Eur. Phys. J. C (2020) 80:121 https://doi.org/10.1140/epjc/s10052-020-7701-8

Regular Article - Theoretical Physics

Hidden-bottom and -charm hexaquark states in QCD sum rules

Bing-Dong Wan1,a , Liang Tang2,b, Cong-Feng Qiao1,3,c 1 School of Physics, University of Chinese Academy of Science, Yuquan Road 19A, Beijing 10049, China 2 College of Physics, Hebei Normal University, Shijiazhuang 050024, China 3 CAS Center for Excellence in Physics, Beijing 10049, China

Received: 2 January 2020 / Accepted: 30 January 2020 / Published online: 14 February 2020 © The Author(s) 2020

Abstract In this paper, we investigate the spectra of the and theoretical progresses can be found in comprehensive prospective hidden-bottom and -charm hexaquark states with reviews like [10Ð14]. quantum numbers J PC = 0++,0−+,1++ and 1−− in the Facing the observations of and framework of QCD sum rules. By constructing appropriate states, it is nature to conjecture that there should exist the interpreting currents, the QCD sum rules analyses are per- hidden-charm and -bottom hexaquark states, and it is time formed up to dimension 12 of the condensates. Results indi- to hunt for them. For the hexaquark states, the deuteron is cate that there exist two possible baryonium states in b- a typical and well-established dibaryon molecular state with P + sector with masses 11.84 ± 0.22 GeV and 11.72 ± 0.26 GeV J = 1 and binding energy EB = 2.225 MeV [15Ð17]. for 0++ and 1−−, respectively. The corresponding hidden- The baryonium states composed of a and an anti- charm partners are found lying respectively at 5.19 ± 0.24 baryon is another special class of heaxquark configuration. In ¯ GeV and 4.78 ± 0.23 GeV. Note that these baryonium Refs. [18,19], the c-c structure was introduced to explain states are all above the dibaryon thresholds, which enables the production and decays of Y (4260). And the heavy bary- their dominant decay modes could be measured at BESIII, was explored as well in the heavy baryon chiral per- BELLEII, and LHCb detectors. turbation theory [20,21]. The method of QCD sum rules [22Ð26] has been applied successfully to many hadronic phenoemena, such as the 1 Introduction spectrum and hadron decays. In QCD sum rules, based on the proper interpreting currents corresponding to a with more than the minimal quark content (qq¯ or hadron of interest, one can construct the two-point or three- qqq) was proposed by Gell-Mann [1] and Zweig [2] in 1964, point correlation functions, which are respectively used to which were named as multiquark exotic states and do not evaluate the mass and the decay property of the related infringe the Quantum Chromodynamics (QCD). Exploring hadron. Then by matching its operator product expansion the existence and properties of such exotic states is one of the (OPE) to its hadronic saturation, the main function for most intriguing research topics of hadronic physics. In past extracting the mass or decay rate of the hadron is estab- few decades, research on the heavy-flavor exotic states has lished. Utilizing this approach, several significant researches made tremendous developments, that is, many charmonium- for the hexaquark states have been done [27Ð31]. In Ref. [27], like/bottomonium-like XYZ states have been observed [3Ð six-quark state d∗(2380) was considering as - structure 7]. In 2015, two hidden-charm Pc(4380) and and the corresponding mass was given. Chen et al. gave an Pc(4450) [8] were observed by the LHCb Collaboration in explicit QCD sum rule investigation for hidden-charm bary- /ψ 0 → /ψ − the J invariant mass spectrum via the b J pK onium states with various relevant local interpreting currents, process. Recently, the LHCb Collaboration reported a new and they found some of these currents can couple to hidden- narrow state Pc(4312), and the previously observed struc- charm baryonium states with the masses around 5.0 GeV ture Pc(4450) appears to be split into two narrower struc- [28]. Reference [29] calculated the mass and coupling con- tures Pc(4440) and Pc(4457) [9]. The recent experimental stant of the scalar hexaquark uuddss. Very recently, the  bound system spectrum with J P = 0+ and 2+ in a molec- a e-mail: [email protected] ular picture were investigated in Ref. [30], and the results b e-mail: [email protected] suggest the existence of two bound  dibaryon states. c e-mail: [email protected] (corresponding author) 123 121 Page 2 of 10 Eur. Phys. J. C (2020) 80 :121

1−− ¯ Wang [31] studied the scalar-diquarkÐscalar-diquarkÐscalar- jμ (x) = abc def [Qd (x)γμ Qc(x)] type hexaquark state with the QCD sum rules, where T   T ×[q (x)Cγ5q (x)][q ¯ (x)γ5Cq¯ (x)] . (7) the three diquarks were arranged as ud, uc, and dc, respec- a b e f tively. Here, a ... f denote color indices, C is the charge conjuga- Besides the above mentioned hexaquark configurations, tion matrix, Q represents the heavy , q stands for the  the QCD theory allows many other possible hexaquark struc- u, and q for the d. tures, which can couple to hidden-charm baryonium states. The correlation function derived from Eqs. (6) and (7) can Moreover, it should be noted that exploring the hidden- be expressed as the following Lorentz covariance form   bottom baryonium states is also significant, which is the main qμqν 2 qμqν 2 μν =− gμν −  (q ) +  (q ), (8) motivation of this work, and they tend to be measurable in the q2 1 q2 0 LHCb experiment. We investigate the hidden-bottom molec- ¯ where the subscripts 1 and 0 denote the quantum numbers ular states in Q-Q configuration with quantum numbers J PC = 0++,0−+,1++ and 1−− in the framework of QCD of the spin 1 and 0 , respectively. However, since the  ( 2) sum rules. Their decay properties, as well as their hidden- leading term 1 q is symmetrical and only contains the  ( 2) charm partners, are also analyzed. spin 1 component, we shall focus on calculating the 1 q The rest of the paper is arranged as follows. After the intro- and employ it to perform the QCD sum rule analyses. duction, some primary formulas of the QCD sum rules in our On the phenomenological side, adopting the usual pole calculation are presented in Sect. 2. The numerical analysis plus continuum parametrization of the hadronic the spectral ( 2) and results are given in Sect. 3. In Sect. 4, possible decay density, we express the correlation function q as  modes of hidden-bottom baryonium states are investigated. (λX )2 1 ∞ ρ(s) PHEN(q2) = + ds , (9) The last part is left for conclusions and discussion. ( X )2 − 2 π − 2 M q s0 s q ¯ where the superscript X denotes the lowest lying Q-Q X ρ( ) 2 Formalism hexaquark state, M is its mass, and s is the spectral density that contains the contributions from higher excited The starting point of the QCD sum rules is the two-point states and the continuum states above the threshold s0.The λX  | | =λX correlation function constructed from two hadronic currents decay constant is defined through 0 j X and  | | =λX with the following form: 0 jμ X μ.  On the OPE side, the correlation function (q2) can be ( ) = 4 iq·x  | { ( ) †( )}| ; written as a dispersion relation form: q i d xe 0 T j x j 0 0 (1)   ∞ ρ OPE( ) OPE 2 s 4 iq·x †  (q ) = ds , (10) μν(q) = i d xe 0|T { jμ(x) jν (0)}|0 , (2) ( + + )2 s − q2 2m Q 2mq 2mq ρ OPE( ) = [OPE( )]/π where, j(x) and jμ(x) are the relevant hadronic currents with where s Im s is the spectral density of J = 0 and 1, respectively. the OPE side, and contains the contributins of the condensates η up to dimension 12, thus We use the notion Q to represent the Dirac baryon fields  2 of Q without free Lorentz indices. It was shown in Ref. [32] ρ OPE(s) = ρ pert(s) + ρ¯qq(s) + ρG (s) η that, Q may take the following quark structure: 2 3 +ρ¯qGq(s) + ρ¯qq (s) + ρG (s) T  η ( ) = [ ( ) γ ( )] ( ), ¯qqqGq ¯  ¯qq2G2 Q x i abc qa x C 5qb x Qc x (3) +ρ (s) + ρ (s) ¯ 2 ¯ 4 where Q = b, c. Therefore, the interpolating currents for +ρ qGq (s) + ρ qq (s). (11) ¯ ++ Q-Q baryounium states with quantum numbers 0 , −+ ++ −− In order to calculate the spectral density of the OPE side, 0 ,1 , and 1 can be respectively constructed as q ( ) Q( ) Eq. (11), the full propagators Sij x and Sij p of a light 0++ ¯ quark (q = u, d or s) and a heavy quark (Q = c or b)are j (x) = abc def [Qd (x)Qc(x)] used: ×[ T ( ) γ  ( )][ ¯ ( )γ ¯ T ( )] , qa x C 5qb x qe x 5Cq f x (4) a a iδ jkx/ δ jkmq itjkGαβ αβ αβ −+ Sq (x) = − − (σ x/ + x/σ ) 0 ¯ jk π 2 4 π 2 2 π 2 2 j (x) = abc def [Qd (x)γ5 Qc(x)] 2 x 4 x 32 x δ δ / δ 2 T   T jk i jkx jkx ×[q (x)Cγ q (x)][q ¯ (x)γ Cq¯ (x)] , (5) − ¯qq+ mq ¯qq− gs q¯σ · Gq a 5 b e 5 f 12 48 192 ++ 1 δ 2 / ta σαβ ( ) = [ ¯ ( )γμγ ( )] i jkx x jk jμ x abc def Qd x 5 Qc x + mq gs q¯σ · Gq− gs q¯σ · Gq   1152 192 ×[ T ( ) γ ( )][ ¯ ( )γ ¯ T ( )] , a qa x C 5qb x qe x 5Cq f x (6) itjk + (σαβ x/ + x/σαβ )mq gs q¯σ · Gq , (12) 768 123 Eur. Phys. J. C (2020) 80 :121 Page 3 of 10 121

Fig. 1 The typical Feynman diagrams related to the (q2) function, where the thick solid line represents the heavy quark, the thin solid line stands for the light quark, and the spiral line denotes the . There is no heavy quark consendsate due to the large heavy quark mass

a a iδ (p/ + m ) i t Gαβ S Q (p) = jk Q − jk 3 Numerical analysis jk 2 − 2 ( 2 − 2 )2 p m Q 4 p m Q αβ αβ ×[σ (p/ + m Q ) + (p/ + m Q )σ ]   In the numerical calculation of QCD sum rules, the values of 2 2 iδ jkm Q g G  m (p/ + m ) + s 1 + Q Q input parameters we take are [35Ð37] ( 2 − 2 )3 2 − 2 12 p m Q p m Q   2 2 2 2 m (m ) = (1.23 ± 0.05) GeV , iδ (p/ + m Q )[p/(p − 3m ) + 2m Q (2p − m )](p/ + m Q ) c c + jk Q Q ( 2 − 2 )6 ( ) = ( . ± . ) , 48 p m Q mb mb 4 24 0 06 GeV × 3 3 , 3 3 gs G (13) ¯qq=−(0.23 ± 0.03) GeV , g2G2=0.88 GeV4 , (17) where, the vacuum condensates are clearly displayed. For s ¯ σ · = 2¯  , more explanation on above propagators, readers may refer to qgs Gq m0 qq  3 3= . 6 , Refs. [33,34]. The Feynman diagrams corresponding to each gs G 0 045 GeV term of Eq. (10) are schematically shown in Fig. (1). m2 = 0.8GeV2 , Applying Borel transform on both Eqs. (9) and (10), using 0 quark-hadron duality, and matching the OPE side with the in which the MS running heavy quark masses are adopted. ( 2) phenomenological side of the correlation function q , For light quarks u and d, we use the chiral limit in our analysis we can finally reach the main function to extract the mass of in which their masses are mu = md = 0. the hexaquark state, which reads The QCD sum rules in Eq. (14) is a function of the Borel 2 parameter M and the continuum threshold s0. To obtain a 2 B L1(s0, M ) reliable mass sum rules result, one should choose suitable M X (s , M2 ) = − B . (14) 0 B ( , 2 ) L0 s0 MB working ranges for these two parameters. As widely used, we employ two criteria to fix the suitable working regions of Here L and L are respectively defined as 2 1 0 MB and s0 [22Ð24,26]. The first one asks for the convergence  s of the OPE, which is to compare the relative contribution of 0 − / 2 ( , 2 ) = ρ OPE( ) s MB each term to the total contribution on the OPE side. The other L0 s0 MB ds s e (15) ( + + )2 2mc 2mq 2mq criterion of QCD sum rules is the pole contribution (PC). As discussed in Refs. [27,29,31], since the large power of s in and the spectral density suppress the PC value, we choose the ∂ pole contribution larger than 15% for hexaquark states. L (s , M2 ) = L (s , M2 ). (16) 1 0 B ∂ 1 0 0 B In order to determine a proper value for s0, we carry 2 MB out a similar analysis in Refs. [38,39]. Since the contin- 123 121 Page 4 of 10 Eur. Phys. J. C (2020) 80 :121

++ 2 Fig. 2 Figures for 0 hidden-bottom and -charm hexaquark molec- tion of the Borel parameter MB . d The OPE convergence of hidden- ular states. a The OPE convergence of hidden-bottom state as function 2 charm state as the function of the√ Borel parameter MB in the region of the Borel parameter M2 in the region 7.0 ≤ M2 ≤ 13.0GeV2 2 2 √ B B 3.0 ≤ M ≤ 5.0GeV with s0 = 5.7GeV.e The pole con- = .  3 ¯ 2 2 B with s0 12 5GeV.The G and qq G terms contribu- tribution of hidden-charm√ state as the function of the Borel param- tions are not displayed, since their magnitudes are tiny, and ¯qq and 2 = . eter MB with s0 5 7GeV.f The mass of hidden-charm hex- ¯   −¯ qGq are 0. b The pole contribution of hidden-bottom√ state as the c c 2 = . aquark molecular state M0++ as the function of the Borel parameter function of the Borel parameter MB with s0 12 5GeV.c The 2 ¯ M b−b B mass of hidden-bottom hexaquark molecular state M0++ as the func- uum√ threshold s0 relates to the mass of the ground state by out the one which yields an optimal window for Borel param- ∼ X + . X 2 s0 M 0 5GeV[26,40√ ], where M denotes the mass eter MB. That is, within the optimal window, the hexaquark X of the ground state, various s0 satisfying this constraint are mass M is somehow independent of the Borel parameter 2 taken into account. Among these values, one needs to select MB as much as possible. In practice, in QCD sum rules cal-

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−− √ Fig. 3 The same caption as in Fig. 2,butfor1 hidden-bottom and -charm hexaquark molecular states, where continuum thresholds s0 are taken as 12.20, 12.40 and 12.60 GeV in c for the hidden-bottom case, and as 5.10, 5.30, and 5.50 GeV in f for the hidden-charm case, from down to up √ > 2 culation, we may vary s0 by 0.2 GeV,which√ gives the lower which indicates for PC 15%√ the upper constraint on MB 2  . 2 = . and upper bounds hence the uncertainties√ of s0. is MB 11 4GeV with s0 12 5 GeV. The masses  −¯ After scanning the values of s , we can obtain the opti- b b 2 √ 0 M ++ as functions of the Borel parameter M are drawn 2 0 B mal s0 together with the suitable window of MB. Quan- ++ in Fig. 2c, where the√ center curve corresponds to the optimal titatively, the OPE convergence of 0 hidden-bottom hex- threshold parameter s0 and the√ upper (lower)√ curve is drawn aquark state is shown in Fig. (2a. According to the first crite- = . to√ display the uncertainty on s0 with s0 12 7GeV rion, we find a strong OPE convergence for M2  9.1GeV2 √ B ( s0 = 12.3 GeV), respectively. = . ++ with s0 12 5 GeV, and then we fix the lower working In the sector, the 0 hidden-charm hex- 2 limit for MB. The curve of the PC is illustrated in Fig. 2b, aquark state may be analyzed similarly, with heavy quark

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= mass m Q mc. The OPE convergence is√ drawn in Fig. Table 1 Typical decay modes of the heavy baryomium for each quan- 2  . 2 = . tum number, where X denotes the baryonium, respectively 2d, where we find MB 3 5GeV with s0 5 7GeV is satisfied under the first criterion. On the other hand, the J PC c-sector b-sector PC is drawn in Fig. 2e, which determines the upper bound √ ++ →  ¯ →  ¯ 2 2  . 2 = . 0 X c c X b b of MB, that is, MB 4 4GeV with s0 5 7 GeV (for ¯ −− →  ¯ →  ¯ c−c 1 X c c X b b PC > 15%). Finally, the masses M ++ as function of the √ 0 → π +π − /ψ → π +π −ϒ 2 = . , . , . X J X Borel parameter MB for s0 5 9 5 7 5 5 GeV are drawn in Fig. 2f, where the upper and lower√ curves correspond to the uncertainties stemming from s0. ++ ¯ decade has passed, the signal for heavy baryonium is still Finally, the mass spectra of the 0 Q-Q hexaquark molecular states are determined to be unclear. To finally ascertain these hidden charm and bottom baryonium, the straightforward procedure is to reconstruct ¯ b−b ++ = ( . ± . ) , them from its decay products, though the detailed characters M0 11 84 0 22 GeV (18) ¯ c−c of them still ask for more exploration. In Ref. [18] the hid- M ++ = (5.19 ± 0.24) GeV . (19) + − 0 den charm baryonium to π π J/ψ decay was thought to be an easy-to-go process, where baryonium mass is below where the errors stems from the uncertainties of the√ quark the threshold, while the DD¯ exclusive process is almost masses, the condensates and the threshold parameter s0. The OPE convergence of 1−− hidden-bottom hexaquark impossible and the final states with should be state is shown in Fig. 3a. Here, we find a strong OPE con- suppressed. According this analysis, the heavy baryonia are √  ¯ vergence for M2  8.9GeV2 with s = 12.4 GeV, which above the c c threshold, and in this case the open-charm B 0  ¯ consist in a good criterion for fixing the value of the lower decay process of c c would be another potential choice. limit of M2 . The curve of the PC is shown in Fig. 3b, which Analogously, the primary decay mode of b-sector heavy bary- B ¯ + − 2 2  . 2 onium should be to bb process, as well the π π ϒ.The determines the upper bound for MB, i.e., MB 11 9GeV √ ¯ b−b typical decay modes of the heavy baryonium for different with s = 12.4. GeV. The masses M −− as functions of 0 √1 quantum numbers are given in Table 1, and these processes M2 s the Borel parameter B for different 0 are drawn in Fig. are expected to be measurable in the running experiments 3c. like the BES III and LHCb. For the 1−− hidden-charm hexaquark state, the OPE con- vergence is drawn in Fig. 3d. According to the first crite- rion, we find the lower limit of M2 , i.e., M2  3.3GeV2 √ B B 5 Conclusions with s0 = 5.3 GeV. The PC is drawn in Fig. 3e, which 2 2  . 2 fix the upper bound for MB, i.e., MB 4 1GeV with √ ¯ In summary, we investigate in this work the hidden-bottom c−c s = 5.3 GeV. The masses M −− as functions of the 0 1√ and -charm hexaquark structures in molecular configuration 2 PC = ++ −+ ++ −− Borel parameter MB for different s0 are drawn in Fig. 3f. with quantum numbers J 0 ,0 ,1 and 1 , −− ¯ Eventually, the masses of the 1 QÐQ hexaquark in the framework of QCD sum rules. After constructing molecular states are determined to be the appropriate interpreting currents, we perform the QCD ¯ b−b sum rules analysis, where the vacuum condensates are con- M −− = (11.72 ± 0.26) GeV , (20) 1 sidered up to dimension 12. Our results indicate that there  −¯ ¯ c c = ( . ± . ) . exist two possible b-b-like baryonium states with masses M1−− 4 78 0 23 GeV (21) 11.84 ± 0.22 GeV and 11.72 ± 0.26 GeV for 0++ and 1−−, Here the errors stems from the uncertainties of the√ quark respectively. The masses of their hidden-charm partners are masses, the condensates and the threshold parameter s0. found to be 5.19 ± 0.24 GeV and 4.78 ± 0.23 GeV, respec- −+ ++ −+ ++ We also analyze the situations of 0 √ and 1 , and find tively. It is found the 0 and 1 currents do not yield 2 that no what values of M and s0 take, no optimal 2 B hadronic√ structures in any reasonable magnitudes of MB and window for stable plateaus exist. That means the currents in s0, say no optimal window for stable plateaus exist. Results Eqs. (5) and (6) do not support the corresponding hexaquark indicate that the Y (4660) reported by Belle Collaboration ¯ molecular states. [41,42] is close in magnitude to our calculation of c-c baryonium state. Moreover, the primary and potential decay modes of baryonium are analyzed, which might serve as a 4 Decay analyses guide for experimental exploration. It should be noted that the hidden-charm baryonium was In Refs. [18,19], the heavy baryonium was introduced to once investigated by Chen, et al. [28]. Analytically we can explain the production and decays of Y (4260), however, a find an agreement with them, however, there exist two main 123 Eur. Phys. J. C (2020) 80 :121 Page 7 of 10 121

 4 2 2 differences between these two analyses. One is the threshold Fαβ (−1 + α + β) 3Fαβ − 5m (−1 + α + β) × Q truncation of non-perturbative condensite, in our calculation 3 × 5 × 219α4β4 the higher dimensional condensate ¯qq2G2 is taken into F 3 m2 (−1 + α + β)4 account, whereas neglected in [28], which turns to be non- + αβ Q −+ ++ 5 × 32 × 221α6β6 negligible in 0 and 1 states analyses. Another differ- × −4m2 α4 + α3(−1 + β) + αβ 3 + (−1 + β)β3 ence lies in the choice of continuum threshold s0 and Borel Q  2 paramter M . In the end the day, we find hidden charm bary- 3 2 2 2 B +Fαβ 2α − 3α (−1 + β) − 3αβ + β (3 + 2β)) , onium states 0++ and 1−− might exist, however according to [28] the existing ones with quantum numbers 0++ and 1++. (A2)   2 α 1−α It it interesting to note that according to QCD sum rule ¯ 2 ¯qq max −− ρ qq (s) = dα dβ analysis the 1 hidden charm baryonium state lies above 3 × 210π 6 α β  min min  the open charm threshold, whereas it might be below the F 4 (1 − α − β) 2F 3 m2 (1 − α − β)2 × − αβ − αβ Q , threshold in heavy quark chiral theory, which deserves further α3β3 α3β3 investigations. Moreover, we notice that our result about the −−  −¯ (A3) 1 c c baryonium state is in consistent with the large-  α  3 max G3 G Nc QCD result of Ref. [43]. ρ (s) = dα 5 × 32 × 222π 10 α  min 1−α Acknowledgements This work was supported in part by the Min- 2 4 × dβFαβ (α + β − 1) istry of Science and Technology of the Peoples’ Republic of China β (2015CB856703); by the National Natural Science Foundation of min 5F 2 + 8Fαβ m2 (2α − 3β + 3) − 24αm4 (α + β − 1) China (NSFC) under the Grants 11975236, 11635009, 11375200 and × αβ Q Q , 11605039; by the Natural Science Foundation of Hebei Province with α3β6 contract No. A2017205124, and by the Science Foundation of Hebei (A4) Normal University under Contract No. L2016B08.  α  −α ¯qqqGq ¯  max 1 ρ¯qqqGq ¯ (s) = dα dβ Data Availability Statement This manuscript has no associated data × 9π 6 3 2 αmin βmin or the data will not be deposited. [Authors’ comment: All data generated 3 2 2 Fαβ − 3m Fαβ (−1 + α + β) or analysed during this study are included in this published article.] − Q , (A5) α2β2 Open Access This article is licensed under a Creative Commons Attri-  bution 4.0 International License, which permits use, sharing, adaptation, 2 α ¯ 2 ¯qGq max distribution and reproduction in any medium or format, as long as you ρ qGq = dα 11π 6 α give appropriate credit to the original author(s) and the source, pro- 2 min    vide a link to the Creative Commons licence, and indicate if changes 2 1−α F 2 Hα αβ m Q were made. The images or other third party material in this article × − − dβ , (A6) 2(1 − α)α β αβ are included in the article’s Creative Commons licence, unless indi- min  α cated otherwise in a credit line to the material. If material is not ¯ 2 2 max ρ¯qq2G2 = qq G α included in the article’s Creative Commons licence and your intended d 3 × 211π 6 α use is not permitted by statutory regulation or exceeds the permit-   min 2 1−α Hα ted use, you will need to obtain permission directly from the copy- × − − dβ right holder. To view a copy of this licence, visit http://creativecomm ( − α)α 2 1 βmin ons.org/licenses/by/4.0/.  2 3 Fαβ m 1 Funded by SCOAP . × Q − (−1 + α + β) αβ 3α3β3  2 3 2 2 2 × Fαβ m α + 3α (−1 + β) + 3αβ + (−3 + β)β Q   ++  ¯ + 4 α4 + α3(− + β) + αβ 3 + (− + β)β3 , Appendix A: The spectral densities of 0 Q- Q hex- m Q 1 1 aquark states (A7)  α ++ ¯ ¯ 4 max Hα The 0 c-c state spectral densities on the OPE side: ρ qq = dα ¯qq4 , (A8) π 2 αmin 48

 α  −α 1 max 1 where MB is the Borel parameter introduced by the Borel ρ pert (s) = dα dβ π 10 transformation, Q = c or b. Here, we also have the following αmin βmin   F 7 (−1 + α + β)4 F 6 m2 (1 − α − β)5 definitions: × αβ + αβ Q , 3 × 5 × 7 × 219α6β6 3 × 52 × 219α6β6 F = (α + β) 2 − αβ , H = 2 − α( − α) , αβ m Q s α m Q 1 s (A1)   (A9) 2 α 1−α   2 G  max ρ G (s) = dα dβ 2 π 10 αmin = 1 − 1 − 4m /s /2,, (A10) αmin βmin Q 123 121 Page 8 of 10 Eur. Phys. J. C (2020) 80 :121   2 2 α 2 ¯ 2 2 ¯qq G  max αmax = 1 + 1 − 4m /s /2 , (A11) ρ qq G = dα Q × 11π 6 3 2 αmin 2 2   β = αm /(sα − m ). (A12) 2 1−α min Q Q Hα × − dβ 2(1 − α)α β  min F 2 −+ αβ m Q 1 Appendix B: The spectral densities of 0  -¯ hex- × + (−1 + α + β) Q Q αβ 3α3β3 aquark states  2 3 2 2 2 × Fαβ m α + 3α (−1 + β) + 3αβ + (−3 + β)β Q  −+  ¯  The 0 cÐ c stateα spectral −α densities on the OPE side: + 4 α4 + α3(− + β) + αβ 3 + (− + β)β3 , 1 max 1 m Q 1 1 (B7) ρ pert (s) = dα dβ 10 π α β  α 2 min min max 2m − 3Hα   ¯qq4 Q 4 F 7 (−1 + α + β)4 F 6 m2 (α + β − 1)5 ρ = dα ¯qq . (B8) αβ αβ Q α 144π 2 × + , (B1) min 3 × 5 × 7 × 219α6β6 3 × 52 × 219α6β6  α  −α  2 max 1 ρG2( ) = G α β s d d −− π 10 α β  ¯ min min Appendix C: The spectral densities of 1 Q- Q hex-  4 2 2 aquark states Fαβ (−1 + α + β) −3Fαβ − 5m (−1 + α + β) × Q 3 × 5 × 219α4β4 −− ¯ The 1 c-c state spectral densities on the OPE side: F 3 m2 (−1 + α + β)4 − αβ Q 5 × 32 × 221α6β6   αmax 1−α pert 1 × 4m2 α4 + α3(−1 + β) + αβ 3 + (−1 + β)β3 ρ (s) = dα dβ Q  π 10 α β min min 3 2 2 2 6 5 2 7 4 +Fαβ 2α − 3α (−1 + β) − 3αβ + β (3 + 2β)) , 7Fαβ (α + β − 1) m + Fαβ (α + β − 1) (α + β + 4) × Q , 3 × 7 × 52 × 219α6β6 (B2)  α  −α (C1) ¯ 2 max 1   ¯qq2 qq 2 2 α 1−α ρ (s) = dα dβ 2 g G  max × 10π 6 ρG ( ) = s α β 3 2 αmin βmin s d d   π 10 α β F 4 ( − α − β) F 3 2 ( − α − β)2 min min αβ 1 2 αβ m Q 1  5 2 × − , (B3) Fαβ (α + β − 1) (α + β + 2) α3β3 α3β3 × × × 19α4β4  α 3 5 2  3 max G3 G 2 4 3 ρ (s) = dα m Fαβ (α + β − 1) 5 × 32 × 222π 10 α + Q  min × 19α4β4 1−α 3 2 × dβF 2 (α + β − )4 F 3 2 (α + β − )4 αβ 1 αβ m Q 1 βmin + 32 × 5 × 221α6β6 −5F 2 − 8Fαβ m2 (8α + 3β − 3) − 24αm4 (α + β − 1)    × αβ Q Q , 3 6 × 2 α4 + α3(β − ) + αβ 3 + β3(β − ) α β 4m Q 1 1   (B4) α 4 2 2 ¯qqqGq ¯  max +Fαβ α + 3α (β − 1) + αβ (β + 3) ρ¯qqqGq ¯ (s) = dα × 9π 6  3 2 αmin  −α 3 2 2 +α3(β + ) + β2(β2 + β − ) , 1 Fαβ + 3m Fαβ (−1 + α + β) 7 7 3 (C2) × dβ Q , (B5) α2β2  α  −α βmin ¯ 2 max 1 ¯qq2 qq  α ρ ( ) = α β ¯ 2 max s d d ¯ 2 qGq 3 × 211π 6 α β ρ qGq = dα min min 211π 6 α min m2 F 3 (α + β − )2 + F 4 [(α + β)2 − ]    4 Q αβ 1 αβ 1 2 1−α F 2 × , (C3) Hα αβ m α3β3 × − dβ Q , ( − α)α αβ (B6) 2 1 βmin

  3 α 1−α  3 G  max ρ G (s) = dα × dβF 2 (α + β − 1)4 × 2 × 22π 10 αβ 5 3 2 αmin βmin 2 2 2 4 (4 + α + β)Fαβ + 8Fαβ m (α + 3(β − 1) + α(7 + β)) + 24αm (α + β − 1) × Q Q , (C4) α3β6

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 α  −α    ¯  ¯  max 1 ρ¯qqqGq ¯ ( ) = qq qGq α β s d d × − 4m2 α4 + α3(β − 1) + αβ 3 + β3(β − 1) 3 × 29π 6 α β Q min min  F 3 (α + β) + 2 F 2 (− + α + β) αβ 3m Q αβ 1 4 2 2 − , (C5) +Fαβ α + 3α (β − 1) + αβ (β − 3) α2β2   2 α 2 ¯qGq max 3 2 2 ρ¯qGq = dα +α (β + 1) + β (β + β + 3) , (D2) 11π 6 2 αmin  α  −α     ¯ 2 max 1 H2 1−α Fαβ 2 F 2 ¯qq2 qq α m Q αβ ρ (s) = dα dβ × − + dβ + , × 11π 6 ( − α)α αβ αβ 3 2 αmin βmin 2 1 βmin 2 − 2 F 3 (α + β − )2 + F 4 [(α + β)2 − ] (C6) 4m Q αβ 1 αβ 1 × , (D3) α3β3

  3 α 1−α  3 G  max ρ G (s) = dα × dβF 2 (α + β − 1)4 × 2 × 22π 10 αβ 5 3 2 αmin βmin 2 2 2 4 (4 + α + β)Fαβ + 8Fαβ m (α + 3(1 − β) + α(1 + β)) − 24αm (α + β − 1) × Q Q , (D4) α3β6

   2 2 α αmax 1−α 2 2 ¯qq G  max ¯  ¯  ¯qqqGq ¯  ρ¯qq G  = dα ρ qq qGq (s) = dα dβ 3 × 211π 6 α 3 × 29π 6 α β  min  min min 2 1−α 3 2 2 Hα F (α + β) − 3m F (−1 + α + β) × − + dβ − αβ Q αβ , ( − α)α 2 2 (D5) 2 1 βmin α β   α F 2 F 2 ¯ 2 max αβ m αβ 1 ¯qGq2 qGq × Q + − (−1 + α + β) ρ = dα αβ 2αβ 3α3β3 211π 6 α   min × 4 α4 + α3(− + β) + αβ 3 + (− + β)β3 H2 m Q 1 1 × − α ( − α)α +F 2 2 1 αβ m Q    −α 2 2 1 Fαβ m Fαβ × α4 + 3α2(−1 + β) + (3 + β)αβ2 + dβ − Q + , (D6)  β αβ 2αβ  min  +α3(β + ) + β2(β2 + β − ) , 2 2 α 4 4 3 (C7) 2 2 ¯qq G  max ρ¯qq G  = dα  × 11π 6 α α H − 2 3 2 min ¯ 4 max α m  ρ qq = α Q ¯ 4 . 2 d qq (C8) Hα α 72π 2 × − min 2(1 − α)α  −α  2 2 1 Fαβ m Fαβ + dβ − Q + ++  ¯ αβ αβ Appendix D: The spectral densities of 1 Q- Q hex- βmin 2 aquark states 1 − (−1 + α + β) 3α3β3  ++  ¯ × 4 α4 + α3(− + β) + αβ 3 + (− + β)β3 The 1 c- c state spectral densities on the OPE side: m Q 1 1   α 1−α +F 2 1 max αβ m Q ρ pert (s) = dα dβ π 10 α β × α4 + 3α2(−2 + β) + (−3 + β)αβ2 min min  F 7 (α + β − 1)4(α + β + 4) − F 6 (α + β − 1)5m2  × αβ αβ Q , +α3(β − 2) + β2(β2 − 2β − 3) , (D7) 3 × 7 × 52 × 219α6β6  α (D1) ¯ 4 max Hα   ρ qq = dα ¯qq4 . (D8) 2 2 αmax 1−α π 2 2 g G  αmin 72 ρG (s) = s dα dβ π 10 αmin βmin  F 5 (α + β − 1)2(α + β + 2) m2 F 4 (α + β − 1)3 × αβ − Q αβ References 3 × 5 × 219α4β4 3 × 219α4β4 F 3 m2 (α + β − 1)4 1. M. Gell-Mann, Phys. Lett. 8, 214 (1964) + αβ Q 32 × 5 × 221α6β6 2. G. Zweig, Report No. CERN-TH-401 3. S.K. Choi et al. [Belle Collaboration], Phys. Rev. Lett. 91, 262001 (2003) 123 121 Page 10 of 10 Eur. Phys. J. C (2020) 80 :121

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