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 Bc = (Ξc0, Ξc+, Λc+), Bc′ = (Σc(++,+,0), Ξc′ (+,0), Ωc0), Bcc = (Ξcc+ +, Ξcc+, Ωc+) or Bcc = Ωccc+ +. With Bn(′) denoted as the baryon octet (decuplet), we find that the Bc → Bn′ℓ+νℓ decays are forbidden, while the Ωc0 → Ω−ℓ+νℓ, Ωcc+ → Ωc0ℓ+νℓ, and Ωccc+ + → Ωcc+ℓ+νℓ decays are the only existing Cabibbo- allowed modes for Bc′ → Bn′ℓ+νℓ, Bcc → Bc′ℓ+νℓ, and Bccc → Bcc(′)ℓ+νℓ, respectively. We predict the rarely studied Bc → Bn(′)M decays, such as ℬΞc0→Λ0K¯0,Ξc+→Ξ0π+=8.3±0.9,8.0±4.1×10−3andℬΛc+→Δ++π−,Ξc0→Ω−K+=5.5±1.3,4.8±0.5×10−3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{\mathcal{B}}\left({\Xi}_c^0\to {\Lambda}^0{\overline{K}}^0,{\Xi}_c^{+}\to {\Xi}^0{\pi}^{+}\right)=\left(8.3\pm 0.9,\ 8.0\pm 4.1\right)\times {10}^{-3}\kern0.5em \mathrm{and}\kern0.5em \mathrm{\mathcal{B}}\left({\Lambda}_c^{+}\to {\Delta}^{++}{\pi}^{-},{\Xi}_c^0\to {\Omega}^{-}{K}^{+}\right)=\left(5.5\pm 1.3,\kern0.5em 4.8\pm 0.5\right)\times {10}^{-3} $$\end{document}. For the observation, the doubly and triply charmed baryon decays of Ωcc+→Ξc+K¯0,Ξcc++→Ξc+π+,Σc++K¯0,andΩccc++→Ξcc++K¯0,Ωcc+π+,Ξc+D+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\Omega}_{cc}^{+}\to {\Xi}_c^{+}{\overline{K}}^0,{\Xi}_{cc}^{++}\to \left({\Xi}_c^{+}{\pi}^{+},{\varSigma}_c^{++}{\overline{K}}^0\right),\kern0.5em \mathrm{and}\kern0.5em {\Omega}_{ccc}^{++}\to \left({\Xi}_{cc}^{++}{\overline{K}}^0,{\Omega}_{cc}^{+}{\pi}^{+},{\Xi}_c^{+}{D}^{+}\right) $$\end{document} 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 [11,12], but not confirmed by the other experiments [13][14][15][16]. 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 ++

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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 [18][19][20] has been demonstrated not to work for the twobody B c → B n M decays [21,22], 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 [23][24][25][26][27], 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 [22,[28][29][30][31][32][33][34][35][36][37], 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 section 2, 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 sections 3 and 4, respectively. In section 5, 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 section 6.

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 ( 3)

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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 |V cd V us |/|V cs V ud | = sin 2 θ c with θ c known as the Cabibbo angle, the operators for the c → sud, c → uqq and c → dus transitions represent the Cabibbo-allowed, Cabibbosuppressed and doubly Cabibbo-suppressed processes, respectively. As the scale-dependent Wilson coefficients, c ± are calculated to be (c + , c − ) = (0.76, 1.78) at the scale µ = 1 GeV in the NDR scheme [38,39]. Based on SU(3) f symmetry, the Lorentz-Dirac structures for the four-quark operators in eq. (2.3) are not explicitly expressed with the quark index q i = (u, d, s) as an SU(3) f triplet (3), such that in eq. (2.1) the quark-current side of (qc) forms an anti-triplet (3), which leads 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. (2.2), the four-quark operators can be presented as (q i q k )(q j c), withq i q kq j being decomposed as3 × 3 ×3 =3 +3 ′ + 6 + 15. Consequently, the operators O  respectively, with s c ≡ sin θ c to include the CKM matrix elements into the tensor notations. Accordingly, the effective Hamiltonian in eq. (2.2) 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.

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 n ) are given by . 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 ( [33]. 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 where and Note that the Wilson coefficients c ± have been absorbed in SU (3) parameters a i , which can relate all possible decay modes. The full expansions of the T amplitudes in eqs. (4.2)-(4.5) are given in tables 2-7.

The doubly charmed
and where B n and B ′ n represent the octet and decuplet of the baryon states in eqs. (2.10) and (2.11), respectively. It is interesting to note that measuring the processes in eq. (4.7) 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. (4.6)-(4.9) are given in tables 8 and 9.

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Bcc → BnMc CA T -amp 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. (4.10) are given in table 10.

Non-leptonic charmed baryon decays
n M decays In the Λ + c → B n M decays, the PDG [1] lists six Cabibbo-favored channels, in addition to two Cabibbo-suppressed ones, whereas no absolute branching fractions for the Ξ 0,+ c decays have been seen [1]. Being demonstrated to well fit the measured values of B(Λ + c → B n M ) [37], SU(3) f symmetry can be used to study the Ξ 0,+ c → B n M decays. For example, according to the data in the PDG [1], it is given that which result in 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. [22] to ignore a 4,5,...,7 from H(15), which are based on (c − /c + ) 2 = 5.5 fromH eff in eq. (2.7), leading to the estimation of

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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.