed derivation in chapter 3 (using the information given in the appendices) the results will be applied to scattering processes where known hadronic molecules as well as hypothetical ones are involved. This is done in order to either search for the Efimov effect or - for the hypothetical molecular states - to predict under which conditions an Efimov trimer is expected. For system without Efimov effect we will calculate in chapter 5 for a selected example S-wave observables of the in this case elastic scattering process. The thesis is completed by a summary of the results and an outlook on further work that can be done based on this work.
The quantum three body problem generally has its own complexities and is difficult to solve. It requires rigorous steps and iterations to find the solution. We often encounter three body system in various branches of physics including Nuclear Physics, Particle, Atomic, Molecular physics, and condensed matter. The underlying physics involved is quite complicated. Faddeev in 1961 made a successful attempt and formulated theory of three particle scattering in a rather rigorous mathematical way. Efimov developed a method that enabled one to obtain the universal properties of three particle system and is quite independent of the two body potentials. This article attempts to introduce Efimov effect. The formalism and the salient features are also discussed.
Inspired by the narrow Feshbach resonance in systems with the two-body interaction, we propose a two-channel model of three-component fermions with a three-body interaction that takes into account finite-range effects in low dimensions. Within this model, the p -wave Efimov-like effect in the four-body sector is predicted in fractional dimensions between d=1 and d=2 . The impact of the finite-range interaction on the formation of the four-body bound states in d=1 is also discussed in detail. Published by the American Physical Society 2024