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arXiv:1806.01999 · Physics Letters B

Fate of the neutron–deuteron virtual state as an Efimov level

Gautam Rupak, Akshay Vaghani, R. Higa, U. van Kolck

Published Sep 7, 2018• 26 citations
arXivHTMLPDF

Abstract

The emergence of Efimov levels in a three-body system is investigated near the unitarity limit characterized by a resonant two-body interaction. No direct evidence of Efimov levels is seen in the three-nucleon system since the triton is the only physical bound state. We provide a model-independent analysis of nucleon–deuteron scattering at low energy by formulating a consistent effective field theory. We show that virtual states evolve into shallow bound states, which emerge as excited triton levels as we drive the system towards unitarity. Even though we consider this specific system, our results for the emergence of the Efimov levels are universal.

DOI
10.1016/j.physletb.2018.08.051

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The emergence of Efimov levels in a three-body system is investigated near the unitarity limit characterized by a resonant two-body interaction. No direct evidence of Efimov levels is seen in the three-nucleon system since the triton is the only physical bound state. We provide a model-independent analysis of nucleon–deuteron scattering at low energy by formulating a consistent effective field theory. We show that virtual states evolve into shallow bound states, which emerge as excited triton levels as we drive the system towards unitarity. Even though we consider this specific system, our results for the emergence of the Efimov levels are universal.
The Efimov phenomenon manifests itself as an emergent discrete scaling symmetry in the quantum three-body problem. In the unitarity limit, it leads to an infinite tower of three-body bound states with energies forming a geometric sequence. In this work, we study the evolution of these so-called Efimov states using relativistic scattering theory. We identify them as poles of the three-particle SSS matrix in the complex energy plane, and we study how they transform from virtual states through bound states to resonances when we change the interaction strength. We dial the scattering parameters toward the unitarity limit and observe the emergence of the universal scaling of energies and couplings---a behavior known from the nonrelativistic case. We additionally find that Efimov resonances follow unusual, cyclic trajectories accumulating at the three-body threshold and then disappear at some values of the two-body scattering length. We propose a partial resolution to this ``missing states'' problem.
The Efimov effect in the SSS wave spin doublet trinucleon system is studied and the virtual state of this system is found to be an Efimov virtual state. In the Efimov limit this state becomes a bound state. But as the strength of the spin triplet nucleon-nucleon interaction is increased, it first moves away from the lowest physical scattering threshold, then stops and turns around and eventually becomes a physical bound state. The movement of other Efimov states of this system were studied and once they become a virtual state they move into the left-hand cut with the increase of the strength of the triplet nucleon-nucleon potential and do not become resonances.NUCLEAR REACTIONS Efimov effect, trinucleon system, Efimov virtual states, Amado model.
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 trajectory of the first excited Efimov state is investigated by using a renormalized zero-range three-body model for a system with two bound and one virtual two-body subsystems. The approach is applied to n–n–18C, where the n–n virtual energy and the three-body ground state are kept fixed. It is shown that such three-body excited state goes from a bound to a virtual state when the n–18C binding energy is increased. Results obtained for the n–19C elastic cross-section at low energies also show dominance of an S-matrix pole corresponding to a bound or virtual Efimov state. It is also presented a brief discussion of these findings in the context of ultracold atom physics with tunable scattering lengths.
Physical systems characterized by a shallow two-body bound or virtual state are governed at large distances by continuous scale invariance, which is broken into discrete scale invariance when three or more particles come into play. This symmetry induces a universal behavior for different systems that is independent of the details of the underlying interaction and rooted in the smallness of the ratio ℓ/ a B ≪ 1, where the length a B is associated with the binding energy of the two-body system [Formula: see text], and ℓ is the natural length given by the interaction range. Efimov physics refers to this universal behavior, which is often hidden by the onset of system-specific nonuniversal effects. In this review, we identify universal properties by providing an explicit link of physical systems to their unitary limit, in which a B → ∞, and we show that nuclear systems belong to this class of universality.
Abstract 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.