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Forward-backward asymmetries in ΛbΛl+l in the Bethe-Salpeter equation approach

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Liang-Liang Liu, Su-Jun Cui, Jing Xu and Xin-Heng Guo. Forward-backward asymmetries in ΛbΛl+l in the Bethe-Salpeter equation approach[J]. Chinese Physics C. doi: 10.1088/1674-1137/ac7041
Liang-Liang Liu, Su-Jun Cui, Jing Xu and Xin-Heng Guo. Forward-backward asymmetries in ΛbΛl+l in the Bethe-Salpeter equation approach[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ac7041 shu
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Forward-backward asymmetries in ΛbΛl+l in the Bethe-Salpeter equation approach

  • 1. College of Physics and information engineering, Shanxi Normal University, Taiyuan 030031, China
  • 2. Department of Physics, Yan-Tai University, Yantai 264005, China
  • 3. College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, China

Abstract: Using the Bethe-Salpeter equation (BSE), we investigate the forward-backward asymmetries (AFB) in ΛbΛl+l(l=e,μ,τ) in the quark-diquark model. This approach provides precise form factors that are different from those of quantum chromodynamics (QCD) sum rules. We calculate the rare decay form factors for ΛbΛl+lb and investigate the (integrated) forward-backward asymmetries in these decay channels. We observe the integrated AlFB, ˉAlFB(ΛbΛe+e)0.1371, ˉAlFB(ΛbΛμ+μ)0.1376, and ˉAlFB(ΛbΛτ+τ)0.1053; the hadron side asymmetries ˉAhFB(ΛbΛμ+μ)0.2315; the lepton-hadron side asymmetries ˉAlhFB(ΛbΛμ+μ)0.0827; and the longitudinal polarization fractions ˉFL(ΛbΛμ+μ)0.5681.

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    I.   INTRODUCTION
    • The decays of hadrons involving the flavor changing neutral current (FCNC) transition such as ΛbΛl+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 decay ΛbΛμ+μ was first observed by the CDF collaboration in 2011 [1]. Some experimental progress on ΛbΛl+l was also achieved [25], and the radiative decay ΛbΛγ was observed in 2019 [3] by the LHCb collaboration. The LHCb collaboration determined the forward-backward asymmetries (AlFB) of the decay ΛbΛμ+μ to be AlFB(ΛbΛμ+μ)=0.05±0.09 (stat) ±0.03 (syst), AhFB(ΛbΛμ+μ)=0.29±0.09 (stat) ±0.03 (syst), and FL(ΛbΛμ+μ)=0.61+0.110.14±0.03 (syst) at the low dimuon invariant mass squared range 15<q2<20 GeV2 in 2015 [4]. However, these numbers were updated in 2018 to ˉAlFB(ΛbΛμ+μ)=0.39±0.04 (stat) ±0.01 (syst), AhFB(ΛbΛμ+μ)=0.3±0.05 (stat) ±0.02 (syst), and ˉAlhFB(ΛbΛμ+μ)=0.25±0.04 (stat) ±0.01 (syst) in the same invariant mass squared region [5]. Note that AlFB is significantly lager than the previous one. In this study, we investigate the AFB of ΛbΛl+l in the Bethe-Salpeter equation (BSE) approach. Theoretically, only a few studies have been conducted on AFB(ΛbΛl+l) [617]. References [6] ([7]) provided the integrated forward-backward asymmetries ˉAlFB(ΛbΛμ+μ)=0.13 (0.12) and ˉAFB(ΛbΛτ+τ)=0.04 (0.03), whereas the results of Ref. [8] were ˉAlFB(ΛbΛe+e)=1.2×108, ˉAlFB(ΛbΛμ+μ)=8×104, and ˉAlFB(ΛbΛτ+τ)=9.6×104. Ref. [10] analyzed the differential ˉAFB(ΛbΛl+l) in the heavy quark limit. Using the nonrelativistic quark model, Ref. [11] investigated the lepton-side forward-backward asymmetries ˉAlFB(ΛbΛl+l). In the quark-diquark model, Ref. [12] investigated the lepton-side forward-backward asymmetries AFB, the hadron-side forward-backward asymmetries AhFB, and the hadron-lepton forward-backward asymmetries AhlFB. In an approach of the light-cone sum rules, Refs. [13, 14] investigated the rare decays of ΛbΛγ and ΛbΛl+l. Ref. [15] investigated the phenomenological potential of the rare decay ΛbΛl+l with a subsequent, self-analyzing ΛbNπ transition. With the form factors (FFs) extracted from a constituent quark model, Ref. [16] investigated the rare weak dileptonic decays of the Λb baryon. Ref. [17] studied B1B2l+l (B1,2 are spin 1/2 baryons) with the SU(3) flavor symmetry. The FFs of ΛbΛ differ in different models. Generally, the number of independent FFs of ΛbΛ 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, and qνσνμγ5, q2 is the square of the transformed momentum. The FF ratio R(q2)=F2(q2)/F1(q2) was considered a constant in many studies assuming the same shape for F1 and F2, 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] the q2 dependence of FF Fi(i=1,2) were given as follows:

      Fi(q2)=Fi(0)1aq2+bq4,

      (2)

      where a and b are constants. Using experimental data for the semileptonic decay ΛcΛe+νe (m2Λq2m2Λc), the CLEO collaboration provided the ratio R=0.35±0.04 (stat) ±0.04 (syst) [19]. In Ref. [20], the authors investigated ΛbΛγ obtaining R=0.25±0.14±0.08. In Refs. [6, 7, 21], the authors investigated the baryonic decay ΛbΛl+l and obtained R=0.25. In Ref. [22], the relation F2(q2)/F1(q2)F2(0)/F1(0) was given. However, according to the pQCD scaling law [2325], the FFs should not have the same shape. Using Stech's approach, Ref. [26] obtained the FF ratio R(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 the Λb 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 of Λb and Λ [28, 29]. In this study, we recalculate the FFs of ΛbΛ in this model.

      The remainder of this paper is organized as follows. In Sec. II, we derive the general FFs and AFB for ΛbΛl+l in the BS equation approach. In Sec. III, the numerical results for AFB and ˉAFB of ΛbΛl+l are provided. Finally, the summary and discussion are presented in Sec. V.

    II.   THEORETICAL FORMALISM

      A.   BSE for Λb(Λ)

    • As shown in Fig. 1, following our previous research, the BS amplitude of Λb(Λ) in momentum space satisfies the integral equation [28, 29, 3339]

      Figure 1.  (color online) BS equation for Λb(Λ) in momentum space (K is the interaction kernel)

      χP(p)=SF(λ1P+p)d4q(2π)4K(P,p,q)χP(q)SD(λ2Pp),

      (3)

      where K(P,p,q) is the kernel, which is defined as the sum of the two particles irreducible diagrams, SF and SD are the propagators of the quark and scalar diquark, respectively. λ1(2)=mq(D)/(mq+mD), where mq(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)=IIV1(p,q)+γμ(p2+q2)μV2(p,q),

      (4)

      where V1 results from the scalar confinement, and V2 is from the one-gluon-exchange diagram. According to the potential model, V1 and V2 have the following forms in the covariant instantaneous approximation (pl=ql) [28, 29, 3739]:

      ˜V1(ptqt)=8πκ[(ptqt)2+μ2]2(2π)2δ3(ptqt)×d3k(2π)38πκ(k2+μ2)2,

      (5)

      ~V2(ptqt)=16π3α2seffQ20[(ptqt)2+μ2][(ptqt)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 as pl=λ1Pvp,pμt=pμ(vp)pμ (vμ=Pμ/M), qμt=qμ(vq)vμ, and ql=λ2Pvq. The second term of ˜V1 is introduced to avoid infrared divergence at the point pt=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)=iv[Λ+qMplωq+iϵ+ΛqMpl+ωqiϵ],

      (7)

      SD(p2)=i2ωD[1plωD+iϵ1pl+ωDiϵ],

      (8)

      where ωq=m2p2tandωD=m2Dp2t, M is the mass of the baryon, and Λ± are the projection operators, which are defined as

      2ωqΛ±q=ωq±v(pt+m),

      (9)

      and satisfy the following relations:

      Λ±qΛ±q=Λ±q,Λ±qΛq=0.

      (10)

      Generally, we require two scalar functions to describe the BS wave function of Λb(Λ) [3335],

      χP(p)=(f1(p2t)+ptf2(p2t))u(P),

      (11)

      where fi,(i=1,2) are the Lorentz-scalar functions of p2t, and u(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)(ωqm)(˜V12ωD˜V2)+pt(pt+qt)˜V24ωDωc(M+ωD+ωq),

      (14)

      M12(pt,qt)=(ωq+m)(qt+pt)qt˜V2+ptqt(˜V12ωD˜V2)4ωDωc(M+ωD+ωc)(mωq)(qt+pt)qt˜V2ptqt(˜V1+2ωD˜V2)4ωDωq(M+ωD+ωq),

      (15)

      M21(pt,qt)=(˜V1+2ωD˜V2)(ωq+m)(1+qtptp2t)˜V24ωDωq(M+ωD+ωq)(˜V12ωD˜V2)+(ωq+m)(1+qtptp2t)˜V24ωDωq(M+ωD+ωq),

      (16)

      M22(pt,qt)=(mωq)(˜V1+2ωD˜V2))ptqtp2t(q2t+ptqt)˜V24p2tωDωq(M+ωD+ωq)(m+ωq)(˜V12ωD˜V2))ptqt+p2t(q2t+ptqt)˜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 to

      SF(p1)=i1+v2(E0+mDpl+iϵ),

      (18)

      where E0=MmmD is the binding energy. Thus, the BS wave function of Λb has the form χP(v)=ϕ(p)uΛb(v,s), where ϕ(p) is the scalar BS wave function [32], and the BS equation for Λb can be replaced by

      ϕ(p)=i(E0+mDpl+iϵ)(p2lω2D)×d4q(2π)4(˜V1+2pl˜V2)ϕ(q).

      (19)

      Generally, we can take E0 to be about 0.14 GeV and κ to be about 0.05 GeV3 [28, 29].

    • B.   Asymmetries of ΛbΛl+l decays

    • In the Standard Model, the ΛbΛl+l (l=e,μ,τ) transitions are described by bsl+l at the quark level. The Hamiltonian for the decay of bsl+l is given by

      H(bsl+l)=GFα22πVtbVts[Ceff9ˉsγμ(1γ5)bˉlγμliCeff7ˉs2mbσμνqνq2(1+γ5)bˉlγμl

      +C10ˉsγμ(1γ5)bˉlγμγ5l],

      (20)

      where GF is the Fermi coupling constant, α is the fine structure constant at the Z mass scale, Vts and Vtb are the CKM matrix elements, q is the total momentum of the lepton pair, and Ci(i=7,9,10) are the Wilson coefficients. Ceff7=0.313, Ceff9=4.334, C10=4.669 [4143]. 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 the Λb(Λ), q2=(PP)2 is the transformed momentum squared, and gi, ti, si, anddi (i=1,2, and 3) are the transition FFs, which are Lorentz scalar functions of q2. The Λb and Λ states can be normalized as follows:

      Λ(P)|Λ(P)=2EΛ(2π)3δ3(PP),

      (22)

      Λb(v,P)|Λb(v,P)=2v0(2π)3δ3(PP).

      (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(r1),d3=F2(r+1),s1=d1=F2mΛb(1+r2rω),

      (24)

      where r=m2Λ/m2Λb and ω=(M2Λb+M2Λq2)/(2MΛbMΛ)=vP/mΛ. The transition matrix for ΛbΛ can be expressed in terms of the BS wave functions of Λb and Λ:

      Λ(P)|ˉdΓb|Λb(P)=d4p(2π)4ˉχP(v)ΓχP(p)S1D(p2).

      (25)

      When ω1, we can obtain the following expression by substituting Eqs. (11) and (19) into Eq. (25):

      F1=k1ωk2,

      (26)

      F2=k2,

      (27)

      where

      k1(ω)=d4p(2π)4f1(p)ϕ(p)S1D(p2),

      (28)

      k2(ω)=11ω2d4p(2π)4f2(p)ptvϕ(p)S1D.

      (29)

      The decay amplitude of ΛbΛl+l can be rewritten as follows:

      M(ΛbΛl+l)=GFλt22π[ˉlγμl{ˉuΛ[γμ(A1+B1+(A1B1)γ5)+iσμνpν(A2+B2+(A2B2)γ5)]uΛb}+ˉlγμγ5l{ˉuΛ[γμ(D1+E1+(D1E1)γ5)+iσμνpν(D2+E2+(D2E2)γ5)+pμ(D3+E3+(D3E3)γ5)]uΛb}],

      (30)

      where Ai, Bi,Dj, andEj (i=1,2 and j=1,2,3) are defined as follows:

      Ai=12{Ceff9(giti)2Ceff7mbq2(di+si)},Bi=12{Ceff9(gi+ti)2Ceff7mbq2(disi)},Dj=12C10(gjtj),Ej=12C10(gj+tj).

      (31)

      In the physical region (ω=(m2Λb+m2Λq2)/(2mΛbmΛ)), the decay rate of ΛbΛl+l is obtained as follows:

      dΓ(ΛbΛl+l)dωdcosθ=G2Fα2214π5mΛb|VtbVts|2vlλ(1,r,s)M(ω,θ),

      (32)

      where s=1+r2rω, λ(1,r,s)=1+r2+s22r2s2rs, vl=14m2lsm2Λb, 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.

      Figure 2.  (color online) Definition of the angle θ in the decay ΛbΛll+.

      M0(ω)=32m2lm4Λbs(1+rs)(|D3|2+|E3|2)+64m2lm3Λb(1rs)Re(D1E3+D3E1)+64m2Λbr(6m2lM2Λbs)Re(D1E1)+64m2lm3Λbr(2mΛbsRe(D3E3)+(1r+s)Re(D1D3+E1E3))+32m2Λb(2m2l+m2Λbs){(1r+s)mΛbrRe(A1A2+B1B2)mΛb(1rs)Re(A1B2+A2B1)2r(Re(A1B1)+m2ΛbsRe(A2B2))}+8m2Λb[4m2l(1rs)+m2Λb((1+r)2s2)](|A1|2+|B1|2)+8m4Λb{4m2l[λ+(1+rs)s]+m2Λbs[(1r)2s2]}(|A2|2+|B2|2)8m2Λb{4m2l(1+rs)m2Λb[(1r)2s2]}(|D1|2+|E1|2)+8m5Λbsv2{8mΛbsrRe(D2E2)+4(1r+s)rRe(D1D2+E1E2)4(1rs)Re(D1E2+D2E1)+mΛb[(1r)2s2](|D2|2+|E2|2)},

      (34)

      M1(ω)=16m4Λbsvlλ{2Re(A1D1)2Re(B1E1)+2mΛbRe(B1D2B2D1+A2E1A1E2)}+32m5Λbsvlλ{mΛb(1r)Re(A2D2B2E2)+rRe(A2D1+A1D2B2E1B1E2)},

      (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 as

      AFB=10dΓdq2dzdz01dΓdq2dzdz11dΓdq2dzdz,

      (37)

      where z=cosθ. The "naively integrated" observables are obtained using [17]

      X=1q2maxq2minq2maxq2minX(q2)dq2.

      (38)

      We define the integrated AFB as

      ˉAFB=ˆqmaxˆqmindˆq2AFB(ˆq2).

      (39)

      where ˆq2=q2/M2Λb. With the aid of the helicity amplitudes of ΛbΛl+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), HmmX(X=U,L,S,P,LP,SP,m=1,2) represent different helicity amplitudes, and Htot is the total helicity amplitude, αΛ=0.642±0.013. The explicit expression for HmmX is provided in Ref. [12].

    III.   NUMERICAL ANALYSIS AND DISCUSSION
    • In this section, we perform a detailed numerical analysis of AFB(ΛbΛl+l). In this study, we take the masses of baryons as mΛb=5.62 GeV and mΛ=1.116 GeV [45], and the masses of quarks as mb=5.02 GeV and ms=0.516 GeV [34, 35, 39]. The variable ω changes from 1 to 2.617,2.614,1.617 for e,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 of αseff for different values of κ for Λ and Λb with E0=0.14 GeV.

      κ/GeV3ΛΛb
      0.0450.5590.775
      0.0470.5550.777
      0.0490.5510.778
      0.0510.5470.780
      0.0530.5440.782
      0.0550.5400.784

      Table 1.  Values of αseff for Λ and Λb 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 ratio R(ω). From this figure, we observe that R(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 of 2.43ω2.52 (corresponding to M2Λq2M2Λc), R(ω) is about 0.25. In the same ω region, assuming the FFs have the same dependence on q2, the CLEO collaboration measured R=0.35±0.04±0.04 in the limit mc+. 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) and R(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, and ˉFL for ΛbΛμ+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 and 0, ˉAlhFB is about 0.1, ˉAhFB is about 0.25, and ˉFL changes from 0.3 to 0.6. Without including the long distance contribution, Ref. [6] provided the integrated forward-backward asymmetry ˉAlBF(ΛbΛμ+μ)=0.1338. The result of Ref. [7] was ˉAlBF(ΛbΛμ+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 result ˉAlBF(ΛbΛμ+μ)=1.7×104(8×104).

      ˉAlFBˉAlhFBˉAhFBˉFL
      [6, 7]0.130.5830
      [8]8.0×104
      [12]0.2860.1010.2880.525
      [13]0.0122+0.01420.0073
      [15]0.29±0.050.13+0.220.030.26±0.030.4±0.1
      [17]0.04+0.000.010.34+0.030.02
      our work0.1376±0.00010.05760.1613±0.00010.3957±0.0002

      Table 2.  Longitudinal polarization fractions and forward-backward asymmetries for ΛbΛμ+μ.

      For q2[15,20] GeV2, the LHCb collaboration provided AlFB(ΛbΛμμ+)=0.05±0.09 in 2015, which was updated to AlFB(ΛbΛμμ+)=0.39±0.04 three years later [4, 5]. In our study, in the same region, the value of AlBF(ΛbΛμμ+) 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 values AlFB(ΛbΛμμ+)=0.344 in the large ςu and small ςd regions (ςu,ςd are model parameters [47]) and AlFB(ΛbΛμμ+)=0.24 in the large ςd and small ςu regions. In Fig. 4, we plot the q2-dependence of AlFB(ΛbΛee+), AlFB(ΛbΛμμ+), and AlFB(ΛbΛττ+). From Fig. 4, we can observe that AlFB(ΛbΛμ+μ) is in good agreement with the lattice QCD calculation in the entire q2 region [48]. The results of other references results are also shown in Table 3. In Fig. 5, we plot the q2-dependence of AhFB(ΛbΛee+), AhFB(ΛbΛμμ+), and AhFB(ΛbΛττ+), respectively. For q2[15,20] GeV2, the LHCb collaboration obtained the value for ΛbΛμμ+ as 0.29±0.07, which is in good agreement our result 0.23040.0685. The results of other references results are also shown in Table 3. In Fig. 6, we plot the q2-dependence of AlhFB(ΛbΛee+), AlhFB(ΛbΛμμ+), and AlhFB(ΛbΛττ+), respectively. Ref. [12] obtained the value AlhFB(ΛbΛμμ+)=0.145, which is agreement with our results 0.12570.1555 in the region q2[15,20] GeV2. In Fig. 7, we plot the q2-dependence of FL(ΛbΛee+), FL(ΛbΛμμ+), and FL(ΛbΛττ+), respectively. In the region q2[15,20] GeV2, the LHCb collaboration obtained the value FL(ΛbΛμμ+)=0.61+0.110.14, which is close to our result of0.33980.4530. 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 and E0 in our model.

      AlFB[15,20]AlhFB[15,20]AhFB[15,20]FL[15,20]
      LHCb [4, 5]0.39±0.040.29±0.070.61+0.110.14
      [6, 7]0.400.250.370.62
      [8]0.240.13>0.308
      [12]0.400.1450.290.38
      [13]0.0750.017
      [17]0.34+0.010.020.4+0.010.02
      [48]0.350(13)0.2710±0.00920.409±0.013
      our work0.440.350.12570.15550.23040.06850.33980.4530

      Table 3.  Longitudinal polarization fractions and forward-backward asymmetries for ΛbΛμ+μ in q2[15,20] GeV2.

      Figure 4.  (color online) Values of AFB(ΛbΛl+l) as a function of q2 for different values of κ as shown in Table 1.

      Figure 5.  (color online) Values of AhFB(ΛbΛl+l) as a function of q2 for different values of κ as shown in Table 1.

      Figure 6.  (color online) Values of AhFB(ΛbΛl+l) as a function of q2 for different values of κ as shown in Table 1.

      Figure 7.  (color online) Values of FL(ΛbΛl+l) as a function of q2 for different values of κ as shown in Table 1.

      Ref. [17] obtained the naively integrated values AlFB=0.19+0.000.01 and FL=0.6±0.02 for ΛbΛμ+μ, whereas in our paper, these values are 0.1976 and 0.5681, respectively. Our results are very close to those of Ref. [17]. In our paper, we obtain ˉAlFB=0.0708±0.0001(0.0590±0.0001) and ˉAhFB=0.1604±0.0001(0.1541±0.0002) for ΛbΛe+e(ΛbΛτ+τ). The values given in Ref. [8] are ˉAlFB=1.2×108(9.6×104) and ˉAhFB=0.321(0.259), and Refs. [13] and [7] provide ˉAlFB=0.0067 and ˉAlFB=0.04 for ΛbΛτ+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.

    IV.   SUMMARY AND CONCLUSIONS
    • In this study, we use the BSE to study the forward-backward asymmetries in the rare decays ΛbΛl+l in a covariant quark-diquark model. In this picture, Λb(Λ) is considered a bound state of a b(s)-quark and a scalar diquark.

      We establish the BSE for the quark and scalar diquark system and then derive the FFs of ΛbΛ. 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 [2325]. Using these FFs, we calculate the forward-backward asymmetries AlFB, AlhFB, andAhFB and longitudinal polarization fractions FL and the integrated forward-backward asymmetries ˉAlFB, ˉAlhFB, andˉAhFB as well asˉFL for ΛbΛl+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.

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