Production of X b in Υ ( 5 S , 6 S ) → γX b Radiative Decays

We investigate the production of in the process , where is assumed to be the counterpart of in the bottomonium sector as molecular state. We use the effective Lagrangian based on the heavy quark symmetry to explore the rescattering mechanism and calculate their production ratios. Our results have shown that the production ratios for are orders of with reasonable cutoff parameter range . The sizeable production ratios may be accessible at the future experiments like forthcoming BelleII, which will provide important clues to the inner structures of the exotic state .

hadronic molecule configuration. Since the mass of X b may be very heavy and its J P C is 1 ++ , it is less likely for a direct discovery at the current electron-positron collision facilities, though the Super KEKB may provide an opportunity in Υ(5S, 6S) radiative decays [17]. In Ref. [18], a search for X b in the ωΥ(1S) final states has been presented and no significant signal is observed for such a state.
The production of X b at the LHC and the Tevatron [19,20] and other exotic states at hadron colliders [21][22][23][24][25][26] have been extensively investigated. In the bottomonium system, the isospin is almost perfectly conserved, which may explain the escape of X b in the recent CMS search [27]. As a result, the radiative decays and isospin conserving decays will be of high priority in searching for X b [28][29][30]. In Ref. [28], we have studied the radiative decays of X b → γΥ(nS) (n = 1, 2, 3), with X b being a candidate for the BB * molecular state, and found that the partial widths into γX b are about 1 keV. In Ref. [29], we studied the rescattering mechanism of the isospin conserving decays X b → Υ(1S)ω, and our results show that the partial width for the X b → Υ(1S)ω is about tens of keVs.
In this work, we will further investigate the X b production in Υ(5S, 6S) → γX b with X b being a BB * molecule candidate. To investigate this process, we calculate the intermediate meson loop (IML) contributions. As well know, IML transitions have been one of the important nonperturbative transition mechanisms been noticed for a long time [31][32][33]. Recently, this mechanism has been used to study the production and decays of ordinary and exotic states  and B decays [61][62][63][64][65][66][67][68], and a global agreement with experimental data were obtained. Thus this approach may be suitable for the process Υ(5S, 6S) → γX b .
The paper is organized as follows. In Sec. II, the effective Lagrangians for our calculation. Then in Sec. III, we present our numerical results. Finally we give the summary in Sec. IV. Based on the heavy quark symmetry, we can write out the relevant effective Lagrangian for the Υ(5S) [68,69]   In order to calculate the process depicted in Fig. 1, we also need the photonic coupling to the bottomed mesons. The magnetic coupling of the photon to heavy bottom meson is described by the Lagrangian [72,73]

II. EFFECTIVE LAGRANGIANS
with where β is an unknown constant, Q = diag{2/3, −1/3, −1/3} is the light quark charge matrix, and Q ′ is the heavy quark electric charge (in units of e). β ≃ 3.0 GeV −1 is determined in the nonrelativistic constituent quark model and has been adopted in the study of radiative D * decays [73].
In the b and c systems, the β value is the same due to heavy quark symmetry [73]. In Eq. (2), the first term is the magnetic moment coupling of the light quarks, while the second one is the magnetic moment coupling of the heavy quark and hence is suppressed by 1/m Q .
At last, assuming that X b is an S-wave molecule with J P C = 1 ++ given by the superposition of B 0B * 0 + c.c and B −B * + + c.c hadronic configurations as As a result, we can parameterize the coupling of X b to the bottomed mesons in terms of the following Lagrangian where x i denotes the coupling constant. Since the X b is slightly below the S-wave BB * threshold, the effective coupling of this state is related to the probability of finding the BB * component in the physical wave function of the bound states and the binding energy, Here, we should also notice that the coupling constant x i in Eq. (6) is based on the assumption that X b is a shallow bound state where the potential binding the mesons is short-ranged.
Based on the relevant Lagrangians given above, the decay amplitudes in Fig. 1 can be generally expressed as follows, where where n = 1, 2 corresponds monopole and dipole form factor, respectively. Λ ≡ m 2 + αΛ QCD and the QCD energy scale Λ QCD = 220 MeV. This form factor is supposed and many phenomenological studies have suggested α ≃ 2 ∼ 3. These two form factors can help us explore the dependence of our results on the form factor.
The explicit expression of transition amplitudes can be found in Appendix (A.2) in Ref. [77], where radiative decays of charmonium are studied extensively based on effective Lagrangian approach.

III. NUMERICAL RESULTS
Before proceeding the numerical results, we first briefly review the predictions on mass of X b .
The existence of the X b is predicted in both the tetraquark model [78] and those involving a molecular interpretation [79][80][81]. In Ref. [78], the mass of the lowest-lying 1 ++bq bq tetraquark is predicated to be 10504 MeV , while the mass of the BB * molecular state is predicated to be a few tens of MeV higher [79][80][81]. For example, in Ref. [79], the mass was predicted to be 10562 MeV, which corresponds to a binding energy to be 42 MeV, while the mass was predicted to be (10580 +9 −8 ) MeV, which corresponds to a binding energy (24 +8 −9 ) MeV in Ref. [81]. As can be seen from the theoretical predictions, it might be a good approximation and might be applicable if the binding energy is less than 50 MeV. In order to cover the range the previous molecular and tetraquark predictions on Ref. [78][79][80][81], we present our results up to a binding energy of 100 MeV, and we will choose several illustrative values: ǫ X b = (5, 10, 25, 50, 100) MeV.
In Table II,  In Fig. 2 (a), we plot the the branching ratios for Υ(5S) → γX b in terms of the binding energy in the large ǫ X b region. As a result, the behavior of the branching ratios is relatively sensitive at small ǫ X b , while it becomes smooth at large ǫ X b . Results with the dipole form factors α = 2.0, 2.5, and 3.0 are shown in Fig. 2 (b) as solid, dash, and dotted curves, respectively. The behavior is similar to that of Fig. 2 (a).
We also predict the branching ratios of Υ(6S) → γX b and present the relevant numerical results in Table III  In Ref. [51], authors introduced a nonrelativistic effective field theory method to study the meson loop effects of ψ ′ → J/ψπ 0 . Meanwhile they proposed a power counting scheme to estimate the contribution of the loop effects, which is used to judge the impact of the coupled-channel effects. For the diagrams in Fig. 1, the vertex involving the initial bottomonium is in P -wave. The momentum in this vertex is contracted with the final photon momentum q, and thus should be counted as q. The decay amplitude scales as follows, where v is understood as the average velocity of the intermediate bottomed mesons.
As a cross-check, we also present the branching ratios of the decays in the framework of NREFT.
The relevant transition amplitudes are similar to that given in Ref. [36] with only different masses and coupling constants. The obtained numerical results for Υ(5S) → γX b and Υ(6S) → γX b in terms of the binding energy are listed in the last column of Table II and III, respectively. As shown in Table II, except for the largest binding energy ǫ X b = 100 MeV, the NREFT predictions of Υ(5S) → γX b are about 1 order of magnitude smaller than the ELA results at the commonly accepted range. For Υ(6S) → γX b shown in Table III, the NREFT predictions are several times smaller than the ELA results in small binding energy range, while the predictions of these two methods are comparable at large binding energy. These difference may give some sense of the theoretical uncertainties for the predicted rates and indicates the viability of our model to some extent.
Here we should notice, for the isoscalar X b , the pion exchanges might be nonperturbative and produce sizeable effects [81][82][83]. In Ref. [81], their calculations show that the relative errors of C 0X are about 20% for the X b . Even if we take into account this effect, the estimated order of the magnitude for the branching ratio Υ(5S, 6S) → γX b may also be sizeable, which may be measured in the forthcoming BelleII experiments.

IV. SUMMARY
In this work, we have investigated the production of X b in the radiative decays of Υ(5S, 6S).
Based on the BB * molecular state picture, we considered its production through the mechanism with intermediate bottom meson loops. Our results have shown that the production ratios for the Υ(5S, 6S) → γX b are about orders of 10 −5 with a commonly accepted cutoff range α = 2 ∼ are several times smaller than the ELA results in small binding energy range, while the predictions of these two methods are comparable at large binding energy. In Ref. [28,29], we have studied the radiative decays and the hidden bottomonium decays of X b . If we consider that the branching ratios of the isospin conserving process X b → ωΥ(1S) are relatively large, a search for Υ(5S) → γX b → γωΥ(1S) may be possible for the updated BelleII experiments. These studies may help us investigate the X b deeply. The experimental observation of X b will provide us with further insight into the spectroscopy of exotic states and is helpful to probe the structure of the states connected by the heavy quark symmetry.