Predictive CP Violating Relations for Charmless Two-body Decays of Beauty Baryons Ξ −,0 b and Λ 0 b With Flavor SU(3) Symmetry
aa r X i v : . [ h e p - ph ] J un Predictive
C P
Violating Relations for Charmless Two-body Decays ofBeauty Baryons Ξ − , b and Λ b With Flavor SU (3) Symmetry
Xiao-Gang He , , ∗ and Guan-Nan Li † INPAC, SKLPPC and Department of Physics,Shanghai Jiao Tong University, Shanghai, China CTS, CASTS and Department of Physics,National Taiwan University, Taipei, Taiwan and Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan (Dated: July 3, 2018)
Abstract
Several baryons containing a heavy b-quark have been discovered. The decays of these states provide newplatform for testing the standard model (SM). We study CP violation in SM for charmless two-body decaysof the flavor SU (3) anti-triplet beauty baryon (b-baryon) B = (Ξ − b , Ξ b , Λ b ) 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 B decays, such as A CP (Ξ − b → K Ξ − ) /A CP (Ξ − b → ¯ K Σ − ) = − Br (Ξ − b → ¯ K Σ − ) /Br (Ξ − b → K Σ − ), A CP (Λ b → π − p ) /A CP (Ξ b → K − Σ + ) = − Br (Ξ b → K − Σ + ) τ Λ b /Br (Λ b → π − p ) τ Ξ b , and A CP (Λ b → K − p ) /A CP (Ξ b → π − Σ + ) = − Br (Ξ b → π − Σ + ) τ Λ b /Br (Λ b → K − p ) τ Ξ b . Futuredata from LHCb can test these relations and also other relations found. ∗ [email protected] † [email protected] B , have been dis-covered [1]. The study of heavy mesons containing a b-quark, the B mesons, provided crucialinformation [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 newplatform to further test the CKM model of CP violation [3–5]. New data on B b-baryon will con-tinue 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 inthese decays both tree and loop level contributions are substantial, providing the possibility ofhaving large CP asymmetries [3, 4]. We will consider such decays. Due to our poor understandingof 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 obtainrelations among different decays for particles containing a b-quark [6]. Several relations obtainedfor B meson decays have been tested to good precisions, in particular for two-body charmless B meson decays [7–10]. With more particles in the final states, the analysis become more complicatedand large flavor SU (3) breaking uncertainties become difficult to control [11]. In this letter we willstudy CP violating relations for low-lying
12 + B b-baryon states decay into two charmless lightparticles using flavor SU (3) symmetry.The low-lying
12 + B b-baryons contain a flavor SU (3) anti-triplet and a sextet [12]. We concen-trate on the anti-triplet decays. The anti-triplet B b-baryons will be indicated by( B ¯3 ) ij = b Ξ b − Λ b − b − Ξ b − Ξ − b (1)Their quark compositions are [12]Λ b = 1 √ ud − du ) b ; Ξ b = 1 √ us − su ) b ; Ξ − b = 1 √ ds − sd ) b . (2)The two charmless states in the final state of B decay are the
12 + baryon P in the octet F andthe pseudoscalar meson M in the octet M , respectively. They are M = π √ + η √ π + K + π − − π √ + η √ K K − ¯ K − η √ , F = Σ √ + Λ √ Σ + p Σ − − Σ √ + Λ √ n Ξ − Ξ − √ . (3)The B → M + F decay can be induced by weak interaction in the SM and can have both parityconserving A c and violating A v amplitudes in the form M ¯ F ( A v + iA c γ ) B . This leads to a decaywidth given by Γ = 2 | p c | ( |S| + |P| ) , (4)where | p c | = p E F − m F . m B and m M , m F are the masses of the initial and final particles. E F isthe energy of the final baryon F . S and P are referred as S and P wave amplitudes with S = A v s ( m B + m F ) − m M πm B , P = A c s ( m B − m F ) − m M πm B . (5)2n the SM there are tree and penguin contributions to S and P for ∆ S = 0 and ∆ S = − S and P amplitudes can be written as: S ( q ) = V ub V ∗ uq T ( q ) + V tb V ∗ tq P ( q ) , P ( q ) = V ub V ∗ uq T ( q ) + V tb V ∗ tq P ( q ) , (6)where q can be d or s . The sub-indices 0 , S and P wave amplitudes. V ij is the CKMmatrix 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 symmetryrelate CP violation in some ∆ S = 0 and ∆ S = − H qeff = 4 G F √ V ub V ∗ uq ( c O + c O ) − X i =3 ( V ub V ∗ uq c uci + V tb V ∗ tq c tci ) O i ] , (7)where q can be d or s , the coefficients c , and c jki = c ji − c ki , with j and k indicate the internal quark,are the Wilson Coefficients (WC) for the operators composed of quarks, photon and gluon fields. O , , O , , , and O , , , are the tree, penguin and electroweak penguin operators. O , are thephotonic and gluonic dipole penguin operators. Details of the operators and their associated WChave been studied by several groups and can be found in Ref. [13]. 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 A = hF M| H qeff |Bi = V ub V ∗ uq T ( q ) + V tb V ∗ tq P ( q ) . (8)The operators O i contains 3, 6 , 15 of flavor SU (3) irreducible representations. Indicating theserepresentations by matrices H (3), H (6), H (15) [6, 7]. The non-zero entries of the matrices H ( i )are given as the followed [6, 7].For ∆ S = 0, H (3) = 1 , H (6) = H (6) = 1 , H (6) = H (6) = − ,H (15) = H (15) = 3 , H (15) = − , H (15) = H (15) = − , (9)and for ∆ S = − H (3) = 1 , H (6) = H (6) = 1 , H (6) = H (6) = − ,H (15) = H (15) = 3 , H (15) = − , H (15) = H (15) = − . (10)For an initial B b-baryon, it is understood that the Hamiltonian will annihilate the b-quark andcontract SU (3) indices in an appropriate way with final states F and M to obtain SU (3) invariantamplitudes. 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 T tri ( q ) = a (3) hF kl M lk | H (3) i |B i ′ i ′′ i ǫ ii ′ i ′′ + b (3) hF kj M ik | H (3) j |B i ′ i ′′ i ǫ ii ′ i ′′ b (3) hF ik M kj | H (3) j |B i ′ i ′′ i ǫ ii ′ i ′′ + a (6) hF kl M lj | H (6) ijk |B i ′ i ′′ i ǫ ii ′ i ′′ + a (6) hF lj M kl | H (6) ijk |B i ′ i ′′ i ǫ ii ′ i ′′ + b (6) hF lk M ij | H (6) jkl |B i ′ i ′′ i ǫ ii ′ i ′′ + b (6) hF ij M lk | H (6) jkl |B i ′ i ′′ i ǫ ii ′ i ′′ + a (15) hF kl M lj | H (15) ijk |B i ′ i ′′ i ǫ ii ′ i ′′ + a (15) hF lj M kl | H (15) ijk |B i ′ i ′′ i ǫ ii ′ i ′′ + b (15) hF lk M ij | H (15) jkl |B i ′ i ′′ i ǫ ii ′ i ′′ + b (15) hF ij M lk | H (15) jkl |B i ′ i ′′ i ǫ ii ′ i ′′ + c (3) hM ij F i ′ j ′ | H (3) i ′′ |B jj ′ i ǫ ii ′ i ′′ + d (3) hM ij F i ′ j ′ | H (3) j |B i ′′ j ′ i ǫ ii ′ i ′′ + d (3) hF ij M i ′ j ′ | H (3) j |B i ′′ j ′ i ǫ ii ′ i ′′ + e (3) hM ij ′ F i ′ j | H (3) j |B i ′′ j ′ i ǫ ii ′ i ′′ + e (3) hF ij ′ M i ′ j | H (3) j |B i ′′ j ′ i ǫ ii ′ i ′′ + c (6) hM ij F i ′ j ′ | H (6) jj ′ k |B i ′′ k i ǫ ii ′ i ′′ + d (6) hM ij F i ′ j ′ | H (6) i ′′ jk |B j ′ k i ǫ ii ′ i ′′ + d (6) hF ij M i ′ j ′ | H (6) i ′′ jk |B j ′ k i ǫ ii ′ i ′′ + e (6) hM ij F i ′ j ′ | H (6) i ′′ j ′ k |B jk i ǫ ii ′ i ′′ + e (6) hF ij M i ′ j ′ | H (6) i ′′ j ′ k |B jk i ǫ ii ′ i ′′ + f (6) hM ij F kj ′ | H (6) i ′ i ′′ k |B jj ′ i ǫ ii ′ i ′′ + g (6) hM kj F ij ′ | H (6) i ′ i ′′ k |B jj ′ i ǫ ii ′ i ′′ + m (6) hM kj F jk | H (6) ii ′ l |B i ′′ l i ǫ ii ′ i ′′ + n (6) hM kj F jl | H (6) ii ′ k |B i ′′ l i ǫ ii ′ i ′′ + n (6) hF kj M jl | H (6) ii ′ k |B i ′′ l i ǫ ii ′ i ′′ + c (15) hM ij F i ′ j ′ | H (15) jj ′ k |B i ′′ k i ǫ ii ′ i ′′ + d (15) hM ij F i ′ j ′ | H (15) i ′′ jk |B j ′ k i ǫ ii ′ i ′′ + d (15) hF ij M i ′ j ′ | H (15) i ′′ jk |B j ′ k i ǫ ii ′ i ′′ + e (15) hM ij F i ′ j ′ | H (15) i ′′ j ′ k |B jk i ǫ ii ′ i ′′ + e (15) hF ij M i ′ j ′ | H (15) i ′′ j ′ k |B jk i ǫ ii ′ i ′′ (11)Expanding the above invariant amplitudes, we obtain contributions to individual decay pro-cesses. For example, expressing the tree decay amplitudes in terms of the coefficients in SU (3)invariant amplitudes, we have T (Λ b → π − p ) = − a (6) − a (15) + 2 b (3) + 2 b (6) + 6 b (15) + c (3) + d (3) − e (3) − c (6)+ d (6) − e (6) − f (6) − g (6) + 2 n (6) + 3 c (15) + 2 d (15) − d (15) + 3 e (15) − e (15) ; (12) T (Λ b → K − p ) = 2 a (3) − a (6) − a (15) + 6 a (15) + 2 b (3) + 2 b (6) + 6 b (15) + d (3) − e (3) − c (6) + d (6) − e (6) + 2 n (6) + 3 c (15) + d (15) − e (15) ; T (Λ b → π − Σ + ) = 2 a (3) + 2 a (6) − a (6) − a (15) + 6 a (15) − c (3) + b (6) − b (6) + c (6) − c (6) + 2 d (6) + 2 e (6) + 2 g (6) − g (6) − d (15) + 3 d (15) − e (15) + e (15) . We have[5] T (Λ b → K − p ) − T (Λ b → π − Σ + ) = T (Λ b → π − p ) . (13)We find several relations among the decay amplitudes shown below T (Ξ − b → K − n ) = T (Ξ − b → π − Ξ ) , T (Ξ b → ¯ K n ) = − T (Λ b → K Ξ ) ,T (Ξ − b → K Ξ − ) = T (Ξ − b → ¯ K Σ − ) , T (Ξ b → K Ξ ) = − T (Λ b → ¯ K n ) ,T (Ξ b → π − Σ + ) = − T (Λ b → K − p ) , T (Λ b → π − p ) = − T (Ξ b → K − Σ + ) ,T (Ξ b → π + Σ − ) = − T (Λ b → K + Ξ − ) , T (Λ b → K + Σ − ) = − T (Ξ b → π + Ξ − ) ,T (Ξ b → K − p ) = − T (Λ b → π − Σ + ) , T (Ξ b → K + Ξ − ) = − T (Λ b → π + Σ − ) . (14)4he full results are listed in Tables I to VI. The expressions for penguin and also P wave amplitudesare similar. Due to mixing between η and η , the decay modes with η in the final sates is not asclean as those with π and K in the final state to study. We do not list decay amplitudes with η in the final states for completeness.It is interesting to note that the pair of decays related by U -spin Λ b → π − p and Ξ b → K − Σ + ,and, Λ b → K − p and Ξ b → π − Σ + , respectively, have the same tree and penguin amplitudes, thatis T ( d ) j = T ( s ) j , and P ( d ) j = P ( s ) j . For these decays, although the absolute values of the decaywidths are different, the rate difference ∆( i ) = Γ( i ) − ¯Γ(¯ i ) are simply related by∆( d ) = − ∆( s ) (15)In obtaining the above relation, we have used the identity: Im ( V ub V ∗ ud V ∗ tb V td ) = − Im ( V ub V ∗ us V ∗ tb V ts ).We list those U-spin related decay rate differences pair with ∆ S = 0 and ∆ S = − − b → K − n ) = − ∆(Ξ − b → π − Ξ ) , (2) ∆(Ξ b → ¯ K n ) = − ∆(Λ b → K Ξ ) , (3) ∆(Ξ − b → K Ξ − ) = − ∆(Ξ − b → ¯ K Σ − ) , (4) ∆(Ξ b → K Ξ ) = − ∆(Λ b → ¯ K n ) , (5) ∆(Ξ b → π − Σ + ) = − ∆(Λ b → K − p ) , (6) ∆(Λ b → π − p ) = − ∆(Ξ b → K − Σ + ) , (16)(7) ∆(Ξ b → π + Σ − ) = − ∆(Λ b → K + Ξ − ) , (8) ∆(Λ b → K + Σ − ) = − ∆(Ξ b → π + Ξ − )(9) ∆(Ξ b → K − p ) = − ∆(Λ b → π − Σ + ) , (10) ∆(Ξ b → K + Ξ − ) = − ∆(Λ b → π + Σ − ) . The above relations imply relations for CP asymmetries A CP ( B a → MF ) ∆ S =0 A CP ( B b → MF ) ∆ S = − = − Br ( B b → MF ) ∆ S = − Br ( B a → MF ) ∆ S =0 · τ B a τ B b , (17)where τ a,b indicate the lifetimes of b-baryons B a,b , Br indicates branching ratio, and A CP indicatesthe CP asymmetry defined as A CP ( B → MF ) = Γ(
B → MF ) − Γ( ¯
B → ¯ M ¯ F )Γ( B → MF ) + Γ( ¯
B → ¯ M ¯ F ) . (18)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 forthe relevant decays, A CP ( ¯ B s → K + π − ) A CP ( ¯ B → K − π + ) = − Br ( ¯ B → K − π + ) τ ¯ B s Br ( ¯ B s → K + π − ) τ ¯ B , A CP ( ¯ B → π + π − ) A CP ( ¯ B s → K + K − ) = − Br ( ¯ B s → K + K − ) τ ¯ B Br ( ¯ B → π + π − ) τ ¯ B s . (19)The present data [1, 3, 14] give: − . ± .
55 and 3 . ± .
40 for the left and right hand sidesof the first equation above. These two values agree with the prediction very well. For the secondequation, the left hand side is − . ± .
78 and the right hand side is 5 . ± .
59. The centralvalues do not agree with the prediction, but agree within allowed error bars at 1 σ level.Corresponding to the above relations, for each of them there are two pairs. For the first one,the two pairs are: A CP (Λ b → π − p ) A CP (Ξ b → K − Σ + ) = − Br (Ξ b → K − Σ + ) τ Λ b Br (Λ b → π − p ) τ Ξ b , A CP (Λ b → K + Σ − ) A CP (Ξ b → π + Ξ − ) = − Br (Ξ b → π + Ξ − ) τ Λ b Br (Λ b → K + Σ − ) τ Ξ b . A CP (Λ b → K − p ) A CP (Ξ b → π − Σ + ) = − Br (Ξ b → π − Σ + ) τ Λ b Br (Λ b → K − p ) τ Ξ b , A CP (Λ b → K + Ξ − ) A CP (Ξ b → π + Σ − ) = − Br (Ξ b → π + Σ − ) τ Λ b Br (Λ b → K + Ξ − ) τ Ξ b . (21)We expect that the similar relations will hold at the same level as their B → MM counterparts. One expects that SU (3) symmetry holds are 20 to 30 percent level as seen in Kaon andHyperon decays. To estimate the level of SU (3) breaking effects, we define r c = − ( A CP ( ¯ B s → K + π − ) /A CP ( ¯ B → K − π + )) / ( Br ( ¯ B → K − π + ) τ ¯ B s /Br ( ¯ B s → K + π − ) τ ¯ B ) as the measure. In the SU (3) limit, r c = 1. Using experimental data, we have r c = 0 . ± .
19. The central value is about5% away from 1. The 1 σ level error bar is about 20%. This is an indication that SU (3) may workbetter in systems with a b quark than that for Kaon and Hyperon systems. Whether this is anaccidental or SU (3) works better for B decays needs to be understood. The relations found forb-baryons above can provide important clues.At present, only Λ b → π + p and Λ b → K − p charmless two body decays have been measuredexerimentally[1, 3, 15]. These data points cannot complete relations predicted in eq.(21). Onlywhen charmless two body Ξ b decays are also measured, the predictions can be tested. We urge ourexperimental colleagues to carry out related measurements to test the SM further.We would like to point out a particularly interesting relation that A CP (Ξ − b → K Ξ − ) A CP (Ξ − b → ¯ K Σ − ) = − Br (Ξ − b → ¯ K Σ − ) Br (Ξ − b → K Ξ − ) . (22)This relation does not involve the lifetimes of the decaying particle. This fact makes it a potentiallygood test with less error sources.Before concluding, we would like to make a comment on the approximate relation existed between¯ B → π − π + and ¯ B → K − π + when annihilation contributions are neglected and the possiblecorresponding one relating Λ b → π − p and Λ b → K − p .We refer the contributions proportional to a ( i ) α as annihilation contributions in view of thefact that the flavor indices of the initial states are contracted by the indices in the Hamiltonianas if the flavor structure of the initial states are annihilated by the Hamiltonian. Since the initialflavor structures are annihilated by the Hamiltonian, no flavor information flow directly to thefinal states implying that the flavor structure of the final states have to be created completelyby the weak interaction, the probability is smaller than those other terms where the initial stateflows 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. Theoreticalcalculations also agree with the assumption of smallness of annihilation contributions [10]. Moreover experimental data support the assumption that the annihilation contributions are small [1, 14].Under the small annihilation contribution assumption, one has [7, 8] A CP ( ¯ B → π − π + ) A CP ( ¯ B → K − π + ) ≈ − Br ( ¯ B → K − π + ) Br ( ¯ B → π − π + ) , A CP ( ¯ B s → K + π − ) A CP ( ¯ B s → K + K − ) ≈ − Br ( ¯ B s → K + K − ) Br ( ¯ B s → K + π − ) . (23)6or the first equation above, using PDG data [1], we find that the left side is given by − . ± . . ± .
17. For the second equation, the left side is given by − . ± . . ± .
60. The predicted relations are in agreementwith data within error bars. In particular the first equation above gives additional confidence onour assumption.Naivly, one might identify the corresponding decays of ¯ B → π − π + , K − π + with Λ b → π − p, K − p , respectively. One therefore might expect that A CP (Λ b → K − p ) /A CP (Λ b → π − p )to be approximately equal to − Br (Λ b → π − p ) /Br (Λ b → K − p ) when annihilation contributionsare neglected. This is, however, not true. One can easily see this by inspecting the relation ineq.(13) and Λ b → π − Σ + is not purely annihilation contribution induced decay, unlike ¯ B s → π − π + for in the case of B → MM decays. The difference can be traced back to the fact that althoughboth ( B − , ¯ B , ¯ B s ) and (Ξ − b , Ξ b , Λ b ) are SU (3) anti-triplet, the b-baryon has two light quarksand there are more ways to pass the initial light quarks to the final states allowing non-annihilationcontributions to induce Λ b → π − Σ + , but not for ¯ B s → π − π + . Therefore even annihilation con-tributions are neglected, A CP (Λ b → K − p ) /A CP (Λ b → π − p ) is not expected to be approximatelyequal to − Br (Λ b → π − p ) /Br (Λ b → K − p ).In summary we have studied CP violating relations for flavor SU (3) anti-triplet B b-baryonsdecay 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 wait more precise experimental datafrom LHCb to further test these relations. Similar analysis can be carried out for sixtet b-baryonto charmless two-body decays. Detailed analysis on this will be presented elsewhere. Acknowledgments
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High Energy Phys. 10 (2012) 037.TABLE I: SU (3) decay amplitudes for ∆ S = 0 , Ξ − b → M P processes.Decay mode a (3) a (6) a (6) a (15) a (15) b (3) b (3) b (6) b (6) b (15) b (15) c (3) d (3) d (3) e (3) e (3) c (6) d (6) d (6) e (6) e (6) f (6) g (6) m (6) n (6) n (6) c (15) d (15) d (15) e (15) e (15) Ξ − b → K − n − b → K Ξ − − b → η Σ − √ (0 1 1 3 3 1 1 1 -3 3 3) √ ( 2 1 1 -1 -1 2 -1 -1 1 14 4 0 -2 -2 0 3 3 -3 -3)Ξ − b → π − Λ √ (0 1 1 3 3 1 1 -3 1 3 3) √ (2 1 1 -1 -1 2 -1 -1 1 1-4 -4 0 -2 -2 0 3 3 -3 -3)Ξ − b → π − Σ √ (0 1 -1 3 -3 -1 1 -1 1 5 3) √ ( 0 1 -1 1 -1 0 1 -1 1 -10 0 0 2 -2 -2 1 -1 1 -1 )Ξ − b → π Σ − √ (0 -1 1 -3 3 1 -1 1 -1 3 5) √ (0 -1 1 -1 1 0 -1 1 -1 10 0 0 -2 2 2 -1 1 -1 1 ABLE II: SU (3) decay amplitudes for ∆ S = − , Ξ − b → M P processes.Decay mode a (3) a (6) a (6) a (15) a (15) b (3) b (3) b (6) b (6) b (15) b (15) c (3) d (3) d (3) e (3) e (3) c (6) d (6) d (6) e (6) e (6) f (6) g (6) m (6) n (6) n (6) c (15) d (15) d (15) e (15) e (15) Ξ − b → π − Ξ − b → ¯ K Σ − − b → η Ξ − √ (0 -2 1 -6 3 1 -2 1 0 3 6) √ (1 -2 1 -1 2 -1 -1 2 -2 1-2 -2 0 -2 4 3 -3 0 0 3)Ξ − b → K − Λ √ (0 1 -2 3 -6 -2 1 0 1 6 3) √ (-1 1 -2 2 -1 1 2 -1 1 -22 2 0 4 -2 -3 0 -3 3 0)Ξ − b → K − Σ √ (0 1 0 3 0 0 1 -2 1 4 3) √ ( 1 1 0 0 -1 1 0 -1 1 0-2 -2 0 0 -2 -1 2 1 -1 -2)Ξ − b → π Ξ − √ (0 0 1 0 3 1 0 1 -2 3 4) √ (1 0 1 -1 0 1 -1 0 0 12 2 0 -2 0 1 1 2 -2 -1) ABLE III: SU (3) decay amplitudes for ∆ S = 0 , Ξ b → M P processes.Decay mode a (3) a (6) a (6) a (15) a (15) b (3) b (3) b (6) b (6) b (15) b (15) c (3) d (3) d (3) e (3) e (3) c (6) d (6) d (6) e (6) e (6) f (6) g (6) m (6) n (6) n (6) c (15) d (15) d (15) e (15) e (15) Ξ b → K + Ξ − -2 (1 -1 1 3 -1 0 0 0 0 0 0)(1 0 0 0 0 0 1 -1 1 -12 2 0 2 -2 0 -3 1 -1 3)Ξ b → π + Σ − -2 (1 -1 0 3 -2 1 0 1 0 3 0)(0 0 0 0 0 1 0 -1 1 00 0 0 0 -2 3 0 -1 1 0)Ξ b → K − p -2( 1 1 -1 -1 3 0 0 0 0 0 0)(1 0 0 0 0 0 -1 1 -1 1-2 -2 0 -2 2 0 1 -3 3 -1)Ξ b → π − Σ + -2( 1 0 -1 -2 3 0 1 0 1 0 3)( 0 -1 0 0 1 1 -1 0 0 10 0 0 -2 0 -3 -1 0 0 1)Ξ b → K Ξ -2( 1 0 1 -2 -1 0 1 0 -1 0 -1)(0 -1 0 1 0 -1 1 0 0 -10 0 0 2 0 1 3 0 0 -3 )Ξ b → ¯ K n -2( 1 1 0 -1 -2 1 0 -1 0 -1 0)( 0 0 -1 1 0 -1 0 1 -1 00 0 0 0 2 -1 0 3 -3 0)Ξ b → η Λ − (6 3 3 -3 -3 1 1 -3 -3 3 3) ( 4 -1 -1 1 1 6 3 3 -3 -30 0 0 6 6 0 3 3 -3 -3)Ξ b → η Σ − √ (0 -1 -1 5 5 -1 -1 -1 3 5 -3) √ (2 1 1 -1 -1 2 -1 -1 1 14 4 0 -2 -2 8 3 -13 13 -3)Ξ b → π Λ − √ (0 -1 -1 5 5 -1 -1 3 -1 -3 5) √ ( 2 1 1 -1 -1 2 -1 -1 1 1-4 -4 0 -2 -2 -8 -13 3 -3 13)Ξ b → π Σ -(2 -1 -1 1 1 1 1 1 1 -5 -5) ( 0 -1 -1 1 1 2 -1 -1 1 10 0 0 -2 -2 0 -1 -1 1 1 ) ABLE IV: SU (3) decay amplitudes for ∆ S = − , Ξ b → M P processes.Decay mode a (3) a (6) a (6) a (15) a (15) b (3) b (3) b (6) b (6) b (15) b (15) c (3) d (3) d (3) e (3) e (3) c (6) d (6) d (6) e (6) e (6) f (6) g (6) m (6) n (6) n (6) c (15) d (15) d (15) e (15) e (15) Ξ b → π + Ξ − -2(0 0 -1 0 -1 1 0 1 0 3 0)(-1 0 -1 1 0 1 -1 0 0 1-2 -2 0 -2 0 3 3 -2 2 -3)Ξ b → K − Σ + -2(0 -1 0 -1 0 0 1 0 1 0 3)(-1 -1 0 0 1 1 0 -1 1 02 2 0 0 -2 -3 -2 3 -3 2)Ξ b → η Ξ − √ (0 2 -1 2 -1 1 -2 -1 0 -1 6) √ (1 2 -1 1 -2 -1 -1 2 -2 1-2 -2 0 -2 4 -7 -9 8 -8 9)Ξ b → ¯ K Λ − √ (0 -1 2 -1 2 -2 1 0 -1 6 -1) √ (1 -1 2 -2 1 -1 2 -1 1 -22 2 0 4 -2 7 8 -9 9 -8)Ξ b → ¯ K Σ − √ (0 1 0 1 0 0 -1 -2 1 4 1) √ (1 1 0 0 -1 -1 0 1 -1 02 2 0 0 2 3 2 -3 3 -2)Ξ b → π Ξ − √ (0 0 1 0 1 -1 0 1 -2 1 4) √ (1 0 1 -1 0 -1 1 0 0 -1-2 -2 0 2 0 -3 -3 2 -2 3) ABLE V: SU (3) decay amplitudes for ∆ S = 0 , Λ b → M P processes.Decay mode a (3) a (6) a (6) a (15) a (15) b (3) b (3) b (6) b (6) b (15) b (15) c (3) d (3) d (3) e (3) e (3) c (6) d (6) d (6) e (6) e (6) f (6) g (6) m (6) n (6) n (6) c (15) d (15) d (15) e (15) e (15) Λ b → K + Σ − b → π − p b → η n √ (0 -1 2 -1 2 -2 1 2 -3 2 3) √ ( -1 1 -2 2 -1 1 -2 1 -1 22 2 0 -4 2 1 0 1 -1 0Λ b → K Λ √ (0 2 -1 2 -1 1 -2 -3 2 3 2) √ ( -1 -2 1 -1 2 1 1 -2 2 -1-2 -2 0 2 -4 -1 1 0 0 -1)Λ b → K Σ √ (0 0 1 0 1 -1 0 -1 0 5 0) √ (-1 0 -1 1 0 1 -1 0 0 1-2 -2 0 -2 0 -5 -5 6 -6 5)Λ b → π n √ (0 1 0 1 0 0 -1 0 -1 0 5) √ ( -1 -1 0 0 1 1 0 -1 1 02 2 0 0 -2 5 6 -5 5 -6) ABLE VI: SU (3) decay amplitudes for ∆ S = − , Λ b → M P processes.Decay mode a (3) a (6) a (6) a (15) a (15) b (3) b (3) b (6) b (6) b (15) b (15) c (3) d (3) d (3) e (3) e (3) c (6) d (6) d (6) e (6) e (6) f (6) g (6) m (6) n (6) n (6) c (15) d (15) d (15) e (15) e (15) Λ b → K + Ξ − b → π + Σ − b → K − p b → π − Σ + b → K Ξ b → ¯ K n b → π Σ b → η Λ (3 0 0 -3 -3 2 2 0 0 -6 -6) (1 2 2 -2 -2 0 0 0 0 00 0 0 0 0 0 -3 -3 3 3)Λ b → η Σ √ (0 -1 -1 2 2 0 0 2 0 -4 0) √ ( 0 0 0 0 0 -2 1 1 -1 -12 2 0 2 2 4 6 -2 2 -6)Λ b → π Λ √ (0 -1 -1 2 2 0 0 0 2 0 -4) √ (0 0 0 0 0 -2 1 1 -1 -1-2 -2 0 2 2 -4 2 6 -6 2)(0 0 0 0 0 -2 1 1 -1 -1-2 -2 0 2 2 -4 2 6 -6 2)