Identifying Glueball at 3.02 GeV in Baryonic $B$ Decays

We examine the nature of the unknown enhancement around 3 GeV observed by the BABAR collaboration in the $m_{p\bar p}$ spectrum of the $\bar B^0\to p\bar p D^0$ decay. Suspecting that the peak is a resonance, which can be neither identified as a charmonium state, such as $\eta_c$ or $J/\psi$, nor classified as one of the light-flavor mesons, we conclude that it corresponds to a glueball fitted as X(3020) with $(m_X,\;\Gamma_X)=(3020\pm 8,\; 107\pm 30)\;\text{MeV}$, which could be the first glueball state above 3 GeV. This state also appears in the $m_{p\bar p}$ spectrum of the $\bar B^0\to p\bar p D^{*0}$ decay.

meson decays are not regarded as the gluon-rich processes, they can be more beneficial to offer accesses to a wider detecting range of heavy glueball productions. We note that the three-body baryonic decay of B → ppM with a two-step process B → (G → pp)M could be an ideal channel, where M is the recoiled meson. In particular, one can think of the G → pp transition as an inverse process of the pp annihilation, which has been used at LEAR and PANDA as a gluon-rich process to search for glueballs. In fact, the process of B → ξK → ppK has been applied to constrain the narrow resonant state ξ, known as the glueball candidate f J (2220) [19,20]. Recently, the BABAR collaboration has observed an unknown enhancement at 3.0 − 3.1 GeV in the m pp spectrum ofB 0 → ppD 0 [21]. We shall take that the peak is a sign for a resonant state as it is unable to be reproduced by the perturbative QCD (pQCD) calculations. Since the charmonium states, such as η c and J/ψ as well as the light-flavor mesons are not favored, we introduce the glueball state at a mass above 3 GeV as the resonant state.
Data Analysis-Before analyzing the unknown peak at 3 GeV in the m pp spectrum of B 0 → ppD 0 [21], one should emphasize that the sharp peak around the threshold area of m pp = (m p + mp) ≃ 2 GeV is commonly observed in B → ppM, which is known as the threshold effect [22]. As this threshold effect dominates the branching ratio, it may shadow the sign of any new resonance. However, in the BABAR's manipulation, the threshold effect has been isolated in Fig. 9c of Ref. [21] with respect to m Dp > 3 GeV, while Fig. 9d of Ref. [21] with respect to m Dp < 3 GeV reveals a resonance even more obviously. As stated by the LHCb collaboration [23], the B − → ppK − decay is able to offer a clean environment to study charmonium states and search for glueballs or exotic states as pp allows intermediate states of any quantum numbers. In fact, the LHCb in Ref. [23] has claimed the peaks observed above 2.85 GeV as resonances, which are further recognized as a serious of charmonium states. This clearly helps us to find the true nature of the enhancement at 3.0 − 3.1 GeV in the m pp spectrum ofB 0 → ppD 0 [21].
In order to explain all data points adopted from Figs. 9c and 9d in Ref. [21], we start with the amplitude based on pQCD counting rules forB 0 → ppD 0 depicted in Fig. 1a. The amplitude is given by [24] where G F is the Fermi constant, V cb and V ud represent the CKM matrix elements for the b → cūd transition at the quark level, and (q 1 q 2 ) V (A) stands forq 1 γ µ (γ 5 )q 2 . For the D meson production, we have where f D is the decay constant of D. The matrix elements for theB 0 → pp transition are parameterized as the most general form [25]: where p = p B − p p − pp and q = p p + pp with p i (i = B, p,p) representing the momenta of the particles. The momentum dependences for the form factors f j (g j ) (j = 1, 2, · · · , 5) based on pQCD counting rules are [26] f . By setting n = 3 to count the number of the hard gluons for the B → pp transition [28], the form of 1/t n that peaks at t → (m p + mp) 2 and decreases with increasing t corresponds with the threshold enhancement. It is interesting to note that we have succeeded in explaining the experimental data observed in baryonic B decays, in particular the branching ratios [24,25,29,30] the predicted values of B(B → ΛΛK(π)) [31] and B(B − → ΛpD ( * )0 ) [24] are approved to agree with the latest measurements [32]. are taken from Ref. [21].
In this study, we use the χ 2 fitting with the values of G F , V cb , V ud , and f D from Ref. [33].
We note that the BABAR's manipulation can be realized by cutting the Dalitz plot of  Fig. 2b about 4 times higher too. This is obviously unacceptable to the data points, such that the existence of the resonance at 3 GeV can be established. In addition, it is interesting to note that the Dalitz plot densities in accordance with the areas I, II and III in Fig. 2a have been measured in Fig. 8a of Ref. [21]. It is clear that the suppression of the decay rate for the area II also implies the similar smallness for the area III. Nonetheless, the area III shows a more condense density converted to be the peak in Fig. 2c, which is unable to be traced back to the non-resonant amplitude (dashed line) in Eq. (1).
We now proceed the second-step identification for the resonance at 3 GeV. AsB Since the dashed line in Fig. 2c from the pQCD effect has been demonstrated to be small, we can estimate the resonant contribution to the total branching ratio. As a result, we are allowed to test the first possibility of the charmonium M(cc) as the resonant state at 3 GeV in terms of a simple relation, given by with B(J/ψ → pp) ≃ 2 × 10 −3 [33] as a new input. It turns out that B(B 0 → J/ψD 0 ) ≃ 4 × 10 −3 , which strongly disagrees with the predicted B(B 0 → J/ψD 0 ) of order 10 −6 [35,36] as well as the experimental upper bound B(B 0 → J/ψD 0 ) < 1.3 × 10 −5 [33]. In addition, it is stated in Ref. [21] that the decay width Γ(J/ψ) = 93 keV is not consistent with the broad 100-200 MeV in the m pp spectrum. Similarly, we also obtain B(B 0 → η c D 0 ) ≃ 6.5 × 10 −3 , which is much larger than the predicted B(B 0 → η c D 0 ) of order 10 −5 [36]. Clearly, the resonance cannot be the charmonium.
As seen in Fig. 1b forB 0 → (X → pp)D 0 with X to be M(dd) or G, the relevant amplitude is the same as that in Eq. (1), while the matrix element of theB 0 → pp transition is given by where m X and Γ X are the mass and the decay width, respectively. Consequently, the relevant amplitude ofB 0 → (X → pp)D 0 now reads with the constants a and b. We note that, no matter what spin the X particle has, the parameterization for theB 0 → (X → pp) transition can be factored into a and b. Although a and b are in principle energy-dependent, their values can only be slightly changed with the deviation for the decay width around 100-200 MeV compared to the energy range at 3 GeV.
Since the parity determination for the X particle is uncertain, we set |a| = |b|. By taking 20 data points as our inputs to the combined amplitude , we fit |a| = |b| and the mass and decay width of the X particle to be respectively. Our result with the above resonance is presented as the solid line in Fig. 2c.
From the figure, we observe that it can fully explain the peak. Moreover, compared to .95 without the resonant amplitude A R , we obtain χ 2 /d.o.f ≃ 1.17 to represent a good fitting by identifying the peak at 3 GeV as the resonant X(3020). To fully consider the errors for the fitted mass and decay width of the X resonance, both the uncertainties from the data points and the theoretical inputs [24] as the background contributions from the pQCD effect are taken into account, whereas the solid line in Fig. 2c corresponds to the best fit. The parameters |a| and |b| fitted to be 4.4 ± 1.0 can be considered as the size of this process, showing the significance to be around 4σ. By integrating over m pp =2.8-3.2 GeV in the m pp spectrum, we give the ratio of the non-resonant and resonant contributions to be (6.7 +3.7 −3.0 )%, indicating a small background size. Due to its mass, X(3020) is unlikely to be M(dd). In fact, there is no observation of any light-flavor meson heavier than f 6 (2510) in the literature [33], and the predicted spectrum of the excited mesons does not span above 2.8 GeV [37]. This agrees with the study of the hadronic Regge trajectories [38], where the mass limits are given to be (2.86 ± 0.11) and (3.10 ± 0.11) GeV for nn and ss mesons, respectively.
Discussions and Conclusions-We remark that, via dd, the resonance at 3 GeV can be also explained by a bound state, such as the excited N * N * bound state with N * being one of the states N(1440), N(1520), and N(1535), provided that it is allowed to release energy to turn itself into pp, and the mass relation of m X ≃ m N * + mN * can be simply satisfied.
Note that Λ c (2800) and Λ c (2940) as excited charmed baryon states are proposed to be DN and D * p bound states [47,48], respectively. However, at present, it is impossible for us to distinguish whether the resonance is the bound state or the glueball state as they carry the same quantum numbers [49].
In sum, we have identified the existence of the glueball state at 3.02 GeV based on the peak in the m pp spectrum ofB 0 → ppD for m Dp < 3 GeV observed by the BABAR collaboration, which could be the first glueball state above 3 GeV. Explicitly, it has been fitted to be X(3020) with (m X , Γ X ) = (3020 ± 8, 107 ± 30) MeV.