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Lattice perspectives on doubly heavy tetraquarks

T0 review · 3 major / 3 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Lattice QCD now predicts a family of doubly heavy tetraquarks: the $T_{bb}$ states are bound, while the $T_{cc}$ appears as a virtual bound state at non-physical quark masses.

desk verdict Solid review of doubly heavy tetraquark lattice results; the abstract overstates the Tcc virtual-state claim relative to the paper's own left-hand-cut caveat. read the letter →

arxiv 2502.04701 v1 pith:IN4HEQKZ submitted 2025-02-07 hep-lat hep-phnucl-th

Anthony Francis This is my paper · ORCID
classification hep-lathep-phnucl-th PACS 12.38.Gc14.40.Rt
keywords doublyheavytetraquarkslatticeQCDhadronspectroscopyfinite-volumescatteringexotichadronsTccTbbboundstates
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This review argues that lattice QCD has reached the point where it can make robust statements about doubly heavy tetraquarks. Its central claims are that the $J^P=1^+$ $T_{bb}^{ud}$ and $T_{bb}^{us}$ states are QCD bound states, and that the most recent $J^P=1^+$ $T_{cc}^{ud}$ studies, at slightly non-physical input quark masses, find a virtual bound state rather than a real one. Because these states are ground states in finite-volume lattice calculations, the same machinery that gives their binding energies can also deliver scattering parameters. The review's practical point is that careful control of finite-volume effects, operator bases, and the left-hand cut is what separates these results from earlier inconclusive attempts.

What carries the argument

The load-bearing tool is the finite-volume quantization condition, which converts a few precisely measured finite-volume energy levels into infinite-volume scattering phase shifts; from those, an effective range expansion yields the scattering length and effective range, and the pole condition $p\cot\delta = -\sqrt{-p^2}$ decides whether the near-threshold state is real, virtual, or resonant. On the spectroscopy side, the generalized eigenvalue problem (GEVP) applied to a matrix of correlators built from di-meson and diquark-antidiquark interpolating operators extracts the needed energy levels, with distillation providing the all-to-all propagators that make momentum-projected operators practical. The potential-based alternative identifies a lattice correlation function with a two-body wave function to define a local potential, which is then inserted into a Schr\"odinger equation. The review's methodological core is the interplay between these two routes, plus the warning that the $D^*\to D\pi$ left-hand cut can break the analyticity assumptions behind the standard quantization condition.

What would settle it

A lattice calculation at physical light-quark masses that includes the $DD\pi$ three-particle channel, or an equivalent treatment of the left-hand cut, and recomputes the $T_{cc}^{ud}$ pole would settle it: if the pole becomes a real bound state below threshold, the virtual-state conclusion is an artifact of the truncated formalism; if it remains virtual, the review's interpretation survives.

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Extended reading notes

Core claim

On the paper's own terms, lattice QCD has evolved from studying inter-meson potentials to performing full scattering analyses of doubly heavy tetraquarks. The established result is a family prediction: the $I(J^P)=0(1^+)$ $T_{bb}^{ud}$ and $T_{bb}^{us}$ are deeply enough bound that their ground states can be read off directly from lattice correlation functions, with binding energies in the range roughly $-190$ to $-70$ MeV and $-100$ to $-40$ MeV respectively across studies. For the experimentally observed $T_{cc}^{ud}$, which is bound by only about $0.3$ MeV in nature, the lattice cannot yet work at physical light-quark masses with full control; instead, scattering analyses at slightly heavier light quarks and slightly off physical charm masses produce scattering parameters whose pole is a virtual bound state, with the expectation that the pole trajectory crosses threshold as parameters move toward physical values. The review treats the $T_{bc}^{ud}$ channels as the open frontier, where recent studies hint at shallow bound states but do not yet agree.

Load-bearing premise

The load-bearing premise is that the finite-volume energy levels used for the $T_{cc}^{ud}$ conclusion can be read with the standard two-particle scattering formalism, even though the nearby decay $D^*\to D\pi$ creates a non-analyticity (the left-hand cut) that the formalism assumes away.

Editorial extensions

If this is right

  • If the lattice results are right, the $J^P=1^+$ $T_{bb}^{ud}$ and $T_{bb}^{us}$ tetraquarks are stable under QCD, with binding energies that grow as the heavy quarks become heavier and the light diquark becomes lighter.
  • The $T_{cc}^{ud}$ is not ruled out as a real bound state at the physical point; the lattice data place it on a pole trajectory that becomes a virtual state at the masses studied, so experiment and theory can be connected by tracking that trajectory.
  • The $T_{bc}^{ud}$ channels become the next test: a deep analysis of $DB$ and $DB^*$ scattering should settle whether the $0(1^+)$ and $0(0^+)$ candidates are shallow bound states.
  • A robust scattering formalism that includes the nearby $DD\pi$ channel will be required before the $T_{cc}^{ud}$ binding energy can be quoted at physical quark masses.
  • The lattice mass-dependence scans act as constraints on phenomenological models: any model that reproduces the binding-energy pattern across quark masses is credible, and any that does not is excluded.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: if the left-hand cut is treated properly, the $T_{cc}^{ud}$ virtual-state conclusion could shift; the standard two-particle quantization analysis of the current data is the assumption most likely to change.
  • Editorial inference: the same finite-volume machinery, once validated on doubly heavy tetraquarks, is a direct template for accessing the deuteron and other multi-hadron systems that suffer from the same left-hand-cut problem.
  • Editorial inference: the contradictory trial-state overlap results for the $T_{bb}$ structure suggest that compositeness should be inferred from scattering parameters via a Weinberg-type criterion rather than from operator overlaps, and the current scattering data are not yet precise enough to decide between molecular and compact.
  • Editorial inference: a concrete testable extension would be a coordinated multi-ensemble study at physical light-quark masses and multiple lattice spacings for $T_{bc}$, which would either confirm the shallow-bound-state hints or show them to be lattice artifacts.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. This manuscript is a review of lattice QCD studies of doubly heavy tetraquarks, covering the methodological toolkit (correlators, effective masses, GEVP, static-potential and HALQCD approaches, heavy-quark actions), a chronological survey from early static-potential work through recent scattering analyses, and a summary of current physics status. The headline claims are that the J^P=1+ T_bb^{ud} and T_bb^{us} are firmly predicted as QCD bound states, that recent studies approaching the J^P=1+ T_cc^{ud} find it to be a virtual bound state at slightly non-physical input quark masses, and that the T_bc channels are an active and less settled frontier. The review includes comprehensive tables of lattice results and discusses the left-hand-cut problem for the Tcc scattering analysis.

Significance. If its status claims hold, this review will be a valuable reference for the lattice and hadron-spectroscopy communities. Its strengths are the explicit treatment of systematic uncertainties, the candid reporting of conflicting lattice results (for example the Tbb binding-energy range from -189(13) MeV to -83(10) MeV), the pedagogical derivation of the finite-volume scattering recipe, and the presence of comprehensive summary tables. The manuscript makes no new lattice calculation or model-dependent derivation, so its contribution is evaluative and synoptic. That evaluation is mostly careful, but the central Tcc virtual-state claim is made in the abstract and in Sec. 5.1.1 while Sec. 5.3 itself identifies a limitation that directly affects the validity of the analysis on which that claim is based. The Tbb bound-state claim is not similarly affected because it is supported by multiple independent spectrum and potential calculations.

major comments (3)
  1. [Abstract; Sec. 5.1.1; Sec. 5.3] The abstract and Sec. 5.1.1 state that recent lattice studies find the T_cc^{ud} to be a virtual bound state at non-physical masses, and Table 5.2 lists the result of [68] as a virtual bound state. This conclusion rests on converting finite-volume energies into p cot(delta) via the Luescher quantization condition (Eqs. 3.8-3.9) and fitting the effective range expansion (Eq. 3.11). Sec. 5.3 states that the nearby D* -> D pi left-hand cut produces a non-analyticity, that the derivation of the Luescher formula relies on analyticity assumptions on the phase shifts, and that a three-particle DD pi treatment is ultimately required. The manuscript therefore contains the evidence that its headline Tcc claim is provisional. Since this is one of the two central status claims of the review, the abstract and the corresponding summary sentences should either be qualified (for example, 'within the standard two-particle Luescher/ERE analysis') or supported by a citation to a quantitative check showing that the ERE fit of [68] is stable despite the left-hand cut.
  2. [Sec. 4.5.1; Table 5.2] The plural wording 'studies ... find it to be a virtual bound state' overstates the consistency of the cited lattice results. The spectrum-based study [68] does find a virtual bound state for its two charm-mass choices at m_pi = 280 MeV, but the HALQCD study [147] quoted in the same section and in Table 5.2 gives scattering parameters whose interpretation changes between virtual and shallow bound states depending on the pion-mass rescaling of the potential ansatz, with binding energies whose asymmetric errors span zero (for example -59(+53,-99) MeV for the lattice-m_pi case). The review should state explicitly which studies support the virtual-state interpretation and how the conflict with [147] is assessed.
  3. [Abstract; Sec. 4.5.1] The phrase 'slightly non-physical input quark masses' in the abstract mischaracterizes the simulation parameters of [68], which is the primary basis of the Tcc claim. That study uses m_pi = 280 MeV, which is more than twice the physical pion mass, with charm-quark masses slightly below and slightly above the physical spin-averaged mass. The abstract should be more precise, for example 'at m_pi = 280 MeV and charm masses near the physical value', so that readers do not infer that the calculation sits close to the physical light-quark point.
minor comments (3)
  1. [Sec. 3.2.5; Sec. 3.2.6] The upper bound on the heavy-quark mass usable with Wilson-type and Domain Wall actions is quoted inconsistently: Sec. 3.2.5 gives m_Q <~ 3.4 or 4.5 GeV, while Sec. 3.2.6 gives m_Q <~ 3.4(4.5) GeV or 2.2(2.9) GeV for the Wilson-type and Domain Wall (brackets) actions. Please harmonize the notation and clarify which lattice spacings correspond to which values.
  2. [Eq. (4.1)] In the discussion of the potential-model Schroedinger equation used in [3], the text describes mu as 'the energy of a heavy quark'; in the two-body Hamiltonian it should be the reduced mass of the two-meson system. Please correct this terminology.
  3. [Sec. 5.1.3] There is a typographical error in 'ansäetze'; the intended word is 'Ansätze'. In addition, the caption of Fig. 3.1 contains a grammatically awkward sequence ('From center to right: In sequence, the free, bound and resonant state scenarios') that should be rephrased.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the review compiles independent lattice results; the Tcc caveat in Sec. 5.3 is a correctness risk, not a circular reduction.

full rationale

This is a review article with no new derivations or fitted predictions of its own. The Tbb status claim is grounded in multiple independent spectrum and potential calculations ([3], [4], [69], [41], [140], [152], [162]), so the author's own works ([4], [78], [133], [41]) are corroborating, not load-bearing. The Tcc virtual-state conclusion is inherited from [68], whose ERE fit to p cot δ does convert finite-volume energies into scattering parameters; but the review does not perform that fit, and it explicitly discloses the key vulnerability: in Sec. 5.3 it states that the Lüscher quantization condition 'builds upon assumptions on analyticity properties of the phase shifts' and that 'ultimately one needs to include the three-particle channel DDπ in some way.' That admission is a limitation of the underlying external analysis, not a self-referential reduction: the review's claim is not defined in terms of its own inputs, nor is any Eq. X identical to Eq. Y by construction. Potential-fit ansätze and HALQCD derivative expansions are presented with their model-dependence flagged rather than hidden. No self-citation is invoked to forbid alternatives or to supply a uniqueness theorem. The finding is therefore no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

This review introduces no new free parameters, axioms beyond standard QCD and lattice methodology, or invented entities. The ledger lists the domain assumptions on which the status claims depend; they are all inherited from the primary literature being reviewed.

assumptions (5)
  • domain assumption Lüscher finite-volume quantization condition and effective range expansion are valid for the extracted finite-volume energy levels.
    Invoked throughout Sec. 3.1.1 and used to convert the [68] energies into scattering parameters for Tcc; the paper notes in Sec. 5.3 that a nearby left-hand cut can invalidate this.
  • domain assumption Ground-state energy from a lattice correlator with exponential volume dependence signals a QCD bound state.
    Used to interpret Tbb binding and to extrapolate to infinite volume, per Eq. 3.7; requires that the level is not a threshold scattering state with power-law volume dependence.
  • domain assumption The lattice results reviewed have controlled systematic uncertainties, including chiral extrapolation, discretization, and finite-volume effects.
    The review's status claim aggregates primary lattice studies; Sec. 5.1.3 lists many unresolved systematics, especially for Tcc and physical pion mass.
  • domain assumption NRQCD and effective relativistic heavy quark actions provide reliable valence heavy quarks within their parameter windows.
    Used in [4], [69], [78], [133], [140], [41], [152] for bottom or charm quarks; Sec. 3.2.6 notes NRQCD lacks a continuum limit and charm is at the edge of validity.
  • domain assumption Isospin-symmetric light quarks and no QED are adequate; isospin breaking is negligible except for the very shallow Tcc.
    Sec. 5.1.2 states all reviewed studies use m_u=m_d and neglect QED; for Tcc (EB about 0.3 MeV) isospin breaking could matter.

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Cite this review

Pith. "Pith review of Lattice perspectives on doubly heavy tetraquarks." pith.science (2026). https://pith.science/paper/IN4HEQKZ

@misc{pith2026250204701,
  author       = {Pith},
  title        = {Pith review of: Lattice perspectives on doubly heavy tetraquarks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IN4HEQKZ}},
  note         = {Machine review of arXiv:2502.04701}
}
abstract

Doubly heavy tetraquarks have emerged as new probes to study the heavy hadron spectrum. With the experimental observation of the $J^P=1^+$ $T_{cc}^+$, they pose a unique opportunity to bring together efforts in experiment, phenomenology, and lattice QCD. In lattice calculations they are accessible as ground states, unlike hidden flavor tetraquarks, and this enables accurate determinations of the scattering parameters alongside the binding energies of these tetraquarks. Today, lattice calculations firmly predict $J^P=1^+$ $T_{bb}^{ud}$ and $T_{bb}^{us}$ as QCD bound states, while recent studies approaching the $J^P=1^+$ $T_{cc}^{ud}$ find it to be a virtual bound state at slightly non-physical input quark masses. Studies of the $J^P=1^+$ $T_{bc}^{ud}$ are ongoing and a new focus area. In light of these developments the evolution of this field until this point is reviewed. Emphasis is put on the methods in lattice spectroscopy that enable a robust evaluation of the lattice studies gathered. They are further reviewed towards their limitations and achievements. Current challenges and opportunities are discussed, including possibilities to approach the left-hand cut in the scattering analysis of the charm candidates and towards understanding the structure of those including two bottom quarks.

Figures

Figures reproduced from arXiv: 2502.04701 by the authors.

Figure 2.1
Figure 2.1. Collected masses of new hadrons observed at LHC [25] at the time of writing. 2.1. A new family of tetraquarks This review focuses on doubly heavy tetraquarks as a new family of heavy hadrons. They are exciting because they are accessible by experiment, QCD phenomenology, and lattice QCD in controlled setups with relatively good statistical signal and without having to perform many difficult extrapolations to control… view at source ↗
Figure 2.2
Figure 2.2. Subtracted distribution of the D0D0π + mass as reported in [1]. The D∗+D0 and D∗0D+ thresholds are shown as vertical dashed lines and the Breit-Wigner peaked line-shape is given in red. The insert shows a zoom of the peak area. in lattice QCD and is found to be 198(4) MeV in the ud-channel at the physical pion mass, as is shown in Sec. 5.4. Further details presented there broadly confirm this picture and the attract… view at source ↗
Figure 3.1
Figure 3.1. Left: Finite volume energies at the same reference volume for the free, bound and resonant scenarios in the toy model. The dotted lines denote the free energy levels in this setup. Three figures from center to right: In sequence, the free, bound and resonant state scenarios for different reference volumes. The volume dependence in the free case is given as the grey lines in all three figures. For more details, see t… view at source ↗
Figures from the paper (26 more)
Figure 3.2
Figure 3.2. Figure 3.2: Left: Scattering phase shifts p cot δ converted from the finite volume energies in [PITH_FULL_IMAGE:figures/full_fig_p013_3_2.png]
Figure 3.3
Figure 3.3. Figure 3.3: Illustration of the diagrams in a static-static-light-light system, the two diagrams correspond to the BB channel relevant to doubly heavy tetraquarks and have been taken from [92]. [92, 93, 94]. Specifically, the tetraquark correlation function reads: WBB(t, ⃗r) = X…
Figure 3.4
Figure 3.4. Figure 3.4: The current parameter window of lattice calculations. Left: Ensemble volumes over pion masses. The horizontal and vertical lines denote the L = 3 fm bound and physical point mπ = 135 MeV, respectively. The colored bands map out the regions of certain mπL where lattic…
Figure 4.1
Figure 4.1. Figure 4.1: Binding energies derived from QQq¯q¯ potentials in [113]. From top left to bottom right, the channels J P = 2+ (top left), 1 +, s = 0 (top right), 1 +, s = 1 (bottom left), and 0 + (bottom right) are shown. The 1 +, s = 0 corresponds to the modern Tbb and already loo…
Figure 4.2
Figure 4.2. Figure 4.2: Inter-meson potentials in the BB-system in Nf = 2 full QCD reported in [93] (top) and [94] (bottom left). Shortly after, a study [118] in Nf = 2 + 1 QCD becomes available on anisotropic lattices (bottom right). lattice with a staggered valence action corresponding to…
Figure 4.3
Figure 4.3. Figure 4.3: Fitted inter-meson static-static-light-light potentials of [3] for the scalar isosinglet channel (left) and vector isotriplet channel (right). The distances are given in units of the lattice spacing a ≃ 0.079 fm. Using these potentials the authors were the first to p…
Figure 4.4
Figure 4.4. Figure 4.4: Extracted finite volume spectra of [123] for the T ud cc in the J P = 0+ (left) and J P = 1+ (right) channels. The spectrum was determined by estimating the splittings between excited and ground states. They are observed to be consistent with a free spectrum. Determi…
Figure 4.5
Figure 4.5. Figure 4.5: Effective energies of I(J P ) = 0(1+) T ud bb (left) and T us bb (right) tetraquarks with the B and B∗ mesons subtracted by forming a correlator ratio [4]. Given in red/blue are the ground/excited state energies determined via a 2 × 2 GEVP. In grey, the results for t…
Figure 4.6
Figure 4.6. Figure 4.6: Chiral extrapolations of the I(J P ) = 0(1+) T ud bb and T us bb tetraquark binding energies performed in [4]. The physical point results using two possible cuts are given as red and blue points. Furthermore, it was hoped that this would reduce excited state contamin…
Figure 4.7
Figure 4.7. Figure 4.7: Finite volume spectra for I(J P ) = 0(1+) T ud cc (left) and T us cc (right) tetraquarks determined in [77]. The energies extracted from large GEVP analyses in different irreps are shown. This study represented the first extensive analysis of the Tcc finite volume sp…
Figure 4.8
Figure 4.8. Figure 4.8: The dependence on the heavy-quark mass ratio, r = mb bare/mb ′ bare, of the binding energies for the bb′u¯d¯, bb′u¯s¯, bbu¯d¯ and bb′u¯s¯ channels with J P = 1+ presented in [133]. configurations used are still the ones of the PACS-CS ensembles, i.e., there is a sing…
Figure 4.9
Figure 4.9. Figure 4.9: Fit window dependence for fitting the GEVP ground state solutions of the correlation matrix made up of the ratio of the tetraquark and two meson correlator data presented in [133]. All correlators were derived using wall-local correlation functions. The final results…
Figure 4.10
Figure 4.10. Figure 4.10: Top: Finite volume energies of the J P = 1+ T ud bb tetraquark determined from local-local and local-smeared correlation functions using a multi-exponential fit approach on the ensemble labeled C005 (mπ = 340 MeV) in [69]. Each column represents a different fit wind…
Figure 4.11
Figure 4.11. Figure 4.11: Contradicting results from trial state overlap analyses for T ud bb at similar physical lattice parameters. Top: Eigenvector component analysis for the ground state of the tetraquark system of [140] from a local-source-local-sink 3 × 3 correlation matrix. The peak a…
Figure 4.12
Figure 4.12. Figure 4.12: Finite volume energies for the J P = 1+ T ud cc tetraquark determined in [68]. Energies were extracted from symmetric GEVPs across a large basis of operators in five irreps, at multiple momenta and in two lattice volumes at mπ ≃ 280 MeV and at slightly lower (left) …
Figure 4.13
Figure 4.13. Figure 4.13: Scattering phase shifts p cot δ and fitted effective range expansion of [68] for the lighter than physical charm quark mass (top left) and the heavier (top right), whereby the pion mass is mπ = 280 MeV. The figures were taken from the supplementary material of the p…
Figure 4.14
Figure 4.14. Figure 4.14: The left panel shows results on the DD∗ inter-meson lattice potential determined in [147] using the HALQCD method. The scattering phase shift δ can be determined by fitting a potential ansatz and solving the resulting Schrödinger equation. Then p cot δ can be evalua…
Figure 4.15
Figure 4.15. Figure 4.15: Results on p cot δ for the J P = 1+ T ud bb tetraquark from a coupled-channel potential analysis using the HALQCD method presented in [152]. The results are shown in descending pion mass from top left to right to bottom left. The shaded bands denote the effective ra…
Figure 4.16
Figure 4.16. Figure 4.16: Left: Finite volume energies determined in [73] the left part of the figure shows the I(J P ) = 0(0+) T ud bc spectrum while the right shows the I(J P ) = 0(1+). Right: The corresponding scattering phase shifts and subsequent fits to the effective range expansion. T…
Figure 4.17
Figure 4.17. Figure 4.17: Left: Continuum extrapolated results for 1/a0 in the I(J P ) = 0(1+) T ud bc channel [153]. Right: The same for the I(J P ) = 0(0+) T ud bc channel [154]. The authors observe bound states in both cases. 5. Challenges and opportunities 5.1. Emerging physics picture 5…
Figure 5.1
Figure 5.1. Figure 5.1: Preliminary pion mass dependence of the binding energies of the J P = 1+ T ud bb and T us bb in [151]. are some indications that there are bound states here, both in direct studies [73, 154], and in a study smoothly varying the heavy quark masses [139], similar to [1…
Figure 5.2
Figure 5.2. Figure 5.2: Results from re-analyzing the data of [68] using an OPE approach in [176]. The OPE form (red) explicitly incorporates the effects of one-pion exchange and adapts the effective range expansion. This incorporates the information of the left-hand cut into the fit. (ζ(⃗p…
Figure 5.3
Figure 5.3. Figure 5.3: Schematic illustration of the approach developed in [178]. From left to right, the pipeline starts with the lattice-determined finite volume energies and evaluates them in an EFT framework to arrive at the scattering phase shifts. These can then be analyzed using the…
Figure 5.4
Figure 5.4. Figure 5.4: Application of the pipeline presented in [PITH_FULL_IMAGE:figures/full_fig_p052_5_4.png]
Figure 5.5
Figure 5.5. Figure 5.5: Left: Diquark-diquark and Diquark-quark mass differences in the spectator embedding approach [188], results shown from [192], see this reference for details. Right: Generic geometry of the density-density correlations was used to study the diquark attractive effect f…
Figure 5.6
Figure 5.6. Figure 5.6: Left: Visualizing the good diquark attractive effect [192]. Top right: Good diquark size r0 versus m2 π. Bottom right: Ratio of radial and tangential sizes over m2 π. an exponential decay with an effective diquark size radii r d 0 as exponent, ∼ exp(−rqq′/rd 0 ). The…

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Forward citations

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Reviewed August 8, 2026 · model on record in the stance chip above.