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$\eta_c\eta_c$ and $J/\psi J/\psi$ scatterings from lattice QCD

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims lattice QCD produces a $J^{PC}=2^{++}$ resonance in $J/\psi J/\psi$ scattering at $(6544(10),\,552(34))$ MeV, matching the fully-charmed $X(6600)$/$X(6400)$ structures seen by ATLAS and CMS.

desk verdict Serious lattice calculation with a plausible but model-dependent 2++ resonance claim; the pole parameters should be read with caution. read the letter →

arxiv 2505.23220 v1 pith:WX7CKRLG submitted 2025-05-29 hep-lat

Geng Li , Chunjiang Shi , Ying Chen , Wei Sun This is my paper
classification hep-lat
keywords latticeQCDfully-charmedtetraquarkJ/ψJ/ψscatteringηcηcLüscherquantizationX(6600)X(6200)virtualstate
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

The paper computes $\eta_c\eta_c$ and $J/\psi J/\psi$ scattering amplitudes directly from lattice QCD at two pion masses and finds a broad $J^{PC}=2^{++}$ resonance in the $^5S_2$ $J/\psi J/\psi$ channel, with mass about 6.54 GeV and width about 550 MeV that barely change when the pion mass is lowered from 420 MeV to 250 MeV. These parameters match the fully-charmed broad structures $X(6600)$ and $X(6400)$ reported by ATLAS and CMS, and the $2^{++}$ assignment matches the quantum numbers preferred by CMS's angular analysis. The same calculation finds a likely $0^{++}$ virtual state 20--40 MeV below the $J/\psi J/\psi$ threshold, which the authors connect to the near-threshold structure $X(6200)$. A sympathetic reader takes this as evidence that the experimentally observed fully-charmed structures arise from the scattering dynamics of two charmonia in QCD itself.

What carries the argument

The engine of the argument is the standard finite-volume method: two-meson correlation functions are computed with the distillation method on two lattice volumes at each pion mass, energy levels are extracted by solving a generalized eigenvalue problem, and L\"uscher's quantization condition assigns each energy level a scattering phase shift $k\cot\delta_0(k)$. For the $^5S_2$ channel these phase shifts cannot be described by an effective-range expansion, but the inverse of $\sqrt{s}\,/\,(k\cot\delta_0)$ is visibly linear in $s$, so the paper parametrizes the $K$-matrix as $K(s)=as+b$ and finds the resonance pole by solving $1-i\rho(s)K(s)=0$ with $\rho(s)=k/\sqrt{s}$; the zero of $K(s)$ at $\sqrt{s_0}\approx6.45$ GeV is a Castillejo--Dalitz--Dyson zero of the amplitude. The dynamics that explains why channels differ is quark rearrangement with one-gluon exchange: the Wick-contraction diagram in which the two mesons exchange quarks dominates over the direct gluonic exchange, and Fierz rearrangement fixes its sign, repulsive for $\eta_c\eta_c$ ($0^{++}$) and attractive for $J/\psi J/\psi$ ($0^{++}$), with the spin structure of the $^5S_2$ state flipping the relative strength to about $-2$ as the lattice data confirm.

What would settle it

A lattice calculation at the physical pion mass on larger volumes, or one that includes the $J/\psi\psi(2S)$ channel in the quantization condition, would settle it: if the $2^{++}$ pole moves outside roughly $6.54\pm0.05$ GeV in mass or $0.55\pm0.10$ GeV in width, the claimed match to $X(6600)$ fails. Independently, extracting the $X(6600)$ pole from a coupled-channel fit to the ATLAS and CMS four-muon spectra would provide an experimental cross-check at the same precision as the lattice pole.

Watch

Extended reading notes

Core claim

Reported most strongly is the observation of a $2^{++}$ resonance in the $^5S_2$ $J/\psi J/\psi$ scattering channel, determined from the pole of a single-channel scattering amplitude fitted to finite-volume energy levels through L\"uscher's quantization condition. Its pole parameters are $(m_R,\Gamma_R)=(6544(10),552(34))$ MeV at $m_\pi\approx420$ MeV and $(6539(13),546(57))$ MeV at $m_\pi\approx250$ MeV, indistinguishable between the two pion masses, and compatible with the parameters of the fully-charmed broad structure $X(6600)$ (or $X(6400)$) reported by ATLAS and CMS. Because the CMS angular distribution analysis of the four-muon final state favors $2^{++}$ and disfavors $0^{++}$ at the 95% confidence level, the paper takes this agreement as evidence that the experimental structures carry these quantum numbers. Alongside the resonance, the $^1S_0$ $J/\psi J/\psi$ amplitude has a positive scattering length and a pole on the second sheet below threshold by 28(10) MeV at $m_\pi=420$ MeV and 38(20) MeV at $m_\pi=250$ MeV, a likely virtual state that may be connected with $X(6200)$.

Load-bearing premise

The claim rests on treating $^5S_2$ $J/\psi J/\psi$ scattering as a single elastic channel up to 6.6 GeV; because the resonance is so wide ($\Gamma_R\approx550$ MeV), a large part of its tail lies above that energy, where $J/\psi\psi(2S)$ and other channels open and the two-body unitarity built into the fit no longer holds.

Editorial extensions

If this is right

  • The structures $X(6600)$ and $X(6400)$ seen by ATLAS and CMS can be identified with a genuine $2^{++}$ resonance in $J/\psi J/\psi$ scattering, supporting the CMS angular-analysis preference for $2^{++}$ over $0^{++}$.
  • The near-threshold $X(6200)$, if it exists, is a $0^{++}$ virtual state 20--40 MeV below the $J/\psi J/\psi$ threshold rather than a bound state, and the $^5S_2$ channel's repulsion rules out a $2^{++}$ assignment for it.
  • Di-charmonium scattering parameters at the two pion masses agree within errors, so the interactions and the resonance are expected to persist essentially unchanged at the physical pion mass.
  • The quark-rearrangement plus one-gluon-exchange mechanism predicts repulsive $\chi_{c0}\chi_{c0}$ interactions and attractive $h_ch_c$ and $\chi_{c1}\chi_{c1}$ interactions, checkable in future lattice calculations.
  • The zero of the amplitude at $\sqrt{s_0}\approx6.45$ GeV coincides with the phenomenological mass of the lowest $2^{++}$ fully-charmed tetraquark, giving the Castillejo--Dalitz--Dyson zero a concrete physical interpretation.

Reading between the lines

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

  • Because the resonance's width is about 550 MeV, a large fraction of its spectral weight lies above 6.6 GeV where $J/\psi\psi(2S)$ and higher channels open; I would expect a coupled-channel re-analysis to move the pole, and the cleanest tests are a lattice calculation that includes those channels explicitly and an experimental pole extraction from the full spectra.
  • If the identification with $X(6600)$ is correct, the pole position from an interference-aware fit to the ATLAS and CMS data should converge near $(6.54\pm0.02)-i(0.27\pm0.04)$ GeV; current published mass-and-width comparisons are too blunt to confirm or refute it.
  • The claimed pion-mass insensitivity makes a concrete prediction: the same $2^{++}$ pole should appear at essentially the same position in $\eta_b\eta_b$ and $\Upsilon\Upsilon$ scattering, testable with current bottom-quark lattice ensembles.
  • A virtual state, unlike a bound state, produces only a threshold enhancement rather than a peak in the invariant-mass spectrum, so the $X(6200)$ interpretation could be tested by fitting the $J/\psi J/\psi$ threshold region in LHCb data with the virtual-state line shape.
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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 / 6 minor

Summary. This manuscript presents an Nf = 2 lattice QCD study of ηcηc and J/ψJ/ψ scattering in the J^PC = 0^{++} and 2^{++} channels, using anisotropic ensembles at mπ ≈ 420 and 250 MeV, two volumes per pion mass, and the distillation method. From the correlation-weight matrices and GEVP analyses with restricted operator sets, the authors conclude that the ηcηc and J/ψJ/ψ channels are nearly decoupled and apply single-channel Lüscher analysis. For the 0^{++} channels they find repulsive ηcηc scattering (a0 ≈ −0.12 fm) and attractive 1S0 J/ψJ/ψ scattering (a0 ≈ +0.2 fm), with a likely virtual state 20–40 MeV below the J/ψJ/ψ threshold. For the 2^{++} channel they find repulsive near-threshold 5S2 J/ψJ/ψ scattering (a0 ≈ −0.16 fm) and, via a two-parameter K-matrix K(s) = as + b fitted below 6.6 GeV and continued to the second sheet, a broad resonance pole with (mR, ΓR) = (6544(10), 552(34)) MeV (mπ ≈ 420 MeV) and (6539(13), 546(57)) MeV (mπ ≈ 250 MeV), which they identify as compatible with the X(6600)/X(6400) structures seen by ATLAS and CMS. The dynamics are interpreted through quark rearrangement plus one-gluon exchange, including a predicted −2 relative strength between the 5S2 and 1S0 channels that is reproduced by the lattice correlation functions.

Significance. If the resonance claim survives a systematic-uncertainty treatment, the paper would provide a first direct lattice-QCD determination of the 2^{++} fully-charmed tetraquark candidate in the J/ψJ/ψ channel, supporting the 2^{++} assignment that CMS favors experimentally. Genuine strengths include: two volumes and two pion masses with consistent results; a careful energy-level analysis with GEVP orthogonality checks, two independent fitting methods, and NV-dependence studies used to exclude untrustworthy levels; explicit reporting of the LHS-smearing energy ceiling; agreement with the independent scattering lengths of Ref. [46]; and a parameter-free internal signature, the OGE prediction of a −2 ratio of quark-rearrangement strengths in the 5S2 and 1S0 channels (Eq. (61), Fig. 22), which the data reproduce. The mπ-insensitivity of the di-charmonium interactions is also notable.

major comments (3)
  1. [Section III.B, Table IV] The central quantitative claim carries only statistical errors, but the paper itself demonstrates model dependence far larger than those errors. The three sum-of-poles fits of Eq. (39) (initial g0 = 10, 20, 40 GeV) give (mR, ΓR) = (6449(21), 761(42)), (6495(12), 681(32)), and (6524(11), 617(35)) MeV, whereas the preferred linear ansatz Eq. (40) gives (6543(10), 548(34)) MeV; the spread is roughly 95 MeV in mass and 210 MeV in width, an order of magnitude above the quoted errors of 10 and 34 MeV. The linear form is justified by parameter economy, but the χ²/d.o.f. values in Fig. 16 do not discriminate: at mπ = 420 MeV (250 MeV) the pole fit with g0 = 10 gives 0.49 (0.19) versus 0.91 (0.40) for the linear fit. A systematic uncertainty reflecting the K-matrix ansatz choice, or a softened claim (a pole in the range ≈ 6.45–6.54 GeV with ΓR ≈ 0.55–0.76 GeV), must be adopted before the abstract's precision statement 'compatible with the parameters of X(6600)' is supported.
  2. [Sections II.B and III.B] The inelastic-channel truncation is stated but not quantified, and it is quantitatively relevant for a pole of this width. The study includes only ηcηc and J/ψJ/ψ below 6.6 GeV, excluding J/ψψ(2S) (threshold ≈ 6.77 GeV). For the quoted pole, a Breit-Wigner spectral function with mR = 6.543 GeV and ΓR = 0.548 GeV has roughly 28% of its weight above that threshold, and the pole's distance in the complex s-plane to the J/ψψ(2S) branch point (≈ 4.8 GeV²) is comparable to its distance to the elastic threshold (≈ 5.8 GeV²). The single-channel K-matrix in Eq. (46) has no branch cut for the omitted channel, so the analytic continuation that defines the pole is performed over a region where the amplitude's analytic structure is incomplete. The J/ψψ(2S) coupling is OZI-allowed and unconstrained by the fitted levels. Without an exploratory estimate of the shift this channel can induce in (mR, ΓR) (e.g., a two-channel fit or a Flatté-type sensitivity check), the quantitative compatibility with X(6600) asserted in the abstract and Sec. IV.C is not established; the paper deserves credit for stating the restriction explicitly in Sec. II.B, but a limitation statement is not a substitute for a systematic estimate.
  3. [Abstract and Section III.B, Fig. 19] The wording 'we observe a 2++ resonance' overstates what the constrained data show. Within the fitted window the phase shift δ0(s) rises from negative values, crosses zero at the CDD zero √s0 ≈ 6.45 GeV (Eq. (40)), and reaches only about π/4 by 6.6 GeV; it does not cross π/2 in the directly constrained region. The resonance is thus a second-sheet pole obtained from extrapolated K-matrix parameters, not a feature unambiguously visible in the finite-volume spectrum, and its interplay with the nearby CDD zero correlates the pole parameters with the functional form. In light of the two preceding comments, the abstract and the concluding compatibility statements should be reworded to something like 'a broad 2++ pole in the extrapolated single-channel amplitude, consistent within the stated systematic uncertainties with X(6600) or X(6400)'.
minor comments (6)
  1. [Abstract / Sections III.A and V] The virtual-state pole position is quoted inconsistently across the paper: the abstract says '20–40 MeV', Sec. III.A gives 28(10) MeV (M420) and 38(20) MeV (M250), and Sec. V says '20–30 MeV'; these values should be harmonized.
  2. [Throughout] There are numerous typos and infelicities: 'a formalism is developped' (Sec. I); 'we construction the OM M operators' (Sec. II.B); 'this this diagram' and 'eneregy' (Sec. II.C); 'Notable, the value' (Sec. III.B); and incomplete reference metadata (Ref. [45] lacks a title, and Ref. [29] lacks the publication year).
  3. [Section V] The statement that the 1D2 ηcηc scattering 'decouples completely' from the 5S2 J/ψJ/ψ scattering is too strong given that Fig. 8 shows small residual off-diagonal weights (the text itself notes 'a tiny deviation from the unit matrix' for L12M420); the claim should be qualified as decoupling within the present statistical precision.
  4. [Section III.B, Eq. (48) and Table IV] The normalization of Γ(J/ψJ/ψ) in Eq. (48), including the 1/2 symmetry factor for identical particles, is only alluded to ('has been considered implicitly'), and the narrow-width approximation is explicitly unsuitable for ΓR ≈ 550 MeV; the Br(J/ψJ/ψ) values in Table IV should be clearly labeled as an approximate internal consistency check, with the convention stated explicitly so the reader can reproduce the 65–75% numbers.
  5. [Section II.G, Tables II and III] The correspondence between the bold-face energy levels in Tables II and III and the actual inputs to the ERE/K-matrix fits is distributed across Sec. II.G and the captions of Figs. 15 and 18; a compact summary listing, for each ensemble and channel, which {En} enter each fit would improve reproducibility.
  6. [Section IV.C, Fig. 23] Figure 23 compares the experimental pole estimates with only the linear-fit poles; given the ansatz spread documented in Table IV, the figure should show the full pole-position range or an uncertainty band to provide an honest visual comparison with the ATLAS/CMS determinations.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the resonance and virtual-state poles are standard analytic continuations of amplitudes fitted to lattice energy levels, with model dependence explicitly quantified; self-citations are methodological only.

full rationale

The paper's central claims are obtained by a standard, self-contained chain: finite-volume energies are extracted via GEVP fits (Sec. II.F), converted to phase shifts through Lüscher's quantization condition (Eq. (31)), and the phase shifts are parametrized by ERE or K-matrix forms (Eqs. (33), (39), (40)) whose parameters are fitted to those lattice-derived points. The quoted 0++ virtual-state and 2++ resonance poles are then solutions of the pole equations (36) and (46), i.e., analytic continuations of the fitted amplitude; they are not prescribed inputs. The paper explicitly reports the model dependence of the 2++ pole (Table IV: (m_R, Gamma_R) moves from (6.449, 0.761) to (6.524, 0.617) GeV under alternative pole-form K(s), with the linear fit giving (6.543, 0.548) GeV), which is a systematic uncertainty rather than a circular definition. The linear K(s)=as+b ansatz is selected because the inverse phase-shift data are nearly linear, and the alternative form is tested. The self-citations (Refs. [39], [54], [59]) are methodological (ensemble setup, ratio-method fitting strategy, QUDA interface) and do not supply the physical result. No equation is defined in terms of the quantity it purports to predict, and no fitted parameter is renamed as an independent prediction.

Assumptions & free parameters 6 free parameters · 7 assumptions · 2 invented entities

The central scattering parameters are not pulled from a separate theory; they are fitted to the lattice energy levels via ERE and K-matrix forms. The main burden is model dependence: the K(s)=a s+b ansatz is chosen after inspecting the data, and the resonance pole is then a function of that ansatz. No ad hoc forces or new mediators are introduced; the reported poles are derived quantities with an external experimental counterpart.

free parameters (6)
  • a0, 1S0 eta_c eta_c scattering length = -0.117(19) fm (M420), -0.120(26) fm (M250)
    Fitted to finite-volume energy levels through Lüscher's formula. Its negative sign yields the repulsive interaction claim for eta_c eta_c.
  • r0, 1S0 eta_c eta_c effective range = -1.30(20) fm (M420), -1.63(40) fm (M250)
    Second ERE parameter from the same fit; fixed by the lattice data, not predicted by a separate theory.
  • a0, 1S0 J/psi J/psi scattering length = 0.25(7) fm (M420), 0.20(8) fm (M250)
    Positive sign indicates the attractive interaction and underpins the virtual-state pole claim.
  • r0, 1S0 J/psi J/psi effective range = 2.31(33) fm (M420), 2.14(41) fm (M250)
    Second ERE parameter from the same fit; used to continue the amplitude to the virtual-state pole.
  • a, K-matrix slope for 5S2 J/psi J/psi = 1.45(12) GeV^-2 (M420), 1.48(20) GeV^-2 (M250)
    Coefficient in K(s)=a s+b fitted to five lattice energy levels; controls the resonance pole position and width.
  • b, K-matrix intercept for 5S2 J/psi J/psi = -60.4(4.9) (M420), -61.6(8.4) (M250)
    Second coefficient of K(s); its zero defines the CDD zero at sqrt(s) about 6.45 GeV and helps determine the resonance pole.
assumptions (7)
  • standard math Lüscher's quantization condition maps finite-volume energy levels to infinite-volume scattering amplitudes.
    Used in Sec. III to convert GEVP energy levels into k cot delta values; this is a proven relation for two-particle states on a torus.
  • domain assumption Charm-quark annihilation diagrams are negligible due to OZI suppression, and chi_c0 decouples from the di-charmonium operator set.
    Sec. II.C argues diagrams (d) and (e) are suppressed and discards the O_chi_c0 operator; if wrong, 0++ energy levels could be contaminated.
  • domain assumption eta_c eta_c and J/psi J/psi channels decouple, so single-channel Lüscher analysis is valid.
    Sec. II.E and Sec. III show weak correlations between the two channels and compatible energies with and without the other channel; the decoupling is supported but not proven, and it is load-bearing for the separate ERE fits.
  • domain assumption Inelastic channels such as J/psi psi(2S) and J/psi psi(3770) can be neglected below the 6.6 to 6.7 GeV cutoff.
    Stated in Sec. II.B; the broad 550 MeV resonance has substantial spectral weight above the cutoff, so the assumption may fail for the pole extraction.
  • domain assumption The LHS subspace truncation at NV=170 or 120 is sufficiently converged for the energy levels used.
    Sec. II.G checks NV=70, 120, and 170 and keeps only levels below 6.6 GeV on L12 or 6.3 GeV on L16; this is an empirical convergence criterion, not a proof.
  • ad hoc to paper The K-matrix of the 5S2 channel is well approximated by K(s)=a s+b over the fit range.
    Eq. (40); chosen because the pole-fit form Eq. (39) has unstable parameters. The resonance pole parameters are functions of this ansatz.
  • ad hoc to paper One-gluon exchange plus Fierz rearrangement explains the signs of the interactions and the observed -2 ratio.
    Sec. IV uses a non-relativistic OGE amplitude and spin wave functions; this is an interpretive model rather than a lattice-derived input, though it is checked against the lattice data.
invented entities (2)
  • 2++ resonance R2++ in 5S2 J/psi J/psi scattering independent evidence
    purpose: Explains the broad X(6600) or X(6400) structure observed by ATLAS and CMS.
    It is a pole of the fitted K-matrix amplitude, so it is not pulled from a hat; experimental X(6600) provides an external handle. Its quoted mass and width depend on the single-channel K(s) model.
  • 0++ virtual state near the J/psi J/psi threshold independent evidence
    purpose: May explain the near-threshold enhancement X(6200).
    Derived from the ERE fit of the 1S0 J/psi J/psi phase shift; X(6200) is suggested by coupled-channel phenomenological fits to LHCb data but is not yet confirmed.

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Cite this review

Pith. "Pith review of $\eta_c\eta_c$ and $J/\psi J/\psi$ scatterings from lattice QCD." pith.science (2026). https://pith.science/paper/WX7CKRLG

@misc{pith2026250523220,
  author       = {Pith},
  title        = {Pith review of: $\eta_c\eta_c$ and $J/\psi J/\psi$ scatterings from lattice QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WX7CKRLG}},
  note         = {Machine review of arXiv:2505.23220}
}
abstract

We investigate the $J^{PC}=(0,2)^{++}$ $\eta_c \eta_c$ and $J/\psi J/\psi$ scatterings in $N_f=2$ lattice QCD at two pion masses $m_\pi\approx 420$\,MeV and 250\,MeV. The quark field smearing scheme used in the distillation method strongly suppresses high-momentum states, thereby limiting the maximum accessible center-of-mass energy in this study to approximately 6.6\,GeV. Given the observed near-decoupling of the $\eta_c\eta_c$ and $J/\psi J/\psi$ channels, we analyze their scattering properties using the single-channel L\"{u}scher's method. In $0^{++}$ channels, $\eta_c\eta_c$ and $J/\psi J/\psi$ have repulsive and attractive interactions, respectively, which are dominated by the quark exchange effects with the Fierz rearrangement determining the interaction characteristic. A likely $0^{++}$ virtual state is observed below the threshold by 20-40\,MeV in the ${}^1S_0$ $J/\psi J/\psi$ scattering and may have connection with $X(6200)$. In $2^{++}$ channels, the near-threshold interaction for the ${}^5S_2$ $J/\psi J/\psi$ scattering is repulsive. The di-charmonium interactions are found to be insensitive to $m_\pi$ and can be consistently understood in terms of quark rearrangement effects combined with a one-gluon-exchange mechanism, both of which are supported by our lattice QCD results. Most importantly, we observe a $2^{++}$ resonance in the ${}^5S_2$ $J/\psi J/\psi$ scattering, whose properties are $(m_R,\Gamma_R)=\big(6544(10),552(34)\big)$\,MeV for $m_\pi\approx 420$\,MeV and $\big(6539(13),546(57)\big)$\,MeV for $m_\pi\approx 250$\,MeV, which are compatible with the parameters of the fully-charmed broad structure $X(6600)$ (or $X(6400)$) reported by ATLAS and CMS.

Figures

Figures reproduced from arXiv: 2505.23220 by the authors.

Figure 1
Figure 1. FIG. 1. The dispersion relations of the charmonium states [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The effective mass extracted from diagram (c) is [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Schematic illustrations of quark Wick contractions. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figures from the paper (25 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Correlation matrix [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The right-hand eigenvector by solving the GEVP, [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Effective energy levels obtained by solving GEVP [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The spectral weight [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Correlation matrix [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The effective energy levels [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Stability of the two fitting methods on the L12M420 ensemble with [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. The stability of the two fitting methods on the L12M420 ensemble with [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Energy level determination of the 0 [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]
Figure 13
Figure 13. Figure 13: , the dashed parabola in the k 2 < 0 region is the real value function ik with the positive (negative) branch corresponding to −i|k| (i|k|). Obviously, the straight line of k cot δ0(k) intersects with ik on the positive branch and indicate the possible existence of a …
Figure 14
Figure 14. Figure 14: FIG. 14. The squared modulus of the transition matrix [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Energy levels with linear fits for the [PITH_FULL_IMAGE:figures/full_fig_p016_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. The scattering phase [PITH_FULL_IMAGE:figures/full_fig_p018_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Pole positions on Riemann Sheet II, obtained via principal component analysis (PCA), shown in both the complex [PITH_FULL_IMAGE:figures/full_fig_p019_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Energy levels with linear fits for the [PITH_FULL_IMAGE:figures/full_fig_p019_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19. The scattering phase shift [PITH_FULL_IMAGE:figures/full_fig_p020_19.png]
Figure 21
Figure 21. Figure 21: FIG. 21. Schematic illustrations of one-gluon exchange be [PITH_FULL_IMAGE:figures/full_fig_p021_21.png]
Figure 22
Figure 22. Figure 22: FIG. 22. Ratio of the quark rearrangement components of [PITH_FULL_IMAGE:figures/full_fig_p022_22.png]
Figure 23
Figure 23. Figure 23: shows the comparison of the pole positions of R2++ in this study with those of X(6600) (or X(6400)) reported by experiments. The J P C quantum numbers of these fully-charmed structures remain a focal point of both experimental and theoretical investigations. The most …
Figure 24
Figure 24. Figure 24: FIG. 24. Stability of the two fitting methods on the L16M420 ensemble with [PITH_FULL_IMAGE:figures/full_fig_p026_24.png]
Figure 25
Figure 25. Figure 25: FIG. 25. Stability of the two fitting methods on the L12M250 ensemble with [PITH_FULL_IMAGE:figures/full_fig_p027_25.png]
Figure 26
Figure 26. Figure 26: FIG. 26. Stability of the two fitting methods on the L16M250 ensemble with [PITH_FULL_IMAGE:figures/full_fig_p028_26.png]
Figure 27
Figure 27. Figure 27: FIG. 27. Stability of the two fitting methods on the L16M420 ensemble with [PITH_FULL_IMAGE:figures/full_fig_p029_27.png]
Figure 28
Figure 28. Figure 28: FIG. 28. Stability of the two fitting methods on the L12M250 ensemble with [PITH_FULL_IMAGE:figures/full_fig_p030_28.png]
Figure 29
Figure 29. Figure 29: FIG. 29. Stability of the two fitting methods on the L16M250 ensemble with [PITH_FULL_IMAGE:figures/full_fig_p031_29.png]

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Forward citations

Cited by 2 Pith papers

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    A lattice QCD computation predicts a 2++ J/psi J/psi resonance with mass 6.54 GeV and width 0.54 GeV, identified with the X(6600) structure.

  2. Next-to-leading order QCD corrections to electromagnetic production and decay of fully charm tetraquarks

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