Charmed Baryon Weak Decays with Vector Mesons

C.Q. Geng, Chia-Wei Liu and Tien-Hsueh Tsai School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China International Centre for Theoretical Physics Asia-Pacific, Hangzhou 310024, China Department of Physics, National Tsing Hua University, Hsinchu 300, Taiwan Physics Division, National Center for Theoretical Sciences, Hsinchu 300, Taiwan (Dated: January 16, 2020)

In this work, we concentrate on the decays of B c → B n V with the SU(3) F flavor symmetry, where B c and B n correspond to the anti-triplet charmed and octet baryons, and V stand for the vector mesons, respectively. In fact, some of the decay branching ratios have been recently explored based on SU(3) F in Ref. [25]. However, the approach in Ref. [25] has ignored the contributions from color-symmetric parts of the effective Hamiltonian and correlations among the SU(3) F parameters. In addition, there should be four independent wave amplitudes [51], but only one is used in Ref. [25]. In this study, we shall include all the wave amplitudes and consider the full effective Hamiltonian. We shall also discuss the decay asymmetry parameters in B c → B n V , such as the up-down and longitudinal polarization asymmetries of B n and asymmetry parameter of V .
This paper is organized as follow. In Sec. II, we present the formalism. In Sec. III, we extract the SU(3) F parameters from the experimental data. We conclude our study in Sec IV.

II. FORMALISM
The most general form of the amplitude for B c → B n V can be written as where ǫ µ is the four vector polarization for the vector meson of V , u i (p i ) and u f (p f ) are the 4-component spinors (momenta) for the initial and final baryons, respectively, and m i represents the initial baryon mass. In general, the physical vector meson with its momentum in the z direction has the vector polarizations of ǫ µ = (0, 1 where λ V is the helicity and m V , p V and E V are the mass, 3-momentum and energy of the vector meson, respectively. In the center of the momentum frame (CMF), the kinematic factors of A 2 and B 2 in Eq. (4) can be further written as Here, we have used p µ i = p µ f + p µ V and ǫ µ p µ V = 0, where p V corresponds to the 4-momentum of the vector meson. It is clear that the terms associated with A 2 and B 2 will only contribute to the decay in the case of λ V = 0, which are suppressed by the factor of p c /m V with p c defined as the magnitude of the 3-momentum in the CMF, so that they can be ignored.
The decay width, up-down asymmetry and longitudinal polarization of B c → B n V are given by where S, P 1,2 and D, corresponding to the orbital angular momenta of l = 0, 1, 2 in the non-relativistic limit, are given by [51] respectively. Here, α and P L are defined by where dΓ is the partial decay width, λ Bn is the helicity of B n and θ is the angle between the spin and momentum directions of B c and B n , respectively.
Since the vector meson of V subsequently decays into two pseudo-scalar mesons, its polarization can be determined. As a result, we can discuss the decay asymmetry parameter of V , defined by [38] with where dΓ V is the partial decay width for the V decay and θ V is the polar angle between p V and the momentum directions of the pseudo-scalar mesons in the CMF of V .
The effective Hamiltonian responsible for the decay processes with ∆c = −1 is where the quark operators are defined as (q 1 q 2 ) = (q 1 γ µ (1 − γ 5 )q 2 ) with summing over the colors, the Wilson coefficient of c 1 (c 2 ) is 1.246(−0.636) at the scale of µ = 1.25 GeV [52] and G F is the Fermi constant. Note that (q, q ′ ) = (d, s), (d, d) or (s, s) and (s, d) correspond to the Cabbibo allowed, singly Cabbibo suppressed and doubly Cabbibo suppressed decays, respectively.
By using the CKM mixing parameters of V cs = V ud ≈ 1 and s c ≡ V us = −V cd ≈ 0.225, the effective Hamiltonian in the flavor basis is given by where (q 1 , q 2 , q 3 ) = (u, d, s), c ± = c 1 ± c 2 , and ǫ lij represents the total antisymmetric tensor with ǫ 123 = 1. Here, the tensor components are given by Two of the creation operators generated by H(15) are symmetric in color. As a result, H (15) does not contribute to the nonfactorizable amplitudes since the charmed baryons are total anti-symmetric in color [53,54].
We separate A 1 and B 1 into 6 and 15 parts under the SU(3) F symmetry: In Eq. (20), A 1 and B 1 are parametrized as where a i and b i are the SU(3) F parameters, while B c,n and V can be written under the tensor components of the SU(3) F representations, given by and respectively.
On the other hand, the contribution from H(15) to c → uq ′q is factorizable, given by for the vector mesons with positive charges, while the creation operators,q ′ andū, are interchanged for the neutral vector mesons. Accordingly, A where V |(qq ′ )|0 = f V m V ǫ * µ and N c is the effective color number. In Eq. (26), we take that f V = 0.215 GeV and the form factors of f i (g i ) are defined by In our calculation, we evaluate the form factors from the MIT bag model [55,56]. The baryon wave functions and form factors are listed in Appendix A.
The factorizable parts in A 2 and B 2 are given by with the "±" signs for mesons with positive and neutral charges, respectively. In general, it is also possible to parametrize the nonfactorizable contributions in A 2 and B 2 according to the SU(3) F symmetry. However, since they are suppressed due to Eq. (5), we will neglect these parts.
To sum up, A ) can be calculated from the factorization approach, while A
The decays of Λ + c → Σ + K * 0 and Ξ + c → pK * 0 share the same coupling strengths in terms of the U−spin symmetry [27] as they are related through interchanging d and s quarks.
Naively, one expects that they should have the same decay widths. However, our results indicate that This hierarchy can be understood by the released energies, given by With a smaller kinematic phase space, the decay of Λ + c → Σ + K * 0 is suppressed compared to Ξ + c → pK * 0 . It can be interpreted as the SU(3) F breaking effect, caused by the mass differences. Meanwhile, the experimental data lead to which is much larger than the value in Eq. (39). Despite this inconsistence, we are still confident that our result in Eq. (39) due to the phase space suppression is correct. We view this result as one of our predictions and suggest the future experiments to revisit the channels.
It is interesting to note that the Cabbibo allowed decays of Λ + c → Λ 0 ρ + and Ξ 0 c → Ξ − ρ + have large branching ratios and decay parameters with small uncertainties as shown in Table 2, so that they can be viewed as the golden modes for the experimental searches.
Similarly, the singly Cabbibo suppressed decays of Λ + c → Λ 0 K * + , Ξ + c → Σ + φ and Ξ 0 c → Ξ − K * + are also recommended to future experiments for the same reasons. In addition, we point out that the decay parameters in Ξ +(0) c → Σ +(0) φ are almost the same in terms of the isospin symmetry. However, the decay branching ratio for the neutral Ξ 0 c mode is suppressed due to the shorter lifetime compared to the Ξ + c one and the factor 2 from the CG coefficient. In Table 5, we compare our results of the Cabbibo allowed decays with those in the literature, where Körner and Krämer (KK) [38],Żenczykowski (Zen) [45] and Hsiao, Yu and Zhao (HYZ) [25] are the studies based on the covariant quark model, pole model and SU(3) F , respectively. In Ref. [38], only the decay widths are provided instead of the branching ratios.
To obtain the branching ratios, we have used the lifetimes in Eq. (1). As seen from Table 5, our results are consistent with those in Ref. [38]. However, the branching ratios of ρ + modes of Λ + c → Λ 0 ρ + , Λ + c → Σ 0 ρ + , Ξ + c → Ξ 0 ρ + and Ξ 0 c → Ξ − ρ + in Ref. [38] are too large compared to our predictions as well as the existing data. Furthermore, most of our results are compatible with those in Ref. [45], whereas differ largely in Λ + c → Λ 0 ρ + , Ξ 0 c → (Ξ − ρ + , Ξ 0 ω) and Ξ + c decays. Note that in Ref. [25], the contributions from H(15) and the correlations between the parameters are not included in their calculations, resulting in larger errors than ours. Except the decays with the existing experimental data, which are also the inputs for the fitting, the predictions in Ref. [25] are quite different from ours even though both of us take the SU(3) F approach. In particular, due to the different treatments of the wave amplitude, the predicted decay branching ratio of Λ + c → Ξ 0 K * + in Ref. [25] is about 8 times larger than ours and the one in the literature [38,45].

IV. CONCLUSIONS
We have explored the charmed baryon decays of B c → B n V based on the SU(3) F flavor symmetry. In these processes, we have calculated the color-symmetric parts of the effective Hamiltonian by the factorization approach assisted with the MIT bag model, while the anti-symmetric ones are extracted from the experimental data. We have systematically obtained all decay branching ratios and parameters in B c → B n V . We have found that our results are consistent with the experimental data except Λ + c → Σ 0 K * + , for which our fitted value of B(Λ + c → Σ 0 K * + ) = (0.38 ± 0.09) × 10 −3 is much smaller than the data of (3.5 ± 1.0) × 10 −3 . As our result contains a very small error, whereas the experimental one is large, we are eager to see the precision measurement of this mode in the future experiments.