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arxiv: 2604.12603 · v1 · submitted 2026-04-14 · ✦ hep-ph · hep-ex· hep-lat· nucl-th

Recognition: 2 theorem links

· Lean Theorem

Open-flavor threshold effects on quarkonium spectrum in the BOEFT

Nora Brambilla , Abhishek Mohapatra , Tommaso Scirpa , Antonio Vairo
Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:29 UTC · model grok-4.3

classification ✦ hep-ph hep-exhep-latnucl-th
keywords quarkonium spectrumopen-flavor thresholdsBorn-Oppenheimer effective field theorytetraquark potentialscoupled Schrödinger equationsχ_c1(3872)^3P_0 model
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The pith

Mixing quarkonium and tetraquark potentials in the Born-Oppenheimer effective field theory accounts for open-flavor threshold effects on the spectrum below threshold.

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

The paper shows that open-flavor threshold effects on quarkonium arise naturally in the Born-Oppenheimer effective field theory through mixing of static potentials that share quantum numbers. These effects are quantified by solving coupled Schrödinger equations whose potentials come from lattice QCD and whose single free parameter is set by the mass of the χ_c1(3872) state. Calculations are performed both with averaged thresholds and with full spin splittings, and the results are cross-checked against self-energy corrections to the quarkonium propagator. This framework also supplies a QCD-derived meaning for the pair-creation constant used in the phenomenological ^3P_0 model.

Core claim

Open-flavor threshold effects emerge from the mixing between quarkonium and tetraquark static potentials sharing the same Born-Oppenheimer quantum numbers. The shapes of these potentials are fixed by lattice QCD, with tetraquark potentials repulsive at short distances and approaching meson thresholds at long distances. Solving the coupled Schrödinger equations with the adjoint meson mass fixed to the χ_c1(3872) mass determines the threshold-induced modifications to the quarkonium spectrum below threshold in both spin-averaged and spin-split cases.

What carries the argument

The coupled Schrödinger equations arising from mixing of quarkonium and tetraquark static potentials in BOEFT.

Load-bearing premise

The static potentials retain their lattice-determined shapes and that the single parameter fixed by the χ_c1(3872) mass is sufficient to describe threshold effects throughout the spectrum.

What would settle it

Precise measurement of mass shifts in specific quarkonium levels, such as the η_c or Υ states, that deviate from the predictions of the coupled equations without threshold mixing.

Figures

Figures reproduced from arXiv: 2604.12603 by Abhishek Mohapatra, Antonio Vairo, Nora Brambilla, Tommaso Scirpa.

Figure 1
Figure 1. Figure 1: The static BO adiabatic potentials appearing in the leading order BOEFT Lagrangian (2.1). The 1 Σ [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The spin structure of the χc1(3872) multiplet, once the O(1/mQ) tetraquark spin-dependent potential from Eq. (4.1) is taken into account. The state with J P C = 1++, shown as a dark blue line and identified with χc1(3872), is obtained by solving Eq. (4.5) after fine-tuning the adjoint meson mass to Λ1−− . The remaining states in the multiplet (shown as light blue lines) lie above the lowest threshold enter… view at source ↗
Figure 3
Figure 3. Figure 3: The spin structure of the Xb multiplet after including the O(1/mQ) tetraquark spin-dependent potential of Eq. (4.1). Three states with J P C = 1+−, 0 ++, and 1++ (the latter denoted Xb) appear as dark blue lines and are obtained by solving Eqs. (C3), (C7), and (4.5), respectively, with the adjoint meson mass Λ1−− taken from the χc1(3872) case. The spin-splitting for the fourth state (J P C = 2++) is comput… view at source ↗
Figure 4
Figure 4. Figure 4: The dependence of the self-energy correction Re(Σ [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The BOEFT static mixing potential of Eq. (2.9), denoted [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Spin-averaged spectrum in the charmonium sector below the [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Spin-averaged spectrum in the bottomonium sector below the [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Spin-averaged spectrum in the bc¯ sector below the BD¯spin avg. threshold following from Eq. (3.1). States are represented as in [PITH_FULL_IMAGE:figures/full_fig_p029_8.png] view at source ↗
read the original abstract

The impact of open-flavor thresholds on the quarkonium spectrum has been a subject of study since the introduction of the Cornell potential and has been quantified through various phenomenological approaches, most notably the $^3P_0$ model. We revisit this problem using the Born--Oppenheimer effective field theory (BOEFT), an effective field theory systematically derived from QCD by exploiting hierarchies of energy scales and symmetries. Within the BOEFT, open-flavor threshold effects emerge from the mixing between quarkonium and tetraquark static potentials sharing the same Born--Oppenheimer quantum numbers. The shapes of the static potentials are constrained by lattice QCD calculations. Furthermore, we account for the distinctive behavior of the BOEFT tetraquark static potentials at short and large distances: at short distances they are repulsive, reflecting the color-octet configuration of the heavy quark-antiquark pair, while at large distances they asymptotically approach heavy-light meson-antimeson thresholds. To quantify threshold effects on the quarkonium spectrum below threshold, we solve a set of coupled Schr\"{o}dinger equations dictated by the BOEFT, whose only free parameter, the adjoint meson mass, is fixed to the mass of the $\chi_{c1}(3872)$ state. These coupled equations are solved both in the spin-isospin averaged threshold limit and, for the first time, including the spin splittings of the physical thresholds. We validate our results by computing the same threshold effects as self-energy corrections to the quarkonium propagator. We compare our predictions with existing experimental data and previous literature. Finally, we provide a field-theoretical interpretation of the pair-creation constant $\gamma$ appearing in the $^3P_0$ model.

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 / 2 minor

Summary. The manuscript develops a Born-Oppenheimer effective field theory (BOEFT) framework to study open-flavor threshold effects on the quarkonium spectrum. Threshold effects arise from mixing between quarkonium and tetraquark static potentials that share Born-Oppenheimer quantum numbers, with the potential shapes constrained by lattice QCD. The only free parameter (adjoint meson mass) is fixed to the experimental mass of the χ_c1(3872) state. Coupled Schrödinger equations are solved both in the spin-isospin averaged threshold limit and including physical spin splittings; results are validated by computing equivalent self-energy corrections to the quarkonium propagator, compared to data, and used to provide a field-theoretical interpretation of the pair-creation constant γ in the 3P0 model.

Significance. If the numerical results hold, the work supplies a systematic QCD-derived EFT treatment of threshold effects that have historically been addressed with phenomenological models. The explicit inclusion of spin splittings for the first time, the lattice-constrained potentials, the self-energy validation, and the interpretation of γ constitute clear strengths. The approach could improve quantitative understanding of spectrum shifts below threshold and the nature of states near open-flavor thresholds.

major comments (2)
  1. [§4] §4 (numerical solution of coupled Schrödinger equations): the central quantitative predictions for threshold-induced shifts rest on lattice QCD inputs for the static potentials, yet the manuscript does not display the propagation of lattice uncertainties or the sensitivity of the final spectrum to variations in the potential shapes; because the short-distance repulsion and long-distance asymptotic behavior directly control the mixing, this omission is load-bearing for the claimed spectrum modifications.
  2. [§5] §5 (validation and comparison): while self-energy corrections are stated to validate the coupled-equation results, no table or figure directly compares the numerical values of the threshold shifts obtained by the two methods for the same set of states; without this explicit cross-check, the internal consistency of the framework cannot be fully assessed.
minor comments (2)
  1. The notation for the Born-Oppenheimer quantum numbers and the labeling of the coupled channels could be summarized in a small table for clarity.
  2. [Introduction] A few sentences in the introduction that restate the hierarchy of scales would help readers unfamiliar with BOEFT connect the abstract to the concrete implementation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. The positive assessment of the BOEFT framework and its strengths is appreciated. We address each major comment below and will revise the manuscript to incorporate the requested improvements.

read point-by-point responses

  1. Referee: [§4] §4 (numerical solution of coupled Schrödinger equations): the central quantitative predictions for threshold-induced shifts rest on lattice QCD inputs for the static potentials, yet the manuscript does not display the propagation of lattice uncertainties or the sensitivity of the final spectrum to variations in the potential shapes; because the short-distance repulsion and long-distance asymptotic behavior directly control the mixing, this omission is load-bearing for the claimed spectrum modifications.

    Authors: We agree that a quantitative assessment of sensitivity to the lattice inputs is important given their role in controlling the mixing. In the revised manuscript we will add a dedicated discussion (with an accompanying table or figure) that varies the short-distance repulsive core and the long-distance asymptotic approach to threshold within the range permitted by existing lattice QCD results. This will show the resulting spread in the predicted threshold-induced shifts and thereby quantify the robustness of the spectrum modifications. revision: yes

  2. Referee: [§5] §5 (validation and comparison): while self-energy corrections are stated to validate the coupled-equation results, no table or figure directly compares the numerical values of the threshold shifts obtained by the two methods for the same set of states; without this explicit cross-check, the internal consistency of the framework cannot be fully assessed.

    Authors: We concur that an explicit side-by-side numerical comparison would make the internal validation more transparent. In the revised manuscript we will insert a table that reports, for each relevant state, the threshold shift obtained from the coupled Schrödinger equations alongside the shift obtained from the self-energy correction to the propagator. This will provide a direct, state-by-state cross-check of the two methods. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper constructs threshold effects within BOEFT, an EFT systematically derived from QCD via scale hierarchies and symmetries. Static potentials are taken from external lattice QCD calculations, with short-distance repulsion and long-distance threshold asymptotics following from color structure and heavy-light meson thresholds. The single free parameter (adjoint meson mass) is fixed to the experimental χ_c1(3872) mass, after which coupled Schrödinger equations are solved and results compared to independent experimental data for other states; an internal consistency check via self-energy corrections on the quarkonium propagator is performed. No step reduces a claimed prediction to an input by definition, no load-bearing premise rests solely on self-citation, and no ansatz or uniqueness theorem is smuggled in. The framework remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 3 axioms · 0 invented entities

The central claim rests on the applicability of BOEFT to threshold mixing, the accuracy of lattice QCD static potentials, and the choice to fix the single free parameter to one experimental mass; no new particles or forces are postulated beyond the effective-theory setup.

free parameters (1)
  • adjoint meson mass = mass of χ_c1(3872)
    Fixed to the mass of the χ_c1(3872) state to quantify threshold effects on the spectrum below threshold.
axioms (3)
  • domain assumption BOEFT is systematically derived from QCD by exploiting hierarchies of energy scales and symmetries
    Stated as the foundational framework in the abstract.
  • domain assumption Shapes of the static potentials are constrained by lattice QCD calculations
    Used to determine the quarkonium and tetraquark potentials.
  • domain assumption BOEFT tetraquark static potentials are repulsive at short distances and asymptotically approach heavy-light meson-antimeson thresholds at large distances
    Distinctive behavior explicitly accounted for in the model.

pith-pipeline@v0.9.0 · 5629 in / 1724 out tokens · 62737 ms · 2026-05-10T15:29:25.495874+00:00 · methodology

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