We investigate the production of Xb in the process Υ(5S,6S)→γXb, where Xb is assumed to be the counterpart of X(3872) in the bottomonium sector as BB¯⁎ 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 Υ(5S,6S)→γXb are orders of 10-5 with reasonable cutoff parameter range α≃2~3. 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 Xb.
National Natural Science Foundation of China112751131157510011505104Natural Science Foundation of Shandong ProvinceZR2015JL0011. Introduction
In the past decades, many so-called XYZ have been observed by the Belle, BaBar, CDF, D0, CMS, LHCb, and BESIII collaborations [1]. Some of them cannot fit into the conventional heavy quarkonium in the quark model [2–5]. Up to now, many studies on the production and decay of these XYZ states have been carried out in order to understand its nature (for a recent review, see [6–8]).
In 2003, the Belle collaboration discovered an exotic candidate X(3872) in the process B+→K++J/ψπ+π- [9] which was subsequently confirmed by the BaBar collaboration [10] in the same channel. It was also discovered in proton-proton/antiproton collisions at the Tevatron [11, 12] and LHC [13, 14]. X(3872) is a particularly intriguing state because on the one hand its total width Γ<1.2 MeV [1] is tiny compared to typical hadronic widths and on the other hand the closeness of its mass to D0D¯∗0 threshold (MX(3872)-MD0-MD∗0=(-0.12±0.24) MeV) and its prominent decays to D0D¯∗0 [1] suggest that it may be an meson-meson molecular state [15, 16].
Many theoretical works have been carried out in order to understand the nature of X(3872) since the first observation of X(3872). It is also natural to look for the counterpart with JPC=1++ (denoted as Xb hereafter) in the bottom sector. These two states are related by heavy quark symmetry which should have some universal properties. The search for Xb may provide us with important information on the discrimination of a compact multiquark configuration and a loosely bound hadronic molecule configuration. Since the mass of Xb may be very heavy and its JPC 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 [18], a search for Xb in the ωΥ(1S) final states has been presented and no significant signal is observed for such a state.
The production of Xb at the LHC and the Tevatron [19, 20] and other exotic states at hadron colliders [21–26] has been extensively investigated. In the bottomonium system, the isospin is almost perfectly conserved, which may explain the escape of Xb in the recent CMS search [27]. As a result, the radiative decays and isospin conserving decays will be of high priority in searching for Xb [28–30]. In [28], we have studied the radiative decays of Xb→γΥ(nS) (n=1,2,3), with Xb being a candidate for BB¯∗ molecular state, and found that the partial widths into γXb are about 1 keV. In [29], we studied the rescattering mechanism of the isospin conserving decays Xb→Υ(1S)ω, and our results show that the partial width for Xb→Υ(1S)ω is about tens of keVs.
In this work, we will further investigate Xb production in Υ(5S,6S)→γXb with Xb being 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 being noticed for a long time [31–33]. Recently, this mechanism has been used to study the production and decays of ordinary and exotic states [34–60] and B decays [61–68], and a global agreement with experimental data was obtained. Thus this approach may be suitable for the process Υ(5S,6S)→γXb.
The paper is organized as follows. In Section 2, we present the effective Lagrangians for our calculation. Then in Section 3, we present our numerical results. Finally we give the summary in Section 4.
2. Effective Lagrangians
Based on the heavy quark symmetry, we can write out the relevant effective Lagrangian for Υ(5S) [68, 69]:(1)LΥ5SB∗B∗=igΥBBΥμ∂μBB¯-B∂μB¯-gΥB∗Bϵμναβ∂μΥν∂αB∗βB¯+B∂αB¯∗β-igΥB∗B∗Υμ∂μB∗νB¯ν∗-B∗ν∂μB¯ν∗+∂μΥνB∗ν-Υν∂μB∗νB¯∗μ+B∗μΥν∂μB¯ν∗-∂μΥνB¯ν∗,where B(∗)=B(∗)+,B(∗)0 and B¯(∗)T=B(∗)-,B¯(∗)0 correspond to the bottom meson isodoublets. ϵμναβ is the antisymmetric Levi-Civita tensor and ϵ0123=+1. Since Υ(5S) is above the threshold of B(∗)B¯(∗), the coupling constants between Υ(5S) and B(∗)B¯(∗) can be determined via experimental data for Υ(5S)→B(∗)B¯(∗) [1]. The experimental branching ratios and the corresponding coupling constants are listed in Table 1. Since there is no experimental information on Υ(6S)→B(∗)B¯(∗) [1], we choose the coupling constants between Υ(6S) and B(∗)B¯(∗), the same values as that of Υ(5S).
The coupling constants of Υ(5S) interacting with B(∗)B-(∗). Here, we list the corresponding branching ratios of Υ(5S)→B(∗)B-(∗).
Final state
B (%)
Coupling
BB-
5.5
1.76
BsB-s
0.5
0.96
BB-∗+c.c.
13.7
0.14 GeV-1
BsB-s∗+c.c.
1.35
0.10 GeV-1
B∗B-∗
38.1
2.22
Bs∗B-s∗
17.6
5.07
In order to calculate the process depicted in Figure 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 [70, 71](2)Lγ=eβQab2FμνTrHb†σμνHa+eQ′2mQFμνTrHa†Haσμν,with(3)H=1+v2B∗μγμ-Bγ5,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≃3.0GeV-1 is determined in the nonrelativistic constituent quark model and has been adopted in the study of radiative D∗ decays [71]. In b and c systems, β value is the same due to heavy quark symmetry [71]. In (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/mQ.
Feynman diagrams for Xb production in Υ(5S)→γXb under BB¯∗ meson loop effects.
At last, assume that Xb is S-wave molecule with JPC=1++ given by the superposition of B0B¯∗0+c.c. and B-B¯∗++c.c. hadronic configurations as(4)Xb=12B0B¯∗0-B∗0B¯0+B+B∗--B-B∗+.As a result, we can parameterize the coupling of Xb to the bottomed mesons in terms of the following Lagrangian:(5)L=12Xbμ†x1B∗0μB¯0-B0B¯∗0μ+x2B∗+μB--B+B∗-μ+h.c.,where xi denotes the coupling constant. Since Xb is slightly below S-wave BB¯∗ threshold, the effective coupling of this state is related to the probability of finding BB¯∗ component in the physical wave function of the bound states and the binding energy, ϵXb=mB+mB∗-mXb [36, 72, 73]:(6)xi2≡16πmB+mB∗2ci22ϵXbμ,where ci=1/2 and μ=mBmB∗/(mB+mB∗) is the reduced mass. Here, we should also notice that the coupling constant xi in (6) is based on the assumption that Xb 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 Figure 1 can be generally expressed as follows:(7)Mfi=∫d4q22π4∑B∗pol.T1T2T3D1D2D3Fm2,q22,where Ti and Di=qi2-mi2(i=1,2,3) are the vertex functions and the denominators of the intermediate meson propagators. For example, in Figure 1(a), Ti(i=1,2,3) are the vertex functions for the initial Υ(5S), final Xb, and photon, respectively. Di(i=1,2,3) are the denominators for the intermediate B+, B-, and B∗+ propagators, respectively.
Since the intermediate exchanged bottom mesons in the triangle diagram in Figure 1 are off-shell, in order to compensate these off-shell effects arising from the intermediate exchanged particle and also the nonlocal effects of the vertex functions [74–76], we adopt the following form factors:(8)Fm2,q22≡Λ2-m22Λ2-q22n,where n=1,2 corresponds to monopole and dipole form factor, respectively. Λ≡m2+αΛQCD and the QCD energy scale ΛQCD=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 [77], where radiative decays of charmonium are studied extensively based on effective Lagrangian approach.
3. Numerical Results
Before proceeding the numerical results, we first briefly review the predictions on mass of Xb. The existence of Xb is predicted in both the tetraquark model [78] and those involving a molecular interpretation [79–81]. In [78], the mass of the lowest-lying 1++b¯q¯bq tetraquark is predicated to be 10504 MeV, while the mass of BB¯∗ molecular state is predicated to be a few tens of MeV higher [79–81]. For example, in [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 (10580-8+9) MeV, which corresponds to a binding energy (24-9+8) MeV in [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 of the previous molecular and tetraquark predictions on [78–81], we present our results up to a binding energy of 100 MeV, and we will choose several illustrative values: ϵXb=(5,10,25,50,100) MeV.
In Table 2, we list the predicted branching ratios by choosing the monopole and dipole form factors and three values for the cutoff parameter in the form factor. As a comparison, we also list the predicted branching ratios in NREFT approach. From this table, we can see that the branching ratios for Υ(5S)→γXb are orders of 10-5. The results are not sensitive to both the form factors and the cutoff parameter we choose.
Predicted branching ratios for Υ(5S)→γXb. The parameter in the form factor is chosen as α=2.0, 2.5, and 3.0. The last column is the calculated branching ratios in NREFT approach.
Binding energy
Monopole form factor
Dipole form factor
NREFT
α=2.0
α=2.5
α=3.0
α=2.0
α=2.5
α=3.0
ϵXb=5 MeV
2.02×10-5
2.06×10-5
2.08×10-5
1.90×10-5
1.99×10-5
2.04×10-5
1.52×10-6
ϵXb=10 MeV
2.58×10-5
2.66×10-5
2.71×10-5
2.32×10-5
2.47×10-5
2.57×10-5
2.12×10-6
ϵXb=25 MeV
3.24×10-5
3.42×10-5
3.54×10-5
2.61×10-5
2.90×10-5
3.09×10-5
3.88×10-6
ϵXb=50 MeV
3.37×10-5
3.65×10-5
3.85×10-5
2.37×10-5
2.75×10-5
3.04×10-5
6.41×10-6
ϵXb=100 MeV
2.91×10-5
3.27×10-5
3.54×10-5
1.65×10-5
2.05×10-5
2.38×10-5
1.20×10-5
In Figure 2(a), we plot the branching ratios for Υ(5S)→γXb in terms of the binding energy ϵXb with the monopole form factors α=2.0 (solid line), 2.5 (dashed line), and 3.0 (dotted line), respectively. The coupling constant of Xb in (6) and the threshold effects can simultaneously influence the binding energy dependence of the branching ratios. With the increasing of the binding energy ϵXb, the coupling strength of Xb increases, and the threshold effects decrease. Both the coupling strength of Xb and the threshold effects vary quickly in the small ϵXb region and slowly in the large ϵXb region. As a result, the behavior of the branching ratios is relatively sensitive at small ϵXb, while it becomes smooth at large ϵXb. Results with the dipole form factors α=2.0, 2.5, and 3.0 are shown in Figure 2(b) as solid, dash, and dotted curves, respectively. The behavior is similar to that of Figure 2(a).
(a) The dependence of the branching ratios of Υ(5S)→γXb on ϵXb using monopole form factors with α=2.0 (solid lines), α=2.5 (dashed lines), and α=3.0 (dotted lines), respectively. (b) The dependence of the branching ratios of Υ(5S)→γXb on ϵXb using dipole form factors with α=2.0 (solid lines), α=2.5 (dashed lines), and α=3.0 (dotted lines), respectively. The results with binding energy up to 100 MeV might make the molecular state assumption inaccurate.
We also predict the branching ratios of Υ(6S)→γXb and present the relevant numerical results in Table 3 and Figure 3 with the monopole and dipole form factors. At the same cutoff parameter α, the predicted rates for Υ(6S)→γXb are a factor of 2-3 smaller than the corresponding rates for Υ(5S)→γXb. It indicates that the intermediate B-meson loop contribution to the process Υ(6S)→γXb is smaller than that to Υ(5S)→γXb. This is understandable since the mass of Υ(6S) is more far away from the thresholds of B(∗)B(∗) than Υ(5S). But their branching ratios are also about orders of 10-5 with a reasonable cutoff parameter α=2~3.
Predicted branching ratios for Υ(6S)→γXb. The parameter in the form factor is chosen as α=2.0, 2.5, and 3.0. The last column is the calculated branching ratios in NREFT approach.
Binding energy
Monopole form factor
Dipole form factor
NREFT
α=2.0
α=2.5
α=3.0
α=2.0
α=2.5
α=3.0
ϵXb=5 MeV
9.71×10-6
1.02×10-5
1.05×10-5
8.16×10-6
9.04×10-6
9.63×10-6
3.38×10-6
ϵXb=10 MeV
1.25×10-5
1.33×10-5
1.38×10-5
9.97×10-6
1.13×10-5
1.22×10-5
4.89×10-6
ϵXb=25 MeV
1.62×10-5
1.76×10-5
1.85×10-5
1.14×10-5
1.34×10-5
1.49×10-5
8.27×10-6
ϵXb=50 MeV
1.76×10-5
1.96×10-5
2.12×10-5
1.08×10-5
1.32×10-5
1.52×10-5
1.30×10-5
ϵXb=100 MeV
1.66×10-5
1.92×10-5
2.12×10-5
8.12×10-6
1.06×10-5
1.28×10-5
2.24×10-5
(a) The dependence of the branching ratios of Υ(6S)→γXb on ϵXb using monopole form factors with α=2.0 (solid lines), α=2.5 (dashed lines), and α=3.0 (dotted lines), respectively. (b) The dependence of the branching ratios of Υ(6S)→γXb on ϵXb using dipole form factors with α=2.0 (solid lines), α=2.5 (dashed lines), and α=3.0 (dotted lines), respectively. The results with binding energy up to 100 MeV might make the molecular state assumption inaccurate.
In [51], authors introduced a nonrelativistic effective field theory method to study the meson loop effects of ψ′→J/ψπ0. 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 Figure 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:(9)v5v23q2~q2v,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 [36] with only different masses and coupling constants. The obtained numerical results for Υ(5S)→γXb and Υ(6S)→γXb in terms of the binding energy are listed in the last column of Tables 2 and 3, respectively. As shown in Table 2, except for the largest binding energy ϵXb=100MeV, the NREFT predictions of Υ(5S)→γXb are about 1 order of magnitude smaller than the ELA results at the commonly accepted range. For Υ(6S)→γXb shown in Table 3, 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 differences may give some sense of the theoretical uncertainties for the predicted rates and indicate the viability of our model to some extent.
Here we should notice, for the isoscalar Xb, the pion exchanges might be nonperturbative and produce sizeable effects [81–83]. In [81], their calculations show that the relative errors of C0X are about 20% for Xb. Even if we take into account this effect, the estimated order of the magnitude for the branching ratio Υ(5S,6S)→γXb may also be sizeable, which may be measured in the forthcoming BelleII experiments.
4. Summary
In this work, we have investigated the production of Xb in the radiative decays of Υ(5S,6S). Based on 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 Υ(5S,6S)→γXb are about orders of 10-5 with a commonly accepted cutoff range α=2~3. As a cross-check, we also calculated the branching ratios of the decays in the framework of NREFT. Except for the large binding energy, the NREFT predictions of Υ(5S)→γXb are about 1 order of magnitude smaller than the ELA results. The NREFT predictions of Υ(6S)→γXb 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 [28, 29], we have studied the radiative decays and the hidden bottomonium decays of Xb. If we consider that the branching ratios of the isospin conserving process Xb→ωΥ(1S) are relatively large, a search for Υ(5S)→γXb→γωΥ(1S) may be possible for the updated BelleII experiments. These studies may help us investigate Xb deeply. The experimental observation of Xb 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.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This work is supported in part by the National Natural Science Foundation of China (Grants nos. 11275113, 11575100, and 11505104) and the Natural Science Foundation of Shandong Province (Grant no. ZR2015JL001).
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