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Few-Body Systems

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

S. Oryu

Published Apr 9, 2018• 9 citations
Publisher

Abstract

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 general particle $$ {transfer}$$ (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: $$a\rightarrow \infty $$ (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 $$n\rightarrow \infty $$ (with $$\zeta _n\rightarrow 0$$ ) 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.

DOI
10.1007/s00601-018-1373-z

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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.

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For a system of three identical bosons interacting via short-range forces, when two of the atoms are about to form a two-body sss-wave dimer, the Efimov effect takes place leading to the formation of an infinite number of three-body (Efimov) states. The lowest Efimov state crosses the three-body break-up threshold when the sss-wave two-body scattering length is $a\ensuremath{\approx}\ensuremath{-}9.73{r}_{\mathrm{vdW}}$, rvdW{r}_{\mathrm{vdW}}rvdW​ being the van der Waals length. This article focuses on a generalized version of this Efimov scenario, where two of the atoms are about to form a two-body ddd-wave dimer, resulting in strong ddd-wave interactions. In his paper [Phys. Rev. A 62, 050702(R) (2000)], Bo Gao predicted that for broad resonances the ddd-wave dimer is always formed when $a\ensuremath{\approx}0.956{r}_{\mathrm{vdW}}$. Here we find that a single universal three-body state associated with the ddd-wave dimer is also formed near the three-body breakup threshold at $a\ensuremath{\approx}1.09{r}_{\mathrm{vdW}}$, or alternatively a2=0.902rvdW{a}_{2}=0.902{r}_{\mathrm{vdW}}a2​=0.902rvdW​, where a2{a}_{2}a2​ is the two-body ddd-wave scattering length. The signature of such a universal three-body state is signaled experimentally by an enhancement of the three-body recombination rate. The three-body effective potential curves that are crucial for understanding the recombination dynamics are also calculated and analyzed. An improved method to calculate the couplings, effective potential curves, and recombination rate coefficients is presented.
Near two-body unitarity, the three-boson system is characterized by an approximate discrete scale invariance manifest in a geometric tower of bound states (the Efimov effect). In the absence of a strong four-body force, the four-boson system has two states associated with each Efimov state, one very nearly unstable, the other several times deeper. We study correlations between the excited- and ground-state properties, such as binding energies and radii, up to next-to-leading order in an effective field theory for short-range forces. We obtain the parameters in these correlations from similar correlations arising from existing precise calculations based on short-range potentials. We also derive correlations among excited-state properties that emerge from the proximity of the state to the break-up threshold into a boson and a three-boson bound state, using an effective field theory for ``halo'' states.
Efimov physics relates to 3-body systems with large 2-body scattering lengths a and small effective ranges r. For many systems in nature the assumption of a small effective range is not valid. The present report shows binding energies E of three identical bosons calculated with 2-body potentials that are fitted to scattering data and momentum cut-offs (L) by inverse scattering. Results agree with previous works in the case of r<0 when L=~10/r. With 1/a=0 the converged energies are given by E(n) =C(n)/r*r with n labeling the energy-branch and calculated values C(0)=0.77, C(1)=.0028. This gives a ratio ~278 thus differing from the value ~515 in the Efimov case. Efimov's angular dependent function is calculated. Good agreement with previous works is obtained when r<< a. With the increased values of effective range the shallow states still appear Efimov-like. For deeper states the angular dependence differs but is independent of the effective range.
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.
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.