Low-energy $DD$ scattering in lattice QCD

Chuan Liu; Feng-Kun Guo; Hanyang Xing; Jia-Jun Wu; Liuming Liu; Pan-Pan Shi; Peng Sun

REVIEW 3 major objections 7 minor 1 cited by

Low-energy $DD$ scattering in lattice QCD

T0 review · 3 major / 7 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The first lattice QCD calculation of single-channel $DD$ scattering finds a weakly repulsive isovector S-wave, with physical-pion scattering length $a_0=-0.26\pm0.05$ fm, and a slightly attractive isoscalar P-wave.

desk verdict First lattice QCD calculation of single-channel DD scattering, with a robust qualitative message but quantitative physical-pion values that rest on a two-point extrapolation. read the letter →

arxiv 2502.07438 v2 pith:YOMOJBL3 submitted 2025-02-11 hep-lat hep-exhep-ph

Pan-Pan Shi , Feng-Kun Guo , Chuan Liu , Liuming Liu , Peng Sun , Jia-Jun Wu , Hanyang Xing This is my paper · ORCID

classification hep-lathep-exhep-ph

keywords latticeQCDDDscatteringLüscherfinite-volumemethodlengtheffectiverangeexpansioncharmedmesonsTcc(3875)+Wilson-Cloverfermions

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 reports the first lattice QCD calculation of single-channel $DD$ scattering for the quantum numbers $I(J^P)=1(0^+)$ and $0(1^-)$, using the Lüscher finite-volume method on $2+1$ flavor Wilson-Clover ensembles with a lattice spacing $a\simeq0.077$ fm and pion masses near 207 and 305 MeV. The central result is that the isovector S-wave interaction is weakly repulsive, with a physical-pion scattering length $a_0=-0.26\pm0.05$ fm and effective range $r_0=-5.5\pm2.1$ fm, while the isoscalar P-wave channel is slightly attractive. If correct, these are the first model-independent numbers for $DD$ final-state interactions below the $D^*D^*$ threshold, and they provide input for calculations of the $T_{cc}^+$ width and for predictions of a possible $DDK$ three-body bound state. The authors state that a more careful chiral and continuum extrapolation will require additional pion masses and lattice spacings.

What carries the argument

The load-bearing object is Lüscher's finite-volume quantization condition, which relates two-particle energy levels in a box to continuum phase shifts; the paper modifies the standard condition by replacing the continuum dispersion relation with $E^2=m_D^2+Zp^2$ for each particle, introducing the factor $(\omega_1+\omega_2)/(Z_1\omega_2+Z_2\omega_1)$ in the zeta-function matrix. The energy levels themselves are obtained with the generalized eigenvalue problem applied to interpolating operators projected onto the $A_1^+$, $E^+$, $T_2^+$ (isovector) and $T_1^-$, $T_2^-$, $A_2^-$ (isoscalar) irreps, and the phase shifts are parameterized by the effective-range expansion $\cot\delta_l=p^{-2l-1}(1/a_l+\tfrac12 r_l p^2)$, which is fitted directly to the spectra. This combination turns a small number of lattice energy levels into scattering lengths and effective ranges.

What would settle it

Compute the same $A_1^+$ spectrum on a third ensemble with $m_\pi\approx 250$ MeV, or at a second lattice spacing, using the identical analysis; if the resulting $a_0$ does not fall on the straight line through the 207 and 305 MeV values within statistical errors, the linear extrapolation used for Eq. (12) is contradicted.

Watch

Extended reading notes

Core claim

Using interacting energy levels extracted from correlation functions in the $A_1^+$ and $T_1^-$ irreps and fitted with the effective-range expansion, the paper determines the S-wave isovector and P-wave isoscalar $DD$ scattering parameters at two pion masses. It finds negative $a_0$ and $r_0$, indicating weak repulsion, and positive $a_1$ and $r_1$, indicating slight attraction; the near-threshold energy levels are close to the free two-$D$-meson levels, so higher partial waves are consistent with negligible interaction. Assuming the continuum dispersion relation, a linear extrapolation in $m_\pi^2$ gives $a_0^{\mathrm{phy}}=-0.26(5)$ fm and $r_0^{\mathrm{phy}}=-5.5(21)$ fm at the physical pion mass, with the same $a_0$ within errors at the chiral limit. The paper does not extrapolate the P-wave parameters because their statistical uncertainties are large. It also notes that a single lattice spacing prevents a continuum extrapolation.

Load-bearing premise

The quoted physical-pion values assume that $a_0$ and $r_0$ are exactly linear functions of $m_\pi^2$ between 207 and 305 MeV; with only two pion masses and one lattice spacing, any curvature, chiral logarithm, or lattice artifact would shift the results beyond the reported uncertainties.

Editorial extensions

If this is right

If the central values hold, the $I=1$ S-wave $DD$ interaction is weakly repulsive, so no $DD$ bound state appears in this channel near threshold.
The physical-pion $a_0$ and $r_0$ provide direct input for quantifying the effect of $DD$ final-state interactions on the predicted $T_{cc}^+$ width.
The same parameters serve as two-body input in continuum and finite-volume studies of a possible $DDK$ three-body bound state.
The modified finite-volume quantization condition, which accounts for a lattice dispersion relation with $Z\neq 1$, can be carried over to other heavy-meson scattering problems where such artifacts matter.
Because only two pion masses and one lattice spacing are used, additional ensembles are needed before the physical-pion result can be considered reliable, as the paper itself notes.

Reading between the lines

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

Because the extrapolation is a straight line through two points, the agreement between the chiral-limit value $a_0^\chi=-0.26(6)$ fm and the physical value is essentially built into the fit; a third pion mass would test whether the true curvature is small enough for this conclusion to survive.
The pattern $|a_0|\ll|r_0|$ matches what is seen in $I=2$ $\pi\pi$ and $I=1$ $KK$ scattering, suggesting a common near-threshold behavior for repulsive hadron-hadron channels that could be probed by collecting more repulsive S-wave systems.
One testable extension is to repeat the extraction with the P-wave parameters included in a coupled-channel $DD$-$D^*D^*$ analysis; the present single-channel values would then serve as a constrained starting point rather than final answers.
A natural next computation is the same analysis on a finer lattice at a different lattice spacing; if the extracted $Z$ factor changes the energy levels significantly, the role of the modified Lüscher formula will be directly visible.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Let me cut to it: this is the first LQCD calculation of single-channel DD scattering in I=1(0+) and I=0(1-), and that alone makes it worth reading. The qualitative result—weak repulsion in the S-wave, weak attraction in the P-wave—is stable across pion masses and across the Z=1 vs Z≠1 treatments. The methodology is sound and honestly executed: distillation, GEVP, modified Lüscher for the noncontinuum dispersion relation, and two volumes at each mass to get more kinematic points. The paper also clearly states its own limitations at the end.

The soft spots are exactly where the reader's report puts them. The headline numbers at the physical pion mass, Eq. (12), come from a two-point linear fit in m_pi^2. With only two pion masses and one lattice spacing, the system is fully constrained; there is no way to test curvature, chiral logs, or continuum effects. The paper acknowledges this, but still presents a0=-0.26(5) and r0=-5.5(21) as central results with statistical errors only. The stress-test point is fair: changing the variable to m_pi or m_pi^3 shifts r0 by roughly 1 fm. So those numbers are preliminary, not determinations.

There are also small internal inconsistencies that a referee would want cleaned up: the abstract's quoted central values with two uncertainties don't match the full-text one-sigma values; the conclusions mention m_pi=350 MeV and I=0(1+) which contradict 305 MeV and 0(1-) elsewhere; and 'isoscaclar' is a typo. More substantively, F32P21's energy levels are excluded from the ERE fits, so the m_pi=207 MeV column in Table IV rests on one volume, which further thins the chiral extrapolation.

Who is this for? People working on Tcc widths and DDK three-body systems. It gives them a first-principles input where previously they had models. The citation pattern looks clean; the prior lattice work on DD* and coupled channels is properly credited. My summary: the qualitative physics is credible, the methodology is right, but the quantitative physical-pion values should be reported with a model-dependence caveat, and ideally with more ensembles before being used as central inputs.

I'd send this to a serious referee; it's a legitimate first result with truthful limitations, and the remaining issues are fixable in revision. If I were citing it, I'd cite it for the qualitative S-wave repulsion and as a first calculation, not for the precise r0.

Referee Report

3 major / 7 minor

Summary. This paper presents a lattice QCD calculation of single-channel DD scattering in the isovector S-wave I(J^P)=1(0+) and isoscalar P-wave 0(1-) channels, using 2+1 flavor Wilson-Clover ensembles at a≈0.077 fm with m_pi≈305 and 207 MeV. The authors extract finite-volume energy levels with distillation and GEVP, apply a Lüscher quantization condition modified for the non-continuum D-meson dispersion relation (Z≠1), fit S- and P-wave effective-range expansion parameters, and extrapolate the S-wave scattering length and effective range to the physical pion mass via a linear form in m_pi^2, obtaining a0=-0.26(5) fm and r0=-5.5(21) fm. The paper concludes that the S-wave interaction is weakly repulsive and the P-wave slightly attractive.

Significance. If the results hold, this is the first lattice QCD determination of single-channel DD scattering in these channels and provides useful model-independent input for the Tcc system and DDK three-body calculations. The treatment of the D-meson dispersion relation in the Lüscher formula is a methodological plus, and the paper is transparent about the limitations of a two-point chiral extrapolation with one lattice spacing. The central qualitative conclusion—weak repulsive S-wave isovector interaction—is stable across pion masses and across the Z=1/Z≠1 treatments. The quantitative physical-pion-mass values, however, currently carry only statistical uncertainties and should be regarded as preliminary.

major comments (3)

[Abstract and Sec. VI, Eq. (12)] The abstract quoted in the submission metadata reports a0=-0.25±0.08±0.12 fm and r0=-5.7±4.5±1.7 fm, while the body abstract and Eq. (12) report a0=-0.26±0.05 fm and r0=-5.5±2.1 fm. These are two different versions of the central quantitative result, with different central values and different error decompositions; they must be reconciled before publication.
[Sec. VI, Eqs. (11)-(12)] The physical-pion-mass values in Eq. (12) are obtained from a two-point linear fit in m_pi^2 with zero degrees of freedom, and the paper explicitly notes that the uncertainties are purely statistical. The extrapolated r0=-5.5(21) fm lies outside the range of the input points and depends entirely on the assumed linearity. Please add a model-variation estimate (e.g., comparing fits in m_pi, m_pi^2, and m_pi^3, or including a curvature term), or state clearly that this value is a preliminary extrapolation rather than a final result.
[Sec. V, Table IV] The evidence for a slightly attractive P-wave interaction is marginal: the P-wave scattering length is a1=1.7(21) fm^3 at 305 MeV and 0.9(17) fm^3 at 207 MeV, each within about 1.5 standard deviations of zero. Please quantify the significance of the attraction or temper the abstract and conclusion wording accordingly.

minor comments (7)

[Sec. I] The first sentence contains a typo: 'Qauntum Chromodynamics' should be 'Quantum Chromodynamics'.
[Sec. V] In the paragraph after Table IV, 'isoscaclar' should be 'isoscalar', and 'differences of of the ERE parameters' has a duplicated 'of'.
[Sec. V] The phrase 'the Z values in Table II bare uncertainties' should read 'bear uncertainties' or 'carry uncertainties'.
[Sec. VII] The conclusions state 'at two different pion masses 207 and 350 MeV', but Table I lists 305 MeV; also the channel '0(1+)' should be '0(1-)'.
[Figs. 4 and 5] The figure labels 'P303' and 'P210' appear inconsistent with the ensemble names P30 and P21; please clarify or rename the labels.
[Fig. 7] The horizontal axes in Fig. 7 are labeled 'm [GeV]' without a subscript; use 'm_pi [GeV]' for clarity.
[Appendix A, Eq. (A10)] The zeta function Zjs is used in Eqs. (8) and (A10) but defined only in Eq. (A11); define it at first use.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ERE parameters are fitted to finite-volume spectra, and the physical-pion values are a transparent two-point linear extrapolation, not an independently verified prediction.

full rationale

The paper's derivation chain is self-contained for its central extraction: finite-volume energy levels are obtained from lattice correlation functions, then related to infinite-volume phase shifts through the standard Luescher quantization condition, with a modified dispersion-relation factor derived in Appendix A from the finite-volume Lippmann-Schwinger equation. The ERE parameters in Table IV are genuine fits to those energy levels, not renamings of the input spectra. The only potentially questionable step is the physical-pion-mass extrapolation in Eq. (11)-(12): with exactly two pion masses, a0 and r0 are fit as linear functions of m_pi^2 and then evaluated at 135 MeV. This is a model-dependent extrapolation whose quoted uncertainties are purely statistical, and the paper explicitly warns that more pion masses and lattice spacings are needed for a robust chiral and continuum extrapolation. Calling the resulting values 'predicted' overstates their independence, but this is not circularity: the physical point is not among the fitted data, the functional form is stated openly, and the extrapolated numbers are not used to define the inputs or to validate the fit. Self-citations to the CLQCD ensembles and to prior methodology are not load-bearing in a circular sense: the ensembles are external inputs used here for a new analysis, and the Luescher/ERE framework is standard and re-derived where modified. No step reduces, by construction or by self-citation, to its own inputs.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The calculation rests on standard lattice QCD methodology and on model assumptions for the chiral extrapolation and partial-wave isolation. It introduces no new particles, forces, or other invented entities.

free parameters (7)

a0 (S-wave DD scattering length, I=1) = -0.23(3) to -0.25(3) fm at simulated pion masses; -0.26(5) fm at physical pion mass
Fitted to A1+ finite-volume energy levels via ERE and Luscher quantization condition (Table IV, Eqs. 9-10).
r0 (S-wave effective range, I=1) = -1.67(24) to -4.3(14) fm at simulated pion masses; -5.5(21) fm at physical pion mass
Fitted with a0 to the same energy levels; large statistical uncertainty due to sensitivity of Luscher zeta function near free levels.
a1 (P-wave scattering length, I=0) = 0.9(17) to 2.1(22) fm^3
Fitted to T1- energy levels (Table IV).
r1 (P-wave effective range, I=0) = 16.8(52) to 20.4(38) fm^-1
Fitted to T1- energy levels (Table IV); large uncertainty.
Z (dispersion relation coefficient) = 0.926(7) to 0.976(6), Table II
Fitted to five single-D meson energies per ensemble and used in the modified Luscher formula (Eq. 8, Appendix A).
Chiral extrapolation coefficients c_a0 and c_a1 = c_a0 = -0.26(6) fm, c_a1 = 0.010(33) fm^3
Determined by fitting a0 at the two pion masses to a0(m_pi) = c_a0 + c_a1 m_pi^2 (Eq. 11).
Chiral extrapolation coefficients c_r0 and c_r1 = c_r0 = -6.4(27) fm, c_r1 = 1.9(11) fm^3
Determined by fitting r0 at the two pion masses to r0(m_pi) = c_r0 + c_r1 m_pi^2 (Eq. 11).

assumptions (6)

domain assumption Lattice QCD with a = 0.07746 fm and two pion masses approximates continuum QCD up to discretization errors of order a^2.
The results are obtained at a single lattice spacing and no continuum extrapolation is performed (Section VI: 'we are unable to extrapolate our results to the continuum limit').
standard math The Luscher finite-volume quantization condition, including the modified dispersion relation of Appendix A, correctly relates finite-volume spectra to infinite-volume phase shifts.
Central method (Eqs. 6-9, Appendix A); relies on exponentially suppressed finite-volume effects and a single-channel treatment.
domain assumption The effective range expansion cot(delta_l) = p^{-2l-1}(1/a_l + (1/2) r_l p^2) is valid for the low-energy levels used, with O(p^4) terms negligible.
Used to fit the phase shifts (Eq. 10); validity is assumed for the near-threshold levels used in the fits.
domain assumption Higher partial-wave mixing (L=4 in A1+, L=3 in T1-) and D*D* coupled-channel effects can be neglected below the D*D* threshold.
The paper states it ignores mixing from higher partial waves and uses only levels below the D*D* threshold to avoid contamination (Sections III and IV).
ad hoc to paper a0 and r0 depend linearly on m_pi^2 over the range 207 to 305 MeV, allowing a two-point extrapolation to the physical pion mass.
Eq. (11) introduces the polynomial form without a chiral perturbation theory justification; with only two points, curvature cannot be tested.
domain assumption The D meson dispersion relation E^2 = m_D^2 + Z p^2 with fitted Z captures lattice artifacts.
Z is fitted to five momenta and used in the modified Luscher formula; deviations from the continuum dispersion relation are assumed to be captured by this form.

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Pith. "Pith review of Low-energy $DD$ scattering in lattice QCD." pith.science (2026). https://pith.science/paper/YOMOJBL3

@misc{pith2026250207438,
  author       = {Pith},
  title        = {Pith review of: Low-energy $DD$ scattering in lattice QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YOMOJBL3}},
  note         = {Machine review of arXiv:2502.07438}
}

abstract

We present the first lattice QCD calculation of single-channel $DD$ scattering with quantum numbers $I(J^P)=1(0^+)$ and $0(1^-)$. The calculation is performed on the $2+1$ flavor Wilson-Clover ensembles with a lattice spacing $a\simeq 0.077$ fm and two different pion masses, $m_{\pi}\simeq207$ and $305$ MeV. The scattering parameters are determined using the L\"uscher's finite volume method. Our results indicate a weak repulsive interaction in the $1(0^+)$ channel and a slightly attractive interaction in the $0(1^-)$ channel. The $S$-wave isovector $DD$ scattering length and effective range, extrapolated to the physical pion mass, are $(-0.25\pm0.08\pm 0.12)$ fm and $(-5.7\pm4.5\pm 1.7)$ fm, respectively.

Figures

Figures reproduced from arXiv: 2502.07438 by the authors.

**Figure 2.** FIG. 2. Effective energy splittings ∆ [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. Energy levels for the [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: FIG. 5. Energy levels for the [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: FIG. 7. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

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