f_0(1710) production in exclusive B decays

Production of $f_0(1710)$, a theoretical endeavor of pure scalar glueball state, is studied in detail from exclusive rare B decay within the framework of perturbative QCD. The branching fraction for $B^\pm \to K^{*\pm} f_0(1710) \to K^{*\pm} (K \bar K)$ is estimated to be about $8 \times 10^{-6}$, while for $B^\pm \to K^\pm f_0(1710) \to K^\pm (K \bar K)$ it is smaller by roughly an order of magnitude. With the accumulation of almost 1 billion $B \bar B$ pairs from the B{\tiny A}B{\tiny AR} and Belle experiments to date, hunting for a scalar glueball via these rare decay modes should be attainable.

From the modern point of view, properties of pseudoscalar mesons can be understood as Nambu-Goldstone bosons due to the spontaneous symmetry breaking of chiral symmetry.
Their low energy dynamics can be described by the chiral lagrangian.On the other hand, scalar mesons are not governed by any low energy symmetry like chiral symmetry and thus they can not take advantage of the power of a symmetry.Indeed, their SU(3) classification, the quark content of their composition, as well as their spectroscopy are not well understood for scalar mesons [1].Moreover, possible mixings of the q q states with a pure glueball state [2,3,4,5,6,7,8] must be taken into consideration.
Recent quenched lattice simulation [9] predicted the lowest pure glueball state has a mass equals 1710 ± 50 ± 80 MeV and J P C = 0 ++ .The first error is statistical while the second is due to approximate anisotropy of the lattice.This suggests that before mixing, a glueball mass should be closed to 1710 MeV, instead of the earlier lattice result of 1500 MeV [2].This makes f 0 (1710) a strong candidate for a lowest pure glueball state as advocated in [10] based on argument of chiral suppression in f 0 (1710) decays into pair of pseudoscalar mesons.The next two pure glueball states predicted by the quenched approximation [9] have masses 2390 ±30 ±120 MeV and 2560 ±35 ±120 MeV with J P C = 2 ++ and 0 −+ respectively.Mixings between the nearby three isosinglet scalars f 0 (1370), f 0 (1500), and f 0 (1710) and the isovector scalar a 0 (1450) have been studied in detail in [2] with the following main result: In the SU(3) symmetry limit, f 0 (1500) is a pure SU(3) octet and degenerate with the isovector scalar meson a 0 (1450), whereas f 0 (1370) is mainly a SU(3) singlet with a small mixing with f 0 (1710) which is composed predominantly by a scalar glueball.
Important production mechanism of glueballs is the decay of heavy quarkonium [11,12,13].In fact, the observed enhancement of the mode J/ψ → f 0 (1710)ω relative to f 0 (1710)φ and the copious production of f 0 (1710) in the radiative J/ψ decays are strong indication that f 0 (1710) is mainly composed of glueball [2].Another interesting mechanism is the direct production from e + e − → γ * → G J H [14], where G J stands for a glueball state of spin J = 0 or 2 and H denotes a J/ψ or Υ.Recently, glueball production from inclusive rare B decay [15] has also been studied.Ironically, scalar glueball state has never been observed in the gluon-rich channels of J/ψ(1S) decays or γγ collisions1 .
In this article, we will study glueball production via exclusive B decay using perturbative QCD (PQCD).Firstly, we will ignore mixing effects and treat f 0 (1710) as a pure scalar glueball suggested by the quenched lattice data.At the end of the paper, we will demonstrate the mixing effects are minuscule.At quark level, the effective Hamiltonian for the decay b → sq q can be written as [17] where V q = V * qs V qb denotes the product of Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and the operators O 1 -O 10 are defined as with α and β being the color indices and C 1 -C 10 the corresponding Wilson coefficients.
In addition, the gluonic penguin vertex for b(p) → s(p ′ )g * (q) with next-to-leading QCD corrections is given by [18] Γ where g s is the strong coupling constant, m b is the b-quark mass, T a is the generator for the color group, and , and C eff 8g can be found in Ref. [19].Since the ground state scalar glueball is composed of two gluons, the effective interaction between a scalar glueball and gluons can be written as [10] where f 0 stands for an unknown effective coupling constant, G 0 denotes the scalar glueball field, and G a µν is the gluon field strength tensor.With these 4-quarks operators O 1 − O 10 (2) and the two effective couplings (3) and (4), we can embark upon the computation of the decay rates for B → K ( * ) G 0 using PQCD.The flavor diagrams for B → K ( * ) G 0 decays are displayed in Fig.  3).Both diagrams are of the same order in α s .In the heavy quark limit, the production of light meson is supposed to respect color transparency [20], i.e., final state interactions are subleading effects and negligible.We will work under this assumption in what follows.Moreover, diagrams like Fig. 2 that are of higher order in α s will be ignored.
FIG. 1: Flavor diagrams for the B → K ( * ) G 0 .s b FIG.2: Other flavor diagrams for the B → K ( * ) G 0 at higher order in α s .
To deal with the transition matrix elements for exclusive B decays, we employ PQCD [21,22] factorization formalism to estimate the hadronic effects.By the factorization theorem, the transition amplitude can be written as the convolution of hadronic distribution amplitudes and the hard amplitude of the valence quarks, in which the distribution amplitudes absorb the infrared divergences and represent the effects of nonperturbative QCD.
As usual, the hard amplitudes can be calculated perturbatively by following the Feynman rules.The nonperturbative objects can be described by the nonlocal matrix elements and are expressed as [23,24,25] for B, K, and K * mesons, respectively, where N c is the number of color, n ± are two light-like vectors satisfying n + • n − = 2, and ǫ L is the longitudinal polarization vector of K * .φ B (x, b), the distribution amplitude of B meson, is constructed as follows [25] φ are the twist-2 and 3 distribution amplitudes of K ( * ) mesons with the argument x stands for the momentum fraction.Finally, m B and m K ( * ) are the masses for the B and where m q and m s denote the light quark masses.The meson distribution amplitudes are subjected to the following normalization conditions where φ B (x) = φ B (x, 0) and f B(K ( * ) ) and f (T ) K ( * ) are the decay constants.We do not introduce transverse momenta for the light mesons K and K * here which we will justify later when we discuss the end-point singularities of the decay amplitudes.
In the light-cone coordinate system, we can pick the two light-like vectors to be n + = (1, 0, 0 ⊥ ) and n − = (0, 1, 0 ⊥ ), and the momenta of the B and K mesons can be written as with r G 0 = m G 0 /m B .For the vector meson K * , we take with r K * = m K * /m B in which the physical condition ǫ L •p K * = 0 is satisfied for massive vector particle.The momenta of the spectator quarks with their transverse momenta included are given by With these light-cone coordinates and distribution amplitudes defined, we can study the transition matrix elements for B → MG 0 (M = K, K * ).We first analyze Fig. 1(a).Within the PQCD approach, we find that Fig. 1(a) is directly proportional to x 1 .Since x 1 is the momentum fraction of the valence quark inside the B meson and its value is expected to be ), its contribution belongs to higher power in heavy quark expansion.As an illustration, we can use the operator O 4 in Eq. ( 2) to demonstrate this effect.Thus, one finds where ).It has been shown in [25] that under Sudakov suppression arising from k ⊥ and threshold resummations, the average transverse momenta of valence quarks are k ⊥ ∼ 1.5 GeV and the end point singularities at x 1,2 → 0 in Eq.( 11) can be effectively removed.With an explicit factor of x 1 appearing in the numerator of Eq.( 11), this contribution is regarded as a higher power effect in 1/m B and therefore can be neglected.We note that this situation is quite similar to the flavor singlet mechanism to the B → η ′ form factor [26].According to the PQCD analysis in Ref. [27], contribution from the possible gluonic component inside η ′ to the B → η ′ form factor also has similar behavior.Its numerical value is two orders of magnitude smaller than the B → π form factor.Similarly, other operators O 1−3 and O 5−10 give the same behavior.Therefore, to the leading power in Λ QCD /m B , the contributions from Fig. 1(a) can be neglected.We will concentrate on the contribution of Fig. 1(b) in what follows.
By using the introduced nonlocal matrix elements for mesons and the light-cone coordinates given above, the transition matrix element for B → MG 0 (M = K, K * ) can be obtained from Fig. 1

(b) as
with the decay amplitude function M M given by (1) M φ p M (x 2 ) + e (3) for the pseudoscalar K, and e K , e K , e for the vector meson K * .Here we have introduced the dimensionless variables 13) is given by with The evolution factor E(t) in Eq.( 13) is defined by where exp(−S B(K) ) is the Sudakov exponents that resummed large logarithmic corrections to the B(K) meson wave functions [28,29].Their explicit forms are given by where γ(α s (µ)) is the anomalous dimension.To leading order in α s , γ(α s (µ)) equals −α s /π.
The function s(Q, b) in Eq.( 18) is given by [30,31] where with f = 4 being the active flavor number and γ E is the Euler constant.As mentioned before, x 1 ≈ Λ/m B ≪ 1, we have dropped all terms related to x 1 in the above expressions for {e M }.Since r K ( * ) ≪ 1, we have retained only those terms in the above formulas for {e M } that are at most linear in r K ( * ) .The scale t where the strong coupling α s (t) in (17), the Sudakov exponents in (18), and the ∆F 1 (t) and F 2 (t) in ( 14) are evaluated will be discussed later.For comparison, we also present the formula of the decay amplitude function M M with k ⊥ = 0 in Appendix A.
For estimating our numerical results, we take the values of theoretical parameters to be: For the B meson distribution amplitude, we use [28] φ with N B = 111.2GeV and ω B = 0.38 GeV.For the distribution amplitudes of the light pseudoscalar K and vector mesons K * , we refer to their results derived by the light-cone QCD sum rules in [32,33,34].Their explicit expressions and relevant values of parameters are collected in the Appendix B for convenience.
According to the results of light-cone QCD sum rules, at small x 2 , the behavior of twist-2 distribution amplitude obeys the asymptotic form φ K ( * ) (x 2 ) ∝ x 2 (1 − x 2 ), whilst those of twist-3 distribution amplitudes approach a constant φ p,σ K ( * ) (x 2 ) ∝ const.Consequently, at small x 2 , the decay amplitude function contributed by the twist-2 distribution amplitudes of K ( * ) behaves like .
Obviously, even if one sets r G 0 to be zero, the effects from twist-2 distribution amplitudes of K ( * ) are well-defined at the end point x 2 → 0. Similarly, the contribution from twist-3 distribution amplitudes to the decay amplitude function at small x 2 behaves like .
Whence r G 0 → 0, one will suffer logarithmic divergences from the twist-3 distribution amplitudes.In practice, r G 0 ∼ 0.32, the divergence will not occur.This implies that the influence of k ⊥ can only be mild.As a common practice, we do not introduce transverse momenta for the valence quarks to suppress large effects from end point singularities.
Since the Wilson coefficients are µ scale dependence, for smearing its dependence, we include the values of Wilson coefficients with the next-to-leading QCD corrections [19].
However, even so, the C eff 4,6,8g are still slightly µ-dependence.Due to this reason, determination of the scale of exchanged hard gluons in Fig. 1 is also one of the origins of theoretical uncertainties.For the gluon that attached to the penguin vertex b → sg * , it carries a typical say x 2 = 0.5, we get q 2 ∼ 3.9 GeV.However, the gluon attached to the spectator quark carries roughly a typical momentum of GeV.We note that a suitable range of x 2 in PQCD is often taken as ∼ 0.3 − 0.7.For definiteness, we take the democratic average value t = ( q 2 + √ −k 2 )/2 as the hard scale, in which the allowed value is within the range t ≈ 2.45 ± 0.45 GeV.This justifies somewhat the validity of the PQCD approach and we will take this range of t as our theoretical uncertainties.For illustration, we present the involving Wilson coefficients at different values of µ scale in Table I.Effective interactions between a scalar glueball G 0 and the pseudoscalars have been studied using chiral perturbation theory [15,35].By using the current experimental data [16] Γ total (f 0 (1710)) = 137 ± 8 MeV and BR(f 0 (1710) → K K) = 0.38 +0.09 −0.19 , this allows us to get an estimate of the unknown coupling f 0 = 0.07 +0.009 −0.018 GeV −1 [15].This result of f 0 should be taken as a crude estimation.For one thing, the data of the branching ratio BR(f 0 (1710) → K K) was not used for averages, fits, limits, etc. by the PDG [16].Instead the following two ratios were used in the PDG analysis: Within the approach of chiral perturbation theory [15], it would be difficult to accommodate these two ratios of Eqs.( 24) and ( 25), since the leading term in the chiral Lagrangian is flavor blind.Here we will present another approach to estimate f 0 .At quark level, the amplitude for G 0 → q q is proportional to the quark mass m q and therefore chirally suppressed.Its explicit form is given by [10] A(G 0 → q q) = −f 0 α s 16π √ 2 3 where β denotes the velocity of the quark and u q (v q ) is the quark (anti-quark) spinor.It has been argued in [10] that the chiral suppression of the amplitude A(G 0 → q q) ∝ m q persist in all order of α s .One may treat the coefficient of this decay amplitude as the short-distance coefficient of the strong decay G 0 → P P where P stands for a pseudoscalar meson like π, K, or η etc, as illustrated in Fig. 3. Thus, with, to leading order in α s , and Flavor diagram for the G 0 → P P with P being a pseudoscalar.
In passing, we note that, using light-cone distribution amplitudes, it has been argued in Ref. [35] that G 0 → ππ, K K might be dominated by the 4-quark process of G 0 → q qq q which is not chirally suppressed.Using this 4-quark mechanism and PQCD factorization scheme, one would predict a large ratio of R π/K ≈ (f π /f K ) 4 ≈ 0.48.For further discussion of this issue, we refer our reader to Refs.[35,40,41].
Using the matrix element defined by Eq. ( 13) with the above chosen values of parameters, the values of M K ( * ) are given in Table II for f 0 = 0.086 GeV −1 and three different values of µ scale.For comparisons, we also present the results with k ⊥ = 0 in Table II.
The branching fractions for B + → (K + , K * + )G 0 decays are tabulated in Table III.From Table III, we find that the branching fraction for B + → K * + G 0 is about one order of magnitude larger than that for B + → K + G 0 .The difference arises not only from the values of the decay constants f K and f K * entered in the distribution amplitudes, but also from the  effects of e (2) ) and e K φ σ K (x 2 ) in the K + G 0 mode, which are switched to e (2) ) and e (3) ) respectively in the K * + G 0 mode.We also find that the k ⊥ influence on B + → K * + G 0 decay is stronger than B + → K + G 0 .In addition, when µ is smaller, k ⊥ has lesser effects on the decay B + → K * + G 0 .The branching fractions for the decay chains IV, where the errors are coming from the experimental data of BR(f 0 (1710) → K K).From Table IV, we learn that one has a better chance to look for the ground state of glueball through the three-body decays B → K * K K, since its branching fraction could be more than a factor of 10 larger than B → KK K.
Recently, BABAR had reported the following branching ratio for B ± → (K + K − )K ± where the (K + K − ) pair coming from the f 0 (1710) [42] BR non-vanishing and they can be derived as for B + → K + (N, S) and B + → K * + (N, S) decays, respectively, where ρ K , a 1 , and a u 4,6 are defined by e q is the electric charge of quark q and F BM 0 with M = N, S and A BK * 0 correspond to the [32]:    The coefficients {b i } are defined as with m q being the mass of m u or m d since m u ≈ m d is assumed.Since m q ≪ m s , in our numerical estimations, we take ρ K + = ρ K − = ρ K .We display the values of decay constant, mass of strange quark, and relevant coefficients of distribution amplitudes for K meson at µ = 1 GeV in Table V.Similarly, the distribution amplitude for K * can be expressed as [33,34] The values of the decay constants and relevant coefficients of the distribution amplitudes for the K * meson are shown in Table VI.

TABLE I :
The involving Wilson coefficients at various values of µ scale.

TABLE II :
Decay amplitude M M (in units of 10 −4 ) for B + → (K + , K * + )G 0 with and without

TABLE V :
The decay constant, mass of strange quark (in units of MeV) and coefficients of distribution amplitudes for K meson at µ = 1 GeV.

TABLE VI :
The decay constants (in units of MeV) and coefficients of distribution amplitudes for K * meson at µ = 1 GeV.