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arXiv:1710.11228 · Few-Body Systems

Few-Body Techniques Using Momentum Space for Bound and Continuum States

Marcelo Takeshi Yamashita, D. S. Rosa, J. H. Sandoval

Published Mar 6, 2018• 1 citation
arXivHTMLPDF
DOI
10.1007/s00601-018-1347-1

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These notes were written for a set of three lectures given in a school at the Max Planck Institute for the Physics of Complex Systems in October/2017 before the workshop Critical Stability of Quantum Few-Body Systems. These lectures are primarily dedicated to the students and represent a very idiosyncratic vision of the author, mainly in the last part of the text related to applications. These notes are only a tentative to show a technique, among many others, to solve problems in a very rich area of the contemporary physics - the Few-Body Physics - many times unknown by a considerable part of the students.
Abstract Few-body scattering and loosely bound states in the vicinity of an isolated narrow Feshbach resonance are well described by a conceptually simple contact interaction two-channel model, where an open atomic continuum channel is coupled to a closed molecular channel. However, real systems are often characterized by the multiplicity of both continuum and molecular channels. Here we develop a systematic framework to deal with multi-channel problems within a contact interaction approximation. We demonstrate the differences and similarities of adding a molecular or atomic channel to the two-channel model. By means of direct comparison, we show that while an additional atomic channel makes loosely bound states shallower, an additional molecular channel makes them more deeply bound. This approach facilitates the study of the influence of multi-channel environments on few-body physics in a systematic manner by gradually increasing the level of complexity. Furthermore, we account for real atomic systems whose understanding can benefit from this study.
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.