Testing SU(3) Flavor Symmetry in Semileptonic and Two-body Nonleptonic Decays of Hyperons

The semileptonic decays and two-body nonleptonic decays of light baryon octet ($T_8$) and decuplet ($T_{10}$) consisting of light $u,d,s$ quarks are studied with the SU(3) flavor symmetry in this work. We obtain the amplitude relations between different decay modes by the SU(3) irreducible representation approach, and then predict relevant branching ratios by present experimental data within $1 \sigma$ error. We find that the predictions for all branching ratios except $\mathcal{B}(\Xi\rightarrow \Lambda^0\pi)$ and $\mathcal{B}(\Xi^*\rightarrow \Xi\pi)$ are in good agreement with present experimental data, that implies the neglected $C_+$ terms or SU(3) breaking effects might contribute at the order of a few percent in $\Xi\rightarrow \Lambda^0\pi$ and $\Xi^*\rightarrow \Xi\pi$ weak decays. We predict that $\mathcal{B}(\Xi^{-}\rightarrow \Sigma^0\mu^-\bar{\nu}_\mu)=(1.13\pm0.08)\times10^{-6}$, $\mathcal{B}(\Xi^{-}\rightarrow\Lambda^0\mu^-\bar{\nu}_\mu)=(1.58\pm0.04)\times10^{-4}$, $\mathcal{B}(\Omega^-\rightarrow\Xi^0\mu^-\bar{\nu}_\mu)=(3.7\pm1.8)\times10^{-3}$, $\mathcal{B}(\Sigma^-\rightarrow \Sigma^0e^-\bar{\nu}_e)=(1.35\pm0.28)\times10^{-10}$, $\mathcal{B}(\Xi^-\rightarrow \Xi^0e^-\bar{\nu}_e)=(4.2\pm2.4)\times10^{-10}$. We also study $T_{10}\to T_8 P_8$ weak, electromagnetic or strong decays. Some of these decay modes could be observed by the BESIII, LHCb and other experiments in the near future. Due to the very small life times of $\Sigma^0$, $\Xi^{*0,-}$, $\Sigma^{*0,-}$ and $\Delta^{0,-}$, the branching ratios of these baryon weak decays are only at the order of $\mathcal{O}(10^{-20}-10^{-13}$), which are too small to be reached by current experiments. Furthermore, the longitudinal branching ratios of $T_{8A} \to T_{8B} \ell^- \bar{\nu}_\ell~(\ell=\mu,e)$ decays are also given.

Theoretically, the factorization does not work well for s, d quark decays since s, d quarks are very light and can not use the heavy quark expansion. There is no reliable method to calculate these decay matrix elements at present. In the lack of reliable calculations, the symmetry analysis can provide very useful information about the decays. SU (3)  The SU(3) flavor symmetry works well in heavy hadron decays, for instance, the b-hadron decays [13][14][15][16][17][18][19][20][21][22][23][24] and the chadron decays [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. The experimental data of some semileptonic hyperon decays are well explained by the Cabibbo theory [10], which assumes the SU(3) symmetry breaking effects are neglected. The SU(3) flavor symmetry breaking effects are also studied in the hyperon beta-decays [40][41][42][43], where it is found that the SU(3) symmetry breaking effects in these decays are small. In this paper, we will systematically study T 8,10 → T 8 −ν and T 8,10 → T 8 P decays by the SU(3) irreducible representation approach (IRA). We will firstly construct the SU(3) irreducible representation amplitudes for different kinds of T 8 and T 10 decays, secondly obtain the decay amplitude relations between different decay modes, then use the available data to extract the SU(3) irreducible amplitudes, and finally predict the not-yetmeasured modes for further tests in experiments.
This paper is organized as follows. In Sec. II, the semileptonic weak decays of the T 8,10 hyperons are studied. In Sec.
III, we will explore the two-body nonleptonic decays of hyperons which are through weak interaction, electromagnetic or strong interaction. Our conclusions are given in Sec. IV.

II. Semileptonic decays of hyperons
The light baryons T 8 (T 10 ), which are octet (decuplet) under the SU(3) flavor symmetry of u, d, s quarks, can be written as In this section, we focus on ∆S = 0 and ∆S = 1 semileptonic decays of hyperons, which decay through d → ue −ν e or s → u −ν transitions, respectively. Since ∆S = 2 semileptonic decays are forbidden, we will not study them in this work.
A. T 8A → T 8B −ν semileptonic decays In the Standard Model (SM), the feynman diagram for T 8A → T 8B −ν decays is shown in Fig. 1, and the amplitudes of T 8A → T 8B −ν can be written as [44] A with Either from parity or from explicit calculation, we have the relations The form factors f i (q 2 ) and g i (q 2 ) are defined by [24] In term of the SU(3) IRA, the helicity amplitudes H where H k n = V q k qn is the CKM matrix element, a ni ≡ (a ni ) For convenience, we setĀ 22 The reparameterization results are given in the last column of Table I, in which we can easily see the helicity amplitude relations between different decay modes.
The differential branching ratios of T 8A → T 8B −ν decays can be written as with  The differential longitudinal branching ratios dB L (T 8A → T 8B −ν )/dq 2 can be obtained from dB(T 8A → The theoretical input parameters and the experimental data within the 1σ error from Particle Data Group [1] will be used in our numerical results. Two cases will be considered in our analysis:    [42,[45][46][47][48][49][50][51][52][53][54]. In this case, we choose the dipole behavior for the form factors as [40,51] where M = 0.97 (1.25) GeV for the vector (axial vector) form factors f i (g i ) in s → u −ν decays, and M = 0.84 ± 0.04 (1.08 ± 0.08) GeV for f i (g i ) in d → ue −ν e decays. For the form factor ratios g 1 (0)/f 1 (0) and f 2 (0)/f 1 (0), they are preferentially taken from experimental measurements. If no relevant experimental measurements are available, they will be taken from Cabibbo theory [51]. The form factor ratios in Tab. II will be used in our results. As a result, the branching ratios only depend on the form factor f 1 (0) and the CKM matrix elemant V uqn . Then these three parameters become for s → u −ν transition, where A ni contains f 1 (0) but without the q 2 dependence. Finally, all experimental data will be considered to constrain these parameters and predict the not-yet-measured branching ratios.
Firstly, we give a comment on the results of the twelve s → u −ν decay modes. In S 1 case, we get A 31 = 5.87±0.21,  The form factor ratios g1(0)/f1(0) and f2(0)/f1(0) from PDG2018 [1] unless otherwise specified. a denotes that the values are obtained from the SU(3)-favour parametrization F and D given in Refs. [40,51] and the measured form factor ratios in Ref. [1], and b denotes that the values are taken from Cabibbo theory [51].

Decay modes
we consider q 2 -dependence of the form factors and all relevant experimental constraints. We get A 31 = 1.04 ± 0.04, A 32 = 0.98 ± 0.03, |δ A 32 | ≤ 28 • , and the branching ratio predictions are given in the third column of Tab. III. We can see that the experimental data of give the finally effective constraints on the relevant parameters, and the SU(3) IRA predictions in S 2 case are quite consistent with the present data within 1σ error. We predict that B(Ξ − → Σ 0 µ −ν µ ) is at 10 −6 order of magnitude, which is promising to be observed by the BESIII and LHCb experiments.
Then we comment the results of the six d → ue −ν e decay modes. Three branching ratios B(Σ − → Λ 0 e −ν e ), B(Σ + → Λ 0 e + ν e ) and B(n → pe −ν e ) are precisely measured, which can be used to constrain on A We obtain A 21 = 4.61 ± 0.01 andĀ 22 = 5.85 ± 0.16 in S 1 case as well as A 21 = 4.50 ± 0.02 andĀ 22 = 0.36 ± 0.36 in are at the order of 10 −10 in S 2 case, which should be tested by the future experiments.
The longitudinal branching ratios of T 8A → T 8B −ν decays are also predicted in S 2 case, which are listed in the last column of Tab. III. Noted that the life time of Σ 0 is very small, so the relevant decay branching ratios are also very small, and the same things happen in latter Ξ * 0,− , Σ * 0,− and ∆ 0,− semileptonic decays. The experimental data and the SM predictions with the ±1σ error bar of branching ratios of T8A → T8B ν . ‡ denotes which experimental data give the finally effective constraints on the parameters, and † denotes the predictions depend on the relative phase, which is not constrained well from present data.
And the differential branching ratios of T 10A → T 8B −ν decays can be written as with The S 1 case given in Sec. II A will be considered in ) has been measured. The experimental datum is listed in the second column of Tab. V. We use B(Ω − → Ξ 0 e −ν e ) to constrain b 31 , and then give the predictions for other relevant decay branching ratios. The results are given in the third column of Tab. V. We obtain B(Ω − → Ξ 0 µ −ν µ ) = (3.7±1.8)×10 −3 , which is very promising to be measured by the BESIII and LHCb. For d → ue −ν e transition, no decay mode has been measured yet. We use H(

III. Nonleptonic two-body decays of light baryons
In this section, we discuss the two-body nonleptonic decays of light baryons T A. Weak decays of light baryons In the SM, as shown in Fig. 2, there are two kinds of diagrams for the nonleptonic s quark decays, the tree level diagram in Fig. 2 (a) and the penguin diagram in Fig. 2 (b). The effective Hamiltonian for nonleptonic s quark decays at scales µ < m c can be written as [55] H where V uq is the CKM matrix element, z i (µ) and y i (µ) are Wilson coefficients. The four-quark operators Q i are where Q 1,2 are current-current operators corresponding to Fig. 2 (a), Q 3−6 (Q 7−10 ) are QCD (electroweak) penguin operators corresponding to Fig. 2 (b). In Eq. (18), the magnetic penguin operators are ignored since their Compared with tree-level contributions related to C 1,2 , the penguin contributions are suppressed by smaller Wilson coefficients C 3,···,10 and can be ignored in these decays.
The four-quark operators Q i can be rewritten as (q i q k )(q j s) with q i = (u, d) as the doublet of 2 under the SU (2) symmetry by omitting the Lorentz-Dirac structure. Since (q i q k )(q j s) can be decomposed as the irreducible representations (IR) of (2 ⊗ 2 ⊗2)s = (2 p ⊕2 t ⊕ 4)s, one may obtain that and we have the relation O(4) 1 21 = −O(4) 2 22 = O(4) 1 12 by the traceless condition. Then Q 1,2 , Q 3−6 and Q 7,10 can be transformed under SU(2) symmetry as2 p ⊕2 t ⊕ 4,2 p /2 t and2 p ⊕2 t ⊕ 4, respectively, From Eq. (20), one can see that the contributions from current-current operators related to C 1,2 are much larger than others related to C 3,···,10 . So we will only consider current-current operator contributions in the following analysis.
After neglecting C 3,···,10 , the effective Hamiltonian in Eq. (23) can be rewritten as where C ± ≡ (C 2 ± C 1 )/2, and H ij k is related to (q i q k )(q j s) operators. From Eq. (20), one gets C 2 + /C 2 − ≈ 13.7%, so C − term related to H(2 t ) − H(2 p ) gives the dominant contribution to the decay branching ratios. The non-zero entries of H ij k corresponding to current-current operators in SU (2) Noted that H(4) 22 2 = − 1 3 only contributes to the penguin operators and we ignore it. In Eq. (25), the2 irreducible representation is linear combinations of2 p,t , so we need only consider a single2 when computing amplitudes from the invariants and reduced matrix elements [25].
The amplitudes of the T 8,10 → T 8 M 8 decays can be written via the effective Hamiltonian in Eq. (18) as These amplitudes may be divided into the S wave and P wave amplitudes, which have been analysed, for instance, in heavy baryon chiral perturbation theory [56][57][58][59] and by using a relativistic chiral unitary approach based on coupled channels [60]. Moreover, since H IR ef f is irreducible in the SU(2) symmetry, and the initial and final state baryons (T 8 , T 10 , M 8 ) are irreducible in the SU(3) symmetry, the amplitudes of T 8,10 → T 8 M 8 can be further written as 1.
where the coefficients a i , b i , c i , d i , e i , f i are constants which contain the Wilson coefficients, CKM matrix elements and information about QCD dynamics. Using H(4) ab c is symmetric in upper indices, b i and d i terms can be simplified by In addition, using i, j antisymmetric in T Finally, Eq. (32) can be simplified as In Tab. VI, we list the IRA amplitudes of T 8 → T 8 P 8 weak decays, which include the H(4) 12 1 , H(4) 22 2 and H(2) 2 terms. The corresponding T 8 → T 8 V 8 weak decays have the same relations as T 8 → T 8 P 8 weak decays. If only considering the dominant contributions from H(2) 2 and redefining the parameters the IRA amplitudes can be greatly simplified as listed in the last column of Tab. VI, in which we can easily see the relations of different decay amplitudes.
For Λ 0 → pπ − , nπ 0 decays, there is only one parameter A 4 . We first get the value of |A 4 | from the data of The slight difference between the prediction and datum comes from the experimental constraint of B(Λ 0 → nπ 0 ).

T 10 → T 8 M 8 weak decays
Feynman diagrams for T 10 → T 8 M 8 nonleptonic decays are also displayed in Fig. 3, and the SU(3) IRA amplitudes Considering H(4) lk m and (T 10 ) nij is symmetric in upper indices, we have the relations Then Eq. (38) can be simplified as The IRA amplitudes for T 10 → T 8 P 8 weak decays are listed in Tab. VIII, and the IRA amplitudes for T 10 → T 8 V 8 weak decays have similar relations. If neglecting H(4) 22 2 terms and c i terms in H(4) 12 1 , and redefining the parameters A 1 = 2(ā 1 +ē 1 ), the six decay amplitudes can be given in simpler forms, which are shown in the last column of Tab.VIII. Furthermore, The branching ratios of T 10 → T 8 P 8 can be obtained in terms of IRA amplitudes and the mass difference in A(T 10A → T 8B P 8 ), which is similar to Eq. (35), is also considered.

B. Electromagnetic or strong decays of light baryons
The light baryons T 10 can also decay through electromagnetic or strong interactions. The Feynman diagram of electromagnetic or strong (ES) decays of T 10 is shown in Fig. 4. In this case, we only need consider the SU (3) symmetry between initial and final states. The SU(3) IRA amplitude of T 10 → T 8 M 8 ES decay is There is only one parameter β 1 for these IRA amplitude. The IRA amplitudes of all the ES T 10 → T 8 P 8 decays are given in Tab. IX. For these ES decays, only three branching ratios are measured, which are given in Tab. X. We first get |β 1 | from the data of B(Σ * → Σπ), then also consider the experimental constraint from B(Σ * → Λπ), and finally give the    give the effective constraints on the parameters, and ⊗ denotes which experimental data are not used to constrain parameters.
Note that the ES T 8 → T 8 P 8 decays and the ES T 10 → T 8 K decays are not allowed by the phase space, since the sum of final hadron masses is larger than the mass of initial state.

IV. SUMMARY
Light baryon decays play very important role in testing the SM and searching for new physics beyond the SM.
Many decay modes have been measured and some decays can be studied at BESIII and LHCb experiments now.
Motivated by this, we have analyzed the semileptonic decays and two-body nonleptonic decays of light baryon octet and decuplet by using the irreducible representation approach to test the SU(3) flavor symmetry. Our main results can be summarized as follows: • Semileptonic light baryon decays: We find that all branching ratio predictions of octet and decuplet baryons through s → u −ν and d → ue −ν e transitions with SU(3) IRA in S 2 case are quite consistent with present experimental measurements within 1σ error. We predict that B(Ξ − → Σ 0 µ −ν µ ) and B(Ω − → Ξ 0 µ −ν µ ) are at the order of magnitudes of 10 −6 and 10 −3 , respectively, and B(Σ − → Σ 0 e −ν e , Ξ − → Ξ 0 e −ν e ) are at the order of 10 −10 . These decays are promising to be observed by the BESIII and LHCb experiments or the future experiments. However, other branching ratios, which are in the range of 10 −20 − 10 −13 , may not be measured for a long time. Moreover, the longitudinal branching ratios of decays of T 8A → T 8B −ν are also predicted in this work.
• Nonleptonic two-body light baryon decays: We obtain the relations of different decay amplitudes by the SU(3) IRA and isospin symmetry. In T 8 → T 8 P 8 weak decays, we find that SU(3) IRA predictions of the branching ratios of Σ, Λ baryons are consistent with present experimental data, B(Σ 0 → pπ − , nπ 0 ) are at the order of 10 −10 by the SU(3) IRA or isospin symmetry, and the neglected C + terms or SU(3) symmetry breaking effects might give a contribution of a few percent to the two branching ratios of Ξ → Λπ. In T 10 → T 8 P weak decays, we predict that B(Ξ * − → Λ 0 π − ), B(Ξ * 0 → Λ 0 π 0 ) and B(Ξ * 0 → Σ + π − ) are at the orders of 10 −12 , 10 −13 and 10 −14 , respectively. In T 10 → T 8 P 8 ES decays, when IRA predictions are consistent with the data of B(Σ * → Σπ) and B(Σ * → Λπ), the prediction of B(Ξ * → Ξπ) is slightly larger than experimental data, which imply that the SU(3) symmetry breaking effects could give visible contributions to B(Ξ * → Ξπ). In addition, we given all the specific branching ratio predictions for these T 10 → T 8 P 8 ES decays.
Although flavor SU(3) symmetry is approximate, it can still provide us very useful information about these decays.
According to our predictions, some branching ratios are accessible to the experiments at BESIII and LHCb. Our results in this work can be used to test SU(3) flavor symmetry approach in light baryon decays by the future experiments..