Possible Hints and Search for Glueball Production in Charmless Rare B Decays

Recent data on B->p pbar K, K0 pi pi and K Kbar K hint at a $\sim 2.3$ GeV object recoiling against a kaon. This could be the narrow state observed in J/\psi ->gamma xi. Nonobservation in p pbar annihilation implies ${\cal B}(\xi \to p\bar p) \sim$ few $\times 10^{-3}$, consistent with eta_c and J/\psi decays, but there are actual hints in p pbar ->phi phi and p p->p pi^+ pi^- pi^+ pi^- p. Simple modeling shows ${\cal B}(B \to \xi K){\cal B}(\xi \to p\bar p) \sim 1\times 10^{-6}$, appearing as a spike in the $p\bar p$ spectrum, with $\sim$ 30 events per 100 fb$^{-1}$; modes such as K K_s K_s, K phi phi, K4 pi (K f_2 pi pi) etc. should be explored. The underlying dynamics of g* ->g xi is analogous to g* ->g etaprime or gluon fragmentation. Discovery of sizable B ->xi K could be useful for CP violation studies.

The existence of glueballs as bound states of gluons, the gauge bosons of QCD, has been conjectured ever since the advent of QCD as the fundamental theory of the strong interaction. Alas, it is a unique feature of nonabelian gauge theories that has yet to be unequivocally tested. The main obstacle to identifying glueballs is their possible qq admixture, which allows the candidates to hide in the richness of qq resonances. Advances in lattice gauge theories suggest the lowest lying glueballs to be the 0 ++ scalar with m G ∼ 1.4-1.8 GeV and 2 ++ tensor with m ξ ∼ 1.9-2.3 GeV, while the 0 −+ glueball P is another 150 MeV heavier [1].
With this impasse, it is desirable to open up new avenues for exploration. The charmless b → sg * process could be [9,10] viable ground for glueball search. This was stimulated in part by the CLEO observation of large B → η ′ K ∼ 8 × 10 −5 [11] and η ′ + X s > ∼ 6 × 10 −4 [12], which were interpretted [13,14] as related to the large glue content of η ′ via the gluon anomaly. The b → sg * transition, followed by the anomaly inspired effective g * → gη ′ coupling, could account for [12,14] the semi-inclusive m Xs spectrum. Replacing η ′ by a glueball may be even more effective [9,10]. In this Letter we point out possible hints for B → ξK decay in the B → ppK, K S π + π − and K + K + K − modes newly observed by Belle, and discuss directions for further study.
Let us first present the case for charmless B decays. The B → ppK decay [15] is the first ever charmless baryonic mode to be observed. While modeling the m pp spectrum by a QCD motivated threshold enhancement, we noted a hint for a ∼ 2.3 GeV peak. The data (fitted B) and our modeling [16] are plotted in Fig. 1(a). Threshold enhancement is apparent, in line with our prediction [17]   Events/ GeV  for B → ρpn before the discovery of B → ppK. However, some excess ∼ 7-10 events is noticeable in the third, i.e. 2.2-2.4 GeV bin [18], amounting to ∼ 0.6-1 × 10 −6 in rate, which we could not accommodated in our simple threshold model. Motivated by this, we find evidence in a few (but not all) other 3-body channels as well.
The B → K + π + π − mode observed by Belle [19] is plotted in Fig. 1(b), with a cut of m K + π − > 2 GeV to suppress background. Despite some activity above 2 GeV, there is not much excess at 2.2-2.3 GeV.
The B → K S π + π − , K S K + K − modes, also observed by Belle [20], are plotted in Figs. 1(c) and (d), respectively. The spectrum for m π + π − > 2 GeV is very clean, with a striking cluster at 2.3 GeV, albeit with only 5 events. The m K + K − spectrum has ∼ 2 events in the same region (but a prominent cluster at ∼ 1.95 GeV). In all, h + h − has about 7 events, and folding in efficiencies, we find the average over K S π + π − , K S K + K − rates in the cluster region is ∼ 2.5 × 10 −6 . The comparison with ppK case is consistent with the BES observation.
Turning Figs. 1(e) and (f). The m min K + K − spectrum above 2 GeV is quite sizable and rich with structure, like Fig. 1(b) amplified but with much less background. This decay is expected to arise solely from the b → sss penguin. One has ∼ 10 events each at 2.3, 2.45 and 2.65 GeV, and ∼ 20 events at 1.9-2.15 GeV, the latter similar to K S K + K − . For m max K + K − one has ∼ 11, 14, 6 events respectively at 2.1, 2.45 and 2.65 GeV, but no 2.3 GeV cluster. Folding in efficiencies, we find a rate of 1.7 to 3.4 ×10 −6 , again consistent with BES and with K S h + h − . We caution, however, that identical particle effects, reflected in two possible K + K − pairings, smear the plots.
To summarize, there is some evidence for a 2.2-2.3 GeV "state" recoiling against a kaon in ppK, K S h + h − and K + K + K − channels, which could be the ξ glueball candidate. The ∼ 2.45 or 2.65 GeV objects might be the pseudoscalar P (or a scalar excitation [1]); there is also some excess in these regions for ppK ( Fig. 1(a)). The absence in K + π + π − is worrisome, but, besides larger background (hence extra cut), there are also amplitude level complications, such as a slower fall-off in m ππ vs. m pp , the tree contribution (in contrast to K S h + h − ), and multiple interfering resonances. We conclude that glueballs may emerge in higher statistics studies of charmless rare B decays, and wish to survey what we know about, and how to gain access to, such glueballs.
It is the pp annihilation experiments which cast doubt on the existence of ξ. These experiments were stimulated by the BES observation of ξ → pp to scan around 2230 MeV, before CERN Lower Energy Antiproton Ring (LEAR) shutdown in 1996. The results were all negative. The conservative conclusion is that ξ → π + π − , K + K − , K 0 S K 0 S , pp, φφ, π 0 π 0 , ηη are all < ∼ 1%. But, together with the narrow Γ ξ ∼ 20 MeV, the stated doubt [8] grew with time. We offer a critique of the situation.
First, two body decays of ξ < ∼ 1% is not surprising. The η c and J/ψ decays via gg and ggg, and their pp rates are 0.12% and 0.21% [2], respectively. If the ξ is the 2 ++ two-gluon glueball, having B(ξ → pp) ∼ few ×10 −3 seems just right. Second, a 20 MeV width for a lowest lying 2.2-2.3 GeV two-gluon glueball is also not unreasonable. On one hand, the " √ OZI" rule [23], i.e. taking the geometric mean of the few MeV width of η c (scaled down to 2 GeV) and the few hundred MeV width of a typical 2 GeV meson gives 10-50 MeV. On the other hand, the near ideal mixing of f 2 (1270)-f ′ 2 (1525) system implies [24] that the relevant lowest lying glueball, the ξ, would be relatively free of qq content, hence the above narrowness argument holds. Third, the lower bound of Eq. (1) is not unreasonable if ξ is really a glueball, but the large B(J/ψ → γξ) is a bit overstated. It arises from combining the BES result on J/ψ → π 0 π 0 [4] with the nonobservation of pp → π 0 π 0 [7]. As we noted, the BES result for π 0 π 0 is likely a factor of 2 to 3 too large.
With these points, it should be clear that ξ is still viable. We now argue that there is in fact some evidence coming from pp annihilation or pp collisions.
Although the JETSET experiment did not observe a narrow ξ in pp → φφ channel, they did find [6] a broad structure just above threshold. In fact, further partial wave analysis [25] found 2 ++ dominance, and a resonance TABLE I. Branching ratios (×10 −6 ) of theB → ξK − , ppK − , ppK 0 and ppπ − modes with Γ ξ = 23 MeV. The first two numbers for B(ppK − ) correspond to the upper and the lower curves of Fig. 1 (a) [16], respectively.  Fig. 6 of Ref. [25], comparing 2 + D0 with 2 + D2 , 2 + S2 waves, we note that it may be better to fit with two Breit-Wigner resonances (or one resonance with a broad underlying structure). We believe the JETSET data does not preclude a narrow resonance at 2.2 GeV.
There is another hint in central hadron production. The empirical "dP T " glueball filter [26] is defined as the difference between the transverse momenta of e.g. the outgoing protons in pp → pXp; dP T → 0 enhances glueball probability of X. Using data from WA102 experiment with X = π + π − π + π − , it was shown that the f 1 (1285) prominent for larger dP T all but disappeared for dP T < 0.2 GeV, while the glueball candidate f 0 (1500) is retained. From Fig. 3(c) of Ref. [26], however, we find a remarkable single-bin (2320-2340 MeV) spike, absent for dP T > 0.2 GeV, but popping up for dP T < 0.2 GeV. With ≃ 100 events on ≃ 360, it constitutes a > 5σ fluctuation. The detector resolution is ∼ 12 MeV [27] hence the spike seems genuine. A broader structure exists at 2430 MeV. Subsequent spin analysis (Fig. 3(f) of second paper of Ref. [27]) also show a "spike" at 2240-2280 MeV, and a broader structure at 2400 MeV, all in the 2 ++ channel of f 2 ππ. By analogy with the large η c → η (′) ππ ∼ (4-5)% [2], ξ → f 2 ππ could be a major decay mode. These features should be investigated further.
We now turn to simple modeling of the B → ppK "bump" assuming a 2 ++ glueball state. That is, we have a B → ξK(π) transition governed by where we factor out the quark mixing factor appropriate for the underlying b → s(d) penguin, The ξ → pp transition is governed by −g ξpp 1 ε µνū γ µ p ν p v, where a less effective p µ p p ν p term is dropped. For given Γ ξ , g ξpp 1 is fixed by B(ξ → pp) ∼ 5 × 10 −3 . Together with the fits in Fig. 1(a), f BξK is fixed (its sign determines interference; we ignore relative strong phase) to reproduce B(B → ppK) = 4.3 × 10 −6 . The results for the B → ξK − , ppK − , ppK 0 and ppπ − modes are given in Table I for Γ ξ = 23 MeV, and their spectra in Fig. 2 for Γ ξ = 70 MeV for sake of illustration. The ppK 0 case depends on the threshold dynamics, while the ξ is far less prominent in ppπ − because it is tree dominant.
The underlying dynamics of B → ξK is rather analogous to that proposed for B → η ′ K and η ′ X s . We have factored out in Eq. (2) G F / √ 2 and the V tb V * ts quark mixing factor coming from the penguin loop, as illustrated in Fig. 3. The g * → gξ coupling is the heavy blob, similar to g * → gη ′ via the gluon anomaly [13,14], which gives rise to the glue-content of η ′ . As argued in Ref. [10] (see also [9]) for the case of P , the production of a bona fide glueball in a similar fashion may be even more effective. How the sgq system evolves into a kaon is not of concern here since B → η ′ K is observed [12], and its large strength, recently confirmed by both Belle and BaBar [28], is still not explained by theory. Thus, it is plausible that B → ξK > B → η ′ K and could be > ∼ 10 −4 . There has been some perturbative arguments for 1/q 2 damping of the effective g * gξ vertex [29], but since nonperturbative effects -which generate m 2 ξ ≫ m 2 ρ -are bound to enter, we advocate [14]  in gluon fragmentation. The g * → gξ process advocated here can be viewed as such, but at only a few GeV energy. This illustrates further the futility to discard the g * gξ vertex by perturbative arguments. One uniquely interesting feature for studying glueball production in charmless B decays is the potential it offers for studying CP violation [9,14]. On one hand, the penguin loop implies sensitivity for new physics beyond the Standard Model, e.g. via the dipole bsg coupling. On the other hand, if B → ξK is really at a few ×10 −4 and ξ is a narrow state, one could accumulate a large number of modes and gain in statistics. CP asymmetries could be at 10-30% level even if new physics contributes only 10% in amplitude [9,14].
From our survey, additional search modes are: B → KK S K S , Kφφ, K4π (e.g. Kf 2 ππ), and perhaps ppK * 2 , beyond the ones given in Fig. 1. Semi-inclusive studies, i.e. B → ξ(→ pp, etc.)+X s , can also be considered. One can also search for other glueballs such as P and G, e.g. B → P K, GK via P → η (′) ππ, KKπ. At the same time, the η ′ study, both exclusive and inclusive, including CP violation effects, should be pursued further.
In summary, we find indication for a narrow state in B → ppK, K S ππ and K + K + K − recoiling against a kaon. This could be the 2 ++ glueball candidate found in radiative J/ψ decays with mass supported by lattice calculations, and with tantalizing hints in pp → φφ and pp → pπ + π − π + π −p . Glueballs may emerge in the study of charmless rare B decays, with confirming evidence from Υ → γpp. Search for ξ (and P ) in B → ppK, K + K + K − , K S h + h − , K + K S K S , Kφφ, K4π should be vigorously pursued, with an eye towards uncovering new physics sources of CP violation.