PRDPRVDAQPhysical Review DPhys. Rev. D2470-00102470-0029American Physical Society10.1103/PhysRevD.103.013007ARTICLESElectroweak interactionsStudying B1(12+)→B2(12+)ℓ+ℓ- semileptonic weak baryon decays with the SU(3) flavor symmetrySTUDYINGWANG, XU, HUA, AND CHENGhttps://orcid.org/0000-0001-5204-3422WangRu-Min1,*https://orcid.org/0000-0002-1779-1684XuYuan-Guo1,†HuaChong1,‡https://orcid.org/0000-0002-1732-6926ChengXiao-Dong2,§College of Physics and Communication Electronics, JiangXi Normal University, NanChang, JiangXi 330022, ChinaCollege of Physics and Electronic Engineering, XinYang Normal University, XinYang, Henan 464000, China
ruminwang@sina.com
yuanguoxv@163.com
huachongccc@163.com
chengxd@mails.ccnu.edu.cn
15January20211January202110310130075December20204January2021Published by the American Physical Society2021authorsPublished by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.
Motivated by recent anomalies in flavor changing neutral current b→sℓ+ℓ- transitions, we study B1→B2ℓ+ℓ-(ℓ=e,μ,τ) semileptonic weak decays with the SU(3) flavor symmetry, where B1,2 are the spin-12 baryons of single bottomed antitriplet Tb3, single charmed antitriplet Tc3, or light baryons octet T8. Using the SU(3) irreducible representation approach, we first obtain the amplitude relations among different decay modes and then predict the relevant not-yet measured observables of Tb3→T8ℓ+ℓ-, Tc3→T8ℓ+ℓ-, and T8→T8′ℓ+ℓ- decays. (a) We calculate the branching ratios of the Tb3→T8μ+μ- and Tb3→T8τ+τ- decay modes in the whole q2 region and in the different q2 bins by the measurement of Λb0→Λ0μ+μ-. Many of them are obtained for the first time. In addition, the longitudinal polarization fractions and the leptonic forward-backward asymmetries of all Tb3→T8ℓ+ℓ- decays are very similar to each other in certain q2 bins due to the SU(3) flavor symmetry. (b) We analyze the upper limits of B(Tc3→T8ℓ+ℓ-) by using the experimental upper limits of B(Λc+→pμ+μ-) and B(Λc+→pe+e-), and find the experimental upper limit of B(Λc+→pμ+μ-) giving the effective bounds on the relevant SU(3) flavor symmetry parameters. The predictions of B(Ξc0→Ξ0e+e-) and B(Ξc0→Ξ0μ+μ-) will be different between the single-quark transition dominant contributions and the W-exchange dominant ones. (c) As for T8→T8′ℓ+ℓ- decays, we analyze the single-quark transition contributions and the W-exchange contributions by using the two experimental measurements of B(Ξ0→Λ0e+e-) and B(Σ+→pμ+μ-), and give the branching ratio predictions by assuming either single-quark transition dominant contributions or the W-exchange dominant contributions. According to our predictions, some observables are accessible to the experiments at BESIII, LHCb and Belle-II.
National Natural Science Foundation of China10.13039/50110000180911675137INTRODUCTION
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 P5′ of B0→K*0μ+μ-[1–4] and the lepton flavor universality observables RK+ and RK*0 with RM=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 Λb decays. The Λb→Λμ+μ- baryon decay is the only one measured among the Tb3→T8ℓ+ℓ- decays at present. B(Λb→Λμ+μ-) was first measured by the CDF Collaboration [7] and then greatly improved by LHCb [8,9]. For Tc3→T8ℓ+ℓ- decays, only B(Λc→pe+e-) and B(Λc→pμ+μ-) have been upper limited by BABAR and LHCb [10,11]. As for T8→T8′ℓ+ℓ- decays, Ξ0→Λ0e+e- and Σ+→pμ+μ- have been measured by NA48 [12] and HyperCP [13], respectively. With the experiment development, some B1→B2ℓ+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 B1→B2ℓ+ℓ- decays theoretically.
The theoretical challenge in the study of B1→B2ℓ+ℓ- decays is calculating the hadronic B1→B2 form factors in the hadronic matrix elements. Form factors for Λb→Λ have been estimated in lattice QCD [14], QCD light cone sum rules [15], the soft-collinear effective theory [16], and perturbative QCD [17]. Form factors for Λb→n have been estimated in the relativistic quark diquark picture [18] and the context of light cone QCD sum rules [19]. Nevertheless, other form factors of Tb3→T8ℓ+ℓ-, such as the ones for Ξb0→Ξ0, Ξb-→Ξ-, Ξb0→Λ0, Ξb0→Σ0, and Ξb-→Σ-, have not been calculated yet. Similarly in Tc3→T8ℓ+ℓ- and T8→T8′ℓ+ℓ- decays, only some form factors are calculated, for example, ones for Λc+→p transition [19–22].
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 B1(12+)→B2(12+)ℓ+ℓ- semileptonic baryon decays have been well studied, for instance, semileptonic Λb0 decays in Refs. [14,23–28], semileptonic Λc+ decays in Refs. [20,21,29], and semileptonic Σ+ decays in Refs. [30–34]. In this work, we will study all weak B1(12+)→B2(12+)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 12-spin and the observable expressions of relevant baryon decays. In Sec. III, we will analyze the semileptonic weak decays of Tb3→T8ℓ+ℓ-, Tc3→T8ℓ+ℓ-, and T8→T8′ℓ+ℓ-. Our conclusions are given in Sec. IV.
THEORETICAL FRAMEBaryon multiplets with <inline-formula><mml:math display="inline"><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:math></inline-formula> spin
The light baryons octet T8 under the SU(3) flavor symmetry of u, d, s quarks can be written as T8=(Λ06+Σ02Σ+pΣ-Λ06-Σ02nΞ-Ξ0-2Λ06).The single charmed antitriplet Tc3 is given as Tc3=(Ξc0,-Ξc+,Λc+).The antitriplet Tb3 with a heavy b quark is Tb3=(Ξb-,-Ξb0,Λb0).
Helicity amplitudes for semileptonic decays
In the SM, the low energy effective Hamiltonians for b→s/dℓ+ℓ-, c→uℓ+ℓ-, and s→dℓ+ℓ- FCNC transitions have similar forms and can be written as [22,30,35–37]H(q1→q2ℓ+ℓ-)=-αeGF2πλq1q2(C9effq¯2γμPLq1ℓ¯γμℓ+C10q¯2γμPLq1ℓ¯γμγ5ℓ-2mq1q2C7effq¯2iqνσμνPRq1ℓ¯γμℓ),where GF denotes the Fermi constant, the fine structure constant αe=e24π, the chiral projection operators PL,R=(1∓γ5)/2, σμν=i[γμ,γν]2, λq1q2 denotes the Cabibbo-Kobayashi-Maskawa (CKM) elements, and Ci denote Wilson coefficients. For the b→s/dℓ+ℓ- transitions via the uu¯, cc¯ loops, λbs(bd)=VtbVts* (VtbVtd*), and the expressions of Wilson coefficients C7,9eff and C10 are given in Ref. [37]. For the c→uℓ+ℓ- transition via dd¯, ss¯ loops, λcuC7,9eff=4παs[Vcd*VudC7,9eff(d)(q2)+Vcs*VusC7,9eff(s)(q2)] as well as C10=0 and the expressions of C7,9eff(s,d)(q2) can be found in Refs. [22,35]. For the s→dℓ+ℓ- transition via uu¯ loop, λsd=VusVud*, C7eff≈VcsVcd*2VusVud*c7γc, C9eff=(z7V-VtsVtd*VusVud*y7V)2παe, and C10=-VtsVtd*VusVud*2παey7A with z7V=-0.046αe, y7V=0.735αe, y7A=-0.700αe as well as c7γc=0.13αe from Refs. [30,36,37]. For Tb3→T8ℓ+ℓ- and Tc3→T8ℓ+ℓ- decays, the Wilson coefficient C9eff receives not only from the four quark operators but also from the long distance (LD) contributions coming from cc¯ for Tb3→T8ℓ+ℓ- and dd¯, ss¯ for Tc3→T8ℓ+ℓ-. Note that the Tc3→T8ℓ+ℓ- decays are dominated by LD contributions. For T8→T8′ℓ+ℓ- decays, the LD contribution arises mainly from the photon-mediated process T8→T8′γ*→T8′ℓ+ℓ-[31,34].
The helicity amplitudes for B1→B2ℓ+ℓ- can be obtained from Eq. (4), MVAλ1,λ2(sp,sk)=-GF2αe4πλq1q2∑ληλ[HVA,λL,sp,skLL,λλ1,λ2+HVA,λR,sp,skLR,λλ1,λ2],with LL(R),λλ1λ2=ε¯μ(λ)⟨ℓ¯(λ1)ℓ(λ2)|ℓ¯γμ(1∓γ5)ℓ|0⟩,HVA,λL(R),sp,sk=ε¯μ*(λ)⟨B2(k,sk)|[(C9eff∓C10)q¯2γμ(1-γ5)q1-2mq1q2C7effq¯2iqνσμν(1+γ5)q1]|B1(p,sp)⟩,where q2≡(p-k)2 bounded in physical region as (2mℓ)2≤q2≤(mB1-mB2)2, the polarization of the gauge boson λ=t,±1, 0, the helicities of the final state leptons are λ1,2, and ηt=1, η±1,0=-1.
The nonvanishing leptonic helicity amplitudes LL(R),λλ1λ2 are LL,+1-12+12=-LR,-1+12-12=q22(1+βℓ)(1+cosθℓ),LR,+1+12-12=-LL,-1-12+12=-q22(1+βℓ)(1-cosθℓ),LL,0-12+12=LR,0+12-12=q2(1+βℓ)sinθℓ,with βℓ=1-4mℓ2q2.
The B1→B2 hadronic matrix elements are calculated in the frameworks of soft-collinear effective theory [16] and lattice QCD [14]. The helicity-based definition of the form factors are presented as [14]⟨B2(k,sk)|q¯2γμq1|B1⟩=u¯(k,sk)[f0(q2)(mB1-mB2)qμq2+f+(q2)mB1+mB2s+{pμ+kμ-qμq2(mB1-mB2)}+f⊥(q2){γμ-2mB2s+pμ-2mB1s+kμ}]u(p,sp),⟨B2(k,sk)|q¯2γμγ5q1|B1(p,sp)⟩=-u¯(k,sk)γ5[g0(q2)(mB1+mB2)qμq2+g+(q2)mB1-mB2s-{pμ+kμ-qμq2(mB1-mB2)}+g⊥(q2){γμ+2mB2s-pμ-2mB1s-kμ}]u(p,sp),⟨B2(p′,s′)|q¯2iσμνqνq1|B1(p,s)⟩=-u¯B2(p′,s′){h+(q2)q2s+(pμ+pμ′-qμq2(mB12-mB22))+(mB1+mB2)h⊥(q2)(γμ-2mB2s+pμ-2mB1s+pμ′)}uB1(p,s),⟨B2(p′,s′)|q¯2iσμνγ5qνq1|B1⟩=-u¯B2(p′,s′)γ5{h˜+(q2)q2s-(pμ+pμ′-qμq2(mB12-mB22))+(mB1-mB2)h˜⊥(q2)(γμ+2mB2s-pμ-2mB1s-pμ′)}uB1(p,s),where s±=(mB1±mB2)2-q2 and f0,⊥V,A,T,T5 are the form factors. And then we obtain the nonvanishing hadronic helicity amplitudes HVA,λL(R),sp,sk, HVA,0L(R),+12+12=f+(q2)(mB1+mB2)s-q2CVAL(R)-g+(q2)(mB1-mB2)s+q2CVAL(R)+2mbq2(h+(q2)q2s--h˜+(q2)q2s+)C7eff,HVA,0L(R),-12-12=f+(q2)(mB1+mB2)s-q2CVAL(R)+g+(q2)(mB1-mB2)s+q2CVAL(R)+2mbq2(h+(q2)q2s-+h˜+(q2)q2s+)C7eff,HVA,+L(R),-12+12=-f⊥(q2)2s-CVAL(R)+g⊥(q2)2s+CVAL(R)-2mbq2(h⊥(q2)(mB1+mB2)2s--h˜⊥(q2)(mB1-mB2)2s+)C7eff,HVA,-L(R),+12-12=-f⊥(q2)2s-CVAL(R)-g⊥(q2)2s+CVAL(R)-2mbq2(h⊥(q2)(mB1+mB2)2s-+h˜⊥(q2)(mB1-mB2)2s+)C7eff,with CVAL(R)≡(C9eff∓C10).
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 ε¯*(λ) and leptonic helicity amplitudes LL(R),λλ1,λ2 are invariant under SU(3) flavor symmetry. The hadronic helicity amplitude relations of B1→B2ℓ+ℓ- are similar to ones of B1→B2γ as given in Ref. [38] and will be given in next section for convenience.
Observables for <inline-formula><mml:math display="inline"><mml:msub><mml:mi mathvariant="script">B</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:msub><mml:mi mathvariant="script">B</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msup><mml:mo>ℓ</mml:mo><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mo>ℓ</mml:mo><mml:mo>-</mml:mo></mml:msup></mml:math></inline-formula>
In the rest frame of the baryon B1, the double differential decay branching ratio is [39]dB(B1→B2ℓ+ℓ-)dq2dcosθℓ=τB12mΛb32λ(mB12,mB22,q2)(8π)312sp+1∑λ1,λ2∑sp,sk|MVAλ1,λ2(sp,sk)|2=N2(q2)[(1-cos2θℓ)(|HVA,0L,+12+12|2+|HVA,0L,-12-12|2+|HVA,0R,+12+12|2+|HVA,0R,-12-12|2)+12(1-cosθℓ)2(|HVA,-L,+12-12|2+|HVA,+R,-12+12|2)+12(1+cosθℓ)2(|HVA,-R,+12-12|2+|HVA,+L,-12+12|2)],with N(q2)=GFλq1q2αe(1+1-4mℓ2q2)τB1q2λ(mB1,mB2,q2)215mB13π5,where λ(a,b,c)=a2+b2+c2-2(ab+bc+ca). And the differential decay branching ratio is dB(B1→B2ℓ+ℓ-)dq2=43N2(q2)HM(q2),with HM(q2)=(|HVA,-L,+12-12|2+|HVA,+R,-12+12|2)+(|HVA,-R,+12-12|2+|HVA,+L,-12+12|2)+(|HVA,0L,+12+12|2+|HVA,0L,-12-12|2+|HVA,0R,+12+12|2+|HVA,0R,-12-12|2).
The longitudinal polarization fraction can be obtained by Eq. (13), FL(q2)=∫-1+1dcosθℓ(2-5cos2θℓ)d2Bdq2dcosθℓ∫-1+1dcosθℓd2Bdq2dcosθℓ,and the concrete expression is FL(q2)=(|HVA,0L,+12+12|2+|HVA,0L,-12-12|2+|HVA,0R,+12+12|2+|HVA,0R,-12-12|2)[HM(q2)]-1.
The leptonic forward-backward asymmetry, AFBℓ(q2)=∫0+1dcosθℓd2Bdq2dcosθℓ-∫-10dcosθℓd2Bdq2dcosθℓ∫-10dcosθℓd2Bdq2dcosθℓ+∫0+1dcosθℓd2Bdq2dcosθℓ,and the concrete expression is AFBℓ(q2)=34[(|HVA,-R,+12-12|2+|HVA,+L,-12+12|2)-(|HVA,-L,+12-12|2+|HVA,+R,-12+12|2)][HM(q2)]-1.
The lepton flavor universality in baryon weak decays Tb3→T8ℓ+ℓ- is defined in a manner identical RK(*) as RTb3→T8≡∫qminqmaxdB(Tb3→T8μ+μ-)/ds∫qminqmaxdB(Tb3→T8e+e-)/ds.
For q2 integration of X(q2)=FL(q2) and AFBℓ(q2), following Ref. [40], two ways of integration are considered. The normalized q2-integrated observables ⟨X⟩ are calculated by separately integrating the numerators and denominators with the same q2 bins. The “naively integrated" observables are obtained by X¯=1qmax2-qmin2∫qmin2qmax2dq2X(q2).
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 B1→B2γ[41–43], may also contribute to the semileptonic baryon decays B1→B2ℓ+ℓ-. In some decays, for example, Σ+→pℓ+ℓ- decays, the W-exchange contributions with the two-quark transition will play a major role [34]. So we will consider these W-exchange contributions in the later analysis of SU(3) flavor symmetry.
RESULTS AND ANALYSIS
The theoretical input parameters and the experimental data within the 1σ error from the Particle Data Group [44] will be used in our numerical results. To obtain SU(3) irreducible representation approach (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 [45]. 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 Tb3 semileptonic decays, there are enough phase spaces to allow for e+e-, μ+μ-, and τ+τ- decays. Tc3 and T8 semileptonic decays only have enough phase spaces to allow for both e+e- and μ+μ- decays. The results of Tb3→T8e+e-, T8μ+μ-, T8τ+τ- decays, Tc3→T8e+e-, T8μ+μ- decays, and T8→T8′e+e-, T8′μ+μ- decays are given in the following subsections A, B, and C, respectively.
For Tb3→T8ℓ+ℓ- decays, the single-quark transition contributions are strongly dominant so that other contributions like the W-exchange contributions are usually omitted. The SU(3) flavor structure Hamiltonian with b→s,d transitions can been found, for instance, in Refs. [46–48], and the SU(3) IRA hadronic helicity amplitudes for Tb3→T8ℓ+ℓ- via b→s/dℓ+ℓ- can be parametrized as H(Tb3→T8ℓ+ℓ-)VA,λL(R),sp,sk=e1(Tb3)[ij]T(3¯)k(T8)[ij]k+e2(Tb3)[ij]T(3)k(T8)[ik]j,which are similar to the decay amplitudes of corresponding Tb3→T8γ modes in Ref. [49]. In Eq. (23), T(3¯)=(0,1,1) denoted the transition operators (q¯2b) with q2=s, d, and the model as well as scale independent parameters ei≡(ei)VA,λL(R),sp,sk(q2). The parameters ei contain information about QCD dynamics and could include the long distance (LD) contributions from hadron resonances. The SU(3) IRA amplitudes of the Tb3→T8ℓ+ℓ- weak decays are given in Table I, and for a better understanding, the information of relevant CKM matrix elements are also listed in Table I. From Table I, one can see that Λb0→Σ0ℓ+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≡e1+e2.
I10.1103/PhysRevD.103.013007.t1
The SU(3) IRA amplitudes of the Tb3→T8ℓ+ℓ- weak decays by the b→s/dℓ+ℓ- transitions, and E≡e1+e2.
Decay modesA(Tb3→T8ℓ+ℓ-)Tb3→T8ℓ+ℓ- via the b→sℓ+ℓ- transition:Λb0→Λ0ℓ+ℓ--2λbsE/6Λb0→Σ0ℓ+ℓ-0Ξb0→Ξ0ℓ+ℓ--λbsEΞb-→Ξ-ℓ+ℓ-λbsETb3→T8ℓ+ℓ- via the b→dℓ+ℓ- transition:Λb0→nℓ+ℓ-λbdEΞb0→Λ0ℓ+ℓ--λbdE/6Ξb0→Σ0ℓ+ℓ--λbdE/2Ξb-→Σ-ℓ+ℓ-λbdE
Among Λb0→Λ0ℓ+ℓ-, Ξb0→Ξ0ℓ+ℓ-Ξb-→Ξ-ℓ+ℓ-, Λb0→nℓ+ℓ-, Ξb0→Λ0ℓ+ℓ-, Ξb0→Σ0ℓ+ℓ-, and Ξb-→Σ-ℓ+ℓ- decays, only Λb0→Λ0μ+μ- decay has been measured, and its branching ratios in the whole q2 region and in different q2 bins are listed in Tables II and 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 Tb3→T8ℓ+ℓ- decays.
II10.1103/PhysRevD.103.013007.t2
Branching ratios for Tb3→T8ℓ+ℓ- decays with 1σ error in the whole q2 region within S1 and S2 cases.
Decay modesExperimental data [44]Our results in S1Our results in S2Other predictionsB(Λb0→Λ0μ+μ-)(×10-6)1.08±0.281.08±0.281.08±0.281.05 [50]B(Ξb0→Ξ0μ+μ-)(×10-6)…1.55-0.43+0.451.77-0.53+0.49B(Ξb-→Ξ-μ+μ-)(×10-6)…1.65-0.46+0.491.87-0.54+0.56B(Λb0→Λ0τ+τ-)(×10-7)…2.30±0.602.74-0.71+0.852.60 [50]B(Ξb0→Ξ0τ+τ-)(×10-7)…3.23-0.89+0.944.42-1.21+1.36B(Ξb-→Ξ-τ+τ-)(×10-7)…3.42-0.95+1.014.76-1.36+1.44B(Λb0→nμ+μ-)(×10-8)…8.15-2.30+2.447.77-2.28+2.42(4.1−1.2+5.4)3.75[50,51]B(Ξb0→Λ0μ+μ-)(×10-8)…1.34-0.39+0.431.45-0.45+0.44B(Ξb0→Σ0μ+μ-)(×10-8)…3.77-1.10+1.224.13-1.24+1.36B(Ξb-→Σ-μ+μ-)(×10-8)…8.00-2.40+2.568.61-2.52+3.06B(Λb0→nτ+τ-)(×10-8)…2.07-0.58+0.622.46-0.70+0.78(2.9−0.8+3.7)1.21[50,51]B(Ξb0→Λ0τ+τ-)(×10-9)…3.42-1.00+1.114.63-1.35+1.71B(Ξb0→Σ0τ+τ-)(×10-9)…8.97-2.62+2.9112.23-3.62+4.12B(Ξb-→Σ-τ+τ-)(×10-8)…1.91-0.57+0.612.60-0.77+0.87III10.1103/PhysRevD.103.013007.t3
Branching ratios for Tb3→T8μ+μ- weak decays in different q2 bins with 1σ error in S1 and S2 cases (in the unit of 10-7).
S1: The SU(3) flavor symmetry parameters without the baryonic momentum-transfer q2 dependence. We treat the SU(3) flavor parameters (E)VA,λL(R),sp,sk(q2) as constants without q2 dependence, which will lead HM(q2) in Eq. (15) to a constant, too. We use the 1σ error experimental data of B(Λb0→Λ0μ+μ-) to constrain HM(q2) (i.e., |E|2) and then predict other B(Tb3→T8ℓ+ℓ-) by the amplitude relations in Table I.
S2: The SU(3) flavor symmetry parameters with the baryonic momentum-transfer q2 dependence. In order to obtain more precise predictions, we use the hadronic helicity amplitude expressions in Eq. (12), which are q2 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. [52]. As for the q2 dependent form factors involving the Tb3→T8 transitions, we use the recent lattice QCD results of Λb0→Λ0[14], in which the form factors are parametrized by f(q2)=f(0)1-q2/(mpolef)2[1+a1fa0fz(q2)+a2fa0f[z(q2)]2],where f=f+,f⊥,f0,g+,g⊥,g0,h+,h⊥,h˜+,h˜⊥, and the details of z(q2) and mpolef can be found in Ref. [14]. We keep f+(0) as an undetermined constant without q2 dependence, and other f(0) can be expressed as a0fa0f+f+(0). The central values of aif in Table 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 Table I will be used for f+(0). We use the 1σ error experimental data of B(Λb0→Λ0μ+μ-) to constrain f+(0) and then predict other B(Tb3→T8ℓ+ℓ-).
Using the experimental data of B(Λb0→Λ0μ+μ-) in the whole q2 region, one can obtain the branching ratios for Tb3→T8μ+μ- and Tb3→T8τ+τ- weak decays in the whole q2 region, which are listed in the third and forth columns of Table II for S1 case and S2 case, respectively. Noted that, the amplitude relations listed in Table 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 Table II for comparing. Since the results of Tb3→T8e+e- decays are quite similar to ones of Tb3→T8μ+μ- decays, we only show Tb3→T8μ+μ- in this work. We have the following remarks for the results in Table II.
Comparing the branching ratios in S1 and S2 cases, one can see that the predictions are slightly different between S1 and S2 cases, which are mainly due to the q2 dependence of the hadronic helicity amplitudes.
Comparing our predictions for B(Λb0→Λ0τ+τ-), B(Λb0→nτ+τ-) and B(Λb0→nμ+μ-) with previous ones in the relativistic quark model [50] and the Bethe-Salpeter equation approach [51], one can see that our predicted B(Λb0→Λ0τ+τ-) and B(Λb0→nμ+μ-) are quite consistent with previous ones,; nevertheless, our central value of B(Λb0→nτ+τ-) is about 2 times larger than theirs.
Many branching ratio predictions for Tb3→T8ℓ+ℓ- are obtained for the first time. The not-yet-measured B(Tb3→T8ℓ+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(Λb0→Λ0μ+μ-) in different q2 bins, one can get the branching ratios of Tb3→T8μ+μ- and Tb3→T8τ+τ- weak decays in different q2 bins within S1 and S2 cases, which are collected for reference in Table III and Table IV, respectively.
IV10.1103/PhysRevD.103.013007.t4
Branching ratios for Tb3→T8τ+τ- weak decays in different q2 bins with 1σ error in S1 and S2 cases (in the unit of 10-7).
[qmin2,qmax2](GeV2)[14.18, 16.0] in S1[14.18, 16.0] in S2[18.0, 20.0] in S1[18.0, 20.0] in S2[15.0, 20.0] in S1[15.0, 20.0] in S2B(Λb0→Λ0τ+τ-)0.83±0.250.84-0.25+0.271.51±0.361.52±0.373.41±0.763.44-0.80+0.86B(Ξb0→Ξ0τ+τ-)1.21-0.38+0.391.40-0.45+0.462.00-0.50+0.532.32-0.59+0.664.79-1.14+1.205.65-1.44+1.43B(Ξb-→Ξ-τ+τ-)1.29-0.40+0.421.49-0.48+0.502.11-0.54+0.572.50±0.675.07-1.22+1.305.92-1.48+1.58B(Λb0→nτ+τ-)0.060-0.019+0.0200.053±0.0180.147-0.039+0.0410.133-0.034+0.0380.282-0.070+0.0740.257-0.064+0.068B(Ξb0→Λ0τ+τ-)0.010-0.003+0.0040.0095-0.0030+0.00390.024±0.0070.026±0.0070.047-0.012+0.0130.048-0.013+0.014B(Ξb0→Σ0τ+τ-)0.029-0.009+0.0110.030-0.010+0.0110.064-0.017+0.0190.071-0.019+0.0210.129-0.033+0.0360.140-0.037+0.041B(Ξb-→Σ-τ+τ-)0.061-0.020+0.0220.063-0.021+0.0230.137-0.038+0.0410.152-0.041+0.0470.273-0.070+0.0800.301-0.078+0.086
The longitudinal polarization fractions and the leptonic forward-backward asymmetries with two ways of integration for Tb3→T8ℓ+ℓ- decays could also be obtained in the S2 case. As shown in Eq. (18) and Eq. (20), the N(q2) 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 Tb3→T8ℓ+ℓ- decays. So the longitudinal polarization fractions and the leptonic forward-backward asymmetries of all Tb3→T8μ+μ- (Tb3→T8τ+τ-) decays are very similar to each other in certain q2 bins. We take ones of Λb0→Λ0μ+μ-, Λ0τ+τ- as examples, which are given in Table V. Excepting in [0.1, 2.0], [0.1, 4.3], [0.1, 16.0], and the whole q2 regions, fL¯ and ⟨fL⟩ (AFBℓ¯ and ⟨AFBℓ⟩) with different q2 integration ways are quite similar in other certain q2 bins. So the obvious differentiation between fL¯ and ⟨fL⟩ (AFBℓ¯ and ⟨AFBℓ⟩) mainly appears in the quite low q2 regions. Note that the normalized longitudinal polarization fraction and normalized leptonic forward-backward asymmetry of Λb0→Λ0μ+μ- in q2∈[15,20]GeV2 have been measured by the LHCb experiment [44], ⟨fL⟩(Λb0→Λ0μ+μ-)[15,20]=0.61-0.14+0.11,⟨AFBℓ⟩(Λb0→Λ0μ+μ-)[15,20]=-0.39±0.04±0.01.We do not impose the above experimental bounds but leave them as predictions. Comparing with the experimental results for ⟨fL⟩(Λb0→Λ0μ+μ-)[15,20] and ⟨AFBℓ⟩(Λb0→Λ0μ+μ-)[15,20], our prediction of ⟨fL⟩(Λb0→Λ0μ+μ-)[15,20] and ⟨AFBℓ⟩(Λb0→Λ0μ+μ-)[15,20] are agreeable with their experimental data within 1.5σ and 1σ error ranges, respectively.
V10.1103/PhysRevD.103.013007.t5
Longitudinal polarization fractions and forward-backward asymmetries for Λb0→Λ0μ+μ-, Λ0τ+τ- decays in different q2 bins with 1σ error in the S2 case.
In addition, the lepton flavor universality RTb3→T8 in three q2 bins within the S2 case are given in Table VI. One can see that all predictions in three q2 bins are virtually indistinguishable from unity; i.e., the lepton mass effects on all RB1→B2 are small in both the low-q2 region and high-q2 region in the SM.
VI10.1103/PhysRevD.103.013007.t6
Lepton flavor universality of Tb3→T8ℓ+ℓ- baryon weak decays in different q2 bins with 1σ error in the S2 case.
Similar to Tc3,8→T8′γ radiative decays [41–43,53], Tc3→T8ℓ+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 Tc3→T8ℓ+ℓ- may be parametrized as H(Tc3→T8ℓ+ℓ-)VA,λL(R),sp,sk=f1(Tc3)[ij]T′(3¯)k(T8)[ij]k+f2(Tc3)[ij]T′(3)k(T8)[ik]j+(f˜1H(6¯)jlk+f˜4H(15)jlk)(Tc3)[ij](T8)[il]k+(f˜2H(6¯)jlk+f˜5H(15)jlk)(Tc3)[ij](T8)[ik]l+(f˜3H(6¯)jlk+f˜6H(15)jlk)(Tc3)[ij](T8)[lk]i,where the SU(3) flavor symmetry parameters fi≡(fi)VA,λL(R),sp,sk and f˜i≡(f˜i)VA,λL(R),sp,sk. The fi terms in Eq. (26) and the later gi 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¯2c) with q2=u, and H(6¯)jlk (H(15)jlk) related to the (q¯lqj)(q¯kc) operator is antisymmetric (symmetric) in upper indices. The nonvanishing H(6¯)jlk and H(15)jlk for c→sud¯,dus¯,ud¯d,us¯s transitions can be found in Ref. [54]. Using l, k antisymmetric in H(6¯)jlk and l, k symmetric in H(15)jlk, we have f˜2=-f˜1,f˜5=f˜4,f˜6=0,which will be used in the following discussion.
For the W-exchange transitions, there are three kinds of charm quark decaying into light quarks, d+c→u+s+ℓ+ℓ-,d+c→u+d+ℓ+ℓ-(s+c→u+s+ℓ+ℓ-),s+c→u+d+ℓ+ℓ-,which are related to H(6¯,15)213, H(6¯,15)212[H(6¯,15)313], and H(6¯,15)312, and they are proportional to Vcs*Vud≈1, Vcd*Vud(Vcs*Vus)≈-sc(sc), and Vcd*Vus≈-sc2 with sc≡sinθC≈0.22453, 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 Tc3→T8ℓ+ℓ- weak decays are given in the third column of Table VII, and for a better understanding, the information of relevant CKM matrix elements is also listed in this table. From Table VII, one can see that singly Cabibbo suppressed Λc+→pℓ+ℓ-, Ξc+→Σ+ℓ+ℓ-, Ξc0→Λ0ℓ+ℓ-, Ξc0→Σ0ℓ+ℓ- decays receive both the single-quark transition and the W-exchange contributions; nevertheless, Cabibbo allowed Λc+→Σ+ℓ+ℓ-, Ξc0→Ξ0ℓ+ℓ- decays and doubly Cabibbo suppressed Ξc+→pℓ+ℓ-, Ξc0→nℓ+ℓ- decays only receive the W-exchange contributions.
VII10.1103/PhysRevD.103.013007.t7
The SU(3) IRA amplitudes of the Tc3→T8ℓ+ℓ- weak decays in the S1 case, F1≡f1+f2, F˜1≡f˜1-f˜3+f˜4, F˜2≡f˜1-f˜3-f˜4, and F˜≡f˜1-f˜3.
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 the Wilson coefficient suppressed; for example, see Refs. [55,56]. If ignoring the Wilson coefficient suppressed H(15) term contributions, there are only two parameters, F1 and F˜≡f˜1-f˜3, in the decay amplitudes of Tc3→T8ℓ+ℓ-. The simplified results are listed in the last column of Table VII. One can see that the Cabibbo allowed and doubly Cabibbo suppressed eight decay modes of Tc3→T8ℓ+ℓ- are related by only one parameter F˜; nevertheless, the singly Cabibbo suppressed eight decays are related by two parameters F˜ and F1. Moreover, there are amplitude relations A(Ξc+→Σ+ℓ+ℓ-)=2A(Ξc0→Σ0ℓ+ℓ-).
In these Tc3→T8ℓ+ℓ- decays, only B(Λc+→pe+e-) and B(Λc+→pμ+μ-) are upper limited by experiment. We list their experimental upper limits (exp. UL) in the second column of Table VIII. Due to the lack of the experiment in Tc3→T8ℓ+ℓ- decays and the complex amplitude expressions included the W-exchange contributions, we will only analyze the Tc3→T8ℓ+ℓ- decays in the S1 case. We assume that W-exchange contributions noted by F˜ or the single-quark transition contributions noted by F1 play a dominant role in these decays. They are separately discussed as follows.
Only considering the W-exchange contributions by setting F1=0, all the Tc3→T8ℓ+ℓ- decays are related by one parameter F˜ as shown in Table VII. The upper limit predictions of Tc3→T8e+e- and Tc3→T8μ+μ- decays in the S1 case are listed in the third column of Table VIII. One can see that B(Λc+→pμ+μ-) gives an effective constraint on these upper limit predictions of the branching ratios.
Only considering the single-quark transition contributions by setting F˜=0, as shown in Table VII, all eight singly Cabibbo suppressed Tc3→T8e+e- and Tc3→T8μ+μ- decays are related by the parameter F1. Then the six upper limit predictions of B(Λc+→pe+e-), B(Ξc+→Σ+e+e-), B(Ξc0→Σ0e+e-), B(Λc+→pμ+μ-), B(Ξc+→Σ+μ+μ-), and B(Ξc0→Σ0μ+μ-) in the S1 case have the same predictions as the ones listed in the third column of Table VIII. Nevertheless, the predictions of other two singly Cabibbo suppressed B(Ξc0→Λ0e+e-) and B(Ξc0→Λ0μ+μ-) are different from the ones listed in Table VIII. We obtain that B(Ξc0→Λ0e+e-)≤9.66×10-9 and B(Ξc0→Λ0μ+μ-)≤9.36×10-9. The previous other predictions for B(Λc+→pe+e-) and B(Λc+→pμ+μ-), which considered the single-quark contributions with/without LD contributions, are also listed in the last two columns of Table VIII for comparing. The predicted upper limits of B(Λc+→pℓ+ℓ-) are about 5 orders larger than the predictions with the SD contributions, smaller than the LD contributing ones, which might mean that the W-exchange transitions cancel out with the LD contributions.
VIII10.1103/PhysRevD.103.013007.t8
Branching ratios of Tc3→T8ℓ+ℓ- decays within 1σ theoretical error in the S1 case.
Decay modesExp. UL [44]Our predictions without F1Others without LDOthers with LDB(Λc+→Σ+e+e-)(×10-6)…≤2.63B(Ξc0→Ξ0e+e-)(×10-6)…≤2.35B(Λc+→pe+e-)(×10-8)≤550≤7.95(3.8±0.5)×10−4(4.05±2.37)×10−6[20,21]37±8420±73[20,21]B(Ξc+→Σ+e+e-)(×10-7)…≤1.29B(Ξc0→Λ0e+e-)(×10-8)…≤8.69B(Ξc0→Σ0e+e-)(×10-8)…≤2.22B(Ξc+→pe+e-)(×10-8)…≤5.55B(Ξc0→ne+e-)(×10-8)…≤1.92B(Λc+→Σ+μ+μ-)(×10-6)…≤2.50B(Ξc0→Ξ0μ+μ-)(×10-6)…≤2.25B(Λc+→pμ+μ-)(×10-8)≤7.7≤7.7(2.8±0.4)×10−4(3.77±2.28)×10−6[20,21]37±8230±66[20,21]B(Ξc+→Σ+μ+μ-)(×10-7)…≤1.25B(Ξc0→Λ0μ+μ-)(×10-8)…≤8.42B(Ξc0→Σ0μ+μ-)(×10-8)…≤2.15B(Ξc+→pμ+μ-)(×10-8)…≤5.41B(Ξc0→nμ+μ-)(×10-8)…≤1.87<inline-formula><mml:math display="inline"><mml:msub><mml:mi>T</mml:mi><mml:mn>8</mml:mn></mml:msub></mml:math></inline-formula> semileptonic weak decays
Similar to T8→T8′γ radiative decays, T8→T8′ℓ+ℓ- semileptonic decays receive the single-quark transition contributions and the W-exchange contributions [34]. The SD contributions come from the Z0 and electromagnetic penguin diagrams as well as the Z0 box diagrams, and the LD contributions arise from an intervirtual photon in the T8→T8′γ* processes. The LD contributions are much larger than the SD ones in the single-quark transition contributions in the T8→T8′ℓ+ℓ- decays. So the LD contributions and the W-exchange contributions might play the major roles in T8→T8′ℓ+ℓ- decays [30,34]. The SU(3) flavor structure of the s→d Hamiltonian can be found in Ref. [49]. The SU(3) IRA hadronic helicity amplitudes for T8→T8′ℓ+ℓ- decays via s→dℓ+ℓ- are H(T8→T8′ℓ+ℓ-)VA,λL(R),sp,sk=g1(T8)[ij]nT′′(3¯)k(T8′)[ij]k+g2(T8)[ij]nT′′(3)k(T8)[ik]j+g3(T8)[in]jT′′(3¯)k(T8′)[ij]k+g4(T8)[in]jT′′(3¯)k(T8′)[ik]j+g5(T8)[in]jT′′(3¯)k(T8′)[jk]i+g˜1(T8)[ij]n(T8′)[il]kH(4)jlk+g˜2(T8)[in]j(T8′)[il]kH(4)jlk+g˜3(T8)[jn]i(T8′)[il]kH(4)jlk,where the model and scale independent parameters gi≡(gi)VA,λL(R),sp,sk and g˜i≡(g˜i)VA,λL(R),sp,sk, T′′(3¯)=(0,1,0) related to the transition operator (d¯s), and H(4)jlk related to (q¯lqj)(q¯kqn) operator with n≡3 for s quark is symmetric in upper indices [49]. The SU(3) IRA hadronic helicity amplitudes of T8→T8′ℓ+ℓ- weak decays are given in Table IX, in which the information of the same CKM matrix elements VusVud* is not shown.
IX10.1103/PhysRevD.103.013007.t9
The SU(3) IRA amplitudes of the T8→T8′ℓ+ℓ- weak decays, G1≡g1+g2+g3-g5, G2≡g4+g5, G˜A≡g˜1-g˜3, and G˜B≡g˜2+g˜3.
There are four complex parameters G1, G2, G˜A, and G˜B in T8→T8′ℓ+ℓ- weak decays. Since the initial baryon Ξ- does not contain u quark and the W-exchange contributions are canceled in Ξ0→Λ0ℓ+ℓ- decays, the amplitudes of Ξ-→Σ-ℓ+ℓ- and Ξ0→Λ0ℓ+ℓ- listed in Table IX only contain coefficients G1,2, which means that the W-exchange transitions do not contribute to the Ξ-→Σ-ℓ+ℓ- and Ξ0→Λ0ℓ+ℓ- decays. Therefore, Ξ-→Σ-ℓ+ℓ- and Ξ0→Λ0ℓ+ℓ- decays could be used to explore the LD contributions. Other decay amplitudes contained both G1,2 and G˜A,B could proceed from the LD contributions and the W-exchange contributions.
Only two branching ratios of Ξ0→Λ0e+e- and Σ+→pμ+μ- decays have been measured at present, which are listed in the second column of Table X. We may constrain |G1+2G2| and |G2+G˜B| from the experimental data of B(Ξ0→Λ0e+e-) and B(Σ+→pμ+μ-), respectively, and we obtain that |G1+2G2||G2+G˜B|≈12. B(Σ+→pe+e-) is obtained by using the constrained |G2+G˜B|, B(Σ+→pe+e-)=(1.60-0.87+1.14)×10-7,which is 1 order smaller than its experimental upper limits B(Σ+→pe+e-)≤7×10-6 at the 90% confidence level [44] and is also smaller than its SM predictions with the single-quark transition LD contributions, 9.1×10-6≤B(Σ+→pe+e-)≤10.1×10-6 in Ref. [30].
X10.1103/PhysRevD.103.013007.t10
Branching ratios of T8→T8′ℓ+ℓ- decays within 1σ theoretical error in the S1 case.
Decay modesExperimental data [44]Our predictions without G˜A,B in S1B(Ξ-→Σ-e+e-)(×10-6)…2.49-0.29+0.31B(Ξ0→Λ0e+e-)(×10-6)7.6±0.67.6±0.6B(Ξ0→Σ0e+e-)(×10-6)…2.05±0.17B(Λ0→ne+e-)(×10-5)…2.06-0.23+0.25B(Σ0→ne+e-)(×10-17)…7.61-4.44+6.59B(Σ+→pe+e-)(×10-7)<701.60-0.87+1.14B(Σ0→nμ+μ-)(×10-17)…1.22-0.71+1.05B(Σ+→pμ+μ-)(×10-8)2.4-1.3+1.72.40-1.30+1.70
It is difficult to estimate which term gives the main contribution among G1, G2, G˜A, and G˜B now. Nevertheless, the LD contributions noted by G1,2 can not been entirely ignored via the experimental measurement of Ξ0→Λ0ℓ+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 |G1G2+2|≈12; i.e., G1≈10G2 or -14G2. After ignoring the small G2 terms in G1+2G2 and 2G1+G2, one gets all branching ratios of relevant T8→T8′ℓ+ℓ- weak decays in S1 case, which are given in the last column of Table X.
If |G˜B|≫|G2|, |G1+2G2||G2+G˜B|≈|G1||G˜B|≈12, the predictions of B(Ξ-→Σ-e+e-), B(Ξ0→Λ0e+e-), B(Σ0→ne+e-), B(Σ+→pe+e-), B(Σ0→nμ+μ-), and B(Σ+→pμ+μ-) are the same as given in Table X. Nevertheless, B(Ξ0→Σ0ℓ+ℓ-) and B(Λ0→nℓ+ℓ-) can not be predicted in this case.
In the case of G˜B≈G2, the predicted results are similar to above ones with |G˜B|≫|G2| except B(Σ0→nℓ+ℓ-)≈0.
If G˜B≈-G2, i.e., the contributions between G2 and G˜B are largely canceled in G2+G˜B term, the situations are complex, which is beyond the scope of this paper.
All predicted branching ratios except B(Σ0→ne+e-) and B(Σ0→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(Ξ0→Σ0e+e-) in the future could further help us to understand the LD contributions and the W-exchange contributions, respectively.
SUMMARY
Semileptonic baryon decays induced by flavor changing neutral current transitions play very important roles in testing the SM and probing new physics. We have studied the semileptonic decays of baryons with 12 spin via the single-quark transitions b→s/dℓ+ℓ-, c→uℓ+ℓ-, and s→dℓ+ℓ- as well as relevant W-exchange transitions by using the SU(3) flavor symmetry, which is a powerful tool to test the physics and to connect the physical quantities without knowing the underlying dynamics. Our main results can be summarized as follows.
Tb3→T8ℓ+ℓ- decays: Decay Λb0→Σ0ℓ+ℓ- is not allowed by the SU(3) irreducible representation approach, and all other 21 decay amplitudes of the Tb3→T8ℓ+ℓ- decay modes via the b→s/dℓ+ℓ- transitions could be related by only one SU(3) flavor symmetry parameter, which could be constrained by the present experimental data of B(Λb0→Λ0μ+μ-). Using the constrained parameter, we have predicted the not-yet-measured observables in the whole q2 region and in different q2 bins within the S1 and S2 cases. The predicted branching ratios are on the order of O(10-8–10-6), many of them are obtained for the first time, and some of them could be reached by the LHCb or Belle-II experiments. The longitudinal polarization fractions and the leptonic forward-backward asymmetries of all Tb3→T8ℓ+ℓ- decays are very similar to each other in certain q2 bins by the SU(3) flavor symmetry. The predictions of ⟨fL⟩(Λb0→Λ0μ+μ-)[15,20] and ⟨AFBℓ⟩(Λb0→Λ0μ+μ-)[15,20] are agreeable with their experimental data within 1.5σ and 1σ error ranges, respectively.
Tc3→T8ℓ+ℓ- decays: Tc3→T8ℓ+ℓ- decays are quite different from Tb3→T8ℓ+ℓ- decays, since the former may receive both the single-quark c→uℓ+ℓ- transition contributions and the W-exchange contributions. After ignoring the Wilson coefficient suppressed H(15) terms, all decay amplitudes of Tc3→T8ℓ+ℓ- have been related by two SU(3) flavor symmetry parameters. Using the 90% experimental upper limit of B(Λc+→pμ+μ-), we have obtained the upper limit predictions of the not-yet-measured B(Tc3→T8ℓ+ℓ-) by considering only one kind of dominant contributions from either single-quark transition LD contributions or the W-exchange contributions.
T8→T8′ℓ+ℓ- decays: Decays T8→T8′ℓ+ℓ- are more complicated than both Tb3→T8ℓ+ℓ- and Tc3→T8ℓ+ℓ- ones, since the quarks are antisymmetric in both the initial states T8 and the final states T8′. We have predicted B(Σ+→pe+e-) in the S1 case. Moreover, we have analyzed the single-quark transition LD contributions and the W-exchange contributions, and found that B(Ξ-→Σ-e+e-), B(Ξ0→Σ0e+e-), and B(Λ0→ne+e-) are on the order of O(10-6–10-5) in most cases except G˜B≈-G2.
According to our predictions, many results in this work can be tested by the experiments at BESIII, LHCb, and Belle-II. And these results can be used to test SU(3) flavor symmetry approach in Tb3,c3,8→T8′ℓ+ℓ- by the future experiments.
ACKNOWLEDGMENTS
The work was supported by the National Natural Science Foundation of China (Contract No. 11675137).
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