PHYSICAL REVIEW D 98, 074011 (2018)
Nonleptonic two-body decays of single heavy baryons ΛQ, ΞQ, and ΩQ (Q = b, c) induced by W emission in the covariant confined quark model
Thomas Gutsche,1 Mikhail A. Ivanov,2 Jürgen G. Körner,3 and Valery E. Lyubovitskij1,4,5,6 1Institut für Theoretische Physik, Universität Tübingen, Kepler Center for Astro and Particle Physics, Auf der Morgenstelle 14, D-72076 Tübingen, Germany 2Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia 3PRISMA Cluster of Excellence, Institut für Physik, Johannes Gutenberg-Universität, D-55099 Mainz, Germany 4Departamento de Física y Centro Científico Tecnológico de Valparaíso-CCTVal, Universidad T´ecnica Federico Santa María, Casilla 110-V, Valparaíso, Chile 5Department of Physics, Tomsk State University, 634050 Tomsk, Russia 6Laboratory of Particle Physics, Tomsk Polytechnic University, 634050 Tomsk, Russia
(Received 2 July 2018; published 12 October 2018)
We have made a survey of heavy-to-heavy and heavy-to-light nonleptonic heavy baryon two-body decays and have identified those decays that proceed solely via W-boson emission, i.e., via the tree graph Ω− → Ωð Þ0ρ− π− Ω− → Ω− ψ η Ξ0;− → Ξ0;− ψ η contribution. Some sample decays are b c ð Þ, b J= ð cÞ, b J= ð cÞ, ð Þ 0 − þ þ Λb → ΛJ=ψðηcÞ, Λb → ΛcDs , Ωc → Ω ρ ðπ Þ, and Λc → pϕ. We make use of the covariant confined quark model previously developed by us to calculate the tree graph contributions to these decays. We calculate rates, branching fractions and, for some of these decays, decay asymmetry parameters. We compare our results to experimental findings and the results of other theoretical approaches when they are available. Our main focus is on decays to final states with a lepton pair because of their clean experimental signature. For these decays we discuss two-fold polar angle decay distributions such as in the cascade decay Ω− → Ω− → Ξπ Λ − ψ → lþl− b ð ; K ÞþJ= ð Þ. Lepton mass effects are always included in our analysis.
DOI: 10.1103/PhysRevD.98.074011
I. INTRODUCTION quarks participating in the weak interaction. For complete- ness we also analyze those nonleptonic modes which are In the present paper we focus on the study of nonleptonic contributed to by both tree graph contributions and by a W-emission (also referred to as tree graph contributions [1]) specific class of W-exchange contribution which vanish in two-body decays of single heavy baryons the SUð3Þ limit. 0 0 0 Our investigation is limited to the decays of the lowest B1ðq1q2q3Þ → B2ðq q q ÞþMðq q¯ ¯ Þð1Þ 1 2 3 m m lying ground states. The higher lying ground-state bottom 3 2 where M stands for a pseudoscalar meson P or a vector and charm baryons such as the spin = partners decay meson V. One of the quarks in the initial baryon is heavy. A strongly or electromagnetically. For these the weak decays necessary condition for the contribution of the tree graph do occur but have tiny branching ratios and are therefore not so interesting from an experimental point of view. In class of decays is that a light quark pair q q ¼ q0q0 is i j i j this work we therefore concentrate on the weak nonleptonic shared by the parent and daughter baryon B1 and B2, decays of the lowest lying spin 1=2 ground state bottom respectively. A sufficient condition for the tree graph class Λ0 Ξ0 Ξ− Ω− Λþ Ξþ Ξ0 Ω0 baryons b, b, b , b and c , c , c, c. of decays is that (i) qm is not among q1, q2, q3 and (ii) qm¯ is 0 0 0 Based on the necessary and sufficient conditions for- not among q , q , q when q , q ¯ are not among the 1 2 3 m m mulated above we have made a search for heavy-to-heavy and heavy-to-light nonleptonic heavy baryon decays that are solely contributed to by the tree graph contributions. We Published by the American Physical Society under the terms of have identified a number of decays in this class. Some of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to these decays have been seen or are expected to be seen in the author(s) and the published article’s title, journal citation, the near future, or upper bounds have been established for and DOI. Funded by SCOAP3. them. Although some of these tree graph decays are singly
2470-0010=2018=98(7)=074011(19) 074011-1 Published by the American Physical Society GUTSCHE, IVANOV, KÖRNER, and LYUBOVITSKIJ PHYS. REV. D 98, 074011 (2018) or doubly Cabibbo–Kobayashi–Maskawa (CKM) sup- The paper is structured as follows. In Sec. II we provide a pressed the huge data sample available at the LHC on list of nonleptonic heavy baryon decays that proceed solely heavy baryon decays will hopefully lead to the detection of by the tree graph contribution. We classify the decays the remaining decay modes in the near future. according to their CKM and color structure. In Sec. III we Nonleptonic baryon and meson decays are interesting discuss the parameters of the effective weak Lagrangians from the point of view that they provide a testing ground for that induce the current-induced transitions used in this our understanding of QCD. They involve an interplay of paper. In Sec. IV we define the matrix elements for the short-distance effects from the operator product expansion nonleptonic decays 1=2þ → 1=2þ0−ð1−Þ and 1=2þ → of the current-current Hamiltonian and long-distance 3=2þ0−ð1−Þ and relate the helicity amplitudes to the effects from the evaluation of current-induced transition invariant amplitudes of these decay processes. In this matrix elements. On the phenomenological side one can way we obtain very compact expressions for the total 1 investigate the role of =Nc color loop suppression effects. rates. In Sec. V we discuss the structure of the interpolating The analysis of nonleptonic two-body heavy baryon decays currents that describe the interactions of light and heavy can also contribute to the determination of CKM matrix baryons with their constituent quarks. In Sec. VI we briefly elements if the theoretical input can be brought under describe the main features of the CCQM model and present control. the values of the model parameters such as the constituent An analysis of the tree graph decays provides important quark masses that have been determined by a global fit to a information on the value of the heavy-to-heavy and heavy- multitude of decay processes. In Sec. VII we present our μ 2 to-light form factors hB2jJ jB1i at one particular q -value numerical results on the partial branching fractions of the corresponding to the mass of the final state meson. A small various nonleptonic baryon decays and compare them to final-state meson mass implies that the form factors are experimental results when they are available. We also probed close to the maximum recoil end of the q2-spectrum discuss angular decay distributions for the most interesting where one has reliable predictions from light cone sum cascade decay processes and present numerical results on rules. The decay constants describing the mesonic part of the spin density matrix elements that describe the angular the transition hMjJμj0i are either experimentally available decay distributions. Finally, in Sec. VIII, we summarize from weak or electromagnetic decays of the mesons, or are our results. constrained from further theoretical model analysis, lattice calculations or QCD sum rule analysis. II. CLASSIFICATION OF FACTORIZING DECAYS In this paper we present a detailed analysis of the nonleptonic tree graph heavy baryon decays in the frame- In this section we list all nonleptonic two-body decays work of our covariant confined quark model (CCQM) of the lowest lying spin-1=2 ground state heavy bottom Λ Ξ Ω proposed and developed in Refs. [2–20]. The CCQM and charm baryon decays b;c, b;c, b;c which proceed model is quite flexible in as much as it allows one to solely via the tree graph. We exclude the higher-lying study mesons, baryons and even exotic states such as ground-state baryons which decay either strongly or tetraquark states as bound states of their constituent quarks. electromagnetically. In this approach particle transitions are calculated through The quark content of the baryon and meson states Feynman diagrams involving quark loops. In the present participating in the nonleptonic two-body decays is application the current-induced baryon transitions B1 → B2 given by are described by a two-loop diagram which requires a → 0 0 0 ¯ genuine two-loop calculation. The vacuum-to-meson tran- B1ðq1q2q3Þ B2ðq1q2q3ÞþMðqmqm¯ Þ: ð2Þ sition involves a one-loop calculation. The high energy behavior of quark loops is tempered by nonlocal Gaussian- One can formulate necessary and sufficient conditions for type vertex functions with a Gaussian-type fall-off behavior the decays which solely proceed via the tree graph con- characterized by a size parameter specific to the meson and tribution. These read baryon in question. The nonlocal hadron-quark vertices ∶ 0 0 1 2 3 have an interpolating current structure. In a recent refine- necessary condition qiqj ¼ qiqji; j ¼ ; ; ment of our model we have incorporated quark confine- 0 0 0 sufficient condition∶ qm ∋ q1;q2;q3; qm¯ ∋ q1;q2;q3; ment in an effective way [21–25]. Quark confinement is introduced at the level of Feynman diagrams and is based qm;qm¯ not part of the effective on an infrared regularization of the relevant quark-loop current-current Lagrangian ð3Þ diagrams. In this way quark thresholds corresponding to free quark poles in the Green functions are removed (see We have made a survey of two-body nonleptonic bottom details in Refs. [21–25]). Tree graph contributions to some and charm baryon decays which are solely contributed to of the decays treated in this paper have been discussed by the tree diagram. The results of this search are given as before in [26–28]. follows.
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Bottom baryon decays:
→ → ∶ Λ0 → Λþ ð Þ− b cc s color favored b c Ds Λ0 → Λ0η ψ color suppressed b cðJ= Þ Ξ0;− → Ξ0;−η ψ b cðJ= Þ Ω− → Ω−η ψ b cðJ= Þ → → ∶ Ω− → Ωð Þ0π− ρ− b cu d color favored b c ð Þ Ξ− → Σ− ð Þ0 color suppressed b D Ω− → Ξð Þ− ð Þ0 b D ∶ → → Ξ0 → Ξþ ð Þ− X b cc d color favored b c D Ω− → Ωð Þ0 ð Þ− b c D Λ0 → η ψ color suppressed b n cðJ= Þ Ξ− → Σ−η ψ b cðJ= Þ Ξ0 → Λ0 Σ0 η ψ b ð Þ cðJ= Þ − ð Þ− Ω → Ξ ηcðJ=ψÞ b ð4Þ ∶ → → Ξ− → Ξ− ð Þ0 X b cu s color suppressed b D Ω− → Ω− ð Þ0 b D ∶ → → Λ0 → ð Þ− X b uc s color favored b pDs Ω− → Ξ 0 ð Þ− b Ds ð Þ Ω− → Ω− ¯ ð Þ0 color suppressed b D ∶ → → Ξ− → Σ−π0 η η0 ρ0 ω X b uu d color suppressed b ð ; ; ; Þ Ω− → Ξð Þ−π0 η η0 ρ0 ω b ð ; ; ; Þ ∶ → → Ξ0 → Σþ ð Þ− XX b uc d color favored b D Ξ− → Σ0 ð Þ− b D ð Þ Ω− → Ξð Þ0 ð Þ− b D Ω− → Ξð Þ− ¯ ð Þ0 b D ∶ → → Ξ− → Ξ−π0 η η0 ρ0 ω XX b uu s color suppressed b ð ; ; ; Þ Ω− → Ω−π0 η η0 ρ0 ω b ð ; ; ; Þ Charm baryon decays:
þ 0 þ þ c → sd→ u∶ color favored Ξc → Ξ π ðρ Þ ð Þ 0 − þ þ Ωc → Ω π ðρ Þ ð5Þ 0 0 ¯ ð Þ0 color suppressed Ωc → Ξ K ð Þ þ X∶ c → ss→ u∶ color suppressed Λc → pϕ
We denote singly Cabibbo suppressed decays by one are marked by an asterix ( ). We also do not list cross (X) and doubly Cabibbo suppressed decays by two W–emission contributions where the flavor symmetries crosses (XX). There are a number of decays which are of the initial and final spectator quarks do not match, also admitted by a W-exchange contribution which are, q q → q0q0 Λ0 → Σ0J=ψ however, dynamically suppressed due to the Körner- i.e., where one has ½ i j f i jg such as in b 3 Ξ0 → Ξ0 ð Þþ ð Þ− Pati-Woo theorem [29,30] in the SUð Þ limit. These decays or in b c D .
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III. EFFECTIVE NONLEPTONIC WEAK We are dealing with the quark level transitions (i) b → cud¯ , LAGRANGIANS b → ccs¯ , c → sdu¯ (CKM favored), (ii) b → ccd¯ , b → ucs¯ , b → cus¯ , b → uud¯ , c → ssu¯ (CKM singly suppressed), and In this section we list the effective nonleptonic weak → ¯ → ¯ Lagrangians, which will be used for the calculation of (iii) b uus, b ucd (CKM doubly suppressed). (1) b → cuq¯ andb → ucq¯ transitionwith q ¼ d, s(CKM heavy baryon decays in present paper. Here and in the → ¯ → ¯ following V are Cabibbo-Kabayashi-Maskawa (CKM) favored: b cud, CKM suppressed: b cus, q1q2 b → ucs¯ , CKM doubly suppressed: b → ucd¯ ): matrix elements. In our manuscript we use the following set of CKM matrix elements: GF † L ¼ pffiffiffi ½V V ðC1Q1 þ C2Q2Þ eff 2 cb uq 0 97425 0 2252 0 00389 Vud ¼ . ;Vus ¼ . ;Vub ¼ . ; † ˜ ˜ þ V VcqðC1Q1 þ C2Q2Þ þ H:c: ð7Þ 0 230 0 974642 0 0406 ub Vcd ¼ . ;Vcs ¼ . ;Vcb ¼ . : ˜ where Qi and Qi is the set offlavor-changing effective ð6Þ four-quark b → cud¯ ðsÞ and b → ucd¯ ðsÞ operators
a1 μ a2 a2 a1 a1 μ a1 a2 a2 Q1 ¼ðc¯ O b Þðq¯ Oμu Þ;Q2 ¼ðc¯ O b Þðq¯ Oμu Þ;
˜ a1 μ a2 a2 a1 ˜ a1 μ a1 a2 a2 Q1 ¼ðu¯ O b Þðq¯ Oμc Þ; Q2 ¼ðu¯ O b Þðq¯ Oμc Þ; ð8Þ
μ μ 5 O ¼ γ ð1 − γ Þ, Ci is the set of Wilson coefficients [31–33]:
C1 ¼ −0.25;C2 ¼ 1.1: ð9Þ
The quark-level matrix elements contributing to the bottom-charm baryon and bottom-up baryon transitions are given by
GF ðbcÞ † μ GF ðbqÞ † μ Mðb → cuq¯ Þ¼pffiffiffi C V V ðcO¯ bÞðqO¯ μuÞ;Mðb → quc¯ Þ¼pffiffiffi C V V ðqO¯ bÞðcO¯ μuÞð10Þ 2 eff cb uq 2 eff cb uq
and
GF ˜ ðbuÞ † μ GF ˜ ðbcÞ † μ Mðb → ucq¯ Þ¼pffiffiffi C V V ðuO¯ bÞðqO¯ μcÞ;Mðb → cuq¯ Þ¼pffiffiffi C V V ðuO¯ bÞðqO¯ μcÞ; ð11Þ 2 eff ub cq 2 eff ub cq
where
ðbcÞ ˜ ðbuÞ ξ ðbqÞ ˜ ðbqÞ ξ Ceff ¼ Ceff ¼ C1 þ C2;Ceff ¼ Ceff ¼ C1 þ C2: ð12Þ
(2) b → ccq¯ transition with q ¼ d, s (CKM favored: b → ccs¯ , CKM suppressed: b → ccd¯ ):
X6 L † eff ¼ VcbVcq CiQi þ H:c:; ð13Þ i¼1
where the Qi are the set of effective four-quark flavor-changing b → q operators
a1 μ a2 a2 a1 a1 μ a2 a2 a1 Q1 ¼ðc¯ O b Þðq¯ Oμc Þ;Q4 ¼ðq¯ O b Þðc¯ Oμc Þ;
a1 μ a1 a2 a2 a1 μ a1 a2 ˜ a2 Q2 ¼ðc¯ O b Þðq¯ Oμc Þ;Q5 ¼ðs¯ O b Þðc¯ Oμc Þ;
a1 μ a1 a2 a2 a1 μ a2 a2 ˜ a1 Q3 ¼ðq¯ O b Þðc¯ Oμc Þ;Q6 ¼ðq¯ O b Þðc¯ Oμc Þ; ð14Þ
with O˜ μ ¼ γμð1 þ γ5Þ. The respective Wilson coefficients are [32]:
C1 ¼ −0.257;C2 ¼ 1.009;C3 ¼ −0.005;C4 ¼ −0.078;C5 ≃ 0;C6 ¼ 0.001: ð15Þ
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The quark-level matrix elements contributing to the bottom-charm baryon and bottom-down(strange) baryon transitions are given by
GF ðbcÞ † μ Mðb → ccq¯ Þ¼pffiffiffi C V V ðcO¯ bÞðqO¯ μcÞ; ð16Þ 2 eff cb cq
GF ðbqÞ † μ Mðb → qcc¯ Þ¼pffiffiffi C V V ðqO¯ bÞðcO¯ μcÞ; ð17Þ 2 eff cb cq
where
ðbcÞ ξ Ceff ¼ ðC1 þ C3 þ C5ÞþðC2 þ C4 þ C6Þ: ð18Þ
and
ðbqÞ ξ Ceff ¼ C1 þ C3 þ C5 þ ðC2 þ C4 þ C6Þ: ð19Þ
(3) c → squ¯ transition with q ¼ d, s (CKM favored: c → sdu¯ , CKM suppressed: c → ssu¯ ):
GF † L ¼ pffiffiffi V V ðC1Q1 þ C2Q2ÞþH:c: ð20Þ eff 2 cs uq
Qi is the set of flavor-changing effective four-quark c → s operators
a1 μ a1 a2 a2 a1 μ a2 a2 a1 Q1 ¼ðs¯ O c Þðu¯ Oμq Þ;Q2 ¼ðs¯ O c Þðu¯ Oμq Þ; ð21Þ
where
C1 ¼ 1.26;C2 ¼ −0.51: ð22Þ
The quark-level matrix elements contributing to the charm-to-nonstrange (e.g., Λc → pϕ) and charm-to-strange þ baryon decay (e.g., Λc → Λπ ) are given by
GF ðcuÞ † μ Mðc → uss¯ Þ¼pffiffiffi C V V ðuO¯ μcÞðsO¯ sÞ; ð23Þ 2 eff cs us
where
ðcuÞ ξ Ceff ¼ C2 þ C1 ð24Þ
and
¯ GF ðcsÞ † μ Mðc → sduÞ¼pffiffiffi C V V ðsO¯ μcÞðuO¯ dÞ; ð25Þ 2 eff cs ud
with
ðcsÞ ξ Ceff ¼ C1 þ C2: ð26Þ
(4) b → uuq¯ transition (CKM singly suppressed: b → uud¯ , doubly suppressed: b → uus¯ ):
GF † L ¼ pffiffiffi V V ðC1Q1 þ C2Q2ÞþH:c: ð27Þ eff 2 ub uq
Qi is the set of flavor-changing effective four-quark b → u operators
074011-5 GUTSCHE, IVANOV, KÖRNER, and LYUBOVITSKIJ PHYS. REV. D 98, 074011 (2018)
a1 μ a1 a2 a2 a1 μ a2 a2 a1 Q1 ¼ðu¯ O b Þðq¯ Oμu Þ;Q2 ¼ðu¯ O b Þðq¯ Oμu Þ; ð28Þ
here
C1 ¼ 1.26;C2 ¼ −0.51: ð29Þ
Ξ− → Σ−π0 η η0 ρ0 ω Ξ− → Ξ−π0 η η0 ρ0 ω The quark-level matrix elements contributing to the b ð ; ; ; Þ and b ð ; ; ; Þ are given by
GF ðbqÞ † μ Mðb → quu¯ Þ¼pffiffiffi C V V ðqO¯ μbÞðuO¯ uÞ; ð30Þ 2 eff ub uq
where
ðbqÞ ξ Ceff ¼ C2 þ C1: ð31Þ
The color factor ξ ¼ 1=Nc will be set to zero in all matrix elements, i.e., we keep only the leading term in the 1=Nc− expansion.
IV. MATRIX ELEMENTS, HELICITY AMPLITUDES, AND RATE EXPRESSIONS
The matrix element of the exclusive decay B1ðp1; λ1Þ → B2ðp2; λ2ÞþMðq; λMÞ is defined by
GF † †μ MðB1 → B2 þ MÞ¼pffiffiffi V V C f M hB2jq¯2Oμq1jB1iϵ ðλ Þ; ð32Þ 2 ij kl eff M M M where M ¼ V, P stands for the vector and pseudoscalar mesons. MM and fM are the respective masses MV, MP and leptonic μ μ μ 5 decay constants fV, fP. The Dirac string O is defined by O ¼ γ ð1 − γ Þ. μ þ þ þ þ The hadronic matrix element hB2jq¯2O q1jB1i is expressed in terms of six (1=2 → 1=2 ) or eight (1=2 → 3=2 ), V=A 2 dimensionless invariant form factors Fi ðq Þ, viz. 1þ 1þ (i) Transition 2 → 2 : V 2 qν V 2 qμ V 2 hB2jq¯2γμq1jB1i¼u¯ðp2;s2Þ γμF1 ðq Þ − iσμν F2 ðq Þþ F3 ðq Þ uðp1;s1Þ M1 M1 A 2 qν A 2 qμ A 2 hB2jq¯2γμγ5q1jB1i¼u¯ðp2;s2Þ γμF1 ðq Þ − iσμν F2 ðq Þþ F3 ðq Þ γ5uðp1;s1Þ M1 M1
1þ 3þ (ii) Transition 2 → 2 : p1α p1αp2μ p1αqμ ¯ γ ¯α V 2 γ V 2 V 2 V 2 γ hB2jq2 μq1jB1i¼u ðp2;s2Þ gαμF1 ðq Þþ μ F2 ðq Þþ 2 F3 ðq Þþ 2 F4 ðq Þ 5uðp1;s1Þ M1 M M 1 1 p1α p1αp2μ p1αqμ ¯ γ γ ¯α A 2 γ A 2 A 2 A 2 hB2jq2 μ 5q1jB1i¼u ðp2;s2Þ gαμF1 ðq Þþ μ F2 ðq Þþ 2 F3 ðq Þþ 2 F4 ðq Þ uðp1;s1Þ M1 M1 M1 where σμν ¼ði=2Þðγμγν − γνγμÞ and all γ matrices are defined as in Bjorken-Drell. We use the same notation for the form V=A factors Fi in all transitions even though their numerical values are different. V=A Next we express the vector and axial helicity amplitudes Hλ2λM in terms of the invariant form factors Fi , where λM ¼ t; 1; 0 and λ2 ¼ 1=2, 3=2 are the helicity components of the meson MðM ¼ P; VÞ and the baryon B2, respectively. We need to calculate the expressions
†μ V A Hλ λ ¼hB2ðp2; λ2Þjq¯2Oμq1jB1ðp1; λ1Þiϵ ðλ Þ¼H − H ð33Þ 2 M M λ2λM λ2λM where we split the helicity amplitudes into their vector and axial parts. For the color enhanced decays the operator q¯2Oμq1 represents a charged current transition while the operator q¯2Oμq1 describes a neutral current transition for the color suppressed decays. We shall work in the rest frame of the baryon B1 with the baryon B2 moving in the positive z-direction:
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⃗ p1 ¼ðM1; 0Þ, p2 ¼ðE2; 0; 0; jp2jÞ and q ¼ðq0; 0; 0; −jp2jÞ. The helicities of the three particles are related by λ1 ¼ λ2 − λM. We use the notation λP ¼ λt ¼ 0 for the scalar (J ¼ 0) contribution to set the helicity label apart from λV ¼ 0 used for the longitudinal component of the J ¼ 1 vector meson. One has (i) Transition 1þ → 1þ: HV ¼þHV and HA ¼ −HA . 2 2 −λ2;−λM λ2;λM −λ2;−λM λ2;λM qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2 q2 V 2 V V A 2 A − A H1 ¼ Qþ=q F1 M− þ F3 H1 ¼ Q−=q F1 Mþ F3 2t M1 2t M1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2 q2 V 2 V V A 2 A − A H10 ¼ Q−=q F1 Mþ þ F2 H10 ¼ Qþ=q F1 M− F2 2 M1 2 M1 pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi M M− V 2 − V − V þ A 2 − A A H11 ¼ Q− F1 F2 H11 ¼ Qþ F1 þ F2 2 M1 2 M1
(ii) Transition 1þ → 3þ: HV ¼ −HV and HA ¼þHA . 2 2 −λ2;−λM λ2;λM −λ2;−λM λ2;λM sffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 Q Q− M M− − q q V − þ V − V V þ V H1 ¼ · 2 F1 M1 F2 Mþ þ F3 þ F4 2t 3 q 2M1M2 2M1 M1 sffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 Q− M M− − q Q M− p2 V − V þ − V þ V j j H10 ¼ · 2 F1 F2 þ F3 2 3 q 2M2 2M1M2 M2 rffiffiffiffiffiffiffi Q Q pffiffiffiffiffiffiffi V − V − V þ V − V H11 ¼ F1 F2 H31 ¼ Q−F1 2 3 M1M2 2 sffiffiffiffiffiffiffiffiffiffiffiffi 2 − 2 2 A Q− Qþ A A A MþM− q A q H1 ¼ · 2 F1 M1 þ F2 M− þ F3 þ F4 2t 3 q 2M1M2 2M1 M1 sffiffiffiffiffiffiffiffiffiffiffiffiffi 2 − 2 p 2 A Qþ A MþM− q A Q−Mþ A j 2j H10 ¼ · 2 F1 þ F2 þ F3 2 3 q 2M2 2M1M2 M2 rffiffiffiffiffiffiffi pffiffiffiffiffiffiffi Q Q− A þ A − A A A H11 ¼ F1 F2 H31 ¼ QþF1 2 3 M1M2 2
2 − 2 We use the abbreviations M ¼ M1 M2, Q ¼ M q , sum over the helicities of the daughter baryon runs over 1=2 2 2 2 λ 1 2 λ 1 2 3 2 1 2þ → 1 2þ jp2j¼λ ðM1;M2;q Þ=ð2M1Þ. 2 ¼ = and 2 ¼ = , = for the ( = = ) For the decay width one finds and (1=2þ → 3=2þ) transitions, respectively. For the vector meson case M ¼ V one sums over λV ¼ 0; 1 and for the 2 p Γ → GF j 2j † 2 2 2 2 H pseudoscalar meson case M ¼ P one has the single value ðB1 B2 þ MÞ¼ 2 jVijVklj CefffMMM N λ 0 32π M1 P ¼ . Angular momentum conservation dictates the constraint jλ2 − λ j ≤ 1=2. ð34Þ M where we denote the sum of the squared moduli of the V. INTERPOLATING THREE-QUARK CURRENTS helicity amplitudes by OF BARYONS AND QUANTUM NUMBERS X 2 We use the following generic notation for the baryonic H ¼ jHλ λ j : ð35Þ N 2; M interpolating currents: λ λ 2; M P 1þ (1) Spin-parity J ¼ 2 : We have chosen to label the sum of the squared moduli of H the helicity amplitudes by the subscript N since N is the J ¼ εa1a2a3 Ja1a2a3 ; common divisor needed for the normalization of the spin a1a2a3 Γ a1 a2 Γ a3 density matrix elements to be discussed in Sec. VII. The J ¼ 1qm1 qm2 C 2qm3 ; ð36Þ
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TABLE I. Interpolating currents and quantum numbers of baryons.
Baryon JP Jabc Mass (MeV) 1þ μ 5 a b c p 2 ð1 − xNÞγ γ d u Cγμu 938.27 μν 5 a b c þxNσ γ d u Cσμνu , xN ¼ 0.8 1þ μ 5 a b c n 2 ð1 − xNÞγ γ u d Cγμd 939.57 μν 5 a b c þxNσ γ u d Cσμνd , xN ¼ 0.8 Λ 1þ a b c 2 s u Cγ5d 1115.68 þ 1þ μ 5 a b c Σ γ γ s u Cγμu 1189.37 2 pffiffiffi Σ0 1þ μ 5 a b c 2 2γ γ s u Cγμd 1192.64 Σ− 1þ μ 5 a b c 2 γ γ s d Cγμd 1197.45 Ξ0 1þ μ 5 a b c 2 γ γ u s Cγμs 1314.86 Ξ− 1þ μ 5 a b c 2 γ γ d s Cγμs 1321.71 þ 3þ 1ffiffi a b c a b c Σ p ðs u Cγμu þ 2u u Cγμs Þ 1382.80 2 qffiffi 3 Σ 0 3þ 2 2 a b c a b c a b c 1383.70 3ðs u Cγμd þ u d Cγμs þ d u Cγμs Þ − 3 1 Σ þ pffiffi a b γ c 2 a b γ c 2 3 ðs d C μd þ d d C μs Þ 1387.20 0 3 1 Ξ þ pffiffi a b γ c 2 a b γ c 2 3 ðu s C μs þ s s C μu Þ 1531.80 − 3 1 Ξ þ pffiffi a b γ c 2 a b γ c 2 3 ðd s C μs þ s s C μd Þ 1535.00 Ω− 3þ a b c 2 s s Cγμs 1672.45 Λ 1þ a b γ c c 2 c u C 5d 2286.46 Ξþ 1þ a b γ c c 2 c u C 5s 2467.87 Ξ0 1þ a b c c 2 c d Cγ5s 2470.87 Ξ0þ 1þ μ 5 a b c c 2 γ γ c u Cγμs 2577.40 Ξ00 1þ μ 5 a b c c 2 γ γ c d Cγμs 2578.80 Ω 1þ μ 5 a b c c 2 γ γ c s Cγμs 2695.20 Ω 3þ a b c c 2 c s Cγμs 2765.90 Λ 1þ a b c b 2 b u Cγ5d 5619.40 Ξ0 1þ a b γ c b 2 b u C 5s 5791.90 Ξ− 1þ a b γ c b 2 b d C 5s 5794.50 Ξ00 1þ γμγ5 a b γ c b 2 b u C μs 5935.02 Ξ0− 1þ μ 5 a b c b 2 γ γ b d Cγμs 5935.02 Ω 1þ μ 5 a b c b 2 γ γ b u Cγμs 6046.10
P 3þ B ¯ (2) Spin-parity J ¼ : Bðq1q2q3Þ∶ L ðxÞ¼g BðxÞ·J ðxÞþH:c:; 2 int ZB Z B Z
εa1a2a3 a1a2a3 a1a2a3 a1 a2 γ a3 ; Jμ ¼ Jμ ;Jμ ¼ qm1 qm2 C μqm3 JBðxÞ¼ dx1 dx2 dx3FBðx x1;x2;x3Þ
ð37Þ εa1a2a3 Γ a1 a2 Γ a3 × 1qm1 ðx1Þqm2 ðx2ÞC 2qm3 ðx3Þ; 38 where Γ1 and Γ2 are Dirac strings; ai and mi are color and ð Þ flavor indices, respectively. The interpolating currents and quantum numbers of the baryons needed in this paper are We treat each of the three constituent quarks as separate summarized in Table I. As indicated in the first entry of dynamic entities, i.e., we do not use the quark-diquark Table I we use a superposition of two possible currents approximation for baryons as done in many applications. 1 − (vector and tensor) for the proton as motivated by nucleon The propagators SqðkÞ¼ =ðmq =kÞ for quarks of all phenomenology. flavors q ¼ u, d, s, c, b are taken in the form of free The Lagrangian describing the coupling of a baryon with fermion propagators. The constituent quark masses mq its constituent quarks is constructed using a nonlocal were fixed in a previous analysis of a multitude of hadronic extension of the interpolating currents in Eqs. (36) and processes (see, e.g., Refs. [8,11]). By analogy we treat (37) (see details in Refs. [2–13], [25,34]). In particular, the mesons as bound states of constituent quarks, i.e., we 1þ interaction Lagrangian for the 2 baryon Bðq1q2q3Þ com- construct respective nonlocal Lagrangians for the mesons posed of constituent quarks q1, q2, and q3 is written as interacting with their constituent quarks (see details in
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constraint equation follows from the calculation of the mass operators for mesons (Fig. 1) and baryons (Fig. 2), As size parameters we use the values listed in Sec. VI for the nucleon, light hyperons, single charm and single heavy ¯ baryons, respectively. FIG. 1. Diagram representing the mass operator of a MðqiqjÞ meson. VI. MODEL PARAMETERS Our covariant constituent quark model (CCQM) [21–25] contains a number of model parameters which have been determined by a global fit to a multitude of decay processes. The best fit values for the constituent quark masses and the universal infrared cutoff parameter are given by FIG. 2. Diagram representing the mass operator of a Bðq1q2q3Þ baryon. mu ms mc mb λ ð39Þ 0.242 0.428 1.672 5.046 0.181 GeV Refs. [21–23]). The compositeness condition of Salam and Weinberg [14] gives a constraint equation between the The nonlocal quark particle vertices are described in terms coupling factors gB and the size parameters ΛB character- of a number of hadronic scale or size parameters ΛH (in izing the nonlocal distribution FBðx; x1;x2;x3Þ. The GeV) for which the best fit values are
πηΛqq¯ Λss¯ η0 Λqq¯ Λss¯ ρωϕ ð η ; η Þ ð η0 ; η0 Þ DD 0.871 ð0.881; 1.973Þð0.257; 2.797Þ 0.610 0.490 0.880 1.600 1.530 Ds Ds BBBs Bs ηc J=ψ 1 750 1 560 1 960 1 710 2 050 1 710 2 200 1 738 ...... ð40Þ p ΛΣΞΣ Ξ Ω 0.3600 0.4925 0.4925 0.4925 0.3201 0.3201 0.3201 Λc Ξc Ωc Ωc Λb Ξb Ωb 0.8675 0.8675 0.8675 0.8675 0.5709 0.5709 0.5709
Note, in the case of pseudoscalar mesons we introduce An important input to the global fit are the mesonic singlet-octet mixing with a mixing angle of θP ¼ −13.34° decay constants. Our global fit results in values for the deduced from Ref. [35] (see details in Refs. [21,23]) while mesonic decay constants which are quite close to the the vector mesons are assumed to be ideally mixed: experimental decay constants [36], results of lattice QCD calculations [37,38] and of nonrelativistic QCD [39] (see Table II). 1 ¯ η → − pffiffiffi sin δΦΛ ðuu¯ þ ddÞ − cos δΦΛ ss;¯ 2 η ηs 1 VII. NUMERICAL RESULTS η0 → ffiffiffi δΦ ¯ ¯ − δΦ ¯ þ p cos Λη0 ðuu þ ddÞ sin Λη0 ss; 2 s Using the numerical values for the model parameters 1 listed in Sec. VI we present our predictions for the δ ¼ θ − θ ; θ ¼ arctan pffiffiffi : ð41Þ P I I 2 branching ratios, helicity amplitudes, and asymmetry parameters of the nonleptonic decays of single heavy baryons in Tables III–XIV. All nonleptonic decays are We used [21,23] the experimental data of the electromag- described by the generic tree diagram shown in Fig. 3.In η η0 netic decays involving and to fit the size parameters our estimate for the branching ratio BrðΛb → ΛJ=ψÞ¼ defining the distribution of nonstrange (q ¼ u, d) and ð8.3 1.1Þ × 10−4 we used the fragmentation fraction of b qq¯ qq¯ ¯ ¯ Λη Λ Λss Λss Λ 7% strange s quarks in these states: , η0 , η , and η0 . quark into baryon of fΛb ¼ from [40].
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TABLE II. Results for the leptonic decay constants fH (in TABLE IV. Branching ratios of nonleptonic two-body decays MeV). of heavy baryons (in units of 10−4): b → c; u → d quark-level transition. fH Our results Data/Lattice QCD Mode Our results Theory fπþ 130.3 130.2 1.7 [36] 221 1 Ω− → Ω0π− fρþ 218.3 [36] b c 18.8 58.1 [34] 198 2 Ω− → Ω0ρ− fω 198.7 [36] b c 54.3 − 0 − fϕ 226.6 227 2 [36] Ω → Ω π b c 17.0 f 204.7 203.7 4.7 0.6 [36] Ω− → Ω 0ρ− D b c 55.8 þ3 − − 0 fD 244.4 245 20−2 [37] Ξ → Σ b D 0.1 − − 0 fD 256.4 257.8 4.1 0.1 [36] Ξ → Σ s b D 0.3 þ3 − − 0 −2 fD 272.6 245 20 [37] Ω → Ξ 1 4 10 s −2 b D . × fη 426.8 387 7 2 [38] Ω− → Ξ− 0 2 6 10−2 c b D . × 346 17 [39] Ω− → Ξ −D0 0.7 × 10−3 416 5 b fJ=ψ 415.1 [36] Ω− → Ξ − 0 1 9 10−3 b D . × fB 192.3 188 17 18 [36] þ39 fB 184.6 196 24−2 [37] fB 238.2 227.2 3.4 [36] TABLE V. Branching ratios of nonleptonic two-body decays of s −4 229 46 heavy baryons (in units of 10 ): b → c; c → d quark-level fBs 217.1 [37] transition.
Mode Our results Ξ0 → Ξþ − 4.5 In the future we plan to present predictions also for the b c D Ξ0 → Ξþ − 9.5 nonleptonic modes of heavy baryons which are also b c D Ω− → Ω0 − 2.4 contributed to by the W–exchange diagrams. In b cD Ω− → Ω0D − 3.0 Ref. [34] we have shown that for some of the heavy-to- b c Ω− → Ω 0D− 1.6 light transitions the total contribution of the W–exchange b c Ω− → Ω 0D − 5.8 diagrams amount up to approximately 60% of the tree b c Λ0 → nη 0.2 diagram contribution in amplitude, and up to approximately b c Λ0 → ψ → b nJ= 0.4 30% for b c transitions. Ξ− → Σ−η b c 0.1 Ξ− → Σ− ψ b J= 0.2 Ξ0 → Λ0η 1 6 10−2 b c . × Ξ0 → Σ0η 2 9 10−2 TABLE III. Branching ratios of nonleptonic two-body decays b c . × 10−4 → ; → Ξ0 → Λ0 ψ 3 1 10−2 of heavy baryons (in units of ): b c c s quark-level b J= . × transition. Ξ0 → Σ0 ψ 7 0 10−2 b J= . × Ω− → Ξ−η 1.3 × 10−2 Our b c Ω− → Ξ−J=ψ 1.8 × 10−2 Mode results Data Theory b Ω− → Ξ −η 6 9 10−2 b c . × Λ0 → Λþ − 147.8 110 10 230þ30 Ω− → Ξ − ψ b c Ds −40 [41]; 110 [42]; b J= 0.2 223 [43];77[44]; 129.1 [45] − 0 − 0 Λ0 → Λþ − 173þ20 A. The decays Ξ ðΞ Þ → Ξ ðΞ ÞJ=ψ and Λb → ΛJ=ψ b c Ds 251.6 −30 [41];91[42]; b b 326 [43]; 141.4 [44]; Ξ− → Ξ− ψ The CKM favored decay b J= was first seen by 198.3 [45] the CDF [57,58] and the D0 collaborations [59] in the chain Λ0 → Λ0η 4.3 1.5 0.9 [26] − − − − − b c of decays Ξ → Ξ J=ψ and J=ψ → μþμ , Ξ → Λπ , and Λ0 → Λ0 ψ 8 3 1 1 b b J= 8.3 . . 1.6 [42]; 2.7 [34]; 6.0 [43]; 2.5 [44]; 2.1 [46]; 3.5 1.8 [47]; TABLE VI. Branching ratios of nonleptonic two-body decays 10−6 → ; → 3.3 2.0 [26] 7.8 [28] of heavy baryons (in units of ): b c u s quark-level 8.4 [48]; 8.2 [49] transition. Ξ− → Ξ−η 2 3 1 4 b c 1.7 . . [26] Ξ0 → Ξ0η Mode Our results b c 1.6 Ξ− → Ξ− ψ 4 9 3 0 Ξ− → Ξ− 0 1.4 b J= 4.6 . . [26] b D Ξ0 → Ξ0 ψ Ξ− → Ξ−D 0 2.1 b J= 4.4 b Ω− → Ω−η Ω− → Ω− 0 b c 1.9 b D 0.7 Ω− → Ω− ψ Ω− → Ω− 0 b J= 8.1 b D 2.0
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TABLE VII. Branching ratios of nonleptonic two-body decays TABLE X. Branching ratios of nonleptonic two-body decays of of heavy baryons (in units of 10−6): b → u; c → s quark-level heavy baryons (in units of 10−9): b → u; u → s quark-level transition. transition.
Mode Our results Data Theory Mode Our results Λ → − 480 18 3 Ξ− → Ξ−π0 b pDs 13.2 < 13.6 [48]; [26] b 0.8 Λ → − 8 8 2 2 Ξ− → Ξ−η b pDs 22.1 6.7 [48]; . . [26] b 0.5 Ω− → Ξ 0 − 4.9 Ξ− → Ξ−η0 b Ds b 1.3 Ω− → Ξ 0 − 11.3 Ξ− → Ξ−ρ0 b Ds b 2.8 Ω− → Ω− ¯ 0 Ξ− → Ξ−ω b D 0.1 b 2.1 Ω− → Ω− ¯ 0 Ω− → Ω−π0 b D 0.3 b 0.7 Ω− → Ω−η b 0.5 Ω− → Ω−η0 b 1.1 Ω− → Ω−ρ0 b 9.1 Ω− → Ω−ω TABLE VIII. Branching ratios of nonleptonic two-body decays b 7.6 of heavy baryons (in units of 10−8): b → u; u → d quark-level transition. TABLE XI. Branching ratios of nonleptonic two-body decays −2 Mode Our results of heavy baryons (in units of 10 ): c → s; d → u quark-level transition. Ξ− → Σ−π0 b 6.2 Ξ− → Σ−η b 4.9 Mode Our results Theory Ξ− → Σ−η0 15.2 b Ξþ → Ξ 0πþ 8.1 × 10−6 Ξ− → Σ−ρ0 19.0 c b Ξþ → Ξ 0ρþ 3.9 × 10−6 Ξ− → Σ−ω 15.8 c b Ω0 → Ω−πþ 0.2 1.0 [42] Ω− → Ξ−π0 c b 0.5 0 − þ − − Ωc → Ω ρ 1.9 3.6 [42] Ω → Ξ η 0.4 0 0 0 b Ω → Ξ K¯ 0.3 Ω− → Ξ−η0 1.4 c b Ω0 → Ξ 0 ¯ 0 2.0 Ω− → Ξ−ρ0 c K b 1.7 Ω− → Ξ−ω b 1.4 Ω− → Ξ −π0 b 2.3 − Ω− → Ξ −η Λ → pπ . These measurements led to a first determination b 2.1 − Ω− → Ξ −η0 of the mass and the lifetime of the Ξ . PDG 2018 quotes a b 9.5 b Ω− → Ξ −ρ0 mass value of 5794.5 1.4 MeV as a weighted average b 257.3 Ω− → Ξ −ω from the two experiments [36]. In a later experiment the b 213.9 LHCb Collaboration collected much higher statistics on the Ξ− same decay chain [60]. The lifetime of the b was τ Ξ− 1 571 0 040 determined to be ð b Þ¼ . . ps [36]. This is Ξ− the most accurate determination of the lifetime of the b to date. Unfortunately a reliable estimate of the branching Ξ− → Ξ− ψ TABLE IX. Branching ratios of nonleptonic two-body decays ratio of the decay b J= is still missing. The decay of heavy baryons (in units of 10−8): b → u; c → d quark-level Ξ0 → Ξ0 ψ Ξ− → Ξ− ψ b J= is closely related to the decay b J= transition. by isospin but has not been seen yet. Mode Our results Ξ0 → Σþ − b D 13.6 Ξ0 → ΣþD − 29.6 TABLE XII. Moduli squared of normalized helicity ampli- b Λ → Ξ− → Σ0D− 7.2 tudes and asymmetry parameters for the cascade decay b b Λ → ¯ 0 → π Ξ− → Σ0 − cð pK ÞþDs ð D Þ transition. b D 15.6 Ω− → Ξ0 − b D 1.6 Our Our Our Ω− → Ξ0 − b D 2.6 Quantity results Quantity results Quantity results Ω− → Ξ 0 − b D 6.7 ˆ 2 −2 ˆ − 0 − H 1 4.46×10 H 0.358 αb −0.364 Ω → Ξ j þ2þ1j U b D 18.5 − − 0 −2 ˆ 2 −3 ˆ Ω → Ξ ¯ 7 3 10 jH 10j 4.98×10 H 0.642 r0 0.642 b D . × þ2 L − − ¯ 0 −2 ˆ 2 ˆ Ω → Ξ D 13.2 × 10 H 1 0.637 H −0.268 r1 −0.632 b j −20j P Ω− → Ξ −D¯ 0 6.8 ˆ 2 ˆ −0 633 −0 901 b jH−1−1j 0.313 HL . (Px, Pz)( . , Ω− → Ξ − ¯ 0 2 P b D 18.9 0.416)
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TABLE XIII. Moduli squared of normalized helicity amplitudes and asymmetry parameters for the Λb → Λð→ pπ−ÞJ=ψð→ lþl−Þ transition.
Quantity Our results LHCb [50] ATLAS [51] CMS [52] ˆ 2 −3 þ0.13 2 H 1 2.34 × 10 −0.10 0.04 0.03 ð0.08 0.06Þ 0.05 0.04 0.02 j þ2þ1j −0.08 ˆ 2 −4 þ0.12 2 H 1 3.24 × 10 0.01 0.04 0.03 ð0.17 0.09Þ −0.02 0.03 0.02 j þ20j −0.17 ˆ 2 þ0.06 2 H 1 0.532 0.57 0.06 0.03 0.59 0.03 0.51 0.03 0.02 j −20j ð −0.07 Þ ˆ 2 þ0.04 2 H 1 0.465 0.51 0.05 0.02 ð0.79 0.02Þ 0.46 0.02 0.02 j −2−1j −0.05 αb −0.069 0.05 0.17 0.07 0.30 0.16 0.06 −0.12 0.13 0.06 r0 0.533 0.58 0.02 0.01 r1 −0.532 −0.56 0.10 0.05 PzðΛÞ −0.995 −1.17 0.26 0.12 PxðΛÞ 0.095
Λ → ϕ 10−4 TABLE XIV. Moduli squared of normalized helicity ampli- TABLE XV. Branching ratios Bð c p Þ in units of . tudes and asymmetry parameters for the cascade decay Ω → b Our result Theoretical predictions Data [36] Ω−ð→ Ξ−p; ΛK−ÞþJ=ψð→ lþl−Þ transitions. 14.0 19.5 [53]; 21.5 [54]; 10.8 1.4 Quantity Our results 9.89 [55]; 4.0 [56]; − 27.3 17.9 [34] BðΩb → Ω J=ψÞ 0.08% Hˆ 0.236 U1=2 ˆ H 0.312 Λ → ϕ lþl− 10−7 U3=2 TABLE XVI. Branching ratio Bð c p ð ÞÞ in units . Hˆ 0.452 L Mode Our results Data [36] þ − 55 Λc → p þ e e 4.11 < Λ → μþμ− 440 It is interesting to note that the neutral current transition c p þ 4.11 < Ξ− → Ξ− Λ → Λ b can be related to the neutral current b and charged current Λb → p transitions by invoking SUð3Þ 2 2 ¯ HU ¼jH11j þjH−1−1j transverse unpolarized; symmetry. This can be seen by using the 3 ⊗ 3 → 8 2 2 2 2 Clebsch-Gordan table listed in Ref. [61]. Based on the H H1 H 1 longitudinal unpolarized; L ¼j 20j þj −20j observation that the antisymmetric ½sd and ½ud diquarks 2 2 H ¼jH11j − jH−1−1j transverse parity-odd polarized; are the (Y ¼ −1=3, I ¼ 1=2) and (Y ¼ 2=3, I ¼ 0) mem- P 2 2 3¯ 2 2 bers of the multiplet one needs the C.G. coefficients [61]. H H1 − H 1 longitudinal polarized; LP ¼j 20j j −20j 2 2 H ¼jH1 j þjH−1 j scalar: ð43Þ 1 1 1 2 1 1 S 2t 2t Ξ− → Ξ−∶ 3¯; − ; ; − ; 3; − ; 0; 0 8; −1; ; − 1 b 3 2 2 3 j 2 2 ¼ Following [8,16,50] we also introduce linear combinations 2 2 pffiffiffiffiffiffiffiffi ˆ 2 ¯ of normalized squared helicity amplitudes jHλ2λ j by Λ → Λ∶ 3; ; 0; 0; 3; − ; 0; 0j8; 0; 0; 0 ¼ 2=3; V b 3 3 writing 2 1 1 1 1 1 Λ → ∶ 3¯ 0 0; 3 8 1 1 b p ; 3 ; ; ; 3 ; 2 ; 2 j ; ; 2 ; 2 ¼ : ð42Þ
The labeling in (42) proceeds according to the sequence jR;Y;I;Izi where R denotes the relevant SUð3Þ represen- tation. One does not expect that SUð3Þ breaking effects are large at the scale of the bottom baryons. The angular decay distribution for the cascade decay Ξ− → Ξ− → Λπ− ψ → lþl− b ð ÞþJ= ð Þ can be adapted from [8] where now one has to use the asymmetry parameter − − αΞ ¼ −0.458 0.012 for the decay Ξ → Λπ [36].Asin Ref. [8] we introduce the following combinations of FIG. 3. Factorizing diagram describing the Bðq1q2q3Þ → 0 0 0 helicity amplitudes B ðq1q2q3ÞþMðq4q¯4 Þ transition.
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ˆ 2 ˆ 2 ˆ 2 ˆ 2 ˆ ˆ α H 1 − H 1 H 1 − H 1 H − H ; We define partial helicity widths in terms of the follow- b ¼j þ20j j −20j þj −2−1j j þ2þ1j ¼ LP P ˆ 2 ˆ 2 ˆ ing bilinear helicity expressions r0 H 1 H 1 H ; ¼j þ20j þj −20j ¼ L 2 ˆ 2 ˆ 2 ˆ G jp2j † 2 2 2 2 r1 ¼jH 10j − jH−10j ¼ HL ; ð44Þ Γ F þ2 2 P I ¼ 2 jVijVklj CefffVMVHI; 32π M1 ˆ 2 2 H where jHλ2λV j ¼jHλ2λV j = N and where the normaliza- I ¼ U; L; P; LP: ð48Þ tion factor HN is given by The total B1 → B2V decay width is given by 2 2 2 2 H ≡ H 1 H 1 H 1 H 1 : 45 N j þ20j þj −20j þj −2−1j þj þ2þ1j ð Þ ΓðB1 → B2VÞ¼ΓU þ ΓL: ð49Þ The full joint angular decay distribution of the cascade þ − → decay B1 → B2ð→ B3 þ PÞþVð→ l l Þ including the We shall also be interested in the cascade decays B1 → → Λ → Λ → ¯ 0 polarization of the parent baryon and including lepton mass B2ð B3PÞþVð PPÞ as, e.g., in b cð pK Þþ → π effects has been given in [8]. The angular decay distribution Ds ð D Þ. The two-fold polar angle distribution can be involves the longitudinal and transverse polarization com- calculated to be ponents for the daughter baryon B2 defined by 4 θ θ 1 α θ Wð 3; PÞ¼ð þ PzðB2Þ B2 cos 3 ˆ 2 ˆ 2 ˆ 2 ˆ 2 P ðB2Þ¼jH 10j − jH−10j þjH 1 1j − jH−1−1j 1 z þ2 2 þ2þ 2 ˆ ˆ 2 þ HU þ HL ð3cos θP − 1Þ ¼ Hˆ þ Hˆ ; 2 LP P ˆ ˆ ˆ ˆ 1 2 P B2 2Re H1 H 1 H1 H 1 : 46 ˆ ˆ xð Þ¼ ð 21 − 0 þ 20 − −1Þ ð Þ þ H þ H α cos θ3ð3cos θ − 1Þ: 2 2 LP 2 P B2 P Note that one can express the longitudinal polarization ð50Þ of the daughter baryon in terms of the linear combinations r1 and αb in (44), i.e., one has PzðB2Þ¼2r1 − αb ¼ Moduli squared of normalized helicity amplitudes and asym- ˆ ˆ Λ → Λ → ¯ 0 HL þ HP. The magnitude of the transverse polarization metry parameters for the cascade decay b cð pK Þþ P → π P ðB2Þ determines the size of the azimuthal correlations Ds ð D Þ are shown in Table XII. x Λ → Λ ψ between the baryon-side and lepton-side decay plains. Next we turn to the decay b J= . The angular Λ → Λ → π− Instead of listing the full angular decay distribution decay distribution of the cascade chain b ð p Þþ − including azimuthal terms we shall only list the two-fold J=ψð→ lþl Þ including polarization effects of the parent Λ polar angle decay distribution given by baryon b has been discussed in detail in [15,16,25]. A new insight has been gained concerning the accessibility of the 1 polarization of the produced parent baryon Λb [51].Ina Wðθ3; θlÞ¼ ð1 þ P ðB2Þα cos θ3Þv · ð1 þ 2εÞ 4 z B2 symmetric collider such as the pp-collider where the Λ ’ 1 2 sample of b s are produced over a symmetric interval þ ðð1 − 3r0Þð3cos θl − 1Þ Λ 4 of pseudorapidity the polarization of the b must average to 2 2 zero. In order to access the polarization of the Λb one must − ðα þ r1Þα ð3cos θl − 1Þ cos θ3Þv · v b B2 e.g., divide the pseudorapidity interval into two forward ð47Þ and backward hemispheres which has not been done in the LHCb [50] and ATLAS [51] experiments. We shall there- α → B2 denotes the asymmetry factor in the decay B2 fore no longer discuss the polarization dependent terms in B3 þ P, a quantity known from experiment in many cases. the angular decay distribution. Such a measurement has to One can reexpress the two angular parameters (1 − 3r0) and wait for a future analysis. ˆ (αb þ r1)in(47) through the components HI by writing We list some numerical results of our previous analysis ˆ ˆ ˆ ˆ Λ → Λ ψ ð1 − 3r0Þ¼ðH − 2H Þ and ðα þ r1Þ¼ð2H − H Þ. on the decay b J= in Table XIII. Table XIII shows U L b LP P Λ Note that we have kept the overall lepton velocity factor that the longitudinal polarization of the daughter baryon qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Λ 2 2 is almost 100% and negative. In fact, one finds Pzð Þ¼ v ¼ 1 − 4ml=q in (47) which results from the momen- ˆ ˆ −0 995 HLP þ HP ¼ . . This leaves very little room for the → lþl− 2 2 tum factor in the decay formula for V . The transverse polarization PxðΛÞ since Pz ðΛÞþPxðΛÞ¼1. 3 2θ − 1 trigonometric function ( cos l ) appearing in We set the T-odd polarization component PxðΛÞ to zero 2 Eq. (47) integrates to zero upon cos θl integration. In since the helicity amplitudes are nearly real in our analysis. 2 the literature one frequently finds ð3cos θl − 1Þ¼ From the numbers in Table XIII and using the definition 2P2ðcos θlÞ written in terms of the Legendre polynomial (46) one finds PxðΛÞ¼0.095. A small transverse polari- P2ðcos θlÞ of second degree. zation PxðΛÞ in turn implies small azimuthal correlations
074011-13 GUTSCHE, IVANOV, KÖRNER, and LYUBOVITSKIJ PHYS. REV. D 98, 074011 (2018)
− þ − 0 + − 0 + − between the two plains defined by (p, π ) and (l , l )as B. The decays Λb → Λc Ds and Λb → Λc D was found in [8] and which has been confirmed by the Λ0 → Λþ − Both the Cabibbo-favored decay b c Ds and the azimuthal measurement in [51]. Cabibbo suppressed decay Λ0 → ΛþD− have been seen by α b c For the asymmetry parameter b we obtained the the LHCb collaboration [64]. The PDG 2018 quotes small negative value α ≃−0.07 [25]. The value of α 0 − b b branching fractions of BðΛ → ΛþD Þ¼ð1.1 0.1Þ × determines the analyzing power of the cascade decay b c s 10−2 B Λ0 → Λþ − 4 6 0 6 10−4 − þ − and ð b c D Þ¼ð . . Þ × for these Λb → Λð→ pπ ÞþJ=ψð→ l l Þ to measure the polari- zation of the parent baryon Λ . A small value of α decays [36]. b b We employ a notation where we label the scalar time implies a poor analyzing power of such an experiment. “ ” λ 0 α component of the current by the suffix t where t ¼ in Our calculated value for the asymmetry parameter b ¼ 0 1 −0.07 agrees within one standard deviation with the order to distinguish between the (J ¼ ) and (J ¼ ) transformation of the two helicity zero cases. We slightly revised value of the asymmetry parameter αb ¼ 0.05 0.17 0.07 reported in [50]. A new measurement shall again define normalized helicity amplitudes by ˆ 2 2 H H on α was reported by the ATLAS collaboration [51]. writing jHλ2tj ¼jHλ2tj = N where N is now given by b 2 2 H H1 H 1 . In this notation the longitudinal Our result is more than two standard deviations away N ¼j 2tj þj −2tj from their reported value of αb ¼ 0.30 0.16 0.06.We polarization of the daughter baryon B2 is given by mention that most of the theoretical model calculations predict small negative values for α ranging from (−0.09) −0 989 Λ → Λ b ˆ 2 ˆ 2 . ; b cD mode P B2 H1 − H 1 to (−0.21) (see Table VI in [8]) except the calculation of zð Þ¼j 2tj j −2tj¼ −0.986; Λb → ΛcDs mode [62,63] who quote a value of αb ¼ 0.777. We believe the calculation of [62,63] to be erroneous. The relation One then has the polar decay distribution between the helicity amplitudes and the HQET form factors F1 and F2 given in [62,63] is not correct. They W θ3 ∝ 1 P B2 λ cos θ3 : 52 also obtain a polarization of the daughter baryon Λ of ð Þ ð þ zð Þ B2 Þ ð Þ PzðΛÞ¼−9% which is much smaller than what is obtained in the other model calculations. The transverse polarization component PxðB2Þ does not come into play in these decays. In Fig. 4 we show a plot of αb over the whole range of accessible q2-values. At the kinematical limits q2 ¼ 0 and 2 2 2 + + − q ¼ðM1 − M2Þ ¼ 20.28 GeV αb takes the values αb ¼ C. The 1=2 → 3=2 +1 factorizing − − −1 and αb ¼ 0 in agreement with the corresponding decay Ωb → Ω J=ψ α limiting values of Eq. (44). The asymmetry parameter b Ω− 2 2 2 The b has a clear signature through the decay 9 59 − − goes through zero between q ¼ mJ=ψ ¼ . GeV and Ω → Ω J=ψ. This decay mode was first observed in 2 2 13 59 2 b q ¼ mψð2SÞ ¼ . GeV as demonstrated in the numeri- [58,65]. The PDG 2018 quotes the average values for 6046 1 1 7 τ Ω− cal results of [25]. We show this plot in order to demon- the mass . . MeV and the life time ð b Þ¼ strate that it is very unlikely that the asymmetry value 1 65þ0.18 . −0.16 ps [60]. We shall take the LHCb result for our reaches a large positive value as in [62,63] between the evaluation of branching rates. above two model independent limiting values. We begin with the decay 1=2þ → 3=2þ1− where we concentrate on the decay chains of the daughter baryon 3=2þ → 1=2þ þ P and the vector meson V → lþl−. 0.2 In Table XIV we list our results on the branching ratio of Hˆ Hˆ 0.0 this decay together with our values for U1 2, U3 2 , and ˆ = = HL. Latter values determine the polar angle distribution of − − 0.2 the subsequent decay Ω → Ξπ, ΛK . We have defined 2 2 2 2 Hˆ ˆ 1 ˆ 1 Hˆ ˆ 3 ˆ 3 2 U1=2 ¼jH 1j þjH− −1j , U3=2 ¼jH 1j þjH− −1j , and q 0.4 2 2 2 2 b ˆ 2 2 2
Α ˆ ˆ ˆ H ¼jH10j þjH−1j , where, as before, jHλ λ j ¼ L 2 2 P 2 V 0.6 2 2 Hλ λ =H and H Hλ λ . A much smaller j 2 V j V V ¼ λ2;λV j 2 V j
0.8 value of the branching fraction has been calculated in [28] − −5 where the authors predict BðΩb → Ω J=ψÞ¼4.5 × 10 . 1.0 The smallness of the branching ratio in the calculation [28] 0 5 10 15 20 2 2 can be traced to the erroneous assumption that the transition q GeV 2 0 2 2 form factors in the process fall from q ¼ to q ¼ mJ=ψ . 2 2 FIG. 4. q dependence of the asymmetry parameter αbðq Þ in We define partial helicity widths in terms of the follow- the Λb → Λ transition. ing bilinear helicity expressions
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2 G jp2j As in Eq. (47) we can represent the angular decay Γ F V V 2C2 f2 M2 H ; I ¼ 32π 2 j ij klj eff V V I distribution in an equivalent form in terms of the second M1 2 order Legendre polynomials P2ðxÞ¼ð3x − 1Þ=2 with I ¼ U1=2;U3=2;L: ð53Þ θ θ x ¼ cos B3 , cosθl . Using the same normalization as in (57) one has The B1 → B2V decay width is given by 1 4 θ θ ∝ 1 2ε 3 2θ − 1 Γ 1 2þ → 3 2þ Γ Γ Γ Wð B3 ; lÞ vð þ Þþ ð cos B3 Þ ðB1ð = Þ B2ð = ÞVÞ¼ U1=2 þ U3=2 þ L: ð54Þ 2 Hˆ − Hˆ Hˆ 1 2ε × ð U1 2 U3 2 þ LÞvð þ Þ The two-fold joint angular decay distribution for the = = − − 1 1=2þ → 3=2þ → 1=2þ0 V → lþl 2 3 cascade decay ð Þþ ð Þ þ ð3cos θl − 1Þv is simplified by the observation that the relevant decays 4 þ þ − 1 of the daughter baryon with B2ð3=2 Þ → B3ð1=2 Þ0 and 2 2 þ ð3cos θ − 1Þð3cos θl − 1Þ Ω− → Ξπ, ΛK− are almost purely parity conserving [36], 8 B3 3 2þ → 1 2þ0− ˆ ˆ 3 i.e., for the decay = = one has Hλ30 ¼ H−λ30. H − H H × ð U1 2 U3 2 þ LÞv ð58Þ This also follows from a simple quark model calculation = = [54,66]. The fact that the vector current induced transition such that the angle dependent parts integrates to zero [16]. 3=2þ → 1=2þ is conserved in the simple quark model can Upon integration one has be traced to the approximation that there is no spin Z interaction between the two light spectator quarks [67]. d cos θld cos θBWðθB; θlÞ θ This means there will be no parity-odd linear cos B3 terms Hˆ Hˆ Hˆ 1 2ε 1 2ε in the angular decay distribution. ¼ð U1=2 þ U3=2 þ LÞvð þ Þ¼vð þ Þ: ð59Þ The two-fold joint polar angle decay distribution can be 1 2þ → 3 2þ0− derived from the master formula Next we discuss the decays = = . Similar to 3 2þ → 1 2þ X the current induced transition = = one finds from θ θ ∝ V 2 1 θ 2 a naive quark model calculation [54] that the vector current Wð l; BÞ jhλ λ − j ½dλ λ −λ − ð lÞ j lþ l V ; lþ l 3 2þ V 1 2þ helicities transition h = jJμ j = i is conserved. The basic mecha- 2 3=2 θ 2 nism is the same as mentioned in the discussion of the × Hλ2λ j ½dλ λ ð BÞ ; ð55Þ þ V þ V 2 3 h1=2 jJμ j3=2 i transition. The crucial assumption is that there is no spin interaction between the spectator quarks. The where the summation extends over all possible helicities 1 conservation of the vector current implies that the helicity λ λ − λ λ λ 0 1 lþ ; l ; 2; 3 ¼ 2, and V ¼ ; .In(55) we have left V V 0 2 amplitudes H λ2;t vanish, i.e., one has H λ2;t ¼ .Wealso out the overall factor jHλ30j . The required table of the μ þ V þ find vector current conservation q h3=2 jJμ j1=2 i¼0 in 3=2 θ Wigner symbol dmm0 ð Þ can be found, e.g., in [68]. our more sophisticated CCQM quark model, i.e., one has The vector current leptonic helicity amplitudes for the V again H λ ¼ 0. The vector current conservation in the → lþl− 2;t decay V are given by (see [25]) CCQM can be traced to the structure of the interpolating Ω Ω− V V currents of the c and of Table I. flip : h 1 1 ¼ h 1 1 ¼ 2m ; − − þ þ l The rate can be calculated from 2 2 2 2 qffiffiffiffiffiffiffi V V 2 2 2 p nonflip : h−1 1 ¼ h 1−1 ¼ q : ð56Þ − GF j 2j † 2 2 2 2 2þ2 þ2 2 Γ 1 2þ → 3 2þ0 H ð = = Þ¼ 2 jVijVklj CefffPMP S; 32π M1 For the angular decay distribution one obtains ð60Þ
3 2 2 2 2 where H ¼jH1 j þjH−1 j ¼ 2jH1 j . The correspond- Wðθ ;θlÞ ∝ vðð1 þ cos θlÞv þ 8εÞ S 2t 2t 2t B3 8 Ω− → Ω−η ing branching ratio for b c is predicted to be 1 −4 2 2 B 5.0 × 10 (see Table III). × ðHˆ ð1 þ 3cos θ ÞþHˆ 3sin θ Þ ¼ 4 U1=2 B3 U3=2 B3 The baryon-side decay distribution in this decay is 3 1 ˆ 2 2 identical to that of the longitudinal contribution in þ H vðð1 þ cos θlÞv þ 8Þ ð1 þ 3cos θ Þ: L 8 4 B3 Eq. (57), i.e., one has ð57Þ 1 1 1 Wðθ Þ¼ ð1 þ 3 cos θ Þ¼ 1 þ ð3 cos θ − 1Þ B3 4 B3 2 2 B3 We have again included an overall factor of v ¼ð1 − 4εÞ1=2 from the momentum factor in the decay formula for the 1 ¼ ð1 þ P2ðcos θ ÞÞ: ð61Þ decay V → lþl−. 2 B3
074011-15 GUTSCHE, IVANOV, KÖRNER, and LYUBOVITSKIJ PHYS. REV. D 98, 074011 (2018) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Λ → p ϕ 2 2 2 2 D. The decay c + 4πα fϕ 4m 2m Γ ϕ → lþl− 1 − l 1 l Λ → ϕ ð Þ¼ 2 þ 2 ð64Þ The Cabibbo-suppressed charm baryon decay c p 27 mϕ mϕ mϕ was first observed by the ACCMOR Collaboration in the NA32 experiment at CERN [69] with a branching ratio of to evaluate the muonic mode. As already remarked above, Λ → ϕ Λ → −πþ 0 04 0 03 Bð c p Þ=Bð c pK Þ¼ . . . Three years muon mass effects are negligible in the rate as Table XVI later the E687 Collaboration at Fermilab reported an upper shows. Note that we give more stringent upper limits on the Λ → ϕ limit for its branching relative to the mode Bð c p Þ= branching ratios of these decays compared to the data [36]. BðΛ → pK−KþÞ < 0.58 at 90% C.L. [70]. In 1996 the ˆ ˆ c The numerical value for the asymmetry (HU − 2HL)is CLEO Collaboration [71] confirmed the previously obser- Λ → ϕ ˆ ˆ ved c p decay mode with significant statistics and HU − 2HL ¼ −0.735: ð65Þ measured the branching ratios BðΛc → pϕÞ=BðΛc → − þ pK π Þ¼0.024 0.006 0.003 and BðΛc → pϕÞ= − þ VIII. SUMMARY BðΛc → pK K Þ¼0.62 0.20 0.12. Finally, in 2002 the Belle Collaboration at KEK [72] measured the ratio of We have calculated the decay properties of a number of Λ → ϕ branching rates with much higher accuracy Bð c p Þ= nonleptonic heavy baryon decays. Of the many possible Λ → −πþ 0 015 0 002 0 002 Bð c pK Þ¼ . . . . nonleptonic heavy baryon decays we have singled out those The good news is that the long-standing problem of the decays which proceed solely through the tree diagram Λ → −πþ absolute branching ratio of the decay c pK has contribution without a possible contamination from been solved by the BELLE [73] and the BES III collab- W-exchange contributions. Note that we have not included orations [74]. Thus the above branching rate ratios can be penguin-type operators in our set of operators as, e.g., converted into absolute branching ratios. the penguin operator governing the quark-level transition In the present case the information on the polarization of b → sss¯ that induces the decay Λb → Λ þ ϕ recently seen the daughter baryon, the proton, is of no particular interest by the LHCb Collaboration [75]. We plan to study decays since the polarization of the proton cannot be probed. The induced by penguin-type operators in the future. For the angular decay distribution on the lepton side can be read off decays dominated by tree diagram contribution we have from Table I in Ref. [8] or can be obtained by integrating calculated decay rates, branching ratios and asymmetry θ ε 2 2 Eq. (47) over cos 3. One has ( ¼ ml=mϕ) parameters associated with the polar angle distributions of 2 their subsequent decay chains. Some of these decays have 1 v θ ∝ 1 2ε 1 ˆ −2 ˆ 3 2θ −1 been seen which has lead to measurements of their lifetimes Wð lÞ vð þ Þ þ41 2εðHU HLÞð cos l Þ þ and their masses. We have also provided results for decays ð62Þ with prominent branching ratios and signatures which have not been observed up to now. Measuring lifetimes and where θl is the polar angle of either of the leptons with masses is only the first step towards a comprehensive ϕ Λ respect to the momentum direction of the in the c rest analysis of heavy baryon decays which would aim for the frame. Note that the second cos θl-dependent term vanishes determination of branching ratios and asymmetry param- after integration. The muon mass corrections to the rate eters in angular decay distributions as has e.g., been done in term in (62) are quite small as is evident from the expansion the decay Λ → ΛJ=ψ. We estimate our errors on branch- 2 3 b vð1 þ 2εÞ¼1–4ε þ Oðε Þ ≈ 0.998. The muon mass cor- ing fractions to be ∼20%. Our errors on the asymmetry rections to the cos θl dependent term, however, are larger parameters are much smaller since the errors cancel out 2 and can be seen to amount to v =ð1 þ 2εÞ¼1–6ε þ in the helicity rate ratios that describe the asymmetry Oðε3Þ ≃ 6%. parameters. We present our results in Tables XV and XVI and It would be interesting to extend our analysis to all compare them with data presented by the Particle Data nonleptonic two-body decays of the heavy baryons. Such a Group [36] and the predictions of other theoretical calculation would involve also contributions from the approaches [34,53–56]. We calculate the width of the W–exchange graphs called color commensurate (C), þ − cascade decay Λc → p þ ϕð→ l l Þ by using the zero exchange (E), and bow-tie (B) graphs in [1]. For the decays width approximation of the bottom baryon states the authors of [1] derived a hierarchy of strength T ≫ C, E ≫ B from a qualitative þ − þ − BðΛc → p þ ϕð→ l l ÞÞ ¼ BðΛc → p þ ϕÞBðϕ → l l Þ: SCET analysis. It would be interesting to find out whether the dominance of the tree graph contribution in the b → c ð63Þ and b → u sector predicted in [1] is borne out by the We take the value of the leptonic decay constant fϕ ¼ phenomenology of these decays. In particular, it would be 226.6 MeV from the eþe− mode calculated in our approach interesting to determine the size of the W–exchange 0 in agreement with data. Using the result Eq. (62) we can contributions to the decays Λb → Ληðη Þ. Within the write down the general ml ≠ 0 decay formula general class of nonleptonic two-body baryon decays
074011-16 NONLEPTONIC TWO-BODY DECAYS OF SINGLE HEAVY … PHYS. REV. D 98, 074011 (2018) one also finds decays which proceed solely by W-exchange the observability of the decays would be to generate MC diagrams. Among these are the decays in which the events according to the described decay chains and to transition between the two spectator quarks is forbidden process them through the detector. due to the fact that one has a transition between a symmetric and antisymmetric spin-flavor configuration ACKNOWLEDGMENTS such as in ½ud → fudg. A few examples are the ¯ þ þþ − þ 0 þ This work was funded by the German c → sdu decays Λc → Δ K , Λc → Σ π and Ξ0 → Ω−Kþ. Then there is a multitude of decays where Bundesministerium für Bildung und Forschung c (BMBF) under Project 05P2015—ALICE at High Rate the flavor composition of the parent and daughter baryon “ preclude a tree graph contribution. In the ðc → s; (BMBF-FSP 202): Jet- and fragmentation processes at þ 0 þ ALICE and the parton structure of nuclei and structure of d → uÞ sector two sample decays are Λc → Ξ K and 0 heavy hadrons”, Carl Zeiss Foundation under Project Λþ → Ξ ð1530ÞKþ which have been observed with the c “Kepler Center für Astro- und Teilchenphysik: sizable branching ratios of B ¼ð5.90 0.86 0.39Þ × 10−3 B 5 02 0 99 0 31 10−3 Hochsensitive Nachweistechnik zur Erforschung des and ¼ð . . . Þ × , respectively unsichtbaren Universums (Gz: 0653-2.8/581/2)”,by b → cud¯ [76]. Among the CKM favored decays are CONICYT (Chile) PIA/Basal FB0821, by the Russian Λ0 → Σ0π0 Ξ− → Σ0 − → ¯ b c and b cK as well as the b ccs decays Federation program “Nauka” (Contract Λ0 → Ξþ 0 → ¯ b c D , the CKM suppressed b uud decays No. 0.1764.GZB.2017), by Tomsk State University com- Ξ− → Λ0π− Λ0 → Λ0 0 b and b K and the doubly Cabibbo petitiveness improvement program under Grant → ¯ Ξ− → Ξ0π− suppressed b uus decays b and No. 8.1.07.2018, and by Tomsk Polytechnic University Λ0 → Ξ0 0 b K . The latter bottom baryon decays are predicted Competitiveness Enhancement Program (Grant No. VIU- to be severely suppressed according to the SCETanalysis of FTI-72/2017). M. A. I. acknowledges the support from [1]. Again, it would be important to find out whether this Precision Physics, Fundamental Interactions and prediction shows up in the experimental rates. Structure of Matter (PRISMA) cluster of excellence A last remark concerns the experimental observability of (Mainz Uni.). M. A. I. and J. G. K. thank the Heisenberg- the decays discussed in this paper. The best way to check on Landau Grant for the partial support.
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