Study of $\Lambda_b\to \Lambda (\phi,\eta^{(\prime)})$ and $\Lambda_b\to \Lambda K^+K^-$ decays

We study the charmless two-body $\Lambda_b\to \Lambda (\phi,\eta^{(\prime)})$ and three-body $\Lambda_b\to \Lambda K^+K^- $ decays. We obtain ${\cal B}(\Lambda_b\to \Lambda\phi)=(3.53\pm 0.24)\times 10^{-6}$ to agree with the recent LHCb measurement. However, we find that ${\cal B}(\Lambda_b\to \Lambda(\phi\to)K^+ K^-)=(1.71\pm 0.12)\times 10^{-6}$ is unable to explain the LHCb observation of ${\cal B}(\Lambda_b\to\Lambda K^+ K^-)=(15.9\pm 1.2\pm 1.2\pm 2.0)\times 10^{-6}$, which implies the possibility for other contributions, such as that from the resonant $\Lambda_b\to K^- N^*,\,N^*\to\Lambda K^+$ decay with $N^*$ as a higher-wave baryon state. For $\Lambda_b\to \Lambda \eta^{(\prime)}$, we show that ${\cal B}(\Lambda_b\to \Lambda\eta,\,\Lambda\eta^\prime)= (1.47\pm 0.35,1.83\pm 0.58)\times 10^{-6}$, which are consistent with the current data of $(9.3^{+7.3}_{-5.3},<3.1)\times 10^{-6}$, respectively. Our results also support the relation of ${\cal B}(\Lambda_b\to \Lambda\eta) \simeq {\cal B}(\Lambda_b\to\Lambda\eta^\prime)$, given by the previous study.

Theoretically, Λ b → p(K * − , π − , ρ − ) decays via b → uū(d, s) at the quark level have been studied in the literature [4][5][6][7][8][9].In particular, it is interesting to point out that the direct CP violating asymmetry in Λ b → pK * − is predicted to be as large as 20%, which is promising to be observed in the future measurements.On the other hand, the decay of Λ b → Λφ via b → sss has not been well explored even though both the decay branching ratio and T-odd triple-product asymmetries [10][11][12] have been examined by the experiment at LHCb [2].According to the newly measured three-body Λ b → ΛK + K − decay by the LHCb Collaboration, given by [13] it implies a resonant Λ b → Λφ, φ → K + K − contribution with the signal seen at the low range of m 2 (K + K − ) from the Dalitz plot.However, to estimate this resonant contribution, one has to understand B(Λ b → Λφ) in Eq. ( 1) first.Such a study is also important for further examinations of the triple-product asymmetries [14].For Λ b → Λη (′) , the relation of [15] seems not to be consistent with the data in Eq. (1).Moreover, the first works on Λ b → Λη ′ with the branching ratios predicted to be O(10 −6 − 10 −5 ) in comparison with the data in Eq. (1) were done before the observations of Λ b → p(K − , π − ), which can be used to extract the Λ b → B n transition form factors from QCD models [8,16].For a reconciliation, we would like to reanalyze Λ b → Λη (′) .
In this work, we will use the factorization approach for the theoretical calculations of with G F the Fermi constant, V q 1 q 2 the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, and α 3 = −V tb V * ts (a 3 + a 4 + a 5 − a 9 /2), where a i ≡ c ef f i + c ef f i±1 /N c for i =odd (even) are composed of the effective Wilson coefficients c ef f i defined in Ref. [17] with the color number N c .As depicted in Fig. 1, the amplitudes of Λ b → Λη (′) are given by with q = u or d, where The matrix elements of the Λ b → Λ baryon transition in Eqs. ( 3) and ( 4) have been parameterized as [18] Λ|sγ µ (1 where f 1 , g 1 , f S , and g P are the form factors, with ]g 1 by virtue of equations of motion.Note that, in Eq. ( 5), we have neglected the form factors related to ūΛ σ µν q ν (γ 5 )u Λ b and ūΛ q µ (γ 5 )u Λ b that flip the helicity [19].With the double-pole momentum dependences, f 1 and g 1 can be written as [8] where we have taken C F (Λ b → Λ) ≡ f 1 (0) = g 1 (0) as the leading approximation based on the SU(3) flavor and SU(2) spin symmetries [20,21].We remark that the perturbative corrections to the Λ b → Λ transition form factors from QCD sum rules have been recently computed in Ref. [22].Clearly, for more precise evaluations of the form factors, these corrections should be included.

III. NUMERICAL RESULTS AND DISCUSSIONS
For our numerical analysis, the CKM matrix elements in the Wolfenstein parameterization are given by [25] ( with (λ, A, ρ, η) = (0.225, 0.814, 0.120 ± 0.022, 0.362 ± 0.013).In Table I, we fix N c = 3 for a i but shift it from 2 to ∞ in the generalized version of the factorization approach to take into account the non-factorizable effects as the uncertainty.For the form factors, we [21] with C F (Λ b → p) = 0.136 ± 0.009 [8].Apart from f φ = 0.231 GeV [26], we adopt the decay constants for η and η ′ from Ref. [23], given by respectively.Subsequently, we obtain the branching ratios, given in Table II.
As seen in Table I, α 3 for Λ b → Λφ is sensitive to the non-factorizable effects.In comparison with the data in Table II and Eq which is much lower than the data of (15.9 ± 4.4) × 10 −6 in Eq. ( 2), leaving some room for other contributions, such as the resonant Λ b → K − N * , N * → ΛK + decay with N * denoted as the higher-wave baryon state.Here, we would suggest a more accurate experimental examination on the ΛK invariant mass spectrum, which depends on the peak around the threshold of m ΛK ≃ m Λ + m K , while the Dalitz plot might possibly reveal the signal [13].
It is known that the gluon content of η (′) can contribute to the flavor-singlet B → Kη (′)   decays in three ways [23]: (i) the b → sgg amplitude related to the effective charm decay constant, (ii) the spectator scattering involving two gluons, and (iii) the singlet weak annihilation.It is interesting to ask if these three production mechanisms are also relevant to the corresponding Λ b → Λη (′) decays.For (i), its contribution to Λ b → Λη (′) has been demonstrated to be small [15] since, by effectively relating b → sgg to b → scc, the cc vacuum annihilation of η (′) is suppressed due to the decay constants (f c η , f c η ′ ) ≃ (−1, −3) MeV [23] being much smaller than f q η (′) in Eq. (12).For (ii), since one of the gluons from the spectator quark connects to the recoiled η (′) , the contribution belongs to the non-factorizable effect, which has been inserted into the effective number of N c (from 2 to ∞) in our generalized factorization approach.For (iii), it is the sub-leading power contribution which does not TABLE II.Numerical results for the branching ratios with the first and second errors from the non-factorizable effects and the form factors, respectively, in comparison with the experimental data [2,3] and the study in Ref. [15].Note that, in column 3, the two values without and with the parenthesis correspond to the form factors in the approach of QCD sum rules and the pole model, respectively.decay mode our results data [2,3] Ref. [15] 10 6 BΛ b → Λφ) 1.77 contribute to Λ b → Λη (′) .
Finally, we remark that, in the b-hadron decays, such as those of B and Λ b , the generalized factorization with the floating N c = 2 → ∞ [17] can empirically estimate the non-factorizable effects, such that it can be used to explain the data as well as make predictions.On the other hand, the QCD factorization [23]