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arxiv: 2606.12309 · v2 · pith:7WEXTZDKnew · submitted 2026-06-10 · ✦ hep-ph

boldsymbol{chi_(c1)}(3872) and its Partners in the Diabatic Born-Oppenheimer Approximation for QCD

Fareed Alasiri , Eric Braaten , Roberto Bruschini This is my paper

Pith reviewed 2026-06-27 09:02 UTC · model grok-4.3

classification ✦ hep-ph
keywords χ_c1(3872)tetraquarksBorn-Oppenheimer approximationheavy quark spin symmetrydiabatic Schrödinger equationhidden-charmhidden-bottom
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The pith

The χ_c1(3872) is the 1++ member of a heavy-quark spin-symmetry multiplet of tetraquarks whose other members are 0++, 1+-, and 2++ states.

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

The paper establishes that in the Born-Oppenheimer approximation for QCD, the χ_c1(3872) is a near-threshold bound state in potentials associated with an isospin-0 adjoint meson. It is the 1++ member of a multiplet with JPC quantum numbers 0++, 1+-, and 2++. A simple interpolation model for the potentials is introduced, and the diabatic Schrödinger equation is solved taking spin splittings nonperturbatively. This allows calculation of the spin splittings and decay widths of the other members into charm-meson pairs, and similar calculations for hidden-bottom tetraquarks. The results provide a template for analyzing all hidden-heavy hadrons.

Core claim

In the diabatic Born-Oppenheimer approximation, the χ_c1(3872) is the 1++ member of a heavy-quark spin-symmetry multiplet of hidden-charm tetraquarks with other members having JPC = 0++, 1+-, 2++. Using an interpolated model for the Born-Oppenheimer potentials and solving the diabatic Schrödinger equation with nonperturbative spin splittings of charm mesons and the adjoint meson, plus a narrow avoided crossing, the energies and decay widths of the multiplet are calculated after tuning to the D* D-bar threshold. The same is done for the hidden-bottom multiplet.

What carries the argument

The interpolation model for Born-Oppenheimer potentials between the adjoint-meson potential at short distances and the triplet-meson-pair potential at large distances, used in the diabatic Schrödinger equation solved with nonperturbative spin splittings.

If this is right

  • The other charm tetraquarks in the multiplet have specific predicted spin splittings and decay widths into charm-meson pairs.
  • The hidden-bottom tetraquarks form an analogous multiplet with calculable energies and decay widths.
  • The approach serves as a template for quantitative analysis of all hidden-heavy hadrons using the Born-Oppenheimer approximation.
  • Spin splittings must be treated nonperturbatively for accurate results in this framework.

Where Pith is reading between the lines

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

  • If the model is accurate, searches for the 0++ and 2++ states near the predicted masses would test the multiplet structure.
  • The framework could be extended to other exotic hadrons by applying similar potential models.
  • Comparison with lattice QCD determinations of the adjoint-meson potential would validate the interpolation.
  • Nonperturbative inclusion of spin effects may resolve discrepancies in other tetraquark calculations.

Load-bearing premise

The simple interpolation model for the Born-Oppenheimer potentials between the adjoint-meson potential at short distances and the triplet-meson-pair potential at large distances accurately captures the dynamics when spin splittings are included nonperturbatively.

What would settle it

A measurement of the mass or decay width of a predicted partner state, such as the 2++ tetraquark, that differs substantially from the calculated value would falsify the central claim.

read the original abstract

In the Born-Oppenheimer approximation for QCD, the exotic hidden-charm tetraquark meson $\chi_{c1}(3872)$ is a near-threshold bound state in Born-Oppenheimer potentials associated with an isospin-0 adjoint meson. The $\chi_{c1}(3872)$ is the $1^{++}$ member of a heavy-quark spin-symmetry multiplet whose other members have $J^{PC}$ quantum numbers $0^{++}$, $1^{+-}$, and $2^{++}$. We introduce a simple model for the Born-Oppenheimer potentials that interpolates between the adjoint-meson potential at short distances and the triplet-meson-pair potential at large distances. We take into account the spin splittings of charm mesons nonperturbatively for the first time by solving the diabatic Schr\"odinger equation. We also take into account the spin splittings of the adjoint meson as well as a narrow avoided crossing with the quarkonium potential. We tune the energy of $\chi_{c1}(3872)$ to the $D^* \bar{D}$ threshold and then calculate the spin splittings of the other members of the multiplet and their decay widths into charm-meson pairs. We also calculate the energies and decay widths of the corresponding multiplet of hidden-bottom tetraquarks. These calculations provide a template for the quantitative analysis of all hidden-heavy hadrons using the Born-Oppenheimer approximation for QCD.

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

2 major / 1 minor

Summary. The manuscript claims that χ_c1(3872) is the 1++ member of a heavy-quark spin-symmetry multiplet (with partners 0++, 1+-, 2++) in the diabatic Born-Oppenheimer approximation for QCD. Using a simple phenomenological interpolation model for the BO potentials (between adjoint-meson at short distance and triplet-meson-pair at long distance), it solves the diabatic Schrödinger equation with nonperturbative spin splittings, tunes the overall energy offset to the D* D-bar threshold, and computes the spin splittings and decay widths into charm-meson pairs for the full multiplet as well as the corresponding hidden-bottom tetraquarks.

Significance. If the interpolation model proves robust, the work supplies a concrete template for quantitative predictions of hidden-heavy hadrons that incorporates spin splittings nonperturbatively for the first time. The diabatic treatment and inclusion of the narrow avoided crossing with the quarkonium channel are positive technical features.

major comments (2)
  1. [potential interpolation model] Model for Born-Oppenheimer potentials (described after the abstract and in the section introducing the interpolation): the functional form of the interpolation is phenomenological and unconstrained by independent lattice data; combined with the single overall energy offset tuned to the χ_c1(3872) mass at the D* D-bar threshold, the reported spin splittings and partial widths for the 0++, 1+-, 2++ states are outputs of this specific choice rather than independent QCD-derived quantities.
  2. [numerical results] Results for charm and bottom multiplets (in the sections presenting numerical energies and widths): no sensitivity analysis or cross-checks against other lattice or experimental data are shown for the interpolation shape or the diabatic treatment of the avoided crossing, so the quantitative reliability of the predicted splittings cannot be assessed independently of the tuning.
minor comments (1)
  1. [spin splittings treatment] The abstract states that spin splittings of the adjoint meson are taken into account, but the explicit implementation (e.g., which equation or potential term) could be stated more clearly for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We respond to the major comments point by point below.

read point-by-point responses

  1. Referee: [potential interpolation model] Model for Born-Oppenheimer potentials (described after the abstract and in the section introducing the interpolation): the functional form of the interpolation is phenomenological and unconstrained by independent lattice data; combined with the single overall energy offset tuned to the χ_c1(3872) mass at the D* D-bar threshold, the reported spin splittings and partial widths for the 0++, 1+-, 2++ states are outputs of this specific choice rather than independent QCD-derived quantities.

    Authors: We agree that the interpolation model is phenomenological, as explicitly stated in the manuscript. The functional form is chosen to connect the short-distance adjoint-meson regime to the long-distance meson-pair regime in a manner consistent with expected QCD behavior. The single tuning to the D* D-bar threshold for the 1++ state is used to predict the remaining states in the multiplet. While the quantitative results necessarily depend on this choice, the primary advance is the nonperturbative diabatic treatment of spin splittings together with the narrow avoided crossing. We will revise the manuscript to add further discussion of the interpolation assumptions and their implications for the predictions. revision: partial

  2. Referee: [numerical results] Results for charm and bottom multiplets (in the sections presenting numerical energies and widths): no sensitivity analysis or cross-checks against other lattice or experimental data are shown for the interpolation shape or the diabatic treatment of the avoided crossing, so the quantitative reliability of the predicted splittings cannot be assessed independently of the tuning.

    Authors: We acknowledge that the manuscript presents no explicit sensitivity analysis with respect to the interpolation parameters or the diabatic handling of the avoided crossing. Such an analysis would help quantify the robustness of the numerical results. We will add a discussion of the sensitivity to reasonable variations in the interpolation width and the location of the avoided crossing, showing the resulting range of predicted splittings and widths. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard model fit to one mass followed by independent dynamical calculations

full rationale

The paper introduces its own interpolation model for the Born-Oppenheimer potentials and solves the diabatic Schrödinger equation with nonperturbative spin splittings. The single overall energy shift is adjusted so that the 1++ state sits at the D* D-bar threshold; the remaining multiplet energies, splittings, and widths are then obtained by direct numerical solution of the model equations. This procedure does not reduce any output to an input by definition or by a self-citation chain. The model choice itself supplies the dynamical content that determines the relative splittings, and no load-bearing step is shown to be equivalent to the fitted value by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The model depends on one explicit tuning parameter and on the domain assumptions that the Born-Oppenheimer separation and heavy-quark spin symmetry remain valid for these near-threshold tetraquarks; no new particles or forces are introduced.

free parameters (1)
  • overall energy offset for the 1++ state
    The model is adjusted so the calculated energy of χ_c1(3872) coincides with the D* D-bar threshold.
axioms (2)
  • domain assumption Born-Oppenheimer approximation is applicable to hidden-charm and hidden-bottom tetraquarks
    Heavy quarks move slowly compared with light degrees of freedom, allowing separation into potentials.
  • domain assumption Heavy-quark spin symmetry organizes the multiplet
    Used to assign the JPC quantum numbers 0++, 1+-, 2++ to the partners of the 1++ state.

pith-pipeline@v0.9.1-grok · 5807 in / 1647 out tokens · 31985 ms · 2026-06-27T09:02:18.753238+00:00 · methodology

discussion (0)

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Reference graph

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