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:nucl-th/0511003

An Effective Theory for the Four-Body System with Short-Range Interactions

Lucas Platter

Published Nov 1, 2005• 0 citations
arXivHTMLPDF

Abstract

We discuss the application of an effective theory with contact interactions to the non-relativistic four-body system with short-range interactions. We have computed the binding energies of the 4He tetramer and the alpha-particle. A well-known linear correlation between three- and four-body binding energies can be understood as a result of the absence of a four-body force at leading order within this framework. Further, we give results for the binding energies of the four-boson system in two spatial dimensions.

DOI
10.48550/arxiv.nucl-th/0511003

Related papers

An effective theory for the four-body system

Lucas Platter

We consider the non‐relativistic four‐body system with large scattering length and short‐range interactions within an effective theory with contact interactions only. We compute the binding energies of the 4He tetramer and of α‐particle. The well‐known linear correlation between the three‐body binding energies and the four‐body binding energies of these physical systems can be understood as a consequence of the absence of a four‐body force at leading order.

arXiv:cond-mat/0410016•AIP conference proceedings•Jan 1, 2005•0 citations

Four-boson system with short-range interactions

Lucas Platter, H.‐W. Hammer, Ulf-G. Meißner

We consider the nonrelativistic four-boson system with short-range forces and large scattering length in an effective quantum mechanics approach. We construct the effective interaction potential at leading order in the large scattering length and compute the four-body binding energies using the Yakubovsky equations. Cutoff independence of the four-body binding energies does not require the introduction of a four-body force. This suggests that two- and three-body interactions are sufficient to renormalize the four-body system. We apply the equations to $^{4}\mathrm{H}\mathrm{e}$ atoms and calculate the binding energy of the $^{4}\mathrm{H}\mathrm{e}$ tetramer. We observe a correlation between the trimer and tetramer binding energies similar to the Tjon line in nuclear physics. Over the range of binding energies relevant to $^{4}\mathrm{H}\mathrm{e}$ atoms, the correlation is approximately linear.

arXiv:cond-mat/0404313•Physical Review A•Nov 5, 2004•201 citations

Four-body systems at large cutoffs in effective field theory

Xincheng Lin

Four-body systems are studied using an effective-field theory with two- and three-body contact interactions. A method to systematically address deep trimers (three-body bound states that are more tightly bound than four-body bound or resonant states) in four-body calculations is developed using a diagrammatic approach. Previous calculations were limited by the existence of deep trimers, which this work overcomes. For cold $^{4}\mathrm{He}$ atoms, binding energies of 526.1(5) and 128.517(1) mK are obtained at leading order for the tetramer ground and excited states, respectively, where errors come from the truncation of three-body partial waves. Tetramer binding energies and decay widths are also computed approaching the unitary limit. In the unitary limit, there are two tetramers associated with each trimer of binding energy ${B}_{3}^{(0)}$. The binding energy and decay width for the associated tetramer ground state are ${E}_{4}^{(0)}=4.60(1){B}_{3}^{(0)}$ and ${\mathrm{\ensuremath{\Gamma}}}_{4}^{(0)}/2=0.0160(1){B}_{3}^{(0)}$, respectively, and for the associated tetramer excited state, ${E}_{4}^{(1)}=1.0022(3){B}_{3}^{(0)}$ and ${\mathrm{\ensuremath{\Gamma}}}_{4}^{(1)}/2=2.57(2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}{B}_{3}^{(0)}$, respectively. This calculation is a gateway to higher-order and/or more-body calculations in nuclear and atomic systems.

arXiv:2304.06172•Physical Review C•Feb 20, 2024•3 citations

Three- and four-boson systems expanded around the unitarity limit: Application to $^4$He molecules

Feng Wu, Xincheng Lin, U. van Kolck +1 more

The three- and four-boson systems with a large scattering length and a short effective range in the two-body sector are studied in the framework of Short-Range Effective Field Theory (SREFT). The starting point (leading order) of the EFT is taken to be the universal unitarity limit, where the two-body sector is parameter-free and only one three-body parameter enters. In this limit, physical systems manifest discrete scale invariance. Deviations from universality arising from finite scattering-length and effective-range corrections, as well as a four-body force required by renormalization, are included perturbatively at next-to-leading order. The three-body ground state and its associated four-body ground and first-excited states are studied using the Faddeev-Yakubovsky (FY) formalism and a complementary diagrammatic approach. By employing techniques to remove contributions from deep trimers in tetramer calculations, we extend our analysis to larger cutoffs than previously accessible within the FY approach to SREFT. Our results for binding energies and radii of $^4$He three- and four-atom systems converge well to results obtained with sophisticated phenomenological potentials. These successes suggest that the physics of $^4$He atomic clusters is governed by only small deviations from discrete scale invariance.

arXiv:2606.00854•May 30, 2026•0 citations

Effective field theory for few-boson systems

Betzalel Bazak, Moti Eliyahu, U. van Kolck

We study universal bosonic few-body systems within the framework of effective field theory at leading order (LO). We calculate binding energies of systems of up to six particles and the atom-dimer scattering length. Convergence to the limit of zero-range two- and three-body interactions is shown, indicating that no additional few-body interactions need to be introduced at LO. Generalizations of the Tjon line are constructed, showing correlations between few-body binding energies and the binding energy of the trimer, for a given dimer energy. As a specific example, we implement our theory for $^{4}\mathrm{He}$ atomic systems and show that the results are in surprisingly good agreement with those of sophisticated $^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}$ potentials. Potential implications for the convergence of the EFT expansion are discussed.

arXiv:1607.01509•Physical Review A•Nov 1, 2016•47 citations

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
We discuss the application of an effective theory with contact interactions to the non-relativistic four-body system with short-range interactions. We have computed the binding energies of the 4He tetramer and the alpha-particle. A well-known linear correlation between three- and four-body binding energies can be understood as a result of the absence of a four-body force at leading order within this framework. Further, we give results for the binding energies of the four-boson system in two spatial dimensions.
We consider the nonrelativistic four-boson system with short-range forces and large scattering length in an effective quantum mechanics approach. We construct the effective interaction potential at leading order in the large scattering length and compute the four-body binding energies using the Yakubovsky equations. Cutoff independence of the four-body binding energies does not require the introduction of a four-body force. This suggests that two- and three-body interactions are sufficient to renormalize the four-body system. We apply the equations to 4He^{4}\mathrm{H}\mathrm{e}4He atoms and calculate the binding energy of the 4He^{4}\mathrm{H}\mathrm{e}4He tetramer. We observe a correlation between the trimer and tetramer binding energies similar to the Tjon line in nuclear physics. Over the range of binding energies relevant to 4He^{4}\mathrm{H}\mathrm{e}4He atoms, the correlation is approximately linear.
Four-body systems are studied using an effective-field theory with two- and three-body contact interactions. A method to systematically address deep trimers (three-body bound states that are more tightly bound than four-body bound or resonant states) in four-body calculations is developed using a diagrammatic approach. Previous calculations were limited by the existence of deep trimers, which this work overcomes. For cold 4He^{4}\mathrm{He}4He atoms, binding energies of 526.1(5) and 128.517(1) mK are obtained at leading order for the tetramer ground and excited states, respectively, where errors come from the truncation of three-body partial waves. Tetramer binding energies and decay widths are also computed approaching the unitary limit. In the unitary limit, there are two tetramers associated with each trimer of binding energy B3(0){B}_{3}^{(0)}B3(0)​. The binding energy and decay width for the associated tetramer ground state are E4(0)=4.60(1)B3(0){E}_{4}^{(0)}=4.60(1){B}_{3}^{(0)}E4(0)​=4.60(1)B3(0)​ and ${\mathrm{\ensuremath{\Gamma}}}_{4}^{(0)}/2=0.0160(1){B}_{3}^{(0)}$, respectively, and for the associated tetramer excited state, E4(1)=1.0022(3)B3(0){E}_{4}^{(1)}=1.0022(3){B}_{3}^{(0)}E4(1)​=1.0022(3)B3(0)​ and ${\mathrm{\ensuremath{\Gamma}}}_{4}^{(1)}/2=2.57(2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}{B}_{3}^{(0)}$, respectively. This calculation is a gateway to higher-order and/or more-body calculations in nuclear and atomic systems.
Three- and four-boson systems expanded around the unitarity limit: Application to 4^44He molecules
The three- and four-boson systems with a large scattering length and a short effective range in the two-body sector are studied in the framework of Short-Range Effective Field Theory (SREFT). The starting point (leading order) of the EFT is taken to be the universal unitarity limit, where the two-body sector is parameter-free and only one three-body parameter enters. In this limit, physical systems manifest discrete scale invariance. Deviations from universality arising from finite scattering-length and effective-range corrections, as well as a four-body force required by renormalization, are included perturbatively at next-to-leading order. The three-body ground state and its associated four-body ground and first-excited states are studied using the Faddeev-Yakubovsky (FY) formalism and a complementary diagrammatic approach. By employing techniques to remove contributions from deep trimers in tetramer calculations, we extend our analysis to larger cutoffs than previously accessible within the FY approach to SREFT. Our results for binding energies and radii of 4^44He three- and four-atom systems converge well to results obtained with sophisticated phenomenological potentials. These successes suggest that the physics of 4^44He atomic clusters is governed by only small deviations from discrete scale invariance.
We study universal bosonic few-body systems within the framework of effective field theory at leading order (LO). We calculate binding energies of systems of up to six particles and the atom-dimer scattering length. Convergence to the limit of zero-range two- and three-body interactions is shown, indicating that no additional few-body interactions need to be introduced at LO. Generalizations of the Tjon line are constructed, showing correlations between few-body binding energies and the binding energy of the trimer, for a given dimer energy. As a specific example, we implement our theory for 4He^{4}\mathrm{He}4He atomic systems and show that the results are in surprisingly good agreement with those of sophisticated $^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}$ potentials. Potential implications for the convergence of the EFT expansion are discussed.
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.