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arXiv:1208.0719 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Wilsonian renormalization group analysis of nonrelativistic three-body systems without introducing dimerons

Koji Harada, Hirofumi Kubo, Issei Yoshimoto

Published Apr 2, 2013• 4 citations
arXivHTMLPDF

Abstract

Low-energy effective field theory describing a nonrelativistic three-body system is analyzed in the Wilsonian renormalization group method. No effective auxiliary field (dimeron) that corresponds to two-body propagation is introduced. The Efimov effect is expected in the case of an infinite two-body scattering length and is believed to be related to the limit cycle behavior in the three-body renormalization group equations (RGEs). If the one-loop property of the RGEs for the nonrelativistic system without the dimeron field, which is essential in deriving RGEs in the two-body sector, persists in the three-body sector, it appears to prevent the emergence of limit cycle behavior. We explain how the multiloop diagrams contribute in the three-body sector without contradicting the one-loop property of the RGEs and derive the correct RGEs, which lead to the limit cycle behavior. The Efimov parameter, ${s}_{0}$, is obtained within a few percent error in the leading orders. We also remark on the correct use of the dimeron formulation. We find rich renormalization group flow structure in the three-body sector. In particular, a novel nontrivial fixed point of the three-body couplings is found when the two-body interactions are absent. We also find, on the two-body nontrivial fixed point, the limit cycle is realized as a loop of finite size in the space of three-body coupling constants when terms with derivatives are included.

DOI
10.1103/physrevd.87.085006

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Theoretical study on a renormalization-group limit cycle in Efimov physics

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arXiv:nucl-th/0406008•Journal of Physics A Mathematical and General•Dec 24, 2004•48 citations
Low-energy effective field theory describing a nonrelativistic three-body system is analyzed in the Wilsonian renormalization group method. No effective auxiliary field (dimeron) that corresponds to two-body propagation is introduced. The Efimov effect is expected in the case of an infinite two-body scattering length and is believed to be related to the limit cycle behavior in the three-body renormalization group equations (RGEs). If the one-loop property of the RGEs for the nonrelativistic system without the dimeron field, which is essential in deriving RGEs in the two-body sector, persists in the three-body sector, it appears to prevent the emergence of limit cycle behavior. We explain how the multiloop diagrams contribute in the three-body sector without contradicting the one-loop property of the RGEs and derive the correct RGEs, which lead to the limit cycle behavior. The Efimov parameter, s0{s}_{0}s0​, is obtained within a few percent error in the leading orders. We also remark on the correct use of the dimeron formulation. We find rich renormalization group flow structure in the three-body sector. In particular, a novel nontrivial fixed point of the three-body couplings is found when the two-body interactions are absent. We also find, on the two-body nontrivial fixed point, the limit cycle is realized as a loop of finite size in the space of three-body coupling constants when terms with derivatives are included.
Physics is often said to be universal when microscopically distinct systems show the same lowenergy behavior.In critical phenomena, the universality of the second-order phase transition is elucidated by the renormalization-group (RG) method, which provides a powerful computational tool for the critical exponents of the power-law behavior of observables.In particular, the attraction of different RG-flow trajectories to a small-dimensional sub-theory space accounts for the universal nature.The attraction structure motivates us to extract universal physical observables by investigating the sub-theory space called the renormalized trajectory, onto which RG flows are attracted.To put the motivation forward, we consider the universality in quantum few-body physics, in which the RG exhibits another characteristic flow than the fixed points.The Efimov physics, which is renowned for self-similar trimer states, shows an RG limit-cycle behavior that refers to a periodic RG-flow trajectory.In this thesis, we investigate the relation between the RG limit cycle and the universality in Efimov physics.Firstly, to verify that the RG limit cycle is the renormalized trajectory in Efimov physics, we demonstrate that the RG-flow trajectories of microscopically distinct systems universally exhibit the RG limit-cycle behavior at sufficiently low energy, thereby giving a numerical evidence of the attraction structure of the RG-flow trajectories to the RG limit cycle; in other words, we show that the RG limit cycle behaves as an attractor of various RG flows.To address this point unbiasedly, we perform an exact RG calculation for various effective models of identical bosons that features Efimov physics at low energy.We apply an exact functional renormalization-group (FRG) method to effective separable models for various systems of identical bosons with short-range resonant interactions.For a given microscopic three-body model, we numerically demonstrate for the first time that the RG-flow trajectory converges at the RG limit cycle in the low-energy regime.Physical observables such as Efimov's scaling factor and the three-body parameter can be extracted from the RG limit cycle.Secondly, we investigate the role of RG limit cycle in a four-body extension of Efimov physics, which has attracted considerable attention in recent years.Although the universality in four-body physics has been established both theoretically and experimentally, the relationship between the RG limit cycle and the four-body physics remains an open question.By performing an FRG calculation for a simple effective field theory, we obtain for the first time the RG limit cycle in the four-body sector that reproduces the universality of the four-body physics.As the first attempt, we develop a simple separable pole approximation of the four-body FRG equation and successfully reproduce qualitative features of the four-body physics.A systematic improvement of the approximation quantitatively reproduces the numbers and the binding energies of tetramer states that emerge universally at low energy.2 The restriction r 0 < ∼ R < ∼ a on the length scale limits the binding energy E t of the Efimov trimers as ℏ ma 2 < ∼ E t < ∼ ℏ mr 2 0 .3 A similar discrete scale-invariant phenomenon is known for the doubly excited H -ion, in which two valence electrons are excited from their ground states [16].In the doubly excited H -ion, an interaction between an electron-
We apply a field-theoretic functional renormalization-group technique to the few-body (vacuum) physics of nonrelativistic atoms near a Feshbach resonance. Three systems are considered: one-component bosons with a U(1) symmetry, two-component fermions with a $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{SU}(2)$ symmetry, and three-component fermions with a $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{SU}(3)$ symmetry. We focus on the scale-invariant unitarity limit of an infinite scattering length. The exact solution for the two-body sector is consistent with the unitary fixed-point behavior of the considered systems. Nevertheless, the numerical three-body solution in the sss-wave sector develops a limit cycle scaling in case of U(1) bosons and SU(3) fermions. The Efimov parameter for the one-component bosons and the three-component fermions is found to be $s\ensuremath{\approx}1.006$, consistent with the result of Efimov.
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.
The Efimov effect is the only experimentally realized universal phenomenon that exhibits the renormalization-group limit cycle with the three-body parameter parametrizing a family of universality classes. Recent experiments in ultracold atoms have unexpectedly revealed that the three-body parameter itself is universal when measured in units of an effective range. By performing an exact functional renormalization-group analysis with various finite-range interaction potentials, we demonstrate that the onset of the renormalization-group flow into the limit cycle is universal, regardless of short-range details, which connects the missing link between the two universalities of the Efimov physics. A close connection between the topological property of the limit cycle and few-body physics is also suggested.
In this thesis nonrelativistic few-body quantum physics is investigated with the method of functional renormalization. In particular, we discuss different theoretical aspects of few-body physics related to the Efimov effect. Our central finding is that this effect manifests itself as a renormalization group limit cycle. First, we treat the problem of a singular, inverse square potential in quantum mechanics. Then we examine the original Efimov problem of three particles interacting via a short-range two-body potential. Subsequently, we present a first step towards the renormalization group solution of the four-body quantum problem for bosons. Finally, a general theoretical feature of the Efimov physics, composite operators with complex scaling dimensions, are discussed.
A distorted-wave version of the renormalization group is applied to scattering by an inverse-square potential and to three-body systems. In attractive three-body systems, the short-distance wavefunction satisfies a Schrödinger equation with an attractive inverse-square potential, as shown by Efimov. The resulting oscillatory behaviour controls the renormalization of the three-body interactions, with the renormalization-group flow tending to a limit cycle as the cut-off is lowered. The approach used here leads to single-valued potentials with discontinuities as the bound states are cut off. The perturbations around the cycle start with a marginal term whose effect is simply to change the phase of the short-distance oscillations, or the self-adjoint extension of the singular Hamiltonian. The full power counting in terms of the energy and two-body scattering length is constructed for short-range three-body forces.