Weak decays of doubly heavy baryons: multi-body decay channels

The European Physical Journal C, Jan 2018

The newly-discovered \(\Xi _{cc}^{++}\) decays into the \( \Lambda _{c}^+ K^-\pi ^+\pi ^+\), but the experimental data has indicated that this decay is not saturated by any two-body intermediate state. In this work, we analyze the multi-body weak decays of doubly heavy baryons \(\Xi _{cc}\), \(\Omega _{cc}\), \(\Xi _{bc}\), \(\Omega _{bc}\), \(\Xi _{bb}\) and \(\Omega _{bb}\), in particular the three-body nonleptonic decays and four-body semileptonic decays. We classify various decay modes according to the quark-level transitions and present an estimate of the typical branching fractions for a few golden decay channels. Decay amplitudes are then parametrized in terms of a few SU(3) irreducible amplitudes. With these amplitudes, we find a number of relations for decay widths, which can be examined in future.

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Weak decays of doubly heavy baryons: multi-body decay channels

Eur. Phys. J. C Weak decays of doubly heavy baryons: multi-body decay channels Yu-Ji Shi 0 Wei Wang 0 Ye Xing 0 Ji Xu 0 0 INPAC, Shanghai Key Laboratory for Particle Physics and Cosmology, MOE Key Laboratory for Particle Physics , Astrophysics and Cosmology , School of Physics and Astronomy, Shanghai Jiao Tong University , Shanghai 200240 , China The newly-discovered c+c+ decays into the c+ K −π +π +, but the experimental data has indicated that this decay is not saturated by any two-body intermediate state. In this work, we analyze the multi-body weak decays of doubly heavy baryons cc, cc, bc, bc, bb and bb, in particular the three-body nonleptonic decays and fourbody semileptonic decays. We classify various decay modes according to the quark-level transitions and present an estimate of the typical branching fractions for a few golden decay channels. Decay amplitudes are then parametrized in terms of a few SU(3) irreducible amplitudes. With these amplitudes, we find a number of relations for decay widths, which can be examined in future. 1 Introduction Nowadays Lattice QCD is the sole approach that can study nonperturbative strong interactions from first principle. Despite the fact that there have been great progresses on Lattice QCD, hadron structures are still often encoded by phenomenological approaches like quark models or QCD sum rules. The quark model can be used to classify the hadrons, in which a baryon is assigned as a three-quark system. Among various baryonic states, doubly heavy baryons are of particular interest since they provide a platform to study the nonperturbative dynamics in the presence of heavy quarks. These states have been searched for a long time [1– 6], and in 2017 the LHCb collaboration has announced an observation of the c+c+(ccu) with the mass m c+c+ = (3621.40 ± 0.72 ± 0.27 ± 0.14)MeV [7]. This analysis is based on the 1.7 f b−1 data accumulated at 13 TeV, and confirmed in the additional sample of data collected at 8 TeV. By all means the observation of c+c+(ccu) is a milestone in hadron physics on both theoretical and experimental sides. One would anticipate that more experimental data on production and decays of doubly heavy baryons will be released based on the larger data sample to be collected by LHCb in future [8]. On the other side, to reveal the internal structures of doubly heavy baryons, more detailed theoretical efforts are needed [ 9–25 ]. To handle weak decays of heavy mesons, factorization approach is widely adopted in order to separate high-energy and low-energy degrees of freedoms. High-energy contributions are calculable using the ordinary perturbation theory. The low-energy degrees, or equivalently the long-distance contributions, are usually parameterized as low energy inputs such as light-cone distribution amplitudes. In terms of heavy baryon decays, neither the low-energy inputs nor the shortdistance coefficients are available in the literature. Only recently the “decay constants” were studied in QCD sum rules [ 26 ]. This work is an extension of a series of previous works [ 10, 11,14,24,26 ]. In Ref. [14], instead of factorization, we have adopted the flavor SU(3) symmetry and classified various decays of doubly heavy baryons. In that work, however, we have limited ourselves to two-body nonleptonic decay modes. The c+c+ baryon has been firstly observed in the mode c+c+ → c+ K −π +π + [7], and experimental data has indicated that this mode is not saturated by two-body intermediate state. This motivates us to study the multi-body decays. The main objective of this work is to do so, and we will focus on the cases where the final states contain one additional light meson, namely three-body nonleptonic decay and four-body semileptonic decays. The rest of this paper is organized as follows. In Sect. 2, we will collect representations for the particle multiplets in the SU(3) symmetry. In Sect. 3, we will give a list of golden channels that can be used to reconstruct the doubly heavy baryons, and we present an estimate of their branching fractions. In Sect. 4, we will analyze the semileptonic decays of the doubly-heavy baryons, in which the final state contains two hadrons. The three-body nonleptonic decays of doublycharmed baryons, doubly-bottom baryons and the baryons with b, c quarks are investigated in Sects. 5, 6 and Sect. 7, respectively. The last section contains a brief summary. 2 Particle multiplets In this section, we start with the representations for the multiplets of the flavor SU(3) group. Quantum numbers of the doubly heavy baryons are derived from the quark model. These baryons can form an SU(3) triplet: (1) (3) 0 1 − √2 + + √16 0 p n ⎞ 0 ⎟⎟ , (2) − 23 0 ⎠ +, 0, +, −, 0, 0, −, c+c+(ccu) ⎞ c+c(ccd) ⎟ , Tbc = ⎜ c+c(ccs) ⎠ ⎝ 0bb(bbu) ⎞ b−b(bbd) ⎠ . b−b(bbs) b+c(bcu) ⎞ 0bc(bcd) ⎟ , ⎠ 0bc(bcs) The light baryons form an SU(3) octet and a decuplet. The octet has the expression: ⎛ 0 Tcc = ⎜ ⎝ Tbb = ⎝ ⎛ ⎛ T8 = ⎜⎜ ⎝ ⎛ √12 0 + √16 − − ++, −, and the light decuplet is given as (T10)111 = (T10)222 = (T10)112 = (T10)121 = (T10)211 = √13 (T10)122 = (T10)212 = (T10)221 = √13 (T10)113 = (T10)131 = (T10)311 = √13 (T10)223 = (T10)232 = (T10)322 = √13 (T10)123 = (T10)132 = (T10)213 = (T10)231 = (T10)312 = (T10)321 = √16 (T10)133 = (T10)313 = (T10)331 = √13 (T10)233 = (T10)323 = (T10)332 = √13 (T10)333 = −. (4) (5) (6) In the meson sector, the light pseudo-scalar meson is an octet, which can be represented as: M8 = ⎜⎜ ⎜ ⎝ and we shall not consider the flavor singlet η1 in this work. This is also applicable to the vector meson octet and other light mesons. Charmed baryons form an anti-triplet or sextet: ⎛ 0 + c 0 c+ ⎞ 0 c ⎠ , Tc6 = ⎝⎜ √√1122 cc++ √12 c0 Tc3¯ = ⎝ − c + − c+ − c0 0 ⎛ c++ √12 c+ √12 c+ ⎞ c0 √12 c0 ⎟ . 0 ⎠ c Charmed mesons forms an SU(3) anti-triplet: Di = D0, D+, Ds+ , Di = D0, D−, Ds− . The above classification is also applicable to bottom mesons. 3 Golden decay channels Before presenting the decay amplitudes for various channels, we will make a list of the golden channels and give an estimate of the decay branching fractions in this section [10]. In the following list we give, a hadron is generic and can be replaced by the states with the same quark structure, for instance one can replace K 0 by K ∗0 which decays into K −π +. Since the π 0, η, ρ+ (decaying into π +π 0), and ω (mainly decaying into π +π −π 0) are difficult to reconstruct at LHC, we have removed the modes involving these hadrons. The Feynman diagrams for the Cabibbo-allowed decays are given in Fig. 1. We only show one type of penguin diagrams. The C, C , B, E diagrams are suppressed by 1/Nc compared to the tree amplitude T . For the cc and cc decays, we collect Cabibbo allowed decays in Table 1. From the D and c decay data, we infer that these Cabibbo allowed decay channels have typical branching fractions at a few percent level. ¯ d s s u s c ¯ d u c d E C s u c c s ¯ d u C P A list of possible modes to reconstruct the bcq baryons is given in Table 2. For the charm quark decay, the typical branching fractions might be a few percents. The final state contains either a bottom meson, or a bottom baryon, whose decay branching fraction is then at the order 10−3. So the branching fraction to reconstruct the bc and bc is very likely at the order 10−5. If the bottom quark decay first in the bcq baryons, the branching fraction might be even smaller than 10−3, since the total width of bc and bc is dominated by charm quark decay. In this case, the branching fraction to reconstruct bc and bc might be even smaller than 10−5. The channels that can be used to reconstruct the bb and bb are collected in Table 3. Their typical branching fractions are at the order 10−3. However in order to reconstruct the bottom meson and bottom baryons in the final state, the price to pay is another factor of 10−3. Including the fraction for J /ψ or D or charmed baryons, we have the largest decay branching fraction for bb and bb at the order of 10−8. 0 bc 0 bc → 0 bc → 0 bc → 0 cc → 0 bc 0 bc → 0 bc → 0 bc → 0 bc → 0 bc → 0 K 0, 0b K ∗0 b +a2(Tcc)i (H3) j (T c3¯ )[k j] Mik ν¯ +a3(Tcc)i (H3) j (T c6)[ik] M kj ν¯ +a4(Tcc)i (H3) j (T c6)[k j] Mik ν¯ . Here the ai are SU(3) irreducible amplitudes. The decay amplitudes for different channels can be deduced from the Hamiltonian in Eq. (8), and given in Table 4. The channels with the CKM factor Vcs can have branching fractions about a few percents, while the c → d induced channels have the branching fractions at the order of 10−3. From these amplitudes, we can find the relations for decay widths in the SU(3) symmetry limit. For decays into a singly charmed baryon (anti-triplet), we have ( c+c → c0 K 0 +ν ) = ( c+c → c+K − +ν ). Relations for decays into a singly charmed baryon (sextet) are given as: ( c+c+ → c++π− +ν ) = 2 ( c+c+ → c+K 0 +ν ) = 2 ( c+c → c+π− +ν ) = ( c+c → c0 K 0 +ν ) = 4 ( c+c → c0π0 +ν ), with q = u, c. The b → c transition is an SU(3) singlet, while the b → u transition forms an SU(3) triplet H3 with (H3)1 = 1 and (H3)2,3 = 0. The hadron level Hamiltonian for semileptonic bb and bb decays is constructed as He f f = a5(Tbb)i (T bc) j Mij ¯ν +a6(Tbb)i (H3) j (T b3¯)[ik] M kj ¯ν +a7(Tbb)i (H3) j (T b3¯)[ jk] Mik ¯ν +a8(Tbb)i (H3) j (T b6){ik} M kj ¯ν +a9(Tbb)i (H3) j (T b6){ jk} Mik ¯ν . (10) The decay amplitudes can be deduced from this Hamiltonian, and the results are given in Table 5. For decays into a bcq, we have the relations for decay widths For the charm quark decays in bc and bc, one can obtain the decay amplitudes from those for cc and cc decays with the replacement of Tcc → Tbc, Tc → Tb and D → B. For the bottom quark decay, one can obtain them from those for bb and bb decays with Tbb → Tbc, Tb → Tc and B → D. Thus we do not repeat the tedious results here. 5 Non-Leptonic Usually the charm quark decays into light quarks are classified into three groups: Cabibbo allowed, singly Cabibbo suppressed, and doubly Cabibbo suppressed: c → sd¯u, c → ud¯d/s¯s, c → ds¯u. (11) Under the flavor SU(3) symmetry, the tree operators like s¯cu¯d transform as 3 ⊗ 3¯ ⊗ 3 = 3 ⊕ 3 ⊕ 6¯ ⊕ 15. So the hadron-level Hamiltonian can be decomposed in terms of a vector (H3), a traceless tensor antisymmetric in upper indices, H6, and a traceless tensor symmetric in upper indices, H15. As we will show in the following, the representation H3 will vanishes from the unitarity of CKM matrix. For the c → sud¯ transition, we have the nonzero matrix element: with all other remaining entries zero. The overall CKM factor is Vc∗s Vus sin(θC ). Since the CKM factors for c → ud¯d and c → us¯s are almost equal in magnitudes, we combine the two transitions. Thus the singly Cabibbo-suppressed channel has the following hadron-level Hamiltonian: (H6)331 = −(H6)133 = (H6)122 = −(H6)221 = sin(θC ), (H15)331 = (H15)313 = −(H15)212 = −(H15)221 = sin(θC ). (16) 5.1 Decays into a charmed baryon and two light mesons With the above expressions, one may derive the effective Hamiltonian for decays involving the anti-triplet heavy baryons as (15) j He f f = b1(Tcc)i (T c3¯)[i j] Mk Mlm (H6)kml + b2(Tcc)i (T c3¯)[ jk] Mij Mlm (H6)kml + b3(Tcc)i (T c3¯)[lm] Mij M kj (H6)lkm + b4(Tcc)i (T c3¯)[lm] Mij Mkm (H6)kjl + b5(Tcc)i (T c3¯)[ jk] Mml Mlm (H6)ijk + b6(Tcc)i (T c3¯)[ jl] Mml Mkm (H6)ijk + b7(Tcc)i (T c3¯)[lm] Mlj Mkm (H6)ijk + b8(Tcc)i (T c3¯)[i j] Mkj Mml (H15)lkm + b9(Tcc)i (T c3¯)[i j] Mml Mkm (H15)ljk + b10(Tcc)i (T c3¯)[ jk] Mij Mlm (H15)kml + b11(Tcc)i (T c3¯)[lm] Mij Mkm (H15)kjl + b12(Tcc)i (T c3¯)[ jl] Mml Mkm (H15)ijk + b1(Tcc)i (T c3¯)[i j] Mml Mkm (H6)ljk . For the sextet baryon, we have the Hamiltonian j He f f = b1(Tcc)i (T c6)[i j] Mk Mlm (H15)kml + b2(Tcc)i (T c6)[i j] Mml Mkm (H15)ljk + b3(Tcc)i (T c6)[ jk] Mij Mlm (H15)kml + b4(Tcc)i (T c6)[lm] Mij M kj (H15)lkm + b5(Tcc)i (T c6)[lm] Mij Mkm (H15)kjl + b6(Tcc)i (T c6)[ jk] Mml Mlm (H15)ijk + b7(Tcc)i (T c6)[ jl] Mml Mkm (H15)ijk + b8(Tcc)i (T c6)[lm] Mlj Mkm (H15)ijk + b9(Tcc)i (T c6)[i j] Mkj Mml (H6)lkm + b10(Tcc)i (T c6)[ jk] Mij Mlm (H6)kml + b11(Tcc)i (T c6)[lm] Mij Mkm (H6)kjl + b12(Tcc)i (T c6)[ jl] Mml Mkm (H6)ijk + b9(Tcc)i (T c6)[i j] Mml Mkm (H6)ljk . (18) We have checked that the b1 and b9 terms give the same contribution as the b1 and b9, and the corresponding amplitudes always contain the factor b1 − b1 for anti-triplet and b9 − b9 for sextet. So we can remove b1 and b9 term in the expanded amplitude. It should be mentioned that the dynamical mechanisms of these terms are not all the same. For the production of final two light mesons, some terms contain one QCD coupling while the others contain two QCD couplings. Expanding the above equations, we will obtain the decay amplitudes given in Tables 6, 7 for the anti-triplet baryon and Tables 8, 9, 10 for the sextet. Based on the expanded amplitudes, we derive the relations for decay widths collected in Appendix A 1. 5.2 Decays into a light baryon, a charmed meson and a light meson The hadron-level Hamiltonian for the decays of Tcc into a light octet baryon, a charmed meson and a light meson is given as He f f = c1(Tcc)i D j i jk M mn (T8)lk (H6)lnm + c2(Tcc)i Dl i jk M mn (T8)lk (H6)njm + c3(Tcc)i Dl i jk Mlm (T8)kn(H6)m nj + c4(Tcc)i Dl i jk Mnj (T8)km (H6)lmn + c5(Tcc)i Dl i jk Mnm (T8)km (H6)ljn c+c+ → c+π+ K0 b2 + b4 + 2b8 − b10 + b11 −2b1+b2−4b3−b4−2b8+2b9+b10−3b11 √6 2 b2 + b4 + b10 − b11 b1−b2−b4+b6−2b7−√b8+b9+b10−b11−b12 2 b2 − 2b3 − 4b5 − b6 + b10 − b12 2b3+b4−b6+√2b7+b11−b12 3 b2 − 2b3 − 4b5 − b6 − b10 + b12 − b2+b4+√b10−b11 2 −b1 − b6 + 2b7 + b8 + b9 − b12 −b1 + b2 − 2b3 − b8 − b9 + b10 b1−2b2+2b3−b4−√b8+b9+2b10−3b11 6 − sin2(θc) b1 − b2 + 2b3 − b8 − b9 + b10 b1+b2+2b3+2b4+√b8−b9+b10 sin2(θc) 6 −2 b2 + b4 + b10 − b11 sin2(θc) − sin2(θc) b1 − b2 + 2b3 + b8 + b9 − b10 2 b2 + b4 − b10 + b11 sin2(θc) −2 2b5 + b6 − b7 sin2(θc) 2 b11−b1√2 sin2(θc) 3 b2 − 2b3 − 4b5 − b6 − b10 + b12 sin2(θc) b1+b6−2b7−b8√+b9−b12 sin2(θc) 2 b1−2b2−2b4+b6−2b7−b8+b9+2b10−2b11−b12 sin2(θc) √6 b1+b6−2b7+b8√−b9+b12 sin2(θc) 2 c+c → c0K+η c+c → c+ηη c+c → c+K0K0 c+c → c+π−K+ c+c → c+K+K− c+c → c0π+π0 + c6(Tcc)l Di ijk Mmn(T8)lk(H6)njm + c7(Tcc)l Dm ijk Mmn(T8)lk(H6)inj + c8(Tcc)l Dm ijk Mni(T8)lk(H6)mjn + c9(Tcc)l Di ijk Mln(T8)km(H6)njm + c10(Tcc)l Dm ijk Mln(T8)km(H6)inj + c11(Tcc)l Dm ijk Mln(T8)kn(H6)imj + c12(Tcc)l Dm ijk Mli(T8)kn(H6)mjn Amplitude Table 9 Doubly charmed baryon decays into a sextet cqq and two light mesons Amplitude + c20(Tcc)i Dl i jk Mlm (T8)kn(H15)m nj + c21(Tcc)i Dl i jk Mnj (T8)km (H15)lmn + c22(Tcc)i Dl i jk Mnm (T8)km (H15)ljn + c23(Tcc)l Di i jk M mn (T8)lk (H15)njm + c24(Tcc)l Dm i jk Mni(T8)lk (H15) mjn + c25(Tcc)l Di i jk Mln(T8)km (H15)njm + c26(Tcc)l Dm i jk Mli (T8)kn(H15) mjn + c27(Tcc)l Di i jk Mnj (T8)km (H15)lmn + c28(Tcc)l Di i jk M mn (T8)lk (H15)ijm + c29(Tcc)l Dm i jk Mni(T8)km (H15)ljn +c30(Tcc)l Dm i jk Mni(T8)km (H15)ljn. + c2(Tcc)l Dm (T10)i jl M mn (H15)n i j + c3(Tcc)l Dm (T10)i jl Mni(H15)m nj + c4(Tcc)l Dm (T10)i jm Mln(H15)n i j + c5(Tcc)l Dm (T10)i jn Mli (H15)m jn + c6(Tcc)l Dm (T10)i jm Mni(H15)ljn + c7(Tcc)l Dm (T10)i jn Mmi (H15)ljn + c8(Tcc)l Dm (T10)iml Mnj (H6)ijn + c9(Tcc)l Dm (T10)i jl Mni(H6)m nj + c10(Tcc)l Dm (T10)i jm Mni(H6)ljn. Expanding the above equations, we will obtain the decay amplitudes given in Tables 11, 12 and 13. This leads to the relations for decay widths: in Appendix A 2. For a light decuplet in the final state, the Hamiltonian is given as He f f = c1(Tcc)l Dm (T10)iml Mnj (H15)ijn The corresponding decay amplitudes are given in Table 14, and it leads to the relations for decay widths also collected in Appendix A 2. (19) (20) Amplitude These decays have the same topology with semileptonic b → s + − decays, and thus the SU(3) relations derived in this subsection are also applicable to semileptonic b → s + − decays. The transition operator b → cc¯d/s can form an SU(3) triplet, which leads to the effective Hamiltonian: He f f = a1(Tbb)i (H3) j M kj (T b3¯)[ik] J /ψ + a2(Tbb)i (H3) j Mik (T b3¯)[ jk] J /ψ + a3(Tbb)i (H3) j M kj (T b6)[ik] J /ψ + a4(Tbb)i (H3) j Mik (T b6)[ jk] J /ψ, (22) with (H3)2 = Vc∗d and (H3)3 = Vc∗s . Decay amplitudes are given in Table 15, from which we derive the relations for decay widths: Appendix A 3. 6.2 b → cc¯d/s: decays into a doubly heavy baryon bcq, an anti-charmed meson and a light meson The b → cc¯d/s transition can induce another type of effective Hamiltonian: He f f = a5(Tbb)i (H3) j (T bc)i Dk M kj + a6(Tbb)i (H3) j (T bc) j Dk Mik + a7(Tbb)i (H3) j (T bc)k D j Mik Table 10 Doubly charmed baryon decays into a sextet cqq and two light mesons Table 11 Doubly charmed baryon decays into a light baryon in the octet, a charmed meson and a light meson This Hamiltonian denotes the decays into doubly heavy The operator to produce a charm quark from the b-quark baryon bcq plus an anti-charmed meson. Decay amplitudes decay, c¯bq¯u, is given by are given in Table 16. Thus we obtain the following relations for decay widths: Appendix A 4. c+c → +D0π0 c+c → 0D+K0 −√12 (c3 − c4 + 2c7 + c8 − c9 + 2c10 − c13 + c15 − c20 + c21 − c24 − c25 − 2c26 + c27 − c29 + 2c30)sin(θc) c+c → 0D+π0 2√13(c1 − c2 + 2c3 − 2c6 + 4c7 + 2c13 + c14 − c15 − 2c16 − c17 − c18 +c19 − 2c21 + 2c22 + 2c23 + 4c24 + 2c27 + c28 − c29 + 3c30)sin(θc) 16+(−53cc141 ++ c31c52+−22cc136−+2cc147++23cc518+−6c36c1−9 −4c67c+234−c86+c252c−9 −3c248c1−0 +3c289c1−1 −3c340c)12si+n(θ4cc)13 −√16(2c3 − c4 + c5 + 4c7 + 2c8 − 2c9 + 4c10 − 2c11 + c12 − c13 + c14 + 2c15 −2c16 − c17 + c21 − c22 − 2c24 − 3c26 + c27 − c28 − 2c29 + 3c30)sin(θc) −√12 (c1 + c2 + c14 + c15 − 2c16 − c17 − c18 − c19 + c28 + c29 − c30)sin(θc) c+c → +D+π− (c4 + c5 + 2c16 + c17 − c21 − c22 + c30)sin(θc) c+c → +Ds+K− (c4 + c5 − 2c11 + c12 + c13 + c14 − 2c16 − c17 − c21 − c22 + c26 − c27 − c28 − c30)(−sin(θc)) Amplitude Table 14 Doubly charmed baryon decays into a light baryon in the decuplet, a charmed meson and a light meson −a2Vc∗d −a1Vc∗d − (a1 + a2) Vc∗s The light quarks in this effective Hamiltonian form an octet 2 with the nonzero entry (H8)1 = Vu∗d for the b → cu¯d transition, and (H8)13 = Vu∗s for the b → cu¯s transition. The hadron-level effective Hamiltonian is then given as Hef f = a9(Tbb)i (T bc)i Mkj Mlj (H8)lk + a10(Tbb)i (T bc) j Mij Mlk(H8)lk + a11(Tbb)i (T bc) j Mil Mkj (H8)lk j + a12(Tbb)i (T bc) j Mil Mlk(H8)k + a13(Tbb)i (T bc) j Mkj Mlk(H8)li + a14(Tbb)i (T bc) j Mlk Mkl (H8)ij . (25) Decay amplitudes are expanded in Table 17, which leads to the relations: Appendix A 5. The effective Hamiltonian from the operator c¯bq¯u gives Hef f = a15(Tbb)i (T b3¯)[i j] D j Mlk(H8)lk + a16(Tbb)i (T b3¯)[i j] Dl Mkj (H8)lk + a17(Tbb)i (T b3¯)[i j] Dl Mlk(H8)k j + a18(Tbb)i (T b3¯)[ jk] D j Mil (H8)lk + a19(Tbb)i (T b3¯)[ jk] Dl Mij (H8)lk + a20(Tbb)i (T b3¯)[ jk] Dl Mlk(H8)ij + a21(Tbb)i (T b3¯)[kl] Dl M kj(H8)ij + a22(Tbb)i (T b6)[i j] D j Mlk(H8)lk + a23(Tbb)i (T b6)[i j] Dl Mkj (H8)lk + a24(Tbb)i (T b6)[i j] Dl Mlk(H8)k j + a25(Tbb)i (T b6)[ jk] D j Mil (H8)lk b−b → b−b → b−b → b+c D0π − 0bc D− K 0 b+c D− K 0 0bc Ds−η b+c Ds− K 0 0bc D−π 0 b−b → b−b → b−b → b−b → b−b → b−b → 0bc D0 K − b+c D−π 0 0bc Ds− K 0 b+c Ds−π 0 0bc D0 K − b−b → b−b → b−b → b+c D0 K − 0bc Ds−π 0 b+c D−η 0bc D0π − b+c Ds−η 0bc D− K 0 0bc D0η 0bc Ds−η 0bc Ds− K + 0bc D0π − 0bc D0η 0bc D−η 0bc Ds− K + b+c Ds−π − b+c Ds−π − 0bc D−η 0bc D−π 0 − a6Vc∗d a6Vc∗s (a6−√2a8)Vc∗s 6 (a6−√2a8)Vc∗s 6 (a6 + a7) Vc∗s a8Vc∗d a7Vc∗s (a5+a6+a7+a8)Vc∗d √6 − (a5+a6+√a72+a8)Vc∗d 23 (a5 + a6 + a7 + a8) Vc∗s l k + a26(Tbb)i (T b6)[ jk] D M j ( H8)l i + a27(Tbb)i (T b6)[ jk] Dl Mlk ( H8)ij + a28(Tbb)i (T b6)[kl] Dl M kj ( H8)ij . Results are given in Table 18 for anti-triplet and Table 19 for sextet, thus we have the relations for decay amplitudes: Appendix A 6 for sextet. Actually, for the anti-triplet case there’s no definite relations between the decay withs. 6.5 b → uc¯d/s: decays into a bottom baryon bqq plus anti-charmed meson and a light meson For the anti-charm production, the operator having the quark contents (u¯ b)(q¯ c) is given by The two light anti-quarks form the 3¯ and 6 representations. The anti-symmetric tensor H3¯ and the symmetric tensor H6 have nonzero components ( H3¯ )13 = −( H3¯ )31 = Vc∗s , ( H6¯)13 = ( H6¯)31 = Vc∗s , for the b → uc¯s transition. For the transition b → uc¯d one requests the interchange of 2 ↔ 3 in the subscripts, and Vcs replaced by Vcd . The effective Hamiltonian is constructed as (26) He f f = b1(Tbb)i (T b 3¯)[i j] Dl Mkj ( H3¯ )kl + b2(Tbb)i (T b 3¯)[i j] Dl Mkl ( H3¯ ) jk + b3(Tbb)i (T b 3¯)[ jk] Di Mlj ( H3¯ )kl + b4(Tbb)i (T b 3¯)[ jk] Dl Mij ( H3¯ )kl + b5(Tbb)i (T b 3¯)[kl] D j Mij ( H3¯ )kl + b6(Tbb)i (T b 3¯)[i j] Dl Mkj ( H6 )kl + b7(Tbb)i (T b 3¯)[i j] Dl Mkl ( H6 ) jk + b8(Tbb)i (T b 3¯)[ jk] Di Mlj ( H6 )kl + b9(Tbb)i (T b 3¯)[ jk] Dl Mij ( H6 )kl + b10(Tbb)i (T b6)[i j] Dl Mkj ( H3¯ )kl + b11(Tbb)i (T b6)[i j] Dl Mkl ( H3¯ ) jk + b12(Tbb)i (T b6)[ jk] Di Mlj ( H3¯ )kl +b13(Tbb)i (T b6)[ jk] Dl Mij ( H3¯ )kl + b14(Tbb)i (T b6)[kl] D j Mij ( H6 )kl + b15(Tbb)i (T b6)[i j] Dl Mkj ( H6 )kl + b16(Tbb)i (T b6)[i j] Dl Mkl ( H6 ) jk b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → 0bc K −η 0bcπ − K 0 0bcπ −η 0bc K 0 K − b+cπ − K − b+c K − K − 0bcπ 0 K − 0bcπ − K 0 0bc K 0 K − 0bc K −η 0bcπ 0 K − 0bcπ − K 0 0bcπ −η 0bc K 0 K − 0bc K −η 0bcηη b+cπ −π − b+cπ − K − 0bcπ 0π − 0bcπ 0 K − 0bcπ − K 0 0bcπ −η 0bc K 0 K − + b18(Tbb)i (T b6)[ jk] Dl Mij ( H6 )kl . (28) Decay amplitudes for different channels are given in Tables 20 and 21. We derive relations for decay amplitudes given in Appendix A 7. The bb can decay into both D0 and D0. The D0 and D0 can form the CP eigenstates D+ and D−. Thus using the bb decays into the D±, one may construct the interference between the b → cu¯ s and b → uc¯s. The CKM angle γ can then be extracted from measuring decay widths of these channels, as in the case of B → D K [ 27–32 ], B → D K0∗,2 [ 33, 34 ] and others. This is also similar for the bb → D± decays and the following bc → D± and bc → D± channels. 6.6 Charmless b → q1q¯2q3 decays: decays into a bottom baryon and two light mesons The charmless b → q (q = d, s) transition is controlled by the weak Hamiltonian He f f : where Oi is a four-quark operator or a moment type operator. At the hadron level, penguin operators behave as the 3 representation while tree operators can be decomposed in terms of a vector H3, a traceless tensor antisymmetric in upper indices, H6, and a traceless tensor symmetric in upper indices, H15. For the S = 0(b → d)decays, the non-zero components + c12(Tbb)i(T b6)[lm]Mij Mkj(H15)lkm + c13(Tbb)i(T b6)[km]Mij Mlm(H15)kjl + c14(Tbb)i(T b6)[jk]Mml Mlm(H15)ijk 6.7 Charmless b → q1q¯2q3 decays: decays into a bottom meson, a light baryon octet and a light meson The effective Hamiltonian is given as Hef f = d1(Tbb)i B j ijk(T8)lkMml(H3)m +d2(Tbb)i Bl ijk(T8)kmMlj(H3)m +d3(Tbb)i Bl ijk(T8)kmMlm(H3)j +d4(Tbb)i Bl ijk(T8)lkMmj (H3)m +d5(Tbb)l Bi ijk(T8)lkMmj (H3)m +d6(Tbb)l Bm ijk(T8)lkMmi(H3)j +d7(Tbb)l Bi ijk(T8)kmMlj(H3)m +d8(Tbb)l Bi ijk(T8)kmMlm(H3)j +d9(Tbb)l Bm ijk(T8)kmMli(H3)j +d10(Tbb)i B j ijk(T8)lkMmn(H6¯)lnm +d11(Tbb)i Bl ijk(T8)lkMmn(H6¯)njm +d12(Tbb)i Bl ijk(T8)knMlm(H6¯)m nj +d13(Tbb)i Bl ijk(T8)kmMnj(H6¯)lmn +d14(Tbb)i Bl ijk(T8)kmMnm(H6¯)ljn +d15(Tbb)l Bi ijk(T8)lkMmn(H6¯)njm +d16(Tbb)l Bm ijk(T8)lkMmn(H6¯)inj +d17(Tbb)l Bm ijk(T8)lkMni(H6¯)mjn +d18(Tbb)l Bi ijk(T8)kmMln(H6¯)njm +d19(Tbb)l Bm ijk(T8)kmMln(H6¯)inj +d20(Tbb)l Bm ijk(T8)knMln(H6¯)imj +d21(Tbb)l Bm ijk(T8)knMli(H6¯)mjn +d22(Tbb)l Bi ijk(T8)kmMnj(H6¯)lmn +d23(Tbb)l Bi ijk(T8)kmMnm(H6¯)ljn +d24(Tbb)l Bm ijk(T8)kmMni(H6¯)ljn +d25(Tbb)l Bm ijk(T8)knMmn(H6¯)lij +d26(Tbb)l Bm ijk(T8)knMmj (H6¯)lin +d27(Tbb)i B j ijk(T8)lkMmn(H15)lnm +d28(Tbb)i Bl ijk(T8)lkMmn(H15)njm +d29(Tbb)i Bl ijk(T8)knMlm(H15)m nj +d30(Tbb)i Bl ijk(T8)kmMnj(H15)lmn +d31(Tbb)i Bl ijk(T8)kmMnm(H15)ljn +d32(Tbb)l Bi ijk(T8)lkMmn(H15)njm +d33(Tbb)l Bm ijk(T8)lkMni(H15)mjn +d34(Tbb)l Bi ijk(T8)kmMln(H15)njm +d35(Tbb)l Bm ijk(T8)knMli(H15)mjn +d36(Tbb)l Bi ijk(T8)kmMnj(H15)lmn +d37(Tbb)l Bi ijk(T8)kmMnm(H15)ljn +d38(Tbb)l Bm ijk(T8)kmMni(H15)ljn +d39(Tbb)l Bm ijk(T8)knMmj (H15)lin. DecayamplitudesfordifferentchannelsaregiveninTables28 and 29 for b → d transition; Tables 30 and 31 for b → s transition respectively. 6.8 Charmless b → q1q¯2q3 Decays: Decays into a bottom meson, a light baryon decuplet and a light meson The effective Hamiltonian is given as Hef f = f1(Tbb)i B j(T10)ijkMlk(H3)l + f2(Tbb)i B j(T10)iklMkj(H3)l + f3(Tbb)i B j(T10)jklMik(H3)l + f4(Tbb)i B j(T10)ijkMlm(H15)kml (33) b−b → b−π+π− b−b → b−π0π0 b−b → b−ηη Amplitude 13(−c1 − 6c2 + 2c3 − 4c4 + 3c5 − 6c6 + 8c7 + 2c8 +12c9 + 5c10 − 4c11 − 3c12 + 3c13 + 6c14 − 6c15 − 3c16) Table 23 Doubly bottom baryon decays into a bqq(anti-triplet) induced by the charmless b → s transition and two light mesons 0bb → b0K0η b−b → b−π+π− b−b → b−π0π0 b−b → b−π+K− 23 (−c3 + c4 + 2c7 − c8 − c10 + 2c11 + 3c14 − 4c15 + c16) Table 24 Doubly bottom baryon decays into a bqq(sextet) Channel induced by the charmless b → d transition and two light mesons 0bb → b+π0π− 0bb → b+π−η 0bb → b+K0K− Amplitude Table 25 Doubly bottom baryon decays into a bqq(sextet) induced by the charmless b → d transitionand two light mesons b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−ηη Decay amplitudes for different channels are given in Tables 32 and 33 for b → d transition; Tables 34 and 35 for b → s transition. We summarize the corresponding relations for decay widths in Appendix A 9. 7 Non-Leptonic Decays of bc and bc can proceed via the b quark decay or the c quark decay. As we have shown in the semileptonic channels, for the charm quark decays, one can obtain the decay amplitudes from those for cc and cc decays with the replacement of Tcc → Tbc, Tc → Tb and D → B. For the bottom quark decay, one can obtain them from those for bb and bb decays with Tbb → Tbc, Tb → Tc and B → D. Thus we do not present the tedious results again. 8 Conclusions Quite recently, the LHCb collaboration has observed the c+c+ in the final state c K −π +π +. Such an important observaAmplitude 4c9 c1−c3−c5−c6+c7−c8+3c9+3c10−3c11+c13−c15+2c16 √3 c1+c3−c5−c6+c7−c8−c9√+3c10−c11+3c13−c15−2c16 2 c1+2c2+c5−c8+√3c9+c10−4c14−c15 2 c1+2c2+c5−c8−√5c9+c10−4c14−c15 2 −c1+c3+c5+c6−c7+c8+c√9+5c10−5c11−c13+c15−2c16 6 2c2+c4−c7+c8+2c10−2c12+3c13−4c14−c15 √2 c1+2c2+c3+c4−c5−c6−c9−3c10−c11−2c12−2c13−4c14−2c15−2c16 √2 c1+6c2−2c3+4c4−3c5+6c6+6c7+3c8+3c9−3c10−6c11−8c12−2c13−12c14−5c15+4c16 3√2 tion will undoubtedly promote the research on both hadron spectroscopy and weak decays of doubly heavy baryons. In this paper, we have analyzed weak decays of doubly heavy baryons cc, cc, (bc), (bc), bb and bb under the flavor SU(3) symmetry, where the final states involve one or two light mesons. This is inspired by the experimental fact that the c+c+ → c K −π +π + is not dominated by any two-body intermediate state. Decay amplitudes for various semileptonic and nonleptonic decays have been parametrized in terms of a few SU(3) irreducible amplitudes. We have found a number of relations or sum rules between decay widths, which can be examined in future measurements at experimental facilities like LHC [8], Belle II [ 38 ] and CEPC [ 39 ]. On the one hand, at first sight the number of relations is desperately large. On the other hand, once a few decay branching fractions were measured in future, these relations can provide richful important clues for the exploration of other decay modes. It should be stressed that our analysis in this work using the flavor SU(3) symmetry is only applicable to non-resonant contributions. For a complete exploration of three-body decays, one should also take into account resonant contributions from two-body states and this has been given in Ref. [14]. Relative phases between them can be obtained in a Dalitz plot analysis or measurements of invariant mass distributions. In addition, SU(3) symmetry breaking effects might also be relevant. Such effects in the phase space can be incorporated once masses of all involved hadrons are known. This will remedy the relations for decay widths we derived. Actually, we have removed the channels kinematically prohibited. Further deviations, if found by experimentalists in future, would have the indications b−ηη Table 27 Doubly bottom baryon decays into a bqq(sextet) induced by the charmless b → s transition and two light mesons Channel 0 K 0 K − b 0 0 b− K K b0π 0 K − b0π − K b0 K −η 0 b−ηη on decay dynamics in the doubly heavy baryon system. We hope this analysis together with experimental measurements in future will help to establish a QCD-rooted approach to handle the production and decays of doubly heavy baryons. Acknowledgements The authors are grateful to Jibo He, Xiao-Hui Hu, Cai-Dian Lü, Fu-Sheng Yu, Zhen-Xing Zhao for useful discussions. W.W. thanks Cai-Dian Lü, and Qiang Zhao for their hospitality when this work is finalized at IHEP, CAS. This work is supported in part by National Natural Science Foundation of China under Grant Nos. 11575110, 11655002, 11735010, Natural Science Foundation of Shanghai under Grant Nos. 15DZ2272100 and 15ZR1423100, Shanghai Key Laboratory for Particle Physics and Cosmology, and by MOE Key Laboratory for Particle Physics, Astrophysics and Cosmology. Appendix A: Relations between nonleptonic decay widths A.1: Doubly charmed baryon decays into a charmed baryon and two light mesons For decays into an anti-triplet baryon, we have thee relations: Amplitude 2 (c3 + c12 + c14 − 2c17) 2 (c3 + c7 + c8 + 3c12 + 3c14 − 2c17) √ 2 (c3 − c7 − c8 − c12 − c14 − 2c17) ( c+c+ → c0π +π +) = c+π + K 0) = c++π +π −) = √12(d4 + d5 + d7 − d9 − d11 + d13 − d15 − d17 − d18 + 2d19 −d28 + 3d30 − d32 − 3d33 − d34 − 6d35 − 2d36 + 2d38) √12(d1 − d5 − d7 − d8 + d10 − d14 − d15 − d17 + 2d20 − d21 +3d27 − d31 − 3d32 − d33 − 6d34 − d35 + 2d36 + 2d37) Table 28 continued Channel ( c+c+ → c++π0K0) = 3 ( c+c+ → c++K0η), ( c+c+ → c0π+π+) = ( c+c+ → c0K+K+), 1 ( c+c+ → c+π+π0) = 4 ( c+c+ → c0π+K+), A.2: cc and c decays into a octet baryon , a charmed meson and a light meson d2 − d6 + d12 − d13 + 2d16 + d17 − d29 + 3d30 − 3d33 − 2d39 d2 + d4 + d5 − d6 + d7 − d9 − d11 + d12 − d15 + 2d16 − d18 + 2d19 −d28 − d29 − 2d30 − d32 + 2d33 − d34 + 2d35 − 2d36 + 2d38 − 2d39 √12(−d1 + d5 + d7 + d8 − d10 + d14 + d15 + d17 − 2d20 + d21 +5d27 + d31 − 5d32 + d33 − 2d34 + d35 − 2d36 − 2d37) √16(d1 − 2d2 − d5 + 2d6 − d7 − d8 − 3d10 − 2d12 − 2d13 − d14 +3d15 − 4d16 + d17 + 2d20 − d21 + 3d27 + 2d29 + 2d30 − d31 −3d32 − 3d33 + 2d34 − d35 + 2d36 + 2d37 + 4d39) −d1 − d4 + d8 + d9 − d10 − d11 − d13 − d14 − d18 + 2d19 +2d20 − d21 − 3d27 − 3d28 − 3d30 − 3d31 + 3d34 + 3d35 − 2d37 − 2d38 d1 + d4 − d8 − d9 − d10 − d11 − d13 − d14 − d18 + 2d19 +2d20 − d21 − d27 − d28 − d30 − d31 + d34 + d35 + 2d37 + 2d38 −2√13(d1 − d2 + d3 − d4 − 2d5 + 2d6 + d10 − d11 + 2d12 − 2d15 +4d16 + 2d22 + d23 − d24 − 2d25 − d26 − 5d27 + 5d28 − 6d30 +6d31 + 10d32 + 12d33 + 2d36 + d37 − d38 + 3d39) For a decuplet baryon, we have ( c+c+ → 0Ds+π+) = ( c+c → +Ds+π−), ( c+c+ → 0Ds+π+) = ( c+c+ → 0D+K+), ( c+c → ++D0π−) = ( c+c → ++D0K−), ( c+c → −Ds+π+) = ( c+c+ → ++Ds+π−), ( c+c → +Ds+π−) = ( c+c → +D+K−), ( c+c → 0D0π+) = ( c+c → +D+π−), ( c+c → 0Ds+K0) = ( c+c → +Ds+K−), ( c+c → −D+π+) = ( c+c → ++D0π−), ( c+c → −Ds+π+) = ( c+c → +Ds+π−), ( c+c → −Ds+π+) = ( c+c → −D+K+), 1 ( c+c → +D0π0) = 4 ( c+c → 0D+π0), ( c+c+ → +D+K0) = ( c+c+ → pDs+K0), ( c+c+ → pD0π+) = ( c+c+ → +D0K+), ( c+c → +D0K0) = ( c+c → pD0K0), ( c+c → +Ds+π−) = ( c+c → pD+K−), ( c+c → −Ds+π+) = ( c+c → −D+K+), ( c+c → pD+π−) = ( c+c → +Ds+K−), ( c+c → +D+π−) = ( c+c → pDs+K−), ( c+c → −Ds+π+) = ( c+c → −D+K+). Decays into an anti-triplet baryon have the relations: bTahreyofonl:lowing relations are derived for decays into an octet ( b0b → b0π0 J/ψ) = 21 ( b−b → b0π−J/ψ) 1 = 2 ( b−b → b−K0 J/ψ), ( b−b → b0π−J/ψ) = ( b0b → 0bK0 J/ψ) = 2 ( b−b → b−π0 J/ψ), 1 ( b0b → b0π0 J/ψ) = 2 ( b0b → b−π+J/ψ) 1 = 2 ( b−b → b0π−J/ψ) b−b → −B−K+ b−b → −B0K0 b−b → −Bs0π0 b−b → −Bs0η Table 30 Doubly bottom baryon decays into a bottom meson, a light baryon(8) induced by the charmless b → s transition and a light meson Channel −√16(2d2 + d3 − d6 + 2d7 + d8 − d9 + d12 − 2d13 − d14 + 2d16 + d17 21(−d3 − d6 + d8 + d9 − 2d11 + d12 − d14 − 2d15 + 2d16 − d17 − d18 +2d19 − 2d20 + d21 − d23 − d24 − 2d25 − d26 + 4d28 + d29 +d31 + 4d32 + d33 + 3d34 − d35 + 3d37 + 3d38 + 3d39) d7+d8−2d20+d21−d22−d23−2d34+d35+3d36+3d37 −d7+d9+d18−2d19+d22−d24+d34−2d35−3d36+3d38 −d2 − d3 − d13 − d14 − 2d25 − d26 + 2d29 + d30 + d31 − 3d39 √12(d7 − d9 + 2d10 + 2d11 + d18 − 2d19 − d22 + d24 − 4d27 − 4d28 +3d34 + 2d35 + 3d36 − 3d38) √16(2d1 + 2d2 + 2d3 + 2d4 + d7 − d9 + d18 − 2d19 − d22 − 2d23 − d24 +4d25 + 2d26 − 6d27 − 6d28 − 4d29 − 4d30 − 4d31 + 3d34 + 2d35 +3d36 − 6d37 − 9d38 + 6d39) Table 30 continued b−b → 0Bs0K− ( b0b → b−K+J/ψ) = ( b−b → b0K−J/ψ), ( b−b → b0π−J/ψ) = ( b−b → b−K0 J/ψ), ( b0b → b0K0 J/ψ) = ( b−b → b0K−J/ψ), ( b−b → b−K0 J/ψ) = ( b−b → b0K−J/ψ). Decays into a sextet baryon have the relations: ( b0b → b+π−J/ψ) = 2 ( b0b → b0K0 J/ψ) 1 ( b0b → b0π0 J/ψ) = 4 ( b0b → b−K+J/ψ) = ( b−b → b−K0 J/ψ) = 4 ( b−b → b−π0 J/ψ) 1 = 4 ( b−b → b−K0 J/ψ) 1 = 2 ( b0b → b−π+J/ψ) = 3 ( b−b → b−ηJ/ψ) = ( b−b → b−K0 J/ψ) ( b−b → b0π−J/ψ) = 6 ( b0b → b0ηJ/ψ) 2√13(2d2 + d3 − d6 + 2d7 + d8 − d9 − 4d10 − 2d11 − d12 − 2d13 − d14 1 ( b0b → b0K0 J/ψ) = 2 ( b0b → b+K−J/ψ) 1 = 4 ( b−b → b0cD−π0) = 2 ( b−b → b0cDs−π0), ( b−b → b+cDs−π−) = ( b0b → 0bcD0K0) Table 31 Doubly bottom baryon decays into a bottom meson, a light baryon(8) induced by the charmless b → s transition and a light meson Channel d1 + d4 − d8 − d9 + d10 + d11 + d13 + d14 + d18 − 2d19 − 2d20 +d21 + 3d27 + 3d28 + 3d30 + 3d31 − 3d34 − 3d35 + 2d37 + 2d38 √12(−d1 − d4 + d8 + d9 + d10 + d11 − d13 − d14 + d18 − 2d19 +2d20 − d21 + d27 + d28 − 3d30 − 3d31 − d34 + 3d35 − 2d37 − 2d38) √12(−d1 − d4 + d8 + d9 − d10 − d11 + d13 + d14 − d18 + 2d19 − 2d20 +d21 − 3d27 − 3d28 + d30 + d31 + 3d34 − d35 − 2d37 − 2d38) −d1 − d4 + d8 + d9 + d10 + d11 + d13 + d14 + d18 − 2d19 − 2d20 + d21 +d27 + d28 + d30 + d31 − d34 − d35 − 2d37 − 2d38 −d29 − d30 + 4d32 + d33 − 2d39) √16(2d1 − d2 − 2d5 + d6 − 2d7 − 2d8 − d12 − d13 − 2d14 − 2d16 − d17 +4d20 − 2d21 − 6d27 + d29 + d30 − 2d31 + 6d32 − 3d33 + 4d34 − 2d35 +4d36 + 4d37 + 2d39) Table 31 continued b−b → 0Bs0K− 0 0 ( b0b → b0cD K ) = ( b−b → b+cD−K−), ( b0b → b0cD−π+) = ( b−b → b0cD−K0), ( b0b → 0bcD−K+) = ( b−b → b+cD−K−), ( b0b → 0bcDs−K+) = ( b−b → 0bcDs−K0), ( b−b → b+cD−π−) = ( b−b → b0cD−K0), ( b−b → b0cD0π−) = ( b−b → 0bcDs−K0), ( b−b → b0cD−K0) = ( b0b → b+cD0K−), ( b−b → b0cDs−K0) = ( b−b → b+cDs−K−), ( b−b → b0cD−K0) = ( b−b → b0cD0K−), ( b0b → b0cD0η) = ( b−b → b0cD−η), ( b0b → 0bcDs−K+) = ( b−b → b0cD0K−), ( b0b → b+cDs−η) = ( b−b → b0cDs−η). A.5: bb and bb decays into a bcq and two light mesons 1 ( b0b → b+cπ0π−) = 2 ( b−b → b0cπ−K0) 1 = 2 ( b−b → b+cπ−K−) 1 = 4 ( b−b → b+cπ−π−) 1 = 2 ( b−b → b0cK0π−) 1 = 2 ( b−b → b+cK−π−) = ( b−b → b0cπ0π−), ( b0b → b0cπ0K0) = 3 ( b0b → b0cK0η) ( b−b → b0cπ0K−) = 3 ( b−b → 0bcK−η) 0 = 3 ( b0b → 0bcηK ), = 3 ( b−b → 0bcηK−), ( b−b → b0cπ0K−) = 3 ( b−b → 0bcK−η) = ( b−b → b+cπ−K−), ( b0b → b0cπ0K0) = 3 ( b0b → 0bcK0η) 1 ( b0b → b−Ds+π0) = 2 ( b−b → b−Ds+π−), 1 ( b−b → b−D0π0) = 2 ( b−b → b−D+π−). A.7: bb and bb decays into a bottom baryon, an anti-charmed meson and a light meson Channels involving an anti-triplet baryon have ( b0b → b0Ds−π+) = 2 ( b−b → 0bDs−π0), ( b−b → b−Ds−π+) = 2 ( b−b → b0Ds−π0), Table 32 Doubly bottom baryon decays (induced by the b → d transition) into a bottom meson, a light baryon(10) and a light meson Channel b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → ++ B−π− − B0π+ + B− K 0 + Bs0π− 0 B− K + 0 B−η − B0η − Bs0 K 0 f2 − 2 f5 − f6 − 2 f10 − f12 f1+ f3− f4+3 f6−2 f√7+6 f8−2 f9− f11− f12 6 f1+ f3+3 f4− f6+6 f√7−2 f8−2 f9+ f11+ f12 6 f2−2 f5+3 √f6−2 f10+ f12 3 f1+ f2+ f3− f4−2 f5−2 f6√−2 f7−2 f8−2 f9−2 f10− f11 3 − f1+ f3−5 f4− f6−2 f√7−2 f8−2 f9+ f11+ f12 6 f1−2 f2+ f3+3 f4+4 f5+ f6−2 √f7−2 f8−2 f9+4 f10−3 f11+3 f12 3 2 f1+ f3− f4− f6−2 f7−2 f8−2 f9− f11+ f12 √3 16 ( f1 − 2 f2 + f3 + 3 f4 + 4 f5 + f6 + 6 f7 − 2 f8 + 6 f9 − 12 f10 − 3 f11 + 3 f12) Table 32 continued Table 33 Doubly bottom baryon decays into a bottom meson, a light baryon(10) induced by the charmless b → d transition and a light meson b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → 0 B0 K − 0 Bs0 K − − B0π0 − B0η f1 − f4 − f6 − f9 − f11 + f12 − f13 while decay modes involving a sextet baryon have 0 b− D K +), b0 D− K 0), b+ Ds− K −), b0 Ds−π 0), b0 Ds−η). bb decays into a bottom baryon and two A.8: bb and light mesons For the anti-triplet baryon, we have 0π 0 K 0) = 3 ( b0b → b 0b K 0η), b−π 0 K 0) = 3 ( b−b → 0π 0 K 0) = 3 ( b0b → b b−π +π 0) = b−π 0 K 0) = 3 ( b−b → b− K 0η). 0b K 0η), 0π 0π −), b 0 b− K η). Decays into a sextet heavy baryon have b0π 0 K 0) = 3 ( b0b → b+π −π −) = 2 ( b−b → b0π 0 K 0) = 3 ( b0b → b−π +π 0) = 1 b+π − K −) = 2 ( b−b → b0 K 0η), b+π − K −), b0 K 0η), b0π 0π −), b−π +π −), b+ K − K −). Table 34 Doubly bottom baryon decays into a bottom meson, a light baryon(10) induced by the charmless b → s transition and a light meson Channel f1− f4+3 f6+3√f9− f11− f12+ f13 3 f1+3 f4− f6+3√f9+ f11+ f12+ f13 3 f1− f4− f6+3 √f9− f11+ f12+ f13 3 f2+ f3+4 f4+6 f5+3 f6+6 f7+√6 f8+3 f9+6 f10−2 f11+ f12− f13 6 −2 f1+ f2+ f3+6 f4+6 f5−3 f6+6 f7+6 f8−3 f9+6 f10+3 f12−3 f13 3√2 f2+6 f5− f√6+6 f10− f12 3 f1+ f2+3 f4+6 f5−2 √f6+3 f9+6 f10+ f11+ f13 3 f2+ f3−2 f5+3 f6−2 f7+√6 f8+3 f9+6 f10+ f12− f13 6 − f2+ f3+4 f4+2 f5+ f6+6 f7−2 f8+3 f9−6 f10−2 f11+ f12− f13 2√3 f1+ f2− f4−2 f5−2 f√6+63 f9+6 f10− f11+ f13 f3−2 f7−2√f8+3 f9− f13 3 f2+ f3−2 f5+3 f6−2 f7+√6 f8+3 f9+6 f10+ f12− f13 3 b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → ++ B− K − + B− K 0 + B0 K − 0 B0 K 0 − B0π+ 0 B− K + 0 B−η − B0η 0 B− K 0 16 (−2 f1 + f2 + f3 + 6 f4 + 6 f5 − 3 f6 − 2 f7 + 6 f8 + f9 − 2 f10 + 3 f12 + 3 f13) Channel Amplitude b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → b−b → − Bs0π 0 0 B0 K − 0 B−π 0 0 B−η 0 B0π − 0 Bs0 K − − B−π + − B0π 0 − B0η − Bs0π 0 Table 34 continued Table 35 Doubly bottom baryon decays into a bottom meson, a light baryon(10) induced by the charmless b → s transition and a light meson A.9: bb and bb decays into a bottom meson, a light decuplet baryon and a light meson The relations for decay widths are given as: − B−π +) = 3 ( b−b → = 3 ( b0b → = 3 ( b0b → = 3 ( b−b → = 3 ( b−b → 0 B−π +) = 2 ( b0b → − B0 K +) 0 − Bs K +) 0 − Bs π +), − B− K +) − B− K +) − B−π +), 0 B− K +). − B0 K +) 0 − Bs K +) Amplitude f2+ f3−2 f5− f6−2 f7−√2 f8− f9−2 f10− f12+ f13 3 − f3+4 f4+2 f7+√2f8+ f9−2 f11− f13 6 −2 f1−2 f2+ f3+6 f4+4 f5+4√f6−2 f7−2 f8+ f9+4 f10+3 f13 3 2 f2 − 2 f5 + 3 f6 − 2 f10 + f12 f2 − 2 f5 − f6 − 2 f10 − f12 √2 (2 f4 − f11) − 23 ( f1 + f2 + f3 − 3 f4 − 2 f5 − 2 f6 − 2 f7 − 2 f8 − 2 f9 − 2 f10) = ( b0b → − B−π +) = ( b−b → 1 = 3 ( b−b → = = ( b−b → 1 0 B−π +) = 2 ( b0b → 0 − Bs π +), − B− K +) − B− K +) − B−π +), 0 B− K +). 1. 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Yu-Ji Shi, Wei Wang, Ye Xing, Ji Xu. Weak decays of doubly heavy baryons: multi-body decay channels, The European Physical Journal C, 2018, 56, DOI: 10.1140/epjc/s10052-018-5532-7