Heavy Baryonic Decays of \Lambda_b \to \Lambda \eta^{(\prime)} and Nonspectator Contribution

We calculate the branching ratios of the hadronic \Lambda_b decays to \eta and \eta^\prime in the factorization approximation where the form factors are estimated via QCD sum rules and the pole model. Our results indicate that, contrary to B\to K\eta^{(\prime)} decays, the branching ratios for \Lambda_b\to\Lambda\eta and \Lambda_b\to\Lambda\eta^\prime are more or less the same in the hadronic \Lambda_b transitions. We estimate the branching ratio of \Lambda_b\to\Lambda\eta^{(\prime)} to be 10.80 (10.32)\times 10^{-6} in QCD sum rules, and 2.78 (2.96)\times 10^{-6} in the pole model. We also estimate the nonfactorizable gluon fusion contribution to \Lambda_b\to\Lambda\eta^\prime decay by dividing this process into strong and weak vertices. Our results point to an enhancement of more than an order of magnitude due to this mechanism.


I. INTRODUCTION
For the last few years, different experimental groups have been accumulating plenty of data for the charmless hadronic B decay modes. CLEO, Belle and BaBar Collaborations are providing us with the information on the branching ratio (BR) and the CP asymmetry for different decay modes. A clear picture is about to emerge from these information. Among the B → P P (P denotes a pseudoscalar meson) decay modes, the BR for the decay B + → K + η ′ is found to be larger than that expected within the standard model (SM). The observed BR for this mode in three different experiments are [1,2,3] B(B ± → K ± η ′ ) = (80 +10 −9 ± 7) × 10 −6 [CLEO], = (77.9 +6.2+9.3 −5.9−8.7 ) × 10 −6 [Belle], = (67 ± 5 ± 5) × 10 −6 [BaBar]. (1) In order to explain the unexpectedly large branching ratio for B → Kη ′ , different assumptions have been proposed, e.g., large form factors [4], the QCD anomaly effect [5,6], high charm content in η ′ [7,8,9], a new mechanism in the Standard Model [10,11], the perturbative QCD approach [12], the QCD improved factorization approach [13,14], or new physics like supersymmetry without R-parity [15,16,17]. Even though some of these approaches turn out to be unsatisfactory, the other approaches are still waiting for being tested by experiment. Therefore, it would be much more desirable if besides using B meson system, one can have an alternative way to test the proposed approaches in experiment.
Weak decays of the bottom baryon Λ b can provide a fertile testing ground for the SM.
Λ b decays can also be used as an alternative and complimentary source of data to B decays, because the underlying quark level processes are similar in both Λ b and B decays. For example, Λ b → Λη (′) decay involves similar quark level processes as B → Kη (′) , i.e., b → qqs (q = u, d, s). In the coming years, large number of Λ b baryons are expected to be produced in hadron machines, like Tevatron and LHC, and a high-luminosity linear collider running at the Z resonance. For instance, the BTeV experiment, with a luminosity 2 × 10 32 cm −2 s −1 , is expected to produce 2 × 10 11 bb hadrons per 10 7 seconds [18], which would result in the production of 2 × 10 10 Λ b baryons per year of running [19]. One of peculiar properties of Λ b decays is that, unlike B decays, these decays can provide valuable information about the polarization of the b quark. Experimentally the polarization of Λ b has been measured [20].
In this work, we study Λ b → Λη (′) decay. Our goal is two-fold: (i) The calculation of the BR for Λ b → Λη (′) involves hadronic form factors which are highly model-dependent. Using different models for the form factors, we calculate the BR for Λ b → Λη (′) and investigate the model-dependence of the theoretical prediction. (ii) As an alternative test for a possible mechanism explaining the large BR for B + → K + η ′ , we examine the same mechanism using Λ b → Λη ′ decay. Among the mechanisms proposed for understanding the large B(B + → K + η ′ ), we focus on a nonspectator mechanism presented in Refs. [10,11]. In this mechanism, η ′ is produced via the fusion of two gluons: one from the QCD penguin diagram b → sg * and the other one emitted by the light quark inside the B meson. We calculate this nonspectator contribution to the BR for Λ b → Λη ′ in order to examine its validity. If this nonspectator process is indeed the true mechanism responsible for the large B(B + → K + η ′ ), then the same mechanism would affect B(Λ b → Λη ′ ) as well. Thus, one can test the validity of this mechanism in the future experiments such as BTeV, LHC-b, etc., by comparing B(Λ b → Λη ′ ) calculated with/without the nonspectator contribution with the measured results.
We organize our work as follows. In Sec. II, we present the effective Hamiltonian for the usual ∆B = 1 transition and for the nonspectator process. We calculate the BR for Λ b → Λη (′) decay without considering the nonspectator mechanism in Sec. III. The nonspectator contribution to Λ b → Λη ′ is estimated in Sec. IV. We conclude in Sec. V.
The effective Hamiltonian H eff for the ∆B = 1 transition is where q = d or s, and with f = u or c and q ′ = u, d, s, c and L(R) = (1 ∓ γ 5 )/2. The SU(3) generator T a is normalized as Tr(T a T b ) = 1 2 δ ab . α and β are the color indices. G µν a and F µν are the gluon and photon field strength, and c i 's are the Wilson coefficients (WCs). We use the improved effective WCs given in Refs. [21,22]. The renormalization scale is taken to be µ = m b [23]. Considering the gluon splits into two quarks, the chomomagnetic operator is rewritten in the Fierz transformed form as where Here p b and p s are the four-momenta of b-and s-quarks, respectively. N c denotes the effective number of colors and k ≡ p b − p s is the gluon momentum. In the heavy quark limit, , where x is the momentum fraction of η (′) . The average gluon momentum can be estimated [24] as The effective Hamiltonian for the nonspectator contribution can be obtained by considering the dominant chromo-electric component of the QCD penguin diagram [11,25]: where q denotes the spectator quark, and p i (i = 1, 2) are the four-momenta of the two gluons Using the decay mode ψ → η ′ γ, H(0, 0, m 2 η ′ ) is estimated to be approximately 1.8 GeV −1 .
In general, the vector and axial-vector matrix elements for the Λ b → Λ transition can be parameterized as where the momentum transfer q µ = p µ Λ b − p µ Λ and f i and g i (i = 1, 2, 3) are Lorentz invariant form factors. Alternatively, with the HQET, the hadronic matrix elements for the Λ b → Λ transition can be parameterized [31] as where v = p Λ b /m Λ b is the four-velocity of Λ b and Γ denotes the possible Dirac matrix. The relations between f i , g i and F i can be easily given by The decay constants of the η and η ′ mesons, f q η (′) , are defined by Due to the η − η ′ mixing, the decay constants of the physical η and η ′ are related to those of the flavor SU(3) singlet state η 0 and octet state η 8 through the relations [26,27] f where θ 8 and θ 0 are the mixing angles and phenomenologically θ 8 = −21.2 0 and θ 0 = −9.2 0 [27]. We use f 8 = 166 MeV and f 0 = 154 MeV [21].
The decay amplitude of Λ b → Λη ′ is given [24] by where In the above amplitude, we have taken into account the anomaly contribution 1 to the matrix element η ′ |sγ 5 s|0 [8,23,28,29], which leads to The similar expression for the decay amplitude of Λ b → Λη can be obtained by replacing η ′ by η in the above Eq. (16).
The decay amplitude given in Eq. (16) can be rewritten in the general form The averaged square of the amplitude is where Then the decay width of Λ b → Λη ′ in the rest frame of Λ b is given by where For numerical calculations, we need specific values for the form factors in the Λ b → Λ transition which are model-dependent. We use the values of the form factors from both the QCD sum rule approach [30] and the pole model [31,32]. In the QCD sum rule approach, the form factors F 1 and F 2 are given by where Here z = For the other relevant conventions and notation, we refer to Ref. [30]. In Figs. 1 and 2 Fig. 3 and Fig. 4, respectively. Our result shows    (15)). However, in the case of Λ b → Λη ( ′ ) , there is no such interference between terms in the amplitude because the amplitude contains terms proportional to η ( ′ ) |O|0 ∝ f q η ( ′ ) only 2 . In the pole model [31,32], the form factors are given by where Λ QCD ∼ 200 MeV and z = . Using N 1 = 52.32 and N 2 = −13.08 , we obtain the values of the form factors: F 1 (q 2 ) = 0.225 (0.217) and F 2 (q 2 ) = −0.056 (−0.054) for . We note that the magnitudes of these form factors are less than a half of those obtained in the QCD sum rule method. This would result in the fact that the BRs for the case of the QCD sum rule approach. Indeed, the BRs for Λ b → Λη ′ and Λ b → Λη are estimated to be and which are about a quarter of those estimated in the QCD sum rule case. For ξ = 1/3, can be written as: where g Λ b N B and g ΛN K parameterize the strong Λ b -Nucleon-B meson and Λ-Nucleon-K meson vertices, respectively. In fact, an estimate of the product g Λ b N B g ΛN K can be obtained by applying the same approximation method to the experimentally measured Λ b → ΛJ/ψ decay mode where the decay amplitude has a similar form as Eq. (31): Consequently, the ratio of the decay rates for Λ b → ΛJ/ψ and B → KJ/ψ can be expressed  [33] in the above ratio leads to the following estimate: On the other hand, the decay rate for B → Kη ′ via the nonspectator Hamiltonian (7) can be calculated as [25] Γ(B → Kη ′ ) = where | p η ′ | is the three momentum of the η ′ meson, i.e.
and p • is the energy transfer by the gluon emitted from the light quark in the B meson rest frame. As a result, using Eq. (31), one can calculate the ratio of the decay rates for Λ b → Λη ′ and B → Kη ′ in terms of the strong couplings g Λ b N B and g ΛN K : The numerical factor in Eq. which is obtained from the experimental data (1), leads to our estimate of the η ′ production in the Λ b → Λ transition:

V. CONCLUSIONS
In this work, we calculated the BRs for the two-body hadronic decays of Λ b to Λ and η or η ′ mesons. The form factors of the relevant hadronic matrix elements are evaluated by two methods: QCD sum rules and the pole model. In QCD sum rules, the sensitivity of the form factors to the Borel parameter is roughly the same for η and η ′ . The variation of F 1 is around 7% for the Borel parameter in the range between 1.5 and 1.9. F 2 on the other hand, is quite sensitive to this parameter, changing by a factor 2 approximately, in the above range. Also, we have checked the variation of the BRs for Λ b → Λη (′) with the effective number of colors N c in order to extend our results to ξ = 1 Nc ≤ 0.1 range, which is favored in fitting the experimental data on the B(B → Kη ′ ) in the framework of generalized factorization. Our results indicate that the BRs for Λ b → Λη and Λ b → Λη ′ are more or less the same in QCD sum rules, 9.15 × 10 −6 and 8.93 × 10 −6 , respectively, for M = 1.7 GeV and N c = 3.
In the pole model on the other hand, the form factor F 1 turns out to be smaller approximately by a factor 2. However, F 2 is roughly the same as in the sum rule case for the smaller values of the Borel parameter. As a result, the predicted branching ratios in this model, B(Λ b → Λη) = 2.36 × 10 −6 and B(Λ b → Λη ′ ) = 2.56 × 10 −6 for N c = 3, are significantly smaller than those obtained via QCD sum rules.
We also made an estimate of the nonspectator gluon fusion mechanism to the hadronic Λ b → Λη ′ decay. The purpose is to find the enhancement of the BR of this baryonic decay if the same underlying process that leads to an unexpectedly large BR for B → Kη ′ is operative in this case as well. We used a simple approach for this estimate where the amplitude is divided into strong and weak vertices. Our results point to a substantial increase in the BR, from more than a factor 3 to around an order of magnitude, compared to QCD sum rule and the pole model predictions, respectively. Future measurements of this Λ b decay mode will test the extent of the validity of these models.