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REVIEW 3 major objections 5 minor 1 cited by

Hadronic decay of vector charmonium from the lattice

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that a generalized ratio method extracts $\Gamma(\psi(3770)\to \bar{D}D) = (24.2\pm 6.4)\ \mathrm{MeV}$ from a single lattice volume, compatible with the experimental $(27.2\pm 1.0)\ \mathrm{MeV}$.

desk verdict Useful extension of the ratio method to an excited initial state, but the quoted width rests on an untested volume-cancellation assumption and should be read as exploratory. read the letter →

arxiv 2411.10123 v2 pith:GS6N7CQN submitted 2024-11-15 hep-lat hep-ph

Benoît Blossier , Jochen Heitger , Jan Neuendorf , Teseo San José This is my paper · ORCID
classification hep-lathep-ph PACS 12.38.Gc14.40.Pq
keywords latticeQCDcharmoniumhadronicdecaywidthpsi(3770)ratiomethodtwistedboundaryconditionsenergyshiftcorrelationfunctions
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 paper argues that a hadronic decay width can be extracted from lattice QCD using the ratio method rather than the costly multi-volume finite-volume scattering program. The authors generalize the ratio method to an excited initial state and apply it to $\psi(3770)\to \bar{D}D$, using twisted boundary conditions to tune the kinematics to the on-shell point. From ratios of three-point and two-point correlation functions on two $N_f=2$ ensembles they obtain $\Gamma=(24.2\pm 6.4)\ \mathrm{MeV}$, consistent with the experimental $(27.2\pm 1.0)\ \mathrm{MeV}$, together with an energy shift for the $\psi(3770)$ and $\bar{D}D$ levels. The significance is that a single volume and a scan over twist angles may be enough to predict near-threshold decay parameters.

What carries the argument

The carrying object is the ratio $R(t) = |\bar{T}_3(t,t_0)| / \sqrt{P^{\bar{D}D}(t)\,\lambda_3(t,t_0)}$, built from the GEVP-projected charmonium-to-$\bar{D}D$ triangle correlator, the $\bar{D}D$ two-point function, and the $\psi(3770)$ eigenvalue. In the degenerate (on-shell) limit this ratio behaves as $|x_{31}|\,t + A$; off-shell it becomes $|x_{31}|\,\sinh(t\Delta)/\Delta + A e^{-t\Delta}$. The linear-in-$t$ (or hyperbolic-sine) enhancement suppresses other states, so a fit at long times isolates the mixing amplitude $x_{31}=\langle \bar{D}D|\psi(3770)\rangle$. Twist angles on the charm quark continuously vary the $\bar{D}D$ momentum to reach or scan around the on-shell point, and a two-level Hamiltonian with entries $\pm\delta/2$ and $x_{31}$, $x_{31}^*$ converts the same amplitude into an energy shift $\epsilon^2 = |x_{31}|^2 + \delta^2/4$.

What would settle it

Compute the same $|x_{31}|$ on these ensembles after enlarging the GEVP to include the $\bar{D}D$ interpolators, as the paper itself suggests; if the amplitude shifts by more than the quoted errors, the ratio does not isolate the wanted transition. Alternatively, repeat the measurement on a third volume at the same lattice spacing and pion mass: the width from $\Gamma = \frac{L^3}{24\pi} p_i E_i |x_{31}|^2$ should be volume-independent if the cancellation is exact.

Watch

Extended reading notes

Core claim

The paper's central claim is that the ratio $R(t)$ built from the GEVP-isolated charmonium three-point function, the $\bar{D}D$ two-point function, and the charmonium eigenvalue acquires a time enhancement linear in $t$ (or, off-shell, proportional to $\sinh(t\Delta)/\Delta$) that isolates the hadronic transition amplitude $x_{31}=\langle \bar{D}D|\psi(3770)\rangle$. Fitting $R(t)$ at several twist angles yields the momentum dependence of $|x_{31}|$, and its on-shell value inserted into $\Gamma = \frac{L^3}{24\pi}\, p_i E_i |x_{31}|^2$ gives the decay width. The paper reports $\Gamma = (24.2\pm 6.4)\ \mathrm{MeV}$ on the larger E5 ensemble ($26.8\pm 6.2$ on D5), compatible with the experimental $27.2\pm 1.0\ \mathrm{MeV}$, and an energy shift $\epsilon = 28.5\pm 4.9\ \mathrm{MeV}$ ($55.1\pm 2.2$ on D5) that would appear in a fully dynamical simulation. This is offered as evidence that near-threshold decay parameters can be obtained from a single volume and a few momenta, without multiple volumes or irreducible representations.

Load-bearing premise

The result stands on two linked assumptions: the fitted ratio isolates the $\psi(3770)\leftrightarrow \bar{D}D$ mixing with negligible contamination from other $\bar{D}D$ levels and other charmonium states, and the finite-volume matrix element converts to the physical width through $\Gamma = \frac{L^3}{24\pi} p_i E_i |x_{31}|^2$, whose volume cancellation is expected but not proven to be exact.

Editorial extensions

If this is right

  • The hadronic decay width of $\psi(3770)$ can be obtained from one lattice volume and a scan over twist angles, without the multi-volume, multi-representation program of the standard finite-volume method.
  • The energy shift $\epsilon = |x_{31}|$ at the on-shell point predicts how much the charmonium and $\bar{D}D$ levels would separate in a fully dynamical simulation where the mixing is active.
  • The momentum dependence of $|x_{31}|$ is well described by a parabola $|x_{31}|(p)=c_1 - c_2(p-p_0)^2$ that vanishes at zero momentum, giving a compact parametrization of the off-shell behavior.
  • Near-threshold decays in other heavy-quark systems, such as $\phi\to K\bar{K}$ for a lighter effective charm mass and $\Upsilon(4S)\to B\bar{B}$ for a heavier one, are expected to be reachable by the same ratio method.

Reading between the lines

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

  • Finite-volume effects seem to shift the on-shell kinematics substantially (the on-shell twist angle differs by a factor of two between the two ensembles); a three-volume study would test whether the $L^3$ cancellation in the width formula holds at the precision claimed.
  • The approach likely works best near threshold, where the narrow-width approximation and the assumption that the resonance is an asymptotic state are safest; broad resonances far above threshold may need additional control.
  • A direct cross-check on the same ensembles with the standard finite-volume scattering formalism would quantify the narrow-width systematic error that this paper leaves unestimated.
  • The qualitative agreement with the ${}^3P_0$ quark model suggests the momentum dependence of the amplitude is dominated by non-relativistic pair-creation kinematics, which could guide model-informed extrapolations to the on-shell point.
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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 / 5 minor

Summary. The manuscript applies the ratio method of Pennanen/Michael and McNeile/Michael to the hadronic decay ψ(3770)→D̄D on two N_f=2 CLS ensembles. By forming ratios of the triangle and two-point correlation functions and using partially twisted boundary conditions to tune the D-meson momentum to the on-shell point, it extracts the transition amplitude x31 = ⟨D̄D|ψ(3770)⟩ from the slope of the ratio R(t). The on-shell amplitude is then converted through Γ = L^3/(24π) p_i E_i |x31|^2, yielding Γ=(26.8±6.2) MeV on D5 and Γ=(24.2±6.4) MeV on E5, compatible with the experimental Γ(ψ(3770)→D̄D)=(27.2±1.0) MeV. The paper also computes the energy shift ε of the charmonium spectrum and compares the momentum dependence of the amplitude with the 3P0 quark model. The authors explicitly label the study exploratory: no continuum limit is taken, pion masses are unphysical, and the systematic error from the finite-volume conversion is described as difficult to estimate.

Significance. If the extraction is controlled, the paper would demonstrate an attractive alternative to the Lüscher method for near-threshold decay widths: a single volume plus twisted boundary conditions, with the decay width obtained directly from correlator ratios rather than from a scattering phase-shift scan. The generalization of the ratio method to an excited initial state and the explicit treatment of the off-shell momentum dependence are useful method developments. The paper is transparent about fit intervals, energies, and amplitudes, and it clearly identifies several limitations. The strengths are the clean derivation of the ratio formulae in Section 2, the use of PTBCs to reach the on-shell point, and the direct comparison with a simple quark model. The main caveat is that the central finite-volume conversion formula and the selection of fit windows are not yet fully controlled, so the method is promising rather than established.

major comments (3)
  1. [Section 2, Eq. (2.11)] The central result Γ=(24.2±6.4) MeV in Table 7 is obtained by inserting a single-volume lattice matrix element |x31| into Eq. (2.11). This finite-volume-to-infinite-volume conversion is asserted, not derived: the text only states that the cancellation of the L^3 factor with the matrix-element volume dependence is 'expected to be exact for sufficiently large volumes', and Section 2 concedes that the method 'does not attempt to connect the finite-volume matrix elements to an infinite-volume process'. The formula was derived in Refs. [12,13] for ground-state transitions, and this paper does not re-derive it for the excited-state case. The agreement between ensembles D5 and E5 cannot verify the required L^{-3/2} scaling because the two ensembles differ in tuned momentum (329 vs 468 MeV) and in pion mass (449 vs 437 MeV). Unless the conversion is derived from a proper quantization condition or tested with a controlled volume scan, the quoted Γ is not a controlled prediction of the physical decay width but a model-dependent conversion of a matrix element. This is load-bearing for the paper's main claim.
  2. [Section 4, Tables 5 and 6] The values of |x31| are extracted from linear fits to R(t) over short time intervals (t/a=2–7 on D5 and 4–10 on E5), and the text states that the final intervals were chosen after generating 10,000 random choices of fit intervals from the same data and using the resulting distributions of c1, c2 and χ² as a guide. This is a post-hoc selection procedure on the same dataset; it can bias the extracted slopes and understate the uncertainty. Since Γ is quadratic in |x31|, the central number inherits this risk. Please demonstrate stability of the result under a fixed selection rule, cross-validation, or a systematic scan over a range of intervals reported as a systematic error. The current presentation does not allow the reader to assess how much of the quoted error is statistical and how much is selection.
  3. [Section 2, Eqs. (2.4)–(2.6)] Equation (2.4) expresses the correlator as a sum over all D̄D levels β, and the text notes that this sum can be removed by solving a GEVP for the final state. In the analysis, however, only a single D̄D interpolator is used, so the contribution of the un-GEVP'd higher D̄D tower is assumed to be negligible without a quantitative estimate. Given the short fit windows used for R(t), the linear slope that defines |x31| could receive unidentified contamination. The authors should provide an estimate of the size of the neglected terms, for instance by fitting with an additional exponential or by comparing with the xT(t) result over the same windows.
minor comments (5)
  1. [Section 2] The notation x31 is used for ⟨D̄D|ψ(3770)⟩, but the meaning of the subscripts is not defined; please state explicitly that 3 refers to the third charmonium level and 1 to the D̄D ground state.
  2. [Section 4, Eq. (4.3)] The parabola parameters are reported as ac1 and c2/a, but the units of p0 are not explicit; please state the momentum units used in the fit.
  3. [Figure 5] The caption says that the band fits only the dark points, but the figure legend does not clearly distinguish the two sets of points; please clarify the legend or caption.
  4. [Abstract and Section 5] The phrase 'fully compatible' with the experimental result in the abstract is stronger than the body's careful caveats, since no systematic error estimate is included; suggest softening the wording.
  5. [Appendix A] The 3P0 quark-model comparison uses experimental values for the oscillator parameters ω and γ, while the lattice data are at unphysical quark masses; the text notes this, but a direct restatement in the main text would avoid confusion in the comparison of Figs. 6 and 8.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the decay width is obtained from lattice-computed matrix elements via a fixed conversion formula; the quoted compatibility with experiment is a genuine, non-fitted cross-check.

full rationale

The central chain is correlator ratios (eqs. 2.5-2.6) -> |x31| -> parabola (eq. 4.3) -> Gamma (eq. 2.11). Each step's input is lattice data or a stated formula: x31 is the slope of the ratio R(t) fitted to the actual three-point correlator; the on-shell momentum is set by lattice masses through eq. (4.2); and the parabola parameters c1 and c2 parametrize the lattice-extracted momentum dependence and are not taken from experiment. Gamma is then obtained by inserting the on-shell lattice matrix element into the quoted formula Gamma = L^3/(24 pi) p_i E_i |x31|^2. This is a rescaling of a measured matrix element under a stated model assumption, not a quantity fitted to the experimental Gamma; the agreement Gamma = (24.2 +/- 6.4) MeV versus (27.2 +/- 1.0) MeV is therefore an independent check modulo systematics. The paper explicitly flags the real limitations: the method 'does not attempt to connect the finite-volume matrix elements to an infinite-volume process' and the systematic error is 'difficult to estimate'. These are correctness and control concerns about eq. (2.11)'s volume cancellation and the narrow-width approximation, not circularity. Self-citations ([40] GEVP, [41] thesis) are methodological references; the ratio method and conversion formula come from non-overlapping prior work [10-13]. No load-bearing step reduces by construction to its own output.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The central decay-width extraction rests on four fitted parameters (c1, c2, A constants, and the chosen fit windows) and on a set of domain assumptions inherited from the ratio method: the narrow-width approximation, the single-DD-interpolator truncation, the continuum dispersion relation, the plane-wave conversion formula with expected volume cancellation, and the unphysical-mass calibration of the ensembles. No invented entities appear.

free parameters (4)
  • c1 (parabola maximum of a|x31|) = D5: 0.01869(32); E5: 0.01276(56) (lattice units)
    Fitted in Section 4, eq. (4.3): |x31|(p) = c1 - c2 (p-p0)^2. c1 sets the on-shell matrix element, evaluated at the momentum from theta_0, and enters Gamma via eq. (2.11).
  • c2 (parabola curvature) = D5: c2/a = 1.169(44); E5: c2/a = 1.183(52)
    Fitted with c1 in the same correlated fit; controls the width of the bell-shaped momentum dependence and the implied peak momentum p0 = sqrt(c1/c2).
  • A (constant in ratio fit, per twist angle) = Tables 5-6, ranging from +0.0140(63) to -0.0152(79)
    Offset in eq. (2.6), R(t) = |x31|/Delta sinh(t Delta) + A exp(-t Delta); fitted for each twist angle and intended to absorb contributions from other states.
  • Fit time windows for the ratio fits = D5: t/a in [2,5] to [2,7]; E5: t/a in [4,8] to [6,10]
    Chosen post hoc from a 10000-sample random scan guided by chi^2, as described in Section 4; these choices affect every extracted a|x31| value.
assumptions (7)
  • domain assumption Narrow-width approximation: psi(3770) is treated as an asymptotic state and the transition amplitude is extracted from the time-enhanced ratio, leaving the method's systematic error unquantified.
    Section 2, final paragraph and Section 5: 'The ratio method relies on the narrow width approximation... it does not establish a direct connection between the lattice matrix elements and the scattering in infinite volume.'
  • domain assumption Time sums are replaced by integrals over time in the spectral decomposition.
    Section 2: 'the lattice spacing is sufficiently fine to replace sums by integrals over time.' This underlies the linear and sinh time behavior of eqs. (2.5)-(2.6).
  • domain assumption The D D final state is non-interacting plane waves, and the finite-volume matrix element converts to the physical width via Gamma = L^3/(24 pi) p_i E_i |x31|^2 with exact volume cancellation.
    Section 2, eq. (2.11): 'We expect this cancelation to be exact for sufficiently large volumes.' The volume dependence of |x31|^2 is only checked between two volumes.
  • domain assumption The D D sector is represented by a single interpolator with no GEVP; contamination from higher D D states is absorbed by the constant A.
    Section 2: 'we do not solve a GEVP for the D D-system'; the spectral decomposition eq. (2.4) is truncated by hand and A is fitted in eq. (2.6).
  • domain assumption The continuum relativistic dispersion relation E_D^2 = m_D^2 + 3 (theta/L)^2 holds on the lattice.
    Section 4, eq. (4.1), used to fix the on-shell twist angle theta_0; verified only at a single lattice spacing in fig. 4.
  • domain assumption The charm quark mass and the lattice scale are calibrated to experiment (m_Ds = 1968 MeV and f_K), while the light-quark masses are unphysical.
    Section 3: 'The charm-quark mass was fixed such that m_Ds = m_Ds,phys... the scale is set using f_K'; m_pi is about 437-449 MeV, so the setup is not at the physical point.
  • domain assumption The 8x8 GEVP isolates psi(3770) without needing the D D level in the matrix.
    Sections 2 and 4: the GEVP uses only charmonium operators (eq. (3.1)); the paper notes, with ref. [42], that some levels could be missed if final-state correlators are excluded.

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Cite this review

Pith. "Pith review of Hadronic decay of vector charmonium from the lattice." pith.science (2026). https://pith.science/paper/GS6N7CQN

@misc{pith2026241110123,
  author       = {Pith},
  title        = {Pith review of: Hadronic decay of vector charmonium from the lattice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GS6N7CQN}},
  note         = {Machine review of arXiv:2411.10123}
}
abstract

Estimating decay parameters in lattice simulations is a computationally demanding problem, requiring several volumes and momenta. We explore an alternative approach, where the transition amplitude can be extracted from the spectral decomposition of particular ratios built from correlation functions. This so-called ratio method has the advantage of not needing various irreducible representations or volumes, and it allows us to predict the decay width $\Gamma$ and the energy shift $\epsilon$ of the spectrum directly. In this work, we apply this method to study the hadronic decay $\psi(3770)\to \bar{D}D$ on two CLS $N_\text{f}=2$ ensembles. This approach requires close to on-shell kinematics to work, and we employ twisted boundary conditions to precisely tune the on-shell point. Although our study is yet to approach the continuum limit, we find a value of $\Gamma$ fully compatible to the physical result, and $\epsilon$ informs us by how much our spectrum would shift in a fully dynamical simulation. Besides lattice calculations, many analytical tools have been proposed to understand decay processes. A relatively simple, early example is the ${}^3P_0$ quark model, which provides a physical insight of the decay process.

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Forward citations

Cited by 1 Pith paper

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  1. The hadronic decay of vector charmonium

    hep-lat 2024-12 conditional novelty 6.0 of 10

    The authors extract the psi(3770) to D Dbar hadronic mixing and a decay width compatible with experiment using a narrow-width ratio method on two CLS ensembles.

Reference graph

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Reviewed August 12, 2026 · model on record in the stance chip above.