Beauty baryon non-leptonic decays into decuplet baryons and $CP$-asymmetries based on $SU(3)$-Flavor analysis

We consider charmless weak decays of beauty-baryons into decuplet baryons and pseudoscalar mesons in a general framework based on $SU(3)$-flavor decomposition of the decay amplitudes. The dynamical assumption independent $SU(3)$ analysis accounts for the effects of an arbitrarily broken $SU(3)$ symmetry in these decays. An alternative approach in terms of quark diagrams is also provided and compared with the $SU(3)$ decomposition in the limit of exact $SU(3)$-flavor symmetry. Furthermore, the symmetries of the effective Hamiltonian is used to relate or neglect reduced $SU(3)$ amplitudes to derive several sum rule relations between amplitudes and relations between $CP$ asymmetries in these decays and identify those that hold even if $SU(3)$ is broken.


I. INTRODUCTION
In recent times, bottom baryon decays have come under increased scrutiny chiefly due to the large number of b-baryon production at LHCb [1]. Among several other interesting rare decays of b-baryons, LHCb has analyzed CP -asymmetries in bottom baryon charmless decays to two-body and multibody hadronic final states [2][3][4][5][6][7][8][9][10]. The theoretical foundation laid out to study B meson decays  is also useful to study bottom and charmed baryon decay amplitude sum-rules and CP asymmetry relations for various decay modes. In particular, the general framework of SU (3) analysis in beauty mesons as well as charm meson decays [52][53][54][55][56][57][58][59][60][61][62][63] into two pseudoscalars (P P ), pseudoscalar-vector boson (P V ), and two vector mesons (V V ) can be easily extended to study bottom and charmed baryon decaying into an octet baryon and a pseudo-scalar meson .
On the other hand, experimental evidences suggest a b baryon to a decuplet baryon transition alongside the pseudo-scalar(vector) meson is also possible. In this paper, we consider such hadronic beauty-baryon decays featuring a decuplet of light baryons and a pseudoscalar meson based on the SU (3) decomposition of the decay amplitudes in terms of SU (3)-reduced amplitudes. An alternate approach based on flavor flow along quark lines is also presented which is equivalent to the SU (3) decomposition in the limit of exact SU (3)-flavor symmetry. In contrast to previous studies, our approach [19,87,88] facilitates an SU (3) decomposition of the decays in terms of SU (3)-reduced amplitudes without any particular set of assumptions about the underlying dynamics.
The formalism of SU (3) decomposition of the decay amplitudes in terms of reduced SU (3) amplitudes is presented in Sec. II. The results are summarized in Ap-pendix A and B. The quark flow diagrams are indicated in Appendix C. The relations between the amplitudes for beauty baryon decays into decuplet of light baryons and pseudoscalar mesons are derived in Sec. III. The effects of SU (3) breaking on account of s-quark mass are considered in Sec. IV. The corresponding relations between CP asymmetries are derived in Sec. V. We finally conclude in Sec. VI.

II. FORMALISM
In this paper we study the charmless decay of an antitriplet (3) beauty-baryon into a decuplet baryon (B) and an octet pseudoscalar meson (P ), i.e. B b (3) → B(10) P (8). This decay is described by a Hamiltonian with ∆Q = 0 and ∆S = −1, 0 (equivalent to ∆I 3 and ∆Y representation). The possible decays can be divided into two sub classes, namely the ∆S = 0 and ∆S = −1 transitions. The allowed final state SU (3) representations (f ) are; 8, 10, 27, 35. There are twenty physical process possible for ∆S = 0 and another twenty for ∆S = −1.
In Appendix A, each of these decay modes are decomposed in terms of the SU (3) reduced amplitudes that add upto forty. Since the physical η and η mesons are admixtures of octet η 8 and singlet η 1 mesons, a study of B b (3) → B(10) P (1) is also necessary. Therefore one has to take into account four (two each for ∆S = −1 and ∆S = 0) additional independent SU (3) amplitudes. Since no assumption about the particular form of effective Hamilton has been made yet, these forty four reduced amplitudes are all independent of each other and no amplitude relation exist between the decay modes. The expression of the amplitudes in terms of reduced SU (3) amplitudes is concisely given as, where, C a,b,c A,B,C are the SU (2) Clebsch-Gordon coefficients and . ( are the SU (3) isoscalar coefficients [89][90][91][92][93] obtained by coupling the representations R a ⊗ R b → R c . T is the triality of a SU (3) representation of the initial state conjugate i and the phase factor appearing in front ensures that correct signs are assigned to the individual initial b-baryon anti-triplet. Given a form of effective Hamiltonian (H eff ), it can be SU (3) decomposed, where F {Y,I,I3} R depends on the SU (3) CG coefficients appearing in front of the SU (3) representations (R I ).
Moreover F {Y,I,I3} R also contains additional factors entering Eq. (3) in form of Wilson coefficients and CKM elements. It is also important to note that by knowing the dynamical coefficients for different isospin values in a given SU (3) representation, one can drop the isospin Casimir label (I) and express the Wigner-Eckart reduced matrix element f R i , in its usual form, independent of the isospin I label. By using completeness of SU (3) CG coefficients up to a phase factor, The lowest order effective Hamiltonian [94][95][96] for charmless b-baryon decays consists of ∆S = −1 and ∆S = 0 parts. Each part is composed from the operators O 1 ,. . . , O 10 . The complete Hamiltonian can be written as: t are the CKM elements and C i s are the Wilson coefficients. O 1 and O 2 are the "tree" operators: O 3 , . . . , O 6 are the "gluonic penguin" operators: and finally O 7 , . . . , O 10 are the four "Electroweak penguin" (EWP) operators: H eff is a linear combinations of four quark operators of the form (q 1 b)(q 2 q 3 ). These operators transform as 3 3 3 under SU (3)-flavor and can be decomposed into sums of irreducible operators corresponding to irreducible SU (3) representations: 15 , 6 , 3 (6) , 3 (3) where the superscript index: '6' ('3') indicates the origin of 3 out of the two possible representations arising from the tensor product of q 1 and q 2 . The SU (3) triplet representation of quarks (q i ) and its conjugate denoting the anti-quarks (q i ) consist of the flavor states; The unbroken dim-6 effective Hamiltonian contains parts that transform as 3, 6, 15 under SU (3)-flavor. The tree part of the effective Hamiltonian responsible for b → u transition has the following SU (3) decomposition , The penguin part of the effective Hamiltonian is also SU (3) decomposed below, where we have ignored the contributions from O 7 and O 8 Electroweak penguin operators due to the smallness of the Wilson coefficients C 7 and C 8 appearing in front of them. For charmless b-decays, b → qcc, q = d, s contributes as an independent 3 of SU (3) without affecting the SU (3) structure of Eq. (11)- (13). With this particular choice of effective Hamiltonian and allowed final state SU (3) representations, there are five independent SU (3)reduced matrix elements: 10 15 3 , 27 15 3 (14) and the SU (3)-reduced amplitudes arising from H T are conveniently expressed as follows; The decay amplitudes for all possible ∆S = −1 and ∆S = 0 processes are expressed using Eq (15) and are given in Table I and Table II respectively.
An alternate description of decay amplitudes is obtained in terms of topological quark diagrams. The symmetry properties of the final state decuplet baryons allow five possible diagrams starting from a flavor antitriplet b-baryon whose light quarks are in a flavor antisymmetric states. Those five independent topologies con- sist of three W -exchanges (E 1 , E 1 , E 3 ), one tree (T ) and a penguin-like (P q ) (q being the flavor of the quark going in the loop) amplitude where the marked quarks are anti-symmetrized in the initial state baryon. In ad-dition, a sixth diagram (S) appears after including the decay modes containing the singlet η 1 . The mapping between the topological amplitudes and the SU (3)-reduced amplitudes is given below, T are indeed suppressed. We have chosen to include the E 3 and T diagrams throughout the rest of this paper for generality. The two approaches are equivalent and imply that individual topological amplitudes cannot be expressed in terms of a single SU (3)-reduced amplitude and vice-versa in these two basis. It is also important to note that the topology, P u , originate purely from the tree operators i.e. O 1 and O 2 as emphasized in [23][24][25][26][27], even though it is denoted by P u and referred to as a penguin topology. Moreover the W -exchange topologies being a four-quark tree-like structure also contribute to the tree amplitudes.
The transition induced by QCD penguin operators, given in Eq (12) is only a 3 under SU (3) and corresponding SU (3)-reduced amplitude is identified with the 'penguin' amplitude P t , A. Relating EWP and Tree amplitudes We begin this section by noting that the 15 and 6 part of the Hamiltonian described in Eqs. (11) and (13) relate the contributions to the decay from the tree and EWP operators [16,22,30] respectively, so as to effectively obey the following relations, a EW 10 = − a EW 10 = − 3 2 Using numerical values of the Wilson coefficients to leading logarithmic order one obtains, To a good approximation [22] these two ratios of Wilson coefficients can be taken to be a common value [95,96] given by κ; With this additional assumption, Eq. (20) implies that the b EW The equivalence of SU (3)-reduced amplitudes to topological diagrams, allows one to interpret P EW i as Electroweak quark diagrams with one insertion of the Electroweak penguin operator; where, i = {1, 2, 3}. As mentioned earlier, there is no simple relation between the 3 part of the EWP Hamiltonian to the tree part. There are, however, decays where the 3 part of the Hamiltonian cannot contribute to the formation of final states which require a pure ∆I = 1 or ∆I = 3/2 transition. For example, the decay Λ 0 b → ∆ + K − receives contribution from 6 and 15 part of the effective Hamiltonian and in this particular case the ratio of EWP and tree contributions is given entirely by the simple ratio −3/2κ(λ s t /λ s u ). The SU (3)-reduced amplitudes for the penguin operators are provided in Eq. 84 and Eq. 85.

III. AMPLITUDE RELATIONS
The complete decay amplitude is given in terms of tree and penguin SU (3)-reduced amplitudes and the CKM elements, where q = s, d denote the ∆S = −1, 0 process. Since the SU (3) operators appear in EW and tree part of the Hamiltonian in a particular combination, the same amplitude relations between ∆S = −1 and ∆S = 0 processes are satisfied by the EWP part and the tree part. Therefore, the following decay amplitude relations are obtained; The ∆S = −1 triangle relations are given below, The ∆S = 0 triangle relations are also obtained, The SU (3) structure of the unbroken Hamiltonian is modified by this term and to the first order in strange quark mass, the broken Hamiltonian is made of the following SU (3) representations [19,52,54,56,61,[97][98][99], where the subscript i = 1, 2, 3 indicates the origin of that representation from 3 ,6, 15 respectively and is the SU (3)-breaking parameter. SU (3) breaking effects will induce higher SU (3) representations and some of the amplitude relations will cease to hold as there are now thirteen independent SU (3)-reduced matrix elements. The isospin relation, and isospin triangle relations, for ∆S = −1, and for ∆S = 0, (59) continue to hold. In addition, arbitrary SU (3)-breaking but isospin conserving effects still forbid ∆I = 2 and ∆I = 5 2 transitions which results in general amplitude sum rules,

V. CP RELATIONS
The general decay amplitude for a spin-1/2 b-baryon (B) to a spin 0 pseudo-scalar (P) and a spin-3/2 (D) is given by, where u µ D is the Rarita-Schwinger spinor for the spin-3/2 decuplet baryon, q µ is the momentum of the pseudoscalar meson and u B is the spinor for the initial spin-1/2 b-baryon. The two coefficients a and b contain the CKM elements as well as the same flavor structure as A tree and A penguin . The total decay rate for an unpolarized b-baryon is given by, where |p D | is the magnitude of the 3-momentum of the decuplet baryon in the rest frame of the initial b-baryon.
The decay products can be in any one of the two possible relative angular momentum states, l = 1 and l = 2 identified as P-wave and D-wave respectively. It is straightforward to connect the P-wave and D-wave amplitude to the SU (3)-amplitudes where A P and A D are A CP is defined [98] subsequently as, where, A CP is the sum of CP violating contribution from the δ P CP and δ D CP with appropriate phase-space factor multiplied: where τ B is the lifetime of the beauty-baryon.
By definition, J being the well known Jarlskog invariant. Based on amplitude relations for the tree and penguin parts obtained in Eq. (27) following δ a CP relations [100][101][102] are obtained, for both l = P and l = D. Since, A CP depends on the masses of the initial and final baryons as well as the final state meson [71,102], some approximation is needed to obtain A CP relations between various modes. Ignoring p D and m D differences between such modes, CP violation relations can be experimentally verified using the relation, where i, j, k and l, m, n are indices corresponding to the various baryons belonging to the above mentioned δ CP relations. There is a further simplification in case i = l, resulting in where the uncertainties due to lifetime measurement also cancel out [67]. The decay asymmetry parameter (α) can be measured from an angular distribution study of the final states provided that the subsequent decay of the decuplet baryon is parity violating. The relative strength of the D-wave contribution [72,102] is extracted from α; By systematically taking into account both partial wave contributions, a reliable prediction for A CP relations is possible.

VI. CONCLUSION
We have explored hadronic anti-triplet (3) beautybaryon into a decuplet baryon (B) and an octet pseudoscalar meson (P ), i.e. B b (3) → B(10) P (8), based on SU (3) decomposition of the decay amplitudes in a general framework. This extends our previous analysis [88] of the anti-triplet beauty baryon decays into the octet or singlet of a light baryon and a pseudoscalar meson and completes the application of the method to decays involving any non-charmed baryon. We have shown that in the most general case, the forty distinct decay modes require forty independent reduced SU (3) amplitudes to describe all possible ∆S = −1 and ∆S = 0 processes. The dimension-6 effective Hamiltonian and allowed final state SU (3) representations constrain the number of independent SU (3)-reduced matrix elements to five. An alternative approach in terms of quark diagrams is also provided and compared with the SU (3) decomposition in the limit of exact SU (3)-flavor symmetry. We explicitly demonstrate a one to one correspondence between the quark-diagrams and SU (3)-reduced matrix elements. Both the approaches indicate that there exist several amplitude relations between different decay modes. We explicitly derive those sum rules relations between decay amplitudes as well as relations between CP asymmetries. We further probe the SU (3)-breaking effects in the decay amplitudes at leading order in the SU (3)-breaking parameter and identify those amplitude relations that survive even when the SU (3) flavor symmetry is no longer exact.
Appendix A SU (3)-decomposition of ∆S = −1 processes for a generic Hamiltonian without any dynamical assumptions: SU (3)-decomposition of ∆S = 0 processes for a generic Hamiltonian without any dynamical assumptions: Appendix B T matrix for ∆S = 0 processes assuming the particular form of dim-6 tree Hamiltonian described in Eq. (11): T matrix for ∆S = −1 processes assuming the particular form of dim-6 tree Hamiltonian described in Eq. (11): P matrix for ∆S = 0 processes assuming the particular form of dim-6 gluonic and Electroweak penguin Hamiltonian described in Eq. (12) and Eq. (13): P matrix for ∆S = −1 processes assuming the particular form of dim-6 gluonic and Electroweak penguin Hamiltonian described in Eq (12) and Eq (13): The SU (3)-decomposition of decay amplitudes without assuming SU (3)-breaking for 3 → 10 ⊗ 1 decay modes are given below: ∆S = 0 processes: ∆S = −1 processes: We have used the following shorthand notation to express the tree (T ) and penguin (P) matrices in a convenient form: (C 10 ± C 9 ) = C ± 9,10 , Rewriting O 1 and O 2 we get, Using the following identity, one can write O − as, Using Firez transformation [108] one can recast Eq. (95) as [105], where q C = Cq * , C being the charge-conjugation operator. According to diquark mechanism [105,106], the diquark aij (u i )(1 + γ 5 ) (d j ) C has a total spin 0 and it transforms as a 3 and 3 under SU (3) F and SU (3) color respectively. This diquark ending up completely in the final state baryon cannot produce 1 a decuplet baryon as any quark pair inside the decuplet baryon transforms as 6 and 3 under SU (3) F and SU (3) color with a total spin equaling 1. On the other hand, the operator Q + which is a product of SU (3) color sextet currents cannot produce a color singlet baryon. In diagrams T and E 3 , the quark pair antisymmetric in color and flavor [103,104] originating from weak interaction ends up in the decuplet baryon which is forbidden by the above mentioned argument [109][110][111][112][113][114][115]. Since the quark pair (ud) 3 is in isospin 0 state 2 , the total quark transition obeys ∆I = 1/2 rule. In contrast, in diagrams E 1 and E 2 , the diquark argument is not applicable as only one of the quarks from weak interaction form the final state baryon while the other ends up in the meson. As a consequence, the number of independent diagrams reduces to three. If we demand an equivalent description of all possible decays in terms of SU (3)reduced amplitudes, some of the reduced amplitudes can no longer be independent. The relation between the remaining diagrams and the SU (3)-reduced amplitudes is given below, where c 8 , b 8 , a 8 , a 10 , a 27 are defined in Eq. (15). Equivalently, the following relations hold for the SU (3)-reduced matrix elements, a 27 = 0 (C 1 = 0, C 2 = 0) (100) The reduction in number of independent SU (3)-reduced amplitudes also imply additional amplitude relations that are enclosed in the box: