SU(3) symmetry and its breaking effects in semileptonic heavy baryon decays

We employ the flavor SU(3) symmetry to analyze semileptonic decays of anti-triplet charmed baryons ($\Lambda^{+}_{c},\Xi_{c}^{+,0}$) and find that the experimental data on ${\cal B}(\Lambda_c\to\Lambda \ell^+\nu_{\ell})$ implies $\mathcal{B}(\Xi_c^0\to\Xi^-e^+\nu_{e})=(4.10\pm0.46)\%$ and $\mathcal{B}(\Xi_c^0\to\Xi^-\mu^+\nu_{\mu})= (3.98\pm0.57)\%$. When this prediction is confronted with recent experimental results from Belle collaboration $\mathcal{B}(\Xi_c^0\to\Xi^-e^+\nu_e)=1.31(04\pm07\pm38)\%$ and $\mathcal{B}(\Xi_c^0\to\Xi^-\mu^+\nu_{\mu})=1.27(06\pm10\pm 37)\%$, it is found that the SU(3) symmetry is severely broken. We then consider the generic SU(3) breaking effects in these decays and find that the data can be accommodated in different scenarios but the breaking effects are inevitably large. In some interesting scenarios, we also explore the testable implications in these scenarios which can be tested with more data become available. Similar analyses are carried out for semileptonic anti-triplet beauty baryon to octet baryons and anti-triplet charmed baryons. The validity of SU(3) for these decays can also be examined when data become available.


I. INTRODUCTION
Weak decays of heavy baryons carrying a charm quark have been studied extensively on both experimental and theoretical aspects [1], as they supply a platform for the study of strong and weak interactions in the standard model (SM). On the experimental side, data on charmed baryons decays from BESIII [2,3], Belle [4] and ALICE [5] collaborations provided important information to extract the CKM matrix element. Belle collaboration has provided a measurement of the Ξ 0 c branching fractions very recently [4]: B Belle (Ξ 0 c → Ξ − e + ν e ) = (1.31 ± 0.04 ± 0.07 ± 0.38)% , B Belle (Ξ 0 c → Ξ − µ + ν µ ) = (1.27 ± 0.06 ± 0.10 ± 0.37)% , which is about a factor of 2 more precise than the ALICE result: This comes from the ALICE measurement of B(Ξ 0 c → Ξ − e + ν e )/B(Ξ 0 c → Ξ − π + ) = 1.38 ± 0.14 ± 0.22 [5] and Belle data B(Ξ 0 c → Ξ − π + ) = 1.8 ± 0.7% [6]. We anticipate the difference between the above results can be clarified with the improvement of the experimental accuracy and the promising prospects on charmed baryons in the future. The available data on the decays from the anti-triplet heavy baryons to the octet baryons have been collected in Table I, while the branching fraction B(Ξ 0 c → Ξ − e + ν e ) = (1.54 ± 0.35)% listed is obtained by averaging the Belle and ALICE data.
On the theoretical side, one can apply the SU(3) flavor symmetry to analyze the semileptonic decays and obtain some model-independent relations among different decays [7][8][9][10][11][12][13][14][15][16][17][18][19]. For semileptonic charmed baryon decays, we have Since the irreducible amplitude can be extracted by fitting data, the SU(3) analysis bridges experimental data and the dynamical approaches like Lattice QCD [20][21][22][23] and model-dependent calculations [24][25][26][27][28][29][30][31][32]. We adopt the experiment data on Λ + c semileptonic decays and the SU(3) relations with the lifetimes τ Λ + c = 2.024 × 10 −13 s, τ Ξ 0 c = 1.53 × 10 −13 s, τ Ξ + c = 4.56 × 10 −13 s [33]. Then we obtain the branching ratios of Ξ 0,+ c shown in Table I, from which one can find an obvious deviation between experiments and theory.  [4,5] 4.10 ± 0.46 3.98 ± 0.57 It should be noted that the flavor SU(3) symmetry is an approximate symmetry, since u, d, and s quarks have different masses which breaks SU(3) symmetry [34]. For a more accurate analysis, SU(3) breaking effects should be included, which is the main focus of this work. Compared to the strange quark mass m s , the up and down quark masses m u,d are much smaller and thus can be neglected. Therefore the s quark mass is the major source for flavor SU(3) symmetry breaking. In this work, we carry out an analysis with the leading-order SU(3) breaking effects on semileptonic anti-triplet charmed baryons decays and explore the scenarios in which recent experimental measurements can be consistently accommodated.
The rest of this paper is organized as follows. In Sec. II, we give the theoretical framework for SU(3) symmetry and study symmetry breaking in semileptonic decays of anti-triplet heavy baryons for the process of c → d/s. In Sec. III, we also obtain numerical results using the SU(3) symmetry term and analyze the SU(3) symmetry breaking effect for the process of b → c/u. A brief conclusion will be presented in the last section.

II. SU(3) SYMMETRY FOR SEMILEPTONIC ANTI-TRIPLET CHARMED BARYON DECAYS
In the flavor SU(3) symmetry limit, hadron multiplets can be classified according to the SU(3) irreducible representation. Baryons with a charm quark and two light quarks can have 3 ⊗ 3 =3 ⊕ 6 representations. The anti-triplet 3 semileptonic baryon (Λ + c , Ξ + c , Ξ 0 c ) decays are our focus here, whose quark level Feynman diagrams are shown in Fig 1(a). In the SM the low-energy effective Hamiltonian for these decays is given as where q = d, s and G F is the Fermi-constant. V cq is CKM matrix element. With the help of helicity amplitude method [35], the decay transition amplitude can be written as with the decomposition of g µν , where the ǫ µ (λ) is transverse(λ = ±1) or longitudinal(λ = 0) polarization states and ǫ µ (t) is timelike polarization states. The above amplitude can be decomposed into the Lorentz invariant hadronic and leptonic matrix elements: where H λ,λw (L λw ) is hadronic(leptonic) helicity amplitude, λ (W ) (0, ±1, t) corresponds to the helicity of the daughter baryon (W ) and the ǫ µ (λ W ) is the polarization vector of W boson. In the SM, charmed baryons can decay into octet baryons. The SU(3) anti-triplet and octet matrix are denoted by Tree operators of charm quark semileptonic decays into light quarks are categorized into c → d/s. Therefore under the flavor SU(3) symmetry, the low-energy effective Hamiltonian can be decomposed in terms of H 3 shown as: The corresponding helicity amplitude can be written as: where the a λ,λw 1 represents SU(3) irreducible nonperturbative amplitude. The a λ,λw 1 can be expressed by the form factors Here u(λ i ) is the spinor of charmed baryons, u(λ) is the spinor of the final state, and f i (i = 1, 2, 3), f ′ i (i = 1, 2, 3) are the form factors. In the heavy quark limit [36], the f 2 , f 3 , f ′ 2 , f ′ 3 are suppressed by 1/m Bc , and only one independent form factor exists if the large-recoil symmetry is adopted [37,38]. Actually, a previous calculation [25] also indicates that the form factor of vector parameter f 1 and axial-vector parameter f ′ 1 have dominant contributions to the heavy baryon decay processes. Thus we neglect f 2 , f ′ 2 , f 3 , and f ′ 3 in our later calculations. Expanding Eq. (10), one obtains the relations between the helicity amplitudes of different channels of anti-triplet charmed baryons, which are presented in Table II. In the SU(3) symmetry limit, the branching fractions of Ξ + c → Ξ 0 ℓ + ν ℓ and Ξ 0 c → Ξ − ℓ + ν ℓ can be predicted  Table I. To shed further light on the decay dynamics, we take the pole model as an illustration to access the q 2 dependence of form factors [39] where f i = f i (q 2 = 0) and m p = 2.061GeV, which is the average mass of D and D s . The differential decay widths can be expressed by these form factors, Here  Table II. For instance in the first process in Table II Using the amplitudes, we can fit the parameters f 1 and f ′ 1 with experimental data. In the fit, we use the experimental values for the particle masses. The fitted results are shown in Table III. Obviously, the χ 2 in fitting is too large to be considered as a good fit, which implies that the SU(3) symmetry is not a good symmetry for charmed baryon decays. In the previous fit, we have neglected the possible SU(3) breaking effects. Because the light u, d, and s quarks have different masses, the SU(3) symmetry is broken. Neglecting the masses of u and d quark the mass matrix M can be written as: We can obtain the modified helicity amplitude as The a λ,λw Adding SU(3) symmetry breaking term and expanding the above formula in Eq. (15), one can obtain the amplitudes of different channels which are collected in the " amplitude I " column of Table IV. It can be seen that the parameters a λ,λw 1 and a λ,λw 5 always appear together in channels Λ + c → Λ 0 ℓ + ν ℓ , Ξ + c → Ξ 0 ℓ + ν ℓ and Ξ 0 c → Ξ − ℓ + ν ℓ . Therefore the SU(3) symmetry breaking irreducible nonperturbative amplitudes a λ,λw 2 , a λ,λw 3 , a λ,λw 4 , and a λ,λw 5 can be parametrized as where the a λ,λw 2 − a λ,λw 4 is the combination that appears in helicity amplitude Ξ + c → Ξ 0 ℓ + ν ℓ and Ξ 0 c → Ξ − ℓ + ν ℓ . By using the replacement rule: f we can directly fit these parameters from the data. In doing the combination of a λ,λw 1 + a λ,λw 5 together to fit data, the a λ,λw 1 in Λ + c → nℓ + ν ℓ , Ξ + c → Σ 0 ℓ + ν ℓ and Ξ 0 c → Σ − ℓ + ν ℓ will need to be treated as a λ,λw  Table V, one can find that SU(3) symmetry breaking effects to the differential decay width for the Ξ 0 c → Ξ − e + ν e can reach as much as 50%,   Table V since a relatively smaller χ 2 is obtained. The inadequacy of the experimental data at this stage prevents a direct analysis of different individual terms especially the a λ,λw 2 , a λ,λw

5
. We hope that more experimental data can be accumulated to further examine the detailed sources of SU(3) symmetry breaking in the future.  The mixing between anti-triplet charmed baryons to the sextet states is due to the following term expanding to the first order in m s , where the angle θ is at the order O(m s ). To the first order in m s , cos θ ∼ 1, sin θ ∼ θ.
To take into account the mixing effects for physical Ξ c states, one needs to work out the sextet semileptonic decay amplitudes which are given to the first order in ω The helicity amplitude for anti-triplet charmed baryon with mass eigenvalue state Ξ mass can be obtained by using Eq. (15), Eq. (19) and Eq. (20). At the leading order, the helicity amplitudes for the decay channel of mass eigenvalue states Ξ 0mass where we have neglected the O(m 2 s ) and higher order corrections. The helicity amplitudes of other channels are listed in the " amplitude I " column of Table IV. In the table, the states in the first column are understood to be the mass eigenstates for the case with the mixing effect.
It is clear that the existing experimental data is insufficient to determine all these parameters. But one can see that by introducing the effective amplitude a ′λ,λw  Table IV. Therefore, our fit results for the case without mixing effects are still valid, but the form factors δf 1 and δf ′ 1 correspond to the new effective amplitudes a ′λ,λw 2 , a ′λ,λw 4 . Although several other form factors such as ∆f 1 and ∆f ′ 1 cannot be constrained due to the lack of experimental data, in some scenarios, we still estimate the branching ratios of some processes from the results in Table IV. We can estimate the branching fractions of Λ + c → ne + ν e and Λ + c → nµ + ν µ : by assuming a λ,λw 5 giving no contribution. If the process of Λ + c → nℓ + ν ℓ is measured by experiments, the contributions of a λ,λw 5 will be obtained. The branching fractions of Ξ + c → Σ 0 ℓ + ν ℓ , Ξ + c → Λ 0 ℓ + ν ℓ and Ξ 0 c → Σ − ℓ + ν ℓ can also be estimated by assuming a ′λ,λw 2 , a λ,λw 3 , and a λ,λw 5 giving no contributions.
For the processes of Ξ + can also be established.

III. SU(3) SYMMETRY ANALYSIS IN ANTI-TRIPLET BEAUTY BARYONS SEMILEPTONIC DECAYS
The anti-triplet beauty baryon semileptonic decays are governed by the Hamiltonian: where q = u, c.
We write the helicity amplitude in SU(3) analysis in a similar fashion as what has been done for semileptonic charmed anti-triplet decays, as where b λ,λw Expanding the H λ,λw , one can obtain SU(3) amplitudes are listed in Table VI and the SU(3) relations can be given as follows, Using the experimental data B(Λ 0 b → Λ + c ℓ −ν ℓ )=(6.2 +1.4 −1.3 )% and B(Λ 0 b → pµ −ν µ )=(4.1 ± 1.0)%, we give the prediction in third column of Table VI.
For the processes we predicted, we expect them to be measured by Belle II and LHCb. The SU(3) symmetry of these processes will probably be tested. Due to the lack of experimental data at this stage, we can not explore the SU(3) symmetry breaking effects by fitting the form factors. We have also worked out how to include SU(3) symmetry breaking effects. The helicity amplitude including SU(3) symmetry breaking about b quark decays is given as: where the b λ,λw term has no contribution, because (H ′ 3 ) k ω n k is equal to zero. Expanding the formula above, we collected the SU(3) amplitudes in Table VII.
A number of relations for decay widths can be readily deduced from Table VII, It can be seen from Eq.(29) that though the SU(3) symmetry breaking effects caused by the light quark mass are taken into account, there are still relations in these processes. These relations result from isospin symmetry which can only be broken if non-zero u and d quark masses with different values are included. We strongly suggest our experimental colleagues carry out measurements for these decays.

IV. CONCLUSION
We have investigated the semileptonic decay of anti-triplet heavy baryons using SU(3) symmetry based on the latest experimental data. In the SU(3) symmetry limit, when fitting the available experimental data to the SU(3) symmetry analysis, we can only obtain a fit with a least χ 2 /d.o.f = 14.3 which means SU(3) symmetry is not a good symmetry for semileptonic charmed anti-baryon decays. We have then carried out detailed analyses with SU(3) symmetry breaking effect due to mass difference between s quark and u/d quark mass. In one scenario, we obtain a reasonable description of all relevant data with a χ 2 /d.o.f = 1.6. As an estimation, we give the branching ratios for Λ + c → nℓ + ν ℓ , Ξ + c → Σ 0 ℓ + ν ℓ , Ξ + c → Λ 0 ℓ + ν ℓ and Ξ 0 c → Σ − ℓ + ν ℓ in some scenarios. We have also extended the analysis to the semileptonic decays of anti-triplet beauty baryons. However, the lack of experimental data prevents us from an in-depth study. Instead, we find a set of SU(3) relations in Eq. (27) and isospin relation in Eq. (29) between the decay widths of such processes. Our results will help to explore the physics behind these SU(3) symmetry breaking experimental data with more experimental data available in the future.