Forward-backward asymmetries in ${\boldsymbol \Lambda_{\boldsymbol b} \boldsymbol\rightarrow \boldsymbol\Lambda l^+ l^- }$ in the Bethe-Salpeter equation approach

The decays of hadrons involving the flavor changing neutral current (FCNC) transition such as $ \Lambda_b \rightarrow \Lambda l^+ 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 $ \Lambda_b \rightarrow \Lambda \mu^+ \mu^- $ was first observed by the CDF collaboration in 2011 [1]. Some experimental progress on $ \Lambda_b \rightarrow \Lambda l^+ l^- $ was also achieved [2–5], and the radiative decay $ \Lambda_b \rightarrow \Lambda \gamma $ was observed in 2019 [3] by the LHCb collaboration. The LHCb collaboration determined the forward-backward asymmetries ($ A^l_{\rm FB} $) of the decay $ \Lambda_b \rightarrow \Lambda \mu^+ \mu^- $ to be $ {A}^l_{\rm FB}(\Lambda_b \rightarrow \Lambda \mu^+ \mu^-)= -0.05\pm 0.09 $ (stat) $ \pm0.03 $ (syst), $ {A}^h_{\rm FB}(\Lambda_b \rightarrow \Lambda \mu^+ \mu^-)=-0.29\pm0.09 $ (stat) $ \pm0.03 $ (syst), and $ F_{\rm L}(\Lambda_b \rightarrow \Lambda \mu^+ \mu^-)=0.61^{+0.11}_{-0.14} \pm 0.03 $ (syst) at the low dimuon invariant mass squared range $ 15 < q^2<20 $ GeV$ ^2 $ in 2015 [4]. However, these numbers were updated in 2018 to $ \bar{A}^l_{\rm FB}(\Lambda_b \rightarrow \Lambda \mu^+ \mu^-)=-0.39\pm 0.04 $ (stat) $ \pm0.01 $ (syst), $ {A}^h_{\rm FB}(\Lambda_b \rightarrow \Lambda \mu^+ \mu^-)=-0.3\pm0.05 $ (stat) $ \pm0.02 $ (syst), and $ \bar{A}^{lh}_{\rm FB}(\Lambda_b \rightarrow \Lambda \mu^+ \mu^-)=0.25\pm0.04 $ (stat) $ \pm0.01 $ (syst) in the same invariant mass squared region [5]. Note that $ A^l_{\rm FB} $ is significantly lager than the previous one. In this study, we investigate the $ A _{\rm FB} $ of $ \Lambda_b \rightarrow \Lambda l^+ l^- $ in the Bethe-Salpeter equation (BSE) approach. Theoretically, only a few studies have been conducted on $ {A}_{\rm FB}(\Lambda_b\rightarrow \Lambda l^+ l^-) $ [6–17]. References [6] ([7]) provided the integrated forward-backward asymmetries $ \bar{A}^l_{\rm FB}(\Lambda_b\rightarrow \Lambda \mu^+ \mu^-)= -0.13 $ ($ -0.12 $) and $ \bar{A}_{\rm FB}(\Lambda_b\rightarrow \Lambda \tau^+ \tau^-)=-0.04 $ ($ -0.03 $), whereas the results of Ref. [8] were $ \bar{A}^l_{\rm FB}(\Lambda_b\rightarrow \Lambda e^+ e^-)=1.2 \times 10^{-8} $, $ \bar{A}^l_{\rm FB}(\Lambda_b\rightarrow \Lambda \mu^+ \mu^-)= 8 \times 10^{-4} $, and $ \bar{A}^l_{\rm FB}(\Lambda_b\rightarrow \Lambda \tau^+ \tau^-)= 9.6\times 10^{-4} $. Ref. [10] analyzed the differential $ \bar{A}_{\rm FB}(\Lambda_b\rightarrow \Lambda l^+ l^-) $ in the heavy quark limit. Using the nonrelativistic quark model, Ref. [11] investigated the lepton-side forward-backward asymmetries $ \bar{A}^l_{\rm FB}(\Lambda_b\rightarrow \Lambda l^+ l^-) $. In the quark-diquark model, Ref. [12] investigated the lepton-side forward-backward asymmetries $ A_{\rm FB} $, the hadron-side forward-backward asymmetries $ A_{\rm FB}^h $, and the hadron-lepton forward-backward asymmetries $ A^{hl}_{\rm FB} $. In an approach of the light-cone sum rules, Refs. [13, 14] investigated the rare decays of $ \Lambda_b \rightarrow \Lambda \gamma $ and $ \Lambda_b \rightarrow \Lambda l^+ l^- $. Ref. [15] investigated the phenomenological potential of the rare decay $ \Lambda_b \rightarrow \Lambda l^+ l^- $ with a subsequent, self-analyzing $ \Lambda_b \rightarrow N \pi $ transition. With the form factors (FFs) extracted from a constituent quark model, Ref. [16] investigated the rare weak dileptonic decays of the $ \Lambda_b $ baryon. Ref. [17] studied $ {\cal B}_1 \rightarrow {\cal B}_2 l^+ l^- $ ($ {\cal B}_{1,2} $ are spin $ 1/2 $ baryons) with the SU(3) flavor symmetry. The FFs of $ \Lambda_b \rightarrow \Lambda $ differ in different models. Generally, the number of independent FFs of $ \Lambda_b \rightarrow \Lambda $ can be reduced to 2 when working in the heavy quark limit [18],

where $ \Gamma= \gamma_\mu,\; \gamma_\mu \gamma_5,\; q^\nu \sigma_{\nu\mu} $, and $ q^\nu \sigma_{\nu\mu}\gamma_5 $, $ q^2 $ is the square of the transformed momentum. The FF ratio $ R(q^2)= F_2(q^2)/F_1(q^2) $ was considered a constant in many studies assuming the same shape for $ F_1 $ and $ F_2 $, 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 $ q^2 $ dependence of FF $ F_i\; (i=1,2) $ were given as follows:

where a and b are constants. Using experimental data for the semileptonic decay $ \Lambda_c \rightarrow \Lambda e^+ \nu_e $ ($ m^2_{\Lambda} \leq q^2 \leq m^2_{\Lambda_c} $), the CLEO collaboration provided the ratio $ R=-0.35\pm0.04 $ (stat) $ \pm0.04 $ (syst) [19]. In Ref. [20], the authors investigated $ \Lambda_b \rightarrow \Lambda \gamma $ obtaining $ R=-0.25\pm0.14\pm0.08 $. In Refs. [6, 7, 21], the authors investigated the baryonic decay $ \Lambda_b \rightarrow \Lambda l^+ l^- $ and obtained $ R=-0.25 $. In Ref. [22], the relation $ F_2(q^2)/F_1(q^2) \approx F_2(0)/F_1(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 ratio $ R(q^2) \propto -1/q^2 $. 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 $ \Lambda_b $ 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, $ \Lambda_b $ (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 $ \Lambda_b $ and Λ [28, 29]. In this study, we recalculate the FFs of $ \Lambda_b \rightarrow \Lambda $ in this model.

The remainder of this paper is organized as follows. In Sec. II, we derive the general FFs and $ A_{\rm FB} $ for $ \Lambda_b \rightarrow \Lambda l^+ l^- $ in the BS equation approach. In Sec. III, the numerical results for $ A_{\rm FB} $ and $ \bar{A}_{\rm FB} $ of $ \Lambda_b \rightarrow \Lambda l^+ 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 $ \Lambda_b(\Lambda) $ in momentum space satisfies the integral equation [28, 29, 33–39]

Figure 1.  (color online) BS equation for $ \Lambda_b(\Lambda) $ in momentum space (K is the interaction kernel)

where $ K(P,p,q) $ is the kernel, which is defined as the sum of the two particles irreducible diagrams, $ S_{ \rm F} $ and $ S_{ D} $ are the propagators of the quark and scalar diquark, respectively. $ \lambda_{1(2)}=m_{q(D)}/(m_q+m_D) $, where $ m_{q(D)} $ is the mass of the quark (diquark), and P is the momentum of the baryon.

We assume the kernel has the following form:

where $ V_1 $ results from the scalar confinement, and $ V_2 $ is from the one-gluon-exchange diagram. According to the potential model, $ V_1 $ and $ V_2 $ have the following forms in the covariant instantaneous approximation ($ p_l=q_l $) [28, 29, 37–39]:

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 $ \alpha_{\rm seff} $ are related to scalar confinement and the one-gluon-exchange diagram, respectively. $ q_t $ is the transverse projection of the relative momentum along the momentum P, which is defined as $ p_l = \lambda_1 P -v\cdot p,\; p_t^\mu = p^\mu-(v\cdot p)p^\mu $ $ (v^\mu=P^\mu/M ),$ $ q_t^\mu = q^\mu -(v \cdot q) v^\mu $, and $ q_l= \lambda_2 P -v \cdot q $. The second term of $ \tilde{V}_1 $ is introduced to avoid infrared divergence at the point $ p_t=q_t $, and μ is a small parameter to avoid the divergence in numerical calculations. Analyzing the electromagnetic FFs of the proton, $ Q_0^2=3.2 $ GeV$ ^2 $ was observed to provide consistent results with the experimental data [40].

The propagators of the quark and diquark can be expressed as follows:

where $ \omega_q = \sqrt{m^2-p_t^2}\; {\rm{and}}\; \omega_D = \sqrt{m_D^2-p_t^2} $, M is the mass of the baryon, and $ \Lambda^\pm $ are the projection operators, which are defined as

and satisfy the following relations:

Generally, we require two scalar functions to describe the BS wave function of $ \Lambda_b(\Lambda) $ [33–35],

where $ f_i, (i=1,2) $ are the Lorentz-scalar functions of $ p_t^2 $, and $ u(P) $ is the spinor of a baryon.

Defining $ \tilde{f}_{1(2)}=\int \dfrac{{\rm d} p_l}{2 \pi}f_{1(2)} $, and using the covariant instantaneous approximation, the scalar BS wave functions satisfy the following coupled integral equations:

where

When the mass of the b quark approaches infinity [32], the propagator of the b quark satisfies the relation $ \not{ v} S_{ \rm F}(p_1)= S_{ \rm F}(p_1) $ and can be reduced to

where $ E_0=M-m-m_D $ is the binding energy. Thus, the BS wave function of $ \Lambda_b $ has the form $ \chi_P (v) = \phi (p)u_{\Lambda_b}(v,s) $, where $ \phi(p) $ is the scalar BS wave function [32], and the BS equation for $ \Lambda_b $ can be replaced by

Generally, we can take $ E_0 $ to be about $ -0.14 $ GeV and κ to be about $ 0.05 $ GeV$ ^3 $ [28, 29].

In the Standard Model, the $ \Lambda_b\rightarrow \Lambda l^+l^- $ ($ l=e,\mu, \tau $) transitions are described by $ b\rightarrow s l^+l^- $ at the quark level. The Hamiltonian for the decay of $ b\rightarrow s l^+l^- $ is given by

where $ G_{\rm F} $ is the Fermi coupling constant, α is the fine structure constant at the Z mass scale, $ V_{\rm ts} $ and $ V_{\rm tb} $ are the CKM matrix elements, q is the total momentum of the lepton pair, and $ C_i\; (i=7,\; 9,\; 10) $ are the Wilson coefficients. $ C^{\rm eff}_7=-0.313 $, $ C^{\rm eff}_9=4.334 $, $ C_{10}=-4.669 $ [41–43]. The relevant matrix elements can be parameterized in terms of the FFs as follows:

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:

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

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:

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

where

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

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

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:

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]:

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^+ $.

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

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

We define the integrated $ A_{\rm FB} $ as

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

The lepton-hadron side asymmetry has the form

The fraction of the longitudinally polarized dileptons is expressed by

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].

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 $
0.045	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 $\alpha_{\rm seff}$ 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.

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 [23–25]. 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.