Abstract
We present the first lattice investigation of coupled-channel \( D\overline{D} \) and \( {D}_s{\overline{D}}_s \) scattering in the JPC = 0++ and 2++ channels. The scattering matrix for partial waves l = 0, 2 and isospin zero is determined using multiple volumes and inertial frames via Lüscher’s formalism. Lattice QCD ensembles from the CLS consortium with mπ ≃ 280 MeV, a ≃ 0.09 fm and L/a = 24, 32 are utilized. The resulting scattering matrix suggests the existence of three charmonium-like states with JPC = 0++ in the energy region ranging from slightly below 2mD up to 4.13 GeV. We find a so far unobserved \( D\overline{D} \) bound state just below threshold and a \( D\overline{D} \) resonance likely related to χc0(3860), which is believed to be χc0(2P). In addition, there is an indication for a narrow 0++ resonance just below the \( {D}_s{\overline{D}}_s \) threshold with a large coupling to \( {D}_s{\overline{D}}_s \) and a very small coupling to \( D\overline{D} \). This resonance is possibly related to the narrow X(3915)/χc0(3930) observed in experiment also just below \( {D}_s{\overline{D}}_s \). The partial wave l = 2 features a resonance likely related to χc2(3930). We work with several assumptions, such as the omission of J/ψω, ηcη and three-particle channels. Only statistical uncertainties are quantified, while the extrapolations to the physical quark-masses and the continuum limit are challenges for the future.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Belle collaboration, Observation of an alternative χc0(2P) candidate in e+e− → J/\( \psi D\overline{D} \), Phys. Rev. D 95 (2017) 112003 [arXiv:1704.01872] [INSPIRE].
Particle Data collaboration, Review of Particle Physics, Prog. Theor. Exp. Phys. 2020 (2020) 083C01 [INSPIRE].
LHCb collaboration, Amplitude analysis of the B+ → D+D−K+ decay, Phys. Rev. D 102 (2020) 112003 [arXiv:2009.00026] [INSPIRE].
LHCb collaboration, A model-independent study of resonant structure in B+ → D+D−K+ decays, Phys. Rev. Lett. 125 (2020) 242001 [arXiv:2009.00025] [INSPIRE].
BaBar collaboration, Observation of Y(3940) → J/ψω in B → J/ψωK at BABAR, Phys. Rev. Lett. 101 (2008) 082001 [arXiv:0711.2047] [INSPIRE].
Belle collaboration, Observation of a charmonium-like enhancement in the γγ → ωJ/ψ process, Phys. Rev. Lett. 104 (2010) 092001 [arXiv:0912.4451] [INSPIRE].
BaBar collaboration, Evidence for the decay X(3872) → J/ψω, Phys. Rev. D 82 (2010) 011101 [arXiv:1005.5190] [INSPIRE].
BaBar collaboration, Study of X(3915) → J/ψω in two-photon collisions, Phys. Rev. D 86 (2012) 072002 [arXiv:1207.2651] [INSPIRE].
R.F. Lebed and A.D. Polosa, χc0(3915) As the Lightest \( c\overline{c}s\overline{s} \) State, Phys. Rev. D 93 (2016) 094024 [arXiv:1602.08421] [INSPIRE].
D. Gamermann, E. Oset, D. Strottman and M.J. Vicente Vacas, Dynamically generated open and hidden charm meson systems, Phys. Rev. D 76 (2007) 074016 [hep-ph/0612179] [INSPIRE].
D. Gamermann and E. Oset, Hidden charm dynamically generated resonances and the e+e− → J/\( \psi D\overline{D} \), J/\( \psi D\overline{D}\ast \) reactions, Eur. Phys. J. A 36 (2008) 189 [arXiv:0712.1758] [INSPIRE].
M. Lüscher, Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. 2. Scattering States, Commun. Math. Phys. 105 (1986) 153 [INSPIRE].
M. Lüscher, Two particle states on a torus and their relation to the scattering matrix, Nucl. Phys. B 354 (1991) 531 [INSPIRE].
M. Lüscher, Signatures of unstable particles in finite volume, Nucl. Phys. B 364 (1991) 237 [INSPIRE].
Hadron Spectrum collaboration, Resonances in coupled πK − ηK scattering from quantum chromodynamics, Phys. Rev. Lett. 113 (2014) 182001 [arXiv:1406.4158] [INSPIRE].
D.J. Wilson, J.J. Dudek, R.G. Edwards and C.E. Thomas, Resonances in coupled πK, ηK scattering from lattice QCD, Phys. Rev. D 91 (2015) 054008 [arXiv:1411.2004] [INSPIRE].
Hadron Spectrum collaboration, An a0 resonance in strongly coupled πη, \( K\overline{K} \) scattering from lattice QCD, Phys. Rev. D 93 (2016) 094506 [arXiv:1602.05122] [INSPIRE].
R.A. Briceño, J.J. Dudek, R.G. Edwards and D.J. Wilson, Isoscalar ππ, \( K\overline{K} \), ηη scattering and the σ, f0, f2 mesons from QCD, Phys. Rev. D 97 (2018) 054513 [arXiv:1708.06667] [INSPIRE].
G. Moir, M. Peardon, S.M. Ryan, C.E. Thomas and D.J. Wilson, Coupled-Channel Dπ, Dη and \( {D}_s\overline{K} \) Scattering from Lattice QCD, JHEP 10 (2016) 011 [arXiv:1607.07093] [INSPIRE].
CLQCD collaboration, A coupled-channel lattice study on the resonance-like structure Zc(3900), Chin. Phys. C 43 (2019) 103103 [arXiv:1907.03371] [INSPIRE].
HAL QCD collaboration, Fate of the Tetraquark Candidate Zc(3900) from Lattice QCD, Phys. Rev. Lett. 117 (2016) 242001 [arXiv:1602.03465] [INSPIRE].
HAL QCD collaboration, The tetraquark candidate Zc(3900) from dynamical lattice QCD simulations, J. Phys. G 45 (2018) 024002 [arXiv:1706.07300] [INSPIRE].
C.B. Lang, L. Leskovec, D. Mohler and S. Prelovsek, Vector and scalar charmonium resonances with lattice QCD, JHEP 09 (2015) 089 [arXiv:1503.05363] [INSPIRE].
M. Padmanath et al., Identifying spin and parity of charmonia in flight with lattice QCD, Phys. Rev. D 99 (2019) 014513 [arXiv:1811.04116] [INSPIRE].
S. Piemonte, S. Collins, D. Mohler, M. Padmanath and S. Prelovsek, Charmonium resonances with JPC = 1−− and 3−− from \( \overline{D}D \) scattering on the lattice, Phys. Rev. D 100 (2019) 074505 [arXiv:1905.03506] [INSPIRE].
R. Brett, J. Bulava, J. Fallica, A. Hanlon, B. Hörz and C. Morningstar, Determination of s- and p-wave I = 1/2 Kπ scattering amplitudes in Nf = 2 + 1 lattice QCD, Nucl. Phys. B 932 (2018) 29 [arXiv:1802.03100] [INSPIRE].
M. Bruno et al., Simulation of QCD with Nf = 2 + 1 flavors of non-perturbatively improved Wilson fermions, JHEP 02 (2015) 043 [arXiv:1411.3982] [INSPIRE].
RQCD collaboration, Lattice simulations with Nf = 2 + 1 improved Wilson fermions at a fixed strange quark mass, Phys. Rev. D 94 (2016) 074501 [arXiv:1606.09039] [INSPIRE].
M. Bruno, T. Korzec and S. Schaefer, Setting the scale for the CLS 2 + 1 flavor ensembles, Phys. Rev. D 95 (2017) 074504 [arXiv:1608.08900] [INSPIRE].
M. Lüscher and S. Schaefer, Lattice QCD with open boundary conditions and twisted-mass reweighting, Comput. Phys. Commun. 184 (2013) 519 [arXiv:1206.2809] [INSPIRE].
D. Mohler and S. Schaefer, Remarks on strange-quark simulations with Wilson fermions, Phys. Rev. D 102 (2020) 074506 [arXiv:2003.13359] [INSPIRE].
Hadron Spectrum collaboration, A Novel quark-field creation operator construction for hadronic physics in lattice QCD, Phys. Rev. D 80 (2009) 054506 [arXiv:0905.2160] [INSPIRE].
C. Morningstar et al., Improved stochastic estimation of quark propagation with Laplacian Heaviside smearing in lattice QCD, Phys. Rev. D 83 (2011) 114505 [arXiv:1104.3870] [INSPIRE].
R.A. Briceño, Two-particle multichannel systems in a finite volume with arbitrary spin, Phys. Rev. D 89 (2014) 074507 [arXiv:1401.3312] [INSPIRE].
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 (2010) 034508 [arXiv:1004.4930] [INSPIRE].
C.E. Thomas, R.G. Edwards and J.J. Dudek, Helicity operators for mesons in flight on the lattice, Phys. Rev. D 85 (2012) 014507 [arXiv:1107.1930] [INSPIRE].
J.J. Dudek, R.G. Edwards and C.E. Thomas, S and D-wave phase shifts in isospin-2 ππ scattering from lattice QCD, Phys. Rev. D 86 (2012) 034031 [arXiv:1203.6041] [INSPIRE].
R.A. Briceño, J.J. Dudek and R.D. Young, Scattering processes and resonances from lattice QCD, Rev. Mod. Phys. 90 (2018) 025001 [arXiv:1706.06223] [INSPIRE].
A.X. El-Khadra, A.S. Kronfeld and P.B. Mackenzie, Massive fermions in lattice gauge theory, Phys. Rev. D 55 (1997) 3933 [hep-lat/9604004] [INSPIRE].
M.B. Oktay and A.S. Kronfeld, New lattice action for heavy quarks, Phys. Rev. D 78 (2008) 014504 [arXiv:0803.0523] [INSPIRE].
C. Michael, Adjoint Sources in Lattice Gauge Theory, Nucl. Phys. B 259 (1985) 58 [INSPIRE].
M. Lüscher and U. Wolff, How to Calculate the Elastic Scattering Matrix in Two-dimensional Quantum Field Theories by Numerical Simulation, Nucl. Phys. B 339 (1990) 222 [INSPIRE].
B. Blossier, M. Della Morte, G. von Hippel, T. Mendes and R. Sommer, On the generalized eigenvalue method for energies and matrix elements in lattice field theory, JHEP 04 (2009) 094 [arXiv:0902.1265] [INSPIRE].
M. Döring, U.-G. Meißner, E. Oset and A. Rusetsky, Unitarized Chiral Perturbation Theory in a finite volume: Scalar meson sector, Eur. Phys. J. A 47 (2011) 139 [arXiv:1107.3988] [INSPIRE].
M.T. Hansen and S.R. Sharpe, Multiple-channel generalization of Lellouch-Lüscher formula, Phys. Rev. D 86 (2012) 016007 [arXiv:1204.0826] [INSPIRE].
C. Morningstar et al., Estimating the two-particle K-matrix for multiple partial waves and decay channels from finite-volume energies, Nucl. Phys. B 924 (2017) 477 [arXiv:1707.05817] [INSPIRE].
D.J. Wilson, R.A. Briceño, J.J. Dudek, R.G. Edwards and C.E. Thomas, Coupled ππ, \( K\overline{K} \) scattering in P-wave and the ρ resonance from lattice QCD, Phys. Rev. D 92 (2015) 094502 [arXiv:1507.02599] [INSPIRE].
C. Hidalgo-Duque, J. Nieves, A. Ozpineci and V. Zamiralov, X(3872) and its Partners in the Heavy Quark Limit of QCD, Phys. Lett. B 727 (2013) 432 [arXiv:1305.4487] [INSPIRE].
V. Baru, E. Epelbaum, A.A. Filin, C. Hanhart, U.-G. Meißner and A.V. Nefediev, Heavy-quark spin symmetry partners of the X(3872) revisited, Phys. Lett. B 763 (2016) 20 [arXiv:1605.09649] [INSPIRE].
Belle collaboration, Production of New Charmoniumlike States in e+e− → \( J/\psi {D}^{\left(\ast \right)}{\overline{D}}^{\left(\ast \right)} \) at \( \sqrt{s} \) ≃ 10.6 GeV, Phys. Rev. Lett. 100 (2008) 202001 [arXiv:0708.3812] [INSPIRE].
BaBar collaboration, Observation of the χc2(2p) Meson in the Reaction γγ → \( D\overline{D} \) at BaBar, Phys. Rev. D 81 (2010) 092003 [arXiv:1002.0281] [INSPIRE].
LHCb collaboration, Near-threshold \( D\overline{D} \) spectroscopy and observation of a new charmonium state, JHEP 07 (2019) 035 [arXiv:1903.12240] [INSPIRE].
L.R. Dai, J.-J. Xie and E. Oset, B0 → \( {D}^0{\overline{D}}^0{K}^0 \), B+ → \( {D}^0{\overline{D}}^0{K}^{+} \), and the scalar \( D\overline{D} \) bound state, Eur. Phys. J. C 76 (2016) 121 [arXiv:1512.04048] [INSPIRE].
L.R. Dai, G. Toledo and E. Oset, Searching for a \( D\overline{D} \) bound state with the ψ(3770) → \( \gamma {D}^0{\overline{D}}^0 \) decay, Eur. Phys. J. C 80 (2020) 510 [arXiv:2004.05204] [INSPIRE].
E. Wang, H.-S. Li, W.-H. Liang and E. Oset, Analysis of the γγ → \( D\overline{D} \) reaction and the \( D\overline{D} \) bound state, Phys. Rev. D 103 (2021) 054008 [arXiv:2010.15431] [INSPIRE].
C.W. Xiao and E. Oset, Three methods to detect the predicted \( D\overline{D} \) scalar meson X(3700), Eur. Phys. J. A 49 (2013) 52 [arXiv:1211.1862] [INSPIRE].
Belle collaboration, Observation of a \( {\chi}_{c2}^{\prime } \) candidate in γγ → \( D\overline{D} \) production at BELLE, Phys. Rev. Lett. 96 (2006) 082003 [hep-ex/0512035] [INSPIRE].
J.F. Giron and R.F. Lebed, Spectrum of the hidden-bottom and the hidden-charm-strange exotics in the dynamical diquark model, Phys. Rev. D 102 (2020) 014036 [arXiv:2005.07100] [INSPIRE].
W. Chen, H.-X. Chen, X. Liu, T.G. Steele and S.-L. Zhu, Mass spectra for \( qcq\overline{c}, sc\overline{s}\overline{c}, qb\overline{q}\overline{b}, sb\overline{s}\overline{b} \) tetraquark states with JPC = 0++ and 2++, Phys. Rev. D 96 (2017) 114017 [arXiv:1706.09731] [INSPIRE].
S. Heybrock et al., Lattice QCD with Domain Decomposition on Intel® Xeon Phi Co-Processors, in proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, New Orleans, LA, U.S.A., 16–21 November 2014, pp. 69–80 [arXiv:1412.2629] [INSPIRE].
S. Heybrock, M. Rottmann, P. Georg and T. Wettig, Adaptive algebraic multigrid on SIMD architectures, PoS LATTICE 2015 (2016) 036 [arXiv:1512.04506] [INSPIRE].
D. Richtmann, S. Heybrock and T. Wettig, Multiple right-hand-side setup for the DD-αAMG, PoS LATTICE 2015 (2016) 035 [arXiv:1601.03184] [INSPIRE].
P. Georg, D. Richtmann and T. Wettig, pMR: A high-performance communication library, PoS LATTICE 2016 (2017) 361 [arXiv:1701.08521] [INSPIRE].
SciDAC, LHPC and UKQCD collaborations, The Chroma software system for lattice QCD, Nucl. Phys. B Proc. Suppl. 140 (2005) 832 [hep-lat/0409003] [INSPIRE].
P. Arts et al., QPACE 2 and Domain Decomposition on the Intel Xeon Phi, PoS LATTICE 2014 (2015) 021 [arXiv:1502.04025] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2011.02542
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Prelovsek, S., Collins, S., Mohler, D. et al. Charmonium-like resonances with JPC = 0++, 2++ in coupled \( \mathrm{D}\overline{\mathrm{D}} \), \( {\mathrm{D}}_{\mathrm{s}}{\overline{\mathrm{D}}}_{\mathrm{s}} \) scattering on the lattice. J. High Energ. Phys. 2021, 35 (2021). https://doi.org/10.1007/JHEP06(2021)035
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2021)035