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Relativistic Three Particle Scattering

Md Habib E Islam

Published Feb 19, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.25777/5fmr-xk29

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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.
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 states are three-body bound states, which appear universally when the s-wave scattering length between the particles is resonantly large.Since the Efimov states and their characteristic discrete scale invariance have recently been observed in ultracold atoms, the Efimov states and their associated few-body phenomena have attracted widening interest both theoretically and experimentally.In this thesis, I theoretically study the Efimov physics and related phenomena, focusing on the following three topics: (i) Universal three-body physics for a mass-imbalanced Fermi system (Chapter 3).For a system of two identical spinless (i.e., spin-polarized) fermions and one distinguishable particle which interact via a short-range potential with a large s-wave scattering length, two classes of universal three-body bound states have been known to appear in different regimes of the mass ratio: the Efimov trimers and the Kartavtsev-Malykh trimers, which feature the discrete and continuous scale invariance, respectively.I have found the third class of universal three-body bound states, which I call the "crossover trimers".The crossover trimers show neither the discrete nor continuous scale invariance.I have identified the regions of these three classes of trimers as a function of the mass ratio and the s-wave scattering length, and shown that the Kartavtsev-Malykh trimers and the Efimov trimers can continuously transform into each other via the crossover trimers as the mass ratio and the s-wave scattering length are varied.Owing to the presence of the crossover trimers, the Kartavtsev-Malykh trimers dissociate into a particle and a dimer when the s-wave scattering length is varied, inducing resonances in the particle-dimer scattering.I have calculated the elastic particle-dimer scattering lengths in arbitrary angular-momentum channels, and shown that the particle-dimer resonances indeed occur.From the resonance positions, I have found accurate values of the critical mass ratios at which the Kartavtsev-Malykh trimers in the higher angular-momentum channels appear. (ii) Universal three-body parameter (Chapter 4).In the Efimov physics, the three-body parameter fixes the short-range three-body phase and hence the scale of the energy spectrum.Until quite recently, it has been widely held that the threebody parameter should depend on the short-range part of the inter-atomic potential, and hence vary almost randomly between different atomic species and hyperfine states.However, mounting evidence in recent experiments suggests otherwise.While some theoretical studies have succeeded in reproducing the universal behavior in the three-body parameter, the underlying physical mechanism has remained unclear and controversial.In this thesis, I elucidate the physical origin of the universal three-body parameter.I propose that a non-adiabatic deformation of the three-body wave function induced by a universal two-body correlation results in a universal three-body repulsion, which prevents three particles from coming close and renders the three-body parameter universal.This mechanism is verified by reproducing the universal three-body repulsion with a simple model wave function.The mechanism found here suggests that the three-body parameter would also be universal for non-van der Waals types of potentials.For various classes of deep two-body potentials, it is shown that the effective range is the relevant length scale to characterize the two-body correlation and sets the three-body parameter.Depending on the shape of the two-body correlation, there
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 character of non-relativistic large three-body bosonic systems is addressed, with focus on highly interesting situations that occur in low-energy nuclear and atomic physics. Investigations on the trajectory of the first excited Efimov state, in a renormalized zero-range three-body model for a system with two bound and one virtual two-body subsystems, are reported. 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 the real part of the elastic s-wave phase-shift ([Formula: see text]) presents a zero, or a pole in [Formula: see text], when the system has an Efimov excited or virtual state. A brief discussion is given on the relevance of the approach for ultracold atom physics with tunable scattering lengths.
Efimov states are known to have a discrete real-space scale invariance; working in momentum space we identify the relevant discrete scale invariance for the scattering amplitude defining its Weierstrass function as well. Through the use of the mathematical formalism for discrete scale invariance for the scattering amplitude we identify the scaling parameters from the pole structure of the corresponding zeta function; its zeroth-order pole is fixed by the Efimov physics. The corresponding geometrical fractal structure for Efimov physics in momentum space is identified as a ray across a logarithmic spiral. This geometrical structure also appears in the physics of atomic collapse in the relativistic regime connecting it to Efimov physics. Transforming to logarithmic variables in momentum space we map the three-body scattering amplitude into Bloch states and the ladder of energies of the Efimov states are simply obtained in terms of the Bohr-Sommerfeld quantization rule. Thus through the mapping the complex problem of three-body short-range interaction is transformed to that of a noninteracting single particle in a discrete lattice.
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.