Processing math: 61%

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

Figures(7) / Tables(3)

Get Citation
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
Milestone
Received: 2022-03-08
Article Metric

Article Views(1127)
PDF Downloads(33)
Cited by(0)
Policy on re-use
To reuse of Open Access content published by CPC, for content published under the terms of the Creative Commons Attribution 3.0 license (“CC CY”), the users don’t need to request permission to copy, distribute and display the final published version of the article and to create derivative works, subject to appropriate attribution.
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Email This Article

Title:
Email:

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.

    HTML

    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, C_{10}=-4.669 [4143]. The relevant matrix elements can be parameterized in terms of the FFs as follows:

      \begin{aligned}[b] \langle \Lambda(P^\prime) | \bar{s}\gamma_{\mu}b | \Lambda_b(P)\rangle =& \bar{u}_{\Lambda}(P^\prime)(g_1\gamma^\mu+ {\rm i}g_2\sigma^{\mu\nu}q_{\nu}+g_3q_\mu)u_{\Lambda_b}(P),\\ \langle \Lambda(P^\prime) | \bar{s}\gamma_{\mu}\gamma_{5}b | \Lambda_b(P)\rangle = & \bar{u}_{\Lambda}(P^\prime)(t_1\gamma^\mu+{\rm i}t_2\sigma^{\mu\nu}q_{\nu}+t_3q^\mu)\gamma_5u_{\Lambda_b}(P),\\ \langle \Lambda (P^\prime) | \bar{s}i\sigma^{\mu\nu}q^{\nu}b | \Lambda_b(P)\rangle = & \bar{u}_{\Lambda}(P^\prime)(s_1\gamma^\mu+{\rm i}s_2\sigma^{\mu\nu}q_{\nu}+s_3q^\mu)u_{\Lambda_b}(P),\\ \langle \Lambda (P^\prime) | \bar{s}i\sigma^{\mu\nu}\gamma_5q^{\nu}b | \Lambda_b(P)\rangle = & \bar{u}_{\Lambda}(P^\prime)(d_1\gamma^\mu+{\rm i}d_2\sigma^{\mu\nu}q_{\nu}+d_3q^\mu)\gamma_5u_{\Lambda_b}(P), \end{aligned}

      (21)

      where P (P^\prime) is the momentum of the \Lambda_b (Λ), q^2= (P-P^\prime)^2 is the transformed momentum squared, and g_i , t_i , s_i , and d_i ( i=1,2 , and 3) are the transition FFs, which are Lorentz scalar functions of q^2 . The \Lambda_b and Λ states can be normalized as follows:

      \langle \Lambda(P^\prime)|\Lambda(P)\rangle = 2 E_\Lambda (2\pi)^3 \delta^3(P-P^\prime),

      (22)

      \langle \Lambda_b(v^\prime,P^\prime)|\Lambda_b(v,P)\rangle = 2 v_0(2\pi)^3 \delta^3(P-P^\prime).

      (23)

      Comparing Eq. (1) with Eq. (21), we obtain the following relations:

      \begin{aligned}[b] & g_1\; =\; t_1\; =\; s_2\; =\; d_2\; =\; \bigg(F_1+\sqrt{r}F_2\bigg),\\ & g_2\; =\; t_2\; =g_3\; =\; t_3\; =\; \frac{1}{m_{\Lambda_{b}}}F_2, \\ & s_3\; =\; F_2 (\sqrt{r}-1),\quad d_3\; =\; F_2(\sqrt{r}+1), \\ & s_1 \; =\; d_1\; =\; F_2 m_{\Lambda_b} (1+r-2\sqrt{r}\omega), \end{aligned}

      (24)

      where r=m_\Lambda^2/m_{\Lambda_b}^2 and \omega= (M_{\Lambda_b}^2+M_{\Lambda}^2-q^2)/ (2M_{\Lambda_b} M_{\Lambda})= v\cdot P^\prime/m_{\Lambda} . The transition matrix for \Lambda_b\rightarrow \Lambda can be expressed in terms of the BS wave functions of \Lambda_b and Λ:

      \langle \Lambda (P^\prime)|\bar{d}\Gamma b|\Lambda_b(P)\rangle =\int\frac{{\rm d}^4p}{(2\pi)^4} \bar{\chi}_{P^\prime}(v^\prime)\Gamma \chi_P(p)S^{-1}_{ D}(p_2).

      (25)

      When \omega \neq 1 , we can obtain the following expression by substituting Eqs. (11) and (19) into Eq. (25):

      F_1 = k_1- \omega k_2,

      (26)

      F_2 = k_2,

      (27)

      where

      k_1(\omega)=\int \frac{{\rm d}^4p}{(2 \pi)^4} f_1(p^\prime) \phi(p) S^{-1}_{ D}(p_2),

      (28)

      k_2(\omega) = \frac{1}{1-\omega^2} \int \frac{{\rm d}^4 p}{(2\pi)^4} f_2(p^\prime) p^\prime_t \cdot v \phi(p) S^{-1}_{ D}.

      (29)

      The decay amplitude of \Lambda_b \rightarrow \Lambda l^+ l^- can be rewritten as follows:

      \begin{aligned}[b] \mathcal{M}(\Lambda_b\rightarrow \Lambda l^+ l^-)=&\frac{G_{\rm F} \lambda_t}{2\sqrt{2}\pi} \big[\bar{l}\gamma_{\mu}l\{\bar{u}_{\Lambda}[\gamma_{\mu}(A_1+B_1+ (A_1-B_1)\gamma_5 ) \\ & + {\rm i}\sigma^{\mu\nu}p_{\nu}(A_2+B_2+ (A_2-B_2)\gamma_5 )]u_{\Lambda_b}\} \\ &+\bar{l}\gamma_{\mu}\gamma_5l\{\bar{u}_{\Lambda}[\gamma^{\mu}(D_1+E_1+ (D_1-E_1)\gamma_5 ) \\ &+{\rm i}\sigma^{\mu\nu}p_{\nu}(D_2+E_2+ (D_2-E_2)\gamma_5 )\\ &+p^{\mu}(D_3+E_3+ (D_3-E_3)\gamma_5 )]u_{\Lambda_b}\}\big], \end{aligned}

      (30)

      where A_i , B_i , D_j , and E_j ( i=1,2 and j=1,2,3 ) are defined as follows:

      \begin{aligned}[b] &A_i=\frac{1}{2}\bigg\{C^{\rm eff}_{9}(g_i-t_i)-\frac{2C^{\rm eff}_7 m_b}{q^2}(d_i +s_i )\bigg\},\\ & B_i = \frac{1}{2}\bigg\{C^{\rm eff}_{9}(g_i+t_i) - \frac{2C^{\rm eff}_7m_b}{q^2}(d_i -s_i )\bigg\}, \\ & D_j = \frac{1}{2}C_{10}(g_j-t_j), \; E_j=\frac{1}{2}C_{10}(g_j+t_j). \end{aligned}

      (31)

      In the physical region ( \omega = (m_{\Lambda_b}^2 + m_{\Lambda}^2 -q^2)/ (2m_{\Lambda_b}m_{\Lambda}) ), the decay rate of \Lambda_b\rightarrow \Lambda l^+l^- is obtained as follows:

      \frac{{\rm d}\Gamma(\Lambda_b\rightarrow \Lambda l^+l^-)}{{\rm d}\omega {\rm d} \cos \theta}=\frac{G^2_{\rm F}\alpha^2}{2^{14}\pi^5m_{\Lambda_b}} |V_{\rm tb}V^*_{\rm ts}|^2v_l\sqrt{\lambda(1,r,s)} \mathcal{M}(\omega, \theta) ,

      (32)

      where s= 1 +r - 2 \sqrt{r} \omega , \lambda(1,r,s)=1+r^2+s^2-2r- 2s- 2rs, v_l=\sqrt{1-\dfrac{4m^2_l}{s m^2_{\Lambda_b}}} , and the decay amplitude is expressed as follows [44]:

      \mathcal{M}(\omega,\theta) = \mathcal{M}_0(\omega) +\mathcal{M}_1(\omega) \cos \theta +\mathcal{M}_2(\omega) \cos^2 \theta,

      (33)

      where θ is the polar angle, as shown in Fig. 2.

      Figure 2.  (color online) Definition of the angle θ in the decay \Lambda_b \rightarrow \Lambda l^- l^+ .

      \begin{aligned}[b] \mathcal{M}_0(\omega)=&32m^2_l m^4_{\Lambda_b}s(1+r-s)(|D_3|^2+|E_3|^2) +64m^2_lm^3_{\Lambda_b}(1-r-s){\rm Re}(D^*_1E_3+D_3E^*_1)\\ &+64m^2_{\Lambda_b}\sqrt{r}(6m^2_l-M^2_{\Lambda_b}s){\rm Re}(D_1^*E_1) + {64m^2_lm^3_{\Lambda_b}\sqrt{r}\big(2m_{\Lambda_b}s {\rm Re}(D^*_3E_3) +(1-r+s){\rm Re}(D^*_1D_3+E^*_1E_3)\big) }\\ &+32m^2_{\Lambda_b}(2m^2_l+m^2_{\Lambda_b}s)\bigg\{(1-r+s)m_{\Lambda_b}\sqrt{r}{\rm Re}(A^*_1A_2+B^*_1B_2)\\ &-m_{\Lambda_b}(1-r-s){\rm Re}(A^*_1B_2+A^*_2B_1) -2\sqrt{r}\big({\rm Re}(A^*_1B_1)+m^2_{\Lambda_b}s {\rm Re}(A^*_2B_2)\big) \bigg \}\\ & + 8 m^2_{\Lambda_b}\bigg[4m^2_l(1-r-s)+m^2_{\Lambda_b}((1+r)^2- s^2)\bigg](|A_1|^2+|B_1|^2)\\ &+8m^4_{\Lambda_b}\bigg\{4m^2_l[\lambda+(1+r-s)s]+m^2_{\Lambda_b}s[(1-r)^2-s^2]\bigg\}(|A_2|^2+|B_2|^2) \\ & - 8m^2_{\Lambda_b}\bigg\{4m^2_l(1+r-s)-m^2_{\Lambda_b}[(1-r)^2-s^2]\bigg\} (|D_1|^2+|E_1|^2) \\ &+ 8m^5_{\Lambda_b}sv^2\bigg\{-8m_{\Lambda_b}s\sqrt{r}{\rm Re}(D^*_2E_2) +4(1-r+s)\sqrt{r}{\rm Re}(D^*_1D_2+E^*_1E_2)\\ & -4(1-r-s) {\rm Re}(D^*_1E_2+D^*_2E_1)+m_{\Lambda_b}[(1-r)^2-s^2] (|D_2|^2+|E_2|^2)\bigg\}, \end{aligned}

      (34)

      \begin{aligned}[b] {\mathcal M}_1(\omega) =& -16 m_{\Lambda_b}^4 s v_l \sqrt{\lambda} \Big\{ 2 {\rm Re}(A_1^* D_1)-2{\rm Re}(B_1^* E_1)+ 2m_{\Lambda_b} {\rm Re}(B_1^* D_2-B_2^* D_1+A_2^* E_1-A_1^*E_2)\Big\}\\ &+32 m_{\Lambda_b}^5 s v_l \sqrt{\lambda} \Big\{ m_{\Lambda_b} (1-r){\rm Re}(A_2^* D_2 -B_2^* E_2)+ \sqrt{r} {\rm Re}(A_2^* D_1+A_1^* D_2-B_2^*E_1-B_1^* E_2)\Big\}, \end{aligned}

      (35)

      \mathcal{M}_2(\omega) = 8m^6_{\Lambda_b}s v_l^2\lambda(|A_2|^2+|B_2|^2+|E_2|^2+|D_2|^2) - 8 m^4_{\Lambda_b}v_l^2\lambda(|A_1|^2+|B_1|^2+|E_1|^2+|D_1|^2).

      (36)

      The lepton-side forward-backward asymmetry, A_{\rm FB} , is defined as

      A_{\rm FB} = \frac{\displaystyle\int_{0}^{1} \dfrac{{\rm d} \Gamma}{{\rm d} q^2 {\rm d}z} {\rm d}z -\displaystyle\int_{-1}^{0} \dfrac{{\rm d} \Gamma}{{\rm d} q^2 {\rm d}z} {\rm d}z }{\displaystyle\int_{-1}^{1} \dfrac{{\rm d} \Gamma}{{\rm d} q^2 {\rm d}z} {\rm d}z },

      (37)

      where z = \cos \theta . The "naively integrated" observables are obtained using [17]

      \langle {X}\rangle = \frac{1}{q^2_{\rm max}- q^2_{\rm min}} \int_{q^2_{\rm min}}^{q^2_{\rm max}}X(q^2){\rm d} q^2.

      (38)

      We define the integrated A_{\rm FB} as

      \bar{A}_{\rm FB} = \int_{\hat{q}_{\rm min}}^{\hat{q}_{\rm max}} {\rm d} \hat{q}^2 A_{\rm FB}(\hat{q}^2).

      (39)

      where \hat{q}^2= q^2 / M_{\Lambda_b}^2 . With the aid of the helicity amplitudes of \Lambda_b \rightarrow \Lambda l^+ 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

      \begin{aligned}[b]& A_{\rm FB}^h(q^2) \\=&\frac{\alpha_\Lambda}{2} \frac{ \dfrac{v^2_l}{2} ({\cal H}_P^{11}+{\cal H}_P^{22}+{\cal H}_{L_P}^{11}+{\cal H}_{L_P}^{22})+\dfrac{3m_l^2}{q^2}({\cal H}_{P}^{11}+{\cal H}_{L_P}^{11}+{\cal H}_{S_P}^{22})}{{\cal H}_{\rm tot}}.\qquad \end{aligned}

      (40)

      The lepton-hadron side asymmetry has the form

      \begin{equation} A_{\rm FB}^{lh}(q^2) =-\frac{3}{4} \frac{\alpha_\Lambda}{2} \frac{v_l {\cal H}_U^{12} } {{\cal H}_{\rm tot}}.\qquad \end{equation}

      (41)

      The fraction of the longitudinally polarized dileptons is expressed by

      \begin{equation} F_L(q^2)=\frac{\dfrac{v^2_l}{2}({\cal H}_L^{11}+{\cal H}_{L}^{22})+ \dfrac{m_l^2}{q^2}({\cal H}_{U}^{11}+{\cal H}_{L}^{11}+{\cal H}_{S}^{22})}{{\cal H}_{\rm tot}}. \end{equation}

      (42)

      In Eqs. (40–42), {\cal H}_{X}^{m m^\prime} (X= U,\; L,\; S,\; P,\; L_P,\; S_P,\; m=1,2) represent different helicity amplitudes, and {\cal H}_{\rm tot} is the total helicity amplitude, \alpha_\Lambda=0.642\pm0.013 . The explicit expression for {\cal H}^{mm^\prime}_{X} is provided in Ref. [12].

    III.   NUMERICAL ANALYSIS AND DISCUSSION
    • In this section, we perform a detailed numerical analysis of A_{\rm FB}(\Lambda_b \rightarrow \Lambda l^+ l^-) . In this study, we take the masses of baryons as m_{\Lambda_b}=5.62 GeV and m_\Lambda=1.116 GeV [45], and the masses of quarks as m_b=5.02 GeV and m_s=0.516 GeV [34, 35, 39]. The variable ω changes from 1 to 2.617,\; 2.614,\; 1.617 for e,\; \mu,\; \tau , respectively.

      Solving Eqs. (12) and (19) for Λ and \Lambda_b , we can obtain the numerical solutions of their BS wave functions. In Table 1, we provide the values of \alpha_{\rm seff} for different values of κ for Λ and \Lambda_b with E_0=-0.14 GeV.

      κ/GeV ^3 Λ \Lambda_b
      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 \alpha_{\rm seff} for Λ and \Lambda_b for different κ values.

      From Table 1, we observe that the value of \alpha_{\rm seff} is weakly dependent on the value of κ. In Fig. 3, we plot the FFs and FF ratio R(\omega) . From this figure, we observe that R(\omega) varies from -0.75 to -0.25 in our model. In Ref. [27], R(\omega) 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 2.43 \leq \omega \leq 2.52 (corresponding to M_\Lambda^2 \leq q^2 \leq M_{\Lambda_c}^2 ), R(\omega) is about -0.25 . In the same ω region, assuming the FFs have the same dependence on q^2 , the CLEO collaboration measured R=-0.35\pm 0.04\pm0.04 in the limit m_c \rightarrow + \infty . These results are in good agreement with our research in the same ω region.

      Figure 3.  (color online) Values of F_1 (solid line), F_2 (dash line) and R(\omega) (dot line) as a function of ω (the lines become thicker with the increase in κ).

      In Table 2, we provide \bar{A}^l_{\rm BF} , \bar{A}^{lh}_{\rm FB} , \bar{A}^h_{\rm FB} , and \bar{F}_L for \Lambda_b \rightarrow \Lambda \mu^+ \mu^- and compare our results with those of other studies. We can observe that these asymmetries differ significantly in different models. Considering these differences, \bar{A}^l_{\rm FB} changes between -0.30 and 0 , \bar{A}^{lh}_{\rm FB} is about 0.1 , \bar{A}^h_{\rm FB} is about -0.25 , and \bar{F}_L changes from 0.3 to 0.6 . Without including the long distance contribution, Ref. [6] provided the integrated forward-backward asymmetry \bar{A}^l_{\rm BF}(\Lambda_b \rightarrow \Lambda \mu^+ \mu^-)= -0.1338 . The result of Ref. [7] was \bar{A}^l_{\rm BF}(\Lambda_b \rightarrow \Lambda \mu^+ \mu^-)= -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 \bar{A}^l_{\rm BF}(\Lambda_b \rightarrow \Lambda \mu^+ \mu^-)= 1.7\times 10^{-4} (8\times 10^{-4}) .

      \bar{A}^l_{\rm FB} \bar{A}^{lh}_{\rm FB} \bar{A}^{h}_{\rm FB} \bar{F}_L
      [6, 7] -0.13 0.5830
      [8] 8.0\times 10^{-4}
      [12] -0.286 0.101 -0.288 0.525
      [13] -0.0122^{+0.0142}_{-0.0073}
      [15] -0.29\pm0.05 0.13^{+0.22}_{-0.03} -0.26\pm0.03 0.4\pm0.1
      [17] -0.04^{+0.00}_{-0.01} 0.34_{-0.02}^{+0.03}
      our work -0.1376\pm0.0001 0.0576 - 0.1613\pm0.0001 0.3957\pm0.0002

      Table 2.  Longitudinal polarization fractions and forward-backward asymmetries for \Lambda_b \rightarrow \Lambda \mu^+ \mu^- .

      For q^2 \in [15,20] GeV ^2 , the LHCb collaboration provided {A}^l_{\rm FB}(\Lambda_b \rightarrow \Lambda \mu^- \mu^+) = -0.05 \pm 0.09 in 2015, which was updated to {A}^l_{\rm FB}(\Lambda_b \rightarrow \Lambda \mu^- \mu^+) = -0.39 \pm 0.04 three years later [4, 5]. In our study, in the same region, the value of {A}^l_{\rm BF}(\Lambda_b \rightarrow \Lambda \mu^- \mu^+) 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 {A}^l_{\rm FB}(\Lambda_b \rightarrow \Lambda \mu^- \mu^+) = -0.344 in the large \varsigma_u and small \varsigma_d regions ( \varsigma_u,\; \varsigma_d are model parameters [47]) and {A}^l_{\rm FB}(\Lambda_b \rightarrow \Lambda \mu^- \mu^+) =-0.24 in the large \varsigma_d and small \varsigma_u regions. In Fig. 4, we plot the q^2 -dependence of A^l_{\rm FB}(\Lambda_b \rightarrow \Lambda e^- e^+) , A^l_{\rm FB}(\Lambda_b \rightarrow \Lambda \mu^- \mu^+) , and A^l_{\rm FB}(\Lambda_b \rightarrow \Lambda \tau^- \tau^+) . From Fig. 4, we can observe that A^l_{\rm FB}(\Lambda_b\rightarrow \Lambda \mu^+ \mu^-) is in good agreement with the lattice QCD calculation in the entire q^2 region [48]. The results of other references results are also shown in Table 3. In Fig. 5, we plot the q^2 -dependence of A^h_{\rm FB}(\Lambda_b \rightarrow \Lambda e^- e^+) , A^h_{\rm FB}(\Lambda_b \rightarrow \Lambda \mu^- \mu^+) , and A^h_{\rm FB}(\Lambda_b \rightarrow \Lambda \tau^- \tau^+) , respectively. For q^2 \in [15,20] GeV ^2 , the LHCb collaboration obtained the value for \Lambda_b \rightarrow \Lambda \mu^- \mu^+ as -0.29\pm0.07 , which is in good agreement our result -0.2304\sim -0.0685 . The results of other references results are also shown in Table 3. In Fig. 6, we plot the q^2 -dependence of A^{lh}_{\rm FB}(\Lambda_b \rightarrow \Lambda e^- e^+) , A^{lh}_{\rm FB}(\Lambda_b \rightarrow \Lambda \mu^- \mu^+) , and A^{lh}_{\rm FB}(\Lambda_b \rightarrow \Lambda \tau^- \tau^+) , respectively. Ref. [12] obtained the value A^{lh}_{\rm FB}(\Lambda_b \rightarrow \Lambda \mu^- \mu^+) = 0.145 , which is agreement with our results 0.1257\sim0.1555 in the region q^2 \in [15,20] GeV ^2 . In Fig. 7, we plot the q^2 -dependence of F_L(\Lambda_b \rightarrow \Lambda e^- e^+) , F_L(\Lambda_b \rightarrow \Lambda \mu^- \mu^+) , and F_L(\Lambda_b \rightarrow \Lambda \tau^- \tau^+) , respectively. In the region q^2 \in [15,20] GeV ^2 , the LHCb collaboration obtained the value F_L(\Lambda_b \rightarrow \Lambda \mu^- \mu^+)=0.61_{-0.14}^{+0.11} , which is close to our result of 0.3398\sim0.4530 . 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 E_0 in our model.

      A^l_{\rm FB [15,20]} {A}^{lh}_{\rm FB[15,20]} {A}^{h}_{\rm FB[15,20]} {F}_{L[15,20]}
      LHCb [4, 5] -0.39\pm0.04 -0.29\pm0.07 0.61^{+0.11}_{-0.14}
      [6, 7] -0.40\sim-0.25 0.37\sim 0.62
      [8] -0.24\sim -0.13 >-0.308
      [12] -0.40 0.145 -0.29 0.38
      [13] -0.075\sim -0.017
      [17] -0.34_{-0.02}^{+0.01} 0.4^{+0.01}_{-0.02}
      [48] -0.350(13) -0.2710\pm0.0092 0.409\pm0.013
      our work -0.44\sim-0.35 0.1257\sim 0.1555 -0.2304\sim-0.0685 0.3398\sim0.4530

      Table 3.  Longitudinal polarization fractions and forward-backward asymmetries for \Lambda_b \rightarrow \Lambda \mu^+ \mu^- in q^2 \in [15,20] GeV ^2 .

      Figure 4.  (color online) Values of A_{\rm FB}(\Lambda_b\rightarrow \Lambda l^+ l^-) as a function of q^2 for different values of κ as shown in Table 1.

      Figure 5.  (color online) Values of A^h_{\rm FB}(\Lambda_b\rightarrow \Lambda l^+ l^-) as a function of q^2 for different values of κ as shown in Table 1.

      Figure 6.  (color online) Values of A^h_{\rm FB}(\Lambda_b\rightarrow \Lambda l^+ l^-) as a function of q^2 for different values of κ as shown in Table 1.

      Figure 7.  (color online) Values of F_L(\Lambda_b\rightarrow \Lambda l^+ l^-) as a function of q^2 for different values of κ as shown in Table 1.

      Ref. [17] obtained the naively integrated values \langle A_{\rm FB}^l \rangle = -0.19^{+0.00}_{-0.01} and \langle F_L \rangle = 0.6\pm0.02 for \Lambda_b \rightarrow \Lambda \mu^+ \mu^- , 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 \bar{A}^l_{\rm FB}= -0.0708\pm 0.0001(-0.0590\pm0.0001) and \bar{A}^{h}_{\rm FB}=-0.1604\pm 0.0001 (-0.1541\pm0.0002) for \Lambda_b\rightarrow \Lambda e^+ e^-(\Lambda_b\rightarrow \Lambda \tau^+ \tau^-) . The values given in Ref. [8] are \bar{A}^l_{\rm FB}=1.2\times 10^{-8}(9.6\times 10^{-4}) and \bar{A}^{h}_{\rm FB}=-0.321(-0.259) , and Refs. [13] and [7] provide \bar{A}^l_{\rm FB}=-0.0067 and \bar{A}^l_{\rm FB}=-0.04 for \Lambda_b\rightarrow \Lambda \tau^+ \tau^- . 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 \Lambda_b \rightarrow \Lambda l^+ l^- in a covariant quark-diquark model. In this picture, \Lambda_b (\Lambda) 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 \Lambda_b \rightarrow \Lambda . 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 A^l_{\rm FB} , A^{lh}_{\rm FB} , and A^h_{\rm FB} and longitudinal polarization fractions F_L and the integrated forward-backward asymmetries \bar{A}^l_{\rm FB} , \bar{A}^{lh}_{\rm FB} , and\bar{A}^h_{\rm FB} as well as \bar{F}_L for \Lambda_b \rightarrow \Lambda l^+l^- (l=e,\; \mu,\; \tau) . 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.

Reference (48)

目录

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return