In this work, three-body bound states were studied in finite volume using an Effective Field Theory framework. The finite volumes under consideration are cubic volumes with periodic boundary conditions. The Effective Field Theory framework used in this work employs only contact interactions and is particularly well suited for studies of universal properties, i.e. properties independent of the details of the interaction on short distances. A particular example in the three-body sector is the Efimov effect, the emergence of a geometrically spaced bound state spectrum. In the first part, systems of three identical bosons are investigated. As a consequence of the breakdown of the spherical symmetry to cubic symmetry, the partial waves of the bound state amplitude are coupled. An infinite set of coupled integral equations for these partial waves is derived. These equations have to be solved numerically in order to obtain the binding energies in the finite volume. The dependence of the energies on the box size is calculated and the results are explicitly verified to be renormalized. Results for positive and negative scattering lengths are shown. The effects of higher partial waves are investigated. The behavior of shallow trimers near the dimer energy as well as deeply bound trimers is studied. The shallowest state investigated crosses the dimer energy at a certain volume and behaves like a scattering state for smaller volumes. Numerical evidence for a universal scaling of the finite volume corrections is provided. Subsequently, the formalism is extended to systems of three nucleons. This case provides the main motivation for this work due to its applicability to Lattice Quantum Chromodynamics (QCD) calculations of the triton. Such calculations always take place inside a finite volume which makes control over the corresponding effects crucial for an understanding of results from the lattice. For the triton, there are two coupled channels already in the infinite volume corresponding to two different spin-isospin combinations. An infinite set of coupled integral equations for the partial waves of the bound state amplitudes is derived. The renormalization of all results is again explicitly verified. The physical triton inside a finite volume is investigated as well as the triton spectrum for unphysical pion masses. The former case qualitatively shows the same behavior as the three-boson case, and the volume dependence is calculated. The smallest volumes investigated are of the order of magnitude typical for present day Lattice calculations. The motivation for the latter part is twofold. On the one hand, Lattice QCD calculations are performed at pion masses larger than the physical one for computational reasons. On the other hand, it has been conjectured that QCD is close to the critical trajectory for an infrared renormalization group limit cycle, in which case the Efimov effect would occur for a critical pion mass. Close to this critical pion mass, the triton has excited states. The behavior of the ground state and of the excited states inside a finite volume is investigated for various pion masses around the critical one. The excited states cross the energy of the bound di-nucleon, as it was already observed for the shallowest bosonic trimer. The results for the ground state were used to provide strong numerical evidence for a universal scaling of the finite volume corrections.
In this work, three-body bound states were studied in finite volume using an Effective Field Theory framework. The finite volumes under consideration are cubic volumes with periodic boundary conditions. The Effective Field Theory framework used in this work employs only contact interactions and is particularly well suited for studies of universal properties, i.e. properties independent of the details of the interaction on short distances. A particular example in the three-body sector is the Efimov effect, the emergence of a geometrically spaced bound state spectrum. In the first part, systems of three identical bosons are investigated. As a consequence of the breakdown of the spherical symmetry to cubic symmetry, the partial waves of the bound state amplitude are coupled. An infinite set of coupled integral equations for these partial waves is derived. These equations have to be solved numerically in order to obtain the binding energies in the finite volume. The dependence of the energies on the box size is calculated and the results are explicitly verified to be renormalized. Results for positive and negative scattering lengths are shown. The effects of higher partial waves are investigated. The behavior of shallow trimers near the dimer energy as well as deeply bound trimers is studied. The shallowest state investigated crosses the dimer energy at a certain volume and behaves like a scattering state for smaller volumes. Numerical evidence for a universal scaling of the finite volume corrections is provided. Subsequently, the formalism is extended to systems of three nucleons. This case provides the main motivation for this work due to its applicability to Lattice Quantum Chromodynamics (QCD) calculations of the triton. Such calculations always take place inside a finite volume which makes control over the corresponding effects crucial for an understanding of results from the lattice. For the triton, there are two coupled channels already in the infinite volume corresponding to two different spin-isospin combinations. An infinite set of coupled integral equations for the partial waves of the bound state amplitudes is derived. The renormalization of all results is again explicitly verified. The physical triton inside a finite volume is investigated as well as the triton spectrum for unphysical pion masses. The former case qualitatively shows the same behavior as the three-boson case, and the volume dependence is calculated. The smallest volumes investigated are of the order of magnitude typical for present day Lattice calculations. The motivation for the latter part is twofold. On the one hand, Lattice QCD calculations are performed at pion masses larger than the physical one for computational reasons. On the other hand, it has been conjectured that QCD is close to the critical trajectory for an infrared renormalization group limit cycle, in which case the Efimov effect would occur for a critical pion mass. Close to this critical pion mass, the triton has excited states. The behavior of the ground state and of the excited states inside a finite volume is investigated for various pion masses around the critical one. The excited states cross the energy of the bound di-nucleon, as it was already observed for the shallowest bosonic trimer. The results for the ground state were used to provide strong numerical evidence for a universal scaling of the finite volume corrections.
The universal properties of a three-boson system with large scattering length are well understood within the framework of Effective Field Theory. They include a geometric spectrum of shallow three-body bound states called “Efimov states” and log-periodic dependence of scattering observables on the scattering length. We investigate the modification of this spectrum in a finite cubic box using a partial wave expansion. The dependence of the binding energies on the box size is calculated for systems with positive and negative two-body scattering length. We compare the full results to results obtained using an expansion around the infinite volume binding energy. The renormalization of the Effective Field Theory in the finite volume is verified explicitly.
In this dissertation, we develop and apply a relativistic framework for studying three-body scattering amplitudes, their analytic structure, and the emergence of universal phenomena such as the Efimov effect. The work integrates three complementary components. First, we introduce a systematically improvable numerical method for solving relativistic three-body integral equations in momentum space. By discretizing the continuum problem into finite matrix equations and extrapolating to the continuum limit, we obtain stable partial-wave amplitudes in the presence of a two-body bound state. Two complementary treatments of the pole contribution are implemented and shown to reproduce previous finite volume results, with controlled estimates of systematic uncertainties. Second, we provide a general prescription for analytically continuing these integral equations into the complex energy plane. Using contour-deformation techniques guided by the singularity structure of the kernel, we compute amplitudes below elastic thresholds and on multiple Riemann sheets. This allows us to determine three-body bound and virtual states consistent with earlier finite-volume studies, and to identify departures from the standard two-body finite-volume formalism due to left-hand cuts. Finally, we apply this framework to study Efimov physics in a fully relativistic setting. By tuning the interaction toward unitarity, we observe the characteristic discrete scaling symmetry of Efimov trimers and recover the expected geometric energy pattern. Tracking trimer poles across physical and unphysical Riemann sheets, we reveal rich trajectories connecting virtual states, bound states, and resonances. We also uncover cyclic behavior of higher Efimov resonances and propose a partial resolution to the associated “missing states” puzzle in terms of the multi-sheeted structure induced by the logarithmic three-body unitarity cut. Overall, this work establishes a coherent relativistic approach to the three-body problem, clarifies its analytic structure, and demonstrates its ability to capture universal features relevant to nuclear, hadronic, and atomic few-body systems.
Few-body effects play an important role for the understanding of ultracold quantum gases. We make use of an effective field theory approach to investigate various aspects of universal few-body physics close to a Feshbach resonance. That is the regime where the scattering length is large compared to all other length scales of the system and thus determines the observables. It is also the regime where the Efimov effect whose main characteristic is the occurrence of a sequence of three-body bound states becomes important. At unitarity, i.e., for diverging scattering length, the ratio of binding energies of neighboring states approaches a constant. The existence of those trimers can be captured by a single three-body parameter and influences observables such as recombination rates. Starting from adequate Lagrangians, we derive atom-dimer scattering amplitudes that contain all information of interest about the three-particle systems. With this method, we first investigate the scattering of atoms and dimers at finite temperature in the presence of the Efimov effect. We calculate the dimer relaxation rate and obtain good agreement with available experimental data. Furthermore, heteronuclear mixtures exhibiting large interspecies scattering length are studied in detail. If bosons are the majority species, the Efimov effect occurs in an S-wave channel and we are able to compute three-body recombination and dimer relaxation rates. The results are compared to the outcome of two existing experiments. For mainly fermionic systems, the Efimov effect is only present in an overall P-wave and three-body recombination at threshold vanishes. However, in mixtures of atoms and dimers, scattering cross sections and dimer relaxation show the typical Efimov behavior of log-periodicity. We also consider two-dimensional Bose gases, where the Efimov effect does not occur. For those systems, we derive an equation including the first non-universal corrections and deduce three-body observables such as binding energies and atom-dimer scattering parameters.
In the low-energy limit, non-relativistic particles with short-range interactions exhibit universal behavior that is largely independent of microscopic details. This universality is typically described by effective field theory, in which the two-body interaction is renormalized to a single parameter-the scattering length. For systems of identical bosons, the three-body problem reveals the Efimov effect, a novel phenomenon proposed that necessitates the introduction of an additional three-body parameter. However, the exact relation between this three-body parameter, the coupling constants in the effective field theory, and the binding energies of Efimov states remains unresolved. In this Letter, we address this question through a comprehensive analysis of the Skorniakov-Ter-Martirosian equation with a finite cutoff. We establish an exact renormalization relation for the three-body parameter and determine its connection to the energies of Efimov bound states. These results are validated through high-precision numerical simulations. We expect our findings to be of fundamental interest across various fields, including atomic, nuclear, condensed matter, and particle physics, and to have broad applications in both few-body and many-body physics.
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.
We have investigated S-wave bound states composed of three identical bosons interacting via regulated delta function potentials in non-relativistic quantum mechanics. For low-energy systems, these short-range potentials serve as an approximation to the underlying physics, leading to an effective field theory. A method for perturbatively expanding the three-body bound-state equation in inverse powers of the cutoff is developed. This allows us to extract some analytical results concerning the behavior of the system. Further results are obtained by solving the leading order equations numerically to 11 or 12 digits of accuracy. The limit-cycle behavior of the required three-body contact interaction is computed, and the cutoff-independence of bound-state energies is shown. By studying the relationship between the two- and three-body binding energies, we obtain a high accuracy numerical calculation of Efimov's universal function. Equations for the first order corrections, necessary for the study of cutoff dependence, are derived. However, a numerical solution of these equations is not attempted.