Predictive $CP$ Violating Relations for Charmless Two-body Decays of Beauty Baryons $\Xi^{-,\;0}_b$ and $\Lambda_b^0$ With Flavor $SU(3)$ Symmetry

Several baryons containing a heavy b-quark have been discovered. The decays of these states provide new platform for testing the standard model (SM). We study $CP$ violation in SM for charmless two-body decays of the flavor $SU(3)$ anti-triplet beauty baryon (b-baryon) ${\cal B} = (\Xi^-_b,\;\Xi^0_b,\;\Lambda_b^0)$ in a model independent way. We found, in the flavor $SU(3)$ symmetry limit, a set of new predictive relations among the branching ratio $Br$ and $CP$ asymmetry $A_{CP}$ for $\cal B$ decays, such as $A_{CP}(\Xi_b^-\to K^0 \Xi^-)/A_{CP}(\Xi_b^-\to \bar K^0 \Sigma^-) = - Br(\Xi_b^- \to \bar K^0 \Sigma^-)/Br(\Xi_b^- \to K^0 \Sigma^-)$, $A_{CP}(\Lambda_b^0 \to \pi^- p)/A_{CP} (\Xi^0_b \to K^- \Sigma^+) = -Br(\Xi^0_b \to K^- \Sigma^+)\tau_{\Lambda_b^0}/Br(\Lambda_b^0 \to \pi^- p)\tau_{\Xi^0_b}$,and $A_{CP}(\Lambda_b^0 \to K^- p)/ A_{CP} (\Xi^0_b \to \pi^- \Sigma^+) = -Br(\Xi^0_b \to \pi^- \Sigma^+)\tau_{\Lambda_b^0}/ Br(\Lambda_b^0 \to K^- p)\tau_{\Xi^0_b}$. Future data from LHCb can test these relations and also other relations found.

Several baryons containing a heavy b-quark, the beauty baryon (b-baryon) B, have been discovered [1]. The study of heavy mesons containing a b-quark, the B mesons, provided crucial information [1] in establishing the standard model (SM) for CP violation, the Cabibbo-Kobayashi-Maskawa (CKM) model [2]. The decays of the B b-baryons will, with no doubt, provide a new platform to further test the CKM model of CP violation [3,4]. New data on B b-baryon will continue come from the LHCb. It is timely to investigate ways to test CP violation in the SM using B b-baryon decays.
For CP violation studies, rare charmless decays of B can play an important role because in these decays both tree and loop level contributions are substantial, providing the possibility of having large CP asymmetries [3,4]. We will consider such decays. Due to our poor understanding of low energy QCD, the evaluations of the decay amplitudes are pluged with large uncertainties. Flavor SU(3) symmetry has been shown to be an excellent tool in reducing uncertainties by obtain relations among different decays for particles containing a b-quark [5]. Several relations obtained for B meson decays have been tested to good precisions, in particular for two-body charmless B meson decays [6][7][8][9]. With more particles in the final states, the analysis become more complicated and large flavor SU(3) breaking uncertainties become difficult to control [10]. In this letter we will study CP violating relations for low-lying 1 2 + B b-baryon states decay into two charmless light particles using flavor SU(3) symmetry.
The low-lying 1 2 + B b-baryons contain a flavor SU(3) anti-triplet and a sixtet [11]. We concentrate on the anti-triplet decays. The anti-triplet B b-baryons will be indicated by The two charmless states in the final state of B decay are the 1 2 + octet baryons F and the pseudoscalar octet mesons M, respectively. They are The B → M + F decay can be induced by weak interaction in the SM and can have both parity conserving A c and violating A v amplitudes in the form MF(A v + iA c γ 5 )B. This leads to a decay width given by where |p c | = E 2 F − m 2 F . m B and m M , m F are the masses of the initial and final particles. E F is the energy of the final baryon F . S and P are referred as S and P wave amplitudes with In the SM there are tree and penguin contributions to S and P for ∆S = 0 and ∆S = −1 processes and S and P amplitudes can be written as: where q can be d or s. The sub-indices 0, 1 denote the S and P wave amplitudes. V ij is the CKM matrix element. In the SM, there are relations between the decay amplitudes with q = d and q = s in the flavor SU(3) symmetry limit. A particularly interesting set of relations is the one with U-spin symmetry relate CP violation in some ∆S = 0 and ∆S = −1 processes. We now show some details on how to obtain such relations.
In the SM, the effective operator for the decay processes under consideration at one electroweak loop level is given by where q can be d or sthe coefficients c 1,2 and c jk i = c j i − c k i , with j and k indicate the internal quark, are the Wilson Coefficients (WC) for the operators composed of quarks, photon and gluon fields. O 1,2 , O 3,4,5,6 and O 7,8,9,10 are the tree, penguin and electroweak penguin operators. O 11,12 are the photonic and gluonic dipole penguin operators. Details of the operators and their associated WC have been studied by several groups and can be found in Ref. [12]. In the above the factor V cb V * cq has been eliminated using the unitarity property of the CKM matrix.
At the hadron level, the decay amplitude can be generically written as The operators O i contains 3, 6 , 15 of flavor SU(3) irreducible representations. Indicating these representations by matrices H(3), H(6), H(15) [5,6]. The non-zero entries of the matrices H(i) are given as the followed [5,6].
For ∆S = 0, and for ∆S = −1, For an initial B b-baryon, it is understood that the Hamiltonian will annihilate the b-quark and contract SU(3) indices in an appropriate way with final states F and M to obtain SU(3) invariant amplitudes. As far as SU(3) properties are concerned, the S and P amplitudes will have various SU(3) irreducible contributions which can be obtained from the following invariant amplitudes, taking the tree S amplitude as example Expanding the above invariant amplitudes, we obtain contributions to individual decay processes. For example, expressing the tree decay amplitudes in terms of the coefficients in SU (3) invariant amplitudes, we have and The full results are listed in Tables I to VI. The expressions for penguin and also P wave amplitudes are similar. Due to mixing between η 8 and η 1 , the decay modes with η 8 in the final sates is not as clean as those with π and K in the final state to study. We list decay amplitudes with η 8 in the final states for completeness.
It is interesting to note that the pair of decays related by U-spin Λ 0 b → π − p and Ξ 0 b → K − Σ + , and, Λ 0 b → K − p and Ξ 0 b → π − Σ + , respectively, have the same tree and penguin amplitudes, that is T (d) j = T (s) j , and P (d) j = P (s) j . For these decays, although the absolute values of the decay widths are different, the rate difference ∆(i) = Γ(i) −Γ(i) are simply related by In obtaining the above relation, we have used the identity: . We list those U-spin related decay rate differences pair with ∆S = 0 and ∆S = −1 in the following The above relations imply relations for CP asymmetries where Br indicate branching ratio, and A CP indicates the CP asymmetry defined as There are similar relations in B decays into two pseudoscalar octet mesons in flavor SU(3) limit. We take the following two relations for discussion for the reason that there are data available for the relevant decays.
The present data [1,3,13] give: −3.41 ± 0.55 and 3.56 ± 0.40 for the left and right hand sides of the first equation above. These two values agree with the prediction very well. For the second equation, the left hand side is −2.21 ± 1.78 and the right hand side is 5.06 ± 1.91. The central values do not agree with the prediction, but agree within allowed error bars at 1σ level. Corresponding to the above relations, for each of them there are two pairs. For the first one, the two pairs are: Λ 0 b → π − p and Ξ 0 b → K − Σ + , and, Λ 0 b → K + Σ − and Ξ 0 b → π + Ξ − . For the second one, we have the two pairs as: Λ 0 b → K − p and Ξ 0 b → π − Σ + , and, Λ 0 b → K + Ξ − and Ξ 0 b → π + Σ − . We expect that the similar relations will hold at the same level as their B → MM counter parts. Future data from LHCb can test these relations to good precisions.
With moderate assumptions about the smallness of annihilation contributions, there are also new relations. We take Λ 0 b → π − p and Λ 0 b → K − p as examples to explain some details. If one neglects the coefficients a(i) α in the SU(3) invariant amplitudes, one would obtain We refer the contributions proportional to a(i) α as annihilation contributions in view of the fact that the flavor indices of the initial states are contracted by the indices in the Hamiltonian as if the flavor structure of the initial states are annihilated by the Hamiltonian. Since the initial flavor structures are annihilated by the Hamiltonian, no flavor information flow directly to the final states implying that the flavor structure of the final states have to be created completely by the weak interaction, the probability is smaller than those other terms where the initial state flows flavor information directly to the final states. Model calculations agree with this picture [4]. Similar situation happens for B → MM. There have been studied extensively. Theoretical calculations also agree with the assumption of smallness of annihilation contributions [9]. More over experimental data support the assumption that the annihilation contributions are small [1,13]. Under the small annihilation contribution assumption, one has [6,7] .
For the first equation above, using PDG data [1], we find that the left side is given by −3.78 ± 0.67 and the right hand side given by 3.82 ± 0.17. For the second equation, the left side is given by −2.0 ± 1.6 and the right hand side is given by 4.71 ± 0.60. The predicted relations are in agreement with data within error bars. In particular the first equation about gives additional confidence on our assumption. If the annihilation contributions are indeed small, one would obtain At present, data have not converged yet. The CDF [3] and LHCb [14] obtain values for the right handed to be: 0.66 ± 0.14 ± 0.08 and 0.86 ± 0.08 ± 0.05, respectively. The PDG [1] values is: 0.84 ± 0.22. For CP asymmetries, the PDG [1] and CDF [3] give The CP asymmetry for Λ 0 b → K − p from PDG is better than 2σ. If we take that as an input, with the branching ratio also from PDG, we would predict CP asymmetry This is to be compared with 0.03 ± 0.08 given by PDG [1]. It is clear that the predicted relation in eq. (18) does not agree with values given in PDG, not even the signs. The predicted relation agrees in signs with the CDF data and values are consistent within 1σ error bars, but the central values is off. We have to wait future experimental data to confirm the prediction.
There are additional relations beside listed above when annihilation contributions are neglected which can be read off from Tables I to VI and are given below (1) ≈ (2) , (3) ≈ (4) , (5) ≈ (6) , (7) ≈ (8) . (20) The above relations can provide further tests for the smallness of annihilation contributions along with SU(3) flavor symmetry. We have studied CP violating relations for flavor SU(3) anti-triplet B b-baryons decay into two charmless light particles. These relations can provide tests for SM with flavor SU(3) and the mechanism for heavy b-baryons decays. We eagerly waiting more precise experimental data from LHCb to further test these relations. Similar analysis can be carried out for sixtet b-baryon to charmless two-body decays. Detailed analysis on this will be presented elsewhere.