Rare $\Lambda_b \rightarrow \Lambda l^+ l^- $ decay in the Bethe-Salpeter equation approach

We study the rare decays $\Lambda_b \rightarrow \Lambda l^+ l^-~(l=e,\mu, \tau)$ in the Bethe-Salpeter equation approach. We find that depending on the values of parameters in our model the branching ratio $Br(\Lambda_b \rightarrow \Lambda \mu^+ \mu^-)\times 10^{6}$ varies from $0.812$ to $1.445$ when $\kappa = 0.050 \sim 0.060$ GeV$^3$ and the binding energy $E_0=-0.14$ GeV while $Br(\Lambda_b \rightarrow \Lambda \mu^+ \mu^-)\times 10^{6}$ varies from $1.051$ to $1.098$ when $\kappa = 0.055$ Gev$^3$ and the binding energy $E_0$ changes from $-0.19$ to $-0.09$ GeV. These results agree with the experimental data. In the same parameter regions, we find that the branching ratio $Br(\Lambda_b \rightarrow \Lambda e^+ e^-(\tau^+ \tau^-) )\times 10^{6}$ varies in the range $0.660-1.028$ ($0.252-0.392$) and $0.749-1.098$ ($0.286-0.489$), respectively.

In the present work, we will use the Bethe-Salpeter (BS) equation to study this rare decay. In our model, Λ (b) are described as a scalar diquark and quark bound systems, and then using the covariant instantaneous approximation the FFs of Λ b → Λ will be calculated for giving the results for the Λ b → Λl + l − decay branching ratios. This paper is organized as follows. In Section II, we will establish the BS equation for Λ b and Λ. In Section III we will derive the FFs for Λ b → Λ in the BS equation approach. In Section IV the numerical results for the FFs and the decay branching ratios of Λ b → Λl + l − will be given. Finally, the summary and discussion will be given in Section V.

II. BS EQUATION FOR Q(ud) 00 SYSTEM
In our work Λ b can be described as a b(ud) 00 system the first and second subscripts correspond to the spin and the isospin of (ud), respectively) system. The BS wave function of the b(ud) 00 system can be defined as the folowing [33][34][35][36][37][38][39]: where ψ(x 1 ) and ϕ(x 2 ) are the field operators of the b-quark and (ud) 00 diquark, respectively, and P is the momentum of Λ b . We use M, m, and m D to represent the masses of the Λ b , the b-quark and the (ud) diquark, respectively. We define the BS wave function in momentum space: where X = λ 1 x 1 + λ 2 x 2 is the coordinate of mass center, λ 1 = m m+m D , λ 2 = m D m+m D , and x = x 1 − x 2 . In momentum space, the BS equation for the b(ud) 00 system satisfies the homogeneous integral equation [33][34][35][36][37][38][39]  where the quark momentum p 1 = λ 1 P + p and the diquark momentum p 2 = λ 2 P − p, S F (p 1 ) and S D (p 2 ) are propagators of the quark and the scalar diquark, respectively, Γ µ = (p 2 + q 2 ) µ α sef f Q 2 0 Q 2 +Q 2 0 is introduced to describe the structure of the scalar diquark [6,35,40]. By analyzing the electromagnetic FFs of proton, it was found that Q 2 0 = 3.2 GeV 2 can lead to consistent results with the experimental data [6]. V 1 and V 2 are the scalar confinement and one-gluon-exchange terms, respectively. Generally, the b(ud) 00 system needs two scalar functions to describe the BS wave function [33,34,37] where f i , (i = 1, 2) are the Lorentz-scalar functions of p 2 t , u(P ) is the spinor of Λ b , p t is the transverse projection of the relative momenta along the momentum P , Motivated by the potential model, V 1 and V 2 have the following forms in the covariant instantaneous approximation (p l = q l ) [35,36,39,41]: where q t is the transverse projection of the relative momenta along the momentum P and defined as The second term ofṼ 1 is introduced to avoid infrared divergence at the point p t = q t , µ is a small parameter to avoid infrared divergence. The parameters κ and α sef f are related to scalar confinement and the one-gluon-exchange diagram, respectively. The quark and diquark propagators can be written as the following: where , the quark propagator can be written as where E 0 = M − m − m D is the binding energy. In general, E 0 is about −0.14 ± 0.05 GeV [37]. Then we can get κ is about 0.05 ± 0.01 GeV 3 for Λ b [38]. Definingf 1(2) = dp l 2π f 1 (2) , and using the covariant instantaneous approximation, p l = q l , the scalar BS wave functions satisfy the coupled integral equatioñ where The BS wave function of Λ b was given in the previous work [35] and has the form Generally, the BS wave function can be normalized under the condition of the covariant instantaneous approximation [41]: where i 1(2) and j 1(2) represent the color indices of the quark and the diquark, respectively, s (′) is the spin index of the baryon Λ b , I p (p, q) i 1 i 2 j 2 j 1 is the inverse of the four-point propagator written as follows In this section, we derive the matrix element of Λ b → Λl + l − in the BS equation approach. At the quark level, Λ b → Λl + l − is described by the b → sl + l − transition. The effective Hamiltonian describing the electroweak penguin and weak box diagrams related to this transition is given by where G F and α are to the Fermi coupling constant and the electromagnetic coupling constant, respectively, P R,L = (1 ± γ 5 )/2, q is the total momentum of the lepton pair and C i (i = 7, 9, 10, ) are the Wilson coefficients. The amplitude of the decay Λ b → Λl + l − is obtained by calculating the matrix element of effective Hamiltonian for the b → sl + l − transition between the initial and final states, Λ|H|Λ b . The matrix element can be parameterized in terms of the FFs as the following: where q = P − P ′ is the momentum transfer, and g i , t i , s i , d i (i = 1, 2 and 3) are various form factors which are Lorentz scalar functions of q 2 . Considering the spin symmetry on the b quark in the limit m b → ∞, the matrix elements in Eq. (20) can be rewritten as where Γ µ represent γ µ , γ µ γ 5 , iσ µν q ν , and iσ µν γ 5 q ν . F i (i = 1, 2) can be expressed as functions solely of ω = v · P ′ /m Λ , which is the energy of the Λ baryon in the Λ b rest frame. In the pole formulae for the extrapolation to q 2 = 0 in the decay Λ b → Λγ we have F 1 (0) = 0.45 (monopole) and F 1 (0) = 0.22 (dipole) [6], while the author of Ref. [6] combined the CLEO data from Ref. [31] to get F 1 (q 2 max ) = 1.21 ignoring the mass of Λ baryon. Lattice QCD (LQCD) gives F 1 (q 2 max ) ≈ 1.25 at the leading order in the heavy quark effective theory [48]. In Ref. [43] it was assumed F 2 = 0. The QCD sum rules analysis obtained that F 1 = 0.50 ± 0.03 and F 2 = −0.1 ± 0.03 at the point GeV. Therefore, we expect F 1 (q 2 max ) < 1.5, considering the correction of Λ QCD /m b . The ratio R = F 2 /F 1 = −0.35 ± 0.04 (stat) ±0.04 (syst) has been previously measured by the CLEO Collaboration using experimental data for the semileptonic decay Λ c → Λe + ν e with the invariant mass in the range from m Λ to m Λc , assuming the same shape for F 1 and F 2 and ignoring the Λ QCD /m c corrections [42]. In Ref.
Comparing Eq. (20) with Eq. (21), we obtain the following relations: where r = m 2 Λ /m 2 Λ b . The transition matrix for Λ b → Λ can be expressed in terms of the BS wave function of Λ b and Λ, Define where v ′ = P ′ /m Λ , then we find the following relations when ω = 1 and The differential decay rate is obtained as the flowing: where the parameters A i , B i and D j , E j (i = 1, 2 and j = 1, 2, 3) are defined as In the physical region(4m 2 l ≤ q 2 ≤ (m Λ b − m Λ ) 2 ), the decay rate of Λ b → Λl + l − is obtained as , λ(1, r, s) = 1 + r 2 + s 2 − 2r − 2s − 2rs, and v l = 1 − 4m 2 l q 2 is the lepton velocity. The decay amplitude is given as [23] where
Solving Eqs. (10) and (11) for Λ with the parameters we have taken, one can get the numerical solutions of BS wave functions. For Λ b we need to solve Eq. (16). In Table. I, we give the values of α s with different binding energy E 0 and different κ for Λ. In Table. II, we give the values of α s with different binding energy E 0 and different κ for Λ b . It can be seen from Tables. I and II that the dependence of α sef f on the parameters κ and E 0 for Λ is obviously stronger than that for Λ b .   0)) are all about −0.23 for different parameters, this value agrees with the experimental result very well [31]. The value of R varies from −0.8 to −0.23 for different E 0 and κ when ω = 1 ∼ 2.6 (corresponding to q 2 from m 2 e to (m Λ b − m Λ ) 2 ). This range agrees with our result and that in Ref. [12]. Considering the experimental data for R(ω) in Ref. [42] and the values of R(ω = 1) decreases with the increase of values of κ or E 0 , we believe that the optimal range for our model parameters is κ = 0.050 GeV 3 and E 0 from −0.19 to −0.09 GeV, because in this region R(q 2 max ) = −0.8 ∼ −0.7 and R varying from −0.8 to −0.23 agree with our previous results. On the other hand, we find that LQCD also gives the value R(q 2 max ) ≈ −0.8 [48]. In Figs. 9-11, we give the ω-dependent differential decay width of Λ b → Λl − l + (l = e, µ, τ ) for different parameters. In our optimal range of parameters and in the range κ = 0.050 ± 0.005 GeV 3 , and E 0 = −0.14 ± 0.5GeV, we obtain the branching ratios, respectively, which are listed in Table III. From this table, we can see that our results are different from those of HQET and QCD sum rules, but our results are consistent with the When the parameters κ and E 0 vary in their regions, we find that the differential branching ratio of Λ b → Λµ + µ − does not have a pole at about ω = 1.2. In Refs. [5,52] when ω is in the range 1 ∼ 1.4 (corresponding to q 2 in the range 15 ∼ 20GeV 2 ), the experimental data have a pole. Considering this different, there could be new physics in this region.

V. SUMMARY AND DISCUSSION
Theoretical studies of the decay Λ b → Λl + l − require knowledge of the matrix element Λ|sΓb|Λ b . At the leading order in the heavy quark effective theory, this matrix element is given by two FFs. In the past few decades, in most of works the FFs were studied based   on QCD sum rules [12], and by fitting the experimental data [31]. With the progresses of experiments, the data about Λ b rare decay has been updated. In the present work, we have performed the first BS equation calculation of these FFs. In our work, Λ Q (Q = b, s) is regarded as a bound state of a Q-quark and a scalar diquark. In this picture, we established the BS equations for Λ Q , and derived the FFs for Λ b → Λ in the BS equation approach. After solving the BS equations of Λ and Λ Q . We calculated the value of R, and decay branching ratio for Λ b → Λl + l − also compared our results with other theoretical works and the experimental data. We found that the shapes of the differential decay branching ratio for Λ b → Λµ + µ − in our model is similar to the experimental data in most part of the region and in our work the shapes of the decay differential branching ratio of Λ b → Λl + l − (l = e, µ, τ ) agree with those of LQCD [29,48]. The experimental data for the differential decay width of Λ b → Λµ + µ − have a pole when ω ≈ 1.2, but in most of    theoretical works such a pole does not appear. Therefore, in this region there could be  new physics. The experimental data need to be improved for higher accuracy in remeasure this region. Our result for Λ b → Λµ + µ − is very close to the experimental data and we also give the predictions for the decays Λ b → Λl + l − (l = e, τ ), which need to be tested in future experimental measurements. We find that for different values parameters the FFs ratio R(ω) changes from −0.80 to −0.23 in our approach. This result agrees with the experimental data and that in Ref. [12], and agrees with LQCD at q 2 max [48]. In the heavy quark effective theory, the approximation 1/m b → ∞ leads to an uncertainty of about Λ QCD /m b . Considering the uncertainties from the parameters E 0 and κ the maximum uncertainty is about 22% in our optimal data region.
In the future, our model can also be used to study the forward-backward asymmetries, T violation and angular distributions in the decays induced by b → sl + l − to further check our FFs.