Abstract
We study the mass spectra of \( \overline{Q}Q\overline{Q}Q \) (Q = c, b) systems in QCD sum rules with the complete next-to-leading order (NLO) contribution to the perturbative QCD part of the correlation functions. Instead of meson-meson or diquark-antidiquark currents, we use diagonalized currents under operator renormalization. We find that differing from conventional mesons \( \overline{q}q \) and baryons qqq, a unique feature of the multiquark systems like \( \overline{Q}Q\overline{Q}Q \) is the operator mixing or color configuration mixing induced by NLO corrections, which is crucial to understand the color structure of the states. Our numerical results show that the NLO corrections are very important for the \( \overline{Q}Q\overline{Q}Q \) system, because they not only give significant contributions but also reduce the scheme and scale dependence and make Borel platform more distinct, especially for the \( \overline{b}b\overline{b}b \) in the \( \overline{\textrm{MS}} \) scheme. We use currents that have good perturbation convergence in our phenomenological analysis. With the \( \overline{\textrm{MS}} \) scheme, we get three JPC = 0++ states, with masses \( {6.35}_{-0.17}^{+0.20} \) GeV, \( {6.56}_{-0.20}^{+0.18} \) GeV and \( {6.95}_{-0.35}^{+0.21} \) GeV, respectively. The first two seem to agree with the broad structure around 6.2 ~ 6.8 GeV measured by the LHCb collaboration in the J/ψJ/ψ spectrum, and the third seems to agree with the narrow resonance X(6900). For the 2++ states we find one with mass \( {7.03}_{-0.26}^{+0.22} \) GeV, which is also close to that of X(6900), and another one around \( {7.25}_{-0.35}^{+0.21} \) GeV, which has good scale dependence but slightly large scheme dependence.
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References
Particle Data Group collaboration, Review of Particle Physics, Phys. Rev. D 98 (2018) 030001 [INSPIRE].
H.-X. Chen, W. Chen, X. Liu and S.-L. Zhu, The hidden-charm pentaquark and tetraquark states, Phys. Rept. 639 (2016) 1 [arXiv:1601.02092] [INSPIRE].
Y.-R. Liu, H.-X. Chen, W. Chen, X. Liu and S.-L. Zhu, Pentaquark and Tetraquark states, Prog. Part. Nucl. Phys. 107 (2019) 237 [arXiv:1903.11976] [INSPIRE].
N. Brambilla et al., The XYZ states: experimental and theoretical status and perspectives, Phys. Rept. 873 (2020) 1 [arXiv:1907.07583] [INSPIRE].
LHCb collaboration, Observation of structure in the J/ψ -pair mass spectrum, Sci. Bull. 65 (2020) 1983 [arXiv:2006.16957] [INSPIRE].
Y. Iwasaki, A Possible Model for New Resonances-Exotics and Hidden Charm, Prog. Theor. Phys. 54 (1975) 492 [INSPIRE].
K.-T. Chao, The (cc)-(\( \overline{c}c \)) (Diquark-Anti-Diquark) States in e+e− Annihilation, Z. Phys. C 7 (1981) 317 [INSPIRE].
J.P. Ader, J.M. Richard and P. Taxil, Do narrow heavy multi quark states exist?, Phys. Rev. D 25 (1982) 2370 [INSPIRE].
J.l. Ballot and J.M. Richard, Four quark states in additive potentials, Phys. Lett. B 123 (1983) 449 [INSPIRE].
L. Heller and J.A. Tjon, On Bound States of Heavy Q2\( \overline{Q} \)2 Systems, Phys. Rev. D 32 (1985) 755 [INSPIRE].
R.J. Lloyd and J.P. Vary, All charm tetraquarks, Phys. Rev. D 70 (2004) 014009 [hep-ph/0311179] [INSPIRE].
B. Silvestre-Brac, Systematics of Q2 (anti-Q2) systems with a chromomagnetic interaction, Phys. Rev. D 46 (1992) 2179 [INSPIRE].
B. Silvestre-Brac and C. Semay, Systematics of L = 0q2\( \overline{q} \)2 systems, Z. Phys. C 57 (1993) 273 [INSPIRE].
N. Barnea, J. Vijande and A. Valcarce, Four-quark spectroscopy within the hyperspherical formalism, Phys. Rev. D 73 (2006) 054004 [hep-ph/0604010] [INSPIRE].
M. Karliner, S. Nussinov and J.L. Rosner, \( QQ\overline{Q}\overline{Q} \) states: masses, production, and decays, Phys. Rev. D 95 (2017) 034011 [arXiv:1611.00348] [INSPIRE].
J. Wu, Y.-R. Liu, K. Chen, X. Liu and S.-L. Zhu, Heavy-flavored tetraquark states with the \( QQ\overline{Q}\overline{Q} \) configuration, Phys. Rev. D 97 (2018) 094015 [arXiv:1605.01134] [INSPIRE].
M.N. Anwar, J. Ferretti, F.-K. Guo, E. Santopinto and B.-S. Zou, Spectroscopy and decays of the fully-heavy tetraquarks, Eur. Phys. J. C 78 (2018) 647 [arXiv:1710.02540] [INSPIRE].
J.-M. Richard, A. Valcarce and J. Vijande, String dynamics and metastability of all-heavy tetraquarks, Phys. Rev. D 95 (2017) 054019 [arXiv:1703.00783] [INSPIRE].
V.R. Debastiani and F.S. Navarra, A non-relativistic model for the [cc][\( \overline{c}\overline{c} \)] tetraquark, Chin. Phys. C 43 (2019) 013105 [arXiv:1706.07553] [INSPIRE].
M.-S. Liu, Q.-F. Lü, X.-H. Zhong and Q. Zhao, All-heavy tetraquarks, Phys. Rev. D 100 (2019) 016006 [arXiv:1901.02564] [INSPIRE].
X. Jin, Y. Xue, H. Huang and J. Ping, Full-heavy tetraquarks in constituent quark models, Eur. Phys. J. C 80 (2020) 1083 [arXiv:2006.13745] [INSPIRE].
G.-J. Wang, L. Meng, M. Oka and S.-L. Zhu, Higher fully charmed tetraquarks: Radial excitations and P-wave states, Phys. Rev. D 104 (2021) 036016 [arXiv:2105.13109] [INSPIRE].
W. Chen, H.-X. Chen, X. Liu, T.G. Steele and S.-L. Zhu, Hunting for exotic doubly hidden-charm/bottom tetraquark states, Phys. Lett. B 773 (2017) 247 [arXiv:1605.01647] [INSPIRE].
Z.-G. Wang, Analysis of the \( QQ\overline{Q}\overline{Q} \) tetraquark states with QCD sum rules, Eur. Phys. J. C 77 (2017) 432 [arXiv:1701.04285] [INSPIRE].
Z.-G. Wang, Tetraquark candidates in the LHCb’s di-J/ψ mass spectrum, Chin. Phys. C 44 (2020) 113106 [arXiv:2006.13028] [INSPIRE].
Z.-G. Wang and Z.-Y. Di, Analysis of the vector and axialvector \( QQ\overline{Q}\overline{Q} \) tetraquark states with QCD sum rules, Acta Phys. Polon. B 50 (2019) 1335 [arXiv:1807.08520] [INSPIRE].
R.M. Albuquerque, S. Narison, A. Rabemananjara, D. Rabetiarivony and G. Randriamanatrika, Doubly-hidden scalar heavy molecules and tetraquarks states from QCD at NLO, Phys. Rev. D 102 (2020) 094001 [arXiv:2008.01569] [INSPIRE].
B.-C. Yang, L. Tang and C.-F. Qiao, Scalar fully-heavy tetraquark states QQ′\( \overline{Q}\overline{Q} \)′ in QCD sum rules, Eur. Phys. J. C 81 (2021) 324 [arXiv:2012.04463] [INSPIRE].
J.-R. Zhang, 0+ fully-charmed tetraquark states, Phys. Rev. D 103 (2021) 014018 [arXiv:2010.07719] [INSPIRE].
W. Heupel, G. Eichmann and C.S. Fischer, Tetraquark Bound States in a Bethe-Salpeter Approach, Phys. Lett. B 718 (2012) 545 [arXiv:1206.5129] [INSPIRE].
Z.-H. Guo and J.A. Oller, Insights into the inner structures of the fully charmed tetraquark state X(6900), Phys. Rev. D 103 (2021) 034024 [arXiv:2011.00978] [INSPIRE].
X.-K. Dong, V. Baru, F.-K. Guo, C. Hanhart and A. Nefediev, Coupled-Channel Interpretation of the LHCb Double-J/ψ Spectrum and Hints of a New State Near the J/ψJ/ψ Threshold, Phys. Rev. Lett. 126 (2021) 132001 [Erratum ibid. 127 (2021) 119901] [arXiv:2009.07795] [INSPIRE].
R. Tiwari, D.P. Rathaud and A.K. Rai, Spectroscopy of all charm tetraquark states, arXiv:2108.04017 [INSPIRE].
C. Hughes, E. Eichten and C.T.H. Davies, Searching for beauty-fully bound tetraquarks using lattice nonrelativistic QCD, Phys. Rev. D 97 (2018) 054505 [arXiv:1710.03236] [INSPIRE].
H.-W. Ke, X. Han, X.-H. Liu and Y.-L. Shi, Tetraquark state X(6900) and the interaction between diquark and antidiquark, Eur. Phys. J. C 81 (2021) 427 [arXiv:2103.13140] [INSPIRE].
Z. Zhao, K. Xu, A. Kaewsnod, X. Liu, A. Limphirat and Y. Yan, Study of charmoniumlike and fully-charm tetraquark spectroscopy, Phys. Rev. D 103 (2021) 116027 [arXiv:2012.15554] [INSPIRE].
A.V. Berezhnoy, A.V. Luchinsky and A.A. Novoselov, Tetraquarks Composed of 4 Heavy Quarks, Phys. Rev. D 86 (2012) 034004 [arXiv:1111.1867] [INSPIRE].
Y. Bai, S. Lu and J. Osborne, Beauty-full Tetraquarks, Phys. Lett. B 798 (2019) 134930 [arXiv:1612.00012] [INSPIRE].
M. Karliner, J.L. Rosner and T. Skwarnicki, Multiquark States, Ann. Rev. Nucl. Part. Sci. 68 (2018) 17 [arXiv:1711.10626] [INSPIRE].
A. Esposito and A.D. Polosa, A \( bb\overline{b}\overline{b} \) di-bottomonium at the LHC?, Eur. Phys. J. C 78 (2018) 782 [arXiv:1807.06040] [INSPIRE].
M.A. Bedolla, J. Ferretti, C.D. Roberts and E. Santopinto, Spectrum of fully-heavy tetraquarks from a diquark+antidiquark perspective, Eur. Phys. J. C 80 (2020) 1004 [arXiv:1911.00960] [INSPIRE].
P. Lundhammar and T. Ohlsson, Nonrelativistic model of tetraquarks and predictions for their masses from fits to charmed and bottom meson data, Phys. Rev. D 102 (2020) 054018 [arXiv:2006.09393] [INSPIRE].
R. Zhu, Fully-heavy tetraquark spectra and production at hadron colliders, Nucl. Phys. B 966 (2021) 115393 [arXiv:2010.09082] [INSPIRE].
M.-S. liu, F.-X. Liu, X.-H. Zhong and Q. Zhao, Full-heavy tetraquark states and their evidences in the LHCb di-J/ψ spectrum, arXiv:2006.11952 [INSPIRE].
Q.-F. Lü, D.-Y. Chen and Y.-B. Dong, Masses of fully heavy tetraquarks \( QQ\overline{Q}\overline{Q} \) in an extended relativized quark model, Eur. Phys. J. C 80 (2020) 871 [arXiv:2006.14445] [INSPIRE].
J.F. Giron and R.F. Lebed, Simple spectrum of \( c\overline{c}c\overline{c} \) states in the dynamical diquark model, Phys. Rev. D 102 (2020) 074003 [arXiv:2008.01631] [INSPIRE].
G. Huang, J. Zhao and P. Zhuang, Pair structure of heavy tetraquark systems, Phys. Rev. D 103 (2021) 054014 [arXiv:2012.14845] [INSPIRE].
R.N. Faustov, V.O. Galkin and E.M. Savchenko, Heavy tetraquarks in the relativistic quark model, Universe 7 (2021) 94 [arXiv:2103.01763] [INSPIRE].
Q. Li, C.-H. Chang, G.-L. Wang and T. Wang, Mass spectra and wave functions of TQQQ−Q− tetraquarks, Phys. Rev. D 104 (2021) 014018 [arXiv:2104.12372] [INSPIRE].
J. Sonnenschein and D. Weissman, Deciphering the recently discovered tetraquark candidates around 6.9 GeV, Eur. Phys. J. C 81 (2021) 25 [arXiv:2008.01095] [INSPIRE].
B.-D. Wan and C.-F. Qiao, Gluonic tetracharm configuration of X(6900), Phys. Lett. B 817 (2021) 136339 [arXiv:2012.00454] [INSPIRE].
Q.-F. Cao, H. Chen, H.-R. Qi and H.-Q. Zheng, Some remarks on X(6900), Chin. Phys. C 45 (2021) 103102 [arXiv:2011.04347] [INSPIRE].
M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, QCD and Resonance Physics. Theoretical Foundations, Nucl. Phys. B 147 (1979) 385 [INSPIRE].
M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, QCD and Resonance Physics: Applications, Nucl. Phys. B 147 (1979) 448 [INSPIRE].
P. Colangelo and A. Khodjamirian, QCD sum rules, a modern perspective, hep-ph/0010175 [INSPIRE].
S. Narison, SVZ sum rules : 30 + 1 years later, Nucl. Phys. B Proc. Suppl. 207–208 (2010) 315 [arXiv:1010.1959] [INSPIRE].
S. Narison, Mini-review on QCD spectral sum rules, Nucl. Part. Phys. Proc. 258–259 (2015) 189 [arXiv:1409.8148] [INSPIRE].
R.M. Albuquerque et al., QCD sum rules approach to the X, Y and Z states, J. Phys. G 46 (2019) 093002 [arXiv:1812.08207] [INSPIRE].
A.A. Ovchinnikov, A.A. Pivovarov and L.R. Surguladze, Baryonic sum rules in the next-to-leading order in αs, Int. J. Mod. Phys. A 6 (1991) 2025 [INSPIRE].
S. Groote, J.G. Korner and A.A. Pivovarov, Next-to-Leading Order perturbative QCD corrections to baryon correlators in matter, Phys. Rev. D 78 (2008) 034039 [arXiv:0805.3590] [INSPIRE].
S. Groote, J.G. Korner and A.A. Pivovarov, Heavy baryon properties with NLO accuracy in perturbative QCD, Eur. Phys. J. C 58 (2008) 355 [arXiv:0807.2148] [INSPIRE].
C.-Y. Wang, C. Meng, Y.-Q. Ma and K.-T. Chao, NLO effects for doubly heavy baryons in QCD sum rules, Phys. Rev. D 99 (2019) 014018 [arXiv:1708.04563] [INSPIRE].
R.-H. Wu, Y.-S. Zuo, C. Meng, Y.-Q. Ma and K.-T. Chao, NLO effects for Ω QQQ baryons in QCD Sum Rules, Chin. Phys. C 45 (2021) 093103 [arXiv:2104.07384] [INSPIRE].
J. Kublbeck, M. Böhm and A. Denner, Feyn Arts: Computer Algebraic Generation of Feynman Graphs and Amplitudes, Comput. Phys. Commun. 60 (1990) 165 [INSPIRE].
T. Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3, Comput. Phys. Commun. 140 (2001) 418 [hep-ph/0012260] [INSPIRE].
R. Mertig, M. Böhm and A. Denner, FEYN CALC: Computer algebraic calculation of Feynman amplitudes, Comput. Phys. Commun. 64 (1991) 345 [INSPIRE].
V. Shtabovenko, R. Mertig and F. Orellana, New Developments in FeynCalc 9.0, Comput. Phys. Commun. 207 (2016) 432 [arXiv:1601.01167] [INSPIRE].
J.G. Korner, D. Kreimer and K. Schilcher, A Practicable γ5-scheme in dimensional regularization, Z. Phys. C 54 (1992) 503 [INSPIRE].
A. von Manteuffel and C. Studerus, Reduze 2 — Distributed Feynman Integral Reduction, arXiv:1201.4330 [INSPIRE].
A.V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated one loop integrals, Phys. Lett. B 302 (1993) 299 [Erratum ibid. 318 (1993) 649] [hep-ph/9212308] [INSPIRE].
E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].
T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
X. Liu and Y.-Q. Ma, AMFlow: a Mathematica package for Feynman integrals computation via Auxiliary Mass Flow, arXiv:2201.11669 [InSPIRE].
X. Liu, Y.-Q. Ma and C.-Y. Wang, A Systematic and Efficient Method to Compute Multi-loop Master Integrals, Phys. Lett. B 779 (2018) 353 [arXiv:1711.09572] [INSPIRE].
E. Bagan, M. Chabab and S. Narison, Baryons with two heavy quarks from QCD spectral sum rules, Phys. Lett. B 306 (1993) 350 [INSPIRE].
C.A. Dominguez, G.R. Gluckman and N. Paver, Mass of the charm quark from QCD sum rules, Phys. Lett. B 333 (1994) 184 [hep-ph/9406329] [INSPIRE].
C.A. Dominguez, L.A. Hernandez and K. Schilcher, Determination of the gluon condensate from data in the charm-quark region, JHEP 07 (2015) 110 [arXiv:1411.4500] [INSPIRE].
S. Aoki et al., Review of lattice results concerning low-energy particle physics, Eur. Phys. J. C 77 (2017) 112 [arXiv:1607.00299] [INSPIRE].
R.A. Bertlmann, Heavy Quark-Anti-quark Systems From Exponential Moments in QCD, Nucl. Phys. B 204 (1982) 387 [INSPIRE].
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Wu, RH., Zuo, YS., Wang, CY. et al. NLO results with operator mixing for fully heavy tetraquarks in QCD sum rules. J. High Energ. Phys. 2022, 23 (2022). https://doi.org/10.1007/JHEP11(2022)023
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DOI: https://doi.org/10.1007/JHEP11(2022)023