pith. sign in

arxiv: 2606.00854 · v1 · pith:TUGIVSW4new · submitted 2026-05-30 · ❄️ cond-mat.quant-gas · nucl-th· physics.atom-ph

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

Feng Wu , Xincheng Lin , Ubirajara van Kolck , Sebastian K\"onig This is my paper

Pith reviewed 2026-06-28 17:48 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas nucl-thphysics.atom-ph
keywords short-range effective field theoryunitarity limitdiscrete scale invariance^4He clustersFaddeev-Yakubovsky equationsfew-body bosonsperturbative correctionstetramer states
0
0 comments X

The pith

The physics of ^4He atomic clusters is governed by only small deviations from discrete scale invariance.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies short-range effective field theory to three- and four-boson systems by expanding around the unitarity limit. Leading order is parameter-free in the two-body sector with one three-body parameter, enforcing discrete scale invariance. Next-to-leading order adds perturbative corrections from finite scattering length and effective range plus one four-body force required for renormalization. Faddeev-Yakubovsky calculations for ^4He trimers and tetramers, after removing deep-trimer contributions, converge to binding energies and radii obtained with phenomenological potentials.

Core claim

Starting from the universal unitarity limit where the two-body sector is parameter-free and discrete scale invariance holds with one three-body parameter, next-to-leading-order corrections for finite scattering length, effective range, and a four-body force reproduce the binding energies and radii of the ^4He three-atom ground state and its associated four-atom ground and excited states, with results converging to those from sophisticated phenomenological potentials.

What carries the argument

Short-range effective field theory expanded perturbatively around the unitarity limit, solved via the Faddeev-Yakubovsky formalism with techniques to remove deep-trimer contributions.

If this is right

  • The three-body ground state and four-body ground and first-excited states are determined by one three-body parameter at leading order plus NLO corrections.
  • Binding energies and radii of ^4He clusters converge well with increasing cutoff once deep-trimer artifacts are removed.
  • The approach extends previous analyses to larger cutoffs than accessible with the FY method alone.
  • The same framework yields consistent results when cross-checked with a complementary diagrammatic approach.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The success for ^4He suggests the EFT expansion may be tested on other bosonic few-body systems with large scattering length.
  • Radii observables, in addition to energies, provide an independent check on the size of NLO corrections.
  • If the pattern of small deviations persists, the method could be applied to estimate properties of larger helium clusters without new parameters.

Load-bearing premise

The short-range EFT power counting remains valid for ^4He and a single three-body parameter plus one four-body force at NLO suffice without higher-order terms spoiling the reported convergence.

What would settle it

A calculation showing that the NLO EFT predictions deviate substantially from measured or high-precision phenomenological binding energies and radii of ^4He clusters at the reported level of convergence would falsify the claim.

Figures

Figures reproduced from arXiv: 2606.00854 by Feng Wu, Sebastian K\"onig, Ubirajara van Kolck, Xincheng Lin.

Figure 1
Figure 1. Figure 1: Dressed dimer propagator (double solid line) as [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Diagrammatic representation of the integral equation for the boson-dimer amplitude [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Diagrammatic representation of the four-body integral equations. Lines as in Figs. 1 and 2. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Diagrammatic representation of the four-body in [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Running of the dimensionless three-body LEC [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Running of the dimensionless four-body LEC [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Running of the dimensionless components F (1) 0,1 (blue circles), F (1) 0,2 (red squares), and F (1) 0,3 (green diamonds) of the four-body LEC F (1) 0 , as functions of the momentum cutoff Λ (in units of κ3,0). The diagrammatic approach with a sharp cutoff is used here. Solid and hollow symbols are ob￾tained with lmax = 2 and 0, respectively. Vertical lines as in [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Three-body ground-state radius r3,0 (in units of κ −1 3,0 ) as a function of the momentum cutoff Λ (in units of κ3,0) for the quartic super-Gaussian regulator in the Faddeev equa￾tion with lmax = 12. Red squares, blue circles (barely visible under the violet hexagons), and green diamonds represent, re￾spectively, the LO results at unitarity, the (incomplete) NLO results when only the scattering length a2 i… view at source ↗
Figure 10
Figure 10. Figure 10: Four-body ground-state binding energy B4,0 (in units of B3,0) to NLO as a function of the momentum cutoff Λ (in units of κ3,0) obtained by solving the FY equations with lmax = 4 (left panel, linear scale) and from the diagrammatic approach with lmax = 2 (right panel, semilogarithmic scale). Symbols and horizontal orange line are as in [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: shows the four-body ground-state radius as a function of the momentum cutoff for the quartic super￾Gaussian regulator. We again use κ −1 3,0 as the unit of length. The bands are now the envelopes of fits to points with Λ/κ3,0 ≥ 10 and 15. The LO radius exhibits very good convergence with respect to both momentum and angular momentum cut￾offs, leading to an extrapolated value κ3,0r (0) 4,0 = 0.33(1). At ex… view at source ↗
Figure 12
Figure 12. Figure 12: Four-body excited-state binding energy B4,1 (in units of B3,0) to NLO as a function of the momentum cutoff Λ (in units of κ3,0) obtained in the diagrammatic approach with lmax = 2. Symbols and horizontal lines are as in Figs. 9 and 10, and bands as in [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Same as the left panel of Fig. 6 but for the standard Gaussian ( [PITH_FULL_IMAGE:figures/full_fig_p023_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Four-body ground-state binding energy B4,0 (in units of B3,0) as a function of the momentum cutoff Λ (in units of κ3,0) obtained by solving the FY equations with lmax = 2 using different regulators. Red squares, blue circles, and green diamonds are the LO results at unitarity with the standard (n = 1), quartic (n = 2), and sextic (n = 3) (super-)Gaussian regulators, respectively. Full points with the same… view at source ↗
Figure 16
Figure 16. Figure 16: Ground-state radius r4,0 (in ˚A) of the 4He tetramer as a function of the dimensionless projection factor η, for a quartic super-Gaussian regulator in the FY equations with Λ = 50.77κ3,0. Red squares and green diamonds correspond to LO and full NLO, respectively, for lmax = 2. seen and can be quantified by a fit ∆B (0) 4,0,P (η) B3,0 = d1 + d2 η . (D3) We find {d1, d2} = {0.0087, 0.514} and {0.0073, 0.520… view at source ↗
Figure 17
Figure 17. Figure 17: Tetramer ground state binding energy as a func [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Tetramer ground-state radius r4,0 (in units of κ −1 3,0 ) as a function of the momentum cutoff Λ (in units of κ3,0) for the quartic super-Gaussian regulator in the FY equations. Yellow circles, blue triangles, and red squares denote LO re￾sults at unitarity for lmax = 6, 8 and 10, respectively. The full points are the corresponding full NLO values with a2, r2, and the four-body force included. the deviati… view at source ↗
Figure 19
Figure 19. Figure 19: Tetramer ground- (left panel) and excited- (right panel) binding energies at NLO as a function of the sharp cutoff [PITH_FULL_IMAGE:figures/full_fig_p027_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Largest three (dimensionless) eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p027_20.png] view at source ↗
read the original abstract

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. 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 manifests 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 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. 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript develops a short-range EFT for three- and four-boson systems expanded perturbatively around the unitarity limit. At LO the two-body sector is parameter-free and only one three-body parameter appears, enforcing discrete scale invariance. At NLO, finite-a and r0 corrections plus one four-body force are included; binding energies and radii of the ^4He trimer and tetramer states are computed via the Faddeev-Yakubovsky equations and a diagrammatic method after removing deep-trimer contributions to reach larger cutoffs. The results are reported to converge to phenomenological-potential benchmarks, from which the authors conclude that ^4He clusters exhibit only small deviations from discrete scale invariance.

Significance. If the NLO truncation is demonstrably valid, the work supplies concrete numerical support for the applicability of the short-range EFT power counting to ^4He and strengthens the discrete-scale-invariance picture for few-boson systems with large scattering length. The dual-method approach and the technical extension to larger cutoffs via deep-trimer removal constitute clear methodological strengths.

major comments (3)
  1. [Abstract / NLO section] Abstract and § on NLO power counting: the central claim that 'only small deviations from discrete scale invariance' govern ^4He rests on the assumption that the single three-body parameter plus one four-body force at NLO suffice; the manuscript does not report the numerical size of the expansion parameter (r0/a or equivalent) for the physical ^4He system, leaving open whether omitted NNLO terms remain perturbatively small.
  2. [FY / diagrammatic methods] Section describing the deep-trimer removal procedure: while this technique permits larger cutoffs, the paper does not demonstrate that the subtraction preserves the NLO renormalization and does not introduce cutoff dependence that would affect the reported convergence to phenomenological values; an explicit cutoff-variation plot after removal is required to support the claim.
  3. [Results / comparison to phenomenology] Results section on binding energies and radii: the agreement with phenomenological potentials is presented after fitting the three- and four-body parameters; without a separate LO-only comparison or a quantified residual after NLO, it is difficult to isolate the size of the 'small deviations' from the fitted counterterms themselves.
minor comments (2)
  1. [EFT Lagrangian] Notation for the four-body force strength should be introduced with an explicit equation number when first defined.
  2. [Figures] Figure captions for the cutoff-dependence plots should state the precise observable shown and the range of cutoffs used after deep-trimer removal.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract / NLO section] Abstract and § on NLO power counting: the central claim that 'only small deviations from discrete scale invariance' govern ^4He rests on the assumption that the single three-body parameter plus one four-body force at NLO suffice; the manuscript does not report the numerical size of the expansion parameter (r0/a or equivalent) for the physical ^4He system, leaving open whether omitted NNLO terms remain perturbatively small.

    Authors: We agree that an explicit numerical value for the expansion parameter would strengthen the assessment of NLO validity. In the revised manuscript we will report the value of r_0/a for the physical ^4He system (computed from the known two-body parameters) in the NLO section. revision: yes

  2. Referee: [FY / diagrammatic methods] Section describing the deep-trimer removal procedure: while this technique permits larger cutoffs, the paper does not demonstrate that the subtraction preserves the NLO renormalization and does not introduce cutoff dependence that would affect the reported convergence to phenomenological values; an explicit cutoff-variation plot after removal is required to support the claim.

    Authors: We acknowledge that an explicit demonstration is needed. The revised manuscript will include a cutoff-variation plot after deep-trimer removal to confirm that the procedure preserves NLO renormalization and does not introduce spurious cutoff dependence. revision: yes

  3. Referee: [Results / comparison to phenomenology] Results section on binding energies and radii: the agreement with phenomenological potentials is presented after fitting the three- and four-body parameters; without a separate LO-only comparison or a quantified residual after NLO, it is difficult to isolate the size of the 'small deviations' from the fitted counterterms themselves.

    Authors: We agree that a direct LO comparison and quantified NLO residuals would better isolate the size of the deviations. The revised results section will add an LO-only comparison together with a quantification of the residuals at NLO. revision: yes

Circularity Check

0 steps flagged

No circularity: EFT parameters fitted to external benchmarks with independent convergence checks

full rationale

The paper constructs an EFT expansion starting from the parameter-free unitarity limit at LO (one three-body parameter) and adding finite-a, r0, and one four-body force at NLO. These low-energy constants are adjusted to reproduce binding energies and radii, after which the results are compared to independent phenomenological potentials. No equation reduces to its input by construction, no fitted quantity is relabeled as a prediction, and no load-bearing premise rests on self-citation. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The framework rests on the validity of short-range EFT power counting around unitarity, the necessity of one three-body parameter at LO, and the introduction of a four-body force at NLO; these are standard domain assumptions but introduce fitted quantities that are not independently derived.

free parameters (2)
  • three-body parameter
    Single parameter required at leading order in the unitarity limit; fitted to three-body data.
  • four-body force strength
    Introduced at NLO to absorb cutoff dependence; fitted or renormalized to four-body observables.
axioms (2)
  • domain assumption Short-range effective field theory power counting is applicable to ^4He
    Assumed throughout; the expansion is taken around the unitarity limit.
  • domain assumption Discrete scale invariance holds exactly at leading order
    Stated as the starting point of the EFT.

pith-pipeline@v0.9.1-grok · 5765 in / 1311 out tokens · 23706 ms · 2026-06-28T17:48:06.339700+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Universality in strongly interacting bosonic clusters

    cond-mat.quant-gas 2026-06 unverdicted novelty 5.0

    An EFT for bosonic clusters shows cutoff-independent convergence of ground-state energies up to N=15 at LO using dimer/trimer inputs, with NLO corrections from range and four-body force improving agreement with realis...

Reference graph

Works this paper leans on

84 extracted references · 54 canonical work pages · cited by 1 Pith paper · 30 internal anchors

  1. [1]

    (A1) is uniquely determined by the requirement that it is orthogonal to|a 0⟩

    We already showed that a solution of Eq. (A1) is uniquely determined by the requirement that it is orthogonal to|a 0⟩

  2. [2]

    That implies the minimum value attained by the first term in Eq

    We also know, by the consistency assumption, that a solution exists. That implies the minimum value attained by the first term in Eq. (A3) is zero and, consequently, we only need to show that the par- ticular solution|x 0⟩that is orthogonal to|a 0⟩min- imizes∥x∥among all possible solutions

  3. [3]

    (A1) and letλ=⟨x|a 0⟩

    To see this, let|x⟩be any solution of Eq. (A1) and letλ=⟨x|a 0⟩. Ifλ= 0, then|x⟩=|x 0⟩by the established uniqueness of|x 0⟩. Otherwise, ifλ̸= 0, then|x⟩ −λ|a 0⟩=|x 0⟩. From the Cauchy-Schwarz inequality we get ⟨x|x0⟩ ≤ ∥x∥ · ∥x 0∥,(A4) but also it holds that ⟨x|x0⟩=⟨x+λa 0|x0⟩=⟨x 0|x0⟩=∥x 0∥2 .(A5) Together it then follows that∥x 0∥ ≤ ∥x∥for any other sol...

  4. [4]

    Rewriting the integral equation and including the four-body force Defining |Γ′ 3⟩ ≡(1−K 33)|Γ 3⟩,(B1) Eq. (62) can be written as K ′ |Γ′ 3⟩ |Γ2⟩ = |Γ′ 3⟩ |Γ2⟩ ,(B2) where K ′ = P34K33(1−K 33)−1 (1 +P 34)K32 (1 +P d)K23(1−K 33)−1 (1 +P d)K22 .(B3) Now we define |eΓ3⟩ |eΓ2⟩ ≡ 1−K 32 0 1 |Γ′ 3⟩ |Γ2⟩ (B4) and plug it into Eq. (B2), yielding eK |eΓ3⟩ |eΓ2⟩ = |...

  5. [5]

    1 + κ3,0 Λ F (ν) 1 (Λ/Λ⋆) + κ3,0 Λ 2 F (ν) 2 (Λ/Λ⋆) +· · · # , (C1) whereF (ν) i (x) are bounded functions, except perhaps for isolated points (“exceptional cutoffs

    Subtracting deep trimer poles Near its pole atE=−B 3, the boson-dimer amplitude in Fig. 2 can be written as (see,e.g., Refs. [23, 35]) t3(E) = R3 E+B 3 + iϵ +· · · =R 3 −iπδ(E+B 3) + PV 1 E+B 3 +· · ·, (B10) whereEis the boson-dimer center-of-mass energy and R3 is the residue oft 3(E) at the pole. The dependence of the amplitude on the incoming and outgoi...

    2010

  6. [6]

    Universality in Few-body Systems with Large Scattering Length

    E. Braaten and H.-W. Hammer, Universality in few-body systems with large scattering length, Phys. Rept.428, 259 (2006), arXiv:cond-mat/0410417

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  7. [7]

    Hammer, S

    H.-W. Hammer, S. K¨ onig, and U. van Kolck, Nuclear effective field theory: status and perspectives, Rev. Mod. Phys.92, 025004 (2020), arXiv:1906.12122 [nucl-th]. 27 101 102 103 Λ/κ3,0 −2 −1 0 1 2 3 4 5 6 B4,0/B3,0 NLO, a2, lmax = 1 NLO, a2, r2, lmax = 1 NLO, a2, lmax = 2 NLO, a2, r2, lmax = 2 101 102 103 Λ/κ3,0 0.875 0.900 0.925 0.950 0.975 1.000 1.025 B...

    work page arXiv 2020

  8. [8]

    Weinberg, Effective chiral Lagrangians for nucleon - pion interactions and nuclear forces, Nucl

    S. Weinberg, Effective chiral Lagrangians for nucleon - pion interactions and nuclear forces, Nucl. Phys. B363, 3 (1991)

    1991

  9. [9]

    Effective Theory of 3H and 3He

    S. K¨ onig, H. W. Grießhammer, H.-W. Hammer, and U. van Kolck, Effective theory of 3H and 3He, J. Phys. G43, 055106 (2016), arXiv:1508.05085 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  10. [10]

    Nuclear Physics Around the Unitarity Limit

    S. K¨ onig, H. W. Grießhammer, H.-W. Hammer, and U. van Kolck, Nuclear Physics Around the Uni- tarity Limit, Phys. Rev. Lett.118, 202501 (2017), arXiv:1607.04623 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  11. [11]

    P. F. Bedaque, H.-W. Hammer, and U. van Kolck, Renor- malization of the three-body system with short-range in- teractions, Phys. Rev. Lett.82, 463 (1999), arXiv:nucl- th/9809025

    work page arXiv 1999

  12. [12]

    P. F. Bedaque, H.-W. Hammer, and U. van Kolck, The three-boson system with short-range interactions, Nucl. Phys. A646, 444 (1999), arXiv:nucl-th/9811046

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  13. [13]

    van Kolck, Unitarity and Discrete Scale Invariance, Few Body Syst.58, 112 (2017)

    U. van Kolck, Unitarity and Discrete Scale Invariance, Few Body Syst.58, 112 (2017)

    2017

  14. [14]

    Efimov, Energy levels arising from the resonant two- body forces in a three-body system, Phys

    V. Efimov, Energy levels arising from the resonant two- body forces in a three-body system, Phys. Lett. B33, 563 (1970)

    1970

  15. [15]

    The Four-Boson System with Short-Range Interactions

    L. Platter, H.-W. Hammer, and U.-G. Meißner, Four- boson system with short-range interactions, Phys. Rev. A70, 052101 (2004), arXiv:cond-mat/0404313

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  16. [16]

    Effective Field Theory for Few-Boson Systems

    B. Bazak, M. Eliyahu, and U. van Kolck, Effective Field Theory for Few-Boson Systems, Phys. Rev. A94, 052502 (2016), arXiv:1607.01509 [cond-mat.quant-gas]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  17. [17]

    Universal Properties of the Four-Body System with Large Scattering Length

    H.-W. Hammer and L. Platter, Universal Properties of the Four-Body System with Large Scattering Length, Eur. Phys. J. A32, 113 (2007), arXiv:nucl-th/0610105

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  18. [18]

    Efimov physics in bosonic atom-trimer scattering

    A. Deltuva, Efimov physics in bosonic atom-trimer scat- tering, Phys. Rev. A82, 040701 (2010), arXiv:1009.1295 [physics.atm-clus]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  19. [19]

    Universal Five- and Six-Body Droplets Tied to an Efimov Trimer

    J. von Stecher, Universal Five- and Six-Body Droplets Tied to an Efimov Trimer, Phys. Rev. Lett.107, 200402 (2011), arXiv:1106.2319 [cond-mat.quant-gas]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  20. [20]

    Spectra of helium clusters with up to six atoms using soft core potentials

    M. Gattobigio, A. Kievsky, and M. Viviani, Spectra of he- lium clusters with up to six atoms using soft-core poten- tials, Phys. Rev. A84, 052503 (2011), arXiv:1106.3853 [physics.atm-clus]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  21. [21]

    Energy spectra of small bosonic clusters having a large two-body scattering length

    M. Gattobigio, A. Kievsky, and M. Viviani, Energy spectra of small bosonic clusters having a large two- body scattering length, Phys. Rev. A86, 042513 (2012), arXiv:1206.0854 [physics.atm-clus]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  22. [22]

    $N$-boson spectrum from a Discrete Scale Invariance

    A. Kievsky, N. K. Timofeyuk, and M. Gattobigio, N-boson spectrum from a Discrete Scale Invariance, Phys. Rev. A90, 032504 (2014), arXiv:1405.2371 [cond- mat.quant-gas]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  23. [23]

    Weakly Bound Cluster States of Efimov Character

    J. von Stecher, Weakly bound cluster states of Efimov character, J. Phys. B43, 101002 (2010), arXiv:0909.4056 [cond-mat.quant-gas]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  24. [24]

    A. N. Nicholson,N-body Efimov states from two- particle noise, Phys. Rev. Lett.109, 073003 (2012), arXiv:1202.4402 [cond-mat.quant-gas]. 28

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  25. [25]

    Ground-state properties of unitary bosons: from clusters to matter

    J. Carlson, S. Gandolfi, U. van Kolck, and S. A. Vi- tiello, Ground-state properties of unitary bosons: from clusters to matter, Phys. Rev. Lett.119, 223002 (2017), arXiv:1707.08546 [cond-mat.quant-gas]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  26. [26]

    Range Corrections to Doublet S-Wave Neutron-Deuteron Scattering

    H.-W. Hammer and T. Mehen, Range corrections to dou- blet S-wave neutron-deuteron scattering, Phys. Lett. B 516, 353 (2001), arXiv:nucl-th/0105072

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  27. [27]

    P. F. Bedaque, G. Rupak, H. W. Grießhammer, and H.- W. Hammer, Low-energy expansion in the three-body system to all orders and the triton channel, Nucl. Phys. A714, 589 (2003), arXiv:nucl-th/0207034

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  28. [28]

    C. Ji, D. R. Phillips, and L. Platter, The three-boson system at next-to-leading order in an effective field theory for systems with a large scattering length, Annals Phys. 327, 1803 (2012), arXiv:1106.3837 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  29. [29]

    Effective Field Theory Analysis of Three-Boson Systems at Next-To-Next-To-Leading Order

    C. Ji and D. R. Phillips, Effective Field Theory Analy- sis of Three-Boson Systems at Next-To-Next-To-Leading Order, Few Body Syst.54, 2317 (2013), arXiv:1212.1845 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  30. [30]

    Four-Body Scale in Universal Few-Boson Systems

    B. Bazak, J. Kirscher, S. K¨ onig, M. Pav´ on Valderrama, N. Barnea, and U. van Kolck, Four-Body Scale in Uni- versal Few-Boson Systems, Phys. Rev. Lett.122, 143001 (2019), arXiv:1812.00387 [cond-mat.quant-gas]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  31. [31]

    F. Wu, T. Frederico, R. Higa, and U. van Kolck, Four-boson first excited state near two-body unitar- ity, Phys. Rev. A109, 043301 (2024), arXiv:2310.17079 [physics.atom-ph]

    work page arXiv 2024

  32. [32]

    Second-order perturbation theory for 3He and pd scattering in pionless EFT

    S. K¨ onig, Second-order perturbation theory for 3He and pd scattering in pionless EFT, J. Phys. G44, 064007 (2017), arXiv:1609.03163 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  33. [33]

    K¨ onig, Energies and radii of light nuclei around uni- tarity, Eur

    S. K¨ onig, Energies and radii of light nuclei around uni- tarity, Eur. Phys. J. A56, 113 (2020), arXiv:1910.12627 [nucl-th]

    work page arXiv 2020

  34. [34]

    L. D. Faddeev, Scattering Theory for a Three-Particle System, Sov. Phys. JETP12, 1014 (1961)

    1961

  35. [35]

    O. A. Yakubovsky, On the integral equations in the the- ory of N particle scattering, Yadern. Fiz., 5: 1312-20 (1967)

    1967

  36. [36]

    Blume and C

    D. Blume and C. H. Greene, Monte carlo hyperspherical description of helium cluster excited states, The Journal of Chemical Physics112, 8053 (2000)

    2000

  37. [37]

    Description of $^{4}$He tetramer bound and scattering states

    R. Lazauskas and J. Carbonell, Description of 4He tetramer bound and scattering states, Phys. Rev. A73, 062717 (2006), arXiv:physics/0609153 [physics.atom-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  38. [38]

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

    E. Hiyama and M. Kamimura, Variational calculation of 4He tetramer ground and excited states using a re- alistic pair potential, Phys. Rev. A85, 022502 (2012), arXiv:1111.4370 [physics.atom-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  39. [39]

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

    E. Hiyama and M. Kamimura, Linear correlations be- tween 4He trimer and tetramer energies calculated with various realistic 4He potentials, Phys. Rev. A85, 062505 (2012), arXiv:1203.3130 [physics.atom-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  40. [40]

    Lin, Four-body systems at large cutoffs in effec- tive field theory, Phys

    X. Lin, Four-body systems at large cutoffs in effec- tive field theory, Phys. Rev. C109, 024002 (2024), arXiv:2304.06172 [nucl-th]

    work page arXiv 2024

  41. [41]

    Contessi, M

    L. Contessi, M. Sch¨ afer, and U. van Kolck, Improved ac- tion for contact effective field theory, Phys. Rev. A109, 022814 (2024), arXiv:2310.15760 [physics.atm-clus]

    work page arXiv 2024

  42. [42]

    I. V. Brodsky, M. Y. Kagan, A. V. Klaptsov, R. Combescot, and X. Leyronas, Exact diagrammatic approach for dimer-dimer scattering and bound states of three and four resonantly interacting particles, Phys. Rev. A73, 032724 (2006)

    2006

  43. [43]

    Qin and J

    J. Qin and J. Vanasse, Effective field theory analy- sis of boson-trimer bond lengths to next-to-leading or- der, Phys. Rev. A103, 023333 (2021), arXiv:2010.06105 [nucl-th]

    work page arXiv 2021

  44. [44]

    K¨ onig, Perturbation theory for four-body systems near unitarity, invited talk at the Institute for Nuclear Theory, University in Seattle, Washington, October 2024

    S. K¨ onig, Perturbation theory for four-body systems near unitarity, invited talk at the Institute for Nuclear Theory, University in Seattle, Washington, October 2024

    2024

  45. [45]

    F. Wu, Bosonic few-body systems expanded around the unitarity limit, talk presented at The European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*), Trento, June 2025

    2025

  46. [46]

    Mondal, M

    S. Mondal, M. Sch¨ afer, M. Bagnarol, N. Barnea, U. Raha, and J. Kirscher, A practical approach to perturbative cor- rections to few-body observables, arXiv:2509.17366 [nucl- th] (2025)

    work page arXiv 2025

  47. [47]

    S. Lyu, L. Zuo, R. Peng, S. K¨ onig, and B. Long, Simpli- fication of chiral nuclear forces near the unitarity limit, arXiv:2511.12522 [nucl-th] (2025)

    work page arXiv 2025

  48. [48]

    van Kolck, Effective field theory of short-range forces, Nucl

    U. van Kolck, Effective field theory of short-range forces, Nucl. Phys. A645, 273 (1999), arXiv:nucl-th/9808007

    Pith/arXiv arXiv 1999

  49. [49]

    Kamada and W

    H. Kamada and W. Gl¨ ockle, Solutions of the Yakubovsky equations for four-body model systems, Nucl. Phys. A 548, 205 (1992)

    1992

  50. [50]

    Gl¨ ockle,The Quantum Mechanical Few-Body Prob- lem(Springer, Berlin; New York, 1983)

    W. Gl¨ ockle,The Quantum Mechanical Few-Body Prob- lem(Springer, Berlin; New York, 1983)

    1983

  51. [51]

    Gl¨ ockle and H

    W. Gl¨ ockle and H. Kamada, On the inclusion of 3N-forces into the 4N-Yakubovsky equations, Nucl. Phys. A560, 541 (1993)

    1993

  52. [52]

    M. R. Hadizadeh and S. Bayegan, Four-body bound state calculations in three-dimensional approach, Few Body Syst.40, 171 (2007), arXiv:nucl-th/0605063

    Pith/arXiv arXiv 2007

  53. [53]

    M. R. Hadizadeh, L. Tomio, and S. Bayegan, Solutions of the bound state Faddeev-Yakubovsky equations in three dimensions by using NN and 3N potential models, Phys. Rev. C83, 054004 (2011), arXiv:1104.3879 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  54. [54]

    E. P. Wigner, Lower Limit for the Energy Derivative of the Scattering Phase Shift, Phys. Rev.98, 145 (1955)

    1955

  55. [55]

    D. R. Phillips and T. D. Cohen, How short is too short? Constraining contact interactions in nucleon-nucleon scattering, Phys. Lett. B390, 7 (1997), arXiv:nucl- th/9607048

    work page arXiv 1997

  56. [56]

    D. R. Lehman, Excluded bound state in theS 1/2 α−N interaction and the three-body binding energies of 6He and 6Li, Phys. Rev. C25, 3146 (1982)

    1982

  57. [57]

    Nogga, R

    A. Nogga, R. G. E. Timmermans, and U. van Kolck, Renormalization of one-pion exchange and power count- ing, Phys. Rev. C72, 054006 (2005), arXiv:nucl- th/0506005

    work page arXiv 2005

  58. [58]

    Y.-H. Song, R. Lazauskas, and U. van Kolck, Triton bind- ing energy and neutron-deuteron scattering up to next- to-leading order in chiral effective field theory, Phys. Rev. C96, 024002 (2017), [Erratum: Phys.Rev.C 100, 019901 (2019)], arXiv:1612.09090 [nucl-th]

    work page arXiv 2017

  59. [59]

    Platter,From Cold Atoms to Light Nuclei: The Four- Body Problem in an Effective Theory with Contact Inter- actions, Phd thesis, Universit¨ at Bonn (2005)

    L. Platter,From Cold Atoms to Light Nuclei: The Four- Body Problem in an Effective Theory with Contact Inter- actions, Phd thesis, Universit¨ at Bonn (2005)

    2005

  60. [60]

    D. B. Kaplan, More effective field theory for nonrelativis- tic scattering, Nucl. Phys. B494, 471 (1997), arXiv:nucl- th/9610052

    work page arXiv 1997

  61. [61]

    G. V. Skorniakov and K. A. Ter-Martirosian, Three Body Problem for Short Range Forces. I. Scattering of Low Energy Neutrons by Deuterons, Sov. Phys. JETP4, 648 (1957)

    1957

  62. [62]

    Long-range interactions for He($n S$)--He$(n' S)$ and He($n S$)--He$(n' P)$

    J.-Y. Zhang, Z.-C. Yan, D. Vrinceanu, J. F. Babb, and 29 H. R. Sadeghpour, Long-range interactions for He(nS)- He(n′S) and He(nS)-He(n ′P), Physical Review A74, 014704 (2006), arXiv:physics/0603232 [physics.atom-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  63. [63]

    Zelleret al., Imaging the He 2 quantum halo state using a free electron laser, Proc

    S. Zelleret al., Imaging the He 2 quantum halo state using a free electron laser, Proc. Nat. Acad. Sci.113, 4651 (2016), arXiv:1601.03247 [physics.atom-ph]

    Pith/arXiv arXiv 2016

  64. [64]

    R. E. Grisenti, W. Schollkopf, J. P. Toennies, G. C. Hegerfeldt, T. Kohler, and M. Stoll, Determination of the Bond Length and Binding Energy of the Helium Dimer by Diffraction from a Transmission Grating, Phys. Rev. Lett.85, 2284 (2000)

    2000

  65. [65]

    Observation of the Efimov state of the helium trimer

    M. Kunitskiet al., Observation of the Efimov state of the helium trimer, Science348, 551 (2015), arXiv:1512.02036 [physics.atm-clus]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  66. [66]

    R. A. Aziz and M. J. Slaman, An examination of ab initio results for the helium potential energy curve, The Journal of chemical physics94, 8047 (1991)

    1991

  67. [67]

    A. R. Janzen and R. A. Aziz, Modern He–He potentials: Another look at binding energy, effective range theory, retardation, and Efimov states, The Journal of chemical physics103, 9626 (1995)

    1995

  68. [68]

    Universal Relations for Identical Bosons from 3-Body Physics

    E. Braaten, D. Kang, and L. Platter, Universal Relations for Identical Bosons from Three-Body Physics, Phys. Rev. Lett.106, 153005 (2011), arXiv:1101.2854 [cond- mat.quant-gas]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  69. [69]

    Chen and P

    L. Chen and P. Zhang, Exact renormalization relation and binding energies for three identical bosons, Phys. Rev. A112, 033319 (2025), arXiv:2506.12531 [cond- mat.quant-gas]

    work page arXiv 2025

  70. [70]

    L. Chen, F. Wu, X. Lin, S. K¨ onig, U. van Kolck, and P. Zhang, Three-body limit cycle: Universal form for general regulators, Phys. Rev. A113, 013314 (2026), arXiv:2509.04746 [cond-mat.quant-gas]

    arXiv 2026

  71. [71]

    C. Ji, E. Braaten, D. R. Phillips, and L. Platter, Univer- sal Relations for Range Corrections to Efimov Features, Phys. Rev. A92, 030702 (2015), arXiv:1506.02334 [cond- mat.quant-gas]

    Pith/arXiv arXiv 2015

  72. [72]

    Renormalization- group-invariant effective field theory

    A. M. Gasparyan and E. Epelbaum, “Renormalization- group-invariant effective field theory” for few-nucleon systems is cutoff dependent, Phys. Rev. C107, 034001 (2023), arXiv:2210.16225 [nucl-th]

    work page arXiv 2023

  73. [73]

    S. K. Adhikari, A. C. Fonseca, and L. Tomio, Method for resonances and virtual states: Efimov virtual states, Phys. Rev. C26, 77 (1982)

    1982

  74. [74]

    Fate of the neutron-deuteron virtual state as an Efimov level

    G. Rupak, A. Vaghani, R. Higa, and U. van Kolck, Fate of the neutron–deuteron virtual state as an Efimov level, Phys. Lett. B791, 414 (2019), arXiv:1806.01999 [nucl- th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  75. [75]

    A. J. Andis, S. Lyu, B. Long, and S. K¨ onig, Perturbative EFT calculation of the deuteron longitudinal response function, arXiv:2512.12823 [nucl-th] (2025)

    work page arXiv 2025

  76. [76]

    The Triton Charge Radius to Next-to-next-to-leading order in Pionless Effective Field Theory

    J. Vanasse, Triton charge radius to next-to-next-to- leading order in pionless effective field theory, Phys. Rev. C95, 024002 (2017), arXiv:1512.03805 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  77. [77]

    O. Thim, A. Ekstr¨ om, and C. Forss´ en, Perturba- tiveχEFT calculations of the deuteron and triton up to N 2LO, Phys. Rev. C112, 064008 (2025), arXiv:2510.12207 [nucl-th]

    work page arXiv 2025

  78. [78]

    Yang, Further theoretical study on the renormaliza- tion group aspect of perturbative corrections, Phys

    C.-J. Yang, Further theoretical study on the renormaliza- tion group aspect of perturbative corrections, Phys. Rev. C112, 014004 (2025), arXiv:2410.08845 [nucl-th]

    work page arXiv 2025

  79. [79]

    Pavon Valderrama, Regulator constraints for the per- turbative renormalizability of attractive triplets, Phys

    M. Pavon Valderrama, Regulator constraints for the per- turbative renormalizability of attractive triplets, Phys. Rev. C112, 064009 (2025), arXiv:2509.23855 [nucl-th]

    work page arXiv 2025

  80. [80]

    Pavon Valderrama, Reexamining the perturbative renormalizability of coupled triplets, Phys

    M. Pavon Valderrama, Reexamining the perturbative renormalizability of coupled triplets, Phys. Rev. C113, 014001 (2026), arXiv:2510.15789 [nucl-th]

    work page arXiv 2026

Showing first 80 references.