Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1701.01200 · Journal of the Korean Physical Society

A renormalization method for a three-boson system with a triboson field

Shung-Ichi Ando

Published Feb 1, 2017• 1 citation
arXivHTMLPDF

Abstract

A renormalization method that introduces an auxiliary field to represent a three-body bound state is studied in a three-boson system with a triboson field. A cutoff dependence in the threeboson system emerges as a limit cycle, and the cyclic singularity is renormalized by employing two methods: a standard method and the auxiliary field method. For each method, different sets of diagrams are involved for renormalization; thus, we numerically study three quantities: a counter term for renormalization, the scattering length of s-wave boson-diboson scattering, and the normalized wavefunction. We confirm that the two methods lead to the same result.

DOI
10.3938/jkps.70.358

Related papers

Quantum Mechanical Three-Body Problem with Short-Range Interactions

Richard Mohr

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.

arXiv:nucl-th/0306086•Jun 29, 2003•4 citations

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

Koji Harada, Hirofumi Kubo, Issei Yoshimoto

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.

arXiv:1208.0719•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Apr 2, 2013•4 citations

Effective theories of scattering with an attractive inverse-square potential and the three-body problem

Thomas Barford, Michael C. Birse

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.

arXiv:nucl-th/0406008•Journal of Physics A Mathematical and General•Dec 24, 2004•48 citations

Four-boson bound states from a functional renormalization group

Benjamín Raziel Jaramillo Avila, Michael C. Birse

We use the functional renormalization group to study the spectrum of three- and four-body states in bosonic systems around the unitary limit. Our effective action includes all energy-independent contact interactions in the four-atom sector, and we introduce a running trimer field to eliminate couplings that involve the atom-atom-dimer channel. The results show behavior qualitatively similar to those from exact approaches. The truncated action we use leads to overbinding of the two four-body states seen in those treatments. It also generates a third state, although only for a very narrow range of two-body scattering lengths.

arXiv:1506.04949•Physical Review A•Aug 3, 2015•6 citations

Renormalization of a finite-range inverse-cube potential

Daniel Odell, Arnoldas Deltuva, José Luis López-Bonilla +1 more

We study the regularization and renormalization of a finite-range inverse-cube potential in the two- and three-body sectors. Specifically, we compare and contrast three different regulation schemes frequently used to study few-body systems as well as the associated renormalization group (RG) flows. We also calculate bound-state and scattering observables over a wide range of cutoffs, demonstrating the sufficiency of a two-body contact interaction to renormalize two- and three-body observables. We supplement these plots with quantified analyses of the observables' residual cutoff dependences.

arXiv:1903.00034•Physical Review C•Nov 4, 2019•9 citations

Boson-Triboson Scattering with Yamaguchi Potential. I: Restriction to s-Wave Contribution from the 3+1-Subamplitude

Yoshiko Matsui

For the investigation of the S-wave boson-triboson phase shift, the Faddeev-Osborn equation is solved numerically by assuming a rank-one separable potential of the Yamaguchi type for the two-particle interaction. The calculation is performed in the exact approach based on rigorous Faddeev theory of the scattering problem. As a first step of the investigation, to gain insight into the origin of a four-body resonance, the contribution from the 3+1-subamplitude is limited to the s-wave only. The calculated phase shift behaves generally like a standard two-body one for a potential that causes one loosely bound state; this phase shift has further characteristic behavior resulting from fine structure consisting of a valley and a peak. The result of the partial cross section indicates the presence of two resonant states in the four-boson system with the Yamaguchi potential.

Progress of Theoretical Physics•Sep 1, 1999•1 citation
A renormalization method that introduces an auxiliary field to represent a three-body bound state is studied in a three-boson system with a triboson field. A cutoff dependence in the threeboson system emerges as a limit cycle, and the cyclic singularity is renormalized by employing two methods: a standard method and the auxiliary field method. For each method, different sets of diagrams are involved for renormalization; thus, we numerically study three quantities: a counter term for renormalization, the scattering length of s-wave boson-diboson scattering, and the normalized wavefunction. We confirm that the two methods lead to the same result.
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.
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.
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.
We use the functional renormalization group to study the spectrum of three- and four-body states in bosonic systems around the unitary limit. Our effective action includes all energy-independent contact interactions in the four-atom sector, and we introduce a running trimer field to eliminate couplings that involve the atom-atom-dimer channel. The results show behavior qualitatively similar to those from exact approaches. The truncated action we use leads to overbinding of the two four-body states seen in those treatments. It also generates a third state, although only for a very narrow range of two-body scattering lengths.
We study the regularization and renormalization of a finite-range inverse-cube potential in the two- and three-body sectors. Specifically, we compare and contrast three different regulation schemes frequently used to study few-body systems as well as the associated renormalization group (RG) flows. We also calculate bound-state and scattering observables over a wide range of cutoffs, demonstrating the sufficiency of a two-body contact interaction to renormalize two- and three-body observables. We supplement these plots with quantified analyses of the observables' residual cutoff dependences.
For the investigation of the S-wave boson-triboson phase shift, the Faddeev-Osborn equation is solved numerically by assuming a rank-one separable potential of the Yamaguchi type for the two-particle interaction. The calculation is performed in the exact approach based on rigorous Faddeev theory of the scattering problem. As a first step of the investigation, to gain insight into the origin of a four-body resonance, the contribution from the 3+1-subamplitude is limited to the s-wave only. The calculated phase shift behaves generally like a standard two-body one for a potential that causes one loosely bound state; this phase shift has further characteristic behavior resulting from fine structure consisting of a valley and a peak. The result of the partial cross section indicates the presence of two resonant states in the four-boson system with the Yamaguchi potential.