Charmed Baryon Weak Decays with SU(3) Flavor Symmetry

We study the semileptonic and non-leptonic charmed baryon decays with $SU(3)$ flavor symmetry, where the charmed baryons can be ${\bf B}_{c}=(\Xi_c^0,\Xi_c^+,\Lambda_c^+)$, ${\bf B}'_{c}=(\Sigma_c^{(++,+,0)},\Xi_{c}^{\prime(+,0)},\Omega_c^0)$, ${\bf B}_{cc}=(\Xi_{cc}^{++},\Xi_{cc}^+,\Omega_{cc}^+)$, or ${\bf B}_{ccc}=\Omega^{++}_{ccc}$. With ${\bf B}_n^{(\prime)}$ denoted as the baryon octet (decuplet), we find that the ${\bf B}_{c}\to {\bf B}'_n\ell^+\nu_\ell$ decays are forbidden, while the $\Omega_c^0\to \Omega^-\ell^+\nu_\ell$, $\Omega_{cc}^+\to\Omega_c^0\ell^+\nu_\ell$, and $\Omega_{ccc}^{++}\to \Omega_{cc}^+\ell^+\nu_\ell$ decays are the only existing Cabibbo-allowed modes for ${\bf B}'_{c}\to {\bf B}'_n\ell^+\nu_\ell$, ${\bf B}_{cc}\to {\bf B}'_c\ell^+\nu_\ell$, and ${\bf B}_{ccc}\to {\bf B}_{cc}^{(\prime)}\ell^+\nu_\ell$, respectively. We predict the rarely studied ${\bf B}_{c}\to {\bf B}_n^{(\prime)}M$ decays, such as ${\cal B}(\Xi_c^0\to\Lambda^0\bar K^0,\,\Xi_c^+\to\Xi^0\pi^+)=(8.3\pm 0.9,8.0\pm 4.1)\times 10^{-3}$ and ${\cal B}(\Lambda_c^+\to \Delta^{++}\pi^-,\,\Xi_c^0\to\Omega^- K^+)=(5.5\pm 1.3,4.8\pm 0.5)\times 10^{-3}$. For the observation, the doubly and triply charmed baryon decays of $\Omega_{cc}^{+}\to \Xi_c^+\bar K^0$, $\Xi_{cc}^{++}\to (\Xi_c^+\pi^+$, $\Sigma_c^{++}\bar K^0)$, and $\Omega_{ccc}^{++}\to (\Xi_{cc}^{++}\bar K^0,\Omega_{cc}^+\pi^+,\Xi_c^+ D^+)$ are the favored Cabibbo-allowed decays, which are accessible to the BESIII and LHCb experiments.

The spectroscopy of the charmed baryons is built by measuring their decay modes. For example, the existence of the Ξ + cc state was once reported by the SELEX collaboration [10,11], but not confirmed by the other experiments [12][13][14][15]. Until very recently, LHCb has eventually found the doubly charmed Ξ ++ cc state at a mass of (3621.40 ± 0.72 ± 0.27 ± 0.14) MeV [4], which is reconstructed as the two-body Ξ ++ cc → Σ ++ c (2455)K * 0 decay with the resonant strong decays of Σ ++ c → Λ + c π + andK * 0 → K − π + , as shown by the theoretical calculation [16]. Note that the corresponding decay lifetime has not been determined yet.
It should be interesting to perform a full exploration of all possible charmed baryon decays, and single out the suitable decay channels for the measurements.
To study the charmed baryon decays, since the most often used factorization approach in the b-hadron decays [17][18][19] has been demonstrated not to work for the two-body B c → B n M decays [20,21], where B n(c) and M are denoted as the (charmed) baryon and meson, respectively, one has to compute the sub-leading-order contributions or the final state interactions to take into account the non-factorizable effects [22][23][24][25][26], whereas the QCD-based models in the B c decays are not available yet. On the other hand, with the advantage of avoiding the detailed dynamics of QCD, the approach with SU(3) flavor (SU(3) f ) symmetry can relate decay modes in the b and c-hadron decays [21,[27][28][29][30][31][32][33][34][35][36], where the SU(3) amplitudes receive non-perturbative and non-factorizable effects, despite the unknown sources. In this paper, in terms of SU(3) f symmetry, we will examine the semileptonic and non-leptonic two-body B c decays, search for decay modes accessible to experiment, and establish the spectroscopy of the charmed baryon states. The analysis will explore the consequences of neglecting a decay amplitude expected to be small. Our paper is organized as follows. In Sec. II, we develop the formalism, where the Hamiltonians, (charmed) baryon and meson states are presented in the irreducible forms under SU(3) f symmetry. The amplitudes of the semileptonic and non-leptonic decay modes are given in Secs. III and IV, respectively. In Sec. V, we discuss all possible decays and show the relationships among them as well as some numerical results, which are relevant to the experiments. We conclude in Sec. VI.

A. The effective Hamitonian
For the semileptonic c → qℓ + ν ℓ transition with q = (d, s), the effective Hamiltonian at the quark-level is presented as where G F is the Fermi constant and V ij are the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix elements, while ( with the four-quark operators O where V cd V ud = −V cs V us has been used. According to |V cd V ud |/|V cs V ud | = sin θ c and to with the tensor notation of H(3) = (0, V cd , V cs ), where V cs = 1 and V cd = − sin θ c . For the c → sud and c → uqq transitions in Eq. (2), the four-quark operators can be presented as 6,15 (Ô 6,15 ), given by which are in accordance with the tensor notations of H(6) ij and H(15) jk i , with the non-zero entries: respectively, with s c ≡ sin θ c to include the CKM matrix elements into the tensor notations.
Accordingly, the effective Hamiltonian in Eq. (2) is transformed as where the contribution of H(6) to the decay branching ratio can be 5.5 times larger than that of H(15) due to (c − /c + ) 2 ≃ 5.5. The simplifications resulting from the neglect of the 15-plet will be investigated below.

B. The (charmed) baryon states and mesons
For the singly charmed baryon states, which consist of q 1 q 2 c with q 1 q 2 being decomposed as the irreducible representation of 3 × 3 =3 + 6, there exist the charmed baryon anti-triplet and sextet, given by with SU(3) parameters α i (i = 1, 2, 3) associated with the B n ℓ + ν ℓ decays. Note that T (B c → B ′ n ) disappears in Eq. (12). This is due to the fact that the symmetric baryon decuplet (B ′ n ) ijk and the anti-symmetric ǫ ijk coexist in the forms of (B ′ n ) ijk H i (3)(B c ) l ǫ ljk and (B ′ n ) ljk H i (3)(B c ) l ǫ ijk , which identically vanish [32]. We also obtain the T amplitudes of the B cc → B (′) c ℓ + ν ℓ and B ccc → B cc ℓ + ν ℓ decays, given by with SU(3) parameters β 1,2 and δ 1 , where the subscript q refers to the d or s quark in B cc . It is interesting to note that, for T (B ccc → B cc ), B ccc = Ω ++ ccc as the charmed baryon singlet has no SU(3) flavor index to connect to the final states and Hamiltonian. The full expanded T amplitudes in Eqs. (12) and (13), corresponding to the semileptonic charmed baryon decays, can be found in Table I. where and and where B n and B ′ n represent the octet and decuplet of the the baryon states in Eqs. (10) and (11), respectively. It is interesting to note that measuring the processes in Eq. (20) can be a test of the smallness of the 15-plet. For the B cc → B (′) c M decays, the T amplitudes are expanded as and The full expansions of the T amplitudes in Eqs. For the triply charmed baryon decays, there are three types of decay modes, that is, The corresponding T amplitudes are given by where B ccc = Ω ++ ccc as the charmed baryon singlet has no SU(3) flavor index to connect to the final states and H (6,15). The full expansions of the T amplitudes in Eq. (23) are given in Table X.

V. DISCUSSIONS
A. Semileptonic charmed baryon decays as the experimental input, and relating the possible B c → B n ℓ + ν ℓ decays with the SU(3) parameter α 1 in Table I, the branching ratios of the Cabibbo-allowed decays are predicted to be while the Cabibbo-suppressed ones are evaluated as where we have taken ( .42 ± 0.26, 2.00 ± 0.06) × 10 −13 s and s c = 0.2248 [1]. Our result of B(Λ + c → ne + ν e ) in Eq. (25) agrees with that in Ref. [21] by SU(3) f symmetry also. The B c → B ′ n ℓ + ν ℓ decays are forbidden modes, reflecting the fact that the B c and B ′ n states are the uncorrelated anti-symmetric triplet and symmetric decuplet, respectively, which can be viewed as the interesting measurements to test the broken symmetry.
In Table I n ℓ + ν ℓ decays, where B ′ c stands for the singly charmed baryon sextet in Eq. (8). We remark that currently it is hard to observe , as the Σ c and Ξ ′ c decays are dominantly through the strong and electromagnetic interactions, with B(Σ c → Λ c π) ≈ 100% and Ξ ′ c → Ξ c γ, respectively. In contrast, the Ω 0 c state that decays weakly can be measurable. In particular, the Ω 0 c → Ω − ℓ + ν ℓ decay with Ω − = sss becomes the only possible Cabibboallowed Ω 0 c case [32], whereas the Ω 0 c → B n ℓ + ν ℓ decays with the baryon octet are forbidden. This is due to the fact that, via the Cabibbo-allowed c → sℓ + ν ℓ transition, the Ω 0 c baryon consists of ssc transforms as the sss state, and has has no association with the the baryon octet. In the Cabibbo-suppressed css → dss transition, one has the Ω 0 c → Ξ (′)− ℓ + ν ℓ decays with Ξ − and Ξ ′− from both baryon octet and decuplet.
which respect the isospin symmetry. Like the singly charmed Ω 0 c cases, the Cabibbo-allowed Ω + cc (ccs) → css transition forbids the Ω + cc → B c ℓ + ν ℓ decays, but allows Ω + cc → Ω 0 In the B ccc → B cc ℓ + ν ℓ decays, SU(3) f symmetry leads to two possible decay modes, of which the branching ratios are related as suggesting that the Cabibbo-allowed Ω ++ ccc → Ω + cc ℓ + ν ℓ decay is more accessible to experiment. to the data in the PDG [1], it is given that which result in by bringing the predictions of Eq. (24) into the relations. On the other hand, the SU (3) parameters for B c → B n M have been extracted from the observed B(Λ + c → B n M) data, given by [36] (a 1 , a 2 , a 3 ) = (0.257 ± 0.006, 0.121 ± 0.015, 0.092 ± 0.021) GeV 3 , where δ a 2 ,a 3 are the relative phases from the complex a 2 and a 3 parameters, and a 1 is fixed to be real. Besides, we follow Ref. [21] to ignore a 4,5,...,7 from H(15), which are based on (c − /c + ) 2 = 5.5 fromH ef f in Eq. (7), leading to the estimation of B(Λ c → Σ + K 0 ) with the (10 − 15)% deviation from the data [36]. By using SU(3) parameters in Eq. (30), we obtain In Eqs. (29) and (31), B I,II indeed come from semileptonic and non-leptonic SU(3) relations, respectively, even though the data inputs have very different sources. As a result, the good agreements for Ξ + c → Ξ 0 π + and Ξ 0 c → Λ 0K 0 clearly support the approach with the SU(3) f symmetry.
n M decays From Table IV to Table VII where Ω 0 c → Ω − π + and Ω 0 c → Ω − e + ν e are identified from Tables I and VII as Cabibboallowed processes, with Ω − belonging to the baryon decuplet B ′ n . On the other hand, as the only Cabibbo-allowed Ω 0 c → B n M mode, Ω 0 c → Ξ 0K 0 has not been measured yet, which calls for the other accessible decay modes. Although it seems that there is no relation for Table V, if H(15) is ignorable, we have for the Cabibbo-suppressed processes, and for the doubly Cabibbo-suppressed ones, which can be regarded to recover the isospin symmetry.
For Ω 0 c → B ′ n M, as seen in Table VII, it is found that In addition, ignoring H(15), we derive the relations with the recovered isospin symmetry, given by and for the Cabibbo-and doubly Cabibbo-suppressed decays, respectively.
• The B cc → B  Table VIII, the Cabibbo-allowd decay modes can be related to the (doubly) Cabibbo-suppressed ones, given by and c M decays In the B cc → B c M decays, the Cabibbo-allowed amplitudes are composed of SU(3) parameters b 7,8 from H(6), instead of b 9,10 from H (15), which indicate that the decays are measurable. In fact, the decay mode of Ξ ++ cc → Ξ + c π + has been suggested to be worth measuring by the model calculation [16]. Here, we connect these Cabibbo-allowed decays to be Γ(Ξ ++ cc → Ξ + c π + ) = Γ(Ω + cc → Ξ + cK 0 ) , which are the most accessible decay modes to the experiments. We note that the accuracy of the prediction involving η is limited by the assumption that η is a pure octet. Next, the Cabibbo-suppressed decays are related as For the doubly Cabibbo-suppressed ones, only when a 9,10 from H(15) are negligible, we can find that There are three kinds of relations in the B cc → B ′ c M decays, given by Note that, Ξ ++ cc → Σ ++ cK * 0 with the strong decays of Σ ++ c → Λ + c π + and K * 0 → K − π + , corresponds to the observation of Ξ ++ cc → Λ + c K − π + π + [4,16]. Since the vector meson octet (V ) is nearly the same as the pseudo-scalar meson ones (M) in Eq. (10) Table X, the B ccc state is indeed the singlet of Ω ++ ccc , and the B ccc → B cc M decays have two types, given by where T 's are proportional to d 1 − 2d 2 and d 1 + 2d 2 , respectively, with d 1(2) from H(15 (6)).
The Ω ++ ccc → B (′) c M c decays can be simply related, given by Note that the decay modes with B c and B ′ c are in accordance with d 4,3 from H(6) and H (15), respectively, such that it is possible that the Cabibbo-allowed Ω ++ ccc → Ξ + c D + decay can be more accessible to the experiments.

VI. CONCLUSIONS
We have studied the semileptonic and non-leptonic charmed baryon decays with SU(3) f symmetry. By separating the Cabibbo-allowed decays from the (doubly) Cabibbo-suppressed ones, we have provided the accessible decay modes to the experiments at BESIII and LHCb.