Lepton Asymmetries in Heavy Baryon Decays of $\Lambda_b\to\Lambda l^+l^-$

We study the dilepton forward-backward and the longitudinal, normal and transverse lepton polarization asymmetries in the heavy baryon decays of $\Lambda_b\to\Lambda l^+l^-$. We show that the asymmetries have a less dependence on the non-perturbative QCD effects. In the standard model, we find that the integrated forward-backward asymmetries (FBAs) and three components of the polarizations in the QCD sum rule approach (pole model) are $-0.13\ (-0.12)$ and $(58.3,-9.4,-0.07)%$ ($(58.3,-12.6,-0.07)%$) for $\Lambda_b\to\Lambda\mu^+\mu^-$ and $-0.04 (-0.03)$ and $(10.9,-10.0,-0.39)%$ ($(10.9,-0.2,-0.34)%$) for $\Lambda_b\to\Lambda\tau^+\tau^-$, respectively.


Introduction
It is known that the FBAs of the dileptons in the inclusive decays of b → sl + l − provide us with information on the short-distance (SD) contributions, which are dominated by the top quark loops in the standard model [1]. The longitudinal lepton polarizations in b → sl + l − , which are another parity violating observables, are also interesting asymmetries. In particular, the tau polarization in b → sτ − τ − could be accessible to the B-Factories [2,3]. It is noted that the FBAs of the exclusive decays B → Ml + l − are identically zero when M are pseudoscalar mesons such as π and K but nonzero for M being vector mesons such as ρ and K * . However, the longitudinal lepton polarizations [4] as well as other components [5] are nonzero for both types of the exclusive B meson decay modes.
In this paper, we study the dilepton forward-backward and various lepton polarization asymmetries in the heavy baryon decays of Λ b → Λl + l − . To study these baryonic decays, one of the most difficulties is to evaluate the hadronic matrix elements. It is known that there are many form factors for the Λ b → Λ transition, which are hard to be calculated since they are related to the non-perturbative effect of QCD. However, in heavy particle decays, the heavy quark effective theory (HQET) could reduce the number of form factors and supply the information with respect to their relative size [6,7,8]. With the HQET, we shall use the QCD sum rule approach [6] and the pole model [9] in our numerical calculations for the form factors.
The paper is organized as follows. In Sec. 2, we study the effective Hamiltonian for the decays of Λ b → Λl + l − (l = e, µ, τ ) and form factors in the Λ b → Λ transition. In Sec. 3, we derive the general forms of the lepton polarization and dilepton forward-backward asymmetries in Λ b → Λl + l − . We give our numerical analysis in Sec. 4. In Sec. 5, we present our conclusions.

Effective Hamiltonian and Form factors
To study the heavy baryon decays of Λ b → Λl + l − (l = e or µ or τ ), we start with the effective Hamiltonian for the b-quark decay of b → sl + l − , given by where G F is the Fermi constant, V ij are the CKM matrix elements, and C i (µ) and O i (µ) are the expressions for the renormalized Wilson coefficients and operators, whose expressions can be found in Ref. [10], respectively. In terms of the Hamiltonian in Eq. (1), the free quark decay amplitude is written as with P L(R) = (1 ∓ γ 5 )/2. We note that in Eq. (2), only the term associated with the Wilson coefficient C 10 is independent of the µ scale. We also note that the dominant contribution to the decay rate is from the long-distance (LD), such as that from the cc resonant states of Ψ, Ψ ′ ...etc. It is known that to find out the LD effects for the B-meson decays, in the literature [1,3,4,11,12,13], both the factorization assumption (FA) and the vector meson dominance (VMD) approximation have been used. For the LD contributions in baryonic decays, we assume that the parametrization is the same as that in the B meson decays. Hence, we may include the resonant effect (RE) by absorbing it to the corresponding Wilson coefficients. The effective Wilson coefficient of C ef f 9 has the standard form where h(x, s) describes the one-loop matrix elements of operators O 1 =s α γ µ P L b βcβ γ µ P L c α and O 2 =sγ µ P L bcγ µ P L c as shown in Ref. It is clear that one of the main theoretical uncertainties in studying exclusive decays arises from the calculation of form factors. With the HQET, the hadronic matrix elements for the heavy baryon decays could be parametrized as follows [9] < Λ(p, where v = p Λ b /M Λ b is the four-velocity of the heavy baryon, q 2 = (p Λ b − p Λ ) 2 is the square of the momentum transform, and Γ denotes the possible Dirac matrix. Note that in terms of the HQET there are only two independent form factors, F 1 and F 2 , in Eq. (3) for each Γ. In the following, we shall use F 1 and R ≡ F 2 /F 1 as the two independent parameters and adopt the HQET approximation to analyze the behavior of Λ b → Λl + l − . From Eqs. (2) and (3), the transition matrix element for Λ with where one has

Lepton Asymmetries
In this section we present the formulas for the forward-backward and the longitudinal, normal and transverse lepton polarization asymmetries in Λ We shall concentrate on the l + spin for the polarizations. To do this, we write the l + four-spin vector in terms of a unit vector,ξ, along the l + spin in its rest frame, as and choose the unit vectors along the longitudinal, normal, and transverse components of the l + polarization to beê respectively. The partial decay width for Λ b → Λ l + l − is given by with where |M 0 | 2 is related to the decay rate for the unpolarized l + and P i (i = L, N, T ) are the longitudinal, normal and transverse polarizations of l + , respectively. Introducing dimensionless variables of , using the transition matrix element of Eq. (4), and integrating the angle dependence of the lepton, the differential decay width in Eq. (9) becomes dΓ = with and where and In Eqs. (12) and (15), the allowed range ofŝ is Defining the longitudinal, normal and transverse l + polarization asymmetries by from Eq. (11) we find that where We note that the transverse part of the lepton polarization in Eq. (20) is a T-odd observable. The differential and normalized dilepton forward-backward asymmetries (FBAs) for the decay of Λ b → Λl + l − as a function ofŝ are defined by and respectively, where θ is the angle of l + with respect to Λ b in the rest frame of the lepton pair. Explicitly, we obtain and where From Eqs. (15), (18)-(21), and (25)-(26), we see that P i (i = L, N, T ) and A F B depend only on R since the factor F 2 1 is canceled out. Thus, once one gets the value of R, the only uncertainty for the asymmetries is from the Wilson coefficients. It is interesting to note that these asymmetries are sensitive to the chiral structure of electroweak interactions since they are related to the products of C 9 C * 7 , C 10 C * 7 and C 9 C * 10 .

Numerical Analysis
In our numerical calculations, the Wilson coefficients are evaluated at the scale µ ≃ m b and the other parameters are listed in Table 1 of Ref. [8]. For the form factors in the Λ b → Λ transition, we use the results from both the QCD sum rule approach [6] and the pole model [9]. In the QCD sum rule approach we use the form where the parameters in Eq. (27) are shown in Table 1. From the Table, we find that R(0) = F 2 (0)/F 1 (0) = −0.17 and R(q 2 max ) = −0.44 which are consistent with the CLEO result of R = −0.25 ± 0.14 ± 0.08 [14]. In the pole model, we adopt and Λ QCD is chosen around 200 MeV . Assuming the form factors for the transition of Λ c → Λ are similar to that of Λ b → Λ and using R = −0.25 [14] and the branching ratio of Λ + c → Λe + ν e , we obtain that N 1,2 are (52.32, −13.08) [8].

Forward-backward Asymmetries
From Eqs. (24) and (25), we see that the FBAs for the light charged lepton modes of Λ b → Λl + l − (l = e and µ) are close to each other. As a result, we shall not mention the electron mode of Λ b → Λe + e − . In Figures 1 and 2, we show A F B (Λ b → Λl + l − ) as a function of dimensionless variableŝ for l = µ and τ , respectively. From Figure 1(a), we see that A F B (Λ b → Λµ + µ − ) has a zero value atŝ 0 which satisfies the condition Furthermore, we find that the contributions from the pole and QCD sum rule models to FBAs overlap at the low q 2 region so that in both models Eq. (29) can be simplified to which is independent of the hadronic form factors. Explicitly, from Figure 1(a), in the standard model we get thatŝ 0 is 0.109 and 0.114 with and without R terms for excluding LD effects, and 0.098 and 0.102 for including LD effects, respectively. It is clear that the zero point of A F B (Λ b → Λµ + µ − ) is mainly affected by the weak Wilson coefficients of C 7 and C 9 that are sensitive to physics beyond the standard model. For example, if one of C 7 and C 9 has an opposite sign to that in the standard model, the condition for the zero point in Eq. (30) will not be satisfied. Therefore, measuring a sizable value of the FBA aroundŝ 0 is a clear indication of new physics. This result is similar to B → K * l + l − decays mentioned by [15] with large energy effective theory (LEET) [16]. We note that the vanishing of the FBAs in the inclusive decays of b → (s, d)l + l − and the exclusive ones of B → (K * , ρ)l + l − were first studied by Burdman [17]. Our conclusion for the baryonic decays coincides with that in Ref. [17]. From the figures, we find that there is no much difference for the FBAs between the QCD sum rule approach and the pole model at the lower values of q 2 , especially for that in the muon mode. By taking R to be zero, the distributions for both models in Figures 1 and 2 should be identical. Thus, the differences for the FBAs in the different QCD models actually reflect the effects of the ratio R. The insensitivity to the form factors for the FBAs provides us a candidate to test the standard model.
In Figure 3, we show the differential FBA of dA F B (ŝ)/dŝ which, unlike A F B , is insensitive to R. This can be understood that due to Eqs. (24) and (25) it is proportional to R F B (ŝ) in which the terms with F 2 1 are the dominant contributions and those with R are negligible since these terms are related to either R 2 or R √ r, which are small.
We now define the integrated FBA to be Without LD contributions, in the standard model we find that and for the QCD sum rule approach (pole model), respectively.

Polarization Asymmetries
We now discuss the longitudinal, normal and transverse polarization asymmetries of the lepton and their implications. From Eqs. (18)−(20), the distributions of P L , P N and P T with respect to the dimensionless kinematic variableŝ are shown in Figures 4−9, respectively. From the figures, we find that the results of the QCD sum rule and pole models to various polarizations are as follows: (1) they overlap fully for P L ; (2) P N is not sensitive to the models except for the small q 2 region in Λ b → Λµ + µ − ; and (3) the effects of the different QCD models to P T are significant at the large q 2 region. Clearly, P L and P N for the most q 2 region in Λ b → Λl + l − are independent of the QCD models. It is easily seen that outside the resonant states, both polarizations of P L and P N are insensitive to the LD effects. We note that P L for Λ b → Λµ + µ − is close to 1, while that for the tau mode is over 40%, in the most values of q 2 except that around resonant regions. The large asymmetries in Λ b → Λl + l − are good candidates to test the standard model. For P T , since it is proportional to the imaginary parts of the Wilson coefficient products, the LD contributions are important. Note that in the standard model, the effective Wilson coefficients of C ef f 9 contains absorptive parts, while C 7 and C 10 have only real values. From Eq. (20), the part of Im(C ef f 9 C * 10 ) yields a nonzero value of P T , but that of Im(C 7 C * 10 ) vanishes. However, due to the enhanced factor 1/ŝ at smallŝ for the term corresponding to Im(C 7 C * 10 ), one could search for these regions since the contribution from some non-standard CP violation model may not be negligible.
Finally, in Table 2, we list the integrated lepton polarization asymmetries in In the table, the results are calculated in the standard model without LD effects.

Conclusions
We have given a detailed analysis on the dilepton forward-backward and the longitudinal, normal and transverse lepton polarization asymmetries for the decays of Λ b → Λl + l − (l = e, µ, τ ) in the standard model. Based on the HQET, there are only two independent form factors, F 1 and F 2 or F 1 and R, involved in the matrix element of Λ b → Λ.
We have shown that all the asymmetries are related to R and free of the other form factor F 1 . Moreover, we have found that R is always associated with √ r so that by neglecting their contributions, there are only a few percentages lose in the asymmetries. Thus, the asymmetries in the heavy baryonic dilepton decays have a less dependence on the non-perturbative QCD effects . We have also demonstrated that P L (Λ b → Λl + l − ) are QCD model independent quantities. We have pointed out that the FBA for the light lepton mode gets to zero atŝ 0 which is only sensitive to the weak couplings. Finally, since the absolute values of the integrated T-odd observables of the transverse lepton polarizations in Λ b → Λl + l − are less than 10 −2 in the standard model, measuring P T such as in the tau model at a level of 10 −2 would be a clear signal for some new CP violation.