Open Access

$D \bar{D}$ interactions are weak near threshold in QCD

David J. Wilson ^1,*, Jozef J. Dudek ^2,†, Robert G. Edwards ^3,‡, and Christopher E. Thomas ^1,§ (For the Hadron Spectrum Collaboration)

¹DAMTP, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
²Department of Physics, College of William and Mary, Williamsburg, Virginia 23187, USA
³Thomas Jefferson National Accelerator Facility, 12000 Jefferson Avenue, Newport News, Virginia 23606, USA

^*Contact author: d.j.wilson@damtp.cam.ac.uk
^†Contact author: jjdudek@wm.edu
^‡Contact author: edwards@jlab.org
^§Contact author: c.e.thomas@damtp.cam.ac.uk

PDF

Phys. Rev. D 114, 054028 – Published 14 September, 2026

DOI: https://doi.org/10.1103/s3p2-552r

Abstract

We study near-threshold $D \bar{D}$ scattering in $S$ - and $D$ -wave to determine whether or not resonances or bound states are present. Working in the approximation where charm annihilation is forbidden, with two degenerate light-quark flavors and a heavier strange quark, isospin is a good quantum number, and the only other channel that is kinematically open is $η_{c} η$ . Using lattice QCD we compute, as a function of varying light-quark mass, the $S$ -matrix for coupled-channel $η_{c} η - D \bar{D}$ scattering and find only weakly interacting meson pairs. In contrast to several other studies, we find no evidence for any bound-state or resonance singularity in the energy region between the deeply bound $χ_{c 0} (1 P)$ state and the $D_{s} {\bar{D}}_{s}$ threshold.

Physics Subject Headings (PhySH)

Article Text

References (68)

S. Uehara et al. (Belle Collaboration), Observation of a charmoniumlike enhancement in the $γ γ \to ω J / ψ$ process, Phys. Rev. Lett. 104, 092001 (2010).
R. Aaij et al. (LHCb Collaboration), Near-threshold $D \bar{D}$ spectroscopy and observation of a new charmonium state, J. High Energy Phys. 07 (2019) 035.
R. Aaij et al. (LHCb Collaboration), Amplitude analysis of the $B^{+} \to D^{+} D^{-} K^{+}$ decay, Phys. Rev. D 102, 112003 (2020).
R. Aaij et al. (LHCb Collaboration), Observation of a resonant structure near the $D_{s}^{+} D_{s}^{-}$ threshold in the $B^{+} \to D^{+} D^{-} K^{+}$ decay, Phys. Rev. Lett. 131, 071901 (2023).
K. Chilikin et al. (Belle Collaboration), Observation of an alternative $χ_{c 0} (2 P)$ candidate in $e^{+} e^{-} \to J / ψ D \bar{D}$ , Phys. Rev. D 95, 112003 (2017).
R. Molina and E. Oset, The Y(3940), Z(3930) and the X(4160) as dynamically generated resonances from the vector-vector interaction, Phys. Rev. D 80, 114013 (2009).
X.-K. Dong, F.-K. Guo, and B.-S. Zou, A survey of heavy-antiheavy hadronic molecules, Prog. Phys. 41, 65 (2021).
L. Maiani, F. Piccinini, A. D. Polosa, and V. Riquer, Diquark-antidiquarks with hidden or open charm and the nature of X(3872), Phys. Rev. D 71, 014028 (2005).
R. F. Lebed and A. D. Polosa, $χ_{c 0} (3915)$ As the lightest $c \bar{c} s \bar{s}$ state, Phys. Rev. D 93, 094024 (2016).
E. J. Eichten, K. Lane, and C. Quigg, New states above charm threshold, Phys. Rev. D 73, 014014 (2006); 73, 079903(E) (2006).
T. Barnes, S. Godfrey, and E. S. Swanson, Higher charmonia, Phys. Rev. D 72, 054026 (2005).
P. G. Ortega, J. Segovia, D. R. Entem, and F. Fernández, Charmonium resonances in the $3.9 GeV / c^{2}$ energy region and the $X (3915) / X (3930)$ puzzle, Phys. Lett. B 778, 1 (2018).
K. Miyake and Y. Yamaguchi, Hidden-charm tetraquarks in a mixture model: Coupled-channel analysis with $c \bar{c}$ and hadronic molecular components, Phys. Rev. D 112, 036004 (2025).
S. Uehara et al. (Belle Collaboration), Observation of a $χ c 2$ candidate in $γ γ \to D \bar{D}$ production at BELLE, Phys. Rev. Lett. 96, 082003 (2006).
B. Aubert et al. (BABAR Collaboration), Observation of the $χ_{c 2} (2 p)$ meson in the reaction $γ γ \to D \bar{D}$ at BABAR, Phys. Rev. D 81, 092003 (2010).
E. Wang, H.-S. Li, W.-H. Liang, and E. Oset, Analysis of the $γ γ \to D \bar{D}$ reaction and the $D \bar{D}$ bound state, Phys. Rev. D 103, 054008 (2021).
O. Deineka, I. Danilkin, and M. Vanderhaeghen, Dispersive analysis of the $γ γ \to D \bar{D}$ data and the confirmation of the DD bound state, Phys. Lett. B 827, 136982 (2022).
M. Hoferichter, D. R. Phillips, and C. Schat, Roy-Steiner equations for $γ γ \to π π$ , Eur. Phys. J. C 71, 1743 (2011).
L.-Y. Dai and M. R. Pennington, Comprehensive amplitude analysis of $γ γ \to π^{+} π^{-}, π^{0} π^{0}$ and $\bar{K} K$ below 1.5 GeV, Phys. Rev. D 90, 036004 (2014).
I. Danilkin and M. Vanderhaeghen, Dispersive analysis of the $γ γ^{*} \to π π$ process, Phys. Lett. B 789, 366 (2019).
M. Luscher, Two particle states on a torus and their relation to the scattering matrix, Nucl. Phys. B354, 531 (1991).
C. h. Kim, C. T. Sachrajda, and S. R. Sharpe, Finite-volume effects for two-hadron states in moving frames, Nucl. Phys. B727, 218 (2005).
R. A. Briceno, Two-particle multichannel systems in a finite volume with arbitrary spin, Phys. Rev. D 89, 074507 (2014).
D. J. Wilson, C. E. Thomas, J. J. Dudek, and R. G. Edwards (Hadron Spectrum Collaboration), Scalar and tensor charmonium resonances in coupled-channel scattering from lattice QCD, Phys. Rev. Lett. 132, 241901 (2024).
D. J. Wilson, C. E. Thomas, J. J. Dudek, and R. G. Edwards (Hadron Spectrum Collaboration), Charmonium $χ_{c 0}$ and $χ_{c 2}$ resonances in coupled-channel scattering from lattice QCD, Phys. Rev. D 109, 114503 (2024).
S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
S. Prelovsek, S. Collins, D. Mohler, M. Padmanath, and S. Piemonte, Charmonium-like resonances with $J^{PC} = 0^{+ +}, 2^{+ +}$ in coupled $D \bar{D}$ , $D_{s} {\bar{D}}_{s}$ scattering on the lattice, J. High Energy Phys. 06 (2021) 035.
P.-P. Shi, M. Albaladejo, M.-L. Du, F.-K. Guo, and J. Nieves, P-wave charmonium contribution to hidden-charm states from a reanalysis of lattice QCD data, Phys. Rev. D 111, 074043 (2025).
R. G. Edwards, B. Joo, and H.-W. Lin, Tuning for three-flavors of anisotropic clover Fermions with stout-link smearing, Phys. Rev. D 78, 054501 (2008).
H.-W. Lin et al. (Hadron Spectrum Collaboration), First results from $2 + 1$ dynamical quark flavors on an anisotropic lattice: Light-hadron spectroscopy and setting the strange-quark mass, Phys. Rev. D 79, 034502 (2009).
L. Liu, G. Moir, M. Peardon, S. M. Ryan, C. E. Thomas, P. Vilaseca, J. J. Dudek, R. G. Edwards, B. Joo, and D. G. Richards (Hadron Spectrum Collaboration), Excited and exotic charmonium spectroscopy from lattice QCD, J. High Energy Phys. 07 (2012) 126.
R. G. Edwards, N. Mathur, D. G. Richards, and S. J. Wallace (Hadron Spectrum Collaboration), Flavor structure of the excited baryon spectra from lattice QCD, Phys. Rev. D 87, 054506 (2013).
J. J. Dudek, R. G. Edwards, M. J. Peardon, D. G. Richards, and C. E. Thomas, Toward the excited meson spectrum of dynamical QCD, Phys. Rev. D 82, 034508 (2010).
C. E. Thomas, R. G. Edwards, and J. J. Dudek, Helicity operators for mesons in flight on the lattice, Phys. Rev. D 85, 014507 (2012).
J. J. Dudek, R. G. Edwards, and C. E. Thomas, S and D-wave phase shifts in isospin-2 pi pi scattering from lattice QCD, Phys. Rev. D 86, 034031 (2012).
M. Peardon, J. Bulava, J. Foley, C. Morningstar, J. Dudek, R. G. Edwards, B. Joo, H.-W. Lin, D. G. Richards, and K. J. Juge (Hadron Spectrum Collaboration), A Novel quark-field creation operator construction for hadronic physics in lattice QCD, Phys. Rev. D 80, 054506 (2009).
C. Michael, Adjoint sources in lattice gauge theory, Nucl. Phys. B259, 58 (1985).
M. Luscher and U. Wolff, How to calculate the elastic scattering matrix in two-dimensional quantum field theories by numerical simulation, Nucl. Phys. B339, 222 (1990).
F. G. Ortega-Gama, J. J. Dudek, and R. G. Edwards (Hadron Spectrum Collaboration), Timelike meson form factors beyond the elastic region from lattice QCD, Phys. Rev. D 110, 094505 (2024).
D. J. Wilson, R. A. Briceno, J. J. Dudek, R. G. Edwards, and C. E. Thomas, Coupled $π π, K \bar{K}$ scattering in $P$ -wave and the $ρ$ resonance from lattice QCD, Phys. Rev. D 92, 094502 (2015).
G. K. C. Cheung, C. O’Hara, G. Moir, M. Peardon, S. M. Ryan, C. E. Thomas, and D. Tims (Hadron Spectrum Collaboration), Excited and exotic charmonium, $D_{s}$ and $D$ meson spectra for two light quark masses from lattice QCD, J. High Energy Phys. 12 (2016) 089.
S. Okubo, Phi meson and unitary symmetry model, Phys. Lett. 5, 165 (1963).
G. Zweig, An SU(3) model for strong interaction symmetry and its breaking. Version 1, Report No. CERN-TH-401, 1964, 10.17181/CERN-TH-401.
J. Iizuka, Systematics and phenomenology of meson family, Prog. Theor. Phys. Suppl. 37, 21 (1966).
M. T. Hansen and S. R. Sharpe, Multiple-channel generalization of Lellouch-Luscher formula, Phys. Rev. D 86, 016007 (2012).
R. A. Briceno and Z. Davoudi, Moving multichannel systems in a finite volume with application to proton-proton fusion, Phys. Rev. D 88, 094507 (2013).
R. A. Briceno, J. J. Dudek, and R. D. Young, Scattering processes and resonances from lattice QCD, Rev. Mod. Phys. 90, 025001 (2018).
A. J. Woss, D. J. Wilson, and J. J. Dudek (Hadron Spectrum Collaboration), Efficient solution of the multichannel Lüscher determinant condition through eigenvalue decomposition, Phys. Rev. D 101, 114505 (2020).
D. J. Wilson, J. J. Dudek, R. G. Edwards, and C. E. Thomas, Resonances in coupled $π K, η K$ scattering from lattice QCD, Phys. Rev. D 91, 054008 (2015).
S. Piemonte, S. Collins, D. Mohler, M. Padmanath, and S. Prelovsek, Charmonium resonances with $J^{PC} = 1^{- -}$ and $3^{- -}$ from $\bar{D} D$ scattering on the lattice, Phys. Rev. D 100, 074505 (2019).
M. Bayar, A. Feijoo, and E. Oset, X(3960) seen in $D_{s}^{+} D_{s}^{-}$ as the X(3930) state seen in $D + D -$ , Phys. Rev. D 107, 034007 (2023).
D. Guo, J.-Z. Wang, D.-Y. Chen, and X. Liu, Connection between near the $D_{s}^{+} D_{s}^{-}$ threshold enhancement in $B^{+} \to D_{s}^{+} D_{s}^{-} K^{+}$ and conventional charmonium $χ_{c 0} (2 P)$ , Phys. Rev. D 106, 094037 (2022).
E. Wang, W.-H. Liang, and E. Oset, Analysis of the $e^{+} e^{-} \to J / ψ D \bar{D}$ reaction close to the threshold concerning claims of a $χ_{c 0} (2 P)$ state, Eur. Phys. J. A 57, 38 (2021).
F.-K. Guo and U.-G. Meissner, Where is the $χ_{c 0} (2 P)$ ?, Phys. Rev. D 86, 091501 (2012).
R. G. Edwards and B. Joo, The chroma software system for lattice QCD, Nucl. Phys. B, Proc. Suppl. 140, 832 (2005).
M. A. Clark, R. Babich, K. Barros, R. C. Brower, and C. Rebbi, Solving Lattice QCD systems of equations using mixed precision solvers on GPUs, Comput. Phys. Commun. 181, 1517 (2010).
R. Babich, M. A. Clark, and B. Joo, Parallelizing the quda Library for Multi-GPU Calculations in Lattice Quantum Chromodynamics, in SC 10 (Supercomputing 2010) New Orleans, Louisiana, November 13-19, 2010 (IEEE, Louisiana, USA, 2010), 10.1109/SC.2010.40.
K. Clark, B. Joo, A. Strelchenko, M. Cheng, A. Gambhir, and R. Brower, Accelerating Lattice QCD Multigrid on GPUs Using Fine-grained Parallelization, in Proceedings of SC 16 (Supercomputing 2016) Salt Lake City, Utah, November 2016 (IEEE Press, 2016), 10.5555/3014904.3014995.
B. Joó, D. Kalamkar, K. Vaidyanathan, M. Smelyanskiy, K. Pamnany, V. Lee, P. Dubey, and W. Watson, Lattice QCD on Intel Xeon ${Phi}^{TM}$ coprocessors, in Supercomputing, Lecture Notes in Computer Science, edited by J. Kunkel, T. Ludwig, and H. Meuer (Springer, Berlin Heidelberg, 2013), Vol. 7905, pp. 40–54.
B. Joó, mg-proto: A Multigrid Library for QCD, https://github.com/JeffersonLab/mg_proto/ (2010).
J. C. Osborn, R. Babich, J. Brannick, R. C. Brower, M. A. Clark, S. D. Cohen, and C. Rebbi, Multigrid solver for clover fermions, Proc. Sci. LATTICE2010 (2010) 037.
R. Babich, J. Brannick, R. C. Brower, M. A. Clark, T. A. Manteuffel, S. F. McCormick, J. C. Osborn, and C. Rebbi, Adaptive multigrid algorithm for the lattice Wilson-Dirac operator, Phys. Rev. Lett. 105, 201602 (2010).
J. Chen, R. G. Edwards, and W. Mao, Graph Contractions for Calculating Correlation Functions in Lattice QCD, in Platform for Advanced Scientific Computing (Association for Computing Machinery, NY, USA, 2023), 10.1145/3592979.3593409.
S. Prelovsek, E. Ortiz-Pacheco, S. Collins, L. Leskovec, M. Padmanath, and I. Vujmilovic, Doubly heavy tetraquarks from lattice QCD: Incorporating diquark-antidiquark operators and the left-hand cut, Phys. Rev. D 112, 014507 (2025).
A. Stump and J. R. Green, Position-space sampling for local multiquark operators in lattice QCD using distillation and the importance of tetraquark operators for $T_{c c} (3875)^{+}$ , Phys. Rev. D 113, 034513 (2026).
W. I. Jay and E. T. Neil, Bayesian model averaging for analysis of lattice field theory results, Phys. Rev. D 103, 114502 (2021).
S. M. Ryan and D. J. Wilson (Hadron Spectrum Collaboration), Excited and exotic bottomonium spectroscopy from lattice QCD, J. High Energy Phys. 02 (2021) 214.
L. Gayer, S. M. Ryan, and D. J. Wilson (Hadron Spectrum Collaboration), Highly excited B, $B_{s}$ and $B_{c}$ meson spectroscopy from lattice QCD, J. High Energy Phys. 01 (2025) 123.

Phys. Rev. D 114, 054028– Published 14 September, 2026

Vol. 114, Iss. 5 — 1 September 2026

Received 9 March 2026
Accepted 15 July 2026

DOI: https://doi.org/10.1103/s3p2-552r

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society

Physical Review D

Reuse & Permissions

Export citation

Export citation

$D \bar{D}$ interactions are weak near threshold in QCD

Abstract

Physics Subject Headings (PhySH)

Article Text

References (68)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation

DD¯ interactions are weak near threshold in QCD

Abstract

Physics Subject Headings (PhySH)

Article Text

References (68)

$D \bar{D}$ interactions are weak near threshold in QCD