-
The decays of hadrons involving the flavor changing neutral current (FCNC) transition such as
Λb→Λl+l− can provide essential information about the inner structure of hadrons, reveal the nature of the electroweak interaction, and provide model-independent information about physical quantities such as Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. The rare decayΛb→Λμ+μ− was first observed by the CDF collaboration in 2011 [1]. Some experimental progress onΛb→Λl+l− was also achieved [2–5], and the radiative decayΛb→Λγ was observed in 2019 [3] by the LHCb collaboration. The LHCb collaboration determined the forward-backward asymmetries (AlFB ) of the decayΛb→Λμ+μ− to beAlFB(Λb→Λμ+μ−)=−0.05±0.09 (stat)±0.03 (syst),AhFB(Λb→Λμ+μ−)=−0.29±0.09 (stat)±0.03 (syst), andFL(Λb→Λμ+μ−)=0.61+0.11−0.14±0.03 (syst) at the low dimuon invariant mass squared range15<q2<20 GeV2 in 2015 [4]. However, these numbers were updated in 2018 toˉAlFB(Λb→Λμ+μ−)=−0.39±0.04 (stat)±0.01 (syst),AhFB(Λb→Λμ+μ−)=−0.3±0.05 (stat)±0.02 (syst), andˉAlhFB(Λb→Λμ+μ−)=0.25±0.04 (stat)±0.01 (syst) in the same invariant mass squared region [5]. Note thatAlFB is significantly lager than the previous one. In this study, we investigate theAFB ofΛb→Λl+l− in the Bethe-Salpeter equation (BSE) approach. Theoretically, only a few studies have been conducted onAFB(Λb→Λl+l−) [6–17]. References [6] ([7]) provided the integrated forward-backward asymmetriesˉAlFB(Λb→Λμ+μ−)=−0.13 (−0.12 ) andˉAFB(Λb→Λτ+τ−)=−0.04 (−0.03 ), whereas the results of Ref. [8] wereˉAlFB(Λb→Λe+e−)=1.2×10−8 ,ˉAlFB(Λb→Λμ+μ−)=8×10−4 , andˉAlFB(Λb→Λτ+τ−)=9.6×10−4 . Ref. [10] analyzed the differentialˉAFB(Λb→Λl+l−) in the heavy quark limit. Using the nonrelativistic quark model, Ref. [11] investigated the lepton-side forward-backward asymmetriesˉAlFB(Λb→Λl+l−) . In the quark-diquark model, Ref. [12] investigated the lepton-side forward-backward asymmetriesAFB , the hadron-side forward-backward asymmetriesAhFB , and the hadron-lepton forward-backward asymmetriesAhlFB . In an approach of the light-cone sum rules, Refs. [13, 14] investigated the rare decays ofΛb→Λγ andΛb→Λl+l− . Ref. [15] investigated the phenomenological potential of the rare decayΛb→Λl+l− with a subsequent, self-analyzingΛb→Nπ transition. With the form factors (FFs) extracted from a constituent quark model, Ref. [16] investigated the rare weak dileptonic decays of theΛb baryon. Ref. [17] studiedB1→B2l+l− (B1,2 are spin1/2 baryons) with the SU(3) flavor symmetry. The FFs ofΛb→Λ differ in different models. Generally, the number of independent FFs ofΛb→Λ can be reduced to 2 when working in the heavy quark limit [18],⟨Λ(p)|ˉsΓb|Λb(v)⟩=ˉuΛ(F1(q2)+F2(q2)⧸v)ΓuΛb(v),
(1) where
Γ=γμ,γμγ5,qνσνμ , andqνσνμγ5 ,q2 is the square of the transformed momentum. The FF ratioR(q2)=F2(q2)/F1(q2) was considered a constant in many studies assuming the same shape forF1 andF2 , and it was derived from quantum chromodynamics (QCD) sum rules in the framework of the heavy quark effective theory [6]. For example, in Refs. [6, 7] theq2 dependence of FFFi(i=1,2) were given as follows:Fi(q2)=Fi(0)1−aq2+bq4,
(2) where a and b are constants. Using experimental data for the semileptonic decay
Λc→Λe+νe (m2Λ≤q2≤m2Λc ), the CLEO collaboration provided the ratioR=−0.35±0.04 (stat)±0.04 (syst) [19]. In Ref. [20], the authors investigatedΛb→Λγ obtainingR=−0.25±0.14±0.08 . In Refs. [6, 7, 21], the authors investigated the baryonic decayΛb→Λl+l− and obtainedR=−0.25 . In Ref. [22], the relationF2(q2)/F1(q2)≈F2(0)/F1(0) was given. However, according to the pQCD scaling law [23–25], the FFs should not have the same shape. Using Stech's approach, Ref. [26] obtained the FF ratioR(q2)∝−1/q2 . From the data in Ref. [27], we can estimate the value of R and observe that it changes from−0.83 to−0.32 , which is not a constant. In our previous studies [28, 29], we observed that the ratio R is not a constant in theΛb rare decay in a large momentum region in which we did not consider the long distance contributions because they have a small effect on the FFs of this decay [30, 31]. In these studies,Λb (Λ) was considered a bound state of two particles: a quark and a scalar diquark. This model has been used to study many heavy baryons [32]. Using the kernel of the BSE, including scalar confinement and one-gluon-exchange terms and the covariant instantaneous approximation, we obtained the Bethe-Salpeter (BS) wave functions ofΛb and Λ [28, 29]. In this study, we recalculate the FFs ofΛb→Λ in this model.The remainder of this paper is organized as follows. In Sec. II, we derive the general FFs and
AFB forΛb→Λl+l− in the BS equation approach. In Sec. III, the numerical results forAFB andˉAFB ofΛb→Λl+l− are provided. Finally, the summary and discussion are presented in Sec. V. -
As shown in Fig. 1, following our previous research, the BS amplitude of
Λb(Λ) in momentum space satisfies the integral equation [28, 29, 33–39]χP(p)=SF(λ1P+p)∫d4q(2π)4K(P,p,q)χP(q)SD(λ2P−p),
(3) where
K(P,p,q) is the kernel, which is defined as the sum of the two particles irreducible diagrams,SF andSD are the propagators of the quark and scalar diquark, respectively.λ1(2)=mq(D)/(mq+mD) , wheremq(D) is the mass of the quark (diquark), and P is the momentum of the baryon.We assume the kernel has the following form:
−iK(P,p,q)=I⊗IV1(p,q)+γμ⊗(p2+q2)μV2(p,q),
(4) where
V1 results from the scalar confinement, andV2 is from the one-gluon-exchange diagram. According to the potential model,V1 andV2 have the following forms in the covariant instantaneous approximation (pl=ql ) [28, 29, 37–39]:˜V1(pt−qt)=8πκ[(pt−qt)2+μ2]2−(2π)2δ3(pt−qt)×∫d3k(2π)38πκ(k2+μ2)2,
(5) ~V2(pt−qt)=−16π3α2seffQ20[(pt−qt)2+μ2][(pt−qt)2+Q20],
(6) where μ is a small parameter; to avoid the divergence in numerical calculation, this parameter is considered to be sufficiently small such that the results are not sensitive to it. The parameters κ and
αseff are related to scalar confinement and the one-gluon-exchange diagram, respectively.qt is the transverse projection of the relative momentum along the momentum P, which is defined aspl=λ1P−v⋅p,pμt=pμ−(v⋅p)pμ (vμ=Pμ/M), qμt=qμ−(v⋅q)vμ , andql=λ2P−v⋅q . The second term of˜V1 is introduced to avoid infrared divergence at the pointpt=qt , and μ is a small parameter to avoid the divergence in numerical calculations. Analyzing the electromagnetic FFs of the proton,Q20=3.2 GeV2 was observed to provide consistent results with the experimental data [40].The propagators of the quark and diquark can be expressed as follows:
SF(p1)=i⧸v[Λ+qM−pl−ωq+iϵ+Λ−qM−pl+ωq−iϵ],
(7) SD(p2)=i2ωD[1pl−ωD+iϵ−1pl+ωD−iϵ],
(8) where
ωq=√m2−p2tandωD=√m2D−p2t , M is the mass of the baryon, andΛ± are the projection operators, which are defined as2ωqΛ±q=ωq±⧸v(⧸pt+m),
(9) and satisfy the following relations:
Λ±qΛ±q=Λ±q,Λ±qΛ∓q=0.
(10) Generally, we require two scalar functions to describe the BS wave function of
Λb(Λ) [33–35],χP(p)=(f1(p2t)+⧸ptf2(p2t))u(P),
(11) where
fi,(i=1,2) are the Lorentz-scalar functions ofp2t , andu(P) is the spinor of a baryon.Defining
˜f1(2)=∫dpl2πf1(2) , and using the covariant instantaneous approximation, the scalar BS wave functions satisfy the following coupled integral equations:˜f1(pt)=∫d3qt(2π)3M11(pt,qt)˜f1(qt)+M12(pt,qt)˜f2(qt),
(12) ˜f2(pt)=∫d3qt(2π)3M21(pt,qt)˜f1(qt)+M22(pt,qt)˜f2(qt),
(13) where
M11(pt,qt)=(ωq+m)(˜V1+2ωD˜V2)−pt⋅(pt+qt)˜V24ωDωq(−M+ωD+ωq)−(ωq−m)(˜V1−2ωD˜V2)+pt⋅(pt+qt)˜V24ωDωc(M+ωD+ωq),
(14) M12(pt,qt)=−(ωq+m)(qt+pt)⋅qt˜V2+pt⋅qt(˜V1−2ωD˜V2)4ωDωc(−M+ωD+ωc)−(m−ωq)(qt+pt)⋅qt˜V2−pt⋅qt(˜V1+2ωD˜V2)4ωDωq(M+ωD+ωq),
(15) M21(pt,qt)=(˜V1+2ωD˜V2)−(−ωq+m)(1+qt⋅ptp2t)˜V24ωDωq(−M+ωD+ωq)−−(˜V1−2ωD˜V2)+(ωq+m)(1+qt⋅ptp2t)˜V24ωDωq(M+ωD+ωq),
(16) M22(pt,qt)=(m−ωq)(˜V1+2ωD˜V2))pt⋅qt−p2t(q2t+pt⋅qt)˜V24p2tωDωq(−M+ωD+ωq)−(m+ωq)(−˜V1−2ωD˜V2))pt⋅qt+p2t(q2t+pt⋅qt)˜V24p2tωDωq(M+ωD+ωq).
(17) When the mass of the b quark approaches infinity [32], the propagator of the b quark satisfies the relation
⧸vSF(p1)=SF(p1) and can be reduced toSF(p1)=i1+⧸v2(E0+mD−pl+iϵ),
(18) where
E0=M−m−mD is the binding energy. Thus, the BS wave function ofΛb has the formχP(v)=ϕ(p)uΛb(v,s) , whereϕ(p) is the scalar BS wave function [32], and the BS equation forΛb can be replaced byϕ(p)=−i(E0+mD−pl+iϵ)(p2l−ω2D)×∫d4q(2π)4(˜V1+2pl˜V2)ϕ(q).
(19) Generally, we can take
E0 to be about−0.14 GeV and κ to be about0.05 GeV3 [28, 29]. -
In the Standard Model, the
Λb→Λl+l− (l=e,μ,τ ) transitions are described byb→sl+l− at the quark level. The Hamiltonian for the decay ofb→sl+l− is given byH(b→sl+l−)=GFα2√2πVtbV∗ts[Ceff9ˉsγμ(1−γ5)bˉlγμl−iCeff7ˉs2mbσμνqνq2(1+γ5)bˉlγμl
+C10ˉsγμ(1−γ5)bˉlγμγ5l],
(20) where
GF is the Fermi coupling constant, α is the fine structure constant at the Z mass scale,Vts andVtb are the CKM matrix elements, q is the total momentum of the lepton pair, andCi(i=7,9,10) are the Wilson coefficients.Ceff7=−0.313 ,Ceff9=4.334 ,C10=−4.669 [41–43]. The relevant matrix elements can be parameterized in terms of the FFs as follows:⟨Λ(P′)|ˉsγμb|Λb(P)⟩=ˉuΛ(P′)(g1γμ+ig2σμνqν+g3qμ)uΛb(P),⟨Λ(P′)|ˉsγμγ5b|Λb(P)⟩=ˉuΛ(P′)(t1γμ+it2σμνqν+t3qμ)γ5uΛb(P),⟨Λ(P′)|ˉsiσμνqνb|Λb(P)⟩=ˉuΛ(P′)(s1γμ+is2σμνqν+s3qμ)uΛb(P),⟨Λ(P′)|ˉsiσμνγ5qνb|Λb(P)⟩=ˉuΛ(P′)(d1γμ+id2σμνqν+d3qμ)γ5uΛb(P), (21) where
P(P′) is the momentum of theΛb (Λ),q2=(P−P′)2 is the transformed momentum squared, andgi ,ti ,si , anddi (i=1,2 , and 3) are the transition FFs, which are Lorentz scalar functions ofq2 . TheΛb and Λ states can be normalized as follows:⟨Λ(P′)|Λ(P)⟩=2EΛ(2π)3δ3(P−P′),
(22) ⟨Λb(v′,P′)|Λb(v,P)⟩=2v0(2π)3δ3(P−P′).
(23) Comparing Eq. (1) with Eq. (21), we obtain the following relations:
g1=t1=s2=d2=(F1+√rF2),g2=t2=g3=t3=1mΛbF2,s3=F2(√r−1),d3=F2(√r+1),s1=d1=F2mΛb(1+r−2√rω),
(24) where
r=m2Λ/m2Λb andω=(M2Λb+M2Λ−q2)/(2MΛbMΛ)=v⋅P′/mΛ . The transition matrix forΛb→Λ can be expressed in terms of the BS wave functions ofΛb and Λ:⟨Λ(P′)|ˉdΓb|Λb(P)⟩=∫d4p(2π)4ˉχP′(v′)ΓχP(p)S−1D(p2).
(25) When
ω≠1 , we can obtain the following expression by substituting Eqs. (11) and (19) into Eq. (25):F1=k1−ωk2,
(26) F2=k2,
(27) where
k1(ω)=∫d4p(2π)4f1(p′)ϕ(p)S−1D(p2),
(28) k2(ω)=11−ω2∫d4p(2π)4f2(p′)p′t⋅vϕ(p)S−1D.
(29) The decay amplitude of
Λb→Λl+l− can be rewritten as follows:M(Λb→Λl+l−)=GFλt2√2π[ˉlγμl{ˉuΛ[γμ(A1+B1+(A1−B1)γ5)+iσμνpν(A2+B2+(A2−B2)γ5)]uΛb}+ˉlγμγ5l{ˉuΛ[γμ(D1+E1+(D1−E1)γ5)+iσμνpν(D2+E2+(D2−E2)γ5)+pμ(D3+E3+(D3−E3)γ5)]uΛb}],
(30) where
Ai ,Bi ,Dj , andEj (i=1,2 andj=1,2,3 ) are defined as follows:Ai=12{Ceff9(gi−ti)−2Ceff7mbq2(di+si)},Bi=12{Ceff9(gi+ti)−2Ceff7mbq2(di−si)},Dj=12C10(gj−tj),Ej=12C10(gj+tj).
(31) In the physical region (
ω=(m2Λb+m2Λ−q2)/(2mΛbmΛ) ), the decay rate ofΛb→Λl+l− is obtained as follows:dΓ(Λb→Λl+l−)dωdcosθ=G2Fα2214π5mΛb|VtbV∗ts|2vl√λ(1,r,s)M(ω,θ),
(32) where
s=1+r−2√rω ,λ(1,r,s)=1+r2+s2−2r−2s−2rs ,vl=√1−4m2lsm2Λb , and the decay amplitude is expressed as follows [44]:M(ω,θ)=M0(ω)+M1(ω)cosθ+M2(ω)cos2θ,
(33) where θ is the polar angle, as shown in Fig. 2.
M0(ω)=32m2lm4Λbs(1+r−s)(|D3|2+|E3|2)+64m2lm3Λb(1−r−s)Re(D∗1E3+D3E∗1)+64m2Λb√r(6m2l−M2Λbs)Re(D∗1E1)+64m2lm3Λb√r(2mΛbsRe(D∗3E3)+(1−r+s)Re(D∗1D3+E∗1E3))+32m2Λb(2m2l+m2Λbs){(1−r+s)mΛb√rRe(A∗1A2+B∗1B2)−mΛb(1−r−s)Re(A∗1B2+A∗2B1)−2√r(Re(A∗1B1)+m2ΛbsRe(A∗2B2))}+8m2Λb[4m2l(1−r−s)+m2Λb((1+r)2−s2)](|A1|2+|B1|2)+8m4Λb{4m2l[λ+(1+r−s)s]+m2Λbs[(1−r)2−s2]}(|A2|2+|B2|2)−8m2Λb{4m2l(1+r−s)−m2Λb[(1−r)2−s2]}(|D1|2+|E1|2)+8m5Λbsv2{−8mΛbs√rRe(D∗2E2)+4(1−r+s)√rRe(D∗1D2+E∗1E2)−4(1−r−s)Re(D∗1E2+D∗2E1)+mΛb[(1−r)2−s2](|D2|2+|E2|2)}, (34) M1(ω)=−16m4Λbsvl√λ{2Re(A∗1D1)−2Re(B∗1E1)+2mΛbRe(B∗1D2−B∗2D1+A∗2E1−A∗1E2)}+32m5Λbsvl√λ{mΛb(1−r)Re(A∗2D2−B∗2E2)+√rRe(A∗2D1+A∗1D2−B∗2E1−B∗1E2)},
(35) M2(ω)=8m6Λbsv2lλ(|A2|2+|B2|2+|E2|2+|D2|2)−8m4Λbv2lλ(|A1|2+|B1|2+|E1|2+|D1|2).
(36) The lepton-side forward-backward asymmetry,
AFB , is defined asAFB=∫10dΓdq2dzdz−∫0−1dΓdq2dzdz∫1−1dΓdq2dzdz,
(37) where
z=cosθ . The "naively integrated" observables are obtained using [17]⟨X⟩=1q2max−q2min∫q2maxq2minX(q2)dq2.
(38) We define the integrated
AFB asˉAFB=∫ˆqmaxˆqmindˆq2AFB(ˆq2).
(39) where
ˆq2=q2/M2Λb . With the aid of the helicity amplitudes ofΛb→Λl+l− , we can also calculate the hadron forward-backward asymmetry, the lepton-hadron side asymmetry, and the fraction of longitudinally polarized dileptons.The hadron forward-backward asymmetry has the form
AhFB(q2)=αΛ2v2l2(H11P+H22P+H11LP+H22LP)+3m2lq2(H11P+H11LP+H22SP)Htot.
(40) The lepton-hadron side asymmetry has the form
AlhFB(q2)=−34αΛ2vlH12UHtot.
(41) The fraction of the longitudinally polarized dileptons is expressed by
FL(q2)=v2l2(H11L+H22L)+m2lq2(H11U+H11L+H22S)Htot.
(42) In Eqs. (40–42),
Hmm′X(X=U,L,S,P,LP,SP,m=1,2) represent different helicity amplitudes, andHtot is the total helicity amplitude,αΛ=0.642±0.013 . The explicit expression forHmm′X is provided in Ref. [12]. -
In this section, we perform a detailed numerical analysis of
AFB(Λb→Λl+l−) . In this study, we take the masses of baryons asmΛb=5.62 GeV andmΛ=1.116 GeV [45], and the masses of quarks asmb=5.02 GeV andms=0.516 GeV [34, 35, 39]. The variable ω changes from1 to2.617,2.614,1.617 fore,μ,τ , respectively.Solving Eqs. (12) and (19) for Λ and
Λb , we can obtain the numerical solutions of their BS wave functions. In Table 1, we provide the values ofαseff for different values of κ for Λ andΛb withE0=−0.14 GeV.κ/GeV 3 Λ Λb 0.045 0.559 0.775 0.047 0.555 0.777 0.049 0.551 0.778 0.051 0.547 0.780 0.053 0.544 0.782 0.055 0.540 0.784 Table 1. Values of
αseff for Λ andΛb for different κ values.From Table 1, we observe that the value of
αseff is weakly dependent on the value of κ. In Fig. 3, we plot the FFs and FF ratioR(ω) . From this figure, we observe thatR(ω) varies from−0.75 to−0.25 in our model. In Ref. [27],R(ω) varied from−0.42 to−0.83 in the same ω region, which is in agreement with our result and the estimated value from Refs. [28, 29] mentioned in the Introduction. In the range of2.43≤ω≤2.52 (corresponding toM2Λ≤q2≤M2Λc ),R(ω) is about−0.25 . In the same ω region, assuming the FFs have the same dependence onq2 , the CLEO collaboration measuredR=−0.35±0.04±0.04 in the limitmc→+∞ . These results are in good agreement with our research in the same ω region.Figure 3. (color online) Values of
F1 (solid line),F2 (dash line) andR(ω) (dot line) as a function of ω (the lines become thicker with the increase in κ).In Table 2, we provide
ˉAlBF ,ˉAlhFB ,ˉAhFB , andˉFL forΛb→Λμ+μ− and compare our results with those of other studies. We can observe that these asymmetries differ significantly in different models. Considering these differences,ˉAlFB changes between−0.30 and0 ,ˉAlhFB is about0.1 ,ˉAhFB is about−0.25 , andˉFL changes from0.3 to0.6 . Without including the long distance contribution, Ref. [6] provided the integrated forward-backward asymmetryˉAlBF(Λb→Λμ+μ−)=−0.1338 . The result of Ref. [7] wasˉAlBF(Λb→Λμ+μ−)=−0.13(−0.12) in the QCD sum rule approach (pole model). Using the covariant constituent quark model with (without) the long distance contribution, Ref. [8] obtained the resultˉAlBF(Λb→Λμ+μ−)=1.7×10−4(8×10−4) .Table 2. Longitudinal polarization fractions and forward-backward asymmetries for
Λb→Λμ+μ− .For
q2∈[15,20] GeV2 , the LHCb collaboration providedAlFB(Λb→Λμ−μ+)=−0.05±0.09 in 2015, which was updated toAlFB(Λb→Λμ−μ+)=−0.39±0.04 three years later [4, 5]. In our study, in the same region, the value ofAlBF(Λb→Λμ−μ+) changes from−0.44 to−0.35 , which is in good agreement with the most recent experimental data of the LHCb collaboration. With the latest high-precision lattice QCD calculations in the same region, Ref. [46] obtained the valuesAlFB(Λb→Λμ−μ+)=−0.344 in the largeςu and smallςd regions (ςu,ςd are model parameters [47]) andAlFB(Λb→Λμ−μ+)=−0.24 in the largeςd and smallςu regions. In Fig. 4, we plot theq2 -dependence ofAlFB(Λb→Λe−e+) ,AlFB(Λb→Λμ−μ+) , andAlFB(Λb→Λτ−τ+) . From Fig. 4, we can observe thatAlFB(Λb→Λμ+μ−) is in good agreement with the lattice QCD calculation in the entireq2 region [48]. The results of other references results are also shown in Table 3. In Fig. 5, we plot theq2 -dependence ofAhFB(Λb→Λe−e+) ,AhFB(Λb→Λμ−μ+) , andAhFB(Λb→Λτ−τ+) , respectively. Forq2∈[15,20] GeV2 , the LHCb collaboration obtained the value forΛb→Λμ−μ+ as−0.29±0.07 , which is in good agreement our result−0.2304∼−0.0685 . The results of other references results are also shown in Table 3. In Fig. 6, we plot theq2 -dependence ofAlhFB(Λb→Λe−e+) ,AlhFB(Λb→Λμ−μ+) , andAlhFB(Λb→Λτ−τ+) , respectively. Ref. [12] obtained the valueAlhFB(Λb→Λμ−μ+)=0.145 , which is agreement with our results0.1257∼0.1555 in the regionq2∈[15,20] GeV2 . In Fig. 7, we plot theq2 -dependence ofFL(Λb→Λe−e+) ,FL(Λb→Λμ−μ+) , andFL(Λb→Λτ−τ+) , respectively. In the regionq2∈[15,20] GeV2 , the LHCb collaboration obtained the valueFL(Λb→Λμ−μ+)=0.61+0.11−0.14 , which is close to our result of0.3398∼0.4530 . The results of other references results are also shown in Table 3. From these figures, we observe that all these asymmetries are not very sensitive to the parameters κ andE0 in our model.− AlFB[15,20] AlhFB[15,20] AhFB[15,20] FL[15,20] LHCb [4, 5] −0.39±0.04 − −0.29±0.07 0.61+0.11−0.14 [6, 7] −0.40∼−0.25 − − 0.37∼0.62 [8] −0.24∼−0.13 − >−0.308 − [12] −0.40 0.145 −0.29 0.38 [13] −0.075∼−0.017 − − − [17] −0.34+0.01−0.02 − − 0.4+0.01−0.02 [48] −0.350(13) − −0.2710±0.0092 0.409±0.013 our work −0.44∼−0.35 0.1257∼0.1555 −0.2304∼−0.0685 0.3398∼0.4530 Table 3. Longitudinal polarization fractions and forward-backward asymmetries for
Λb→Λμ+μ− inq2∈[15,20] GeV2 .Figure 4. (color online) Values of
AFB(Λb→Λl+l−) as a function ofq2 for different values of κ as shown in Table 1.Figure 5. (color online) Values of
AhFB(Λb→Λl+l−) as a function ofq2 for different values of κ as shown in Table 1.Figure 6. (color online) Values of
AhFB(Λb→Λl+l−) as a function ofq2 for different values of κ as shown in Table 1.Figure 7. (color online) Values of
FL(Λb→Λl+l−) as a function ofq2 for different values of κ as shown in Table 1.Ref. [17] obtained the naively integrated values
⟨AlFB⟩=−0.19+0.00−0.01 and⟨FL⟩=0.6±0.02 forΛb→Λμ+μ− , whereas in our paper, these values are−0.1976 and0.5681 , respectively. Our results are very close to those of Ref. [17]. In our paper, we obtainˉAlFB=−0.0708±0.0001(−0.0590±0.0001) andˉAhFB=−0.1604±0.0001(−0.1541±0.0002) forΛb→Λe+e−(Λb→Λτ+τ−) . The values given in Ref. [8] areˉAlFB=1.2×10−8(9.6×10−4) andˉAhFB=−0.321(−0.259) , and Refs. [13] and [7] provideˉAlFB=−0.0067 andˉAlFB=−0.04 forΛb→Λτ+τ− . Comparing the values in these theoretical approaches, we observe that the asymmetries may vary widely among the theoretical models because the FFs in these models are different. -
In this study, we use the BSE to study the forward-backward asymmetries in the rare decays
Λb→Λl+l− in a covariant quark-diquark model. In this picture,Λb(Λ) is considered a bound state of ab(s) -quark and a scalar diquark.We establish the BSE for the quark and scalar diquark system and then derive the FFs of
Λb→Λ . We solve the BS equation of this system and then provide the values of the FFs and R. We observe that the ratio R is not a constant, which is in agreement with Ref. [26] and the pQCD scaling law [23–25]. Using these FFs, we calculate the forward-backward asymmetriesAlFB ,AlhFB , andAhFB and longitudinal polarization fractionsFL and the integrated forward-backward asymmetriesˉAlFB ,ˉAlhFB , andˉAhFB as well asˉFL forΛb→Λl+l−(l=e,μ,τ) . Comparing with other theoretical studies, we observe that the FFs are different; thus, these asymmetries are different. The long distance contributions are not included in this paper. They will be considered in our future research to compare the experimental data more exactly.
