Production of $X_b$ via $\Upsilon(5S, 6S)$ radiative decays

We investigate the production of $X_b$ in the process $\Upsilon(5S,6S)\to \gamma X_b$, where $X_b$ is assumed to be a $B {\bar B}^*$ molecular state. Two kinds of meson loops of $B^{(*)}{\bar B}^{(*)}$ and $B_1^{\prime}{\bar B}^{(*)}$ were considered. To explore the rescattering mechanism, we calculated the relevant branching ratios using the effective Lagrangian based on the heavy quark symmetry. The branching ratios for the $\Upsilon(5S\,,6S) \to \gamma X_b$ were found to be at the orders of $10^{-7} \sim 10^{-6}$. Such sizeable branching ratios might be accessible at BelleII, which would provide important clues to the inner structures of the exotic state $X_b$.

A lot of theoretical effort has been made to understand the nature of X(3872) since its initial observation.Naturally, it follows to look for the counterpart with J P C = 1 ++ (denoted as X b hereafter) in the bottom sector.These two states, which are related by heavy quark symmetry, should have some universal properties.The search for X b could provide us the discrimination between a compact multiquark configuration and a loosely bound hadronic molecule configuration.Since the mass of X b is very heavy and its J P C are 1 ++ , a direct discovery is unlikely at the current electron-positron collision facilities, though the Υ(5S, 6S) radiative decays are possible in the Super KEKB [19].In Ref. [20], a search for X b in the ωΥ(1S) final states has been presented, but no significant signal is observed.The production of X b at the LHC and the Tevatron [21,22] and other exotic states at hadron colliders [23][24][25][26][27][28] 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 [29].As a result, the radiative decays and isospin conserving decays are of high priority in searching X b [30][31][32][33].In Ref. [30], we have studied the radiative decays X b → γΥ(nS) (n = 1, 2, 3), with X b being a candidate for the B B * molecular state, and the partial widths into γX b were found to be about 1 keV.In this work, we revisit the X b production in Υ(5S, 6S) → γX b using the nonrelativistic effective field theory (NREFT).
The rest of the paper is organized as follows.In Sec.II, we present the theoretical framework used in this work.
Then in Sec.III the numerical results are presented, and a brief summary is given in Sec.IV.Under the assumption that X b is a B B * molecule, its production can be described by the triangle diagrams in Fig. 1.With the quantum numbers of 1 −− , the initial bottomonium can couple to either two S-wave bottomed mesons in a P -wave, or one P -wave and one S-wave bottomed mesons in an S-or D-wave.The X b couples to the B B * pair in an S-wave.Because the states considered here are close to the open bottomed mesons thresholds, the intermediate bottomed and antibottomed mesons are nonrelativistic.We are thus allowed to use a nonrelativistic power counting, the framework of which has been introduced to study the intermediate meson loop effects [45].The three momentum scales as v, the kinetic energy scales as v 2 , and each of the nonrelativistic propagator scales as v −2 .The S-wave vertices are independent of the velocity, while the P -wave vertices scales as v or as the external momentum, depending on the process in question.
For the diagrams (a), (b), and (c) in Fig. 1, the vertices involving the initial bottomonium are in a P -wave.The momentum in these vertices is contracted with the final photon momentum q and thus should be counted as q.The vertices involving the photon are also in a P -wave, which should be counted as q.The decay amplitude scales as where E γ is the external photon energy, N A contains all the constant factors, and v A is the average of the two velocities corresponding to the two cuts in the triangle diagram.While for the diagrams (d) and (e) in Fig. 1, all the vertices are in S-wave.Then the amplitude for the Figs.1(d) and (e) scales as To calculate the diagrams in Fig. 1, we employ the effective Lagrangians constructed in the heavy quark limit.In this limit, the S-wave heavy-light mesons form a spin multiplet H = (P, V ) with s P l = 1/2 − , where P and V denote the pseudoscalar and vector heavy mesons, respectively, i.e., P (V ) = (B ( * )+ , B ( * )0 , B ( * )0 s ).The s P l = 1/2 + states are collected in S = (P * 0 , P 1 ) with P * 0 and P 1 denoting the B * 0 and B 1 states, respectively.In the two-component notation [47,48], the spin multiplets are given by where σ is the Pauli matrix, and a is the light flavor index.The fields for their charge conjugated mesons are Considering the parity, the charge conjugation, and the spin symmetry, the leading order Lagrangian for the coupling of the S-wave bottomonium fields to the bottomed and antibottomed mesons can be written as [47] L Here The field for the S-wave Υ and η b is Υ = Υ • σ + η b .g 1 and g 2 are the coupling constants of Υ(5S) to a pair of 1/2 − bottom mesons and a 1/2 − -1/2 + pair of bottom mesons, respectively.We use g 1 and g 2 for the coupling constants of Υ(6S).Using the experimental branching ratios and widths of Υ(5S, 6S) [1], we get the coupling constants g 1 = 0.1 GeV −3/2 and g 1 = 0.08 GeV −3/2 .On the other hand, we take g 2 = g 2 = 0.05 GeV −1/2 , as used in the previous work [49].
To get the transition amplitude, we also need to know the photonic coupling to the bottomed mesons.The magnetic coupling of the photon to the S-wave bottomed mesons is described by the Lagrangian [48,50] where 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 is an effective coupling constant and, in this work, we take β 3.0 GeV −1 , which is determined in the nonrelativistic constituent quark model and has been adopted in the study of radiative D * decays [50].In Eq. ( 6), 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 .The radiative transition of the 1/2 + bottomed mesons to the 1/2 − states may be parameterized as [51] L where β = 0.42 GeV −1 is the same as used in Ref. [52].
The X b is assumed to be an S-wave molecule with J P C = 1 ++ , which is given by the superposition of B 0 B * 0 + c.c and B − B * + + c.c hadronic configurations: Therefore, 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 B B * threshold, the effective coupling of this state is related to the probability of finding the B B * component in the physical wave function of the bound states and the binding energy, [39,53,54] where Here, it should be pointed out that the coupling constant x i in Eq. ( 10) is based on the assumption that X b is a shallow bound state where the potential binding the mesons is short-ranged.
The decay amplitudes of the triangle diagrams in Fig. 1 can be obtained and the explicit transition amplitudes for Υ(5S, 6S) → γX b are presented in Appendix A. The partial decay widths of Υ(5S, 6S) → γX b are given by where E γ is the photon energies in the Υ(5S, 6S) rest frame.

III. NUMERICAL RESULTS
In Ref. [55], authors predicted a large width of 238 MeV for B 1 .This large width effect for B 1 was taken into account in our calculations by using the Breit-Wigner (BW) parameterization to approximate the spectral function of the 1/2 + bottom meson of width.The explicit formula for B 1 is where is the normalization factor, MB 1 (s) represents the loop amplitude of B 1 calculated using s as the mass squared, Before proceeding to the numerical results, we first briefly review the predictions of the mass of X b .The existence of the X b is predicted in both the tetraquark model [56] and those involving a molecular interpretation [57][58][59].In Ref. [56], the mass of the lowest-lying 1 ++ bqbq tetraquark is predicated to be 10504 MeV, while the mass of the B B * molecular state is predicated to be a few tens of MeV higher [57][58][59].For example, in Ref. [57], the mass was predicted to be 10562 MeV, corresponding to a binding energy of 42 MeV, while with a binding energy of (24 +8 −9 ) MeV it was predicted to be (10580 +9 −8 ) MeV [59].Therefore, 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 for the previous molecular and tetraquark predictions in Refs.[56][57][58][59], we performed the calculations up to a binding energy of 100 MeV and choose several illustrative values of X b = (5, 10, 25, 50, 100) MeV for discussion.
In Table I, we list the contributions of Υ(5S) → γX b from B ( * ) B( * ) loops, B 1 B( * ) loops, and the total contributions.
For the B 1 , we choose the Γ B 1 to be 0, 100 MeV and 200 MeV, respectively.It can be seen that the contributions   10) and the threshold effects can simultaneously influence the binding energy dependence of the partial widths.With increasing the binding energy X b , the coupling strength of X b increases, and the threshold effects decrease.Both the coupling strength of X b and the threshold effects vary quickly in the small X b region and slowly in the large X b region.As a result, the partial width is relatively sensitive to the small X b , while at the large X b region it keeps nearly constant.As seen, at the same binding energy, the partial widths with small Γ B 1 are larger

FIG. 2 :
FIG. 2: The dependence of the decay widths of Υ(5S) → γX b (a) and Υ(6S) → γX b (b) on the binding energy for different B 1 widths as indicated by the numbers in the graph.The right y-axis represents the corresponding branching ratio.

TABLE I :
The predicted decay widths (in units of keV) of Υ(5S) → γX b for different binding energies.Here we choose the Γ B 1 to be 0, 100, and 200 MeV, respectively.