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Minimal renormalon subtraction stabilizes pNRQCD spectroscopy of heavy multi-quark systems and yields baryon masses that undershoot lattice QCD by 125–175 MeV.

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2026-07-13 06:28 UTC pith:G45OABBD

load-bearing objection First MRS-pNRQCD spectra for multi-heavy systems: scale stability is real, lattice offset is the expected size, non-singlet MRS is the only load-bearing caveat. the 2 major comments →

arxiv 2607.08817 v1 pith:G45OABBD submitted 2026-07-09 hep-ph hep-exhep-lat

Renormalon subtracted nonrelativistic QCD for heavy hadron systems

Beno\^it Assi , Andreas S. Kronfeld , Simon Vaiva , Michael L. Wagman This is my paper
classification hep-ph hep-exhep-lat
keywords pNRQCDminimal renormalon subtractionheavy baryonsfully-heavy tetraquarksGreen's function Monte Carlostatic potentialrenormalonsquarkonium spectroscopy
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Heavy multi-quark systems (quarkonia, triply-heavy baryons, fully-heavy tetraquarks) are governed by a hierarchy of scales that lets potential nonrelativistic QCD encode the dynamics in a Schrödinger equation with matched static potentials. Fixed-order expansions of those potentials suffer from factorial growth of coefficients—renormalons—that produce hundred-MeV scale dependence and poor order-by-order convergence. This paper implements minimal renormalon subtraction (MRS) inside a variational/Green’s-function Monte Carlo solver: the leading renormalon is isolated into an explicit power correction and the remaining short-distance series is Borel-resummed, then evaluated with a smooth dynamical scale. Charm and bottom masses are tuned once to spin-averaged 1S quarkonium, after which the same Hamiltonian predicts Ωccc, Ωccb, Ωcbb and Ωbbb masses. At NNLO the MRS results lie 125–175 MeV below lattice QCD, with the fractional gap shrinking as ~1/mQ, while scale-variation bands collapse by factors of three to ten. The same framework maps the binding landscape of unequal-mass fully-heavy tetraquarks and locates a critical heavy-to-light mass ratio below which deeply bound states appear. The practical payoff is a controlled, computationally cheap route to heavy-hadron spectra that can be extended to top-containing systems and dark-sector analogues.

Core claim

When the leading renormalon is subtracted from the static two-body potentials of pNRQCD and the residual series is Borel-resummed, the resulting MRS Hamiltonian, solved by GFMC, produces triply-heavy baryon masses that undershoot lattice-QCD benchmarks by a systematic 125–175 MeV whose fractional size falls as ~1/mQ, while renormalization-scale bands shrink by factors of 3–10 relative to fixed-order results.

What carries the argument

Minimal renormalon subtraction (MRS) applied to the static potential: the factorial asymptotic series is split into a short-distance piece (fixed-order coefficients minus their asymptotic tails) plus a Borel-resummed tail, evaluated at a smooth dynamical scale μ'(r) = √(μ_cut^{2} + λ^{2}/r^{2}), then fed into a few-body Schrödinger equation solved by variational and Green’s-function Monte Carlo.

Load-bearing premise

The same Borel resummation that cancels the leading renormalon in the color-singlet channel is assumed to work without change for the non-singlet color channels that appear in baryons and tetraquarks.

What would settle it

Include the O(1/mQ) and O(1/mQ^{2}) spin-dependent and velocity-dependent potentials already present in lattice NRQCD and recompute the same baryon masses; if the 125–175 MeV gap closes while the MRS scale bands remain narrow, the central claim is confirmed.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Scale variation of bottomonium and Ωbbb binding energies drops from hundreds of MeV to ~10 MeV at NNLO, making residual theory error dominated by missing 1/mQ operators rather than renormalon instability.
  • A single pair of MRS-tuned charm and bottom masses yields parameter-free predictions for all mixed-flavor mesons, triply-heavy baryons, and fully-heavy tetraquarks.
  • Unequal-mass fully-heavy tetraquarks bind only for heavy-to-light mass ratios ≲ 0.085 (MRS NLO); equal-mass systems such as bb̄b̄b and cc̄c̄c remain unbound.
  • The same MRS-pNRQCD-QMC pipeline supplies well-defined QCD benchmarks for top-containing systems and for composite dark-matter spectra without new lattice runs at every mass point.

Where Pith is reading between the lines

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

  • If the non-singlet renormalon structure can be derived and shown to match the singlet case, MRS becomes a universal short-distance scheme for every color channel in multi-quark EFTs.
  • The observed 1/mQ scaling of the lattice discrepancy already supplies a quantitative target for the size of the next-order 1/mQ potentials that must be matched.
  • Because the method is coordinate-space and mass-continuous, it can scan dark-sector hadron spectra across continuous mass ratios far more cheaply than lattice QCD.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper implements minimal renormalon subtraction (MRS) for the static two-body potentials of pNRQCD through NNLO, combines them with fixed-order three-body forces, and solves the resulting few-body Schrödinger problems with VMC/GFMC. Charm and bottom masses are tuned to spin-averaged 1S quarkonium masses; the same masses are then used without retuning to predict triply-heavy baryon masses (Ωccc, Ωccb, Ωcbb, Ωbbb and top-containing analogues) and the binding landscape of fully-heavy tetraquarks QQ¯Q′¯Q′. MRS is shown to shrink renormalization-scale bands by factors of 3–10 relative to fixed-order results and to bring LO/NLO/NNLO bands into partial overlap. NNLO MRS baryon masses undershoot lattice QCD by 125–175 MeV, with fractional differences falling as ∼1/mQ, which the authors attribute to omitted O(1/mQ) operators. For tetraquarks a critical mass ratio mQ/mQ′≃0.085 is obtained at NLO MRS, below which bb¯t¯t, bc¯t¯t and cc¯t¯t are bound.

Significance. If the MRS reorganization and its extension to multi-quark color channels hold, the work supplies a practical, computationally efficient route to renormalon-stable heavy-hadron spectroscopy that complements lattice QCD. The explicit isolation of the leading renormalon into a Borel-resummed tail (Appendix, Eqs. A.6–A.15), the systematic comparison of fixed-order versus MRS scale dependence (Figs. 1–4, Tables I–V), and the falsifiable 1/mQ scaling of the lattice discrepancy are genuine strengths. The tetraquark critical-ratio map and the top-containing benchmarks are new, parameter-free predictions once the quark masses are fixed. The framework is immediately reusable for dark-sector scans and for near-threshold tt¯ phenomenology.

major comments (2)
  1. Sec. II B (paragraph beginning “For multiquark systems…”) and Eq. (6): the MRS Borel resummation is derived for the color-singlet potential, yet is applied unchanged to the color-antisymmetric QQ channel that dominates the baryon Hamiltonian (Eq. 17, CB=2/3). The paper correctly labels this an unproven scheme choice. Because the absolute baryon masses in Table III (the central claim) rest on this channel, a quantitative estimate of the associated systematic—or an explicit statement that the quoted 125–175 MeV discrepancy already absorbs it—is needed before the numbers can be treated as controlled.
  2. Sec. IV B / Table III: the lattice discrepancy is interpreted as missing O(1/mQ) corrections because the fractional difference falls as ∼1/mQ. While the scaling is suggestive, no explicit O(1/mQ) operator is evaluated. A single controlled estimate (e.g., the leading Darwin or spin-orbit shift for Ωbbb) would convert the interpretation from plausible to quantitative and would strengthen the claim that MRS has removed the dominant perturbative uncertainty.
minor comments (5)
  1. Sec. II B and Appendix: the three regularization prescriptions (hard cutoff, frozen αs, µ′) are compared only for bottomonium (Figs. 6–8). A one-sentence statement that the same µcut=1 GeV choice was verified for the baryon and tetraquark systems would remove a minor reproducibility concern.
  2. Table I: the FO LO charmonium and bottomonium masses are listed without uncertainties while MRS LO entries carry them; a uniform convention would improve readability.
  3. Fig. 5 caption: the marker styles for the three scale choices are not fully legible in grayscale; adding a legend or distinct open/filled symbols would help.
  4. Sec. V: tetraquark results stop at NLO because multi-body potentials lack an MRS formula. A brief forward-looking sentence on how the three-body MRS problem might be approached would be useful for readers planning follow-up work.
  5. References: a few arXiv-only entries (e.g., [75], [100]–[102]) could be updated to journal versions if available at proof stage.

Circularity Check

0 steps flagged

No significant circularity: quark masses tuned to experimental 1S quarkonia are fixed inputs; baryon/tetraquark masses and critical ratios are independent GFMC outputs compared to external lattice benchmarks.

full rationale

The derivation chain is self-contained and non-circular. Quark masses m_c and m_b are tuned once to experimental spin-averaged 1S quarkonium masses (Eqs. 14-15, Table I) and then held fixed as inputs; all subsequent baryon masses (Table III), top-containing states (Table IV), mixed mesons (Table II), and tetraquark binding ratios/critical mass ratios (Fig. 5, Table V) are genuine GFMC solutions of the MRS-pNRQCD Hamiltonian and are compared to independent lattice-QCD and experimental benchmarks. The MRS reorganization (Eq. 6, Appendix) is applied order-by-order to the known static potentials; residual scale variation is reported as a diagnostic, not absorbed into the predictions. Self-citations to the authors' prior fixed-order QMC papers [23-25] and Kronfeld's MRS formalism serve only as methodological baselines or FO comparison columns; they do not force the NNLO MRS central values, the 125-175 MeV lattice undershoot, or the ~0.085 critical mass ratio. The openly flagged scheme choice of applying singlet MRS to non-singlet channels is an assumption, not a circular reduction. No equation equates a claimed prediction to a fitted input by construction.

Axiom & Free-Parameter Ledger

5 free parameters · 4 axioms · 0 invented entities

The calculation rests on the standard pNRQCD power counting, the published MRS Borel formula, and a handful of numerical cut-offs chosen by hand. Quark masses are free parameters fixed to data; the non-singlet MRS extension and the absorption of the power correction into m_Q are additional modeling choices without independent external calibration.

free parameters (5)
  • m_c (MRS NNLO) = 1.55489 GeV
    Tuned so that the spin-averaged 1S charmonium mass equals the experimental value 3.06865 GeV; all subsequent charm-containing predictions inherit this value.
  • m_b (MRS NNLO) = 4.80221 GeV
    Tuned so that the spin-averaged 1S bottomonium mass equals the experimental value 9.44498 GeV.
  • µ_cut = 1 GeV
    Infrared cutoff below which the dynamical scale µ'(r) freezes; chosen by hand so that MRS is active for typical bound-state separations.
  • r* = 0.35 GeV^{-1}
    Reference separation used to set the Bohr radius of the Coulomb trial wave-function; chosen inside the perturbative window.
  • λ (central) = 1
    Dimensionless scale parameter that sets the central value of µ'(r); varied by factors of 2 for residual scale bands.
axioms (4)
  • domain assumption pNRQCD static Hamiltonian at O(m_Q^0) plus fixed-order three-body potentials at NNLO is a controlled approximation for systems with m_Q ≫ Λ_QCD.
    Invoked throughout Sec. II A and used to justify omission of all 1/m_Q operators.
  • ad hoc to paper The MRS Borel resummation derived for the color-singlet static potential may be applied unchanged to every two-body color channel (octet, antisymmetric, symmetric).
    Explicitly labeled a “scheme choice” in Sec. II B; no derivation is supplied for non-singlet representations.
  • domain assumption The leading renormalon power correction Λ_R can be absorbed into the definition of the heavy-quark masses when those masses are tuned to experimental quarkonium masses.
    Stated in Sec. II B and Sec. III A; standard in short-distance mass schemes but not independently verified here for multi-quark systems.
  • standard math GFMC imaginary-time projection with the given Trotter step and finite-τ extrapolation converges to the true ground-state energy of the static Hamiltonian.
    Standard Monte-Carlo theory; residual systematics quantified by τ and δτ variations (Sec. II C).

pith-pipeline@v1.1.0-grok45 · 29785 in / 3079 out tokens · 28853 ms · 2026-07-13T06:28:01.412619+00:00 · methodology

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read the original abstract

We present a renormalon-subtracted formulation of potential nonrelativistic QCD (pNRQCD) for precision spectroscopy of heavy hadron systems, combining variational and Green's function Monte Carlo (VMC/GFMC) methods with NNLO static two- and three-body potentials. Minimal renormalon subtraction (MRS) systematically sums leading factorially growing terms, thereby stabilizing perturbative convergence and reducing renormalization-scale dependence. We tune charm and bottom quark masses to spin-averaged $1S$ quarkonium states and predict $\Omega_{ccc}$, $\Omega_{ccb}$, $\Omega_{cbb}$, and $\Omega_{bbb}$ baryon masses, as well as QCD-stable baryons containing top quarks. NNLO MRS results undershoot lattice QCD by 125--175~MeV, with fractional differences decreasing as $\sim 1/m_Q$, consistent with neglected $\mathcal{O}(1/m_Q)$ corrections. Applying these methods to unequal-mass fully-heavy tetraquarks, we determine the critical heavy-to-light mass ratio for binding and compute binding energies across the mass-ratio landscape.

Figures

Figures reproduced from arXiv: 2607.08817 by Andreas S. Kronfeld, Beno\^it Assi, Michael L. Wagman, Simon Vaiva.

Figure 1
Figure 1. Figure 1: FIG. 1. Quarkonium binding energies [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Triply-heavy baryon binding energies [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Binding energy ratio ∆ [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Baryon-to-meson mass ratio [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Binding energy ratios [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: compares the resulting potentials. Hard cut￾off and frozen αs produce kinks in V (r) near r ∼ 1/µcut where the regularization activates, while µ ′ yields a smooth potential with continuous derivatives at all sep￾arations. The choice of µcut affects how much of the integra￾tion region benefits from MRS [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Bottomonium binding energy versus renormalization [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Bottomonium binding energy comparing all three [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗

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Reference graph

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