Hunting for the X_b via Radiative Decays

In this paper, we study radiative decays of X_b, the counterpart of the famous X(3872) in the bottomonium-sector as a candidate for meson-meson molecule, into the \gamma \Upsilon(nS) (n=1, 2, 3). Since it is likely that the X_b is below the B\bar B^* threshold and the mass difference between the neutral and charged bottom meson is small compared to the binding energy of the X_b, the isospin violating decay mode X_b\to \Upsilon (nS)\pi^+\pi^- would be greatly suppressed. This will promote the importance of the radiative decays. We use the effective Lagrangian based on the heavy quark symmetry to explore the rescattering mechanism and calculate the partial widths. Our results show that the partial widths into \gamma \Upsilon(nS) are about 1 keV, and thus the branching fractions may be sizeable, considering the fact the total width may also be smaller than a few MeV like the X(3872). These radiative decay modes are of great importance in the experimental search for the X_b particularly at hadron collider. An observation of the X_b will provide a deeper insight into the exotic hadron spectroscopy and is helpful to unravel the nature of the states connected by the heavy quark symmetry.


I. INTRODUCTION
In the past decades, there has been great progress in hadron spectroscopy thanks to the unprecedented data sample accumulated by the B factories and hadron-hadron colliders. A number of charmonium-like and bottomonium-like states have been discovered on these experimental facilities so far but not all of them can be placed in the ordinaryqq (for reviews, see Refs. [1][2][3][4]).
The X(3872) is the first and perhaps the most renowned exotic candidate. It was first discovered in 2003 by Belle in the B + → K + + J/ψπ + π − final state [5] and subsequently confirmed by the BaBar Collaboration [6]. Complementary observation is also found in proton-proton/antiproton collisions at the Tevatron [7,8] and LHC [9,10]. Though the existence is well established, the nature of the X(3872) is still ambiguous due to a few peculiar properties. First, compared to typical hadronic widths the total width is tiny. Only an upper bound has been measured experimentally: Γ < 1.2 MeV [11]. The mass lies closely to the D 0 D * 0 threshold, M X(3872) − M D 0 − M D * 0 = (−0.12 ± 0.24) MeV [12], which leads to speculations that the X(3872) is presumably a meson-meson molecular state [13,14].
These peculiar features have stimulated considerable research interest in investigating the production and decays of the X(3872) towards understanding its nature. A very important aspect involves the discrimination of a compact multiquark configuration and a loosely bound hadronic molecule configuration. In this viewpoint, it would be also valuable to look for the analogue in the bottom sector, referred to as X b following the notation suggested in Ref. [15], as states related by heavy quark symmetry may have universal behaviours. Since the X b is expected to be very heavy and its J P C of 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 [16].
In Ref. [17], the production of the X b at the LHC and the Tevatron has been investigated, along the same line with the studies on the search for exotic states at hadron colliders [18][19][20][21][22][23][24]. It is shown that the production rates at the LHC and the Tevatron are sizeable [17]. On the other hand, the search for the X b also depends on reconstructing the X b , which motivates us to study the X b decays. Since this meson is expected to be far below threshold, the isospin violating decay mode for instance X b → Υπ + π − is highly suppressed, and this may explain the escape of X b in the recent CMS search [25]. As a consequence, radiative decays of the X b will be of high priority, on which we will focus in this paper. As we will show in the following, these modes have sizeable decay widths.
The paper is organized as follows. In Sec. II, we will introduce the formalism used in this work.
Based on this framework, numerical results are presented in Sec. III and the summary will be given in Sec. IV.

II. RADIATIVE DECAYS
The calculation of contributions from the meson loops requests the leading order effective Lagrangian. Based on the heavy quark symmetry, we employ the relevant effective Lagrangian for the Υ(nS) [54,55] where blets. ǫ µναβ is the anti-symmetric Levi-Civita tensor and ǫ 0123 = −1. Due to the heavy quark symmetry, the following relationships of the couplings are valid [54,55] where g n = √ m Υ(nS) /(2m B f Υ(nS) ); m Υ(nS) and f Υ(nS) denote the mass and decay constant of Υ(nS), respectively. The decay constant f Υ(nS) can be extracted from the Υ(nS) → e + e − : where α = 1/137 is the electromagnetic fine-structure constant. Using the masses and leptonic We consider the iso-scalar X b as a S-wave molecular state with the positive charge parity given by the superposition of B 0B * 0 + c.c and B −B * + + c.c hadronic configurations as The coupling of X b to the bottomed meson is based on the effective Lagrangian where x i denotes the coupling constant.
For a bound state below an S-wave two-hadron threshold, the effective coupling of this state to the two-body channel is related to the probability of finding the two-hadron component in the physical wave function of the bound states and the binding energy, where The magnetic coupling of the photon to heavy bottom meson is described by the Lagrangian [58,59] with where Q = diag{2/3, −1/3, −1/3} is the light quark charge matrix, β is an unknown parameter and Q ′ is the heavy quark electric charge (in units of e). In the nonrelativistic constituent quark model β ≃ 3.0 GeV −1 , which has been adopted in the study of radiative D * decays [59]. Note heavy quark symmetry ensures that β is the same in the b and c systems, so we take the same value as Ref. [59]. 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 decay amplitudes for the transitions in Fig. 1 can be expressed in a generic form in the effective Lagrangian approach as follows, where V i and a i = q 2 i − m where Λ ≡ m 2 + αΛ QCD and the QCD energy scale Λ QCD = 220 MeV. This form factor is supposed to compensate the off-shell effects arising from the intermediate exchanged particle and the nonlocal effects of the vertex functions [60][61][62], and phenomenological studies have suggested α ∼ 2.
The explicit expression of the transition amplitudes can be found in Appendix (A.6) in Ref. [63], where radiative decays of charmonium are studied extensively based on the effective Lagrangian approach.

III. NUMERICAL RESULTS
The existence of the X b was predicted in both the tetraquark model [64] and hadronic molecular calculations [65][66][67]. The mass of the lowest-lying 1 ++bq bq tetraquark was predicted to be 10504 MeV in Ref. [64], while the mass of the BB * molecule based on the mass of the X(3872) is a few tens of MeV higher [66,67]. In Ref. [66], the mass was predicted to be (10580 +9 −8 ) MeV, corresponding to a binding energy of (24 +8 −9 ) MeV. These studies have provided a range for the binding energy, for which in the following we will choose a few illustrative values: Choosing two values for the cutoff parameter α, we have predicted the partial decay widths   Table I. From this table, we can see that the widths for the X b radiative decays are about 1 keV. It is noteworthy to recall that the upper bound for the Γ(X(3872)) is 1.2 MeV [11]. If the X b were similarly narrow, our results would indicate a sizeable branching fractions, at least 10 −3 , for these radiative decay modes.
In Fig. 2, we present the partial widths for the this enhancement structure is independent of the cutoff parameter α.
It would be interesting to further clarify the uncertainties arising from the introduction of the form factors by studying the ratios between different partial decay widths. We define the following ratios which are plotted in Fig. 3 for the dependence on the cutoff parameter and Fig. 4 for the dependence on binding energy. Since the first coupling vertices are the same for those decay channels when taking the ratio, so the ratio only reflects the open threshold effects through the intermediate bottomed meson loops. The ratios are less sensitive to the cutoff parameter, which is a consequence of the fact that the involved loops are the same. As can be seen from this figure, when the cutoff parameter α increases, the ratios decrease. These predictions can be tested by the experimental measurements in future.

IV. SUMMARY
Our understanding of hadron spectroscopy will be greatly improved by studies of exotic states that may defy the conventional models of qq meson spectroscopy, and accordingly great progress has been made in the past decades. One of the most important aspects in the study of exotics is the discrimination of a compact multiquark configuration and a loosely bound hadronic molecule.
Such task requests a large amount of efforts on both experimental and theoretical sides in future.
In this work, we have investigated the radiative decays of the X b , the counterpart of the famous X(3872) in the bottomonium-sector as a candidate for meson-meson molecule, into the γΥ(nS).
Since this state may be far below the BB * threshold, the isospin violating decay mode X b → Υπ + π − would be highly suppressed, and stimulate the importance of the radiative decays. We have made used of the effective Lagrangian based on the heavy quark symmetry, and explore the rescattering mechanism. Our results have shown that the partial widths for the X b → γΥ(nS) are about 1 keV, and thus the branching fractions may be sizeable, taking into account the fact the total width may also be smaller than a few MeV like X(3872). This study of radiative decays and the previous work on production rates in hadron-hadron collisions have indicated a promising prospect to find the X b at hadron collider in particular the LHC, and we suggest our experimental colleagues to perform an analysis. Such attempt will likely lead to the discovery of the X b and thus enrich the exotics garden in the heavy quarknoium sector.