Fully-heavy structures in the invariant mass spectrum of J/ψψ(3686) , J/ψψ(3770) , ψ(3686)ψ(3686) , and J/ψΥ(1S) at hadron colliders
FFully-heavy structures in the invariant mass spectrum of J /ψψ (3686) , J /ψψ (3770) , ψ (3686) ψ (3686) ,and J /ψ Υ (1 S ) at hadron colliders Jun-Zhang Wang , , ∗ Xiang Liu , , † , ‡ and Takayuki Matsuki § School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China Research Center for Hadron and CSR Physics, Lanzhou University & Institute of Modern Physics of CAS, Lanzhou 730000, China Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou, Gansu 730000, China Tokyo Kasei University, 1-18-1 Kaga, Itabashi, Tokyo 173-8602, Japan (Dated: December 8, 2020)Motivated by a recent successful dynamical explanation for the newly observed fully-charm structure X (6900)in the mass spectrum of di- J /ψ by LHCb [J. Z. Wang et al. arXiv:2008.07430], in this work, we extend thesame dynamical rescattering mechanism to predict the line shape of more potential fully-heavy structures in theinvariant mass spectrum of J /ψψ (3686), J /ψψ (3770), ψ (3686) ψ (3686), and J /ψ Υ (1 S ) at high energy proton-proton collisions, whose verification in experiments should be helpful to further clarify the nature of X (6900).The above final states of vector heavy quarkonia can be experimentally reconstructed more e ff ectively by a µ + µ − pair in the muon detector compared with Q ¯ Q meson with other quantum numbers. Furthermore, thecorresponding peak mass positions of each of predicted fully-heavy structures are also given. Our theoreticalstudies here could provide some valuable information for the future measurement proposals of LHCb and CMS,especially based on the accumulated data after completing Run III of LHC in the near future. I. INTRODUCTION
Since the first observation of charmonium J /ψ in 1974[1, 2], the fully-heavy-flavor physics has been one of the mosthottest issues in the field of quantum chromodynamics(QCD).Benefitting from the unique heavy quark symmetry and non-relativistic behavior, the fully-heavy system is usually treatedas an excellent platform to solve the non-perturbative puzzlesof QCD [3, 4]. In the past several decades, within the continu-ous e ff orts from high energy collision experiments, a numberof heavy quarkonium and quarkoniumlike states were discov-ered, especially novel charmoniumlike XYZ structures, whoseproperties have provoked theoretical wide discussions and in-deed largely enrich our knowledge for the color confinementinteraction (see review articles in Refs. [5–9] for detail).Although great progress has been made in the study ofheavy quarkonium physics, both experimental and theoreticalinvestigations for the fully-heavy systems containing beyondtwo heavy flavors are still absent. Very recently, the LHCbCollaboration reported the measurements of double J /ψ pro-duction by using proton-proton data at center of mass ener-gies of 7, 8, and 13 TeV, where a clear peak around 6.9 GeVcalled X (6900) and two underlying structures near the pro-duction threshold of J /ψ J /ψ at 6.2 GeV and 7.3 GeV wereobserved in the invariant mass spectrum of di- J /ψ [10], re-spectively. From the final states of J /ψ J /ψ , the observationof X (6900) together with other two peaks indicates the firstexperimental evidence for fully-charm structures by the inter-action of four charm flavors. Thus, the LHCb’s observationhas recently stimulated theorists with a great deal of enthusi-asm to discuss the nature of X (6900) in Refs. [11–38], most † Corresponding author ∗ Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] of which contribute to the resonance interpretation of compacttetraquark hadronic state.Among the recent theoretical explanations, di ff erent fromthe opinions of fully-charm tetraquark state, the Lanzhougroup has proposed a dynamical mechanism to explore sev-eral new structures observed by LHCb [25]. The key idea isbased on a special dynamical contribution in reaction pp → J /ψ J /ψ X , in which di ff erent combinations of a double char-monium directly produced in high energy proton-proton col-lisions are transferred into final state J /ψ J /ψ . Compared witha continuous distribution in the invariant mass spectrum of a J /ψ -pair, this mechanism has been found to produce an obvi-ous cusp at the corresponding mass threshold of an intermedi-ate double charmonium. By a theoretical analysis for the lineshape of experimental data, three peaks observed by LHCbare well reproduced near 6.5, 6.9, and 7.3 GeV in the invariantmass spectrum of di- J /ψ , which naturally correspond to threerescattering channels of η c (1 S ) χ c (1 P ), χ c (1 P ) χ c (1 P ), and χ c (1 P ) X (3872) [25], respectively. From this point of view,the X (6900) may not be a genuine resonance, which shouldbe emphasized before declaring the discovery of a new exotichadron state.Of course, if the proposed dynamical explanation for X (6900) [25] is reasonable, we can naturally conjecture thatthis mechanism could be universal in the production of adouble charmonium in high energy hadron colliders. Thus,the verification of this novel dynamical mechanism could beachieved by measurements of the di ff erent double charmo-nium production. Based on this motivation, in this work,we study possible fully-charm structures existing in high en-ergy reactions, pp → J /ψψ (3686) X , pp → J /ψψ (3770) X , pp → ψ (3686) ψ (3686) X by the dynamical rescattering mech-anism. In addition, we also predict a specific fully-heavystructures induced by the rescattering channels of a char-monium plus a bottomonium in the production process of J /ψ Υ (1 S ). Here, the reason for choosing vector charmonia ψ (3686) and ψ (3770) is that they belong to the same J /ψ fam-ily and the feasibility of their prompt production has been a r X i v : . [ h e p - ph ] D ec proven in the high energy proton-proton collision experiments[39, 40]. Compared with other charmonium states with dif-ferent quantum numbers, these charmonia together with bot-tomonium Υ (1 S ) can be experimentally reproduced more eas-ily by the final states of µ + µ − . Hence, we expect that the pre-dictions presented in this work should be valuable to searchfor more fully-heavy structures in the invariant mass spectrumof a double charmonium and also could be tested in the futureLHCb and CMS experiment.This paper is organized as follows. After Introduction,we will present our theoretical framework how to calculatethe dynamical rescattering contributions in the hadroproduc-tion of a double heavy quarkonium in Sec. II. In Sec. III,the line shape predictions for potential fully-heavy structureson the invariant mass spectrum of J /ψψ (3686), J /ψψ (3770), ψ (3686) ψ (3686), and J /ψ Υ (1 S ) are shown based on the dy-namical rescattering mechanism, whose behaviors and peakpositions are also discussed. This paper ends with the sum-mary in Sec. IV. II. DYNAMICAL RESCATTERING MECHANISM IN THEHADROPRODUCTION OF A DOUBLE HEAVYQUARKONIUM
The hadroproduction of a double heavy quarkonium in highenergy proton-proton collisions is a very important subject ofheavy quarkonium physics. At present, we know that a dou-ble heavy quarkonium can be directly produced by both thesingle parton scattering (SPS) and double parton scattering(DPS) processes [41–52]. However, the situation may be morecomplicated in a real production process and some unknowndynamical e ff ects may exist, where an available approach totest a possible underlying dynamical mechanism is the mea-surement of the corresponding invariant mass spectrum of adouble heavy quarkonium. Focusing on a general productionprocess pp → H H X , where H H are the studied doubleheavy quarkonium. As shown in the schematic diagrams inFig. 1, in addition to the dominant direct production via SPSand DPS, the rescattering reaction of pp → ( h i h j → H H ) X H Q ¯ Q H Q ¯ Q h Q ¯ Qi h Q ¯ Qj H Q ¯ Q H Q ¯ Q ( a ) Direct production ( b ) Dynamical rescattering mechanism FIG. 1: The schematic diagrams for hadroproduction of a doubleheavy quarkonium marked by H H . Left diagram ( a ): the directproduction process by single and double parton scattering; right dia-gram ( b ): the dynamical rescattering mechanism involving the vari-ous allowed intermediate heavy quarkonium pairs h i h j . The skybulecircle represents direct production of a double heavy quarkonium inhadron collisions. may be an important underlying dynamical mechanism in-volved in the production of H H . Here, the intermediateparticles h i h j are composed of the combination of alternativedouble heavy quarkonium allowed by the system’s quantumnumbers of rescattering process h i h j → H H . It is worth em-phasizing that because of the lack of experimental informa-tion, our present knowledge for inner interaction of process h i h j → H H is still limited, so the coupling among inter-mediate charmonium pairs h i h j and H H has to be absorbedinto a vertex for the convenience of the subsequent theoreticaltreatment.Starting from an S -wave interaction between an interme-diate heavy quarkonium pair h i h j , the production amplitudeof H H by dynamical rescattering mechanism becomes theone proportional to the scalar two-point loop integral, whoseexpression can be given by, in the rest frame of H H , L i j ( m H H ) = (cid:90) dq (2 π ) e − (2 (cid:126) q ) /α ( q − m i + i (cid:15) )(( P − q ) − m j + i (cid:15) ) = i m i m j − µα √ π ) / + µ (cid:112) µ m (cid:32) erfi (cid:34) √ µ m α (cid:35) − i (cid:33) π/ e − µ m α , (1)where µ = ( m i m j ) / ( m i + m j ) and m = m H H − m i − m j . Here, m i and m j are the resonant mass of intermediate charmoniumstates h i and h j , respectively, and m H H = ( p H + p H ) is thesquare of the invariant mass of H H . P = ( m H H , , , H H system and the erfithe imaginary error function. We also introduce an exponen-tial form factor e − (2 (cid:126) q ) /α to avoid the ultraviolet divergence ofscalar two-point loop integral, and α is a cuto ff parameter.In this work, we mainly consider the quantum number com-bination of J PC J PC = −− −− for H H . Then, we can easilyconclude that the system of h i h j must satisfy C = + C parity. Based on this restriction, we y y ' c c 0 c c 0 y y " y y c c 0 c c 1 c c 0 c c 2 c c 1 c c 1 c c 1 c c 2 c c 2 c c 2 h c c 1 ' Line shape of the invariant mass spectrum m J / y y ( 3 6 8 6 ) ( G e V ) h c y ' c c 0 c c 1 ' y ' y ' h c y c c 1 c c 1 ' m J / y y ( 3 6 8 6 ) ( G e V ) c c 2 c c 1 ' y ' y " y ' y y " y " y " y FIG. 2: The predicted line shapes of the invariant mass distributionof J /ψψ (3686) produced in high energy proton-proton collisions us-ing the only contributions from dynamical rescattering mechanism. can select the following allowed channels, 1 −− −− , 0 ++ ++ ,0 ++ ++ , 0 ++ ++ , 1 ++ ++ , 1 ++ ++ , and 2 ++ ++ , etc. for the h i h j with parity P = +
1, and 1 −− + − , 0 − + ++ , 0 − + ++ , 0 − + ++ , etc.for the h i h j with parity P = −
1. For the rescattering processeswith two kinds of P parity, the line shapes on the invariantmass spectrum of m H H can be given by [25] A i j ( m H H ) = g i j L i j ( m H H ) e c m H H (cid:113) λ ( m H H , m H , m H )2 m H H (2)and A (cid:48) i j ( m H H ) = g (cid:48) i j L i j ( m H H ) e c (cid:48) m H H λ ( m H H , m H , m H ) m H H , (3)respectively, where λ ( x , y , z ) = x + y + z − xy − xz − yz is the K¨allen function. The exponential form factors e c ( (cid:48) )0 m H H are introduced when we parameterize a direct production am-plitude of an intermediate double heavy quarkonium, whichrefers to the treatment of experimental analysis of LHCb [10]because of the complexity and di ffi culty in the present theoret-ical calculations [53–57]. Since there are no relevant experi-mental data to determine the magnitude of coupling constants g ( (cid:48) ) i j , we adjust the values of g ( (cid:48) ) i j to normalize the maximum ofthe line shape of A ( (cid:48) )2 i j ( m H H ) to be one.For the one-loop rescattering processes formulated by Eqs.(2-3), there exists a square root branch point in scalar two-point integral L i j ( m H H ), √ m H H − m i − m j , where an inte-gral singularity at the threshold of m i + m j appears at theon-shell of two intermediate heavy quarkonium states. Thethreshold singularity causes a cusp exactly at the correspond-ing threshold in the invariant mass distribution of m H H .However, in an actual process, the sharpness of a thresholdcusp may be weakened by the resonant width of intermedi-ate heavy quarkonium states. This width e ff ect may be im-portant for describing the line shape of the invariant massspectrum for m H H and may even change the peak posi-tion from the threshold. So, the width e ff ect will be con-sidered in the following calculations by replacing m i and m j in Eq. (1) with ( m i − i Γ i /
2) and ( m j − i Γ j / J /ψψ (3686), J /ψψ (3770), ψ (3686) ψ (3686), and J /ψ Υ (1 S ) and the corresponding peakpositions can also be given. In the following, we will discussthem carefully. III. NUMERICAL RESULTS AND DISCUSSIONSA. Fully-charm structures on the invariant mass spectrum of J /ψψ (3686) , J /ψψ (3770) , and ψ (3686) ψ (3686) After a successful non-resonant dynamical explanationon X (6900) observed in the invariant mass spectrum of J /ψ J /ψ [25], we will extend our theoretical framework to y y c c 0 c c 1 y y " c c 0 c c 2 c c 1 c c 1 c c 1 c c 2 c c 2 c c 2 h c y ' c c 0 c c 1 ' Line shape of the invariant mass spectrum m J / y y ( 3 7 7 0 ) ( G e V ) y ' y ' h c y c c 1 c c 1 ' c c 2 c c 1 ' m J / y y ( 3 7 7 0 ) ( G e V ) y ' y " y ' y y " y " y " y FIG. 3: The predicted line shapes of the invariant mass distributionof J /ψψ (3770) produced in high energy proton-proton collisions us-ing the only contributions from dynamical rescattering mechanism. Line shape of the invariant mass spectrum m y ( 3 6 8 6 ) y ( 3 6 8 6 ) ( G e V ) y ' y ' y ' y c c 1 c c 1 ' y " y " c c 2 c c 1 ' y " y y ' y " FIG. 4: The predicted line shapes of the invariant mass distribu-tion of ψ (3686) ψ (3686) produced in high energy proton-proton colli-sions using the only contributions from dynamical rescattering mech-anism. discuss potential fully-charm structures in the hadroproduc-tion of a double charmonium, J /ψψ (3686), J /ψψ (3770) and ψ (3686) ψ (3686). According to the present charmonium spec-troscopy [58, 59], we select ten established charmoniumor charmoniumlike states as intermediate rescattering parti-cles in the dynamical mechanism, which are η c (1 S )(0 − + ), J /ψ , ψ (3686), ψ (3770)(1 −− ), h c (1 P )(1 + − ), χ c (1 P ), χ c (1 P ), χ c (1 P ), X (3872), χ c (1 P )( J ++ with J = , , X (3842)(3 −− ). Most of them have been directly discovered inhigh energy proton-proton experiments [39, 40, 60, 61]. Ad-ditionally, it is worth emphasizing that the direct hadropro-duction rates of η c , X (3872), and P -wave charmonium states χ cJ with J = , , J /ψ by both experiments [39, 60] and theoretical es-timations from nonrelativistic QCD (NRQCD) [62–70]. Inthe following discussions, without any special emphasis, η c , ψ , ψ (cid:48) , ψ (cid:48)(cid:48) , h c , χ cJ , χ (cid:48) c , and ψ refer to η c (1 S ), J /ψ , ψ (3686), ψ (3770), h c (1 P ), χ cJ (1 P ), χ (cid:48) c (2 P ) and ψ (1 D ), respectively,and χ (cid:48) c (2 P ) = X (3872) [71–75], ψ (1 D ) = X (3842) [59].In this work, three model parameters α , c , and c (cid:48) are uni-formly taken as 2.0, -1.5, and -1.0, respectively, which refersto the fitting results of the scenario-I for di- J /ψ mass spec-trum in Ref. [25]. The predicted line shapes of fully-charmstructures of the invariant mass distribution for J /ψψ (3686)from high energy proton-proton collisions are presented inFig. 2. It can be seen that there exist twenty allowed thresh-old cusps at the energy region from 6783 to 7700 MeV. Thesepeak structures can be divided into four energy regions, i.e.,(6 . ∼ . . ∼ . . ∼ .
40) and (7 . ∼ . ψψ (cid:48) , χ c χ c , ψψ (cid:48)(cid:48) , ψψ , and χ c χ c areclustered in a short energy region of 6900 to 6960 MeV, whichare close to the production threshold of 6783 MeV. Simulta-neously, there is also a similar energy region with small rangeof 7370 to 7440 MeV, which include four channels of ψ (cid:48) ψ (cid:48) , h c ψ , χ c χ (cid:48) c and χ c χ (cid:48) c . At present experimental statistics, itis not likely to identify the individual signal from these closepeaks and their contributions may overlap because of the sim-ilar line shapes near peaks. According to the above analyses,we strongly encourage experimentalists to search for the mostpromising two fully-charm structures on the invariant massspectrum of J /ψψ (3686) near 6.9 and 7.4 GeV. As for the re-maining rescattering channels, it is worth noting that the cuspe ff ect from η c χ (cid:48) c channel is relatively weak so its contributionmay be covered by the direct production background.The predicted line shapes of fully-charm structureson the invariant mass distributions of J /ψψ (3770) and ψ (3686) ψ (3686) from high energy proton-proton collisionsare shown in Figs. 3 and 4, respectively. Here, the allowedintermediate double charmonium channels are consistent withthose in J /ψψ (3686). Because of a general suppression of thecontributions from o ff -shell channels, we consider only theintermediate channels above a production threshold and thenthere are seventeen and seven selected rescattering channelsfor hadroproduction of J /ψψ (3770) and ψ (3686) ψ (3686), re-spectively. Influenced by the phase space distribution, theirpeak line shapes will be fatter than those in the invariantmass spectrum of J /ψψ (3686), especially for the channelsnear the production threshold. Similarly to Fig. 2, we cansee from Fig. 3 that there also exist two typical energy re-gions of (6990 ∼ ∼ J /ψψ (3770), which are related to five dou-ble charmonium channels of ψψ (cid:48)(cid:48) , ψψ , χ c χ c , χ c χ c , and χ c χ c and four channels of ψ (cid:48) ψ (cid:48) , h c ψ , χ c χ (cid:48) c , and χ c χ (cid:48) c ,respectively. This means that it is worth expecting to observetwo clear structures near 7.0 and 7.4 GeV in the invariant massspectrum for J /ψψ (3770) in the future LHCb and CMS exper-iments. As for the hadroproduction of ψ (3686) ψ (3686), con-sidering that its production threshold reaches 7372 MeV, wesuggest the experiments to explore possible fully-charm struc-tures near 7.5 GeV, which corresponds to five close thresh-old cusps of double charmonium channels for ψ (cid:48) ψ (cid:48) , χ c χ (cid:48) c , χ c χ (cid:48) c , ψ (cid:48) ψ (cid:48)(cid:48) , and ψ (cid:48) ψ as shown in Fig. 4.In addition to the line shapes of threshold cusps from dif- TABLE I: The peak mass positions of di ff erent rescattering chan-nels in the invariant mass spectrum for J /ψψ (3686), J /ψψ (3770), and ψ (3686) ψ (3686) in high energy proton-proton collisions. The resultsare all in unit of MeV.Rescattering channels m J /ψψ (3686) m J /ψψ (3770) m ψ (3686) ψ (3686) ψψ (cid:48) · · · · · · χ c χ c · · · · · · ψψ (cid:48)(cid:48) · · · ψψ · · · χ c χ c · · · χ c χ c · · · χ c χ c · · · χ c χ c · · · χ c χ c · · · η c χ (cid:48) c · · · · · · h c ψ (cid:48) · · · χ c χ (cid:48) c · · · ψ (cid:48) ψ (cid:48) h c ψ · · · χ c χ (cid:48) c χ c χ (cid:48) c ψ (cid:48) ψ (cid:48)(cid:48) ψ (cid:48) ψ ψ (cid:48)(cid:48) ψ (cid:48)(cid:48) ψ (cid:48)(cid:48) ψ ferent rescattering channels, the corresponding invariant masspositions at peaks are also given and summarized in Table I.For the intermediate channels composed of point-like parti-cles, their peak mass positions should exactly equal to themass summation of intermediate states. However, due to thewidth e ff ects of intermediate resonances and the phase spacedistribution function, the actual peak mass position is gener-ally larger than the threshold position. In addition, we findthat though the line shape of a threshold cusp is obviously de-pendent on the model parameters α and c ( (cid:48) )0 , they have littlee ff ects on the maximum position. Therefore, the predictionsfor the peak mass positions of di ff erent fully-charm structureslisted in Table I should be credible as long as the masses andwidths of intermediate charmonia are determined, which willbe valuable for the future experimental search proposals. B. Fully-heavy structures involved with b-flavor in theinvariant mass spectrum of J /ψ Υ (1 S ) We can continue to extend the dynamical rescatteringmechanism to the predictions of fully-heavy structures in-volved with bottom flavor. The potential fully-bottom struc-tures in the hadroproduction of a double bottomonium ΥΥ have been studied in Ref. [25]. In this subsection, we willfocus on a special case of fully-heavy structures, which willbe hopefully discovered in the invariant mass distribution ofa charmonium J /ψ plus a bottomonium Υ (1 S ) from high en-ergy proton-proton collisions. For the candidates of interme-diate charmonia and bottomonia in the dynamical productionprocesses of J /ψ Υ (1 S ), we consider only low-lying η c , ψ , h c , χ cJ and η b , Υ , h b , χ bJ with J = , ,
2, respectively, where y ¡ c c 0 h b c c 1 h b h c c b 0 h c c b 1 h c c b 2 c c 2 h b h c ¡ y h b Line shape of the invariant mass spectrum m J / y ¡ ( 1 S ) ( G e V ) c c 0 c b 0 c c 0 c b 1 c c 0 c b 2 c c 1 c b 0 m J / y ¡ ( 1 S ) ( G e V ) c c 1 c b 1 c c 2 c b 0 c c 1 c b 2 c c 2 c b 1 c c 2 c b 2 FIG. 5: The predicted line shapes of the invariant mass distributionof J /ψ Υ (1 S ) produced in high energy proton-proton collision usingthe only contributions from dynamical rescattering mechanism.TABLE II: The peak mass positions of di ff erent rescattering channelsin the invariant mass spectrum of J /ψ Υ (1 S ) in high energy proton-proton collision.Rescattering channels m J /ψ Υ (1 S ) (MeV) ψ Υ χ c η b χ c η b η c χ b η c χ b η c χ b χ c η b h c Υ ψ h b χ c χ b χ c χ b χ c χ b χ c χ b χ c χ b χ c χ b χ c χ b χ c χ b χ c χ b unknown widths of several bottomonium states are taken bytheoretical estimations [76]. Here, it is worth emphasizingthat though the present studies on bottomonium production inthe high energy proton-proton experiments are still absent, butthe observation of high excited states χ b (3 P ) and χ b (3 P ) byCMS in 2018 [77] has proven the ability of LHC to producethe b ¯ b states. Thus, the measurements of the hadronic produc-tion of bottomonia are still worth being expected in the future.The predicted line shapes and peak mass positions offully-heavy structures in the invariant mass distribution of J /ψ Υ (1 S ) from high energy proton-proton collisions areshown in Fig. 5 and Table II, respectively. Benefited fromheavy quark symmetry, which causes an approximate massdegeneracy between two S -wave bottomonia η b and Υ as well as among four P -wave bottomonium states χ bJ with J = , , h b , the threshold cusps in the invariant mass spectrum of J /ψ Υ (1 S ) are mainly concentrated in four separate energyregions, i.e., 12 .
65, (12 . ∼ . . ∼ .
40) and(13 . ∼ .
48) GeV as shown in Fig. 5. Here, it can beseen that a near-threshold structure at 12.65 GeV is providedby the channel of ψ Υ . Anyway, the above four energy re-gions are highly recommended for future relevant experimen-tal measurements. IV. SUMMARY
Recently, the LHCb collaboration reported the observationof a new structure X (6900) in the reconstruction events ofdi- J /ψ , which is the first evidence for the existence of fully-heavy structures in the invariant mass distributions of a dou-ble heavy quarkonium [10]. On account of the importanceof X (6900) discovery, its nature has aroused great interestsamong theorists. In Ref. [25], we have proposed a special dy-namical mechanism to explain the peak line shape of X (6900),whose core is a dynamical rescattering process that the al-lowed combinations of an intermediate double charmoniumdirectly produced in high energy proton-proton collisions aretransferred into a final state J /ψ J /ψ . Furthermore, we havefound that these processes could produce the obvious thresh-old cusps near the position of mass summation of the corre-sponding intermediate double charmonium.Motivated by a successful description of experimental lineshapes for di- J /ψ mass spectrum of LHCb by a dynamicalrescattering mechanism [25], in this work, we have extendedour theoretical framework to study more fully-charm struc-tures in the invariant mass spectrum of a di ff erent double char-monium from high energy proton-proton collisions, which are J /ψψ (3686), J /ψψ (3770), and ψ (3686) ψ (3686). According toour theoretical predictions, we have strongly recommendedsome hopefully detectable fully-charm structures in the in-variant mass spectrum of a double charmonium to experimen-talists, whose peak mass positions are 6.9 and 7.4 GeV for J /ψψ (3686), 7.0 and 7.4 GeV for J /ψψ (3770), and 7.5 GeVfor ψ (3686) ψ (3686), respectively. A special case of fully-heavy structures involved with bottom flavor in the hadropro-duction of J /ψ Υ (1 S ) has been also predicted, which are foundto be clustered in four energy regions of 12 .
65, (12 . ∼ . . ∼ . . ∼ .
48) GeV.Just like the observation of a fully-charm structure X (6900)[10], when the couplings between intermediate rescatteringchannels and products of a double charmonium are strongenough, it is easy to distinguish the peak signals of thresholdcusps from the direct production background by SPS and DPSmechanisms, which usually behave like a continuous distribu-tion. Fortunately, in near future, the Run III of LHC will beperformed and then the High-Luminosity-LHC upgrade willachieve a data collection of an integrated luminosity of 300fb − in pp collisions at a CM energy of 14 TeV [78]. There-fore, we greatly expect that these novel fully-heavy structurespredicted in this work can be observed in the future measure-ments, especially at LHCb and CMS. ACKNOWLEDGEMENTS
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