#### 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 +).
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