Studying $\mathcal{B}_1(\frac{1}{2}^+)\to \mathcal{B}_2(\frac{1}{2}^+)\ell^+\ell^-$ Semi-leptonic Weak Baryon Decays with the SU(3) Flavor Symmetry

Motivated by recent anomalies in FCNC $b\to s\ell^+\ell^-$, we study $B_1\to B_2\ell^+\ell^-(\ell=e,\mu,\tau)$ semi-leptonic weak decays with the SU(3) flavor symmetry, where $B_{1,2}$ are the spin-1/2 baryons of single bottomed anti-triplet $T_{b3}$, single charmed anti-triplet $T_{c3}$ or light baryon octet $T_{8}$. Using the SU(3) irreducible representation approach, we first obtain the amplitude relations among different decays, and then predict the relevant not-yet measured observables of these decays. (a) We calculate the branching ratios of the $T_{b3}\to T_8 \mu^+\mu^-$ and $T_{b3}\to T_8 \tau^+\tau^-$ in the whole $q^2$ region and in the different $q^2$ bins by the measurement of $\Lambda^0_b\to \Lambda^0 \mu^+\mu^-$. Many of them are obtained for the first time. In addition, the longitudinal polarization fractions and the leptonic forward-backward asymmetries of all $T_{b3}\to T_{8}\ell^+\ell^-$ decays are very similar to each other in certain $q^2$ bin due to the SU(3) flavor symmetry. (b) We analyze the upper limits of $B(T_{c3}\to T_{8}\ell^+\ell^-)$ by using the experimental upper limits of $B(\Lambda^+_c\to p\mu^+\mu^-)$ and $B(\Lambda^+_c\to pe^+e^-)$, and find the experimental upper limit of $B(\Lambda^+_c\to p\mu^+\mu^-)$ giving effective bounds on the relevant SU(3) flavor symmetry parameters. The predictions of $B(\Xi^0_c \to \Xi^0e^+e^-)$ and $B(\Xi^0_c \to \Xi^0\mu^+\mu^-)$ will be different between the single-quark transition dominant contributions and the W-exchange dominant ones. (c) As for $T_{8}\to T'_8 \ell^+\ell^-$ decays, we analyze the single-quark transition contributions and the W-exchange contributions by using two measurements of $ B(\Xi^0\to \Lambda^0 e^+e^-)$ and $ B(\Sigma^+\to p\mu^+\mu^-)$, and give the branching ratio predictions by assuming either single-quark transition dominant contributions or the W-exchange dominant ones.


I. INTRODUCTION
Flavor changing neutral current (FCNC) processes, such as b → sℓ + ℓ − , give access to important tests of the standard model (SM) and searches for new physics beyond the SM.Recently, some discrepancies with the SM are reported in several observables in B meson decays, for example, the angular-distribution observable P ′ 5 of B 0 → K * 0 µ + µ − [1][2][3][4] and the lepton flavor universality observables R K + and R K * 0 with R M = B(B→Mµ + µ − ) B(B→Me + e − ) [5,6].Semileptonic baryon decays are quite different to B, D, K meson ones, for instance, the initial baryons may be polarized, the transitions involve a diquark system as a spectator rather than a single-quark, and the W -exchange contributions of two-quark and three-quark transitions might appear in baryon decays.Therefore, the baryon decays provide the important additional tests of the SM predictions, which can be used to improve the understanding of recent anomalies in B meson decays.Recently significant experimental progress has been achieved in studying rare Λ b decays.The Λ b → Λµ + µ − baryon decay is the only one measured among the T b3 → T 8 ℓ + ℓ − decays at present.B(Λ b → Λµ + µ − ) was first measured by the CDF Collaboration [7] and then greatly improved by LHCb [8,9].For T c3 → T 8 ℓ + ℓ − decays, only B(Λ c → pe + e − ) and B(Λ c → pµ + µ − ) have been upper limited by BABAR and LHCb [10,11].As for T 8 → T ′ 8 ℓ + ℓ − decays, Ξ 0 → Λ 0 e + e − and Σ + → pµ + µ − have been measured by NA48 [12] and HyperCP [13], respectively.With the experiment development, some B 1 → B 2 ℓ + ℓ − decays will be improved or detected by the BESIII, LHCb, and Belle-II Collaborations in the near future, so it is necessary to study B 1 → B 2 ℓ + ℓ − decays theoretically.
The theoretical challenge in the study of B 1 → B 2 ℓ + ℓ − decays is calculating the hadronic B 1 → B 2 form factors in the hadronic matrix elements.Form factors for Λ b → Λ have been estimated in lattice QCD [14][15][16], QCD light cone sum rules [17], the soft-collinear effective theory [18], and perturbative QCD [19].Form factors for Λ b → n have been estimated in the relativistic quark diquark picture [20] and the context of light cone QCD sum rules [21,22].
Nevertheless, other form factors of T b3 → T 8 ℓ + ℓ − , such as the ones for Ξ 0 b → Σ − , have not been calculated yet.Similarly in T c3 → T 8 ℓ + ℓ − and T 8 → T ′ 8 ℓ + ℓ − decays, only some form factors are calculated, for example, ones for Λ + c → p transition [22][23][24][25].Theoretical calculations of the hadronic matrix elements are not well understood due to our poor understanding of QCD at low energy regions.The SU(3) flavor symmetry approach is independent of the detailed dynamics offering us an opportunity to relate different decay modes.Nevertheless, it cannot determine the sizes of the amplitudes by itself.However, if experimental data are enough, one may use the data to extract the amplitudes, which can be viewed as predictions based on symmetry.Although SU(3) flavor symmetry is only an approximate symmetry because u, d and s quarks have different masses, it still provides some useful information about the decays.One popular way of predicting the SU(3) flavor symmetry is to construct the SU(3) irreducible representation amplitude by decomposing the effective Hamiltonian, in which one only focuses on the SU(3) flavor structure of the initial states and finial states, but does not involve the details about the interaction dynamics.Some B 1 ( 1 2 + ) → B 2 ( 1 2 + )ℓ + ℓ − semileptonic baryon decays have been well studied, for instance, semileptonic Λ 0 b decays in Refs.[14][15][16][26][27][28][29][30][31], semileptonic Λ + c decays in Refs.[23,24,32], and semileptonic Σ + decays in Refs.[33][34][35][36][37].In this work, we will study all weak B 1 ( 1 2 + ) → B 2 ( 1 2 + )ℓ + ℓ − decays by using the SU(3) irreducible representation approach.We first obtain the amplitude relations among different decay modes, then use the available data to extract the SU(3) irreducible amplitudes and finally predict the not-yet-measured modes for further tests in experiments.This paper is organized as follows.In Sec.II, we will collect the representations for the baryon multiplets of 1  2 -spin and the observable expressions of relevant baryon decays.In Sec. III we will analyze the semileptonic weak decays of T b3 → T 8 ℓ + ℓ − , T c3 → T 8 ℓ + ℓ − , and T 8 → T ′ 8 ℓ + ℓ − .Our conclusions are given in Sec.IV.

II. Theoretical Frame
A. Baryon multiplets with 1 2 spin The light baryons octet T 8 under the SU(3) flavor symmetry of u, d, s quarks can be written as The single charmed antitriplet T c3 is given as The antitriplet T b3 with a heavy b quark is
receives not only from the four quark operators but also from the long distance (LD) contributions coming from cc for T b3 → T 8 ℓ + ℓ − and d d, ss for T c3 → T 8 ℓ + ℓ − .Note that the T c3 → T 8 ℓ + ℓ − decays are dominated by LD contributions.
The nonvanishing leptonic helicity amplitudes L λ1λ2 L(R),λ are with The B 1 → B 2 hadronic matrix elements are calculated in the frameworks of soft-collinear effective theory [18] and lattice QCD [14][15][16].The helicity-based definition of the form factors are presented as [14][15][16] where s ± = (m B1 ± m B2 ) 2 − q 2 and f V,A,T,T5 0,⊥ are the form factors.And then we obtain the nonvanishing hadronic helicity amplitudes H with C In addition, in terms of the SU(3) flavor symmetry, baryon states and quark operators can be parametrized into SU(3) tensor forms, while the polarization vectors ǭ * (λ) and leptonic helicity amplitudes L λ1,λ2 L(R),λ are invariant under SU(3) flavor symmetry.The hadronic helicity amplitude relations of B 1 → B 2 ℓ + ℓ − are similar to ones of B 1 → B 2 γ as given in Ref. [41], and will be given in next section for convenience.

C. Observables for
In the rest frame of the baryon B 1 , the double differential decay branching ratio is [42] with where And the differential decay branching ratio is with The longitudinal polarization fraction can be obtained by Eq. ( 13) and the concrete expression is The leptonic forward-backward asymmetry and the concrete expression is The lepton flavor universality in baryon weak decays For q 2 integration of X(q 2 ) = F L (q 2 ) and A ℓ F B (q 2 ), following Ref.[43], two ways of integration are considered.The normalized q 2 -integrated observables X are calculated by separately integrating the numerators and denominators with the same q 2 bins.The "naively integrated" observables are obtained by Note that, besides the single-quark transition contributions, the W -exchange contributions via the two-quark and three-quark transitions as well as the internal radiation transition, which contribute to the radiative baryon decays [44][45][46], may also contribute to the semileptonic baryon decays B 1 → B 2 ℓ + ℓ − .In some decays, for example, Σ + → pℓ + ℓ − decays, the W -exchange contributions with the two-quark transition will play a major role [37].So we will consider these W-exchange contributions in the later analysis of SU(3) flavor symmetry.

III. Results and Analysis
The theoretical input parameters and the experimental data within the 1σ error from the Particle Data Group [47] will be used in our numerical results.To obtain SU(3) IRA amplitudes, one just needs to contract all upper and lower indices of the hadrons and the Hamiltonian to form all possible SU(3) singlets and associate each with a parameter which lumps up the Wilson coefficients and unknown hadronization effects [48].These parameters can be determined theoretically and experimentally.In this work, we will determine these parameters by relevant experimental data, and then give the predictions for other not-yet-measured decay modes.For T b3 semileptonic decays, there are enough phase spaces to allow for e + e − , µ + µ − , and τ + τ − decays.
which are similar to the decay amplitudes of corresponding T b3 → T 8 γ modes in Ref. [52].In Eq. ( 23), T ( 3) = (0, 1, 1) denoted the transition operators (q 2 b) with q 2 = s, d, and the model as well as scale independent parameters e i ≡ The parameters e i contain information about QCD dynamics, and could include the long distance (LD) contributions from hadron resonances.The SU(3) IRA amplitudes of the T b3 → T 8 ℓ + ℓ − weak decays are given in Tab.I, and for a better understanding, the information of relevant CKM matrix elements are also listed in Tab.I.
From Tab.I, one can see that Λ 0 b → Σ 0 ℓ + ℓ − decays are not allowed by the SU(3) flavor symmetry, and other decay modes via b → s/dℓ + ℓ − can be related by only one parameter E ≡ e 1 + e 2 .

Among Λ
decay has been measured, and its branching ratios in the whole q 2 region and in different q 2 bins are listed in Tab.II and Tab.III, respectively.One can constrain the relevant SU(3) flavor parameters by the experimental data within 1σ error bar and then predict other not-yet-measured branching ratios.Two cases will be considered in our analysis of T b3 → T 8 ℓ + ℓ − decays.
S 1 : The SU(3) flavor symmetry parameters without the baryonic momentum-transfer q 2 dependence.
We treat the SU(3) flavor parameters (E) (q 2 ) as constants without q 2 dependence, which will lead

Decay modes
TABLE II: Branching ratios for T b3 → T8ℓ + ℓ − decays with 1σ error in the whole q 2 region within S1 and S2 cases.

Decay modes
Experimental data [47] Our results in S 1 Our results in S 2 Other predictions 12.23 +4.12 −3.62 with 1σ error in S2 case.
[q 2 min , q 2 max ](GeV  H M (q 2 ) in Eq. ( 15) to a constant, too.We use the 1σ error experimental data of B(Λ 0 b → Λ 0 µ + µ − ) to constrain H M (q 2 ) (i.e., |E| 2 ), and then predict other B(T b3 → T 8 ℓ + ℓ − ) by the amplitude relations in Tab.I. S 2 : The SU(3) flavor symmetry parameters with the baryonic momentum-transfer q 2 dependence.In order to obtain more precise predictions, we use the hadronic helicity amplitude expressions in Eq. ( 12), which are q 2 dependent and can be expressed by the Wilson coefficients and the form factors.The expressions of the Wilson coefficients without the LD contributions are taken from Ref. [55].As for the q 2 dependent form factors involving the T b3 → T 8 transitions, we use the recent lattice QCD results of Λ 0 b → Λ 0 [14], in which the form factors are parametrized by where f = f + , f ⊥ , f 0 , g + , g ⊥ , g 0 , h + , h ⊥ , h + , h ⊥ , and the details of z(q 2 ) and m f pole can be found in Ref. [14].We keep f + (0) as an undetermined constant without q 2 dependence, and other f (0) can be expressed as The central values of a f i in Tab.V of Ref. [14] will be used in our analysis.Since these form factors also preserve the SU(3) flavor symmetry, the same relations in Tab.I will be used for f + (0).We use the 1σ error Using the experimental data of B(Λ 0 b → Λ 0 µ + µ − ) in the whole q 2 region, one can obtain the branching ratios for T b3 → T 8 µ + µ − and T b3 → T 8 τ + τ − weak decays in the whole q 2 region, which are listed in the third and forth columns of Tab.II for S 1 case and S 2 case, respectively.Noted that, the amplitude relations listed in Tab.I are obtained from the SU(3) flavor symmetry; nevertheless, the different baryon masses in the same baryon multiplets are considered in the branching ratio predictions, and the below is same.
Previous predictions are also listed in the last column of Tab.II for comparing.Since the results of T b3 → T 8 e + e − decays are quite similar to ones of T b3 → T 8 µ + µ − decays, we only show T b3 → T 8 µ + µ − in this work.We have the following remarks for the results in Tab.II.
• Comparing the branching ratios in S 1 and S 2 cases, one can see that the predictions are slightly different between S 1 and S 2 cases, which are mainly due to the q 2 dependence of the hadronic helicity amplitudes.
• Comparing our predictions for B(Λ ) with previous ones in the relativistic quark model [53] and the Bethe-Salpeter equation approach [54], one can see that our predicted ) is about 2 times larger than theirs.
• Many branching ratio predictions for T b3 → T 8 ℓ + ℓ − are obtained for the first time.The not-yet-measured B(T b3 → T 8 ℓ + ℓ − ) are on the order of O(10 −8 − 10 −6 ), and some of them could be reached by the LHCb or Belle-II experiments.
Using the experimental data of B(Λ 0 b → Λ 0 µ + µ − ) in different q 2 bins, one can get the branching ratios of T b3 → T 8 µ + µ − and T b3 → T 8 τ + τ − weak decays in different q 2 bins within S 1 and S 2 cases, which are collected for reference in Tab.III and Tab.IV, respectively.
The longitudinal polarization fractions and the leptonic forward-backward asymmetries with two ways of integration for T b3 → T 8 ℓ + ℓ − decays could also be obtained in the S 2 case.As shown in Eq. ( 18) and Eq. ( 20), the N (q 2 ) terms are canceled in the ratios; therefore, the longitudinal polarization fractions and the leptonic forward-backward asymmetries only depend on the hadronic helicity amplitudes, which preserve the SU(3) flavor symmetry in T b3 → T 8 ℓ + ℓ − decays.
So the longitudinal polarization fractions and the leptonic forward-backward asymmetries of all T b3 → T 8 µ + µ − (T b3 → T 8 τ + τ − ) decays are very similar to each other in certain q 2 bins.We take ones of Λ 0 b → Λ 0 µ + µ − , Λ 0 τ + τ − as examples, which are given in Tab.V. Excepting in [0.1, 2.0], [0.1, 4.3], [0.1, 16.0] and the whole q 2 regions, f L and f L (A ℓ F B and A ℓ F B ) with different q 2 integration ways are quite similar in other certain q 2 bins.So the obvious differentiation between f L and f L (A ℓ F B and A ℓ F B ) mainly appears in the quite low q 2 regions.Note that the normalized longitudinal polarization fraction and normalized leptonic forward-backward asymmetry of Λ 0 b → Λ 0 µ + µ − in q 2 ∈ [15, 20] GeV 2 have been measured by the LHCb experiment [47], We do not impose the above experimental bounds but leave them as predictions.Comparing with the experimental re- [15,20] and [15,20] and [15,20] are agreeable with their experimental data within 1.5σ and 1σ error ranges, respectively.In addition, the lepton flavor universality R T b3 →T8 in three q 2 bins within the S 2 case are given in Tab.VI.One can see that all predictions in three q 2 bins are virtually indistinguishable from unity; i.e., the lepton mass effects on all R B1→B2 are small in both the low-q 2 region and high-q 2 region in the SM.
TABLE VI: Lepton flavor universality of T b3 → T8ℓ + ℓ − baryon weak decays in different q 2 bins with 1σ error in the S2 case.

B. T c3 semileptonic weak decays
Similar to T c3,8 → T ′ 8 γ radiative decays [44][45][46]56], T c3 → T 8 ℓ + ℓ − decays receive single-quark, two-quark, and three-quark transition contributions with the W -exchange and internal radiation contributions.The internal radiation contributions are suppressed by the two W propagators and can be safely neglected.The SU(3) IRA hadronic helicity amplitudes for T c3 → T 8 ℓ + ℓ − be parametrized as where the SU(3) flavor symmetry parameters . The f i terms in Eq. (26) and the later g i terms in Eq. ( 29) denote the short distance (SD) and the LD contributions via the single-quark transitions.The f i terms in Eq. ( 26) and the later g i terms in Eq. ( 29) denote the W -exchange contributions of the two-quark and three-quark transitions.T ′ ( 3) = (1, 0, 0) denotes the transition operators (q 2 c) with q 2 = u, and H( 6) lk j (H(15) lk j ) related to the (q l q j )(q k c) operator is antisymmetric (symmetric) in upper indices.The nonvanishing H( 6) lk j and H(15) lk j for c → su d, dus, u dd, uss transitions can be found in Ref. [57].Using l, k antisymmetric in H( 6) lk j and l, k symmetric H(15) lk j , we have which will be used in the following discussion.
For the W -exchange transitions, there are three kinds of charm quark decaying into light quarks which are related to H( 6, 15) 22453, respectively.So three kinds decays given in Eq. ( 28) are called Cabibbo allowed, singly Cabibbo suppressed, and doubly Cabibbo suppressed decays, respectively.
The SU(3) IRA amplitudes of the T c3 → T 8 ℓ + ℓ − weak decays are given in the third column of Tab.VII, and for a better understanding, the information of relevant CKM matrix elements is also listed in this Table .From Tab.VII, one can see that singly Cabibbo suppressed Λ In addition, the contribution of H( 6) to the decay branching ratio is about 5.5 times larger than one of H(15) due to Wilson coefficient suppressed; for example, see Refs.[58,59].If ignoring the Wilson coefficient suppressed H (15) term contributions, there are only two parameters, F 1 and F ≡ f 1 − f 3 , in the decay amplitudes of T c3 → T 8 ℓ + ℓ − .The simplified results are listed in the last column of Tab.VII.One can see that the Cabibbo allowed and doubly Cabibbo suppressed eight decay modes of T c3 → T 8 ℓ + ℓ − are related by only one parameter F ; nevertheless, the singly Cabibbo suppressed eight decays are related by two parameters F and F 1 .Moreover, there are amplitude relations ) and B(Λ + c → pµ + µ − ) are upper limited by experiment.We list their experimental upper limits (exp.UL) in the second column of Tab.VIII.Due to the lack of the experiment in T c3 → T 8 ℓ + ℓ − decays and the complex amplitude expressions included the W -exchange contributions, we will only analyze the T c3 → T 8 ℓ + ℓ − decays in the S 1 case.We assume that W -exchange contributions noted by F or the single-quark transition contributions noted by F 1 play a dominant role in these decays.They are separately discussed as follows.
• Only considering the W -exchange contributions by setting F 1 = 0, all the T c3 → T 8 ℓ + ℓ − decays are related by one parameter F as shown in Tab.VII.The upper limit predictions of T c3 → T 8 e + e − and T c3 → T 8 µ + µ −
It is difficult to estimate which term gives the main contribution among G 1 , G 2 , G A , and G B now.Nevertheless, the LD contributions noted by G 1,2 can not been entirely ignored via the experimental measurement of Ξ 0 → Λ 0 ℓ + ℓ − .So we will give the following discussions.
• If only considering the single-quark transition contributions, i.e., G A = G B = 0, one obtains G1 G2 + 2 ≈ 12; i.e., G 1 ≈ 10G 2 or −14G 2 .After ignoring the small G 2 terms in G 1 + 2G 2 and 2G 1 + G 2 , one gets all branching ratios of relevant T 8 → T ′ 8 ℓ + ℓ − weak decays in S 1 case, which are given in the last column of Tab.X. • If G B ≈ −G 2 , i.e., the contributions between G 2 and G B are largely canceled in G 2 + G B term, the situations are complex, which is beyond the scope of this paper.
All predicted branching ratios except B(Σ 0 → ne + e − ) and B(Σ 0 → nµ + µ − ) in above first three cases are on the order of O(10 −8 − 10 −5 ), some of them might be observed by BESIII and Belle-II experiments in the near future.
The measurement of B(Ξ − → Σ − e + e − ) and B(Ξ 0 → Σ 0 e + e − ) in the future could further help us to understand the LD contributions and the W -exchange contributions, respectively.

TABLE III :
Branching ratios for T b3 → T8µ + µ − weak decays in different q 2 bins with 1σ error in S1 and S2 cases ( in unit of 10 −7 ).

TABLE IV :
Branching ratios for T b3 → T8τ + τ − weak decays in different q 2 bins with 1σ error in S1 and S2 cases ( in unit of 10 −7 ).

TABLE V :
Longitudinal polarization fractions and forward-backward asymmetries for Λ 0 b