$f_J(2220)$ and Hadronic $\bar B^0_s$ Decays

We study the hadronic $\bar B^0_s$ decays based on the existence of the resonant state $f_J(2220)$. In particular, we are able to explain the unexpected large experimental result of ${\cal B}(\bar B^0_s\to J/\psi p\bar p) =(3.0^{+1.2}_{-1.1}\pm 0.52\pm 0.03)\times 10^{-6}$ measured recently by the LHCb collaboration due to the resonant contribution in $\bar B^0_s\to J/\psi f_J(2220)$ with $f_J(2220) \to p\bar p$, while it is estimated to be at most of order $10^{-9}$ in terms of the OZI rule without the resonance. In addition, we find that ${\cal B}(\bar B^0_s\to D^{*0}(f_J\to) p\bar p)=(4.05\pm 2.46)\times 10^{-7}$, ${\cal B}(\bar B^0_s\to J/\psi(f_J\to)\pi\pi)=(15.6\pm 15.2)\times 10^{-6}$ and ${\cal B}(\bar B^0_s\to D^{*0}(f_J\to)\pi\pi)=(21.2\pm 20.9)\times 10^{-7}$, while ${\cal B}(\bar B^0_s\to J/\psi(f_J\to)K\bar K)<1.6\times 10^{-5}$ and ${\cal B}(\bar B^0_s\to D^{*0}(f_J\to)K\bar K)<2.2\times 10^{-6}$. Moreover, we predict that the decay branching ratios of $\bar B^0_s\to (J/\psi\,,D^{*0})\Lambda\bar \Lambda$ are $(2.68\pm1.23)\times 10^{-7}$ and $(2.25\pm0.80)\times 10^{-6}$. Some of the predicted $\bar B^0_s$ decays are accessible to the experiments at the LHCb.


I. INTRODUCTION
In some three-body B meson decay of B → BB ′ M, with BB ′ a baryon pair and M a recoiled meson or photon, the partial decay width as the function of m BB ′ = p B + pB′ is observed to have a peak near m BB ′ ≃ m B + mB′ of the threshold area. This is the so-called threshold enhancement, which dominates the decay branching ratio of B → BB ′ M. The examples of these decays include B → ppM with M = (D ( * ) , K ( * ) , π, ρ) and B → ΛpM ′ with M ′ = (π, ρ, γ). Theoretically, the threshold effect has been realized as the result of the perturbative QCD (pQCD) effect [1,2]. Consequently, many experimental data on the baryonic B decays can be well explained [3][4][5].
However, it is not the case forB 0 s → J/ψpp. The branching ratio ofB 0 s → J/ψpp presented by the LHCb collaboration is given by [6]: where the first and second uncertainties are statistical and systematic, respectively, while the third one originates from the control channel branching fraction measurement. Note that B(B 0 → J/ψpp) = (2.0 +1.9 −1.7 ± 0.9 ± 0.1) × 10 −7 has been also given by the LHCb [6]. With B 0 → (cc)(dd) → J/ψpp, the pp production has the direct transition fromB 0 → dd → pp, which associates with the threshold enhancement, such that theoretical prediction of (11.4 ± 5.0) × 10 −7 in Refs. [5,7] can be consistent with the observation. On the contrary,B 0 s → J/ψpp viaB 0 s → ss → pp leads to the OZI suppression, while ss should be first annihilated to produce pp. With the OZI suppression of B(φ → ππ)/B(φ → KK) ≃ 10 −4 [8], one expects [8]. Therefore, to understand the large branching ratio of around 3 × 10 −6 forB 0 s → J/ψpp in Eq. (1), a new theoretical study on this decay is clearly needed.
To explain B(B 0 s → J/ψpp), one possible solution is to have a resonant state between the ss annihilation and pp production inB 0 s → J/ψpp, so that the process through its mass shell allows an on-shell enhancement for the decay branching ratio. Indeed, it is common to observe resonant peaks in B → ppM. For example, one finds the cc mesons, where the resonant η c → pp and J/ψ → pp raise the m pp spectrum of B − → K − pp [9], as well as those identified as the charmed baryons and the glueball from D ( * ) p and pp spectra in [10][11][12], respectively. According to Refs. [8,13], since f J (2220) ≡ f J with the quantum numbers J P C = 2 ++ or 4 ++ has the channel of f J → pp, particularly, with its mass and decay width within the allowed region of the m pp spectrum inB 0 s → J/ψpp, it is reasonable that f J can be our candidate as the resonant state inB 0 s → J/ψpp. The experimental status of f J is reported in Ref. [14], where its evidences come from the Mark III collaboration [15] and the BES collaboration [16], also being supported by π − (K − )p collisions [17]. However, the direct confirmations from pp collisions [18] and 2γ processes [19] are inconclusive. Hence, it leaves the room for theB 0 s meson decays to provide the new scenario for the f J study. Moreover, according to the QCD models [20], such as the Lattice QCD (LQCD) calculation [21], f J has the mass close to that of the tensor glueball (G 2 ++ ) with J P C = 2 ++ . Moreover, the theoretical prediction of [22] agrees with the lower bound of B(J/ψ → γf J ) > 2.5 × 10 −3 [8]. With f J (2220) being identified as G 2 ++ , Eq. (2) can be related to the radiative J/ψ decays by the BABAR collaboration [24], given by [8], such that we obtain where the limit is based on the 1σ error of the measured value on J/ψ → γG 2 ++ . We remark that the results in Eq. (4) are consistent with the ratios: (0.17 ± 0.09, 1.0 ± 0.5) in the PDG [8].
In this paper, we shall explainB 0 s → J/ψpp with f J (2220) as the resonant state to pp. Due to this resonant state, we will also study the other hadronic decays ofB 0 s , such as

II. FORMALISM
In the effective Hamiltonian [23], the amplitude ofB 0 s → V BB ′ with the baryon pair BB ′ = pp or ΛΛ can be factorized as where G F is the Fermi constant, V q 1 q 2 are the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, and a V 2 is the coefficient studied in Ref. [5], while (q 1 q 2 ) V −A denotesq 1 γ µ (1 − γ 5 )q 2 and V stands for the vector meson J/ψ(D * 0 ) with q = c(u). In Eq. (5), the matrix element of the vector meson production is defined by where m V , f V and ε * µ are the mass, decay constant and polarization of the vector meson V , respectively.
ForB 0 s → V ΛΛ, since the ss pair can have a direct transition to be a part of the internal quarks in ΛΛ as seen in Fig. 1a, the most general matrix elements of theB 0 s → ΛΛ transition are given by [4] with p = pB0 s −p Λ −pΛ and the form factors g i and f i (i=1,2, ..., 5). In the approach of pQCD counting rules [1,2], we are able to count the number of the hard gluon propagators within the baryon pair, such that the momentum dependences of g i and f i can be parameterized as [4] with t ≡ m 2 ΛΛ = (p Λ + pΛ) 2 . To the leading order, the counting gives n = 3, in which 2 of them are for the gluons connecting to the valence quarks, while the rest one for the gluon speeding ups inB 0 s to be part ofΛ. As t approaches the threshold area, the increasing value of 1/t 3 creates a peak in the m BB ′ spectrum of B → BB ′ , which interprets the threshold enhancement. Under the SU(3) flavor symmetry, D g i and D f i are related by D g 1 (f 1 ) = D || , and D g k = −D f k = D k || (k = 2, 3, · · · , 5), in which the reduced constants D || , (D || ) and D k || can be fitted through the measured baryonic decays [5].
ForB 0 s → V pp, because the matrix elements of theB 0 s → (ss →)pp transition need the ss annihilation to produce pp, which encounters the OZI suppression, it is not suitable for pQCD counting rules. Consequently, B(B 0 s → J/ψpp) is estimated to be smaller than 10 −9 as mentioned early. On the other hand, for the resonant transition ofB 0 s → f J → pp as shown in Fig. 1b, m f J ≃ 2.23 GeV is in the pp invariant mass (m pp ) spectrum, of which the range of 1.88 GeV< m pp < 2.27 GeV is so confined, such that the resonance has a complete peak, enhancing the decay branching ratio ofB 0 s → J/ψpp. The matrix element of the resonantB 0 s → f J → pp transition is given by where Γ f J (m f J ) stands for the decay width (mass) of f J . In terms of Eqs. (5), (6), and (9), we can write the amplitude ofB 0 GeV due to the heavy J/ψ mass, a and b can only be changed slightly so that they are nearly constants. Moreover, the dominant contribution to the branching ratio comes from the pole effect, which is even narrow, fixing the pole at m f J = 2.23 GeV. In fact, pp|f J as the strong interaction conserves the parity, such that it is in the form of eitherūv or uγ 5 v for the parity to be even or odd. Hence, while f J has been confirmed to have an even parity as the data indicated [8], b = 0. In Eq. (10), since a is unknown, it will be fitted with B(B 0 s → J/ψpp) and then used to predict B(B 0 s → D * 0 pp) as well as those of B 0 s → J/ψ(ππ, KK) andB 0 s → D * 0 (ππ, KK). To integrate over the phase space of the three-body decays, the general equation in the PDG [8] can be referred, which is given by where m 12 = p B + pB′, m 23 = p B + p V and |A| 2 represents the amplitude squared. By from the study of the charmful three-body baryonicB 0 (B − ) decays in Ref. [5]. For the decay constants, we use (f D * , f J/ψ )=(0.23, 0.41) GeV [29]. As a result, with |a| fitted to be 1.04 ± 0.26 we obtain B(B 0 s → J/ψ(f J →)pp) = (3.00 ± 1.74) × 10 −6 to explain the data in Eq. (1). Consequently, we can calculate the branching ratios ofB 0 s → J/ψΛΛ andB 0 s → D 0 * pp (ΛΛ), of which the m BB ′ spectra are drawn in Fig. 2, while the total branching ratios are listed in Table I with the errors coming from the uncertainties in various form factors.
SinceB 0 s → J/ψΛΛ and B − → J/ψΛp are essentially identical, except for the spectator quarks inB 0 s and B − , their branching ratios should be at the same level. Nonetheless, from Table I, we see that B(B 0 s → J/ψΛΛ) ≃ 0.02B(B − → J/ψΛp). The reason for this is that m ΛΛ around the threshold area is smaller than m Λp by 100 MeV, which causes the  [30]. It is interesting to note that the assumption of the constant parameter of a is demonstrated to be insensitive to the data fitting, while the pole effects via the resonant f J → pp inB 0 s → J/ψ(D * 0 )pp are narrow and sharp, as shown in Fig. 2. A further confirmation for the resonantB 0 s → J/ψ(f J →)pp can depend on the future search forB 0 s → D * 0 (f J →)pp, whose decay branching ratio is predicted to be (4.70 ±2.89) ×10 −7 (see Table I are too small to have impacts on the experimental results, due to the fact that the threshold effects in the decays shadow the resonant peaks. Instead, the unexpected large value of (1) would reveal the existence of f J (2220) due to the suppressed threshold effect in the decay. Table I for V = D * 0 . We then let theB 0 s decays be the new scenario to study the f J state.

IV. CONCLUSIONS
We have studied the roles of f J (2220), considered as the tensor glubeball state of G 2 ++ , in the hadronicB 0 s decays. Explicitly, we have shown that the recent measured large branching ratio by the LHCb for the OZI suppressed decay of B(B 0 s → J/ψpp) can be understood due to the resonant contribution of f J (2220). We have also found that B(B