Three-body charmless baryonic $\bar B^0_s$ decays

We study for the first time the three-body charmless baryonic decays $\bar B^0_s\to \bar p \Lambda M^+ (p \bar \Lambda M^-)$, with $M=\pi, K$. We find that the branching ratios of $\bar B^0_s\to (\bar p \Lambda K^+$ and $p\bar \Lambda K^-)$ and $\bar B^0_s\to p\bar \Lambda \pi^-$ are $(5.1\pm 1.1)\times 10^{-6}$ and $(2.8\pm 1.5)\times 10^{-7}$, respectively, which agree with recent experimental results reported by the LHCb collaboration. In addition, we derive the relations \mbox{${\cal B}(\bar B^0_s\to \bar p \Lambda K^+)\simeq (f_K/f_\pi)^2(\tau_{B^0_s}/\tau_{B^0}) {\cal B}(\bar B^0\to \bar p \Lambda \pi^+)$} and ${\cal B}(\bar B^0_s\to p\bar \Lambda \pi^-)/{\cal B}(\bar B^0_s\to p\bar \Lambda K^-)\simeq {\cal B}(B^-\to p\bar p \pi^-)/{\cal B}(B^-\to p\bar p K^-)$ to be confronted to future experimental measurements. The fact that all four processes $B^0_s, \bar B^0_s \to p\bar \Lambda K^-, \bar p \Lambda K^+$ can occur opens the possibility of decay-time-dependent CP violation measurements in baryonic $B$ decays, something that had not been realised before.


I. INTRODUCTION
In contrast with mesonic B decays, the decays of B mesons to baryonic final states have been observed to have unique signatures due to the baryon-pair (B 1B2 ) formations, which reflect rich mechanisms for the hadronizations of the spinors. For example, the BaBar and Belle experiments at the B factories [1] reported typical three-body charmless baryonic B decay branching ratios B(B → B 1B2 M) ≃ O(10 −6 ), and provided evidence for prominent peaks around m B 1B2 ≃ m B 1 +mB 2 in the baryon-antibaryon spectra of baryonic B decays [2], which show that the B 1B2 formations favour the threshold area. However, in two-body decays B → B 1B2 , there is no large energy release from the recoiled meson [3], such that the total energy of B 1B2 is at the m B scale, which definitely deviates from the threshold area [4]. As a result, B(B → B 1B2 ) are seen to be small, around 10 −8 − 10 −7 [5][6][7].
The aforementioned observations inB 0 /B − → B 1B2 (M) decays may also hold for B 0 s → B 1B2 (M) decays now experimentally accessible to the LHCb collaboration [11,12]. Nonetheless, baryonicB 0 s decays are not trivially related to baryonicB 0 and B − decays. For example, replacing (ū,d) bys inB 0 /B − , one may approximately infer that which will be shown to be mostly incorrect, except for the first relation. We will also demonstrate that the recent first observation, made by the LHCb collaboration, of a baryonic B 0 s decay, namelyB 0 s → pΛK − , and the measurement of its branching ratio [13], combines in reality the branching ratios ofB 0 s → pΛK − andB 0 s →pΛK + .

II. FORMALISM
The decayB 0 →pΛπ + is flavour specific, unlike the similar mode of theB 0 s meson, which can decay to bothpΛK + and pΛK − final states. The latter three-body baryonicB 0 s decays proceed through different configurations as demonstrated in the Feynman diagrams in Fig. 1. Specifically, the baryon pairs involve quark currents and B meson transitions as depicted in Figs. 1(a,b) and (c,d), respectively. The amplitudes can be factorized in terms of the effective Hamiltonian at the quark level [14] as [9,[15][16][17][18] A(B 0 respectively, and a 1(4,6) ≡ c ef f 1(4,6) +c ef f 2 (3,5) /N ef f c are composed of the effective Wilson coefficients c ef f i defined in Ref. [14] with N ef f c the effective colour number, ranging between 2 and ∞ to account for the non-factorizable effects in the generalized factorization approach. The (2) replacing the strange quark by the down quark.
In our calculation, the matrix elements ofB 0 s →pΛK + in Eq.
(2) are expressed as [15,16] where F BM 0,1 are the form factors for the B → M transition, and F 1,2 , g A , h A , f S , and g P the timelike baryonic form factors. ForB 0 where factors. The form factors in Eqs. (3) and (4) are momentum dependent. Explicitly, F BM 0,1 are given by [19] F BM In perturbative QCD counting rules, the baryonic form factors depend on 1/t n as the leadingorder expansion [9,17,20,21], given by

III. NUMERICAL RESULTS AND DISCUSSIONS
For the numerical analysis, the theoretical inputs of the CKM matrix elements in the Wolfenstein parameterisation are given by [1]  flavour and SU(2) spin symmetries [15,20], leading to [10] extracted from the data. Here, [22] and h A = C h A /t 2 have both been neglected. Note that C h A is fitted to be in accordance with B(B 0 → pp) = 1.47×10 −8 [4]. On the other hand, theB 0 s → pΛK − decay corresponds toB 0 → ppD ( * )0 , B − → pp(K ( * )− , π − ), B 0 → ppK ( * )0 , and B − → ppe −ν e through the B → BB ′ transition form factors, which are related by the same symmetries [9,17], given by The effective Wilson coefficients for theB 0 s (B 0 s ) decays are given by [14] c ef f Integrating over the phase space of the three-body decays [1] we obtain the spectra with the uncertainties from the form factors, non-factorizable effects, and CKM matrix elements in order. The B(B 0 s →pΛK + ) is calculated to be close to the observed B(B 0 →pΛπ + ) = (3.14 ± 0.29) × 10 −6 [1], which confirms the first relation in Eq. (1).
Nonetheless, using the experimental measurements of , which disproves the other relations in Eq. (1). The reason for this is that theB 0 s → pΛ and B − → pp transitions give different contributions. Consequently, we should revise the relations in Eq. (1) to be From an experimental perspective, the measured branching ratio is B(B 0 s → pΛK − + B 0 s → pΛK − ), given that the flavour of the reconstructed B 0 s meson at production is not determined -the identification of the flavour at production, a procedure known as flavour tagging, requires a decay-time-dependent analysis. Assuming negligible CP violation,  [13]. In contrast, B(B 0 s → pΛπ − ) is estimated to be of order 10 −7 , consistent with its non-observation with the present data sample [13].
All four processes B 0 s ,B 0 s → pΛK − ,pΛK + are possible, just as in the case of thē B 0 s → D ± s K ∓ decays [23]. A B-flavour tagged decay-time-dependent analysis of these baryonic decay modes is necessary to disentangle all contributions. As the ratio of thē B 0 s →pΛK + and B 0 s →pΛK + branching ratios is predicted to be rather large, cf. Eq. (13), sizeable interference due to B 0 s -B 0 s mixing is expected, which hints at possibly large timedependent CP violating asymmetries. Time-dependent analyses require a typical minimum data sample of order 1000 to 1500 signal candidates, see for example the LHCb analysis presented in Ref. [23]. Extrapolating from the 260 ± 21 B 0 s ,B 0 s → pΛK −p ΛK + candidates selected in the recent LHCb analysis [13], assuming (as done in LHCb extrapolations) a two-fold increase in the bb production cross-section between the first data taking period of the LHC, and the present second period started in 2015, we conclude that such an analysis will require the full data sample to be collected by 2018.
Based on the observation ofB 0 s → (pΛK + , pΛK − ), it is promising to study other charm-less baryonicB 0 s decays such asB 0 The presence of extra resonances or neutral particles in the final states of these decay modes makes the experimental searches more demanding, though feasible by both the LHCb experiment and the future Belle II experiment.