Glueball Production via Gluonic Penguin B Decays

We study glueball $G$ production in gluonic penguin decay $B\to G + X_s$, using the next-to-leading order $b\to s g^*$ gluonic penguin interaction and effective couplings of a glueball to two perturbative gluons. Subsequent decays of a scalar glueball are described by using techniques of effective chiral Lagrangian to incorporate the interaction between a glueball and pseudoscalar mesons. Mixing effects between the pure glueball with other mesons are considered. Identifying the $f_0(1710)$ to be a scalar glueball, we find that both the top and charm penguin to be important and obtain a sizable branching ratio for $B\to f_0(1710) + X_s$ of order $1.3\times 10^{-4} (f/0.07\mbox{GeV}^{-1})^2$, where the effective coupling strength $f$ is estimated to be $0.07$ GeV$^{-1}$ using experimental data for the branching ratio of $f_0(1710) \to K \overline K$ based on chiral Lagrangian estimate. An alternative perturbative QCD based estimation of $f$ is a factor of 20 larger, which would imply a much enhanced branching ratio. Glueball production from this rare semi-inclusive $B$ decay can be probed at the LHCb and Belle II to narrow down the allowed parameter space. Similar branching ratio is expected for the pseudoscalar glueball. We also briefly comment on the case of vector and tensor glueballs.


I. INTRODUCTION
Despite the fact that a glueball state has not been confirmed experimentally, its existence is an unavoidable and yet uncannily prediction of QCD. Glueball states may have different Lorentz structures in general, such as a scalar, a pseudoscalar or even a tensor with either normal or exotic J PC assignment. The prediction for the glueball masses is, however, a difficult task. Theoretical calculations indicate that the lowest lying glueball state is a scalar 0 ++ state with a mass in the range of 1.6 to 2 GeV. Previous quenched lattice QCD calculation gave such a glueball mass m G equals 1.71 ± 0.05 ± 0.08 GeV [1]. Recent result from unquenched lattice QCD calculation [2] gives a result of 1.795±0.060 GeV for this state.
Quenched lattice calculation also showed that the lowest lying 0 −+ state for a pure pseudoscalar glueball may have a higher mass around 2.560 ± 0.035 ± 0.120 GeV [1]. QCD sum-rule approach [10][11][12][13] also predicted higher than 1.8 GeV for pseudoscalar glueball mass. All these results do not favored the earlier speculation of the η(1405) being a pseudoscalar glueball. η(1405) is indeed a perfect candidate for a 0 −+ glueball since it is copiously produced from the radiative decay of J/ψ and not seen in the γγ mode. Recently it has been shown that when mixing effects and related data for the transition matrix elements of anomalous axial-vector current between vacuum and the three states |η , |η ′ and |η (1405) are taken into account, the physical pseudoscalar glueball can be as light as 1.4 ± 0.1 GeV [14]. Clearly, a full unquenched lattice calculation with the fermion determinant included must be performed in order to settle this issue satisfactory. Indeed, it has been argued some time ago [13] that due to the dynamical fermion effects the full QCD prediction for the pseudoscalar glueball mass will be substantially departure from its quenched approximation.
However, the mass for the lowest lying 0 −+ state is still missing in the latest unquenched lattice results [2].
Understanding glueball dynamics is therefore important though very challenging. In this work we will study scalar or pseudoscalar glueball production from rare semi-inclusive B me-son decay. The main purpose of this paper is to refocus the attention of experimentalists for undertaking the proposed searches. From the experimental findings we can learn a lot. The quantitative numerical predictions are bound to have significant theoretical uncertainties; nevertheless experimental efforts will be useful.
For a recent review for the phenomenology of scalar and pseudoscalar glueballs, we refer to Ref. [15], and for a recent summary of glueballs on the lattice, see Ref. [16].
We lay out this paper as the following. In section II, we discuss the effective vertex of b → sg * induced by the gluonic penguin and the effective couplings between scalar and pseudoscalar glueballs with the gluons. In section III, we discuss the interaction of scalar glueball with the light pseudoscalar meson octet using the chiral Lagrangian technique. The pseudoscalar glueball case will be briefly mentioned as well. In section IV, the rates of the semi-inclusive B meson decay into scalar and pseudoscalar glueballs are presented. In section V, mixing effects for both scalar and pseudoscalar glueballs will be considered. We summarize our results in section VI.

II. EFFECTIVE COUPLINGS
Since the leading Fock space of a glueball G is made up of two gluons, production of glueball is therefore most efficient at a gluonic rich environment like J/ψ or Υ → (gg)γ → Gγ [17,18]. Direct glueball production is also possible at the e + e − [19] and hadron colliders.
In this work, we show that B → G + X s decay also provides an interesting mechanism to produce and to detect a glueball. The leading contribution for this process is shown in Fig. [1], where the squared vertex refers to the gluonic penguin interaction and the round vertex stands for an effective coupling between a glueball and the gluons. The gluonic penguin b → sg * has been studied extensively in the literature and was used in the context for inclusive decay b → sgη ′ [20]. The effective interaction for b → sg * with next-to-leading QCD correction can be written as [21] where ∆F 1 = 4π(C 4 (q, µ)+C 6 (q, µ))/α s (µ) and F 2 = −2C 8 (µ) with C i (q, µ) (i = 4, 6, and 8) the Wilson's coefficients of the corresponding operators in the ∆B = 1 effective weak Hamiltonian, q = p − p ′ = k + k ′ , and T a is the generator for the color group. We will use the next-to-leading order numerical values of ∆F 1 and F 2 [21]. The top quark contribution gives ∆F top 1 = −4.86 and F top 2 = +0.288 at µ = 5 GeV; whereas the charm quark contribution involves a q 2 dependence through C charm 4 (q, µ) = C charm 6 (q, µ) = P charm and Q(q, m, µ) = 4 Here m c = 1.4 GeV is the charm quark mass and C 2 (µ = 5 GeV) = 1.150. The following effective coupling between a scalar glueball and two gluons can be parametrized as [22][23][24] where G is the interpolating field for the glueball G, G a µν is the gluon field strength, and f is an unknown coupling constant. When combined with the interaction vertex in Eq. (1), a scalar glueball can be produced in rare b quark decay, b → sgG (Fig. (1)). At the hadronic level, this leads to the rare semi-inclusive decay B → G + X s . The above effective coupling has also been used for the study of exclusive decays of B ± → f 0 (1710) + K ( * )± [25].
Analogously one can also study inclusive B decays into a pseudoscalar glueballG, whose leading effective coupling to the gluon can be parameterized as [22][23][24], whereG denoting the interpolating field for the pseudoscalar glueball andf is an unknown coupling. When combined with the interaction vertex in Eq. (1), a pseudoscalar glueball can be produced in rare b quark decay, b → sgG (Fig. (1)). At the hadronic level, this leads to the rare semi-inclusive decay B →G + X s .

III. DECAY RATES
If glueballs are composed of only gluons, one can easily obtain the branching ratios for B → G + X s . With the two effective couplings given in Eqs. (1) and (4), the following decay (1)) can be obtained readily with In Eq. (6), N c is the number of color and c 0,1,2 are given by The decay rate for pseudoscalar glueball case B →G+X s can be deduced from the scalar glueball one by replacing the coupling f withf and the mass m G with the pseudoscalar glueball mass mG in Eq. (6). This also reproduces previous result for the x − y distribution obtained in Ref. [20] for a similar process b → sgη ′ . Thus it is not possible to differentiate the produced glueball is a scalar or pseudoscalar from the x−y distribution of this semi-inclusive B decay.

IV. INTERACTION OF GLUEBALLS WITH LIGHT MESONS
The production rate of a scalar or a pseudoscalar glueball from B decays depend on the strength of the couplings f orf respectively. We first discuss how to use the chiral Lagrangian methods to incorporate the interaction for a scalar glueball with the other light mesons and then employ experimental data to determine f .
The form of effective coupling of a scalar glueball with two gluons, L 0 = f GG a µν G aµν , suggests that glueball couples to the QCD trace anomaly. As briefly mentioned in [26], the interaction of a scalar glueball with light hadrons through the trace anomaly can be formulated systematically by using techniques of chiral Lagrangian. The kinetic energy and the symmetry breaking mass terms for the light pseudoscalar mesons are given by [27] L where f π = 132 MeV being the pion decay constant, and with B 0 = 2031 MeV. Here we have neglected the isospin breaking effects due to small mass difference between the light u and d quarks and usedm = m u = m d . The QCD trace anomaly is well known and given by [27] where b = 11 − 2n f /3 is the QCD one-loop beta function with n f = 3 being the number of light quarks. Treating the effective interaction (4) as a perturbation to the energy momentum stress tensor, one would then modify Θ µ µ to be where f q = f α s m q (16π √ 2/3β) ln [(1 + β)/(1 − β)] being the one-loop induced Gqq coupling [24] with β = (1 − 4m 2 q /m 2 G ) 1/2 . Note that f q is proportional to f . The corresponding chiral Lagrangian is thus modified accordingly where Using the above chiral Lagrangian, one can calculate the decay rates for G → π + π − , π 0 π 0 , With the above chiral Lagragian Eq. (14), one can also obtain information about the size of the coupling f if combined with the branching ratio Br(f 0 (1710) → KK) = 0.36 ± 0.12 [29] and the total width Γ f 0 (1710) = 137 ± 8 MeV for f 0 (1710) [28]. Using a value of the strong coupling constant α s = 0.35 extracted from the experimental data of τ decay, we estimate the central value of the unknown coupling f = 0.07 GeV −1 .
In our estimation of f given above, we have extrapolated low energy theorems to the glueball mass scale which is higher than Λ QCD . The use of chiral Lagrangian is quite questionable. One might overestimate the hadronic matrix elements in due course. This implies that the extracted value of f would be too small.
An alternative approach is to use perturbative QCD. Indeed, this calculation was carried out some time ago for the decay G → KK in [26]. This approach can also give an estimate of the amplitude. The problem facing this approach is that the energy scale may not be high enough to have the perturbative QCD contribution to dominate. Using the asymptotic lightcone wave functions, we find that the branching ratios for G → KK and G → ππ decays are proportional to f 2 K and f 2 π , respectively. This leads to the ratio of Γ(ππ)/Γ(KK) to be about 0.48 in agreement with data within error bar. Fitting branching ratio for G → KK, we find that the value of f would be about 20 times larger than that obtained above using the chiral approach. Incidentally, the value of f derived from the chiral Lagrangian is within a factor of 2 compared with that estimated just by using the free quark decay rate of G → ss.
In our later calculations, we will use the conservative value f = 0.07 GeV −1 determined using the chiral Lagrangian given in Eq. (14). This will give the most conservative estimate for the branching ratio since the chiral approach gives the smallest f . Clearly, a rigourous determination of f is on the lattice. But we do not anticipate to have a lattice evaluation of its value any time soon. We have to contend with its rough estimation from all the theoretical tools that are available so far. Despite the fact one can not have a precise determination of the effective coupling f , we believe that glueball production from the proposed rare semi-inclusive B decay is a very interesting mechanism which is worth searching for experimentally.
Using the value of f determined above from the chiral Lagrangian, we find the branching ratio for b → sgG to be 4.5 × 10 −5 (f · GeV/0.07) 2 with just the leading top penguin contribution to ∆F 1 is taken into account. The correction from the charm penguin is nevertheless substantial and should not be neglected. Inclusion of both top and charm penguins gives rise to an enhancement about a factor of 3 in the branching ratio Since f 0 (1710) has a large branching ratio into KK, the signal of scalar glueball can be identified by looking at the secondary KK invariant mass. The recoil mass spectrum of X s can also be used to extract information. The distribution of dBr(b → sgG)/dM Xs as a function of the recoil mass of X s is plotted in Fig. (2) where M 2 Xs = (p ′ + k) 2 = m 2 b x. Due to parity conservation, a pseudoscalar glueball will not decay into two pseudoscalar mesons. Instead it will decay into one scalar and one pseudoscalar mesons or three pseu-doscalar mesons. If η(1405) is indeed the lowest lying pseudoscalar glueball, kinematics allows it to decay into f 0 (980)η, KKπ, ηππ and so on, as these modes were indeed reported as seen by Particle Data Group [28]. The interaction of pseudoscalar glueball with the light scalar and pseudoscalar meson octets has been studied in Ref. [30] using the chiral Lagrangian techniques for the resonance X(2370) identified as a pseudoscalar glueball. One could apply the chiral Lagrangian given in Ref. [30] to the lighter state η(1405). However the available phase space for η(1405) to decay is much smaller than X(2370). We will not pursuit that further here.

V. MIXING EFFECTS
As is well known, glueballs can mix with other pure qq states with the same quantum numbers, which had complicated the identification of glueballs experimentally. For these mixing effects, we will adopt the mixing models of Ref. [9] for the scalar glueballs and Ref. [14] for the pseudoscalar glueballs. For the scalar case, we have [9] Inverting the above relation, we obtain |G = −0.36|f 0 (1370) + 0.03|f 0 (1500) + 0.93|f 0 (1710) .
up to the corrections from the mass differences among the three states.
Hence one induces B → η(1405)X s , B → η ′ X s and B → ηX s with following relative up to the corrections from the mass differences among the three states.
We note that the above relative branching ratios were derived under the assumption that B → G/G + X s arises purely from the three-body process b → sg G/G with the glueball mixing effects taken into account. There may be two-body process B → G/G + X s via b → s G/G from four-quark operators in the effective Hamiltonian for the G/G production from B decay. These contributions will modify the above predictions of the relative branching ratios. Being a two-body decay type, the energy of G/G is energetic, given by E 2−body Xs )/2m B with m B = 5279.50 ± 0.30 MeV and m Xs typically equals 1 GeV or so. On the other hand, for the G/G production induced by b → sg G/G , the energy of G/G is more spread out. One therefore expects with a suitable cut on the glueball energy the two-body contributions can be removed, leaving the softer G/G production presumably coming from the mechanism we suggested in this paper to satisfy the relative branching ratios given in this section.
To implement the cut, recall that the energy of the glueball E G/G is related to the rescaled Thus to make a cut on the maximum of E G/G corresponds to a cut on the minimum (lower limit in the integration) of x. In the two-body decay, the energy of the glueball is fixed at E 2−body Since m Xs is of order 1 GeV, we will impose a cut on the minimum of x as in the form x min = δ ·r by varying δ in a range from 1 to 3. The effects of these cuts on the branching ratios of Br(B → f 0 (1710)X s ) and Br(B → η(1405)X s ) are shown in Table I. The impact of these cuts reduce the branching ratios by at most a factor of 3.

VI. DISCUSSIONS
Before closing, we would like to make the following comments: (1) In 2005, BESII experiment [31] has observed an enhanced decay in J/ψ → γη ′ ππ with a peak around the invariant mass of η ′ ππ at 1835 MeV. Proposal has been made to interpret this state X(1835) to be due to a pseudoscalar glueball [32]. Later BESIII experiment [33] has further confirmed the observation of X(1835) and its spin is consistent with expectation for a pseudoscalar. Taking X(1835) to be a pseudoscalar glueball, we would obtain a branching ratio of 3.7 × 10 −5 (f · GeV/0.07) 2 with top penguin contribution only, and is enhanced to 1.1 × 10 −4 (f · GeV/0.07) 2 if charm penguin is also included. If the couplingf is of the same order of magnitude as f , the branching ratio for B → X(1835) + X s is also sizable.
(2) Lattice QCD calculations also predicted lowest lying spin 2 tensor states: The 2 ++ and 2 −+ masses are 2390 ± 30 ± 120 MeV and 3040 ± 40 ± 150 MeV respectively from the earlier quenched calculations [1], and 2620 ± 50 MeV and 3460 ± 320 MeV respectively from recent unquenched results [2]. These heavier tensor states can also be produced from the semi-inclusive B decay but with much smaller available kinematic phase space compared with the scalar and pseudoscalar states. Leading operators describing the couplings between these tensor glueballs with the gluons are where G µν andG µν are interpolating fields for the 2 ++ and 2 −+ glueball states respectively. Note thatG µν is not the dual of G µν despite the˜. It is straightforward to extend the present analysis to these tensor states.
(3) Lattice calculations [2] also predicted lowest lying spin 1 vector glueball states which would necessarily decay into three gluons. We can also use effective operators describing interactions between the vector glueballs and gluons. We will leave them as exercises for our readers.
To conclude, we have studied semi-inclusive production of a scalar/pseudoscalar glueball in rare B decay through the gluonic penguin and effective glueball-gluon interactions. The branching ratio for the scalar glueball case is found to be of the order 10 −4 . B decays into a pseudoscalar glueball through gluonic penguin is also expected to be sizable. Note that for the scalar glueball case, we have used a conservative estimate of the effective coupling f . Observation of B → X s f 0 (1710) at a branching ratio of order 10 −4 or larger will provide an strong indication that f 0 (1710) is mainly a scalar glueball.
Recall that we have now more than 600 millions of BB accumulated at Belle and more than 300 millions at BABAR. Enhancement by a factor of 50 is expected at the Super B Factory with a designed luminosity of 8 × 10 35 cm −2 s −1 [34]. Moreover, at the 14 TeV LHCb, the bb cross section can be as large as 500 µb. One would expect about 50 × 10 12 bb pair with an integrated luminosity of 100 fb −1 at the LHCb. Probing the branching ratio Br(b → sgG/G) at the level of 10 −4 is quite feasible. One should emphasis that both chiral Lagrangian and perturbative QCD approaches used to extract the value of the effective coupling f is not precise. Despite this shortcoming, the glueball production mechanism suggested in this work is an interesting one. Measurement of this type may also help to pin down the energy scale where the chiral Lagrangian is valid for hadronic matrix element calculations. We strongly urge our experimental colleagues to rekindle their interests in search for the various glueball states; in particular those produced by the semi-inclusive decay of the B meson proposed in this work.