Production of hidden-heavy and double-heavy hadronic molecules at the 𝑍 factory of CEPC

,

According to heavy quark flavor symmetry (HQFS), the bottom counterparts of these states like the   ,   ,   , and   should also exist and have been predicted in various models [34][35][36][37][38][39][40], but so far only two bottomonium-like   states,  ±  (10610) and  ±  (10650), have been observed by the Belle experiment in the Υ(5) decay processes [36].One of the decisive reasons for the deficiency of the signals from exotic states in the bottom sector is the limitation of the collision energy and detection efficiency of the present high-energy colliders.The  factory of future Circular Electron Positron Collider (CEPC) [41] with high center-of-mass (c.m.) energy (close to the mass of the  boson), clean background, high resolution and detection ability can provide great opportunity to study the heavy exotic states.
The c.m. energy, luminosity and event yields of the  factory are listed in Table I [41,42].
At parton level, the hidden-charm, hidden-bottom, double-charm, and double-bottom exotic states are produced in the processes  →  c,  →  b,  →  c c, and  →  b b, respectively.The  [2].The production of the doubly heavy baryons Ξ  , Ξ  , and Ξ  at the  factory with the underlying processes  →  c c,  →  b c, and  →  b b has been studied in Ref. [43] in the NRQCD framework, and the total production cross sections for Ξ  , in the compact tetraquark configuration at the  factory has also been studied in Refs.[44,45] by employing the Monte Carlo (MC) event generators MadGraph5 aMC@NLO [46] and Pythia6 [47], and the estimated production cross section is about 36 fb, corresponding to 3.6×10 6 events produced at the  factory during a two-year operation.The MC generator Pythia has also been widely used to simulate the production of the multiquark states in the  +  − [48,49],   [50,51],   [52][53][54][55][56][57][58][59], and  p [52,54,55,60] collisions.
In this work, we will employ the Pythia8 [61] to estimate the production cross sections of the charmonium-like states  (3872) and   (3900), the double-charm tetraquark state   (3875) + , the hidden-charm pentaquark states   (4380) + ,   (4440) + ,   (4450) + ,   (4312) + , and   (4459), and the bottomonium-like states   (10610) and   (10650) in the  +  − collisions at the  factory, assuming they are -wave hadronic molecules.The production cross sections of some typical hidden-charm and double-charm hadronic molecules predicted in Refs.[62,63] as well as the   state as a B B * molecule will also be calculated.
The remaining parts of this paper is organized as follows.In Sec.II, we introduce the inclusive production mechanism of the hadronic molecules.The numerical results of the cross sections are presented in Sec.III.A brief summary is given in Sec.IV.

II. LEPTOPRODUCTION
In this section, we introduce the inclusive production mechanism of the hadronic molecule  in  +  − collisions.As shown in Fig. 1, a heavy-hadron pair  ′ is inclusively generated in the  +  − collision, which is of short-distance nature, and then they are bound through the final state interaction (FSI) to form a hadronic molecule.At the  factory, the typical short-distance pair- production process is mediated by a virtual photon and  boson, while the contribution from the intermediate photon is negligible [43].To give an order-of-magnitude estimation for the production cross sections of the hadronic molecules in  +  − collision, we utilize the Pythia8 to simulate the short-distance inclusive production of the  ′ pair and other particles, and the long-distance FSI is derived in the nonrelativistic effective field theory (NREFT) framework.In general, the amplitude for the production of the hadronic molecule state  can be factorized as [54,64,65] where M [ ′ + all] is the short-distance amplitude for the inclusive production process  +  − →  ′ + all,  is the Green function of the intermediate heavy hadron  ′ pair, and   is the longdistance amplitude for  ′ → .Here the Green's function  is UV divergent and is regularized by the Gaussian regulator [66]  (, where  = √︁ 2( −   −   ′ ) is the binding momentum of the  ′ pair with  the reduced mass of  and  ′ , erfi is the imaginary error function, and the cutoff Λ is in the range of 0.5-1.0GeV, following Refs.[51,54,67].
For the -wave shallow hadronic molecule, the amplitude   can be approximated by the effective coupling constant   , which can be extracted from the residues of the low-energy  ′ →  ′ scattering amplitude  () as where  0 is the pole position in the complex -plane, satisfying det[1 −  ( 0 , Λ)] = 0.One has  0 =   for a bound state on the physical Riemann sheet (RS) or virtual state on the unphysical RS, and  0 =   − Γ/2 for a resonance on the unphysical RS with mass   and width Γ.For the nearthreshold hadronic molecules, one can use a constant separateable potential  for the  ′ →  ′ scattering, and the scattering amplitude  () can be solved from the Lippmann-Schwinger equation as The production cross section of the hadronic molecule  can also be factorized into shortdistance and long-distance parts.The short-distance part is given by the differential MC cross section of the inclusive  ′ production, where  is the three-momentum in the c.m. frame of the  ′ system.The overall factor    ′ ∼ O (1) represents the difference between the MC simulation and the experimental data, and can be roughly taken as    ′ ≃ 1 for an order-of-magnitude estimate [51,54].The short-distance production amplitude M [ ′ + all], which is insensitive to the final-state relative momentum  [51,54], can be approximated as a constant and taken outside from the integration of the final-state momentum .Consequently, the differential cross section for  ′ production in the MC event The total cross section for the hadronic molecule  production is given by where the phase-space integration is the same as that in Eq. ( 5) [54].With the use of Eqs. ( 1) and (5), the production cross section of  can be derived as where   and   ′ are the masses of the heavy hadrons  and  ′ , respectively.

III. NUMERICAL RESULTS
In this section, the production cross sections of the typical hidden-heavy and double-heavy hadronic molecules at the  factory are estimated at an order-of-magnitude level using Eq. ( 8).The differential cross sections ([ ′ + all]/) MC of the  ′ pair production in the  +  − collisions are obtained using the MC event generator Pythia [47].Some typical differential cross sections for the production of the charm-anticharm, double-charm, bottom-antibottom, and double-bottom hadron pairs are shown in Fig. 2, and the differential cross sections for other heavy hadron pairs can be found in Appendix A. The formation of a hadronic molecule requires the constituent hadrons move collinearly with a small relative momentum.The choice of the cut of momentum  has a small effect to the cross section and does not change our order-of-magnitude estimate.Therefore we follow the works in Refs.[50,51] and choose a small relative momentum range | | < 350 MeV where and the coefficient  is obtained by fitting the differential pair-production cross sections simulated by the MC event generator.The final expression of the production cross section of  can be written as The predicted cross sections for the hidden-charm, double-charm, and hidden-bottom hadronic molecules with Λ = 0.5 GeV (out of the parentheses) and Λ = 1.0 GeV (in the parentheses) are listed in Tables II, III, and IV, respectively, where the binding energy is defined as with   the mass of the produced hadronic molecule.The binding energies at the outside and inside of the square brackets in these tables correspond to Λ = 0.5 and 1.0 GeV, respectively.
The results in Tables II, III, and IV reveal that: • The production cross sections of the hidden-charm hadronic molecules  (3872) and   (3900) at the  factory are at pb level, and the production cross section of the   (3900) state is about 5 -10 times larger than that of the  (3872), which is comparable with the prediction in the semi-inclusive leptoproduction process [50].Considering the integrated luminosity ∫  = 100 ab −1 as listed in Table I, there will be approximately 3×10 7 -1.6×10 8 and 4×10 8 -8×10 8 events of the  (3872) and   (3900) produced in the two-year operation of the  factory, respectively.• The production cross sections of the hidden-charm pentaquark candidates   and   states at the  factory are at the same level (about a few to tens of fb), two to three orders of magnitude smaller than those of the  (3872) and   (3900).Such cross sections give 10 5 -10 6 production events at the  factory during the two-year operation.
• The production cross sections of the double-charm tetraquark candidates  +  and its heavyquark-spin symmetry (HQSS) partner  * +  [69] are comparable with the cross sections of the   and   states, two to three orders of magnitude smaller than those of the hidden-charm tetraquarks.Such a large gap between the production cross sections of the   and the hiddencharm tetraquarks can be attributed to the parton level where the production of double-charm molecules requires two pairs of  c produced from the  boson decay.The branching ratio of  →  c c is much smaller than that of  →  c, which is the underlying process for the production of the hidden-charm hadronic molecules.There will be about 2.3 × 10 5 -9.7 × 10 5 and 1.3 × 10 5 -5.5 × 10 5 events for the production of   and  *  at the  factory, respectively.The   events produced at the  factory is roughly three times larger than those in the proposed electron-ion colliders in US in the two-year operation [51].Furthermore, assuming Br[ +  →  0 D0  + ] ≃ 59.6% in terms of the leading-order estimation of the XEFT [70] and Br[ 0 →  −  + ] = 3.9% [2], the number of   events reconstructed in the  0  0  + invariant mass distribution can reach O (10 3 ) at the  factory.The event number for  +  →  0 (→  −  + ) D0 (→  −  + ) + observed by the LHCb Collaboration is 117 ± 16 with an integrated luminosity of 9 fb −1 [21].Therefore, the  factory could be a much better platform to study the   in detail and to search for its spin partner  *  .
• The production cross sections of the hidden-charm baryon-antibaryon hadronic molecules predicted in Ref. [63] and the double-charm meson-baryon hadronic molecules predicted in Ref. [62] are at the same order of magnitude, about 0.1 -1.0 fb, one order of magnitude smaller than those of the   ,   , and  ( * ) .An exception is the Λ  Λ molecule, whose production cross section is one magnitude larger than other hidden-charm baryon-antibaryon hadronic molecules, and the resulting events for the Λ  Λ molecule production at the  factory is about 4.9 × 10 5 -6 × 10 6 .Therefore it is purposeful to search for the Λ  Λ molecule at the  factory of CEPC.TABLE IV.Order-of-magnitude estimations of the inclusive production cross sections (in units of pb) for the hidden-bottom hadronic molecules at the  factory.The values of the cross sections at the outside (inside) of the parentheses correspond to the cutoff Λ = 0.5 GeV (1.0 GeV).• In the bottom sector, the production cross sections of the hidden-bottom hadronic molecules   (10610) and   (10650) can reach tens to hundreds of fb, about one to two order(s) of magnitude smaller than those of the  (3872) and   (3900), and one order of magnitude larger than the production cross section of the double-charm states   and  *  .The expected number of   events at CEPC over a two-year period is around 10 7 , indicating promising prospects for its discovery and detailed study.Considering that the branching ratios of the   (10610) decays to the Υ(1), Υ(2), and Υ(3) final states are 5.4 +1.9 −1.5 × 10 −3 , 3.6 all → Υ() + all,  = 1, 2, 3 at the  factory is about two orders of magnitude smaller than the cross sections at Belle2 [36,72], the integrated luminosity at the  factory, roughly three orders of magnitude higher than Belle's (121.4 fb −1 ), results in approximately one order of magnitude larger   event yields compared to Belle.

Constituents
Despite the absence of the experimental signal from the double-bottom tetraquark ( ū d)   state at present, the existence of the   with quantum numbers  (  ) = 0(1 + ) has been approved by the lattice QCD (LQCD) calculation [73] in the HAL QCD method, where the   is predicted to be a deeply bound state with a binding energy   ] ≈ 10 −3 (10 −1 ) fb for Λ = 0.5(1.0)GeV, 2 -5 orders of magnitude smaller than those of the   states and the   state in the compact tetraquark configuration predicted in Refs.[44,45].

IV. SUMMARY
In summary, we have investigated the inclusive differential production cross sections of the  +  − →  ′ + all processes using the Monte Carlo event generator Pythia8 and estimated the production cross sections of typical hidden/double-charm, and hidden/double-bottom hadronic molecules at the  factory by considering the FSI between the hadron pairs  ′ .The predicted production cross sections of the hidden-charm molecules  (3872) and   (3900) are at the pb level, and the expected event yields of these molecules are about 10 7 -10 8 .The production cross sections of the hidden-charm pentaquark candidates   and   , and the double-charm tetraquark candidates   and  *  as -wave hadronic molecules are at the same order of magnitude, about two to three orders of magnitude smaller than the cross sections of  (3872) and   (3900).The production cross sections of some possible hidden-charm baryon-antibaryon and double-charm meson-baryon hadronic molecules predicted in Refs.[62,63] are further smaller than those of the   ,   , and by about one order of magnitude, except the Λ  Λ molecule whose production cross section is one order of magnitude larger than those of other hidden-charm baryon-antibaryon molecules.
As the  boson can decay to one and two pairs of  b in the parton level with sizeable branching ratios, the  factory is a ideal platform for the study of hidden/double bottom exotic states.The estimated production cross sections of the   states can reach tens to hundreds of pb, giving 10 4 -10 5 event yields of  +  − →  →   (10650)/  (10610) + all → Υ() + all,  = 1, 2, 3 with the two-year integrated luminosity ∫  = 100 ab −1 , which is about one order of magnitude larger than the event yields in the Belle experiment.The production cross sections of the double-bottom tetraquark candidate   as a deeply bound -wave B B * molecule is also estimated using the binding energy from LQCD calculation as an input, and the result is about [ ( * )   ] ≈ 10 −3 (10 −1 ) fb for Λ = 0.5(1.0)GeV, 2-5 orders of magnitude smaller than those of the   states and the   state in the compact tetraquark configuration predicted in Refs.[44,45].Our order-of-magnitude estimates indicate appreciable production event yields of these hidden/double-charm and hidden/doublebottom hadronic molecules at the  factory.In this section, we show all the differential cross sections for the constituent hadron pairs of the hadronic molecules considered in the main text.In the charm sector, Fig. 3 shows the differential cross sections of the constituents of  (3872) and   (3900).Fig. 4 shows the differential cross sections of the constituents of   states.Fig. 5 shows the differential cross sections of the constituents of   (4312),   (4380),   (4440), and   (4457).Fig. 6 shows the differential cross sections of the Λ  Λ , Σ  Σ , Ξ  Ξ , Λ  Σ , and Λ  Ξ pairs as constituents of the hidden-charm baryon-antibaryon hadronic molecules predicted in Ref. [63].Fig. 7 shows the differential cross sections of the constituents of   ,  *  , and the Λ  , Λ   * pairs as constituents of double-charm meson-baryon hadronic molecules predicted in Ref. [62].
In the bottom sector, Fig. 8 shows the differential cross sections of the constituents of   (10610) ± and   (10650) ± .Fig. 9 shows the differential cross sections of the constituents of   (10610) 0 and   (10650) 0 .Fig. 10 shows the differential cross sections of the constituents of   and  *  .

FIG. 2 .
FIG. 2. Differential cross sections d/d (in units of pb/GeV) for the process  +  − →  0 →  ′ .The range of the relative momentum between the two produced hadrons is chosen as | | < 350 MeV.The histograms denote the differential cross sections simulated by Pythia8, and the red dashed curves are obtained from the fits using d/d ∝  2 .The subfigures demonstrate the differential production cross sections for (a) Ξ 0  D0 ; single   = 155 ± 17 MeV only considering the B B * single channel and a binding energy  coupled   = 83 ± 10 MeV considering the B B * − B * B * coupled channel treatment relative to the B B * threshold.In addition, the b b and b b tetraquarks have also been predicted by the LQCD calculation in Ref. [74], with binding energies 100 ± 10 +43 −36 MeV and 30 ± 3 +31 −11 MeV relative to the  * and  *  thresholds, respectively.The large binding energies for both the double bottom and anti-bottom tetraquarks indicate the   could not be simply regarded as a pure B B * hadronic molecule.To give an order-of-magnitude estimate, we assume that the   and  *  have the same binding energy (83 MeV), and still calculate their production cross sections in the hadronic molecule picture.The results are about [ ( *)

2 1 B
The exclusive production of   can occur through the process  +  − → Υ(5) →    via the intermediate states  ′ The enhancement of this process, as discussed in Ref.[71], can lead to a large exclusive cross section.
tsqn202103062 and the Higher Educational Youth Innovation Science and Technology Program Shandong Province under Grant No. 2020KJJ004.P.-P.S. also acknowledges the Generalitat valenciana (GVA) for the project with ref.CIDEGENT/2019/015.

Appendix A :
Differential cross sections of the hadron pairs

TABLE I .
[41,42]rgy configurations, instantaneous luminosity (), integrated luminosity ( ∫ ), and event yields of the  factory at CEPC[41,42]. , and Ξ  are 848.03fb, 2260.51 fb, and 41.16 fb, respectively.The production of the   state The inclusive production of the  as a  ′ hadronic molecule in  +  − collisions, where "all" denotes the other particles produced in this process.

TABLE II .
Order-of-magnitude estimations of the inclusive production cross sections (in units of pb) for the hidden-charm hadronic molecules at the  factory.The values of the binding energies at the outside (inside) of the square brackets and the cross sections at the outside (inside) of the parentheses correspond to the cutoff

TABLE III .
Order-of-magnitude estimations of the inclusive production cross sections (in units of pb) for the double-charm hadronic molecules at the  factory.The values of the binding energies at the outside (inside) of the square brackets and the cross sections at the outside (inside) of the parentheses correspond to the cutoff Λ = 0.5 GeV (1.0 GeV).