MITP/21-070

Topological tensor invariants and the current algebra approach:
Analysis of 196 nonleptonic two-body decays of single
and double charm baryons – a review

Stefan Groote1, Jürgen G. Körner2
1Füüsika Instituut, Tartu Ülikool,

W. Ostwaldi 1, EE-50411 Tartu, Estonia

2PRISMA Cluster of Excellence, Institut für Physik,

Johannes-Gutenberg-Universität,

Staudinger Weg 7, D-55099 Mainz, Germany

We shed new light on the standard current algebra approach to the nonleptonic
two-body decays of single and double heavy charm baryons. By making use of the
completeness relation for the flavor wave functions of the ground state baryon
20’ representation we are able to rewrite the results of the current algebra
approach in terms of the seven topological tensor invariants describing the
decays. The representation of the current algebra results in terms of
topological tensor invariants depends only on the initial and final state of
the process. The summation over intermediate states inherent to the current
algebra result is automatically taken care of in the tensor representation. In
this way one arrives at a new, quick and very compact assessment of the
results of the current algebra/pole model calculation. For example, one can
quickly identify the decays in which the p.v. $S$-wave amplitude is predicted
to vanish implying zero polarization asymmetries in these decays. We provide
tables of the values of the seven topological tensor invariants for all
Cabibbo favored and singly and doubly Cabibbo suppressed nonleptonic single
and double charm baryon decays. In total we treat 196 charm baryon decays. We
also discuss the charm preserving $\Delta C=0$ decays of single charm baryons
and the usual hyperon decays.

##  1 Introduction

The experimental landscape of charm baryon decays has considerably changed in
2015 through the detection of $\Lambda_{c}^{+}$ pair production by the BESIII
collaboration at the Beijing $e^{+}\,e^{-}$ collider BEPCII [1]. Not only was
it now possible to determine the absolute branching ratios of various
nonleptonic $\Lambda_{c}$ decays through tagging techniques (see also Ref.
[2]), but the BESIII collaboration could also determine the asymmetry
parameters of the nonleptonic two-body decays $\Lambda^{+}_{c}\to
pK_{S}^{0},\,\,\Lambda^{0}\pi^{+},\,\Sigma^{+}\pi^{0}$ and $\Sigma^{0}\pi^{+}$
[3] (see also the review [4]). The decay asymmetry parameter of the decay
$\Xi_{c}^{0}\to\Xi^{-}\pi^{+}$ was recently measured by the Belle
collaboration with the result $\alpha_{\Xi_{c}^{0}\to\Xi^{-}\pi^{+}}=-0.60\pm
0.04$ [5].

Both the absolute branching ratios and the asymmetry parameters provide
essential input for attempts to theoretically understand the nonleptonic two-
body decays of charm baryons. The first Cabibbo suppressed decay
$\Lambda_{c}\to p\,\phi$ was measured by the CLEO collaboration in 1995 [6],
followed by more precise measurements of this mode later on [7, 8]. At a later
stage the BESIII collaboration measured the singly Cabibbo suppressed decay
$\Lambda_{c}^{+}\to p\,\eta$ [9], later on made more precise in Ref. [10]
including an improved upper limit on the Cabibbo suppressed decay
$\Lambda_{c}^{+}\to p\,\pi^{0}$. By using double tag techniques, the BESIII
collaboration also identified the decay $\Lambda_{c}^{+}\to nK_{s}^{0}\pi^{+}$
involving a neutron [11] which raises the hope that final states including a
neutron can be measured in the future. A considerably larger data sample of
$\Lambda_{c}^{+}$ pairs is expected in the near future at BEPCII. The rate of
$\Lambda_{c}^{+}$ pairs is to be increased by a factor of 16 from a total of
about $10^{5}$ to $1.6\cdot 10^{6}$ $\Lambda_{c}^{+}$ pairs [12] which will
considerably enlarge the data sample of $\Lambda^{+}_{c}$ decays. At the same
time an approved energy upgrade will increase the energy of BEPCII to
$4.9\mathop{\rm GeV}\nolimits$, opening the possibility of $\Sigma_{c}$ pair
production. If the energy of the BEPCII could be further increased to a little
above $4.95\mathop{\rm GeV}\nolimits$ one could also detect the production of
$\Xi_{c}^{+}$ and $\Xi_{c}^{0}$ pairs.

Absolute values of the branching fractions of the $\Xi_{c}^{+}$ and
$\Xi_{c}^{0}$ have recently become available again from the BELLE
collaboration through tagging techniques in the decays
$B^{-}\to\bar{\Lambda}_{c}^{-}\,\Xi_{c}^{0}$ and
$B^{0}\to\bar{\Lambda}_{c}^{-}\,\Xi_{c}^{+}$ [13, 14]. Using similar
techniques one could also determine the absolute values of the branching
fractions of the decays of the $\Omega_{c}^{0}$ e.g. in the kinematically
accessible decay $B^{-}\to\Omega_{c}^{0}\,\overline{\Xi_{c}^{+}}$ [15]. This
would be quite welcome since the absolute branching ratios of the decays of
the $\Omega_{c}^{0}$ are experimentally unknown at present.

The LHCb collaboration has shown that it is now even possible to detect doubly
Cabibbo suppressed (DCS) nonleptonic charm baryon decays by their measurement
of the decay $\Xi_{c}^{+}\to p\phi$ [16]. Furthermore, the LHCb collaboration
recently provided results on a branching ratio measurement of the charm
conserving $\Delta C=0$ singly Cabibbo suppressed (SCS) decay
$\Xi_{c}^{0}\to\Lambda_{c}^{+}\,\pi^{-}$ [17]. Finally, the LHCb collaboration
has extended the scope of possible charm baryon decay measurements by
presenting results on the nonleptonic double charm baryon two-body decay
$\Xi^{++}_{cc}\to\Xi^{+}_{c}\,\pi^{+}$ [18].

It is apparent that the experimentalists have provided us with a rich sample
of results on exclusive charm baryons decays with the hope of much more to
come. This has led to an upsurge of interest in charm baryon decays in the
theoretical community documented by the many recent papers on this subject.

Basic to the current–current (more precisely coined current$\times$current)
quark model description of nonleptonic charm baryon two-body decays are the
five generic quark diagrams depicted in Fig. 1 which are also termed
topological diagrams. Diagrams Ia and Ib are usually referred to as
factorizable diagrams or tree diagrams while diagrams IIa, IIb and III are
referred to as nonfactorizable or $W$-exchange diagrams. In this paper we
concentrate on the structure of the $W$-exchange diagrams IIa, IIb and III
which we relate to the $s$\- and $u$-channel contributions of the current
algebra aproach.

IaIbIIaIIbIII

Figure 1: The topological diagrams contributing to the nonleptonic charm
baryon decays

The theoretical papers on exclusive two-body charm baryons decays roughly fall
into three classes which differ by their dynamical input. In the simplest case
one exploits SU(3) symmetry to relate the amplitudes of different charm baryon
decays [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. An open issue is whether
to apply SU(3) symmetry to the invariant or to the helicity amplitudes of
these decays. This will make a big difference for decays involving the $\eta$
and $\eta^{\prime}$ mesons. The SU(3) approach has been extended to include
SU(3) symmetry breaking effects [20, 27]. Even more insight can be gained by
incorporating the topological diagram approach [30, 31, 32, 33] into the SU(3)
analysis where the topological diagrams are characterized by the five diagrams
Fig. 1 described above. It is, however, not sufficient to simply associate a
given decay with its accompanying topological diagrams, but one has to
calculate the appropiate weight factors for each decay. The SU(3) fit in Ref.
[34] using topological diagrams must therefore be considered to be invalid.

In the constituent quark model approach one attempts to calculate the
nonleptonic transitions represented by Fig. 1 in terms of constituent quark
model transitions. While diagrams Ia, Ib and IIb are directly accessible to a
constituent quark model calculation [35, 36], diagrams IIa and III involve the
creation of a quark pair from the vacuum in addition to the constituent quark
model transitions. In the covariantized constituent quark model (CCQM)
calculations [30, 37] the quark pair creation from the vacuum is effectively
described by the ${}^{3}P_{0}$ model where the strength of the ${}^{3}P_{0}$
interaction is fixed by a fit to the data. Both of the above papers are based
on an approach developed much earlier by F. Hussain and P. Rotelli [38] and by
J. G. Körner and T. Gudehus [39] for studying nonleptonic hyperon decays. In
the covariant confined quark model calculations of Refs. [40, 41, 42] the five
diagrams in Fig. 1 are interpreted as multiloop Feynman diagrams with nonlocal
interactions describing the three- and four-point hadron-quark vertices.

Even more dynamics is added in the standard current algebra plus pole model
approach [21, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54]. This approach
has its roots in the description of nonleptonic hyperon decays developed in
the 1960’s [55, 56, 57, 58] which is based on current algebra and the soft
pion theorem. Even if the criterion of a soft pseudoscalar meson is not always
satisfied in the nonleptonic decays of single and double charm baryons (in
exception are the $\Delta C=0$ single charm baryon decays), the current
algebra approach to charm baryon decays provides a definite calculational
framework which can be tested against experiment. In a recent series of papers
Cheng and collaborators presented a detailed analysis of the nonleptonic
decays of single and double charm baryons using a variant of the current
algebra approach in which the purported current algebra contributions of the
topological diagram III in Fig. 1 are dropped [59, 60, 61, 62]. The authors of
Ref. [36] make use of the current algebra approach in addition to calculating
the diagram I and IIb transitions in the constituent quark model. This
possibly amounts to a double counting of the contribution from the
$W$-exchange diagram IIb.

Our paper is structured as follows. In Sec. 2 we write down the effective
Hamiltonian for the nonleptonic current–current transition. In Sec. 3 we
discuss issues of SU(4) and SU(3) relevant to the nonleptonic single and
double charm baryon decays. In particular, we introduce the 20 ground state
baryons comprising the ${\bf 20^{\prime}}$ representation of SU(4) and write
down the third rank flavor tensors that are associated with them. We present
the orthonormality and completeness relation of the third rank flavor tensors
of the ${\bf 20^{\prime}}$ representation. We introduce a minimal set of seven
topological tensor invariants in flavor space, built from the tensor
representations of the ground state baryons, the ground state pseudoscalar
mesons, and the effective Hamiltonian. The seven tensor invariants are each
associated with one of the corresponding topological quark diagrams.

In Sec. 4 we present tables of the values of the topological tensor invariants
for the antitriplet and sextet single charm baryon decays, the $\Delta C=0$
single charm baryon decays, the triplet double charm baryon decays to the
antitriplet and sextet single charm baryons, and the double charm baryon
decays to the light baryon octet and heavy $D,D_{s}$ mesons, each for the
Cabibbo favored (CF), singly Cabibbo suppressed (SCS) and doubly Cabibbo
suppressed (DCS) decays. Altogether we list the values of the topological
tensor invariants for 196 single and double nonleptonic charm baryon decays.
For each class of decays we explicate the linear relations among the
associated tensor invariants. For completeness we also list the values of the
tensor invariants for the ordinary nonleptonic hyperon decays.

In Sec. 5 we briefly recapitulate the current algebra plus pole model approach
to nonleptonic charm baryon decays where we closely follow the presentation
and notation of Refs. [59, 60, 61, 62]. Using the completeness relation for
the ground state baryons we explicitly derive the relation of the so-called
$s$-channel and $u$-channel contributions of the current algebra approach to
the topological tensor invariants. The representation of the current algebra
results in terms of topological tensor invariants depends only on the initial
and final state of the process. The summation over intermediate states
inherent to the current algebra result is automatically taken care of in the
tensor representation. In Sec. 6 we discuss several explicit examples, leading
to the exposure of general features of the topological tensor and current
algebra approaches in Sec. 7. Sec. 8 contains our summary and outlook, while
technical issues are treated in five Appendices.

In Appendix A we list explicit values of the tensor components of the ground
state baryons, the ground state pseudoscalar mesons, and the effective
Hamiltonian. Appendix B contains the derivation of the completeness relation
for the members of the ${\bf 20^{\prime}}$ representation. In Appendix C we
define a set of flavor tensor invariants describing the strong and weak
transitions $\langle B_{f}M|B_{i}\rangle$ and $\langle B_{f}|{\cal H}_{\rm
eff}|B_{i}\rangle$ involving the charm baryon states, the values of which are
needed for the comparison of the current algebra and the topological diagram
approaches. We provide tables of the values of the strong and weak tensor
invariants needed for the example cases discussed in the main text. In
Appendix D we clarify the implications and shortcomings of the variant of the
current algebra approach introduced in Refs. [59, 60, 61, 62]. In Appendix E
we are dealing with the spin kinematics.

##  2 The effective weak current–current Hamiltonian

The $\Delta C=1$ effective Hamiltonian for the Cabibbo favored (CF) decays is
given by

${\cal H}_{\rm eff}=\frac{G_{F}}{2\sqrt{2}}V_{cs}V^{*}_{ud}\left(c_{+}{\cal
O}_{+}+c_{-}{\cal O}_{-}\right)+H.c.$ (1)

where the four-quark operators read

${\cal O}_{\pm}=(\bar{s}c)(\bar{u}d)\pm(\bar{u}c)(\bar{s}d)$ (2)

with $(\bar{q}_{1}q_{2})=\bar{q}_{1}\gamma_{\mu}(1-\gamma_{5})q_{2}$.

The $\Delta C=1$ singly Cabibbo suppressed (SCS) decays are induced by two
distinct effective Hamiltonians which are labeled by $a$ and $b$. They read

$\displaystyle{\cal H}_{\rm eff}(a)$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{2\sqrt{2}}V_{cs}V^{*}_{us}\left(c_{+}{\cal
O}_{+}(a)+c_{-}{\cal O}_{-}(a)\right)+H.c.$ (3) $\displaystyle{\cal H}_{\rm
eff}(b)$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{2\sqrt{2}}V_{cd}V^{*}_{ud}\left(c_{+}{\cal
O}_{+}(b)+c_{-}{\cal O}_{-}(b)\right)+H.c.$ (4)

with ${\cal O}_{\pm}(a)=(\bar{s}c)(\bar{u}s)\pm(\bar{u}c)(\bar{s}s)$ and
${\cal O}_{\pm}(b)=(\bar{d}c)(\bar{u}d)\pm(\bar{u}c)(\bar{d}d)$, respectively.
Using e.g. the Wolfenstein parametrization one finds that
$V_{cs}V^{*}_{us}=-V_{cd}V^{*}_{ud}+{\cal O}(\lambda^{4})$ from unitarity,
i.e. to a very good approximation one has $V_{cs}V^{*}_{us}=-V_{cd}V^{*}_{ud}$
which we shall always use.

The $\Delta C=1$ doubly Cabibbo suppressed decays (DCS) are induced by the
effective Hamiltonian

${\cal H}_{\rm eff}(c)\ =\
\frac{G_{F}}{2\sqrt{2}}V_{cd}V^{*}_{us}\left(c_{+}{\cal O}_{+}(c)+c_{-}{\cal
O}_{-}(c)\right)+H.c.$ (5)

where ${\cal O}_{\pm}(c)=(\bar{d}c)(\bar{u}s)\pm(\bar{u}c)(\bar{d}s)$.

Finally, the $\Delta C=0$ charm baryon decays are induced by the two SCS
effective Hamiltonians

$\displaystyle{\cal H}_{\rm eff}(a^{\prime})$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{2\sqrt{2}}V_{us}V^{*}_{ud}\left(c_{+}{\cal
O}_{+}(a^{\prime})+c_{-}{\cal O}_{-}(a^{\prime})\right)+H.c.$ (6)
$\displaystyle{\cal H}_{\rm eff}(b^{\prime})$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{2\sqrt{2}}V_{cd}V^{*}_{cs}\left(c_{+}{\cal
O}_{+}(b^{\prime})+c_{-}{\cal O}_{-}(b^{\prime})\right)+H.c.$ (7)

with ${\cal O}_{\pm}(a^{\prime})=(\bar{u}s)(\bar{d}u)\pm(\bar{d}s)(\bar{u}u)$
and ${\cal O}_{\pm}(b^{\prime})=(\bar{d}c)(\bar{c}s)\pm(\bar{c}c)(\bar{d}s)$,
respectively. From unitarity one has $V_{cd}V^{*}_{cs}=-V_{us}V^{*}_{ud}$
which again is true up to ${\cal O}(\lambda^{4})$ in the Wolfenstein
parametrization.

In this paper we will mostly be concerned with the so-called $W$-exchange
contributions which, according to the Körner–Pati–Woo (KPW) theorem [63, 64],
are related to the operators ${\cal
O}_{-}=(\bar{q}_{1}q_{2})(\bar{q}_{3}q_{4})-(\bar{q}_{3}q_{2})(\bar{q}_{1}q_{4})$
only, i.e. induced by the effective Hamiltonian

${\cal H}_{\rm eff}({\cal
O}_{-})=\frac{G_{F}}{2\sqrt{2}}V^{\phantom{*}}_{q_{2}q_{1}}V^{*}_{q_{3}q_{4}}\left(c_{-}{\cal
O}_{-}\right)+H.c.$ (8)

Table 1:  Ground state charm baryon states with $J^{P}=1/2^{+}$. The square and curly brackets $[ab]$ and $\\{ab\\}$ denote antisymmetric and symmetric flavor label combinations. Notation | Quark | SU(3) | $(I,I_{3})$ | $S$ | $C$ |  Mass [MeV]
---|---|---|---|---|---|---
| content |  |  |  |  |
$\Lambda_{c}^{+}$ | $c[ud]$ | $\overline{\bf 3}$ | $(0,0)$ | $0$ | $1$ |  $2286.46\pm 0.14$
$\Xi_{c}^{+}$ | $c[su]$ | $\overline{\bf 3}$ | $(1/2,1/2)$ | $-1$ | $1$ |  $2467.95\pm 0.19$
$\Xi_{c}^{0}$ | $c[sd]$ | $\overline{\bf 3}$ | $(1/2,-1/2)$ | $-1$ | $1$ |  $2470.99\pm 0.40$
$\Sigma_{c}^{++}$ | $cuu$ | ${\bf 6}$ | $(1,1)$ | $0$ | $1$ |  $2453.97\pm 0.14$
$\Sigma_{c}^{+}$ | $\\{cud\\}$ | ${\bf 6}$ | $(1,0)$ | $0$ | $1$ |  $2452.9\pm 0.4$
$\Sigma_{c}^{0}$ | $cdd$ | ${\bf 6}$ | $(1,-1)$ | $0$ | $1$ |  $2453.75\pm 0.14$
$\Xi_{c}^{\prime+}$ | $c\\{su\\}$ | ${\bf 6}$ | $(1/2,1/2)$ | $-1$ | $1$ |  $2578.4\pm 0.5$
$\Xi_{c}^{\prime\,0}$ | $c\\{sd\\}$ | ${\bf 6}$ | $(1/2,-1/2)$ | $-1$ | $1$ |  $2579.2\pm 0.5$
$\Omega_{c}^{0}$ | $css$ | ${\bf 6}$ | $(0,0)$ | $-2$ | $1$ |  $2695.2\pm 1.7$
$\Xi_{cc}^{++}$ | $ccu$ | ${\bf 3}$ | $(1/2,1/2)$ | $0$ | $2$ |  $3621.2\pm 0.7$
$\Xi_{cc}^{+}$ | $ccd$ | ${\bf 3}$ | $(1/2,-1/2)$ | $0$ | $2$ |  $3621$
$\Omega_{cc}^{+}$ | $ccs$ | ${\bf 3}$ | $(0,0)$ | $-1$ | $2$ |  $3710$

##  3 SU(4), SU(3) and topological tensor invariants

The 20 ground state baryons with spin parity quantum numbers $J^{P}=1/2^{+}$
make up the 20’ representation of SU(4) associated with the Young tableaux

$\Young&\cr\cr.$

In the $C=0$ sector one has the usual light baryon octet comprised of the
eight light baryons ($p$, $n$, $\Lambda^{0}$, $\Sigma^{+}$, $\Sigma^{0}$,
$\Sigma^{-}$, $\Xi^{+}$, $\Xi^{0}$). The $C=1$ single charm sector contains
the antitriplet charm baryons ($\Lambda_{c}^{+}$, $\Xi_{c}^{+}$,
$\Xi_{c}^{0}$) and the sextet charm baryons ($\Sigma_{c}^{+}$,
$\Sigma_{c}^{0}$, $\Sigma_{c}^{-}$, $\Xi_{c}^{\prime+}$,
$\Xi_{c}^{\prime\,0}$, $\Omega_{c}^{0}$).

In HQET (or in the spectator quark model) the nine $C=1$ antitriplet and
sextet single charm baryons can be viewed as the 21 members of the ${\bf 21}$
representation of the approximate light spin–flavor symmetry group SU(6) with
the subgroup SU(2)$\times$SU(3). The decomposition reads

${\bf 21}\subset{\bf 1}\otimes\overline{\bf 3}\oplus{\bf 3}\otimes{\bf 6}$ (9)

where ${\bf 1}$ and ${\bf 3}$ denote the $J=0$ and $J=1$ representations of
spin SU(2) with dimensions $2J+1$. The SU(6) spin–flavor wave functions of the
antitriplet and sextet single charm baryons can be found e.g. in Ref. [60].

Finally, the $C=2$ double charm baryons are made up by the triplet of the
double heavy charm baryons ($\Xi_{cc}^{++}$, $\Xi_{cc}^{+}$,
$\Omega_{cc}^{+}$). In Table 1 we list the quark content, the quantum numbers
and, when available, the experimental mass values [65] of the 12 charm baryon
states. The masses of the double charm states $\Xi^{+}_{cc}$ and
$\Omega^{+}_{cc}$ have not been measured yet. For $\Xi^{+}_{cc}$ we assume
equality of the mass value with $\Xi^{++}_{cc}$, dropping, however, the fifth
digit and the error specification. For $\Omega^{+}_{cc}$ we list the mass
prediction of Ref. [37, 66] based on the one-gluon exchange model of de
Rujula, Georgi and Glashow which features a Breit–Fermi spin–spin interaction
term [67]. In as much as the model prediction (dated from 1992) for the mass
of the $\Xi_{cc}$ iso-doublet of $3621\mathop{\rm MeV}\nolimits$ is very close
to the measured value of $3621.2\pm 0.7\mathop{\rm MeV}\nolimits$ [65], we
feel quite confident about the predicted mass of the $\Omega_{cc}^{+}$ listed
in Table 1 based on the same model calculation [37, 66].

In the charm sector the single charm baryons $\Sigma_{c}^{++,+,0}$ decay
dominantly via one-pion emission while the dominant decay modes of the
$\Xi_{c}^{\prime+}$ and $\Xi_{c}^{\prime\,0}$ are the one-photon emission
modes. The remaining seven single and double charm baryons ($\Lambda_{c}^{+}$,
$\Xi_{c}^{+}$, $\Xi_{c}^{0}$, $\Omega_{c}^{0}$, $\Xi_{cc}^{++}$,
$\Xi_{cc}^{+}$, $\Omega_{cc}^{+}$) decay via weak interactions. An important
class of these weak charm baryon decays are their nonleptonic two-body decays
into a ground state baryon with $J^{P}=1/2^{+}$ and a pseudoscalar meson with
$J^{P}=0^{-}$. The two-body nonleptonic decays of the seven single and double
charm baryons ($\Lambda_{c}^{+}$, $\Xi_{c}^{+}$, $\Xi_{c}^{0}$,
$\Omega_{c}^{0}$, $\Xi_{cc}^{++}$, $\Xi_{cc}^{+}$, $\Omega_{cc}^{+}$) are the
subject of this paper.

In this paper we will not be concerned with the 20 $C=0,1,2,3$ $J^{P}=3/2^{+}$
ground state baryons which, in SU(4), belong to the ${\bf 20}$ representation
with the associated Young tableaux

$\Young&\cr\,\,.$

The properties of the ${\bf 20}$ representation are discussed in Ref. [66]. In
principle, the states of the ${\bf 20}$ representation could contribute as
intermediate states in the current algebra plus pole model approach discussed
in Sec. 5 but are prevented from contributing as intermediate states by the
KPW theorem [63, 64] which states that the contraction of the flavor
antisymmetric current–current operator with a flavor symmetric final state
configuration is zero in the SU(3) limit.

It is quite illuminating to represent the baryon flavor wave functions as
third rank tensors $B_{abc}$ in flavor space instead of the second rank
tensors $B_{a}^{b}$ frequently used in the literature. In this way one can
keep track of the quark content of the baryons. Moreover, in the transitions
involving baryons one can identify the flavor flow in the transitions which
directly leads to the concept of topological diagrams and the topological
tensor invariants associated with them. Explicit representations of the third
rank flavor tensors of the twenty $J^{P}=1/2^{+}$ ground state baryons are
given in Appendix A. For the ground state baryons and mesons we have used the
phase conventions of Lichtenberg [68].

The flavor wave functions of the $J^{P}=1/2^{+}$ ground state baryons satisfy
a Jacobi-type identity which reads

$B^{\ell}_{a[bc]}+B^{\ell}_{b[ca]}+B^{\ell}_{c[ab]}=0\,.$ (10)

for each of the flavor wave functions of the ground state multiplet, where
$\ell=1,\ldots,8$ in SU(3) and $\ell=1,\ldots,20$ in SU(4). They are
normalized and orthogonal according to

$\sum_{k,m,n}B_{\ell}^{k[mn]}B^{\ell^{\prime}}_{k[mn]}=\delta^{\ell^{\prime}}_{\ell}.$
(11)

Furthermore, they satisfy the completeness relation

$\sum_{\ell}B^{\ell}_{k[mn]}B_{\ell}^{b[cd]}=\frac{2}{6}(\delta_{k}^{b}\delta_{m}^{c}\delta_{n}^{d}-\delta_{k}^{b}\delta_{m}^{d}\delta_{n}^{c})-\frac{1}{6}(\delta_{m}^{b}\delta_{n}^{c}\delta_{k}^{d}-\delta_{m}^{b}\delta_{n}^{d}\delta_{k}^{c})-\frac{1}{6}(\delta_{n}^{b}\delta_{k}^{c}\delta_{m}^{d}-\delta_{n}^{b}\delta_{k}^{d}\delta_{m}^{c}).$
(12)

Contracting the completeness relation (12) with
$\delta^{k}_{b}\,\delta^{m}_{c}\,\delta^{n}_{d}$ one obtains the dimension
$d=N(N^{2}-1)/3$ of the ground state representation in SU($N$) which agrees
with the dimension of the representation calculated according to the hook rule
with the result

${\rm dim}\left(\kern-9.0pt\raise-12.0pt\hbox{\Young
&\cr\cr}\right)=\frac{N(N+1)(N-1)}{3\cdot 1\cdot 1}=\frac{1}{3}N(N^{2}-1).$
(13)

One thus obtains $d=8$ and $d=20$ for SU(3) and SU(4), respectively, as
expected.

The completeness relation (12) is central to the chain of reasoning in Sec. 5
which, in the limit of SU(3) and the absence of hyperfine interactions, links
the results of the current algebra approach to a linear superposition of
topological reduced matrix elements with coefficients given by the topological
tensor invariants. For the interested reader we provide an explicit proof of
the completeness relation in Appendix B. Using the completeness relation we
never invoke SU(4) symmetry but rather make use of the SU(3) symmetry of the
separate $C=0$, $C=1$ and $C=2$ sectors of the completeness relation (12). For
a given flavor configuration defined by the in and out states, the sum over
intermediate states $\ell$ in Eq. (12) extends either over one of the ground
state baryons or, in the case of the flavor degenerate pairs of states
$(\Lambda^{0},\Sigma^{0})$, $(\Lambda_{c}^{+},\Sigma_{c}^{+})$,
$(\Xi_{c}^{0},\Xi^{\prime\,0}_{c})$ and $(\Xi_{c}^{+},\Xi^{\prime+}_{c})$,
over maximally two ground state baryons. In most of the cases analyzed in this
paper, due to symmetry and constituent quark model considerations only one
state of the flavor degenerate pair of states contribute as intermediate
state. When both flavor degenerate states contribute to a given intermediate
state we assume mass degeneracy of the two states to sum the two
contributions. More details can be found in Sec. 5.

In flavor space the ground state to ground state transitions
$1/2^{+}\rightarrow 1/2^{+}+0^{-}(1^{-})$ induced by the antisymmetric flavor-
changing tensor $H^{[q_{2}q_{4}]}_{[q_{1}q_{3}]}$ representing the effective
Hamiltonian ${\cal H}_{\rm eff}({\cal O}_{-})$ in Eq. (8) are represented by
the seven topological tensor invariants

$\displaystyle
I^{-}_{1}(\ell,\ell^{\prime})=B_{\ell}^{a[bc]}B^{\ell^{\prime}}_{a[bc^{\prime}]}M^{d}_{d^{\prime}}H^{[c^{\prime}d^{\prime}]}_{[cd]}$
$\displaystyle
I^{-}_{2}(\ell,\ell^{\prime})=B_{\ell}^{a[bc]}B^{\ell^{\prime}}_{b[c^{\prime}a]}M^{d}_{d^{\prime}}H^{[c^{\prime}d^{\prime}]}_{[cd]}$
$\displaystyle
I_{3}(\ell,\ell^{\prime})=B_{\ell}^{a[bc]}B^{\ell^{\prime}}_{a[b^{\prime}c^{\prime}]}M^{d}_{c}H^{[c^{\prime}b^{\prime}]}_{[db]}$
$\displaystyle
I_{4}(\ell,\ell^{\prime})=B_{\ell}^{b[ca]}B^{\ell^{\prime}}_{a[b^{\prime}c^{\prime}]}M^{d}_{c}H^{[c^{\prime}b^{\prime}]}_{[db]}$
$\displaystyle\hat{I}_{3}(\ell,\ell^{\prime})=B_{\ell}^{a[bc]}B^{\ell^{\prime}}_{a[b^{\prime}c^{\prime}]}M^{c^{\prime}}_{d}H^{[db^{\prime}]}_{[cb]}$
$\displaystyle\hat{I}_{4}(\ell,\ell^{\prime})=B_{\ell}^{a[bc]}B^{\ell^{\prime}}_{b^{\prime}[c^{\prime}a]}M^{c^{\prime}}_{d}H^{[db^{\prime}]}_{[cb]}$
$\displaystyle
I_{5}(\ell,\ell^{\prime})=B_{\ell}^{a[bc]}B^{\ell^{\prime}}_{a^{\prime}[b^{\prime}c^{\prime}]}M^{c^{\prime}}_{c}H^{[a^{\prime}b^{\prime}]}_{[ab]}$
(14)

for the baryonic transitions $\ell^{\prime}\to\ell$, where the fourth rank
flavor changing tensor $H^{[q_{2}q_{4}]}_{[q_{1}q_{3}]}$ takes the values $\pm
1$ as specified in Appendix A. A summation over doubly occurring indices in
Eq. (3) is implied. In the present application in which we consider various
SU(3) subsector transitions, the summation runs over the three light quarks
($u=1,d=2,s=3$) and the charm quark which we label as $c=4$. After having used
the Jacobi identity (10) and/or by rearranging tensor labels including the
exchange of dummy summation indices, the set of seven tensor invariants
comprise the minimal set of possible tensor contractions. For example, the
tensor invariant
$I_{5}^{\prime}(\ell,\ell^{\prime})=B_{\ell}^{a[bc]}B^{\ell^{\prime}}_{b^{\prime}[c^{\prime}a^{\prime}]}M^{c^{\prime}}_{c}H^{[a^{\prime}b^{\prime}]}_{[ab]}$
is redundant since it can be seen to be equal to $I_{5}(\ell^{\prime},\ell)$
by simply rearranging the tensor labels. The list (3) contains only connected
tensor contractions. We have thus not included the tadpole-type unconnected
tensor contraction
$B_{\ell}^{a[bc]}B^{\ell^{\prime}}_{a[bc]}M^{d}_{d^{\prime}}H^{[cd^{\prime}]}_{[cd]}$
which vanishes for the $\Delta C=1$ transitions since
$H^{[cd^{\prime}]}_{[cd]}=0$ (see Appendix A).

Table 2: SU(3) and SU(2) properties of the effective $\Delta C=1$ and $\Delta C=0$ Hamiltonian ${\cal H}_{\rm eff}({\cal O}_{-})$ in Eq. (8), represented in flavor space by the fourth rank tensor $H^{[q_{2}q_{4}]}_{[q_{1}q_{3}]}$ |  | $Y$ | $I_{3}$ | $SU(3)$ | $\Delta I$ | $\Delta U$ | $\Delta U_{3}$ | $\Delta V$
---|---|---|---|---|---|---|---|---
CF | $H^{[su]}_{[cd]}$ | $2/3$ | $-1$ | ${\bf 6}$ | $1$ | $1$ | $1$ | $0$
SCS | $H^{[su]}_{[cs]}$ | $-1/3$ | $-1/2$ | ${\bf 6}\oplus\overline{\bf 3}$ | $1/2$ | $1,0$ | $0,\,0$ | $1/2$
| $H^{[du]}_{[cd]}$ | $-1/3$ | $-1/2$ | ${\bf 6}\oplus\overline{\bf 3}$ | $1/2$ | $1,0$ | $0,0$ | $1/2$
| $H^{[su]}_{[cs]}-H^{[du]}_{[cd]}$ | $-1/3$ | $-1/2$ | ${\bf 6}$ | $1/2$ | $1$ | $0$ | $1/2$
DCS | $H^{[du]}_{[cs]}$ | $-4/3$ | $0$ | ${\bf 6}$ | $0$ | $1$ | $-1$ | $1$
$\Delta C=0$ | $H^{[ud]}_{[su]}$ | $-1$ | $1/2$ | ${\bf 8}$ | $1/2$ | $1$ | $-1$ | $1/2$
| $H^{[dc]}_{[cs]}$ | $-1$ | $1/2$ | ${\bf 8}$ | $1/2$ | $1$ | $-1$ | $1/2$

In Table 2 the SU(3) and SU(2) properties of the fourth rank flavor changing
tensor are listed. Note that each of the tensors $H^{[su]}_{[cs]}$ and
$H^{[du]}_{[cd]}$ inducing the SCS transition transforms separately as ${\bf
6}\oplus\overline{\bf 3}$ in SU(3). Only their difference
$H^{[su]}_{[cs]}-H^{[du]}_{[cd]}$ transforms as the ${\bf 6}$ representation
in SU(3) as can be explicitly verified by using the Clebsch–Gordan tables of
Kaeding [71]. In Fig. 2 we display the weight diagrams of the ${\bf 6}$ and
$\overline{\bf 3}$ representations where we have marked the locations of the
CF, SCSa, SCSb and DCS effective quark transitions in the two weight diagrams.
When taking the difference of the two SCS contributions one remains with a
$U$-spin triplet at the left boundary of the ${\bf 6}$ weight diagram. We have
somewhat dwelt on this point because we have not found an adequate discussion
concerning this cancellation in the literature.

Figure 2: Weight diagrams of the sextet (left) and antitriplet (right)
representation of the effective weak Hamiltonian. The locations of the CF, SCS
and DCS transitions are marked by a circle dot symbol $\odot$.

We mention that the reduction to a minimal set of tensor invariants has not
been attained in Refs. [31, 33]. As an example we take the Cabibbo enhanced
decays of the antitriplet charm baryons. The authors of Ref. [33] introduce
six and three topological tensor invariants each for the topology classes IIa
and III where only two and one are needed ($I_{3}$, $I_{4}$ and $I_{5}$,
resp.) while Kohara [31] defines four and one topological tensor invariants
for the same topology classes IIa and III where, as before, only the two
invariants $I_{3}$, $I_{4}$ and the invariant $I_{5}$ are needed. This makes
it quite cumbersome to compare mutual results. Note that the distinction of
whether the strange quark from the charm quark transition ends up in the meson
or the baryon for the topology class IIa, as done in Ref. [33], is not
warranted in SU(3).

Let us briefly pause to discuss the charge conjugation properties of the seven
tensor invariants (3). In flavor space the charge conjugation operation raises
or lowers the tensor indices, i.e. one has
$\bar{B}_{\ell}^{a[bc]}=B^{\ell}_{a[bc]}$,
$\bar{B}^{\ell}_{a[bc]}=B_{\ell}^{a[bc]}$, $\bar{M}^{b}_{a}=M^{a}_{b}$, and
$\bar{H}^{[cd]}_{[ab]}=H^{[ab]}_{[cd]}$. As an example we consider the charge
conjugation operation on the tensor invariant $I_{3}$. One has

$\bar{I}_{3}(\ell,\ell^{\prime})=B^{\ell}_{a[bc]}B_{\ell^{\prime}}^{a[b^{\prime}c^{\prime}]}M^{c}_{d}H^{[db]}_{[c^{\prime}b^{\prime}]}=\hat{I}_{3}(\ell^{\prime},\ell).$
(15)

Similarly one has
$\bar{I}_{4}(\ell,\ell^{\prime})=\hat{I}_{4}(\ell^{\prime},\ell)$,
$\bar{I}^{-}_{1,2}(\ell,\ell^{\prime})=I^{-}_{1,2}(\ell^{\prime},\ell)$, and
$\bar{I}_{5}(\ell,\ell^{\prime})=I_{5}(\ell^{\prime},\ell)$. Therefore, there
are two antisymmetric and five symmetric tensor combinations under the flavor
charge conjugation operation. The two antisymmetric combinations are

$(\bar{I}_{3}-\bar{\hat{I}}_{3})(\ell,\ell^{\prime})=-(I_{3}-\hat{I}_{3})(\ell^{\prime},\ell),\qquad(\bar{I}_{4}-\bar{\hat{I}}_{4})(\ell,\ell^{\prime})=-(I_{4}-\hat{I}_{4})(\ell^{\prime},\ell).$
(16)

They would contribute to the parity violating (p.v.) amplitude $A$ of Sec. 5.
The five symmetric combinations read

$\displaystyle(\bar{I}_{3}+\bar{\hat{I}}_{3})(\ell,\ell^{\prime})=+(I_{3}+\hat{I}_{3})(\ell^{\prime},\ell)\qquad\quad(\bar{I}_{4}+\bar{\hat{I}}_{4})(\ell,\ell^{\prime})=+(I_{4}+\hat{I}_{4})(\ell^{\prime},\ell)$
$\displaystyle\quad\bar{I}^{-}_{1}(\ell,\ell^{\prime})=I^{-}_{1}(\ell^{\prime},\ell)\qquad\bar{I}^{-}_{2}(\ell,\ell^{\prime})=I^{-}_{2}(\ell^{\prime},\ell)\qquad\bar{I}_{5}(\ell,\ell^{\prime})=I_{5}(\ell^{\prime},\ell).$
(17)

They would contribute to the parity conserving (p.c.) amplitude $B$ of Sec. 5.
Note, however, that the dynamics of the tree diagram contributions leads to an
extra factor of $(m_{1}-m_{2})$ in the case of the p.v. amplitude $A$ which
implies that the tree diagram invariants also contribute to the amplitude $A$.

Returning to the topological analysis one can associate each of the tensor
invariants (3) with one of the topological diagrams in Fig. 1 by following the
flavor flow in the diagrams. It is important to realize that it is not
sufficient to just associate a given decay with a given set of topological
diagrams. Instead, one needs to calculate the projection of that given decay
onto the topological diagrams in terms of the topological tensor invariants.
The association reads

$\displaystyle(I^{-}_{1},I^{-}_{2})\,\,\longleftrightarrow\,\,\mbox{diagrams
Ia, Ib}\qquad\qquad(I_{3},I_{4})\,\,\longleftrightarrow\,\,\mbox{diagram IIa}$
(18)
$\displaystyle(\hat{I}_{3},\hat{I}_{4})\,\,\longleftrightarrow\,\,\mbox{diagram
IIb}\qquad\qquad\qquad I_{5}\,\,\longleftrightarrow\,\,\mbox{diagram III}$
(19)

Note that this relation is not one-to-one. For example, from the flavor flow
the decay $\Lambda_{c}^{+}\to\Xi^{0}K^{+}$ can be seen to be contributed to by
the topology IIa. However, from an explicit calculation one finds that only
the topological tensor invariant $I_{4}$ becomes populated. This has important
ramifications for the structure of the decay as will be discussed in Sec. 4.

To conclude, in flavor space a general amplitude ${\cal A}_{fki}$ describing
the decay $B_{i}\to B_{f}+M_{k}$ can be expanded along seven topological
reduced amplitudes $\mathbfcal{T}_{j}$ with coefficients given by the seven
topological tensor invariants $I^{-}_{1},\ldots,I_{5}$. One has

${\cal A}(B_{i}\stackrel{{\scriptstyle H}}{{\longrightarrow}}B_{f}M_{k})={\cal
A}_{fki}=\sum_{j}I^{j}_{fki}\,\mathbfcal{T}_{j}\qquad
j=1^{-},2^{-},\,3,\,4,\,\hat{3},\,\hat{4},\,5$ (20)

where the $B_{i}$ and $B_{f}$ belong to some given SU(3) representation. The
$\mathbfcal{T}_{j}$ are the topological invariant amplitudes of the process
while the tensor invariants $I^{j}_{fki}$ project onto these invariant
amplitudes. Put in a different language, the topological tensor invariants
$I^{j}_{fki}$ would correspond to SU(3) Clebsch–Gordan coefficients while the
reduced matrix elements $\mathbfcal{T}_{j}$ would correspond to the SU(3)
invariant amplitudes. All SU(3) relations between given transition amplitudes
including the subclasses of $I$, $U$ and $V$ sum rules are implicit in the
expansion (20).

In the general case one is overcounting the number of tensor invariants in the
expansion (20), i.e. the number of significant SU(3) invariants can generally
be less than the number of seven SU(3) invariants in Eq. (20). In general, the
rank of the coefficient matrix $I^{j}_{fki}$ linking ${\cal A}_{fki}$ with
$\mathbfcal{T}_{j}$ is less than seven. This implies that one cannot, in
general, determine the reduced topological amplitudes $\mathbfcal{T}_{j}$ from
a given set of experimentally measured amplitudes ${\cal A}_{fki}$. In such a
case there will be a number of linear relations among the seven tensor
invariants $I^{j}_{fki}$ which will be commented on and written down for the
different SU(3) transitions treated in this paper. Some of the linear
relations involve tensor invariants of the same topology class. In such a case
the linear relations can be obtained by combining the information on the
symmetry or antisymmetry of the ($u,d,s$) light quark components of a given
class of charm baryons with the use of the Jacobi identity. The remaining
linear relations between tensor invariants involving different topology
classes cannot be obtained by mere tensor manipulations. We conjecture that
one needs Schouten type identities to derive these additional tensor
identities. In this paper we use linear algebra methods to construct these
additional linear relations explicitly. With the help of the linear relations
between tensor invariants one can recombine the topological invariant
amplitudes to a minimal set of significant (nontrivial) topological invariant
amplitudes. However, it should be clear that this set of minimal reduced
amplitudes is not unique.

The flavor space tensor contractions represent a convenient way of calculating
Clebsch–Gordon coefficients entering in nonleptonic charm baryon decays. They
were first introduced in Ref. [30] which also contains some supplementary
background material. The values of the topological tensor invariants for all
Cabibbo favored (CF), singly Cabibbo suppressed (SCS) and doubly Cabibbo
suppressed (DCS) nonleptonic charm baryon decays needed in this paper are for
instance given in Tables 3, 4, 5, 6 and 7.

The transition to the isoscalar states are written in terms of the ideally
mixed states $\eta_{\omega}=\frac{1}{\sqrt{2}}(u\bar{u}+d\bar{d})$ and
$\eta_{\phi}=s\bar{s}$. For each decay we have factored out the products of
the denominator factors appearing in the flavor space wave functions, the
inverse of which appear as factors multiplying the amplitude of the respective
decays.

##  4 Tables of the topological tensor invariants

In this section we proceed to discuss the various classes of charm baryon
decays. They are classified according to the SU(3) transitions that specify
the decays. Even though we do not discuss the factorizable contributions, we
also list the values of the tensor invariants $I^{-}_{1}$ and $I^{-}_{2}$
since they enter in the linear relations between the seven ${\cal O}^{-}$
induced tensor invariants. The corresponding tensor invariants $I^{+}_{1}$ and
$I^{+}_{2}$ induced by the operator ${\cal O}^{+}$ can be easily obtained from
$I^{-}_{1}$ and $I^{-}_{2}$ as described in Appendix A.

For the transitions involving the isospin zero neutral meson states we
separately list results for the octet state
$\eta_{8}=(u\bar{u}+d\bar{d}-2s\bar{s})/\sqrt{6}$ and the singlet state
$\eta_{1}=(u\bar{u}+d\bar{d}+s\bar{s})/\sqrt{3}$. The physical $\eta$ and
$\eta^{\prime}$ states are linear superpositions of these two states according
to

$\eta=\cos\theta\eta_{8}-\sin\theta\eta_{1}\qquad\eta^{\prime}=\sin\theta\eta_{8}+\cos\theta\eta_{1}$
(21)

with $\theta=-15.4^{\circ}$ [69, 70]. We also list the transitions into the
ideally mixed states $\eta_{\omega}=(u\bar{u}+d\bar{d})/\sqrt{2}$ and
$\eta_{\phi}=s\bar{s}$ with obvious applications to the corresponding
$1/2^{+}\to 1/2^{+}\,+1^{-}$ transitions involving the ideally mixed vector
states $\omega$ and $\phi$.

Table 3: Values of the seven topological tensor invariants for the antitriplet charm baryon decays $B_{c}(\overline{\bf 3})\to B({\bf 8})+M({\bf 8},{\bf 1})$ induced by the CF flavor transitions $(c\to s;\,d\to u)$. We have always factored out the products of the normalizing denominator factors appearing in the flavor space quark model wave functions. The product of the denominator factors appear as overall factors of the process specification in column 2. |  |  |  | $I_{1}^{-}$ | $I^{-}_{2}$ | $I_{3}$ | $I_{4}$ | $\hat{I}_{3}$ | $\hat{I}_{4}$ | $I_{5}$
---|---|---|---|---|---|---|---|---|---|---
CF | $12\,\Lambda_{c}^{+}$ | $\to$ | $\Lambda^{0}\pi^{+}$ | $-2$ | $-2$ | $-2$ | $+4$ | $-2$ | $+4$ | $+1$
| $4\sqrt{3}\,\Lambda_{c}^{+}$ | $\to$ | $\Sigma^{0}\pi^{+}$ | $0$ | $0$ | $+2$ | $0$ | $-2$ | $+4$ | $+1$
| $4\sqrt{3}\,\Lambda_{c}^{+}$ | $\to$ | $\Sigma^{+}\pi^{0}$ | $0$ | $0$ | $-2$ | $0$ | $+2$ | $-4$ | $-1$
| $4\sqrt{3}\,\Lambda_{c}^{+}$ | $\to$ | $\Sigma^{+}\eta_{\omega}$ | $0$ | $0$ | $-2$ | $0$ | $-2$ | $+4$ | $-1$
| $2\sqrt{6}\Lambda_{c}^{+}$ | $\to$ | $\Sigma^{+}\eta_{\phi}$ | $0$ | $0$ | $-2$ | $+2$ | $0$ | $0$ | $0$
| $12\Lambda_{c}^{+}$ | $\to$ | $\Sigma^{+}\eta_{8}$ | $0$ | $0$ | $+2$ | $-4$ | $-2$ | $+4$ | $-1$
| $6\sqrt{2}\Lambda_{c}^{+}$ | $\to$ | $\Sigma^{+}\eta_{1}$ | $0$ | $0$ | $-4$ | $+2$ | $-2$ | $+4$ | $-1$
| $2\sqrt{6}\,\Lambda_{c}^{+}$ | $\to$ | $p\bar{K}^{0}$ | $+1$ | $+1$ | $+2$ | $-2$ | $0$ | $0$ | $0$
| $2\sqrt{6}\,\Lambda_{c}^{+}$ | $\to$ | $\Xi^{0}K^{+}$ | $0$ | $0$ | $0$ | $-2$ | $0$ | $0$ | $-1$
| $2\sqrt{6}\,\Xi_{c}^{+}$ | $\to$ | $\Sigma^{+}\bar{K}^{0}$ | $+1$ | $+1$ | $0$ | $0$ | $+2$ | $-4$ | $0$
| $2\sqrt{6}\,\Xi_{c}^{+}$ | $\to$ | $\Xi^{0}\pi^{+}$ | $-1$ | $-1$ | $0$ | $0$ | $-2$ | $+4$ | $0$
| $12\,\Xi_{c}^{0}$ | $\to$ | $\Lambda^{0}\bar{K}^{0}$ | $-1$ | $-1$ | $-4$ | $+2$ | $+2$ | $-4$ | $-1$
| $4\sqrt{3}\,\Xi_{c}^{0}$ | $\to$ | $\Sigma^{0}\bar{K}^{0}$ | $+1$ | $+1$ | $0$ | $-2$ | $+2$ | $-4$ | $-1$
| $2\sqrt{6}\,\Xi_{c}^{0}$ | $\to$ | $\Sigma^{+}K^{-}$ | $0$ | $0$ | $0$ | $+2$ | $0$ | $0$ | $+1$
| $4\sqrt{3}\,\Xi_{c}^{0}$ | $\to$ | $\Xi^{0}\pi^{0}$ | $0$ | $0$ | $+2$ | $-2$ | $-2$ | $+4$ | $0$
| $4\sqrt{3}\,\Xi_{c}^{0}$ | $\to$ | $\Xi^{0}\eta_{\omega}$ | $0$ | $0$ | $+2$ | $-2$ | $+2$ | $-4$ | $0$
| $2\sqrt{6}\,\Xi_{c}^{0}$ | $\to$ | $\Xi^{0}\eta_{\phi}$ | $0$ | $0$ | $+2$ | $0$ | $0$ | $0$ | $+1$
| $12\,\Xi_{c}^{0}$ | $\to$ | $\Xi^{0}\eta_{8}$ | $0$ | $0$ | $-2$ | $-2$ | $+2$ | $-4$ | $-2$
| $6\sqrt{2}\,\Xi_{c}^{0}$ | $\to$ | $\Xi^{0}\eta_{1}$ | $0$ | $0$ | $+4$ | $-2$ | $+2$ | $-4$ | $+1$
| $2\sqrt{6}\,\Xi_{c}^{0}$ | $\to$ | $\Xi^{-}\pi^{+}$ | $-1$ | $-1$ | $-2$ | $+2$ | $0$ | $0$ | $0$
Table 4:  Values of the seven topological tensor invariants for the SCS antitriplet charm baryon decays $B_{c}(\overline{\bf 3})\to B({\bf 8})+M({\bf 8},{\bf 1})$ induced by the flavor transition $(c\to s;\,s\to u)$. |  |  |  | $I^{-}_{1}$ | $I^{-}_{2}$ | $I_{3}$ | $I_{4}$ | $\hat{I_{3}}$ | $\hat{I}_{4}$ | $I_{5}$
---|---|---|---|---|---|---|---|---|---|---
SCSa | $12\,\Lambda^{+}_{c}$ | $\to$ | $\Lambda^{0}K^{+}$ | $+2$ | $+2$ | $0$ | $0$ | $+2$ | $-4$ | $0$
| $4{\sqrt{3}}\,\Lambda^{+}_{c}$ | $\to$ | $\Sigma^{0}K^{+}$ | $0$ | $0$ | $0$ | $0$ | $+2$ | $-4$ | $0$
| $2\sqrt{6}\,\Lambda^{+}_{c}$ | $\to$ | $\Sigma^{+}K^{0}$ | $0$ | $0$ | $0$ | $0$ | $-2$ | $+4$ | $0$
| $2\sqrt{6}\,\Lambda^{+}_{c}$ | $\to$ | $n\pi^{+}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $4{\sqrt{3}}\,\Lambda^{+}_{c}$ | $\to$ | $p\pi^{0}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $4{\sqrt{3}}\,\Lambda^{+}_{c}$ | $\to$ | $p\eta_{\omega}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\sqrt{6}\,\Lambda^{+}_{c}$ | $\to$ | $p\eta_{\phi}$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$
| $12\,\Lambda^{+}_{c}$ | $\to$ | $p\eta_{8}$ | $+2$ | $+2$ | $0$ | $0$ | $0$ | $0$ | $0$
| $6\sqrt{2}\,\Lambda^{+}_{c}$ | $\to$ | $p\eta_{1}$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\sqrt{6}\,\Xi^{+}_{c}$ | $\to$ | $\Xi^{0}K^{+}$ | $+1$ | $+1$ | $0$ | $-2$ | $+2$ | $-4$ | $-1$
| $4{\sqrt{3}}\,\Xi^{+}_{c}$ | $\to$ | $\Sigma^{+}\pi^{0}$ | $0$ | $0$ | $-2$ | $0$ | $0$ | $0$ | $-1$
| $4{\sqrt{3}}\,\Xi^{+}_{c}$ | $\to$ | $\Sigma^{+}\eta_{\omega}$ | $0$ | $0$ | $-2$ | $0$ | $0$ | $0$ | $-1$
| $2{\sqrt{6}}\,\Xi^{+}_{c}$ | $\to$ | $\Sigma^{+}\eta_{\phi}$ | $-1$ | $-1$ | $-2$ | $+2$ | $-2$ | $+4$ | $0$
| $12\,\Xi^{+}_{c}$ | $\to$ | $\Sigma^{+}\eta_{8}$ | $+2$ | $+2$ | $+2$ | $-4$ | $+4$ | $-8$ | $-1$
| $6{\sqrt{2}}\,\Xi^{+}_{c}$ | $\to$ | $\Sigma^{+}\eta_{1}$ | $-1$ | $-1$ | $-4$ | $+2$ | $-2$ | $+4$ | $-1$
| $2\sqrt{6}\,\Xi^{+}_{c}$ | $\to$ | $p\bar{K}^{0}$ | $0$ | $0$ | $+2$ | $-2$ | $0$ | $0$ | $0$
| $12\,\Xi^{+}_{c}$ | $\to$ | $\Lambda^{0}\pi^{+}$ | $0$ | $0$ | $-2$ | $+4$ | $0$ | $0$ | $+1$
| $4{\sqrt{3}}\,\Xi^{+}_{c}$ | $\to$ | $\Sigma^{0}\pi^{+}$ | $0$ | $0$ | $+2$ | $0$ | $0$ | $0$ | $+1$
| $2\sqrt{6}\,\Xi^{0}_{c}$ | $\to$ | $\Xi^{0}{K}^{0}$ | $0$ | $0$ | $0$ | $0$ | $+2$ | $-4$ | $-1$
| $2\sqrt{6}\,\Xi^{0}_{c}$ | $\to$ | $\Xi^{-}K^{+}$ | $+1$ | $+1$ | $0$ | $-2$ | $0$ | $0$ | $0$
| $2\sqrt{6}\,\Xi^{0}_{c}$ | $\to$ | $\Sigma^{+}\pi^{-}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $-1$
| $2\sqrt{6}\,\Xi^{0}_{c}$ | $\to$ | $\Sigma^{-}\pi^{+}$ | $0$ | $0$ | $+2$ | $0$ | $0$ | $0$ | $0$
| $4{\sqrt{6}}\,\Xi^{0}_{c}$ | $\to$ | $\Sigma^{0}\pi^{0}$ | $0$ | $0$ | $-2$ | $0$ | $0$ | $0$ | $+1$
| $4{\sqrt{6}}\,\Xi^{0}_{c}$ | $\to$ | $\Sigma^{0}\eta_{\omega}$ | $0$ | $0$ | $-2$ | $0$ | $0$ | $0$ | $-1$
| $4{\sqrt{3}}\,\Xi^{0}_{c}$ | $\to$ | $\Sigma^{0}\eta_{\phi}$ | $-1$ | $-1$ | $-2$ | $+2$ | $-2$ | $+4$ | $0$
| $12{\sqrt{2}}\,\Xi^{0}_{c}$ | $\to$ | $\Sigma^{0}\eta_{8}$ | $+2$ | $+2$ | $+2$ | $-4$ | $+4$ | $-8$ | $-1$
| $12\,\Xi^{0}_{c}$ | $\to$ | $\Sigma^{0}\eta_{1}$ | $-1$ | $-1$ | $-4$ | $+2$ | $-2$ | $+4$ | $-1$
| $12{\sqrt{2}}\,\Xi^{0}_{c}$ | $\to$ | $\Lambda^{0}\pi^{0}$ | $0$ | $0$ | $-2$ | $+4$ | $0$ | $0$ | $+1$
| $12{\sqrt{2}}\,\Xi^{0}_{c}$ | $\to$ | $\Lambda^{0}\eta_{\omega}$ | $0$ | $0$ | $-2$ | $+4$ | $0$ | $0$ | $-1$
| $12\,\Xi^{0}_{c}$ | $\to$ | $\Lambda^{0}\eta_{\phi}$ | $+1$ | $+1$ | $-2$ | $-2$ | $-2$ | $+4$ | $0$
| $12\sqrt{6}\,\Xi^{0}_{c}$ | $\to$ | $\Lambda^{0}\eta_{8}$ | $-2$ | $-2$ | $+2$ | $+8$ | $+4$ | $-8$ | $-1$
| $12\sqrt{3}\,\Xi^{0}_{c}$ | $\to$ | $\Lambda^{0}\eta_{1}$ | $+1$ | $+1$ | $-4$ | $+2$ | $-2$ | $+4$ | $-1$
| $2\sqrt{6}\,\Xi^{0}_{c}$ | $\to$ | $p{K}^{-}$ | $0$ | $0$ | $0$ | $-2$ | $0$ | $0$ | $0$
| $2\sqrt{6}\,\Xi^{0}_{c}$ | $\to$ | $n\bar{K}^{0}$ | $0$ | $0$ | $+2$ | $0$ | $0$ | $0$ | $0$
Table 5:  Values of the seven topological tensor invariants for the SCS antitriplet charm baryon decays $B_{c}(\overline{\bf 3})\to B({\bf 8})+M({\bf 8},{\bf 1})$ induced by the flavor transition $(c\to d;\,d\to u)$. |  |  |  | $I^{-}_{1}$ | $I^{-}_{2}$ | $I_{3}$ | $I_{4}$ | $\hat{I_{3}}$ | $\hat{I}_{4}$ | $I_{5}$
---|---|---|---|---|---|---|---|---|---|---
SCSb | $12\,\Lambda^{+}_{c}$ | $\to$ | $\Lambda^{0}K^{+}$ | $0$ | $0$ | $-2$ | $-2$ | $0$ | $0$ | $-2$
| $4{\sqrt{3}}\,\Lambda^{+}_{c}$ | $\to$ | $\Sigma^{0}K^{+}$ | $0$ | $0$ | $+2$ | $-2$ | $0$ | $0$ | $0$
| $2\sqrt{6}\,\Lambda^{+}_{c}$ | $\to$ | $\Sigma^{+}K^{0}$ | $0$ | $0$ | $-2$ | $+2$ | $0$ | $0$ | $0$
| $2\sqrt{6}\,\Lambda^{+}_{c}$ | $\to$ | $n\pi^{+}$ | $-1$ | $-1$ | $0$ | $+2$ | $-2$ | $+4$ | $+1$
| $4{\sqrt{3}}\,\Lambda^{+}_{c}$ | $\to$ | $p\pi^{0}$ | $+1$ | $+1$ | $0$ | $-2$ | $+2$ | $-4$ | $-1$
| $4{\sqrt{3}}\,\Lambda^{+}_{c}$ | $\to$ | $p\eta_{\omega}$ | $-1$ | $-1$ | $-4$ | $+2$ | $-2$ | $+4$ | $-1$
| $2\sqrt{6}\,\Lambda^{+}_{c}$ | $\to$ | $p\eta_{\phi}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $12\,\Lambda^{+}_{c}$ | $\to$ | $p\eta_{8}$ | $-1$ | $-1$ | $-4$ | $+2$ | $-2$ | $+4$ | $-1$
| $6\sqrt{2}\,\Lambda^{+}_{c}$ | $\to$ | $p\eta_{1}$ | $-1$ | $-1$ | $-4$ | $+2$ | $-2$ | $+4$ | $-1$
| $2\sqrt{6}\,\Xi^{+}_{c}$ | $\to$ | $\Xi^{0}K^{+}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $4{\sqrt{3}}\,\Xi^{+}_{c}$ | $\to$ | $\Sigma^{+}\pi^{0}$ | $+1$ | $+1$ | $0$ | $0$ | $0$ | $0$ | $0$
| $4{\sqrt{3}}\,\Xi^{+}_{c}$ | $\to$ | $\Sigma^{+}\eta_{\omega}$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2{\sqrt{6}}\,\Xi^{+}_{c}$ | $\to$ | $\Sigma^{+}\eta_{\phi}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $12\,\Xi^{+}_{c}$ | $\to$ | $\Sigma^{+}\eta_{8}$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$
| $6{\sqrt{2}}\,\Xi^{+}_{c}$ | $\to$ | $\Sigma^{+}\eta_{1}$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\sqrt{6}\,\Xi^{+}_{c}$ | $\to$ | $p\bar{K}^{0}$ | $0$ | $0$ | $0$ | $0$ | $+2$ | $-4$ | $0$
| $12\,\Xi^{+}_{c}$ | $\to$ | $\Lambda^{0}\pi^{+}$ | $-1$ | $-1$ | $0$ | $0$ | $-4$ | $+8$ | $0$
| $4{\sqrt{3}}\,\Xi^{+}_{c}$ | $\to$ | $\Sigma^{0}\pi^{+}$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\sqrt{6}\,\Xi^{0}_{c}$ | $\to$ | $\Xi^{0}{K}^{0}$ | $0$ | $0$ | $+2$ | $0$ | $0$ | $0$ | $0$
| $2\sqrt{6}\,\Xi^{0}_{c}$ | $\to$ | $\Xi^{-}K^{+}$ | $0$ | $0$ | $-2$ | $0$ | $0$ | $0$ | $0$
| $2\sqrt{6}\,\Xi^{0}_{c}$ | $\to$ | $\Sigma^{+}\pi^{-}$ | $0$ | $0$ | $0$ | $+2$ | $0$ | $0$ | $0$
| $2\sqrt{6}\,\Xi^{0}_{c}$ | $\to$ | $\Sigma^{-}\pi^{+}$ | $-1$ | $-1$ | $0$ | $+2$ | $0$ | $0$ | $0$
| $4{\sqrt{6}}\,\Xi^{0}_{c}$ | $\to$ | $\Sigma^{0}\pi^{0}$ | $+1$ | $+1$ | $0$ | $-4$ | $0$ | $0$ | $0$
| $4{\sqrt{6}}\,\Xi^{0}_{c}$ | $\to$ | $\Sigma^{0}\eta_{\omega}$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$
| $4{\sqrt{3}}\,\Xi^{0}_{c}$ | $\to$ | $\Sigma^{0}\eta_{\phi}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $12{\sqrt{2}}\,\Xi^{0}_{c}$ | $\to$ | $\Sigma^{0}\eta_{8}$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$
| $12\,\Xi^{0}_{c}$ | $\to$ | $\Sigma^{0}\eta_{1}$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$
| $12{\sqrt{2}}\,\Xi^{0}_{c}$ | $\to$ | $\Lambda^{0}\pi^{0}$ | $-1$ | $-1$ | $0$ | $0$ | $-4$ | $+8$ | $0$
| $12{\sqrt{2}}\,\Xi^{0}_{c}$ | $\to$ | $\Lambda^{0}\eta_{\omega}$ | $+1$ | $+1$ | $+8$ | $-4$ | $+4$ | $-8$ | $0$
| $12\,\Xi^{0}_{c}$ | $\to$ | $\Lambda^{0}\eta_{\phi}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $+2$
| $12\sqrt{6}\,\Xi^{0}_{c}$ | $\to$ | $\Lambda^{0}\eta_{8}$ | $+1$ | $+1$ | $+8$ | $-4$ | $+4$ | $-8$ | $-4$
| $12\sqrt{3}\,\Xi^{0}_{c}$ | $\to$ | $\Lambda^{0}\eta_{1}$ | $+1$ | $+1$ | $+8$ | $-4$ | $+4$ | $-8$ | $+2$
| $2\sqrt{6}\,\Xi^{0}_{c}$ | $\to$ | $p{K}^{-}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $+1$
| $2\sqrt{6}\,\Xi^{0}_{c}$ | $\to$ | $n\bar{K}^{0}$ | $0$ | $0$ | $0$ | $0$ | $+2$ | $-4$ | $-1$
Table 6:  Values of the seven topological tensor invariants for the SCS antitriplet charm baryon decays $B_{c}(\overline{\bf 3})\to B({\bf 8})+M({\bf 8},{\bf 1})$ induced by both flavor transitions (a) and (b). |  |  |  | $I^{-}_{1}$ | $I^{-}_{2}$ | $I_{3}$ | $I_{4}$ | $\hat{I_{3}}$ | $\hat{I}_{4}$ | $I_{5}$
---|---|---|---|---|---|---|---|---|---|---
SCS | $12\,\Lambda^{+}_{c}$ | $\to$ | $\Lambda^{0}K^{+}$ | $+2$ | $+2$ | $+2$ | $+2$ | $+2$ | $-4$ | $+2$
| $4{\sqrt{3}}\,\Lambda^{+}_{c}$ | $\to$ | $\Sigma^{0}K^{+}$ | $0$ | $0$ | $-2$ | $+2$ | $+2$ | $-4$ | $0$
| $2\sqrt{6}\,\Lambda^{+}_{c}$ | $\to$ | $\Sigma^{+}K^{0}$ | $0$ | $0$ | $+2$ | $-2$ | $-2$ | $+4$ | $0$
| $2\sqrt{6}\,\Lambda^{+}_{c}$ | $\to$ | $n\pi^{+}$ | $+1$ | $+1$ | $0$ | $-2$ | $+2$ | $-4$ | $-1$
| $4{\sqrt{3}}\,\Lambda^{+}_{c}$ | $\to$ | $p\pi^{0}$ | $-1$ | $-1$ | $0$ | $+2$ | $-2$ | $+4$ | $+1$
| $4{\sqrt{3}}\,\Lambda^{+}_{c}$ | $\to$ | $p\eta_{\omega}$ | $+1$ | $+1$ | $+4$ | $-2$ | $+2$ | $-4$ | $+1$
| $2\sqrt{6}\,\Lambda^{+}_{c}$ | $\to$ | $p\eta_{\phi}$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$
| $12\,\Lambda^{+}_{c}$ | $\to$ | $p\eta_{8}$ | $+3$ | $+3$ | $+4$ | $-2$ | $+2$ | $-4$ | $+1$
| $6\sqrt{2}\,\Lambda^{+}_{c}$ | $\to$ | $p\eta_{1}$ | $0$ | $0$ | $+4$ | $-2$ | $+2$ | $-4$ | $+1$
| $2\sqrt{6}\,\Xi^{+}_{c}$ | $\to$ | $\Xi^{0}K^{+}$ | $+1$ | $+1$ | $0$ | $-2$ | $+2$ | $-4$ | $-1$
| $4{\sqrt{3}}\,\Xi^{+}_{c}$ | $\to$ | $\Sigma^{+}\pi^{0}$ | $-1$ | $-1$ | $-2$ | $0$ | $0$ | $0$ | $-1$
| $4{\sqrt{3}}\,\Xi^{+}_{c}$ | $\to$ | $\Sigma^{+}\eta_{\omega}$ | $+1$ | $+1$ | $-2$ | $0$ | $0$ | $0$ | $-1$
| $2{\sqrt{6}}\,\Xi^{+}_{c}$ | $\to$ | $\Sigma^{+}\eta_{\phi}$ | $-1$ | $-1$ | $-2$ | $+2$ | $-2$ | $+4$ | $0$
| $12\,\Xi^{+}_{c}$ | $\to$ | $\Sigma^{+}\eta_{8}$ | $+3$ | $+3$ | $+2$ | $-4$ | $+4$ | $-8$ | $-1$
| $6{\sqrt{2}}\,\Xi^{+}_{c}$ | $\to$ | $\Sigma^{+}\eta_{1}$ | $0$ | $0$ | $-4$ | $+2$ | $-2$ | $+4$ | $-1$
| $2\sqrt{6}\,\Xi^{+}_{c}$ | $\to$ | $p\bar{K}^{0}$ | $0$ | $0$ | $+2$ | $-2$ | $-2$ | $+4$ | $0$
| $12\,\Xi^{+}_{c}$ | $\to$ | $\Lambda^{0}\pi^{+}$ | $+1$ | $+1$ | $-2$ | $+4$ | $+4$ | $-8$ | $+1$
| $4{\sqrt{3}}\,\Xi^{+}_{c}$ | $\to$ | $\Sigma^{0}\pi^{+}$ | $+1$ | $+1$ | $+2$ | $0$ | $0$ | $0$ | $+1$
| $2\sqrt{6}\,\Xi^{0}_{c}$ | $\to$ | $\Xi^{0}{K}^{0}$ | $0$ | $0$ | $-2$ | $0$ | $+2$ | $-4$ | $-1$
| $2\sqrt{6}\,\Xi^{0}_{c}$ | $\to$ | $\Xi^{-}K^{+}$ | $+1$ | $+1$ | $+2$ | $-2$ | $0$ | $0$ | $0$
| $2\sqrt{6}\,\Xi^{0}_{c}$ | $\to$ | $\Sigma^{+}\pi^{-}$ | $0$ | $0$ | $0$ | $-2$ | $0$ | $0$ | $-1$
| $2\sqrt{6}\,\Xi^{0}_{c}$ | $\to$ | $\Sigma^{-}\pi^{+}$ | $+1$ | $+1$ | $+2$ | $-2$ | $0$ | $0$ | $0$
| $4{\sqrt{6}}\,\Xi^{0}_{c}$ | $\to$ | $\Sigma^{0}\pi^{0}$ | $-1$ | $-1$ | $-2$ | $+4$ | $0$ | $0$ | $+1$
| $4{\sqrt{6}}\,\Xi^{0}_{c}$ | $\to$ | $\Sigma^{0}\eta_{\omega}$ | $+1$ | $+1$ | $-2$ | $0$ | $0$ | $0$ | $-1$
| $4{\sqrt{3}}\,\Xi^{0}_{c}$ | $\to$ | $\Sigma^{0}\eta_{\phi}$ | $-1$ | $-1$ | $-2$ | $+2$ | $-2$ | $+4$ | $0$
| $12{\sqrt{2}}\,\Xi^{0}_{c}$ | $\to$ | $\Sigma^{0}\eta_{8}$ | $+3$ | $+3$ | $+2$ | $-4$ | $+4$ | $-8$ | $-1$
| $12\,\Xi^{0}_{c}$ | $\to$ | $\Sigma^{0}\eta_{1}$ | $0$ | $0$ | $-4$ | $+2$ | $-2$ | $+4$ | $-1$
| $12{\sqrt{2}}\,\Xi^{0}_{c}$ | $\to$ | $\Lambda^{0}\pi^{0}$ | $+1$ | $+1$ | $-2$ | $+4$ | $+4$ | $-8$ | $+1$
| $12{\sqrt{2}}\,\Xi^{0}_{c}$ | $\to$ | $\Lambda^{0}\eta_{\omega}$ | $-1$ | $-1$ | $-10$ | $+8$ | $-4$ | $+8$ | $-1$
| $12\,\Xi^{0}_{c}$ | $\to$ | $\Lambda^{0}\eta_{\phi}$ | $+1$ | $+1$ | $-2$ | $-2$ | $-2$ | $+4$ | $-2$
| $12\sqrt{6}\,\Xi^{0}_{c}$ | $\to$ | $\Lambda^{0}\eta_{8}$ | $-3$ | $-3$ | $-6$ | $+12$ | $0$ | $0$ | $+3$
| $12\sqrt{3}\,\Xi^{0}_{c}$ | $\to$ | $\Lambda^{0}\eta_{1}$ | $0$ | $0$ | $-12$ | $+6$ | $-6$ | $+12$ | $-3$
| $2\sqrt{6}\,\Xi^{0}_{c}$ | $\to$ | $p{K}^{-}$ | $0$ | $0$ | $0$ | $-2$ | $0$ | $0$ | $-1$
| $2\sqrt{6}\,\Xi^{0}_{c}$ | $\to$ | $n\bar{K}^{0}$ | $0$ | $0$ | $+2$ | $0$ | $-2$ | $+4$ | $+1$
Table 7:  Values of the seven topological tensor invariants for the DCS antitriplet charm baryon decays $B_{c}(\overline{\bf 3})\to B({\bf 8})+M({\bf 8},{\bf 1})$ induced by the flavor transition $(c\to d;\,s\to u)$. |  |  |  | $I^{-}_{1}$ | $I^{-}_{2}$ | $I_{3}$ | $I_{4}$ | $\hat{I}_{3}$ | $\hat{I}_{4}$ | $I_{5}$
---|---|---|---|---|---|---|---|---|---|---
DCS | $2\sqrt{6}\,\Lambda_{c}^{+}$ | $\to$ | $pK^{0}$ | $-1$ | $-1$ | $0$ | $0$ | $-2$ | $+4$ | $0$
| $2\sqrt{6}\,\Lambda_{c}^{+}$ | $\to$ | $nK^{+}$ | $+1$ | $+1$ | $0$ | $0$ | $+2$ | $-4$ | $0$
| $12\,\Xi_{c}^{+}$ | $\to$ | $\Lambda^{0}K^{+}$ | $+1$ | $+1$ | $-2$ | $-2$ | $+4$ | $-8$ | $-2$
| $4\sqrt{3}\,\Xi_{c}^{+}$ | $\to$ | $\Sigma^{0}K^{+}$ | $+1$ | $+1$ | $+2$ | $-2$ | $0$ | $0$ | $0$
| $2\sqrt{6}\,\Xi_{c}^{+}$ | $\to$ | $\Sigma^{+}K^{0}$ | $-1$ | $-1$ | $-2$ | $+2$ | $0$ | $0$ | $0$
| $4\sqrt{3}\,\Xi_{c}^{+}$ | $\to$ | $p\pi^{0}$ | $0$ | $0$ | $0$ | $-2$ | $0$ | $0$ | $-1$
| $2\sqrt{6}\,\Xi_{c}^{+}$ | $\to$ | $n\pi^{+}$ | $0$ | $0$ | $0$ | $+2$ | $0$ | $0$ | $+1$
| $4\sqrt{3}\,\Xi_{c}^{+}$ | $\to$ | $p\eta_{\omega}$ | $0$ | $0$ | $-4$ | $+2$ | $0$ | $0$ | $-1$
| $2\sqrt{6}\,\Xi_{c}^{+}$ | $\to$ | $p\eta_{\phi}$ | $0$ | $0$ | $0$ | $0$ | $-2$ | $+4$ | $0$
| $12\,\Xi_{c}^{+}$ | $\to$ | $p\eta_{8}$ | $0$ | $0$ | $-4$ | $+2$ | $+4$ | $-8$ | $-1$
| $6\sqrt{2}\,\Xi_{c}^{+}$ | $\to$ | $p\eta_{1}$ | $0$ | $0$ | $-4$ | $+2$ | $-2$ | $+4$ | $-1$
| $12\,\Xi_{c}^{0}$ | $\to$ | $\Lambda^{0}K^{0}$ | $+1$ | $+1$ | $-2$ | $-2$ | $+4$ | $-8$ | $-2$
| $4\sqrt{3}\,\Xi_{c}^{0}$ | $\to$ | $\Sigma^{0}K^{0}$ | $-1$ | $-1$ | $-2$ | $+2$ | $0$ | $0$ | $0$
| $2\sqrt{6}\,\Xi_{c}^{0}$ | $\to$ | $\Sigma^{-}K^{+}$ | $+1$ | $+1$ | $+2$ | $-2$ | $0$ | $0$ | $0$
| $2\sqrt{6}\,\Xi_{c}^{0}$ | $\to$ | $p\pi^{-}$ | $0$ | $0$ | $0$ | $-2$ | $0$ | $0$ | $-1$
| $4\sqrt{3}\,\Xi_{c}^{0}$ | $\to$ | $n\pi^{0}$ | $0$ | $0$ | $0$ | $+2$ | $0$ | $0$ | $+1$
| $4\sqrt{3}\,\Xi_{c}^{0}$ | $\to$ | $n\eta_{\omega}$ | $0$ | $0$ | $-4$ | $+2$ | $0$ | $0$ | $-1$
| $2\sqrt{6}\Xi_{c}^{0}$ | $\to$ | $n\eta_{\phi}$ | $0$ | $0$ | $0$ | $0$ | $-2$ | $+4$ | $0$
| $12\,\Xi_{c}^{0}$ | $\to$ | $n\eta_{8}$ | $0$ | $0$ | $-4$ | $+2$ | $+4$ | $-8$ | $-1$
| $6\sqrt{2}\,\Xi_{c}^{0}$ | $\to$ | $n\eta_{1}$ | $0$ | $0$ | $-4$ | $+2$ | $-2$ | $+4$ | $-1$

###  4.1 The charm baryon decays $B_{c}(\overline{\bf
3})\stackrel{{\scriptstyle\\!\\!H(\bf 6)}}{{\longrightarrow}}B({\bf 8})+M({\bf
8,1})$

The nonleptonic two-body decays of the three antitriplet charm baryons
($\Lambda_{c}^{+},\Xi_{c}^{+},\Xi_{c}^{0}$) belong to this class of decays.
The SU(3) decomposition of the transitions read (we use the notation of
Kaeding for multiple SU(3) representations [71])

$\overline{\bf 3}\to{\bf 6}\otimes{\bf 8}\otimes{\bf 8}=3\cdot\overline{\bf
3}\oplus 4\cdot{\bf 6}\oplus 5\cdot\overline{\bf 15}\oplus\overline{\bf
15}\,^{\prime}\oplus\overline{\bf 21}\oplus\overline{\bf 24}\oplus
2\cdot\overline{\bf 42}\oplus\overline{\bf 60}$ (22)

As explained in Sec. 3, the effective Hamiltonian inducing the $W$-exchange
contributions transforms as a sextet in SU(3). For the SCS decays this comes
about by a cancellation of the antitriplet contributions when the two
effective Hamiltonians ${\cal H}_{\rm eff}(c\to s;\,s\to u)$ and ${\cal
H}_{\rm eff}(c\to d;\,d\to u)$ are subtracted from one another. The
$\overline{\bf 3}$ representation appears three times in the decomposition.
Therefore, there are three SU(3) invariant amplitudes. This in turn implies
that there are four linear relations among the seven topological tensor
invariants. They read

$I^{-}_{1}=I^{-}_{2},\qquad 2\hat{I}_{3}+\hat{I}_{4}=0,\qquad
I_{3}+I_{4}=2I_{5},\qquad 2I^{-}_{1}=I_{3}+\hat{I}_{3}.$ (23)

The relation $I^{-}_{1}=I^{-}_{2}$ can be obtained by rewriting the tensor
labels of the parent baryon in the tensor invariant $I^{-}_{2}$ in the way
$B^{1}_{b[c^{\prime}a]}\to-B^{1}_{a[c^{\prime}b]}\to B^{1}_{a[bc^{\prime}]}$
by realizing that the indices $a$ and $b$ refer to light quarks. Similarly one
can proof $2\hat{I}_{3}+\hat{I}_{4}=0$ by realizing that $a$ and $c^{\prime}$
are the light quarks in the parent baryon $B^{1}_{b^{\prime}[c^{\prime}a]}$
and using the Jacobi identity
$B^{1}_{b^{\prime}[c^{\prime}a]}+B^{1}_{c^{\prime}[ab^{\prime}]}+B^{1}_{a[b^{\prime}c^{\prime}]}=0$.

A more direct access to a minimal set of independent tensor contractions is to
switch to second rank tensor representations of the antitriplet charm baryons,
the octet baryons and the effective Hamiltonian [19, 29]. As a result one now
has only three independent tensor contractions, the number of which agrees
with the above number of SU(3) invariants. The linear relations (23) can be
seen to be in agreement with the analysis of Jia et al. [29].

Eqs. (23) are very useful when checking the results for the topological tensor
invariants $I_{j}(fki)$ that will be listed later on. Note in particular that
the linear relations (23) do not separately hold for the contributions of the
two transitions ($a$) $c\\!\to\\!s;\,\,s\\!\to\\!u$ and ($b$)
$c\\!\to\\!d;\,\,d\\!\to\\!u$. They can be seen to hold true only for the sum
of the two contributions. Note that the last equation does not hold for all
decay processes listed in Tables 3 to 7, in particular not for decays into
$\eta_{1}$ and, related to that, into $\eta_{\omega}$ and $\eta_{\phi}$, as
those are members of (a mixture of $M({\bf 8})$ and) $M({\bf 1})$.

Therefore, we also need to discuss the transition $B_{c}(\overline{\bf
3})\stackrel{{\scriptstyle\\!\\!H(\bf 6)}}{{\longrightarrow}}B({\bf 8})+M({\bf
1})$ involving the singlet meson state $M({\bf 1})$. In this case the SU(3)
decomposition reads

$\overline{\bf 3}\to{\bf 6}\otimes{\bf 8}\otimes{\bf 1}=\overline{\bf
3}\oplus{\bf 6}\oplus\overline{\bf 15}\oplus\overline{\bf 24}.$ (24)

Thus there is only one SU(3) invariant amplitude describing the transition
$B_{c}(\overline{\bf 3})\stackrel{{\scriptstyle\\!\\!H(\bf
6)}}{{\longrightarrow}}B({\bf 8})+M({\bf 1})$ which means that all transitions
involving a singlet meson state $M({\bf 1})$ are proportional to one another
which is manifest in the decays into $\eta_{1}$. In the corresponding rows of
Tables 3 to 7 one has

$I^{-}_{1}=I^{-}_{2}=0,\qquad I_{3}=-2I_{4}=2\hat{I}_{3}=-\hat{I}_{4}=4I_{5}.$
(25)

The vanishing of the tree diagram invariants can be argued again directly, as
the single quark line transition ${\bf 1}\to{\bf 6}\otimes{\bf 1}\otimes{\bf
3}={\bf 8}\oplus{\bf 10}$ into singlet meson states vanishes in SU(3).

We list the values of the seven topological tensor invariants separately for
the Cabibbo favored (CF) (Table 3), singly Cabibbo suppressed (SCS) (Tables 4,
5 and 6), and doubly Cabibbo suppressed (DCS) (Table 7) antitriplet charm
baryon decays. The SCS decays are induced by the two SCS transitions ($a$)
$c\to s;\,s\to u$ and ($b$) $c\to d;\,d\to u$ which are listed separately in
Tables 4 and 5. As explained in Sec. 2, in the SU(3) limit dealt with in this
paper one has to subtract the two contributions ($a$) and ($b$). The result of
this subtraction is shown in Table 6.

As discussed in Sec. 3, Kohara has introduced four reduced matrix elements
$d_{1},d_{2},d_{3},d_{4}$ for the topology class IIa where only two are needed
[31]. Using the results of e.g. Table 3 for the decays involving octet mesons
one finds that the four topological reduced matrix elements of Kohara are
related to our topological reduced matrix elements by
$d_{1}=2\mathbfcal{T}_{3}$, $d_{2}=-2(\mathbfcal{T}_{3}-\mathbfcal{T}_{4})$,
$d_{3}=2\mathbfcal{T}_{4}$, and $d_{4}=2\mathbfcal{T}_{3}-\mathbfcal{T}_{4}$.
This shows again that the set $d_{1},\,d_{2},\,d_{3},\,d_{4}$ is redundant
since one has $d_{1}=d_{3}+d_{4}$ and $d_{2}=-d_{4}$. We do not agree with
Ref. [31] on the contributions of the $d_{i}$ to the decays involving the
SU(3) singlet meson $\eta_{1}$.

Table 8:  Values of the seven topological tensor invariants for the nonleptonic CF, SCS and DCS decays of the sextet charm baryon state $\Omega^{0}_{c}$ belonging to the class of decays $B_{c}({\bf 6})\to B({\bf 8})+M({\bf 8},{\bf 1})$. The SCS decays are induced by the flavor transition ($a$) $(c\to s;\,s\to u)$ and ($b$) $(c\to d;\,d\to u)$. The subscripts $a$ and $b$ of the tensor invariants identify the origin of the respective flavor transitions where the relative sign of the two transitions has been accounted for. |  |  |  | $I^{-}_{1}$ | $I^{-}_{2}$ | $I_{3}$ | $I_{4}$ | $\hat{I}_{3}$ | $\hat{I}_{4}$ | $I_{5}$
---|---|---|---|---|---|---|---|---|---|---
CF | $2\Omega_{c}^{0}$ | $\to$ | $\Xi^{0}\bar{K}^{0}$ | $-1$ | $+1$ | $0$ | $0$ | $-2$ | $0$ | $0$
SCS | $2\sqrt{2}\Omega_{c}^{0}$ | $\to$ | $\Xi^{0}\pi^{0}$ | $+1_{b}$ | $-1_{b}$ | $+2_{a}$ | $-2_{a}$ | $0$ | $0$ | $0$
| $2\sqrt{2}\Omega_{c}^{0}$ | $\to$ | $\Xi^{0}\eta_{\omega}$ | $-1_{b}$ | $+1_{b}$ | $+2_{a}$ | $-2_{a}$ | $0$ | $0$ | $0$
| $2\Omega_{c}^{0}$ | $\to$ | $\Xi^{0}\eta_{\phi}$ | $+1_{a}$ | $-1_{a}$ | $+2_{a}$ | $0$ | $+2_{a}$ | $0$ | $+1_{a}$
| $2\sqrt{6}\Omega_{c}^{0}$ | $\to$ | $\Xi^{0}\eta_{8}$ | $-2_{a}-1_{b}$ | $+2_{a}+1_{b}$ | $-2_{a}$ | $-2_{a}$ | $-4_{a}$ | $0$ | $-2_{a}$
| $2\sqrt{3}\Omega_{c}^{0}$ | $\to$ | $\Xi^{0}\eta_{1}$ | $+1_{a}-1_{b}$ | $-1_{a}+1_{b}$ | $+4_{a}$ | $-2_{a}$ | $+2_{a}$ | $0$ | $+1_{a}$
| $2\sqrt{6}\Omega_{c}^{0}$ | $\to$ | $\Lambda^{0}\bar{K}^{0}$ | $0$ | $0$ | $-4_{a}$ | $+2_{a}$ | $+4_{b}$ | $0$ | $-1_{a}$
| $2\sqrt{2}\Omega_{c}^{0}$ | $\to$ | $\Sigma^{0}\bar{K}^{0}$ | $0$ | $0$ | $0$ | $-2_{a}$ | $0$ | $0$ | $-1_{a}$
| $2\Omega_{c}^{0}$ | $\to$ | $\Sigma^{+}K^{-}$ | $0$ | $0$ | $0$ | $+2_{a}$ | $0$ | $0$ | $+1_{a}$
| $2\Omega_{c}^{0}$ | $\to$ | $\Xi^{-}\pi^{+}$ | $-1_{b}$ | $+1_{b}$ | $-2_{a}$ | $+2_{a}$ | $0$ | $0$ | $0$
DCS | $2\Omega_{c}^{0}$ | $\to$ | $pK^{-}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $+1$
| $2\Omega_{c}^{0}$ | $\to$ | $n\bar{K}^{0}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $-1$
| $4\sqrt{3}\Omega_{c}^{0}$ | $\to$ | $\Lambda^{0}\pi^{0}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $4\sqrt{3}\Omega_{c}^{0}$ | $\to$ | $\Lambda^{0}\eta_{\omega}$ | $0$ | $0$ | $+8$ | $-4$ | $0$ | $0$ | $0$
| $2\sqrt{6}\Omega_{c}^{0}$ | $\to$ | $\Lambda^{0}\eta_{\phi}$ | $0$ | $0$ | $0$ | $0$ | $+4$ | $0$ | $+2$
| $12\Omega_{c}^{0}$ | $\to$ | $\Lambda^{0}\eta_{8}$ | $0$ | $0$ | $+8$ | $-4$ | $-8$ | $0$ | $-4$
| $6\sqrt{2}\Omega_{c}^{0}$ | $\to$ | $\Lambda^{0}\eta_{1}$ | $0$ | $0$ | $+8$ | $-4$ | $+4$ | $0$ | $+2$
| $4\Omega_{c}^{0}$ | $\to$ | $\Sigma^{0}\pi^{0}$ | $0$ | $0$ | $0$ | $-4$ | $0$ | $0$ | $0$
| $4\Omega_{c}^{0}$ | $\to$ | $\Sigma^{0}\eta_{\omega}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\sqrt{2}\Omega_{c}^{0}$ | $\to$ | $\Sigma^{0}\eta_{\phi}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $4\sqrt{3}\Omega_{c}^{0}$ | $\to$ | $\Sigma^{0}\eta_{8}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\sqrt{6}\Omega_{c}^{0}$ | $\to$ | $\Sigma^{0}\eta_{1}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\Omega_{c}^{0}$ | $\to$ | $\Sigma^{+}\pi^{-}$ | $0$ | $0$ | $0$ | $+2$ | $0$ | $0$ | $0$
| $2\Omega_{c}^{0}$ | $\to$ | $\Sigma^{-}\pi^{+}$ | $0$ | $0$ | $0$ | $+2$ | $0$ | $0$ | $0$
| $2\Omega_{c}^{0}$ | $\to$ | $\Xi^{-}K^{+}$ | $-1$ | $+1$ | $-2$ | $0$ | $0$ | $0$ | $0$
| $2\Omega_{c}^{0}$ | $\to$ | $\Xi^{0}K^{0}$ | $+1$ | $-1$ | $+2$ | $0$ | $0$ | $0$ | $0$

###  4.2 The charm baryon decays $B_{c}({\bf 6})\stackrel{{\scriptstyle H({\bf
6})}}{{\longrightarrow}}B({\bf 8})+M({\bf 8,1})$

The nonleptonic two-body decays of the charm baryon $\Omega_{c}^{0}$ belong to
this class of decays. The SU(3) decomposition of the direct product ${\bf
6}\otimes{\bf 8}\otimes{\bf 8}$ has been written down before in Eq. (22). One
notes that the ${\bf 6}$ representation appears four times in the
decomposition. The transition ${\bf 6}\to{\bf 6}\otimes{\bf 8}\otimes{\bf 8}$
is thus described by four SU(3) invariants. This implies that there are
$7-4=3$ linear relations among the 7 tensor invariants. These are

$I^{-}_{1}+I^{-}_{2}=0,\qquad\hat{I_{4}}=0,\qquad
2I^{-}_{1}=I_{3}+\hat{I}_{3}.$ (26)

The first relation $I^{-}_{1}+I^{-}_{2}=0$ follows from the Jacobi identity
rewriting $I^{-}_{2}$ as

$I^{-}_{2}(\ell,\ell^{\prime})=B_{\ell}^{a[bc]}B^{\ell^{\prime}}_{b[c^{\prime}a]}M^{d}_{d^{\prime}}H^{[c^{\prime}d^{\prime}]}_{[cd]}=B_{\ell}^{a[bc]}(-B^{\ell^{\prime}}_{c^{\prime}[ab]}-B^{\ell^{\prime}}_{a[bc^{\prime}]})M^{d}_{d^{\prime}}H^{[c^{\prime}d^{\prime}]}_{[cd]}$
(27)

and noting that the first term on the r.h.s. of Eq. (27) vanishes due to the
fact that $c^{\prime}$ must be the charm quark, and that the light quarks
$a,b$ are symmetric in the ${\bf 6}$ representation. With the same reasoning
one can show that $\hat{I}_{4}=0$. The relation $2I^{-}_{1}=I_{3}+\hat{I}_{3}$
cannot be obtained by tensor label manipulations but can be derived from the
values of the three respective tensor invariants in Table 8. In the case of
the SCS transitions the suffixes $a$ and $b$ identify the origin of the
respective flavor transitions ($a$) and ($b$) including the relative sign.
Note that we always keep the ($a$)- and ($b$)-type contributions in the SCS
contribution apart for two reasons: in order to incorporate some SU(3)
breaking effects in this way, and for comparison with parts of the literature
in which the two contributions have been kept apart. Note that the tensor
relation $2I^{-}_{1}=I_{3}+\hat{I}_{3}$ does not hold separately for the
transitions ($a$) and ($b$) but only for their sum, and it does not hold if
the singlet meson state is involved.

The transitions involving the singlet meson state $\eta_{1}$ are again
described by a single SU(3) invariant, as can be read off from the
decomposition (24). For these transitions on finds
$I^{-}_{1}=I^{-}_{2}=\hat{I}_{4}=0$ and $I_{3}=-2I_{4}=2\hat{I}_{3}=4I_{5}$.

Table 9:  Values of the seven tensor invariants for the $\Delta C=0$ SCS decays of the single charm baryons $\Xi_{c}^{+}$, $\Xi_{c}^{0}$, and $\Omega_{c}^{0}$ into antitriplet charm baryons induced by the flavor transitions ($a^{\prime}$) $s\to u;\quad u\to d$ and ($b^{\prime}$) $c\to d;\quad s\to c$. The subscripts $a^{\prime}$ and $b^{\prime}$ of the tensor invariants identify the origin of the respective flavor transitions where the relative sign of the two transitions has been accounted for. |  |  | $I^{-}_{1}$ | $I^{-}_{2}$ | $I_{3}$ | $I_{4}$ | $\hat{I}_{3}$ | $\hat{I}_{4}$ | $I_{5}$
---|---|---|---|---|---|---|---|---|---
$12\sqrt{2}\Xi_{c}^{+}$ | $\to$ | $\Lambda_{c}^{+}\pi^{0}$ | $-5_{a^{\prime}}$ | $+4_{a^{\prime}}$ | $-2_{b^{\prime}}$ | $4_{b^{\prime}}$ | $-8_{a^{\prime}}$ | $4_{a^{\prime}}$ | $1_{b^{\prime}}$
$12\Xi_{c}^{0}$ | $\to$ | $\Lambda_{c}^{+}\pi^{-}$ | $-5_{a^{\prime}}$ | $+4_{a^{\prime}}$ | $-2_{b^{\prime}}$ | $4_{b^{\prime}}$ | $-8_{a^{\prime}}$ | $4_{a^{\prime}}$ | $1_{b^{\prime}}$
$2\sqrt{6}\,\Omega_{c}^{0}$ | $\to$ | $\Xi_{c}^{+}\pi^{-}$ | $1_{a^{\prime}}$ | $-2_{a^{\prime}}$ | $+2_{b^{\prime}}$ | $-4_{b^{\prime}}$ | $0$ | $0$ | $0$
$4\sqrt{3}\,\Omega_{c}^{0}$ | $\to$ | $\Xi_{c}^{0}\pi^{0}$ | $-1_{a^{\prime}}$ | $+2_{a^{\prime}}$ | $-2_{b^{\prime}}$ | $+4_{b^{\prime}}$ | $0$ | $0$ | $0$

###  4.3 The $\Delta C=0$ singly Cabibbo suppressed charm baryon decays
$B_{c}(\overline{\bf 3})\stackrel{{\scriptstyle H({\bf
8})}}{{\longrightarrow}}B_{c}(\overline{\bf 3})+M({\bf 8})$ and $B_{c}({\bf
6})\stackrel{{\scriptstyle H({\bf 8})}}{{\longrightarrow}}B_{c}(\overline{\bf
3})+M({\bf 8})$

The $\Delta C=0$ charm baryon decays are induced by the SCS quark flavor
transitions ($a^{\prime}$) $s\to u;\,u\to d$ and ($b^{\prime}$) $c\to d;\,s\to
c$. The kinematical constraints of the $\Delta C=0$ decays only allow for the
pionic modes. Therefore, there is no need to discuss the decays into the
$\eta$ and $\eta^{\prime}$ states or the $K$-meson states. As concerns the
application of the current algebra approach to charm baryon decays, the
$\Delta C=0$ charm baryon decays are the favorites of all charm baryon decays
since the emitted pion has very little energy, i.e. the pion satisfies the
requirement of the soft pion theorem. The $\Delta C=0$ charm baryon decays
have been discussed before in Refs. [72, 73, 74, 75, 76].

There are two isospin decays in each the two classes $B_{c}(\overline{\bf
3})\stackrel{{\scriptstyle H({\bf 8})}}{{\longrightarrow}}B_{c}(\overline{\bf
3})+M({\bf 8})$ and $B_{c}({\bf 6})\stackrel{{\scriptstyle H({\bf
8})}}{{\longrightarrow}}B_{c}(\overline{\bf 3})+M({\bf 8})$. The four
kinematically accessible decays are listed in Table 9 together with the values
of the topological tensor invariants. As mentioned in the Introduction, the
decay $\Xi_{c}^{0}$ $\to$ $\Lambda_{c}^{+}\pi^{-}$ has recently been observed
[17].

For the $\Xi_{c}^{+,0}$ and $\Omega^{0}$ decays the SU(3) decomposition reads

$\displaystyle\overline{\bf 3}\to{\bf 8}\otimes\overline{\bf 3}\otimes{\bf 8}$
$\displaystyle=$ $\displaystyle 3\cdot\overline{\bf 3}\oplus 3\cdot{\bf
6}\oplus 4\cdot\overline{\bf 15}\oplus\overline{\bf 15}\,^{\prime}\oplus
2\cdot\overline{\bf 24}\oplus\overline{\bf 42}$ (28) $\displaystyle{\bf
6}\to{\bf 8}\otimes\overline{\bf 3}\otimes{\bf 8}$ $\displaystyle=$
$\displaystyle 3\cdot\overline{\bf 6}\oplus 3\cdot{\bf 6}\oplus
4\cdot\overline{\bf 15}\oplus\overline{\bf 15}\,^{\prime}\oplus
2\cdot\overline{\bf 24}\oplus\overline{\bf 42}$ (29)

One concludes that one should have four linear relations each among the seven
tensor invariants. With the help of tensor manipulations alone one finds the
relations $2I_{3}=-I_{4}$ for both transitions $\overline{\bf
3}\stackrel{{\scriptstyle H({\bf 8})}}{{\longrightarrow}}\overline{\bf 3}+{\bf
8}$ and ${\bf 6}\stackrel{{\scriptstyle H({\bf
8})}}{{\longrightarrow}}\overline{\bf 3}+{\bf 8}$. The data base in Table 9 is
not large enough to identify the remaining tensor identities. From a group
theoretical point of view one could, of course, enlarge the data base by
including also off-shell decays. We have not pursued this possible avenue.

While there are three SU(3) reduced matrix elements each for the two classes
of decays there is only one isospin SU(2) reduced matrix element for each of
the two classes of decays. This becomes evident when doing the same exercise
as in Eq. (28) but now for isospin SU(2). One finds

$\displaystyle{\bf 2}\to{\bf 2}\otimes{\bf 1}\otimes{\bf 3}={\bf 2}\oplus{\bf
4}$ $\displaystyle\mbox{(in spin notation $\tfrac{1}{2}\to\tfrac{1}{2}\otimes
0\otimes 1=\tfrac{1}{2}\oplus\tfrac{3}{2}$)},$ $\displaystyle{\bf 1}\to{\bf
2}\otimes{\bf 2}\otimes{\bf 3}={\bf 1}\oplus 2\cdot{\bf 3}\oplus{\bf 5}$
$\displaystyle\mbox{($0\to\tfrac{1}{2}\otimes\tfrac{1}{2}\otimes 1=0\oplus
2\cdot 1\oplus 2$)}.$ (30)

This leads to the $\Delta I=1/2$ isospin relations

$\displaystyle\sqrt{2}\,M(\Xi_{c}^{+}\to\Lambda_{c}^{+}\pi^{0})$
$\displaystyle=$ $\displaystyle M(\Xi_{c}^{0}\to\Lambda_{c}^{+}\pi^{-})$ (31)
$\displaystyle\sqrt{2}\,M(\Omega_{c}^{0}\to\Xi_{c}^{0}\pi^{0})$
$\displaystyle=$ $\displaystyle-M(\Omega_{c}^{0}\to\Xi_{c}^{+}\pi^{-})$ (32)

in agreement with the entries in Table 9.

Table 10:  Values of the seven topological tensor invariants for the CF, SCS and DCS decays of the triplet double charm baryon states $\Xi_{cc}^{++}$, $\Xi_{cc}^{+}$ and $\Omega_{cc}^{+}$ into the antitriplet single charm baryon states $\Lambda_{c}^{+}$, $\Xi_{c}^{+}$ and $\Xi_{c}^{0}$ belonging to the class of decays $B_{cc}({\bf 3})\to B_{c}(\overline{\bf 3})+M({\bf 8},{\bf 1})$. The notation of the subscripts $a$ and $b$ is explained in Table 8. |  |  |  | $I^{-}_{1}$ | $I^{-}_{2}$ | $I_{3}$ | $I_{4}$ | $\hat{I}_{3}$ | $\hat{I}_{4}$ | $I_{5}$
---|---|---|---|---|---|---|---|---|---|---
CF | $2\sqrt{6}\,\Xi_{cc}^{++}$ | $\to$ | $\Xi_{c}^{+}\pi^{+}$ | $-2$ | $+1$ | $0$ | $0$ | $-4$ | $+4$ | $0$
| $2\sqrt{6}\,\Xi_{cc}^{+}$ | $\to$ | $\Xi_{c}^{0}\pi^{+}$ | $-2$ | $+1$ | $-4$ | $+2$ | $0$ | $0$ | $0$
| $4\sqrt{3}\,\Xi_{cc}^{+}$ | $\to$ | $\Xi_{c}^{+}\pi^{0}$ | $0$ | $0$ | $+4$ | $-2$ | $-4$ | $+4$ | $0$
| $4\sqrt{3}\,\Xi_{cc}^{+}$ | $\to$ | $\Xi_{c}^{+}\eta_{\omega}$ | $0$ | $0$ | $+4$ | $-2$ | $+4$ | $-4$ | $0$
| $2\sqrt{6}\,\Xi_{cc}^{+}$ | $\to$ | $\Xi_{c}^{+}\eta_{\phi}$ | $0$ | $0$ | $+4$ | $-2$ | $0$ | $0$ | $0$
| $12\,\Xi_{cc}^{+}$ | $\to$ | $\Xi_{c}^{+}\eta_{8}$ | $0$ | $0$ | $-4$ | $+2$ | $+4$ | $-4$ | $0$
| $6\sqrt{2}\,\Xi_{cc}^{+}$ | $\to$ | $\Xi_{c}^{+}\eta_{1}$ | $0$ | $0$ | $+8$ | $-4$ | $+4$ | $-4$ | $0$
| $2\sqrt{6}\,\Xi_{cc}^{+}$ | $\to$ | $\Lambda_{c}^{+}\bar{K}^{0}$ | $-2$ | $+1$ | $-4$ | $+2$ | $0$ | $0$ | $0$
| $2\sqrt{6}\,\Omega_{cc}^{+}$ | $\to$ | $\Xi_{c}^{+}\bar{K}^{0}$ | $-2$ | $+1$ | $0$ | $0$ | $-4$ | $+4$ | $0$
SCS | $2\sqrt{6}\,\Xi_{cc}^{++}$ | $\to$ | $\Xi_{c}^{+}K^{+}$ | $+2_{a}$ | $-1_{a}$ | $0$ | $0$ | $+4_{a}$ | $-4_{a}$ | $0$
| $2\sqrt{6}\,\Xi_{cc}^{++}$ | $\to$ | $\Lambda_{c}^{+}\pi^{+}$ | $+2_{b}$ | $-1_{b}$ | $0$ | $0$ | $+4_{b}$ | $-4_{b}$ | $0$
| $2\sqrt{6}\,\Xi_{cc}^{+}$ | $\to$ | $\Xi_{c}^{0}K^{+}$ | $+2_{a}$ | $-1_{a}$ | $+4_{b}$ | $-2_{b}$ | $0$ | $0$ | $0$
| $4\sqrt{3}\,\Xi_{cc}^{+}$ | $\to$ | $\Lambda_{c}^{+}\pi^{0}$ | $+2_{b}$ | $-1_{b}$ | $0$ | $0$ | $+4_{b}$ | $-4_{b}$ | $0$
| $4\sqrt{3}\,\Xi_{cc}^{+}$ | $\to$ | $\Lambda_{c}^{+}\eta_{\omega}$ | $-2_{b}$ | $+1_{b}$ | $-8_{b}$ | $+4_{b}$ | $-4_{b}$ | $+4_{b}$ | $0$
| $2\sqrt{6}\,\Xi_{cc}^{+}$ | $\to$ | $\Lambda_{c}^{+}\eta_{\phi}$ | $+2_{a}$ | $-1_{a}$ | $0$ | $0$ | $0$ | $0$ | $0$
| $12\,\Xi_{cc}^{+}$ | $\to$ | $\Lambda_{c}^{+}\eta_{8}$ | $-4_{a}-2_{b}$ | $+2_{a}+1_{b}$ | $-8_{b}$ | $+4_{b}$ | $-4_{b}$ | $+4_{b}$ | $0$
| $6\sqrt{2}\,\Xi_{cc}^{+}$ | $\to$ | $\Lambda_{c}^{+}\eta_{1}$ | $+2_{a}-2_{b}$ | $-1_{a}+1_{b}$ | $-8_{b}$ | $+4_{b}$ | $-4_{b}$ | $+4_{b}$ | $0$
| $4\sqrt{3}\,\Omega_{cc}^{+}$ | $\to$ | $\Xi_{c}^{+}\pi^{0}$ | $+2_{b}$ | $-1_{b}$ | $+4_{a}$ | $-2_{a}$ | $0$ | $0$ | $0$
| $4\sqrt{3}\,\Omega_{cc}^{+}$ | $\to$ | $\Xi_{c}^{+}\eta_{\omega}$ | $-2_{b}$ | $+1_{b}$ | $+4_{a}$ | $-2_{a}$ | $0$ | $0$ | $0$
| $2\sqrt{6}\,\Omega_{cc}^{+}$ | $\to$ | $\Xi_{c}^{+}\eta_{\phi}$ | $+2_{a}$ | $-1_{a}$ | $+4_{a}$ | $-2_{a}$ | $+4_{a}$ | $-4_{a}$ | $0$
| $12\,\Omega_{cc}^{+}$ | $\to$ | $\Xi_{c}^{+}\eta_{8}$ | $-4_{a}-2_{b}$ | $+2_{a}+1_{b}$ | $-4_{a}$ | $+2_{a}$ | $-8_{a}$ | $+8_{a}$ | $0$
| $6\sqrt{2}\,\Omega_{cc}^{+}$ | $\to$ | $\Xi_{c}^{+}\eta_{1}$ | $+2_{a}-2_{b}$ | $-1_{a}+1_{b}$ | $+8_{a}$ | $-4_{a}$ | $+4_{a}$ | $-4_{a}$ | $0$
| $2\sqrt{6}\,\Omega_{cc}^{+}$ | $\to$ | $\Xi_{c}^{0}\pi^{+}$ | $-2_{b}$ | $+1_{b}$ | $-4_{a}$ | $+2_{a}$ | $0$ | $0$ | $0$
| $2\sqrt{6}\,\Omega_{cc}^{+}$ | $\to$ | $\Lambda_{c}^{+}\bar{K}^{0}$ | $0$ | $0$ | $-4_{a}$ | $+2_{a}$ | $+4_{b}$ | $-4_{b}$ | $0$
DCS | $2\sqrt{6}\,\Xi_{cc}^{++}$ | $\to$ | $\Lambda_{c}^{+}K^{+}$ | $+2$ | $-1$ | $0$ | $0$ | $+4$ | $-4$ | $0$
| $2\sqrt{6}\,\Xi_{cc}^{+}$ | $\to$ | $\Lambda_{c}^{+}K^{0}$ | $+2$ | $-1$ | $0$ | $0$ | $+4$ | $-4$ | $0$
| $2\sqrt{6}\,\Omega_{cc}^{+}$ | $\to$ | $\Xi_{c}^{0}K^{+}$ | $-2$ | $+1$ | $-4$ | $+2$ | $0$ | $0$ | $0$
| $2\sqrt{6}\Omega_{cc}^{+}$ | $\to$ | $\Xi_{c}^{+}K^{0}$ | $+2$ | $-1$ | $+4$ | $-2$ | $0$ | $0$ | $0$
| $4\sqrt{3}\,\Omega_{cc}^{+}$ | $\to$ | $\Lambda_{c}^{+}\pi^{0}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $4\sqrt{3}\,\Omega_{cc}^{+}$ | $\to$ | $\Lambda_{c}^{+}\eta_{\omega}$ | $0$ | $0$ | $+8$ | $-4$ | $0$ | $0$ | $0$
| $2\sqrt{6}\,\Omega_{cc}^{+}$ | $\to$ | $\Lambda_{c}^{+}\eta_{\phi}$ | $0$ | $0$ | $0$ | $0$ | $+4$ | $-4$ | $0$
| $\,12\,\Omega_{cc}^{+}$ | $\to$ | $\Lambda_{c}^{+}\eta_{8}$ | $0$ | $0$ | $+8$ | $-4$ | $-8$ | $+8$ | $0$
| $6\sqrt{2}\,\Omega_{cc}^{+}$ | $\to$ | $\Lambda_{c}^{+}\eta_{1}$ | $0$ | $0$ | $+8$ | $-4$ | $+4$ | $-4$ | $0$

###  4.4 The double charm baryon decays $B_{cc}({\bf
3})\stackrel{{\scriptstyle H({\bf 6})}}{{\longrightarrow}}B_{c}(\overline{\bf
3})+M({\bf 8,\,1})$

The nonleptonic two-body decays of the triplet of double charm states
$\Xi_{cc}^{++}$, $\Xi_{cc}^{+}$ and $\Omega_{cc}^{+}$ into the antitriplet
single charm baryons belong to this class of decays. Of the many possible two
body decays only the decay $\Xi^{++}_{cc}\to\Xi_{c}^{+}\,\pi^{+}$ has been
identified to date [18].

The ${\bf 3}$ representation occurs twice in the decomposition

${\bf 3}\to{\bf 6}\otimes\overline{\bf 3}\otimes{\bf 8}=2\cdot{\bf 3}\oplus
2\cdot{\bf\overline{6}}\oplus 3\cdot{\bf 15}\oplus{\bf 15^{\prime}}\oplus{\bf
24}\oplus{\bf 42}$ (33)

There are thus $7-2=5$ relations for the tensor invariants which read

$I^{-}_{1}+2I^{-}_{2}=0,\qquad
I_{3}+2I_{4}=0,\qquad\hat{I}_{3}+\hat{I}_{4}=0,\qquad I_{5}=0,\qquad
2I^{-}_{1}=I_{3}+\hat{I}_{3}$ (34)

As before, the last relation does not hold for the decay involving the octet
singlet state. For the transitions involving the singlet meson state one finds

${\bf 3}\to{\bf 6}\otimes\overline{\bf 3}\otimes{\bf 1}={\bf 3}\oplus{\bf 15}$
(35)

which implies that all transitions involving the singlet meson are
proportional to each other with $I^{-}_{1}=I^{-}_{2}=I_{5}=0$ and
$I_{3}=-2I_{4}=2\hat{I}_{3}=-2\hat{I}_{4}$. In Table 10 we list the values of
the tensor invariants for all the decays of this class.

Table 11:  Values of the seven tensor invariants for the CF, SCS and DCS decays of the triplet double charm baryon states $\Xi_{cc}^{++}$, $\Xi_{cc}^{+}$, and $\Omega_{cc}^{+}$ into sextet single baryon states, $B_{cc}({\bf 3})\to B_{c}({\bf 6})+M({\bf 8})$. The subscripts $a$ and $b$ for the SCS decays has been explained in Table 8. Decays into $\eta_{1}$ states are not listed since they vanish identically. |  |  |  | $I^{-}_{1}$ | $I^{-}_{2}$ | $I_{3}$ | $I_{4}$ | $\hat{I}_{3}$ | $\hat{I}_{4}$ | $I_{5}$
---|---|---|---|---|---|---|---|---|---|---
CF | $2\sqrt{2}\,\Xi_{cc}^{++}$ | $\to$ | $\Xi_{c}^{\prime+}\pi^{+}$ | $0$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\,\Xi_{cc}^{++}$ | $\to$ | $\Sigma_{c}^{++}\bar{K}^{0}$ | $0$ | $+1$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\sqrt{2}\,\Xi_{cc}^{+}$ | $\to$ | $\Xi_{c}^{\prime\,0}\pi^{+}$ | $0$ | $-1$ | $0$ | $+2$ | $0$ | $0$ | $0$
| $4\,\Xi_{cc}^{+}$ | $\to$ | $\Xi_{c}^{\prime+}\pi^{0}$ | $0$ | $0$ | $0$ | $-2$ | $0$ | $0$ | $0$
| $4\,\Xi_{cc}^{+}$ | $\to$ | $\Xi_{c}^{\prime+}\eta_{\omega}$ | $0$ | $0$ | $0$ | $-2$ | $0$ | $0$ | $0$
| $2\sqrt{2}\,\Xi_{cc}^{+}$ | $\to$ | $\Xi_{c}^{\prime+}\eta_{\phi}$ | $0$ | $0$ | $0$ | $+2$ | $0$ | $0$ | $0$
| $4\sqrt{3}\,\Xi_{cc}^{+}$ | $\to$ | $\Xi_{c}^{\prime+}\eta_{8}$ | $0$ | $0$ | $0$ | $-6$ | $0$ | $0$ | $0$
| $2\,\Xi_{cc}^{+}$ | $\to$ | $\Sigma_{c}^{++}K^{-}$ | $0$ | $0$ | $0$ | $+2$ | $0$ | $0$ | $0$
| $2\sqrt{2}\,\Xi_{cc}^{+}$ | $\to$ | $\Sigma_{c}^{+}\bar{K}^{0}$ | $0$ | $+1$ | $0$ | $-2$ | $0$ | $0$ | $0$
| $2\,\Xi_{cc}^{+}$ | $\to$ | $\Omega_{c}^{0}K^{+}$ | $0$ | $0$ | $0$ | $-2$ | $0$ | $0$ | $0$
| $2\sqrt{2}\,\Omega_{cc}^{+}$ | $\to$ | $\Xi_{c}^{\prime+}\bar{K}^{0}$ | $0$ | $+1$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\,\Omega_{cc}^{+}$ | $\to$ | $\Omega_{c}^{0}\pi^{+}$ | $0$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$
SCS | $2\sqrt{2}\,\Xi_{cc}^{++}$ | $\to$ | $\Xi_{c}^{\prime+}K^{+}$ | $0$ | $+1_{a}$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\sqrt{2}\,\Xi_{cc}^{++}$ | $\to$ | $\Sigma_{c}^{+}\pi^{+}$ | $0$ | $+1_{b}$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\sqrt{2}\,\Xi_{cc}^{++}$ | $\to$ | $\Sigma_{c}^{++}\pi^{0}$ | $0$ | $-1_{b}$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\sqrt{2}\,\Xi_{cc}^{++}$ | $\to$ | $\Sigma_{c}^{++}\eta_{\omega}$ | $0$ | $+1_{b}$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\,\Xi_{cc}^{++}$ | $\to$ | $\Sigma_{c}^{++}\eta_{\phi}$ | $0$ | $-1_{a}$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\sqrt{6}\,\Xi_{cc}^{++}$ | $\to$ | $\Sigma_{c}^{++}\eta_{8}$ | $0$ | $+2_{a}+1_{b}$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\sqrt{2}\,\Xi_{cc}^{+}$ | $\to$ | $\Xi_{c}^{\prime\,0}K^{+}$ | $0$ | $+1_{a}$ | $0$ | $+2_{b}$ | $0$ | $0$ | $0$
| $2\,\Xi_{cc}^{+}$ | $\to$ | $\Sigma_{c}^{0}\pi^{+}$ | $0$ | $+1_{b}$ | $0$ | $-2_{b}$ | $0$ | $0$ | $0$
| $4\,\Xi_{cc}^{+}$ | $\to$ | $\Sigma_{c}^{+}\pi^{0}$ | $0$ | $-1_{b}$ | $0$ | $+4_{b}$ | $0$ | $0$ | $0$
| $4\,\Xi_{cc}^{+}$ | $\to$ | $\Sigma_{c}^{+}\eta_{\omega}$ | $0$ | $+1_{b}$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\sqrt{2}\,\Xi_{cc}^{+}$ | $\to$ | $\Sigma_{c}^{+}\eta_{\phi}$ | $0$ | $-1_{a}$ | $0$ | $0$ | $0$ | $0$ | $0$
| $4\sqrt{3}\,\Xi_{cc}^{+}$ | $\to$ | $\Sigma_{c}^{+}\eta_{8}$ | $0$ | $+2_{a}+1_{b}$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\sqrt{2}\,\Omega_{cc}^{+}$ | $\to$ | $\Sigma_{c}^{+}\bar{K}^{0}$ | $0$ | $0$ | $0$ | $-2_{a}$ | $0$ | $0$ | $0$
| $2\sqrt{2}\,\Omega_{cc}^{+}$ | $\to$ | $\Xi_{c}^{\prime\,0}\pi^{+}$ | $0$ | $+1_{b}$ | $0$ | $+2_{a}$ | $0$ | $0$ | $0$
| $2\,\Omega_{cc}^{+}$ | $\to$ | $\Omega_{c}^{0}K^{+}$ | $0$ | $+1_{a}$ | $0$ | $-2_{a}$ | $0$ | $0$ | $0$
DCS | $2\sqrt{2}\,\Xi_{cc}^{++}$ | $\to$ | $\Sigma_{c}^{+}K^{+}$ | $0$ | $+1$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\,\Xi_{cc}^{++}$ | $\to$ | $\Sigma_{c}^{++}K^{0}$ | $0$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\,\Xi_{cc}^{+}$ | $\to$ | $\Sigma_{c}^{0}K^{+}$ | $0$ | $+1$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\sqrt{2}\,\Xi_{cc}^{+}$ | $\to$ | $\Sigma_{c}^{+}K^{0}$ | $0$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\sqrt{2}\,\Omega_{cc}^{+}$ | $\to$ | $\Xi_{c}^{\prime\,0}K^{+}$ | $0$ | $+1$ | $0$ | $-2$ | $0$ | $0$ | $0$
| $2\sqrt{2}\,\Omega_{cc}^{+}$ | $\to$ | $\Xi_{c}^{\prime+}K^{0}$ | $0$ | $-1$ | $0$ | $+2$ | $0$ | $0$ | $0$
| $4\,\Omega_{cc}^{+}$ | $\to$ | $\Sigma_{c}^{+}\pi^{0}$ | $0$ | $0$ | $0$ | $-4$ | $0$ | $0$ | $0$
| $4\,\Omega_{cc}^{+}$ | $\to$ | $\Sigma_{c}^{+}\eta_{\omega}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\sqrt{2}\,\Omega_{cc}^{+}$ | $\to$ | $\Sigma_{c}^{+}\eta_{\phi}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $4\sqrt{3}\,\Omega_{cc}^{+}$ | $\to$ | $\Sigma_{c}^{+}\eta_{8}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$

###  4.5 The double charm baryon decays $B_{cc}({\bf
3})\stackrel{{\scriptstyle H({\bf 6})}}{{\longrightarrow}}B_{c}({\bf
6})+M({\bf 8,\,1})$

The nonleptonic two-body decays of the double charm states $\Xi_{cc}^{++}$,
$\Xi_{cc}^{+}$ and $\Omega_{cc}^{+}$ into the sextet single charm baryons
belong to this class of decays. Of the many possible decay modes none has been
identified to date.

The ${\bf 3}$ representation occurs twice in the decomposition

${\bf 3}\to{\bf 6}\otimes{\bf 6}\otimes{\bf 8}=2\cdot{\bf 3}\oplus
2\cdot\overline{\bf 6}\oplus 4\cdot{\bf 15}\oplus 2\cdot{\bf
15^{\prime}}\oplus 2\cdot{\bf 24}\oplus 2\cdot{\bf 42}\oplus{\bf 48}$ (36)

Therefore, there are $7-2=5$ relations among the seven tensor invariants which
can be derived by either tracking the charm quark flavor flow in the
respective topological diagrams or by taking into account that the single
charm baryons $B_{c}({\bf 6})$ are symmetric in the light quark indices. The
relations read

$I^{-}_{1}=I_{3}=\hat{I}_{3}=\hat{I}_{4}=I_{5}=0,$ (37)

i.e. the only nonvanishing tensor invariants are $I^{-}_{2}$ and $I_{4}$. In
particular, the tensor invariants $\hat{I}_{3}$ and $\hat{I}_{4}$ are zero
because of the KPW theorem. The KPW theorem holds only in the SU(3) limit.
Violations of the KPW theorem due to the SU(3) constituent mass breaking
effect $m_{s}\neq m_{u}$ were calculated in Ref. [77] for the decays
$\Xi_{cc}^{++}\to\Xi_{c}^{\prime+}\pi^{+}$ and
$\Omega_{cc}^{+}\to\Xi_{c}^{\prime+}\bar{K}^{0}$ and turned out to be of the
order of $(1-4)\%$. The authors of Ref. [59] obtained a mass breaking effect
of the KPW theorem of the order of 2$\%$ in their bag model calculation of the
same decays.

The singlet meson state ${\bf 1}$ does not contribute to this class of decays
as can be seen from the decomposition

${\bf 3}\to{\bf 6}\otimes{\bf 6}\otimes{\bf 1}=\overline{\bf 6}\oplus{\bf
15}\oplus{\bf 15^{\prime}}.$ (38)

In Table 11 we list the values of the seven topological flavor invariants.

Table 12: Values of the seven topological tensor invariants for the nonleptonic CF, SCS and DCS decays $B_{cc}({\bf 3})\to B({\bf 8})+M(\overline{\bf 3})$ of the triplet double charm baryon states $\Xi_{cc}^{++}$, $\Xi_{cc}^{+}$ and $\Omega^{+}_{cc}$ into the light baryon octet and the charm mesons $D^{+}$, $D^{0}$ and $D_{s}^{+}$. The notation for the SCS decays is explained in the caption of Table 8. |  |  |  | $I_{1}^{-}$ | $I^{-}_{2}$ | $I_{3}$ | $I_{4}$ | $\hat{I}_{3}$ | $\hat{I}_{4}$ | $I_{5}$
---|---|---|---|---|---|---|---|---|---|---
CF | $2\Xi_{cc}^{++}$ | $\to$ | $\Sigma^{+}D^{+}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $-2$ | $0$
| $2\Xi_{cc}^{+}$ | $\to$ | $\Sigma^{+}D^{0}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $+1$
| $2\sqrt{2}\Xi_{cc}^{+}$ | $\to$ | $\Sigma^{0}D^{+}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $-2$ | $-1$
| $2\sqrt{6}\Xi_{cc}^{+}$ | $\to$ | $\Lambda^{0}D^{+}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $-2$ | $+1$
| $2\Xi_{cc}^{+}$ | $\to$ | $\Xi^{0}D_{s}^{+}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $+1$
| $2\Omega_{cc}^{+}$ | $\to$ | $\Xi^{0}D^{+}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $-2$ | $0$
SCS | $2\Xi_{cc}^{++}$ | $\to$ | $pD^{+}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $+2_{b}$ | $0$
| $2\Xi_{cc}^{+}$ | $\to$ | $pD^{0}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $-1_{b}$
| $2\Xi_{cc}^{+}$ | $\to$ | $nD^{+}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $+2_{b}$ | $+1_{b}$
| $2\Xi_{cc}^{++}$ | $\to$ | $\Sigma^{+}D_{s}^{+}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $+2_{a}$ | $0$
| $\sqrt{2}\Xi_{cc}^{+}$ | $\to$ | $\Sigma^{0}D_{s}^{+}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $+2_{a}$ | $0$
| $2\sqrt{6}\Xi_{cc}^{+}$ | $\to$ | $\Lambda^{0}D_{s}^{+}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $+2_{a}$ | $-2_{b}$
| $2\Omega_{cc}^{+}$ | $\to$ | $\Sigma^{+}D^{0}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $+2_{a}$
| $2\sqrt{2}\Omega_{cc}^{+}$ | $\to$ | $\Sigma^{0}D^{+}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $-1_{a}$
| $2\sqrt{6}\Omega_{cc}^{+}$ | $\to$ | $\Lambda^{0}D^{+}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $+4_{b}$ | $-1_{a}$
| $2\Omega_{cc}^{+}$ | $\to$ | $\Xi^{0}D_{s}^{+}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $+2_{a}$ | $+1_{a}$
DCS | $2\Xi_{cc}^{++}$ | $\to$ | $pD_{s}^{+}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $+2$ | $0$
| $2\Xi_{cc}^{+}$ | $\to$ | $nD_{s}^{+}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $+2$ | $0$
| $2\Omega_{cc}^{+}$ | $\to$ | $pD^{0}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $+1$
| $2\Omega_{cc}^{+}$ | $\to$ | $nD^{+}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $+1$
| $2\sqrt{2}\Omega_{cc}^{+}$ | $\to$ | $\Sigma^{0}D_{s}^{+}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2\sqrt{6}\Omega_{cc}^{+}$ | $\to$ | $\Lambda^{0}D_{s}^{+}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $+4$ | $+2$

###  4.6 The double charm baryon decays $B_{cc}({\bf
3})\stackrel{{\scriptstyle H({\bf 6})}}{{\longrightarrow}}B({\bf
8})+M(\overline{\bf 3})$

None of this class of decays into a light baryon and a charm meson in the
final state has been observed. The final states in these decays have a very
distinct signature which could lead to their discovery in the not too distant
future. To our knowledge the only theoretical paper dealing with these decays
is Ref. [78]. As a seed for the $B_{cc}({\bf 3})\to B({\bf 8})+M(\overline{\bf
3})$ decay, Li et al. first consider the short distance decay of a double
charm baryon state into a single charm baryon and a $C=0$ meson [78]. The
charm quantum number is then transferred from the baryon to the meson by long
distance final state interactions. The authors of Ref. [78] estimate that the
CF decays of this class of decays could be of the order of $1\%$ with the SCS
and DCS decays suppressed by the respective CKM suppression factors.

For the SU(3) decomposition one obtains

${\bf 3}\to{\bf 6}\otimes{\bf 8}\otimes\overline{\bf 3}=2\cdot{\bf 3}\oplus
2\cdot{\bf\overline{6}}\oplus 3\cdot{\bf 15}\oplus{\bf 15^{\prime}}\oplus{\bf
24}\oplus{\bf 42}$ (39)

The decomposition shows that there are two independent SU(3) invariants.
Therefore, one expects $7-2=5$ relations among the seven topological tensor
invariants. By visual inspection of the five topologies one finds that
$I^{-}_{1}=I^{-}_{2}=I_{3}=I_{4}=0$. The topological invariant
$\hat{I}_{3}(\ell,\ell^{\prime})=B_{\ell}^{a[bc]}B^{\ell^{\prime}}_{a[b^{\prime}c^{\prime}]}M^{c^{\prime}}_{d}H^{[c^{\prime}b^{\prime}]}_{[db]}$
vanishes since $b^{\prime}$ and $c^{\prime}$ are heavy charm quark labels
which are necessarily symmetric such that the initial state tensor
$B^{\ell^{\prime}}_{a[b^{\prime}c^{\prime}]}$ vanishes. Altogether one has

$I^{-}_{1}=I^{-}_{2}=I_{3}=I_{4}=\hat{I}_{3}=0.$ (40)

The only nonvanishing topological tensor invariants are $\hat{I}_{4}$ and
$I_{5}$. In Table 12 we list the values of the nonzero tensor invariants
$\hat{I}_{4}$ and $I_{5}$ for the altogether 22 CF, SCS and DCS double charm
decays $B_{cc}\to B\,D^{+/0},B\,D_{s}^{+}$. Note that the decay
$\Omega_{cc}^{+}\to\Sigma^{0}D_{s}^{+}$ is forbidden in the SU(3) limit
because of the KPW theorem.

Table 13: Values of the seven SU(3) tensor invariants for the nonleptonic hyperon decays induced by the transitions $(s\to u;\,u\to d)$. |  |  | $I^{-}_{1}$ | $I^{-}_{2}$ | $I_{3}$ | $I_{4}$ | $\hat{I}_{3}$ | $\hat{I}_{4}$ | $I_{3}-\hat{I}_{3}$ | $I_{4}-\hat{I}_{4}$ | $I_{5}$
---|---|---|---|---|---|---|---|---|---|---|---
$2\sqrt{6}\,\Lambda^{0}$ | $\to$ | $p\pi^{-}$ | $+1$ | $+1$ | $0$ | $-2$ | $+2$ | $-4$ | $-2$ | $+2$ | $-1$
$4\sqrt{3}\,\Lambda^{0}$ | $\to$ | $n\pi^{0}$ | $-1$ | $-1$ | $0$ | $+2$ | $-2$ | $+4$ | $+2$ | $-2$ | $+1$
$2\sqrt{2}\,\Sigma^{+}$ | $\to$ | $p\pi^{0}$ | $-1$ | $+1$ | $0$ | $-2$ | $-2$ | $0$ | $+2$ | $-2$ | $-1$
$2\,\Sigma^{+}$ | $\to$ | $n\pi^{+}$ | $0$ | $0$ | $0$ | $+2$ | $0$ | $0$ | $0$ | $+2$ | $+1$
$2\,\Sigma^{-}$ | $\to$ | $n\pi^{-}$ | $-1$ | $+1$ | $0$ | $0$ | $-2$ | $0$ | $+2$ | $0$ | $0$
$2\sqrt{6}\,\Xi^{-}$ | $\to$ | $\Lambda^{0}\pi^{-}$ | $-2$ | $+1$ | $0$ | $0$ | $-4$ | $+4$ | $+4$ | $-4$ | $0$
$4\sqrt{3}\,\Xi^{0}$ | $\to$ | $\Lambda^{0}\pi^{0}$ | $-2$ | $+1$ | $0$ | $0$ | $-4$ | $+4$ | $+4$ | $-4$ | $0$

###  4.7 The hyperon decays $B(8)\stackrel{{\scriptstyle
H(8)}}{{\longrightarrow}}B(8)+M(8)$

In complete the picture obtained so far, we also consider the decays of
hyperons into light baryons. The decomposition

${\bf 8}\to{\bf 8}\otimes{\bf 8}\otimes{\bf 8}=2\cdot{\bf 1}\oplus 8\cdot{\bf
8}\oplus 4\cdot{\bf 10}\oplus 4\cdot{\bf\overline{10}}\oplus 6\cdot{\bf
27}\oplus 2\cdot{\bf 35}\oplus 2\cdot\overline{\bf 35}\oplus{\bf 64}$ (41)

suggests that there are eight SU(3) invariant couplings which outnumber the
seven invariant tensor couplings by one. However, one with the help of a
Fierz-type identity one can show that the eight SU(3) couplings are not
independent.

To proof this, consider the usual second rank tensor baryon representation

$B=\left(\begin{array}[]{ccc}-\frac{\Lambda^{0}}{\sqrt{6}}+\frac{\Sigma^{0}}{\sqrt{2}}&-\Sigma^{+}&p\\\
\Sigma^{-}&-\frac{\Lambda^{0}}{\sqrt{6}}-\frac{\Sigma^{0}}{\sqrt{2}}&n\\\
\Xi^{-}&-\Xi^{0}&\frac{2\Lambda^{0}}{\sqrt{6}}\\\ \end{array}\right),$ (42)

together with the representations for mesons and the weak transition tensor

$\bar{M}=\left(\begin{array}[]{ccc}\frac{\eta_{8}}{\sqrt{6}}+\frac{\pi^{0}}{\sqrt{2}}&\pi^{-}&K^{-}\\\
-\pi^{+}&\frac{\eta_{8}}{\sqrt{6}}-\frac{\pi^{0}}{\sqrt{2}}&-\bar{K}^{0}\\\
K^{+}&K^{0}&-\frac{2\eta_{8}}{\sqrt{6}}\\\ \end{array}\right),\qquad
S=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&1\\\ 0&0&0\\\ \end{array}\right).$
(43)

In SU(3) a Levi-Civita tensor with four indices in three flavor dimensions
vanishes. Therefore, one can write down a Fierz-type identity which reads

$\epsilon_{abcd}\epsilon^{efgh}=\left|\begin{array}[]{cccc}\delta_{a}^{e}&\delta_{a}^{f}&\delta_{a}^{g}&\delta_{a}^{h}\\\
\delta_{b}^{e}&\delta_{b}^{f}&\delta_{b}^{g}&\delta_{b}^{h}\\\
\delta_{c}^{e}&\delta_{c}^{f}&\delta_{c}^{g}&\delta_{c}^{h}\\\
\delta_{d}^{e}&\delta_{d}^{f}&\delta_{d}^{g}&\delta_{d}^{h}\\\
\end{array}\right|=0.$ (44)

When one contracts the Fierz-type identity with
$B^{a}_{e}\bar{B}^{b}_{f}\bar{M}^{c}_{g}S^{d}_{h}$ one obtains the relation

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\mathop{\rm
tr}\nolimits(B\bar{B}\bar{M}S)+\mathop{\rm
tr}\nolimits(B\bar{B}S\bar{M})+\mathop{\rm
tr}\nolimits(B\bar{M}S\bar{B})+\mathop{\rm
tr}\nolimits(B\bar{M}\bar{B}S)+\mathop{\rm tr}\nolimits(BS\bar{M}\bar{B})+$
(45) $\displaystyle+\mathop{\rm tr}\nolimits(BS\bar{B}\bar{M})-\mathop{\rm
tr}\nolimits(B\bar{B})\mathop{\rm tr}\nolimits(\bar{M}S)-\mathop{\rm
tr}\nolimits(B\bar{M})\mathop{\rm tr}\nolimits(\bar{B}S)-\mathop{\rm
tr}\nolimits(BS)\mathop{\rm tr}\nolimits(\bar{B}\bar{M}).$

We conjecture that the use of this Fierz-type identity would provide for the
missing linear relations between tensor invariants of differing topologies. In
Table 13 one can identify the $\Delta I=1/2$ (or octet) rules [79, 80]

$\displaystyle{\cal A}(\Lambda^{0}\to p\pi^{-})$ $\displaystyle=$
$\displaystyle-\sqrt{2}{\cal A}(\Lambda^{0}\to n\pi^{0})$ (46)
$\displaystyle\sqrt{2}{\cal A}(\Sigma^{+}\to p\pi^{0})$ $\displaystyle=$
$\displaystyle-{\cal A}(\Sigma^{+}\to n\pi^{+})+{\cal A}(\Sigma^{-}\to
n\pi^{-})$ (47) $\displaystyle{\cal A}(\Xi^{-}\to\Lambda^{0}\pi^{-})$
$\displaystyle=$ $\displaystyle\sqrt{2}{\cal
A}(\Xi^{0}\to\Lambda^{0}\pi^{0}).$ (48)

However, for some reason the Lee–Sugawara relation [81, 82] is not satisfied,

$2A(\Xi^{-}\to\Lambda^{0}\pi^{-})+A(\Lambda^{0}\to
p\pi^{-})\neq\sqrt{3}A(\Sigma^{+}\to p\pi^{0})$ (49)

Topological tensor invariants for this class of processes have been considered
in Ref. [83].

##  5 The current algebra description of
nonleptonic charm baryon decays

In the second part of this general survey we add elements of the current
algebra approach to the information contained in the tables. The current
algebra approach was originally developed for the description of nonleptonic
hyperon decays (see Refs. [84, 85, 86, 87] and references therein) and later
applied to nonleptonic decays of charmed baryons [88, 89]. As will be
described in the following it turns out that the flavor invariants of the
$W$-exchange contributions to the current algebra amplitudes can be expressed
in terms of the topological flavor invariants $I_{i}$ ($i=3,4,5$) and
$\hat{I}_{i}$ ($i=3,4$) introduced in Sec. 3. We briefly recapitulate the
standard current algebra plus soft pion approach to nonleptonic two-body charm
baryon decays $B(1/2^{+})\to B(1/2^{+})+M(0^{-})$ where we will stay quite
close to the presentation and notation of Refs. [54, 59, 60, 61, 62].

We define $S$\- and $P$-wave amplitudes $A_{fki}$ and $B_{fki}$, resp., by
writing

$\langle B_{f}M_{k}|{\cal H}_{\rm
eff}|B_{i}\rangle=\bar{u}_{f}(A_{fki}-B_{fki}\gamma_{5})u_{i}.$ (50)

We follow the convention of Ref. [54, 59, 60, 61, 62] in that $A_{fki}$ and
$B_{fki}$ are defined with a relative minus sign in Eq. (50). Note that the
choice of this sign will affect the sign of the calculated asymmetry parameter
of the nonleptonic two-body decays.

Again in the notation of Ref. [54, 59, 60, 61, 62] the p.v. and p.c.
amplitudes $A$ and $B$ are given by

$A_{fki}^{\phantom{a}}=A_{fki}^{\rm fac}+A_{fki}^{\rm pole}+A_{fki}^{\rm
com},\qquad B_{fki}^{\phantom{a}}=B_{fki}^{\rm fac}+B_{fki}^{\rm
pole}+B_{fki}^{\rm com}.$ (51)

The factorizing contributions $A_{fki}^{\rm fac}$ and $B_{fki}^{\rm fac}$
related to the topological invariants $I^{-}_{1}$ and $I^{-}_{2}$ are
complemented by the nonfactorizing contributions, consisting of a pole and a
commutator part.

###  5.1 The parity violating $S$-wave amplitude $A_{fki}^{\rm com}$

For the commutator contribution one obtains

$A_{fki}^{\rm com}=\frac{\sqrt{2}}{f_{k}}\langle B_{f}|[M_{k},{\cal H}^{\rm
pc}_{\rm eff}]|B_{i}\rangle,$ (52)

where ${\cal H}^{\rm pc}_{\rm eff}$ is the p.c. part of the effective
Hamiltonian, $M_{k}$ is the SU(3) vector charge associated with the
pseudoscalar meson $k$, and where the $f_{k}$ are the pseudoscalar coupling
constants, with e.g. $f_{\pi}=0.95\,m_{\pi}$. One then introduces a sum over
intermediate baryon states $\sum_{\ell}|B_{\ell}\rangle\langle B_{\ell}|$ and
$\sum_{\ell^{\prime}}|B_{\ell^{\prime}}\rangle\langle B_{\ell^{\prime}}|$ to
rewrite Eq. (52) as

$\displaystyle A^{\rm com}_{fki}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{f_{k}}\Big{(}\sum_{\ell}\langle
B_{f}|M_{k}|B_{\ell}\rangle\langle B_{\ell}|{\cal H}^{\rm pc}_{\rm
eff}|B_{i}\rangle-\sum_{\ell^{\prime}}\langle B_{f}|{\cal H}^{\rm pc}_{\rm
eff}|B_{\ell^{\prime}}\rangle\langle
B_{\ell^{\prime}}|M_{k}|B_{i}\rangle\Big{)}\ =$ (53) $\displaystyle=$
$\displaystyle A_{fki}^{\rm com}(s)-A_{fki}^{\rm com}(u)$

Since $M_{k}$ is a conserved vector charge operator, the sum over intermediate
states extends only over the twenty $J^{p}=1/2^{\,+}$ ground state baryons of
the SU(4) ${\bf 20^{\prime}}$ representation.

The notation “$s$” and “$u$” stands for the $s$\- and $u$-channel
contributions in the current algebra approach where the $s$-channel
contribution refers to the process where one has a weak transition at the
first stage followed by meson emission and vice-versa for the $u$-channel
contribution, as depicted in Fig. 3. One can associate the topological
diagrams with the $s$-channel and $u$-channel contributions by cutting the
diagrams IIa, IIb and III in Fig. 1 at the appropriate places. Diagrams IIa
and IIb are clearly associated with the $s$\- and $u$-channel contributions,
respectively, whereas diagram III can be seen to be associated with both $s$\-
and $u$-channel contributions depending on where diagram III is cut. As it
turns out, diagram III contributes in equal amounts to the $s$\- and
$u$-channel parts for both $S$\- and $P$-wave transitions. Altogether the
$s$-channel contribution correspond to the contribution of diagram IIa and one
half of diagram III, while the $u$-channel contribution corresponds to diagram
IIb and one half of diagram III.

Figure 3:  $s$-channel contribution (left) and $u$-channel contribution
(right)

Our aim is to proof that the commutator contributions $A_{fki}^{\rm com}(s)$
and $A_{fki}^{\rm com}(u)$ can be rewritten in terms of a linear superposition
of the topological tensor invariants $I_{3}$, $I_{4}$, $\hat{I}_{3}$,
$\hat{I}_{4}$ and $I_{5}$ introduced in Sec. 3. This is achieved by making use
of the completeness relation (12). The same can be done for the pole term
contributions $A_{kfi}^{\rm pole}(s)$ and $A_{fki}^{\rm pole}(u)$ in the
absence of hyperfine mass splittings for the flavor degenerate members of the
${\bf 20^{\prime}}$ representation, as will be written down later.

In Appendix C we demonstrate that the $f$-type baryon matrix element $\langle
B_{f}|M_{k}|B_{\ell}\rangle$ of the conserved vector charge can be expressed
in terms of two basic SU(3) contractions $\tilde{I}_{1}$ and $\tilde{I}_{2}$.
One has

$\langle
B_{f}|M_{k}|B_{\ell}\rangle:=(I^{f})_{fk\ell}=4(\tilde{I}_{1})_{fk\ell}+2(\tilde{I}_{2})_{fk\ell}$
(54)

and similarly for $\langle B_{\ell^{\prime}}|M_{k}|B_{i}\rangle$. The two
basic tensor contractions $\tilde{I}_{1}$ and $\tilde{I}_{2}$ are given by

$(\tilde{I}_{1})_{fk\ell}=B_{f}^{a[bc]}B^{\ell}_{a[bc^{\prime}]}(M_{k})^{c^{\prime}}_{c}\quad(\tilde{I}_{2})_{fk\ell}=B_{f}^{a[bc]}B^{\ell}_{b[c^{\prime}a]}(M_{k})^{c^{\prime}}_{c}.$
(55)

The matrix element $\langle B_{\ell}|{\cal H}^{\rm pc}_{\rm eff}|B_{i}\rangle$
of the effective Hamiltonian (called $a_{\ell\,i}$ in Ref. [60]) splits into a
dynamical piece and a symmetry factor. In the bag model calculation the
factors $a_{\ell\,i}$ can be expressed in terms of a single tensor
contraction. In the normalization of Ref. [60] one writes

$\langle B_{\ell}|{\cal H}^{\rm pc}_{\rm
eff}|B_{\ell^{\prime}}\rangle:=a_{\ell\ell^{\prime}}=6\bar{X}_{2}(4\pi)(I^{\rm
pc})_{\ell\ell^{\prime}}$ (56)

with

$(I^{\rm
pc})_{\ell\ell^{\prime}}=B_{\ell}^{a[bc]}B^{\ell^{\prime}}_{a[b^{\prime}c^{\prime}]}H^{[b^{\prime}c^{\prime}]}_{[bc]}.$
(57)

In addition, one has

$A_{fki}^{\rm com}(s)-A_{fki}^{\rm
com}(u)=\frac{1}{f_{k}}6\bar{X}_{2}(4\pi)\left(\hat{A}_{fki}^{\rm
com}(s)-\hat{A}_{fki}^{\rm com}(u)\right)$ (58)

with

$\hat{A}_{fki}^{\rm com}(s)-\hat{A}_{fki}^{\rm
com}(u)=\sum_{\ell}(I^{f})_{fk\ell}(I^{\rm
pc})_{\ell\,i}-\sum_{\ell^{\prime}}(I^{\rm
pc})_{f\ell^{\prime}}(I^{f})_{\ell^{\prime}ki}.$ (59)

In Ref. [60] the values of the bag integrals as e.g. $X_{2}$ are given as
dimensionless numbers. This is puzzling at first sight since the amplitude
$A^{\rm com}_{fki}$ is dimensionless. This conundrum is resolved after a
literature search. As stated some time ago in Ref. [43], the values of the bag
integrals are given in units of $c_{-}\,G_{F}\,\mathop{\rm GeV}\nolimits^{3}$
which should be augmented by the CKM factor $C_{\rm CKM}$.

The relation to the topological tensor invariants is obtained by using the
completeness relation (12) in order to perform the sum over the intermediate
baryon states in Eq. (53). We begin with the $s$-channel contribution
$A_{fki}^{\rm com}(s)$ for which we go through the derivation step by step. As
discussed in Appendix C, the flavor invariant contribution $I^{f}$ can be
expressed as a linear superposition of the two building blocks $\tilde{I}_{1}$
and $\tilde{I}_{2}$ in the form $I^{f}=4\tilde{I}_{1}+2\tilde{I}_{2}$. Let us
write out the contributions of the two building blocks $\tilde{I}_{1}$ and
$\tilde{I}_{2}$ to the sum over intermediate states in $A_{fki}^{\rm com}(s)$.
The result of taking the sum over the intermediate states via the completeness
relation (12) can be expressed in terms of the topological tensor invariants
$I_{3}$, $I_{4}$ and $I_{5}$. One has

$\displaystyle\sum_{\ell}(\tilde{I}_{1})_{fk\ell}I^{\rm pc}_{\ell\,i}$
$\displaystyle=$ $\displaystyle
B_{f}^{a[bc]}(M_{k})^{c^{\prime}}_{c}\Big{(}\sum_{\ell}B^{\ell}_{a[bc^{\prime}]}B_{\ell}^{r[st]}\Big{)}B^{i}_{r[a^{\prime}b^{\prime}]}H^{[a^{\prime}b^{\prime}]}_{[st]}=\tfrac{2}{3}I_{3}-\tfrac{1}{3}I_{4}+\tfrac{2}{3}I_{5},$
$\displaystyle\sum_{\ell}(\tilde{I}_{2})_{fk\ell}I^{\rm pc}_{\ell\,i}$
$\displaystyle=$ $\displaystyle
B_{f}^{a[bc]}(M_{k})^{c^{\prime}}_{c}\Big{(}\sum_{\ell}B^{\ell}_{b[c^{\prime}a]}B_{\ell}^{r[st]}\Big{)}B^{i}_{r[a^{\prime}b^{\prime}]}H^{[a^{\prime}b^{\prime}]}_{[st]}=-\tfrac{1}{3}I_{3}+\tfrac{2}{3}I_{4}+\tfrac{2}{3}I_{5}.\qquad$
(60)

The contribution of $I_{4}$ cancels in the sum
$I^{f}=4\tilde{I}_{1}+2\tilde{I}_{2}$ of the two contributions (5.1), and one
arrives at

$\hat{A}_{fki}^{\rm com}(s)=4\,\sum_{\ell}(\tilde{I}_{1})_{fk\ell}I^{\rm
pc}_{\ell\,i}+2\,\sum_{\ell}(\tilde{I}_{2})_{fk\ell}I^{\rm
pc}_{\ell\,i}=2I_{3}+4I_{5}.$ (61)

Doing the same exercise for the $u$-channel contribution $A_{fki}^{\rm
com}(u)$ in Eq. (53) one obtains

$\displaystyle\sum_{\ell^{\prime}}(I^{\rm
pc})_{f\ell^{\prime}}(\tilde{I}_{1})_{\ell^{\prime}ki}$ $\displaystyle=$
$\displaystyle
B_{f}^{a[bc]}H^{b^{\prime}c^{\prime}}_{bc}\Big{(}\sum_{\ell^{\prime}}B^{\ell}_{a[b^{\prime}c^{\prime}]}B_{\ell}^{r[st]}\Big{)}B^{i}_{r[sa^{\prime}]}(M_{k})^{a^{\prime}}_{t}\
=\ \tfrac{2}{3}\hat{I}_{3}-\tfrac{1}{3}\hat{I}_{4}+\tfrac{2}{3}I_{5},$
$\displaystyle\sum_{\ell^{\prime}}(I^{\rm
pc})_{f\ell^{\prime}}(\tilde{I}_{2})_{\ell^{\prime}ik}$ $\displaystyle=$
$\displaystyle
B_{f}^{a[bc]}H^{[a^{\prime}b^{\prime}]}_{[st]}\Big{(}\sum_{\ell}B^{\ell}_{b[c^{\prime}a]}B_{\ell}^{r[st]}\Big{)}B^{i}_{s[a^{\prime}r]}(M_{k})^{a^{\prime}}_{t}\
=\ -\tfrac{1}{3}\hat{I}_{3}+\tfrac{2}{3}\hat{I}_{4}+\tfrac{2}{3}I_{5},$

leading to

$\hat{A}_{fki}^{\rm com}(u)=4\sum_{\ell^{\prime}}(I^{\rm
pc})_{f\ell^{\prime}}(\tilde{I}_{1})_{\ell^{\prime}ki}+2\sum_{\ell^{\prime}}(I^{\rm
pc})_{f\ell^{\prime}}(\tilde{I}_{2})_{\ell^{\prime}ki}=2\hat{I}_{3}+4I_{5}.$
(63)

Calculating the difference of the $s$\- and $u$-channel contributions
$\hat{A}_{fki}^{\rm com}(s)$ and $\hat{A}_{fki}^{\rm com}(u)$ according to Eq.
(53), one obtains the remarkable result that the contribution of the
topological tensor invariant $I_{5}$ cancels out. The final result is

$\hat{A}_{fki}^{\rm com}=\hat{A}_{fki}^{\rm com}(s)-\hat{A}_{fki}^{\rm
com}(u)=2(I_{3}-\hat{I}_{3}).$ (64)

We mention that the result $A_{fki}^{\rm com}\sim I_{3}-\hat{I}_{3}$ was
already been derived in the early paper [30].

The p.v. pole contributions not dealt with in detail in this paper read

$A_{fki}^{\rm
pole}=-\sum_{\ell}\frac{g_{fk\ell}\,b_{\ell\,i}}{m_{i}-m_{\ell}}-\sum_{\ell^{\prime}}\frac{b_{f\ell^{\prime}}\,g_{\ell^{\prime}ki}}{m_{f}-m_{\ell^{\prime}}}\
=\ A_{fki}^{\rm pole}(s)+A_{fki}^{\rm pole}(u),$ (65)

where $b_{\ell\ell^{\prime}}=\langle B_{\ell}|{\cal H}^{\rm pv}_{\rm
eff}|B_{\ell^{\prime}}\rangle$ are the p.v. matrix elements which are much
smaller than the p.c. matrix elements $a_{\ell\ell^{\prime}}$ [90, 91]. For
the same reason, one also skips

$B^{\rm com}_{fki}=\frac{\sqrt{2}}{f_{k}}\Big{(}\sum_{\ell}\langle
B_{f}|M_{k}|B_{\ell}\rangle\langle B_{\ell}|{\cal H}^{\rm pv}_{\rm
eff}|B_{i}\rangle-\sum_{\ell^{\prime}}\langle B_{f}|{\cal H}^{\rm pv}_{\rm
eff}|B_{\ell^{\prime}}\rangle\langle
B_{\ell^{\prime}}|M_{k}|B_{i}\rangle\Big{)}$ (66)

###  5.2 The parity conserving $P$-wave amplitude $B_{fki}^{{\rm pole}}$

Using again the notation of Refs. [54, 59, 60, 61, 62], the pole contribution
to the p.c. $P$-wave amplitude $B$ is given by

$B_{fki}^{\rm
pole}=\sum_{\ell}\frac{g_{fk\ell}\,a_{\ell\,i}}{m_{i}-m_{\ell}}+\sum_{\ell^{\prime}}\frac{a_{f\ell^{\prime}}\,g_{\ell^{\prime}ki}}{m_{f}-m_{\ell^{\prime}}}=B_{fki}^{\rm
pole}(s)+B_{fki}^{\rm pole}(u).$ (67)

Again, the first and second terms in Eq. (67) represent the $s$\- and
$u$-channel pole contributions, respectively. The sum over the intermediate
state labels $\ell$ and $\ell^{\prime}$ extends over all 20 $J^{P}=1/2^{+}$
ground state baryons that have the correct quantum numbers to contribute to
the $s$\- and $u$-channel pole contributions (67). In the present application
one has zero, one or maximally two contributions in the sum over intermediate
ground state baryons. The latter case occurs when pairs of flavor degenerate
states contribute to the sum as e.g. the states ($\Lambda^{0}$, $\Sigma^{0}$)
in the $C=0$ sector or ($\Lambda_{c}^{+}$, $\Sigma_{c}^{+}$), ($\Xi_{c}^{0}$,
$\Xi\,^{\prime\,0}_{c}$) and ($\Xi_{c}^{+}$, $\Xi\,^{\prime\,+}_{c}$) in the
$C=1$ sector. The members of the ground state $J^{P}=3/2^{+}$ 20-plet do not
contribute to the sum over intermediate states since the weak transitions
$a_{\ell\,i}$ and $a_{f\ell^{\prime}}$ with $\ell,\ell^{\prime}$ from the
$J^{P}=3/2^{+}$ 20-plet vanish in the SU(3) limit because of the KPW theorem.

The $s$\- and $u$-channel pole contributions involve the strong coupling
coefficients denoted by $g_{fk\ell}$ and the weak parity conserving transition
matrix element $a_{\ell\,i}$ which also appeared in $A^{\rm com}_{fki}$. One
then uses the generalized Goldberger–Treiman relation

$g_{fk\ell}=\frac{\sqrt{2}}{f_{k}}(m_{f}+m_{\ell})g^{A}_{fk\ell}$ (68)

to express the strong coupling coefficients $g_{fk\ell}$ by the appropriate
axial vector elements $g^{A}_{fk\ell}$. One has

$B_{fki}^{\rm pole}(s)+B_{fki}^{\rm
pole}(u)=\frac{\sqrt{2}}{f_{k}}\left(\sum_{\ell}\,g^{A}_{fk\ell}\,\frac{m_{f}+m_{\ell}}{m_{i}-m_{\ell}}\,a_{\ell\,i}+\sum_{\ell^{\prime}}a_{f\ell^{\prime}}\frac{m_{i}+m_{\ell^{\prime}}}{m_{f}-m_{\ell^{\prime}}}\,g^{A}_{\ell^{\prime}ki}\right).$
(69)

As in the p.v. case we factor out the parameters depending on the dynamic
model to remain with a purely flavor dependent contribution $\hat{B}$. In the
bag model calculation of Ref. [54, 59, 60, 61, 62] this is achieved by writing

$\displaystyle B_{fki}^{\rm pole}(s)+B_{fki}^{\rm pole}(u)$ $\displaystyle=$
$\displaystyle\frac{1}{f_{k}}6\bar{X}_{2}(4\pi)\frac{2}{3}\bar{Z}(4\pi)\left(\hat{B}_{fki}^{\rm
pole}(s)+\hat{B}_{fki}^{\rm pole}(u)\right)$ (70)

where

$\hat{B}_{fki}^{\rm pole}(s)+\hat{B}_{fki}^{\rm pole}(u)=\sum_{\ell}I^{\rm
CQM}_{fk\ell}I^{\rm pc}_{\ell\,i}R_{fi}(B_{\ell})+\sum_{\ell}I^{\rm
pc}_{f\ell^{\prime}}I^{\rm CQM}_{\ell^{\prime}ki}R_{if}(B_{\ell^{\prime}})$
(71)

with the compact notation for the two mass ratio expressions

$R_{fi}(B_{\ell})=\frac{m_{f}+m_{\ell}}{m_{i}-m_{\ell}}=R_{s}(B_{\ell}),\qquad
R_{if}(B_{\ell^{\prime}})=\frac{m_{i}+m_{\ell^{\prime}}}{m_{f}-m_{\ell^{\prime}}}=R_{u}(B_{\ell^{\prime}}),$
(72)

the second ones used as pseudonyms. The flavor invariant $I^{\rm CQM}_{fki}$
denotes the flavor structure of the matrix element $\langle
B_{\ell^{\prime}}M_{k}|B_{i}\rangle$ in the constituent quark model. In the
bag model the flavor structure of the strong coupling is given by a $d/f$
ratio of $d/f=3/2$ (cf. Appendix C). In terms of the two building blocks
$\tilde{I}_{1}$ and $\tilde{I}_{2}$ introduced in Eq. (55) the strong matrix
element is given by

$I^{\rm
CQM}_{fki}=(4\tilde{I}_{1}+5\tilde{I}_{2})_{fki}=B_{f}^{a[bc]}\Big{(}4B^{i}_{a[bc^{\prime}]}+5B^{i}_{b[c^{\prime}a]}\Big{)}(M_{k})^{c^{\prime}}_{c}$
(73)

Different from the p.v. $S$-wave case the sum over intermediate states cannot
be taken because of the mass ratio factors which differ for the contributions
of the hyperfine doublet partners. However, if one neglects the hyperfine mass
splitting, the sum over intermediate states can be performed as in the p.v.
$S$-wave case. To proceed, we define average values of the masses of the set
of hyperfine doublet partners $\\{B_{\ell}\\}$ denoted by $\bar{m}_{\ell}$ and
correspondingly average values for the mass ratio factors by writing

$\bar{R}_{fi}=\frac{m_{f}+\bar{m}_{\ell}}{m_{i}-\bar{m}_{\ell}}=\bar{R}_{s}(\\{B_{\ell}\\}),\qquad\bar{R}_{if}=\frac{m_{i}+\bar{m}_{\ell^{\prime}}}{m_{f}-\bar{m}_{\ell^{\prime}}}=\bar{R}_{u}(\\{B_{\ell^{\prime}}\\}).$
(74)

The average mass ratio factors $\bar{R}_{fi}$ and $\bar{R}_{if}$ can then be
factored out from the sum over intermediate states and one obtains

$\hat{B}_{fki}^{\rm pole}(s)+\hat{B}_{fki}^{\rm
pole}(u)=\bar{R}_{fi}\sum_{\ell}I^{\rm CQM}_{fk\ell}I^{\rm
pc}_{\ell\,i}+\bar{R}_{if}\sum_{\ell^{\prime}}I^{\rm
pc}_{f\ell^{\prime}}I^{\rm CQM}_{\ell^{\prime}ki}.$ (75)

The sum over the intermediate states can now be performed, using again the
completeness relation (12). The task is simplified by the fact that the strong
transition $I^{\rm CQM}=4\tilde{I}_{1}+5\tilde{I}_{2}$ is a linear
superposition of the same two building blocks $\tilde{I}_{1}$ and
$\tilde{I}_{2}$ that were used in the evaluation of the corresponding sums in
the amplitude $A$. The result of calculating the sum (75) is then given by

$\displaystyle\sum_{\ell}I^{\rm CQM}_{fk\ell}I^{\rm pc}_{\ell\,i}$
$\displaystyle=$ $\displaystyle I_{3}+2I_{4}+6I_{5},$
$\displaystyle\sum_{\ell^{\prime}}I^{\rm pc}_{f\ell^{\prime}}I^{\rm
CQM}_{\ell^{\prime}ki}$ $\displaystyle=$
$\displaystyle\hat{I}_{3}+2\hat{I}_{4}+6I_{5}.$ (76)

The flavor content of the pole model contributions is thus given by

$\hat{B}_{fki}^{\rm pole}(s)+\hat{B}_{fki}^{\rm
pole}(u)=(I_{3}+2I_{4}+6I_{5})\bar{R}_{fi}+(\hat{I}_{3}+2\hat{I}_{4}+6I_{5})\bar{R}_{if}.$
(77)

We have found again the quite remarkable fact that, in the absence of
hyperfine mass splittings for the flavor degenerate members of the ${\bf
20^{\prime}}$ representation, the pole model representation of the p.c.
$P$-wave amplitude $B$ depends only on the initial and final states of the
nonleptonic decays and not on the detailed structure of the intermediate
states.

Eqs. (5.2) can be generalized to the case where the strong $d/f$ ratio takes
on general values (see Appendix C). One obtains

$\displaystyle\sum_{\ell}I^{\rm gen}_{fk\ell}I^{\rm pc}_{\ell\,i}(s)$
$\displaystyle=$
$\displaystyle\tfrac{5}{3}d(I_{3}+2I_{4}+6I_{5})-(\tfrac{5}{3}d-f)(2I_{3}+4I_{5}),$
$\displaystyle\sum_{\ell^{\prime}}I^{\rm pc}_{f\ell^{\prime}}I^{\rm
gen}_{\ell^{\prime}ki}(u)$ $\displaystyle=$
$\displaystyle\tfrac{2}{3}d(\hat{I}_{3}+2\hat{I}_{4}+6I_{5})-(\tfrac{2}{3}d-f)(2\hat{I}_{3}+4I_{5}).$
(78)

One can check that one recovers Eqs. (5.2) for $d/f=3/2$, i.e. for $d=3/5$ and
$f=2/5$.

##  6 Some more sample decays

Initiated by a thorough analysis of the methods applied in Refs. [59, 60, 61,
62] with details found in Appendix D, in this section we present explicit
results for some sample decays on the connection of the current algebra
results with the topological tensor invariants. We emphasize again that this
connection holds true in the SU(3) limit. The relation between the two
approaches derived in Sec. 5 is explicitly verified in these examples. We
include at least one decay of each of the classes of decays discussed in Sec.
4. For each decay we include in the header the values of the seven tensor
invariants in the same sequence
$(I^{-}_{1},I^{-}_{2},I_{3},I_{4},\hat{I}_{3},\hat{I}_{4},I_{5})$ as in the
tables. For a better perception we have underlined the nonvanishing
contributions of the topological tensor invariants.

###  6.1 The CF decay $\Lambda_{c}^{+}\to\Lambda^{0}\,\pi^{+}$
$12(I_{i})=(-2,-2,-2,4,-2,4,1)$

In this case, all seven tensor invariants are non-zero. Together with the SCS
decay $\Lambda_{c}^{+}\to\Lambda^{0}\,K^{+}$, this decay is the only one in
which $I_{3}=\hat{I}_{3}\neq 0$. Since one has $I_{3}-\hat{I}_{3}=0$, one
concludes that in these decays the $W$-exchange contribution to the p.v.
amplitude $A$ is zero. Nevertheless, there are also tree diagram contributions
which will contribute to the amplitude $A$. Expressed in terms of the
topological tensor invariants, the $W$-exchange contributions to the reduced
amplitudes $\hat{A}$ and $\hat{B}$ are given by

$\displaystyle\hat{A}^{\rm com}$ $\displaystyle=$
$\displaystyle(\underline{2I_{3}}+\underline{4I_{5}})-(\underline{2\hat{I}_{3}}+\underline{4I_{5}})=0-0,$
$\displaystyle\hat{B}^{\rm pole}$ $\displaystyle=$
$\displaystyle(\underline{I_{3}}+\underline{2I_{4}}+\underline{6I_{5}})\,R_{s}(\Sigma^{+})+(\underline{\hat{I}_{3}}+\underline{2\hat{I}_{4}}+\underline{6I_{5}})\,R_{u}(\Sigma_{c}^{0})\
=$ (79) $\displaystyle=$ $\displaystyle 1\cdot R_{s}(\Sigma^{0})+1\cdot
R_{u}(\Sigma_{c}^{0}).$

The same result is obtained by explicit summation over the intermediate
states. The intermediate states are $\Sigma^{+}$ in the $s$-channel and
$\Sigma_{c}^{0}$ in the $u$-channel. The intermediate states do not contribute
to the p.v. amplitude $A$ since $F_{\pi^{+}}$ is a conserved charge operator
which implies $I^{f}_{\Lambda^{0}\pi^{+}\Sigma^{+}}=0$ and
$I^{f}_{\Sigma_{c}^{0}\pi^{+}\Lambda_{c}^{+}}=0$. The contribution of the
intermediate states to the p.c. amplitude $B$ is given in terms of the flavor
factors multiplying the relavent mass ratio factors $R_{s}(\Sigma^{0})$ and
$R_{u}(\Sigma_{c}^{0})$. One has

$\displaystyle\hat{B}^{\rm pole}_{s}(\Sigma^{0})$ $\displaystyle:$
$\displaystyle I^{\rm CQM}_{\Lambda^{0}\pi^{+}\Sigma^{+}}I^{\rm
pc}_{\Sigma^{+}\Lambda_{c}^{+}}=1,$ $\displaystyle\hat{B}^{\rm
pole}(u;\Sigma_{c}^{0})$ $\displaystyle:$ $\displaystyle I^{\rm
pc}_{\Lambda^{0}\Sigma_{c}^{0}}I^{\rm
CQM}_{\Sigma_{c}^{0}\pi^{+}\Lambda_{c}^{+}}=1,$ (80)

where $I^{\rm pc}_{\Sigma^{+}\Lambda_{c}^{+}}=-4/2\sqrt{6}$, $I^{\rm
pc}_{\Lambda^{0}\Sigma_{c}^{0}}=4/2\sqrt{6}$, $I^{\rm
CQM}_{\Lambda^{0}\pi^{+}\Sigma^{+}}=-6/2\sqrt{6}$, and $I^{\rm
CQM}_{\Sigma_{c}^{0}\pi^{+}\Lambda_{c}^{+}}=6/2\sqrt{6}$ (see Tables 16 and 18
in Appendix C). The mass ratios read

$R_{s}(\Sigma^{0})=\frac{m_{\Lambda^{0}}+m_{\Sigma^{0}}}{m_{\Lambda_{c}^{0}}-m_{\Sigma^{0}}}=2.11\qquad
R_{u}(\Sigma_{c}^{0})=\frac{m_{\Lambda_{c}^{0}}+m_{\Sigma_{c}^{0}}}{m_{\Lambda^{0}}-m_{\Sigma_{c}^{0}}}=-3.54,$
(81)

Our results are in agreement with the results in Ref. [60] up to an overall
sign difference. Note that there is a contribution to $\hat{B}^{\rm pole}$ in
Eq. (6.1) from the topology III given by the term proportional to $I_{5}$. We
conclude that this term was in fact included in the analysis of Ref. [60]
despite the claim of the authors that the contributions from the type III
diagram were omitted.

###  6.2 The CF decay $\Lambda_{c}^{+}\to\Sigma^{0}\pi^{+}$
$4\sqrt{3}(I_{i})=(0,2,2,0,-2,4,1)$

The second decay we discuss is the CF decay
$\Lambda_{c}^{+}\to\Sigma^{0}\pi^{+}$. Again, this decay proceeds only by
$W$-exchange. The asymmetry parameter has been measured and is given by
$\alpha(\Lambda_{c}\to\Sigma^{0}\pi^{+})=-0.73\pm 0.17\pm 0.07$. This implies
that the $S$-wave amplitude must be nonvanishing. And in fact, a glance at
Table 3 shows that one has a $s$-channel contribution while the $u$-channel
contribution vanishes since $2\hat{I}_{3}+4I_{5}=0$. Let us check this in more
detail. The $W$-exchange contributions to the reduced amplitudes $\hat{A}$ and
$\hat{B}$ are given by

$\displaystyle\hat{A}^{\rm com}$ $\displaystyle=$
$\displaystyle(\underline{2I_{3}}+\underline{4I_{5}})-(\underline{2\hat{I}_{3}}+\underline{4I_{5}})\
=\ 2/\sqrt{3}-0(u),$ $\displaystyle\hat{B}^{\rm pole}$ $\displaystyle=$
$\displaystyle(\underline{I_{3}}+2I_{4}+\underline{6I_{5}})\,R_{s}(\Sigma^{+})+(\underline{\hat{I}_{3}}+\underline{2\hat{I}_{4}}+\underline{6I_{5}})\,R_{u}(\Sigma_{c}^{0})\
=$ (82) $\displaystyle=$ $\displaystyle
2/\sqrt{3}\,R_{s}(\Sigma^{+})+3/\sqrt{3}\,R_{u}(\Sigma_{c}^{0})$

A direct summation over the intermediate state results in

$\displaystyle\hat{A}^{\rm com}(s;\Sigma^{+})$ $\displaystyle:$ $\displaystyle
I^{f}_{\Sigma^{0}\pi^{+}\Sigma^{+}}I^{\rm
pc}_{\Sigma^{+}\Lambda_{c}^{+}}=2/\sqrt{3},$ $\displaystyle\hat{A}^{\rm
com}(u;\Sigma_{c}^{0})$ $\displaystyle:$ $\displaystyle I^{\rm
pc}_{\Sigma^{0}\Sigma_{c}^{0}}I^{f}_{\Sigma_{c}^{0}\pi^{+}\Lambda_{c}^{+}}=0,$
$\displaystyle\hat{B}^{\rm pole}(s;\Sigma^{+})$ $\displaystyle:$
$\displaystyle I^{\rm CQM}_{\Sigma^{0}\pi^{+}\Sigma^{+}}I^{\rm
pc}_{\Sigma^{+}\Lambda_{c}^{+}}=2/\sqrt{3},$ $\displaystyle\hat{B}^{\rm
pole}(u;\Sigma_{c}^{0})$ $\displaystyle:$ $\displaystyle I^{\rm
pc}_{\Sigma^{0}\Sigma_{c}^{0}}I^{\rm
CQM}_{\Sigma_{c}^{0}\pi^{+}\Lambda_{c}^{+}}=3/\sqrt{3}$ (83)

where we have used $I^{f}_{\Sigma^{0}\pi^{+}\Sigma^{+}}=-4/2\sqrt{2}$, $I^{\rm
pc}_{\Sigma^{+}\Lambda_{c}^{+}}=-4/2\sqrt{6}$,
$I^{f}_{\Sigma_{c}^{0}\pi^{+}\Lambda_{c}^{+}}=0$, $I^{\rm
CQM}_{\Sigma^{0}\pi^{+}\Sigma^{+}}=-4/2\sqrt{2}$ and $I^{\rm
CQM}_{\Sigma_{c}^{0}\pi^{+}\Lambda_{c}^{+}}=6/2\sqrt{6}$ from Tables 16 and 18
in Appendix C.

###  6.3 The SCS decay $\Xi_{c}^{0}\to pK^{-}$
$2\sqrt{6}(I_{i})=(0,0,0,-2_{a},0,0,-1_{b})$

This decay is interesting from the point of view that there are altogether
four states that contribute as intermediate states. These are
$(\Lambda^{0},\Sigma^{0})$ in the $s$-channel and
$(\Lambda_{c}^{+},\Sigma_{c}^{+})$ in the $u$-channel. The structure of the
tensor invariants can be seen to be identical for this decay and the decay
$\Lambda_{c}^{+}\to\Xi^{0}\,K^{+}$ treated in the previous section. We
anticipate that the treatment in terms of tensor invariants will be a much
simpler undertaking than the explicit summation of the four contributing
intermediate states. In our analysis we shall retain the identification of the
weak transitions in terms of the ($a$) and ($b$) type contributions.

The reduced commutator term $\hat{A}^{\rm com}$ and pole term $\hat{B}^{\rm
pole}$ are given by

$\displaystyle\hat{A}^{\rm com}$ $\displaystyle=$
$\displaystyle(2I_{3}+4I_{5})-(2\hat{I}_{3}+4I_{5})\ =\
(-4_{b}/2\sqrt{6})-(-4_{b}/2\sqrt{6})=0,$ $\displaystyle\hat{B}^{\rm pole}$
$\displaystyle=$
$\displaystyle(I_{3}+\underline{2I_{4}}+\underline{6I_{5}})\bar{R}_{s}(\Lambda^{0},\Sigma^{0})+(\hat{I}_{3}+2\hat{I}_{4}+\underline{6I_{5}})R_{u}(\Sigma_{c}^{0})\
=$ (84) $\displaystyle=$
$\displaystyle-(4_{a}+6_{b})/2\sqrt{6}\,\bar{R}_{s}(\Lambda^{0},\Sigma^{0})-6_{b}/2\sqrt{6}\,R_{u}(\Sigma_{c}^{0}).$

When taking the intermediate state route, one has to take into account that
$I^{f}_{\Sigma_{c}^{+}K^{-}\Xi_{c}^{0}}=0$ and $I^{\rm
CQM}_{\Lambda_{c}^{+}K^{-}\Xi_{c}^{0}}=0$ (see Appendix C). For the relevant
products of flavor factors one obtains

$\displaystyle\hat{A}^{\rm com}(s;\Lambda^{0},\Sigma^{0})$ $\displaystyle:$
$\displaystyle I^{f}_{pK^{-}\Lambda^{0}}I^{\rm
pc}_{\Lambda^{0}\,\Xi_{c}^{0}}+I^{f}_{pK^{-}\Sigma^{0}}I^{\rm
pc}_{\Sigma^{0}\,\Xi_{c}^{0}}=\big{(}-(2_{a}+4_{b})+2_{a}\big{)}/2\sqrt{6},$
$\displaystyle\hat{A}^{\rm com}(u;\Lambda_{c}^{+})\,$ $\displaystyle:$
$\displaystyle I^{\rm
pc}_{p\,\Lambda_{c}^{+}}I^{f}_{\Lambda_{c}^{+}K^{-}\Xi_{c}^{0}}=(-4_{b})/2\sqrt{6},$
$\displaystyle\hat{B}^{\rm pole}(s;\Lambda^{0},\Sigma^{0})$ $\displaystyle:$
$\displaystyle I^{\rm CQM}_{pK^{-}\Lambda^{0}}I^{\rm
pc}_{\Lambda^{0}\Xi_{c}^{0}}+I^{\rm CQM}_{pK^{-}\Sigma^{0}}I^{\rm
pc}_{\Sigma^{0}\Xi_{c}^{0}}=\big{(}-(3_{a}+6_{b})-1_{a}\big{)}/2\sqrt{6},$
$\displaystyle\hat{B}^{\rm pole}(u;\Sigma_{c}^{+})$ $\displaystyle:$
$\displaystyle I^{\rm pc}_{p\,\Sigma_{c}^{+}}I^{\rm
CQM}_{\Sigma_{c}^{+}K^{-}\Xi_{c}^{0}}=-6_{b}/2\sqrt{6}.$ (85)

Keeping in mind that the two $S$-wave contributions have to be subtracted
while the two $P$-wave contributions have to be added, one finds agreement of
the two calculational routes. The net result is that the commutator
contribution to the decay $\Xi_{c}^{0}\to pK^{-}$ vanishes.

###  6.4 The CF decay $\Omega_{c}^{0}\to\Xi^{0}\bar{K}^{0}$
$2(I_{i})=(-1,-1,0,0,-2,0,0)$

The only $W$-exchange contribution to this decay is from the tensor invariant
$\hat{I}_{3}=-1$ (see Table 8). Therefore, the decay proceeds only via the
$u$-channel, which becomes obvious from

$\displaystyle\hat{A}^{\rm com}$ $\displaystyle=$
$\displaystyle\,(2I_{3}+4I_{5})-(\underline{2\hat{I}_{3}}+4I_{5})\ =\
\,0(s)-(-2),$ $\displaystyle\hat{B}^{\rm pole}$ $\displaystyle=$
$\displaystyle
0(s)+(\underline{\hat{I}_{3}}+2\hat{I}_{4}+6I_{5})\,R_{u}(\Xi_{c}^{0},\Xi_{c}^{\prime\,0})\
=\ 0(s)-\bar{R}_{u}(\Xi_{c}^{0},\Xi_{c}^{\prime\,0}).\qquad$ (86)

where $0(s)$ stands for the absence of an $s$-channel contribution. The
interest in this decay is caused by the fact that both intermediate states
$\Xi_{c}^{0}$ and $\Xi_{c}^{{}^{\prime}0}$ contribute to the $P$-wave
amplitude $B^{\rm pole}(u)$. In terms of the flavor invariants $I^{f}$,
$I^{\rm pc}$ and $I^{\rm CQM}$ one obtains

$\displaystyle\hat{A}^{\rm com}(u;\Xi_{c}^{{}^{\prime}0})$ $\displaystyle:$
$\displaystyle I^{\rm
pc}_{\Xi^{0}\Xi_{c}^{{}^{\prime}0}}\,I^{f}_{\Xi_{c}^{{}^{\prime}0}\bar{K}^{0}\Omega_{c}^{0}}\
=\ -2,$ $\displaystyle\hat{B}^{\rm pole}(u;\Xi_{c}^{0},\Xi_{c}^{\prime\,0})$
$\displaystyle:$ $\displaystyle I^{\rm pc}_{\Xi^{0}\Xi_{c}^{0}}\,I^{\rm
CQM}_{\Xi_{c}^{0}\bar{K}^{0}\Omega_{c}^{0}}+I^{\rm
pc}_{\Xi^{0}\Xi_{c}^{{}^{\prime}0}}\,I^{\rm
CQM}_{\Xi_{c}^{{}^{\prime}0}\bar{K}^{0}\Omega_{c}^{0}}\ =\ -1.\qquad$ (87)

The results are in agreement with Ref. [62]. There is no need to differentiate
between the current algebra approach and the modified current algebra result
in this case since $I_{5}=0$.

In the limit of vanishing hyperfine splitting
$m_{\Xi_{c}^{0}}=m_{\Xi_{c}^{\prime 0}}$ one can sum the two $P$-wave
$u$-channel contributions to obtain

$\hat{B}^{\rm
pole}=2(\underline{\hat{I}_{3}}+2\hat{I}_{4}+6I_{5})\frac{m_{\Xi_{0}}-\bar{m}_{\Xi_{c}^{0}}}{m_{\Omega_{c}^{0}}-\bar{m}_{\Xi_{c}^{0}}}=-2\frac{m_{\Xi_{0}}-\bar{m}_{\Xi_{c}^{0}}}{m_{\Omega_{c}^{0}}-\bar{m}_{\Xi_{c}^{0}}},$
(88)

where $\bar{m}_{\Xi_{c}^{0}}$ denotes a suitable mass average of the two
flavor degenerate states $\Xi_{c}^{0}$ and $\Xi_{c}^{\prime\,0}$. The decay
was analyzed in Ref. [50], again in the context of the current algebra
approach. In the first version of Ref. [50] there was the claim that the
contributions of the intermediate states $\Xi^{0}_{c}$ and
$\Xi^{\prime\,0}_{c}$ cancel each other for
$m_{\Xi_{c}^{0}}=m_{\Xi^{\prime\,0}_{c}}$. This statement was corrected later
on in an Erratum to Ref. [50] with a result in agreement with Eq. (88). That
the two contributions do not, in fact, cancel can be easily checked by a mere
visual inspection of the tensor invariants in Table 8. We have provided this
example to illustrate the power of the current algebra approach when expressed
in terms of topological tensor invariants because the compactness of the
tensor invariant expressions allows one to easily trace mistakes in a current
algebra result.

###  6.5 The SCS decay $\Omega_{c}^{0}\to\Sigma^{0}\bar{K}^{0}$
$2\sqrt{2}(I_{i})=(0,0,0,-2_{a},0,0,-1_{a})$

The decay proceeds only through $W$ exchange and via the ($a$)-type weak
interaction. The interest in this decay is based on the fact that up to
normalization factors the topological invariant structure is identical to the
one of the previous decay $\Lambda_{c}^{+}\to\Xi^{0}\,K^{+}$. The same
statement holds true for the decays

$\displaystyle{\rm CF}$ $\displaystyle:$
$\displaystyle\Lambda_{c}^{+}\to\Xi^{0}K^{+};\,\,\quad\quad\qquad\qquad\Xi_{c}^{0}\to\Sigma^{+}K^{-}$
$\displaystyle{\rm SCS}$ $\displaystyle:$
$\displaystyle\Omega_{c}^{0}\to\Sigma^{+}K^{-},\,\Sigma^{0}\bar{K}^{0};\,\quad\qquad\Xi_{c}^{0}\to\Sigma^{+}\pi^{-},\,pK^{-}$
$\displaystyle{\rm DCS}$ $\displaystyle:$ $\displaystyle\Xi_{c}^{+}\to
p\pi^{0},\,n\pi^{+};\quad\qquad\qquad\Xi_{c}^{0}\to p\pi^{-},\,n\pi^{0}$ (89)

One obtains

$\displaystyle\hat{A}^{\rm com}$ $\displaystyle=$
$\displaystyle\,(2I_{3}+4I_{5})-(2\hat{I}_{3}+4I_{5})\ =\
(-4_{b}/2\sqrt{2})-(-4_{b}/2\sqrt{2})=0,$ $\displaystyle\hat{B}^{\rm pole}$
$\displaystyle=$
$\displaystyle(I_{3}+\underline{2I_{4}}+\underline{6I_{5}})\bar{R}_{s}(\Xi^{0})+(\hat{I}_{3}+2\hat{I}_{4}+\underline{6I_{5}})\bar{R}_{u}(\Xi_{c}^{0},\Xi_{c}^{\prime\,0})\
=$ (90) $\displaystyle=$
$\displaystyle-(10_{a})/2\sqrt{2}\,\bar{R}_{s}(\Xi^{0})-6_{a}/2\sqrt{2}\,\bar{R}_{u}(\Xi_{c}^{0},\Xi_{c}^{\prime\,0})$

or

$\displaystyle\hat{A}^{\rm com}(s;\Xi^{0})$ $\displaystyle:$ $\displaystyle
I^{f}_{\Sigma^{0}\bar{K}^{0}\Xi^{0}}I^{H}_{\Xi^{0}\Omega_{c}^{0}}=-4_{a}/2\sqrt{2}$
$\displaystyle\hat{A}^{\rm com}(u;\Xi_{c}^{0})$ $\displaystyle:$
$\displaystyle
I^{H}_{\Sigma^{0}\Xi_{c}^{0}}I^{f}_{\Xi_{c}^{0}\bar{K}^{0}\Omega_{c}^{0}}=-4_{a}/2\sqrt{2}$
$\displaystyle\hat{B}^{\rm pole}(s;\Xi^{0})$ $\displaystyle:$ $\displaystyle
I^{\rm
CQM}_{\Sigma^{0}K^{0}\Xi^{0}}I^{H}_{\Xi^{0}\Omega_{c}^{0}}=-10_{a}/2\sqrt{2}$
$\displaystyle\hat{B}^{\rm pole}(u;\Xi_{c}^{0},\Xi_{c}^{{}^{\prime}0})$
$\displaystyle:$ $\displaystyle I^{H}_{\Sigma^{0}\Xi_{c}^{0}}I^{\rm
CQM}_{\Xi_{c}^{0}\bar{K}^{0}\Omega_{c}^{0}}+I^{H}_{\Sigma^{0}\Xi_{c}^{{}^{\prime}0}}I^{\rm
CQM}_{\Xi_{c}^{{}^{\prime}0}\bar{K}^{0}\Omega_{c}^{0}}=(-2_{a}-4_{a})/2\sqrt{2},\qquad$
(91)

where

$R_{u}(\Xi_{c}^{0})=\frac{m_{\Omega_{c}^{0}}+m_{\Xi_{c}^{0}}}{m_{\Sigma^{0}}-m_{\Xi_{c}^{0}}}=4.83,\qquad
R_{u}(\Xi_{c}^{\prime\,0})=\frac{m_{\Omega_{c}^{0}}+m_{\Xi_{c}^{\prime\,0}}}{m_{\Sigma^{0}}-m_{\Xi_{c}^{\prime\,0}}}=4.53.$
(92)

###  6.6 The $\Delta C=0$ SCS decays $\Xi_{c}^{+}\to\Lambda_{c}^{+}\,\pi^{0}$
and $\Xi_{c}^{0}\to\Lambda_{c}^{+}\,\pi^{-}$