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:2304.06172 · Physical Review C

Four-body systems at large cutoffs in effective field theory

Xincheng Lin

Published Feb 20, 2024• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevc.109.024002

Related papers

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

Small Clusters of $^4$He Atoms in Finite-Cutoff Effective Field Theory

Betzalel Bazak

Small clusters of $^4$He atoms are benchmark systems for universal few-body physics near the unitary limit. We study these systems using a finite-cutoff effective field theory calibrated to low-energy observables from the realistic LM2M2 potential. The chosen two-body cutoff reproduces both the atom--atom scattering length and effective range, while a regulated three-body interaction is adjusted to the trimer and tetramer ground-state energies. We distinguish total binding energies from the one-atom separation energies $S_A^*=B_A^*-B_{A-1}$ of the shallow excited states. Ground-state energies through $A=8$ are reproduced at the few-percent level, but threshold-sensitive observables are less accurate. In particular, the calculated tetramer separation energy is $S_4^*=2.85$~mK, compared with $0.92$--$0.96$~mK in converged LM2M2 calculations, and the atom--trimer scattering length differs substantially from modern LM2M2 benchmarks. For $A\ge5$, the available excited-state benchmarks are sparse and strongly method dependent. Comparison with another soft-core interaction suggests that the discrepancies primarily reflect short-distance physics and higher-order operators omitted from the finite-cutoff Hamiltonian rather than numerical convergence of the stochastic variational calculation. Finite-cutoff EFT therefore provides an efficient description of ground-state systematics, whereas shallow excited states and atom--cluster scattering require a controlled higher-order and cutoff-dependence analysis.

cond-mat.quant-gasCross-listedarXiv:2511.12538•Scientific Reports•Nov 18, 2025•0 citations

Linear correlations between 4 He trimer and tetramer energies calculated with various realistic 4 He potentials

Emiko Hiyama, M. Kamimura

In a previous work [Phys. Rev. A 85, 022502 (2012)] we calculated, with the use of the Gaussian expansion method for few-body systems, the energy levels and spatial structures of the ${}^{4}$He trimer and tetramer ground and excited states using the LM2M2 potential, which has a very strong short-range repulsion. In this work, we calculate the same quantities using the currently most accurate ${}^{4}$He-${}^{4}$He potential [Przybytek et al., Phys. Rev. Lett. 104, 183003 (2010)] that includes the adiabatic, relativistic, QED, and residual retardation corrections. Contributions of the corrections to the tetramer ground-(excited-) state energy $\ensuremath{-}573.90\phantom{\rule{0.16em}{0ex}}(\ensuremath{-}132.70)$ mK are respectively $\ensuremath{-}4.13\phantom{\rule{0.16em}{0ex}}(\ensuremath{-}1.52)$ mK, $+9.37\phantom{\rule{0.16em}{0ex}}(+3.48)$ mK, $\ensuremath{-}1.20\phantom{\rule{0.16em}{0ex}}(\ensuremath{-}0.46)$ mK, and $+0.16\phantom{\rule{0.16em}{0ex}}(+0.07)$ mK. Further including other realistic ${}^{4}$He potentials, we calculated the binding energies of the trimer and tetramer ground and excited states, ${B}_{3}^{(0)},\phantom{\rule{0.16em}{0ex}}{B}_{3}^{(1)},\phantom{\rule{0.16em}{0ex}}{B}_{4}^{(0)}$, and ${B}_{4}^{(1)}$, respectively. We found that the four kinds of the binding energies for the different potentials exhibit perfect linear correlations between any two of them over the range of binding energies relevant for ${}^{4}$He atoms (namely, six types of the generalized Tjon lines are observed). The dimerlike-pair model for ${}^{4}$He clusters, proposed in the previous work, predicts a simple interaction-independent relation $\frac{{B}_{4}^{(1)}}{{B}_{2}}=\frac{{B}_{3}^{(0)}}{{B}_{2}}+\frac{2}{3}$, which precisely explains the correlation between the tetramer excited-state energy and the trimer ground-state energy, with ${B}_{2}$ being the dimer binding energy.

arXiv:1203.3130•Physical Review A•Jun 6, 2012•52 citations

Variational calculation of 4 He tetramer ground and excited states using a realistic pair potential

Emiko Hiyama, M. Kamimura

We calculated the ${}^{4}$He trimer and tetramer ground and excited states with the LM2M2 potential using our Gaussian expansion method for ab initio variational calculations of few-body systems. The method has been extensively used for a variety of three-, four-, and five-body systems in nuclear physics and exotic atomic and molecular physics. The trimer (tetramer) wave function is expanded in terms of symmetric three- (four-) body Gaussian basis functions, ranging from very compact to very diffuse, without assumption of any pair correlation function. The calculated results for the trimer ground and excited states are in excellent agreement with values reported in the literature. The binding energies of the tetramer ground and excited states are obtained as 558.98 and 127.33 mK (0.93 mK below the trimer ground state), respectively. We found that precisely the same shape of the short-range correlation (${r}_{ij}\ensuremath{\lesssim}4$ \AA{}) in the dimer appears in the ground and excited states of the trimer and tetramer. The overlap function between the trimer excited state and the dimer ground state and that between the tetramer excited state and the trimer ground state are almost proportional to the dimer wave function in the asymptotic region (up to $\ensuremath{\sim}$1000 \AA{}). Also, the pair correlation functions of trimer and tetramer excited states are almost proportional to the squared dimer wave function. We then propose a model which predicts the binding energy of the first excited state of ${}^{4}$He${}_{N}$ ($N\ensuremath{\ge}3$) measured from the ${}^{4}$He${}_{N\ensuremath{-}1}$ ground state to be nearly $\frac{N}{2(N\ensuremath{-}1)}{B}_{2}$ where ${B}_{2}$ is the dimer binding energy.

arXiv:1111.4370•Physical Review A•Feb 2, 2012•58 citations

Binding and structure of tetramers in the scaling limit

M. R. Hadizadeh, Marcelo Takeshi Yamashita, Lauro Tomio +2 more

The momentum-space structure of the Faddeev-Yakubovsky (FY) components of weakly bound tetramers is investigated at the unitary limit using a renormalized zero-range two-body interaction. The results, obtained by considering a given trimer level with binding energy ${B}_{3}$, provide further support to a universal scaling function relating the binding energies of two successive tetramer states. The correlated scaling between the tetramer energies comes from the sensitivity of the four-boson system to a short-range four-body scale. Each excited $N$th tetramer energy ${B}_{4}^{(N)}$ moves as the short-range four-body scale changes, while the trimer properties are kept fixed, with the next excited tetramer ${B}_{4}^{(N+1)}$ emerging from the atom-trimer threshold for a universal ratio ${B}_{4}^{(N)}/{B}_{3}={B}_{4}^{(N)}/{B}_{4}^{(N+1)}\ensuremath{\simeq}4.6$, which does not depend on $N$. We show that both channels of the FY decomposition [atom-trimer ($K$ type) and dimer-dimer ($H$ type)] present high-momentum tails that reflect the short-range four-body scale. We also found that the $H$ channel is favored over the $K$ channel at low momentum, when the four-body momentum scale largely overcomes the three-body scale.

arXiv:1110.5214•Physical Review A•Feb 8, 2012•33 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 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 \ensuremathΓ4(0)/2=0.0160(1)B3(0){\mathrm{\ensuremath{\Gamma}}}_{4}^{(0)}/2=0.0160(1){B}_{3}^{(0)}\ensuremathΓ4(0)​/2=0.0160(1)B3(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 \ensuremathΓ4(1)/2=2.57(2)\ifmmode×\else\texttimes\fi10\ensuremath−4B3(0){\mathrm{\ensuremath{\Gamma}}}_{4}^{(1)}/2=2.57(2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}{B}_{3}^{(0)}\ensuremathΓ4(1)​/2=2.57(2)\ifmmode×\else\texttimes\fi10\ensuremath−4B3(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.
Small Clusters of 4^44He Atoms in Finite-Cutoff Effective Field Theory
Small clusters of 4^44He atoms are benchmark systems for universal few-body physics near the unitary limit. We study these systems using a finite-cutoff effective field theory calibrated to low-energy observables from the realistic LM2M2 potential. The chosen two-body cutoff reproduces both the atom--atom scattering length and effective range, while a regulated three-body interaction is adjusted to the trimer and tetramer ground-state energies. We distinguish total binding energies from the one-atom separation energies SA∗=BA∗−BA−1S_A^*=B_A^*-B_{A-1}SA∗​=BA∗​−BA−1​ of the shallow excited states. Ground-state energies through A=8A=8A=8 are reproduced at the few-percent level, but threshold-sensitive observables are less accurate. In particular, the calculated tetramer separation energy is S4∗=2.85S_4^*=2.85S4∗​=2.85~mK, compared with 0.920.920.92--0.960.960.96~mK in converged LM2M2 calculations, and the atom--trimer scattering length differs substantially from modern LM2M2 benchmarks. For A≥5A\ge5A≥5, the available excited-state benchmarks are sparse and strongly method dependent. Comparison with another soft-core interaction suggests that the discrepancies primarily reflect short-distance physics and higher-order operators omitted from the finite-cutoff Hamiltonian rather than numerical convergence of the stochastic variational calculation. Finite-cutoff EFT therefore provides an efficient description of ground-state systematics, whereas shallow excited states and atom--cluster scattering require a controlled higher-order and cutoff-dependence analysis.
In a previous work [Phys. Rev. A 85, 022502 (2012)] we calculated, with the use of the Gaussian expansion method for few-body systems, the energy levels and spatial structures of the 4{}^{4}4He trimer and tetramer ground and excited states using the LM2M2 potential, which has a very strong short-range repulsion. In this work, we calculate the same quantities using the currently most accurate 4{}^{4}4He-4{}^{4}4He potential [Przybytek et al., Phys. Rev. Lett. 104, 183003 (2010)] that includes the adiabatic, relativistic, QED, and residual retardation corrections. Contributions of the corrections to the tetramer ground-(excited-) state energy $\ensuremath{-}573.90\phantom{\rule{0.16em}{0ex}}(\ensuremath{-}132.70)$ mK are respectively $\ensuremath{-}4.13\phantom{\rule{0.16em}{0ex}}(\ensuremath{-}1.52)$ mK, +9.37(+3.48)+9.37\phantom{\rule{0.16em}{0ex}}(+3.48)+9.37(+3.48) mK, $\ensuremath{-}1.20\phantom{\rule{0.16em}{0ex}}(\ensuremath{-}0.46)$ mK, and +0.16(+0.07)+0.16\phantom{\rule{0.16em}{0ex}}(+0.07)+0.16(+0.07) mK. Further including other realistic 4{}^{4}4He potentials, we calculated the binding energies of the trimer and tetramer ground and excited states, B3(0),B3(1),B4(0){B}_{3}^{(0)},\phantom{\rule{0.16em}{0ex}}{B}_{3}^{(1)},\phantom{\rule{0.16em}{0ex}}{B}_{4}^{(0)}B3(0)​,B3(1)​,B4(0)​, and B4(1){B}_{4}^{(1)}B4(1)​, respectively. We found that the four kinds of the binding energies for the different potentials exhibit perfect linear correlations between any two of them over the range of binding energies relevant for 4{}^{4}4He atoms (namely, six types of the generalized Tjon lines are observed). The dimerlike-pair model for 4{}^{4}4He clusters, proposed in the previous work, predicts a simple interaction-independent relation B4(1)B2=B3(0)B2+23\frac{{B}_{4}^{(1)}}{{B}_{2}}=\frac{{B}_{3}^{(0)}}{{B}_{2}}+\frac{2}{3}B2​B4(1)​​=B2​B3(0)​​+32​, which precisely explains the correlation between the tetramer excited-state energy and the trimer ground-state energy, with B2{B}_{2}B2​ being the dimer binding energy.
We calculated the 4{}^{4}4He trimer and tetramer ground and excited states with the LM2M2 potential using our Gaussian expansion method for ab initio variational calculations of few-body systems. The method has been extensively used for a variety of three-, four-, and five-body systems in nuclear physics and exotic atomic and molecular physics. The trimer (tetramer) wave function is expanded in terms of symmetric three- (four-) body Gaussian basis functions, ranging from very compact to very diffuse, without assumption of any pair correlation function. The calculated results for the trimer ground and excited states are in excellent agreement with values reported in the literature. The binding energies of the tetramer ground and excited states are obtained as 558.98 and 127.33 mK (0.93 mK below the trimer ground state), respectively. We found that precisely the same shape of the short-range correlation (${r}_{ij}\ensuremath{\lesssim}4$ \AA{}) in the dimer appears in the ground and excited states of the trimer and tetramer. The overlap function between the trimer excited state and the dimer ground state and that between the tetramer excited state and the trimer ground state are almost proportional to the dimer wave function in the asymptotic region (up to $\ensuremath{\sim}$1000 \AA{}). Also, the pair correlation functions of trimer and tetramer excited states are almost proportional to the squared dimer wave function. We then propose a model which predicts the binding energy of the first excited state of 4{}^{4}4HeN{}_{N}N​ ($N\ensuremath{\ge}3$) measured from the 4{}^{4}4He${}_{N\ensuremath{-}1}$ ground state to be nearly $\frac{N}{2(N\ensuremath{-}1)}{B}_{2}$ where B2{B}_{2}B2​ is the dimer binding energy.
The momentum-space structure of the Faddeev-Yakubovsky (FY) components of weakly bound tetramers is investigated at the unitary limit using a renormalized zero-range two-body interaction. The results, obtained by considering a given trimer level with binding energy B3{B}_{3}B3​, provide further support to a universal scaling function relating the binding energies of two successive tetramer states. The correlated scaling between the tetramer energies comes from the sensitivity of the four-boson system to a short-range four-body scale. Each excited NNNth tetramer energy B4(N){B}_{4}^{(N)}B4(N)​ moves as the short-range four-body scale changes, while the trimer properties are kept fixed, with the next excited tetramer B4(N+1){B}_{4}^{(N+1)}B4(N+1)​ emerging from the atom-trimer threshold for a universal ratio ${B}_{4}^{(N)}/{B}_{3}={B}_{4}^{(N)}/{B}_{4}^{(N+1)}\ensuremath{\simeq}4.6$, which does not depend on NNN. We show that both channels of the FY decomposition [atom-trimer (KKK type) and dimer-dimer (HHH type)] present high-momentum tails that reflect the short-range four-body scale. We also found that the HHH channel is favored over the KKK channel at low momentum, when the four-body momentum scale largely overcomes the three-body scale.
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.