-
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 ,C_{10}=-4.669 [41–43]. 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, andg_i ,t_i ,s_i , andd_i (i=1,2 , and 3) are the transition FFs, which are Lorentz scalar functions ofq^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 , andE_j (i=1,2 andj=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 asA_{\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]. -
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 asm_{\Lambda_b}=5.62 GeV andm_\Lambda=1.116 GeV [45], and the masses of quarks asm_b=5.02 GeV andm_s=0.516 GeV [34, 35, 39]. The variable ω changes from1 to2.617,\; 2.614,\; 1.617 fore,\; \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 withE_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 ratioR(\omega) . From this figure, we observe thatR(\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 of2.43 \leq \omega \leq 2.52 (corresponding toM_\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 onq^2 , the CLEO collaboration measuredR=-0.35\pm 0.04\pm0.04 in the limitm_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) andR(\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 and0 ,\bar{A}^{lh}_{\rm FB} is about0.1 ,\bar{A}^h_{\rm FB} is about-0.25 , and\bar{F}_L changes from0.3 to0.6 . 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 theq^2 -dependence ofA^l_{\rm FB}(\Lambda_b \rightarrow \Lambda e^- e^+) ,A^l_{\rm FB}(\Lambda_b \rightarrow \Lambda \mu^- \mu^+) , andA^l_{\rm FB}(\Lambda_b \rightarrow \Lambda \tau^- \tau^+) . From Fig. 4, we can observe thatA^l_{\rm FB}(\Lambda_b\rightarrow \Lambda \mu^+ \mu^-) is in good agreement with the lattice QCD calculation in the entireq^2 region [48]. The results of other references results are also shown in Table 3. In Fig. 5, we plot theq^2 -dependence ofA^h_{\rm FB}(\Lambda_b \rightarrow \Lambda e^- e^+) ,A^h_{\rm FB}(\Lambda_b \rightarrow \Lambda \mu^- \mu^+) , andA^h_{\rm FB}(\Lambda_b \rightarrow \Lambda \tau^- \tau^+) , respectively. Forq^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 theq^2 -dependence ofA^{lh}_{\rm FB}(\Lambda_b \rightarrow \Lambda e^- e^+) ,A^{lh}_{\rm FB}(\Lambda_b \rightarrow \Lambda \mu^- \mu^+) , andA^{lh}_{\rm FB}(\Lambda_b \rightarrow \Lambda \tau^- \tau^+) , respectively. Ref. [12] obtained the valueA^{lh}_{\rm FB}(\Lambda_b \rightarrow \Lambda \mu^- \mu^+) = 0.145 , which is agreement with our results0.1257\sim0.1555 in the regionq^2 \in [15,20] GeV^2 . In Fig. 7, we plot theq^2 -dependence ofF_L(\Lambda_b \rightarrow \Lambda e^- e^+) ,F_L(\Lambda_b \rightarrow \Lambda \mu^- \mu^+) , andF_L(\Lambda_b \rightarrow \Lambda \tau^- \tau^+) , respectively. In the regionq^2 \in [15,20] GeV^2 , the LHCb collaboration obtained the valueF_L(\Lambda_b \rightarrow \Lambda \mu^- \mu^+)=0.61_{-0.14}^{+0.11} , which is close to our result of0.3398\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 andE_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^- inq^2 \in [15,20] GeV^2 .Figure 4. (color online) Values of
A_{\rm FB}(\Lambda_b\rightarrow \Lambda l^+ l^-) as a function ofq^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 ofq^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 ofq^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 ofq^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 and0.5681 , 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 ab(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 asymmetriesA^l_{\rm FB} ,A^{lh}_{\rm FB} , andA^h_{\rm FB} and longitudinal polarization fractionsF_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.
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