Baryonic rare decays of Lambda(b)->Lambda l(+)l(-)

arXiv:hep-ph/0106193v1 18 Jun 2001

Baryonic Rare Decays of

_Λ

_{→ Λℓ}

ℓ

Chuan-Hung Chen

and C. Q. Geng

a

Department of Physics, National Cheng Kung University

Tainan, Taiwan, Republic of China

b

Department of Physics, National Tsing Hua University

Hsinchu, Taiwan, Republic of China

Abstract

We present a systematic analysis for the rare baryonic exclusive decays of Λb → Λl+l−(l =

e, µ, τ ). We study the differential decay rates and the di-lepton forward-backward, lepton polarization and various CP asymmetries with a new simple set of form factors inspired by the heavy quark effective theory. We show that most of the observables are insensitive to the non-perturbative QCD effects. To illustrate the effect of new physics, we discuss our results in an explicit supersymmetric extension of the standard model, which contains new CP violating phases and therefore induces sizable CP violating asymmetries.

1

Introduction

A priority in current particle physics research is to determine the parameters of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements [1] in the standard model (SM). Due to the CLEO measurement of the radiative b → sγ decay [2], some interest has been focused on the rare decays related to b → sl+_l− _{induced by the flavor changing}

neutral currents (FCNCs). In the SM, these rare decays occur at the loop level and depend on the CKM elements. In the literature, most of studies have been concentrated on the corresponding exclusive rare B-meson decays such as B → K(∗)_l+_l− _{[3]. However,}

these exclusive modes contain several unknown hadronic form factors, which cannot be measured in the present B-meson facilities unlike the kaon cases. Recently, we have examined the exclusive rare baryonic decays of Λb → Λl ¯l (l = ν, e, µ, τ) [4, 5, 6] and

found that some of physical quantities are insensitive to the hadronic uncertainties. In this paper, we give a systematic study on the baryonic decays of Λb → Λl+l−. We

will explore various possible CP even and odd asymmetries to show how the hadronic unknown parameters are factored out in most of cases. To illustrate CP violating effect, we will also discuss an explicit CP violating model with SUSY.

The paper is organized as follows. In Sec. 2, we give the effective Hamiltonian for the decays of Λb → Λl¯l and the most general form factors in the Λb → Λ transition. In

Sec. 3, we derive the general forms of the differential decay rates. In Sec. 4, we study the di-lepton forward-backward, lepton polarization and various CP violating asymmetries. We perform our numerical analysis in Sec. 5. We present our conclusions in Sec. 6.

2

Effective Hamiltonian and form factors

In the SM, the effective Hamiltonian for b → sl+_l− _{is given by}

H = −4G√F 2VtbV ∗ ts 10 X i=1 Ci(µ) Oi(µ) (1)

where the expressions of the renormalized Wilson coefficients Ci(µ) and operators Oi(µ)

can be found in Ref. [7]. From Eq. (1), the free quark decay amplitude is written as Mb → sl+l− ₌ G_√Fαem 2π VtbV ∗ ts " ¯ s C9ef f(µ) γµPL− 2mb q2 C ef f 7 (µ) iσµνqνPR ! b ¯lγµl +¯sC10γµPLb ¯lγµγ5l i (2) with PL(R) = (1 ∓ γ5)/2. We note that in Eq. (2), only the term associated with the

Wilson coefficient C10 is independent of the µ scale. Besides the short-distance (SD)

contributions, the long-distance (LD) ones such as that from the c¯c resonant states of Ψ, Ψ′_{...etc are also important for the decay rate. It is known that for the LD effects}

in the B-meson decays [8, 9, 10, 11, 12, 13], both the factorization assumption (FA) and the vector meson dominance (VMD) approximation have been used. In baryonic decays, we assume that the parametrization of LD contributions 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 coefficient. In this paper as a more complete analysis we also include the LD contributions to the decay of b → sγ, induced by the nonfactorizable

effects [14, 15]. The effective Wilson coefficients of C₉ef f and C₇ef f can be expressed as the standard form C9ef f(µ) = C9(µ) + Y (z, s′) , (3) C7ef f(µ) = C7(µ) + C7′  µ, q2 , (4) where Y (z, s′_{) =}  h(z, s′) + 3 α2 em X j=Ψ,Ψ′ kjπΓ (j → l +_l−_{) M} j q2_{− M}2 j + iMjΓj   −1 2h(1, s ′_{) (4C} 3+ 4C4+ 3C5+ C6) − 1 2h(0, s ′_{) (C} 3 + 3C4) , C′ 7  µ, q2 = C′ b→sγ(µ) + ω  h(z, s′) + 3 α2 em X j=Ψ,Ψ′ kjπΓ (j → l +_l−_{) M} j q2_{− M}2 j + iMjΓj   , (5) with h(z, s′_{) = −}8 9ln z + 8 27+ 4 9x − 2 9(2 + x) |1 − x| 1/2 ×    ln  √ 1−x+1 √ 1−x−1   − iπ for x ≡ 4z 2_/s′ _{< 1} 2 arctan_√1 x−1 for x ≡ 4z 2_/s′ _{> 1} , C′ b→sγ = iαs 2 9η 14/23_(G 1(xt) − 0.1687) − 0.03C2(µ)  , G1(x) = x (x2_{− 5x − 2)} 8 (x − 1)3 + 3x2_{ln x} 4 (x − 1)4 . (6)

Here Y (z, s′_{) combines the one-loop matrix elements and the LD contributions of operators}

O1- O6, C_b→sγ′ is the absorptive part of b → sγ [16] with neglecting the small contribution

from VubVus∗, z = mc/mb, s′ = q2/m2b, η = αs(mW) /αs(µ), xt = mt2/m2W, Mj (Γj) are

the masses (widths) of intermediate states, |ω| ≤ 0.15 describing the nonfactorizable contributions to b → sγ decay at q2 _{= 0 [14, 15], and the factors k}

j are phenomenological

parameters for compensating the approximations of the FA and VMD and reproducing the correct branching ratios of B(Λb → ΛJ/Ψ → Λl+l−) = B(Λb → ΛJ/Ψ) × B(J/Ψ →

l+_l−_{) when we study the Λ}

b decays. We note that by taking kΨ ≃ −1/(3C1+ C2) and

B(Λb → ΛJ/Ψ) = (4.7 ± 2.8) × 10−4, the kj factors in the Λb case are almost the same as

that in the B-meson one [5]. The Wilson coefficients (WCs) at the scale of µ ∼ mb ∼ 4.8

GeV are shown in Table 1.

Table 1: Wilson coefficients for mt= 170 GeV, µ = 4.8 GeV.

W C C1 C2 C3 C4 C5

−0.226 1.096 0.01 _−0.024 0.007

W C C6 C7 C8 C9 C10

Using the form factors given in Appendix A, we write the amplitude of Λb → Λl+l− as MΛb → Λl+l−  = G_√Fαem 2π VtbV ∗ ts n H₁µLµ+ H2µL5µ o (7) where H1µ = ¯Λγµ(A1PR+ B1PL) Λb+ ¯Λiσµνqν(A2PR+ B2PL) Λb, H₂µ = ¯Λγµ(D1PR+ E1PL) Λb+ ¯Λiσµνqν(D2PR+ E2PL) Λb (8) +qµΛ (D¯ 3PR+ E3PL) Λb, Lµ = ¯lγµl , L5_µ = ¯lγµγ5l (9) with Ai = C9ef f fi− gi 2 − 2mb q2 C ef f 7 fT i + gTi 2 , Bi = C9ef f fi+ gi 2 − 2mb q2 C ef f 7 fT i − gTi 2 , Di = C10 fi− gi 2 , Ei = C10 fi+ gi 2 . (10) and i = 1, 2, 3.

The processes for the heavy to light baryonic decays such as those with Λb → Λ have

been studied based on the heavy quark effective theory (HQET) in Ref. [17] and it is found that hΛ(pΛ)| ¯sΓb |Λb(pΛb)i = ¯uΛ  F1(q2)+ 6 vF2(q2)  ΓuΛb (11)

where Γ denotes the Dirac matrix, v = pΛb/MΛb is the four-velocity of Λb, q = pΛb − pΛ

is the momentum transfer, and F1,2 are the form factors. Clearly, there are only two

independent form factors F1,2 in the HQET. Comparing with the general forms of the

form factors in Appendix A, we get the relations among the form factors as follows: g1 = f1 = f2T = gT2 = F1+√rF2, g2 = f2 = g3 = f3 = gTV = fTV = F2 MΛb , g_TS = f_TS = 0, gT₁ = f₁T = F2 MΛb q2, gT₃ = F2 MΛb (MΛb + MΛ) , f T 3 = − F2 MΛb (MΛb − MΛ) , (12) where r = M2

Λ/MΛ2b. From the CLEO result of R = −0.25±0.14±0.08 [18], we know that

they are large, whereas all the others are small since they are related to the small form factor F2. Furthermore, from Eq. (10), we find that {fT} and {gT} are associated with

C7which is about one order of the magnitude smaller than C9and C10so that their effects

to the deviation of the results in the HQET are small. Hence, with the information of the HQET, we can make a good approximation for the general form factors of transition matrix elements given in Eqs. (7) and (10). Altogether, we have the following relations:

¯ f ≡ f1+ g1 2 , fT 2 + g2T f1+ g1 ≃ 1 f1 − g1 f1+ g1 ≃ δ, g2 f2 ≃ gT 1 fT 1 ≃ gT 2 fT 2 ≃ 1, fT 1 + g1T f1+ g1 1 q2 ≃ f2+ g2 f1+ g1 . (13)

In the HQET, it is easy to show that δ = 0 , ρ ≡ MΛb f2+ g2 f1+ g1 ! = F2 F1+√rF2 . (14)

3

Differential decays rates

In this section we first present the formulas by including the lepton mass for the double differential decay rates with respect to the angle of the lepton and the invariant mass of the di-lepton. In the following we only show the results of the SM with the form factors in Eq. (13). The general ones with including right-handed couplings are presented in Appendix B.

Introducing dimensionless variables of t = pΛb · pΛ/M

2 Λb, r = M 2 Λ/MΛ2b, ˆml = ml/MΛb, ˆ mb = mb/MΛb, and s = q 2_/M2

Λb, the double partial differential decay rates for Λb → Λ l

+_l− (l = e, µ, τ ) can be written as d2_Γ dsdˆz = G2 Fαem2 λ2t 768π5 M 5 Λb q φ (s) v u u t1 −4m 2 l q2 f¯ 2_R Λb(s, ˆz) , (15) where RΛb(s, ˆz) = I0(s, ˆz) + ˆzI1(s, ˆz) + ˆz 2_I 2(s, ˆz) (16) and I0(s, ˆz) = −6√rs " −2 ˆmbρ 1 + 2 m2 l q2 !

ReCeff₉ Ceff∗₇ +δ 1 + 2m 2 l q2 !   C ef f 9    2 + _{1 − 6}m 2 l q2 ! |C10|2 !# +3 4  (1 − r)2− s2  (2 ˆmbρ)2   C ef f 7    2 + C ef f 9    2 + |C10|2  +6 ˆm2_lt  (2 ˆmbρ)2   C ef f 7    2 + C ef f 9    2 − |C10|2 

+6√_{r (1 − t)} ( 4 1 + 2m 2 l q2 ! ˆ m2_bρ C ef f 7    2 +ρs " 1 + 2m 2 l q2 !   C ef f 9    2 + _{1 − 2}m 2 l q2 ! |C10|2 #) +12 1 + 2m 2 l q2 ! ˆ mb(t − r) 

1 + sρ2ReCeff₉ Ceff∗₇ +12 1 + 2m 2 l q2 ! ˆ

mb√rsρReCeff9 Ceff∗7

−6  s (1 − t) (t − r) − 1 8  (1 − r)2− s2  × " 4 ˆm2b s   C ef f 7    2 + sρ2   C ef f 9    2 + |C10|2 # −6 ˆm2_{l (2r − (1 + r) t)}   2 ˆmb s !2   C ef f 7    2 + ρ2   C ef f 9    2 − |C10|2   , (17) I1(s, ˆz) = 3 v u u t1 − 4m2 l q2 φ (s) n sh_{1 − 2}√_{rρ − (1 − r) ρ}2i ReCeff 9 C∗10 +2 ˆmb  1 − sρ2ReCeff₇ C∗ 10 o , (18) I2(s, ˆz) = − 3 4φ (s) 1 − 4 m2 l q2 !_ (2 ˆmbρ)2   C ef f 7    2 + C ef f 9    2 + |C10|2  +3 4φ (s) 1 − 4 m2 l q2 ! " 4 ˆm2 b s   C ef f 7    2 + sρ2   C ef f 9    2 + |C10|2 # , (19)

with ˆz = ˆpB · ˆpl+ being the angle between the momenta of Λ_b and l+ in the di-lepton

invariant mass frame and φ (s) = (1 − r)2 − 2s (1 + r) + s2_{. Here, for the simplicity, we}

have not displayed the dependence of the µ scale in effective Wilson coefficients. We note that the main nonperturbative QCD effect from ¯f has been factored out in Eq. (15). The function RΛb(s, ˆz) is only related to the two parameters of δ and ρ which become one in

the HQET. Since ρ is the ratio of form factors and insensitive to the QCD models, the QCD effects in the baryonic di-lepton decays are clearly less significant. Therefore, these decay modes are good physical observable to test the SM.

After integrating the angular dependence, the invariant mass distributions as function of s are given by dΓ (Λb → Λl+l−) ds = G2 Fα2emλ2t 384π5 M 5 Λb q φ (s) v u u t1 − 4m 2 l q2 f¯ 2_R Λb(s) , (20) where RΛb(s) = Γ1(s) + Γ2(s) + Γ3(s) (21) with Γ1(s) = −6√rs " −2 ˆmbρ 1 + 2 m2 l q2 !

+δ 1 + 2m 2 l q2 !   C ef f 9    2 + _{1 − 6}m 2 l q2 ! |C10|2 !# + " −2r 1 + 2m 2 l q2 ! − 4t2 1 − m 2 l q2 ! + 3 (1 + r) t # ×  (2 ˆmbρ)2   C ef f 7    2 + C ef f 9    2 + |C10|2  +6 ˆm2_lt  (2 ˆmbρ)2   C ef f 7    2 + C ef f 9    2 − |C10|2  , (22) Γ2(s) = 6√r (1 − t) ( 4 1 + 2m 2 l q2 ! ˆ m2_bρ C ef f 7    2 +ρs " 1 + 2m 2 l q2 !   C ef f 9    2 + _{1 − 2}m 2 l q2 ! |C10|2 #) +12 1 + 2m 2 l q2 ! ˆ mb(t − r) 

1 + sρ2ReCeff₉ Ceff∗₇ , (23) Γ3(s) = 12 1 + 2 m2 l q2 ! ˆ

mb√rsρReCeff9 Ceff∗7

− " 2t2 1 + 2m 2 l q2 ! + 4r _{1 −} m 2 l q2 ! − 3 (1 + r) t # × " 4 ˆm2 b s |C7| 2 + sρ2   C ef f 9    2 + |C10|2 # −6 ˆm2_{l (2r − (1 + r) t)}   2 ˆmb s !2   C eff 7    2 + ρ2   C eff 9    2 − |C10|2    . (24) The limits for s are given by

4 ˆm2_{l ≤ s ≤}_{1 −}√r2. (25)

From Eqs. (22)-(24), we see that ρ appears either as √rρ or ρ2 _{which is small since}

r ∼ 0.04 and |ρ| ∼ 0.25.

4

Lepton and CP asymmetries

4.1

Forward-backward asymmetries

The differential and normalized forward-backward asymmetries (FBAs) for the decays of Λb → Λl+l− as a function of s are defined by

dAF B(s) ds = " Z 1 0 dˆz d2_{Γ (s, ˆz)} dsdˆz − Z 0 −1 dˆz d 2_{Γ (s, ˆz)} dsdˆz # (26) and AF B(s) = 1 dΓ (s) /ds " Z 1 0 dˆz d2_{Γ (s, ˆz)} dsdˆz − Z 0 −1 dˆzd 2_{Γ (s, ˆz)} dsdˆz # , (27)

respectively. Explicitly, using Eq. (15), we obtain dAF B(s) ds = G2 Fα2emλ2t 28_π5 M 5 Λbφ (s) 1 − 4 ˆ m2 l ˆ s ! ¯ f2RF B(s) (28) and AF B(s) = 3 2 q φ (s) s 1 −4 ˆm 2 l s RF B(s) RΛb(s) (29) where RF B(s) = s h 1 − 2√rρ − (1 − r) ρ2iReCeff₉ C∗ 10 +2 ˆmb  1 − sρ2ReCeff₇ C∗ 10. (30)

It is known that the FBA is a parity-odd but CP-even observable, which depends on the chirality of the leptonic and hadronic currents. In order to obtain one power of ˆz dependence, the related differential decay rate should be associated with T rLµL5ν. This

explains why the FBAs depend on ReC9ef fC10∗ and ReC ef f

7 C10∗ . However, unlike that in the

decays of B → Kl+_l− _{where the FBAs are always zero since they only involve vector and}

tensor types of currents, the transition matrix elements in the baryonic decays preserve the chirality of free quark interaction.

Similar to the B-meson decays [3, 19] the FBA in Eq. (29) vanishes at s0 which satisfies

with the relation

ReCeff₉ C∗ 10= − 2 ˆmb s0 1 − s0ρ2 1 − 2√rρ − (1 − r) ρ2ReC eff 7 C∗10. (31)

We will see later that the vanishing point is only sensitive to the effects of weak interaction.

4.2

Lepton polarization asymmetries

To display the spin effects of the lepton, we choose the four-spin vector of l+ _{in terms of}

a unit vector, ˆξ, along the spin of l+ _{in its rest frame, as}

s0₊ = ~p+· ˆξ ml , ~s+ = ˆξ + s0 + El+ + m_l ~p+, (32)

and the unit vectors along the longitudinal and transverse components of the l+

polariza-tion to be ˆ eL = ~p+ |~p+| , ˆ eT = ~pΛ× ~p+ |~pΛ× ~p+| , ˆ eN = ˆeL× ˆeT , (33) respectively.

Defining the longitudinal and transverse l+ _{polarization asymmetries by} Pi(ˆs) = dΓeˆi · ˆξ = 1  − dΓeˆi· ˆξ = −1  dΓeˆi· ˆξ = 1  + dΓeˆi· ˆξ = −1 , (34)

with i = L and T , we find that

PL = v u u t1 − 4m 2 l q2 RL(s) RΛb(s) , (35) PT = 3 4π ˆml v u u t1 − 4m2 l q2 q sφ (s)RT(s) RΛb(s) , (36) where RL = −ReCeff9 C∗10 h (1 − r)2+ s (1 + r) − 2s2+ 6√_{rρs (1 − r + s)} +ρ2s_{2 (1 − r)}2_{− s (1 + r) − s}2i −6 ˆmbReCeff7 C∗10 h (1 − r − s)1 + ρ2s+ 4√rρsi, (37) RT = h 1 − 2√rρ − ρ2(1 − r)iImCeff₉ C∗ 10 +2 ˆmb s  1 − ρ2sImCeff7 C∗10. (38)

Here we do not discuss the normal polarization (PN) because the nonperturbative effects

from the form factors are large at the small s region and moreover, the dependence of Wilson coefficients is similar to the invariant mass distribution [6]. We note that the longitudinal lepton polarization of PL in Eq. (35) is also a parity-odd and CP-even

observable just like the FBA, whereas PT in Eq. (36) a T-odd one which is related to the

triple correlation of ~s+· (~pΛ× ~p+). In general, PT can be induced without CP violation as

the cases in B-meson [20] and kaon [21] decays. However, we expect that they are small. Moreover, such effects can be extracted away while we consider the difference between the particle and anti-particle as discussed in the next section.

4.3

CP asymmetries

In this subsection, we define the following interesting direct CP asymmetries (CPAs) by ∆Γ = dΓ − d¯Γ dΓ + d¯Γ, (39) ∆F B = dΓF B− d¯ΓF B dΓ + d¯Γ , (40) ∆Pi = dΓ_{ξ · ~e}~ i  − d¯Γ~ξ · ~ei  dΓ + d¯Γ , i = L, T (41)

where we have used dΓ + d¯Γ as the normalization. The above four CPAs are CP-odd quantities and they are CP violating observables. For ∆Γ,F B,PL in Eqs. (39)-(41), to

display the difference of the physical observable between the particle and anti-particle, it is necessary to have the strong and weak phases simultaneously in the processes. In the decays of b → sl+_l− _{(l = e, µ, τ ), the strong phases are generated by the absorptive}

parts of one-loop matrix elements in operators O1 ∼ O6 and LD contributions. However,

since PT is a T -odd observable and only related to the imaginary couplings, even without

strong phases, we still can have nonzero values of ∆T(Λb → Λl+l−). For b → sl+l− and

¯b → ¯sl−_l+ _{decays the Wilson coefficients C}ef f

9 (µ) and C ef f

7 (µ) in Eqs. (3) and (4) can be

rewritten as C9ef f(µ) = C90(µ) + iC9abs(µ) , ¯ C9ef f(µ) = C90∗(µ) + iC9abs(µ) , C7ef f(µ) = C70(µ) + iC7abs(µ) , ¯ C7ef f(µ) = C70∗(µ) + iC7abs(µ) , (42) with C₉0(µ) = C9(µ) + ReY (z, s′) , C0 7(µ) = C7(µ) + ReC7′(µ, q2) , C₉abs(µ) = ImY (z, s′_{) ,} C₇abs(µ) = ImC′ 7(µ, q2) , (43)

where we have assumed that the strong phases are all from the SM and there are no weak phases in absorptive parts. We note that there is no strong phase in C10.

According to Eqs. (20), (29), (35), and (36), the CP asymmetries are all related to the following combinations:

ReC9ef fC7ef f ∗− Re ¯C9ef fC¯7ef f ∗ = 2C9absImC70+ 2C7absImC90,

ReC7,9ef fC10∗ − Re ¯C ef f 7,9 C¯10∗ = 2C7,9absImC10, ImC_7,9ef fC∗ 10− Im ef f

7,9C¯10∗ = 2ImC7,90 C10∗ + 2C7,9absImC10,

  C ef f 7,9    2 − C¯ ef f 7,9    2 = 4C_7,9absImC_7,90 . (44) Explicitly, the CP asymmetries in Eqs. (39), (40), and (41) are found to be

∆Γ = 2 RΛb(s) ( 6 ˆmb 1 + 2 m2 l q2 ! h

2√_{rρs + (t − r)}1 + sρ2i hC₉absImC0₇ + Cabs₇ ImC0₉i + " −2r 1 + 2m 2 l q2 ! − 4t2 _{1 −}m2l q2 ! + 3 (1 + r) t + 6 ˆm2 lt # ×h4 ˆm2

bρ2C7absImC07+ Cabs9 ImC09

i + " −2t2 1 + 2m 2 l q2 ! − 4r 1 −m 2 l q2 ! + 3 (1 + r) t − 6mˆ 2 l s (2r − t − tr) # × " 4 ˆm2 b s C abs

7 ImC07 + sρ2Cabs9 ImC09

# + 6√_{rρ (1 − t)} 1 + 2m 2 l q2 !

×h4 ˆm2_bC₇absImC₇0+ sC₉absImC₉0i_{− 6}√rsδ 1 + 2m

2 l q2 ! C₉absImC₉0 ) , (45)

∆F B = 3 2RΛb(s) v u u t1 − 4m2 l q2 φ(s)ImC10 h s_{1 − 2}√_{rρ − (1 − r) ρ}2C9abs + 2 ˆmb  1 − sρ2C₇absi , (46) ∆PL = − 1 RΛb(s) v u u t1 −4m 2 l q2 ImC10 n C₉absh6√_{rρs (1 − r + s) +}1 + 2sρ2_{(1 − r)}2 +s_{1 − sρ}2_{(1 + r) − s}22 + sρ2i +6 ˆmbC7abs h (1 − r − s)1 + sρ2+ 4√rρsio , (47) ∆PT = 3π ˆml 4RΛb(s) v u u t1 − 4m 2 l q2 q sφ (s)nImC₉0C∗ 10+ C9absImC10  ×1 − 2√rρ − (s + 2t − 2r) ρ2 +2 ˆmb s  1 − sρ2 ImC₇0C∗ 10+ C7absImC10  ) . (48)

As seen from the above equations, ∆Γ is related to ImC7 and ImC9, while ∆F B, ∆PL and

∆PT depend on ImC10. Moreover, for small values of C

abs

9 ImC10 and C7absImC10, ∆PT

would still be sizable because ImC0

9C10∗ or ImC70C10∗ would be large.

5

Numerical analysis

In our numerical calculations, the Wilson coefficients are evaluated at the scale µ ≃ mb

and the other parameters are listed in Table 1 of Ref. [5]. From Eq. (13), we know that the main effects to the deviation of the HQET are from δ. By using a proper nonzero value of δ, we will see later that the deviations of the decay branching ratios of Λb → Λl+l−

are only a few percent. Since there is no complete calculation for the form factors of the Λb → Λ transition in the literature, we use the form factors derived from QCD sum rule

under the assumption of the HQET, given by Fi(q2) =

Fi(0)

1 + aq2_{+ bq}4 , (49)

with the parameters shown in Table 1 of Ref. [6]. In order to illustrate the contributions of new physics, we adopt the results of the generic supersymmetric extension of the SM [22] in which CSU SY 7 = −1.75 (δu23)LL− 0.25 (δ23u )LR− 10.3  δd 23  LR, C₉SU SY = 0.82 (δu₂₃)_LR, C₁₀SU SY _{= −9.37 (δ}u₂₃)_LR+ 1.4 (δ₂₃u )_LR(δ₃₃u )_RL+ 2.7 (δ₂₃u )_LL, (50) and take the following values instead of scanning the whole allowed parameter space:

(δu 23)LL ∼ 0.1 , (δ₃₃u )_{RL ∼ 0.65 ,}  δ₂₃d  LR ∼ 3 × 10 −2_ei2π 5 , (δ₂₃u )_{LR ∼ −0.8e}iπ4 , (51)

where (δ_ijq)AB (i, j = 1, 2, 3 and A, B = L, R) denote the parameters in the mass insertion

method, which describe the effects of the flavour violation. The set of the parameters in Eq. (51) satisfies with the constraint from B → Xsγ on C7 = C7SM+ C7SU SY [22]. Hence,

the numerical values of the SUSY contributions to the relevant Wilson coefficients are as follows:

Re C₇SU SY _{≃ 0.06 ,} Im C₇SU SY _{≃ −0.29 ,} Re C₉SU SY _{≃ −0.46 ,} Im C₉SU SY _{≃ −0.46 ,}

Re C₁₀SU SY _{≃ 4.78 ,} Im C₁₀SU SY _{≃ 4.50 .} (52) We note that the contributions of the minimal supersymmetric standard model (MSSM) to b → sl+_l− _{can be found in Refs. [23] and [24].}

To show the typical values of various asymmetries, we define the integrated quantities as ¯ Q = Z smax smin Q(s)ds (53)

where Q denote the physical observables with smin = 4 ˆml and smax= (1 −√r)2.

5.1

Decay rates and invariant mass distributions

We now discuss the influences of δ, ρ, and ω on the branching ratios (BRs) of Λb → Λl+l−

decays in detail. The effects of kj for compensating the assumption of the FA and VMD

have been analyzed in [5]. In Table 2, we show the BRs by choosing different sets of parameters. Our results are given as follows:

Table 2: BRs (in the unit of 10−6_{) for various parameters with ω = 0 and neglecting LD}

effects. Parameter Λb → Λe+e− Λb → Λµ+µ− Λb→ Λτ+τ− HQET 2.23 2.08 _{1.79 × 10}−1 δ = 0.05 2.36 2.21 _{1.86 × 10}−1 δ = −0.05 2.09 1.96 _{1.71 × 10}−1 ρ = 0, δ = 0 2.52 2.38 _{2.66 × 10}−1 C7 = 0, δ = 0 2.36 2.34 2.23 × 10−1 C7 = −C7SM, δ = 0 3.34 3.19 2.76 × 10−1

1. By taking |δ| = 0.05 which means 10% away from that in to the HQET, we clearly see that the deviations of the BRs are only 4 − 6%. It is a good approximation to neglect the explicit δ term in Eqs. (17), (22) and (45). Hence, ¯f = (f1+ g1) /2,

which also owns the δ effect, is the main nonperturbative part.

2. If ρ = 0, the effects are about 10% for e and µ modes but 48% for τ one.

3. If one neglects the contribution from C7, the influences on B(Λb → Λl+l−) for e,

µ and τ modes are about 5%, 12% and 24%, respectively. However, taking the magnitude of C7 is the same as the SM but with an opposite sign, the deviations

The contributions of the parameter ω to the BRs of Λb → Λl+l− are listed in Table

3 and the invariant mass distributions are shown in Figure 1. From Table 3, it is clear that the nonfactorizable effects are small on the BRs. However, those directly related to ω effects such as PT and CP asymmetries will have large influences.

Table 3: BRs (in the unit of 10−6_{) without LD effects with different values of ω}

Mode Λb → Λe+e− Λb → Λµ+µ− Λb → Λτ+τ−

ω = 0.15 2.24 2.12 _{1.89 × 10}−1

ω = 0. 2.23 2.08 _{1.79 × 10}−1

ω = −0.15 2.25 2.06 _{1.71 × 10}−1

As for the new physics contributions, using the values of the SUSY model in Eq. (52), we show the results in Table 4. Although the deviations of the BRs to the SM are not significant, they have a large effect on the lepton and CP asymmetries which will be shown next.

Table 4: BRs (in unit of 10−6_{) in the generic SUSY model.}

Model Λb → Λe+e− Λb → Λµ+µ− Λb → Λτ+τ−

SUSY 2.47 2.24 _{1.79 × 10}−1

5.2

Forward-backward and lepton polarization asymmetries

From Eq. (14) and R = F2/F1 ≃ −0.25 in the HQET, we have that ρ ≃ −0.26. We note

that ρ is defined by the ratio of the form factors and it is expected to be insensitive to the QCD models. With smax≃ 0.64, we obtain smaxρ2 ≃ 0.04, (1 − r)ρ2 ≃ 0.06 and 2√rρ

≃ 0.2. Using these values, one can simplify Eqs. (30) and (31) to RF B(s) ≃ s  1 − 2√rρReCeff₉ C∗ 10+ 2 ˆmbReCeff7 C∗10 (54) and ReCeff₉ C∗ 10≃ − 2 ˆmb s0(1 − 2√rρ) ReCeff₇ C∗ 10, (55)

respectively. It is easy to see that s0 is only sensitive to the Wilson coefficients. The result

is similar to the case in B → K∗_l+_l− _{decay [3, 19] where the approximation of the large}

energy effective theory (LEET) [25] is used. As for the lepton polarization asymmetries, with the same approximation, Eqs. (37) and (38) can also be reduced to

RL ≃ −ReCeff9 C∗10 h 1 + s − 2s2+ 6√rρs (1 + s)i −6 ˆmbReCeff7 C∗10 h 1 − s + 4√rρsi, (56) RT ≃  1 − 2√rρImCeff 9 C∗10+ 2 ˆmb s ImC eff 7 C∗10, (57)

respectively. Hence, the lepton asymmetries are all more sensitive to the Wilson coeffi-cients than the nonperturbative QCD effects.

It is worth to mention that the effects of ω, introduced for the LD contributions to b → sγ and absorbed to C7ef f, will change ReCeff7 in the SM such that s0 is also shifted.

Therefore, in terms of s0, we can also theoretically determine ω by comparing the result

with that of ω = 0. Another interesting quantity is T-odd observable of PT which is

proportional to C10ImCeff7 in the SM. Due to the enhancement of C10, a nonzero value of

ω will modify PT enormously. As for the other asymmetries, the effects are insignificant.

The estimations of integrated lepton asymmetries with different values of ω in the SM are displayed in Table 5 and the corresponding distributions are shown in Figures 2 − 4.

Table 5: Integrated lepton asymmetries in the SM without LD effects.

Parameter Mode 102_A¯ F B 102P¯L 102P¯T ω = 0.15 Λb → Λµ+µ− −14.37 59.50 0.11 Λb → Λτ+τ− −3.98 10.70 0.53 ω = 0. Λb → Λµ+µ− −13.38 58.30 0.07 Λb → Λτ+τ− −3.99 10.84 0.39 ω = −0.15 Λb → Λµ+µ− −12.24 56.70 0.04 Λb → Λτ+τ− −4.00 10.94 0.23

To illustrate the new physics effects, the integrated lepton asymmetries in the generic SUSY model with ω are listed in Table 6 and their distributions as a function of q2_/M

Λb

are shown in Figures 5 − 7. From the figures, we see that SUSY effects make the shapes of lepton asymmetries quite differ from that in the SM. We summary the results as follows: 1. Since the SD contributions to C9C10∗ and ReC7C10∗ are −1.40 and −1.35, respectively,

which violate the condition in Eq. (55), the vanishing point is removed.

2. Due to the factor of ˆmb/s, from Figure 7, we see that ImC7ef fC10∗ has a large effect

on PT in the small s region.

3. In the SUSY model, PT could reach 1% and 10% for the light lepton and τ modes,

which are only 0.2% and 3% at most in the SM, respectively.

Table 6: Integrated lepton asymmetries in the generic SUSY model with ω = 0.

Mode 102_A¯

F B 102P¯L 102P¯T

Λb → Λµ+µ− −10.53 24.46 −0.57

Table 7: CP asymmetries in the generic SUSY model for different values of ω Parameter Mode 102_∆¯ Γ 102∆¯F BA 102∆¯PL 10 2_∆¯ PT ω = 0.15 Λb → Λµ+µ− 2.05 −2.62 6.48 −0.53 Λb → Λτ+τ− 1.83 −0.79 1.94 −2.01 ω = 0. Λb → Λµ+µ− 1.59 −1.89 5.00 −0.47 Λb → Λτ+τ− 1.38 −0.59 1.53 −2.21 ω = −0.15 Λb → Λµ+µ− 1.05 −1.06 3.34 −0.40 Λb → Λτ+τ− 0.89 −0.37 1.06 −2.41

5.3

CP asymmetries

In the SM, for b → sl+_l−_{, the relevant CKM matrix element is V}

tbVts∗ which is real under

the Wolfenstein’s parametrization. Nonzero CPAs will indicate clearly the existence of new physics. We remark that the CPAs can be in fact induced by the complex CKM matrix element VubVus∗ which is also the source of the direct CPA in B → Xsγ in the SM.

However, we expect that such effects to the CPAs in b → sl+_l− _{are smaller than that in}

B → Xsγ where the CPA is less than 1%. The main reason for the smallness is because

of the presences of C9 and C10 contributions to the rates of b → sl+l−, which are absent

in B → Xsγ.

With the values in Eq. (52), the averaged CPAs in the generic SUSY model for Λb → Λl+l− are listed in Table 7 and their distributions as a function of s = q2/MΛ2b are

shown in Figures 8 − 11. The results are given as follows:

1. From Eqs. (45) and (48), we see that the terms corresponding to Cabs

7 ImC70 and

ImC0

7C10∗ + C7absImC10 are associated with a factor of ˆmb/s. If sizable imaginary

parts exist, in the small s region the distributions will be significant. Due to this reason, in Figure 8a one finds that ∆Γ(s) for Λb → Λl+l− (l = e, µ) increase as s

decreases. On the other hand, if the term with ˆmb/s in Eq. (48) is dropped, the

distributions of ∆PT(s) for e and µ modes do not contain zero value. We note that

with the values in Eq. (52), the main effect on ∆Γ(s) in the small s region is from

C′ b→sγ.

2. ∆PL(s) for all lepton channels and ∆PT(s) for the τ one could be over 10%, while

the remaining CP asymmetries are at the level of a few percent. We remark that if we can scan all the allowed SUSY parameters, the asymmetries except ∆PT(s) for

lighter lepton modes would reach up 10%.

3. It is known that ∆PT(s) is a T-odd observable and the other CPAs belong to the

direct CP violation which needs absorptive parts in the processes. This is the reason why the distributions of ∆Γ(s), ∆F BA(s) and ∆PL(s) around the RE region have

the similar shapes but are different from that of ∆PT(s). Moreover, all the direct

CPAs are sensitive to ω unlike the cases of the CP conserving lepton asymmetries discussed in Sec. 5.2.

6

Conclusions

We have given a systematic study on the rare baryonic decays of Λb → Λl+l−(l = e, µ, τ ).

For the Λb → Λ transition, we have related all the form factors with F1 and F2, and we

have found that δ = 0 and ρ ≃ R ≡ F2/F1, in the limit of the HQET. Inspired by the

HQEF, we have presented the differential decay rates and the di-lepton forward-backward, lepton polarization and four possible CP violating asymmetries in terms of the parameters

¯

f , δ and ρ. We have shown that the non-factorizable effects for the BRs and CP-even lepton asymmetries are small but large for PT and the direct CPAs. We have also

demon-strated that most of the observables such as AF B, PL,T and ∆α (α = Γ, F B, PL andPT).

are insensitive to the non-perturbative QCD effects. We have illustrated our results in the specific CP violating SUSY model. We have found that all the direct CP violating asymmetries are in the level of 1 − 10%. To measure these asymmetries at the nσ level, for example, in the tau mode, at least 0.5n2 _{× (10}9 _{− 10}10_{) Λ}

b decays are required. It

could be done in the second generation B-physics experiments, such as LHCb, ATLAS, and CMS at the LHC, and BTeV at the Tevatron, which produce ∼ 1012_{b¯b pairs per year}

[26] Finally we remark that measuring these CPAs at a level of 10−2 _{is a clear indication}

of new CP violation mechanism beyond the SM. Acknowledgments

This work was supported in part by the National Science Council of the Republic of China under Contract Nos. NSC-89-2112-M-007-054 and NSC-89-2112-M-006-033 and the National Center for Theoretical Science.

Appendix

A. Form factors and decay amplitudes

For the exclusive decays involving Λb(pΛb) → Λ(pΛ), the transition form factors can

be parametrized generally as follows:

hΛ| ¯s γµ b |Λbi = f1u¯ΛγµuΛb+ f2u¯Λiσµν q ν_u Λb + f3qµu¯ΛuΛb, (58) hΛ| ¯s γµγ5 b |Λbi = g1u¯Λγµγ5uΛb+ g2u¯Λiσµν q ν_γ 5uΛb+ g3qµu¯Λγ5uΛb, (59) hΛ| ¯siσµνb |Λbi = fTu¯ΛiσµνuΛb + f V T u¯Λ(γµqν− γνqµ) uΛb +f_TS(Pµqν− Pνqµ) ¯uΛuΛb, (60) hΛ| ¯siσµνγ5b |Λbi = gTu¯Λiσµνγ5uΛb + g V Tu¯Λ(γµqν − γνqµ) γ5uΛb +gTS(Pµqν − Pνqµ) ¯uΛγ5uΛb, (61)

where P = pΛb+ pΛ , q = pΛb− pΛ and form factors, {fi} and {gi}, are all functions of

q2_{. Using the equations of the motion, we have}

(MΛ+ MΛb) ¯uΛγµuΛb = (pΛb + pΛ)_µu¯ΛuΛb + i¯uΛσµνq ν_u Λb, (62) (MΛ− MΛb) ¯uΛγµγ5uΛb = (pΛb + pΛ)_µu¯Λγ5uΛb + i¯uΛσµνq ν_γ 5uΛb. (63)

The form factors for dipole operators are derived as hΛ| ¯siσµνqνb |Λbi = f1Tu¯ΛγµuΛb+ f T 2 u¯ΛiσµνqνuΛb + f T 3 qµu¯ΛuΛb, (64) hΛ| ¯siσµνqνγ5b |Λbi = g1Tu¯Λγµγ5uΛb + g T 2u¯Λiσµνqνγ5uΛb+ g T 3qµu¯Λγ5uΛb. (65) with f₂T = fT − fTSq2, f₁T = hf_TV + f_TS(MΛ+ MΛb) i q2, f₁T _{= −} q 2 (MΛb− MΛ) f₃T , g₂T = gT − gTSq2, g₁T = hg_TV + gS_T (MΛ− MΛb) i q2, g₁T = q 2 (MΛb+ MΛ) gT₃ . (66)

We now give the most general formulas by including the right-handed coupling in the effective Hamiltonian with a complete set of form factors. The free quark decay amplitudes for b → sl+_l− _{are given by}

Hb → sl+l− ₌ G_√Fαem 2π VtbV ∗ ts h ¯ sγµC₉LPL+ C9RPR  b ¯lγµl +¯sγµC₁₀LPL+ C10RPR  b ¯lγµγ5l −2m_q2b¯siσµνq ν C₇LPR+ C7RPL  b ¯lγµl # (67)

where CL

i and CiR(i = 7, 9, 10) denote the effective Wilson coefficients of left- and

right-handed couplings, respectively. With the most general form factors in Eqs. (58), (59), (64) and (65), and the effective Hamiltonian in Eq. (67), the transition matrix elements for the decays of Λb → Λl+l− are expressed as

MΛb → Λl+l−  = GF√αem 2 VtbV ∗ ts nh ¯ Λγµ(A1PR+ B1PL) Λb +¯Λiσµνqν(A2PR+ B2PL) Λb i ¯lγµl +hΛγ¯ µ(D1PR+ E1PL) Λb+ ¯Λiσµνqν(D2PR+ E2PL) Λb +qµΛ (D¯ 3PR+ E3PL) Λb i ¯lγµγ5l o (68) where Ai = C9R fi+ gi 2 − 2mb q2 C R 7 fT i − giT 2 + C L 9 fi− gi 2 − 2mb q2 C L 7 fT i + giT 2 , Bi = C9L fi+ gi 2 − 2mb q2 C L 7 fT i − giT 2 + C R 9 fi− gi 2 − 2mb q2 C R 7 fT i + giT 2 , Di = C10R fi+ gi 2 + C L 10 fi− gi 2 , Ei = C10L fi+ gi 2 + C R 10 fi− gi 2 . (69)

B. Differential decay rates

Using the transition matrix elements in Eq. (68), the double differential decay rates can be derived as dΓ dsdˆz = G2 Fα2emλ2t 768π5 M 5 Λb q φ (s) v u u t1 − 4m 2 l q2 f¯ 2_R Λb(s, ˆz) , (70) where RΛb(s, ˆz) = I0(s, ˆz) + ˆzI1(s, ˆz) + ˆz 2_I 2(s, ˆz) (71) with I0(s, ˆz) = −6√rˆs " 1 + 2m 2 l q2 ! ReA1B∗1 + 1 − 6 m2 l q2 ! ReD1E∗1 # +3 4  (1 − r)2− s2 |A1|2+ |B1|2+ |D1|2+ |E1|2  +6 ˆm2_lt_|A1|2+ |B1|2− |D1|2− |E1|2  +12 ˆm2_lMΛb √ r (1 − t) (ReD1D′∗3 + ReE1E′∗3) +12 ˆm2 lMΛb(t − r) (ReD1E ′∗ 3 + ReD3E′∗1)

+6MΛb √ rs (1 − t) " 1 + 2m 2 l q2 ! (ReA1A∗2+ ReB1B∗2) + _{1 − 2}m 2 l q2 ! (ReD1D∗2+ ReE1E∗2) # −6MΛbs (t − r) " 1 + 2m 2 l q2 ! (ReA1B∗2+ ReA2B∗1) + _{1 − 6}m 2 l q2 ! (ReD1E∗2+ ReD2E∗1) # −6MΛ2b √ rs2 1 + 2m 2 l q2 ! ReA2B∗2− 6M2Λb √ rs2 _{1 − 6}m 2 l q2 ! ReD2E∗2 −6MΛ2b  s (1 − t) (t − r) − 1₈(1 − r)2− s2 _ |A2|2 + |B2|2+ |D2|2+ |E2|2  −6MΛ2bmˆ 2 l (2r − (1 + r) t)  |A2|2+ |B2|2− |D2|2− |E2|2  +12 ˆm2lMΛ2bst (ReD2D ′∗ 3 + ReE2E′∗3) + 12 ˆm2lMΛ2b √ rs (ReD2E′∗3 + ReD′∗3E2) , I1(s, ˆz) = 3sφ (s) n − (ReA1D∗1− ReB1E∗1) + MΛb h√ r (ReA1D∗2− ReB1E∗2) + (ReA1E∗2− ReB1D∗2) + √

r (ReA2D∗1− ReB2E∗1) − (ReA2E∗1− ReB2D∗1)

i +MΛb(1 − r) (ReA2D ∗ 2− ReB2E∗2)} , I2(s, ˆz) = − 3 4φ (s) 1 − 4 m2 l q2 !  |A1|2+ |B1|2+ |D1|2+ |E1|2  +3 4M 2 Λbφ (s) 1 − 4 m2 l q2 !  |A2|2+ |B2|2+ |D2|2+ |E2|2  (72) where D′

3 = D3− D2 and E3′ = E3− E2, and ˆz = ˆpB· ˆpl+ denotes the angle between the

momentum of Λb and that of l+ in the di-lepton invariant mass frame.

C. Forward-backward and lepton asymmetries

From Eq. (68), the functions of RΛb and RF B in the differential and normalized FBAs

for Λb → Λl+l− in Eqs. (28) and (29) are given by

RΛb(s) = 1 2 Z 1 −1 dˆzRΛb(s, ˆz) (73) and RF B(s) = −Re (A1D∗1− B1E∗1) + MΛb h√ rRe (A1D∗2− B1E∗2) + Re (A1E∗2− B1D∗2) +√rRe (A2D∗1− B2E1∗) − Re (A2E∗1− B2D∗1) i +M_Λ2b(1 − r)Re (A2D ∗ 2− B2E∗2) , (74)

respectively. We can also define the longitudinal and transverse lepton polarization asym-metries. Explicitly, by the definition of Eq. (34), we get

PL = − 1 RΛb(s) s 1 − 4 ˆm 2 l s n s (1 + r − s) + (1 − r)2− s2Re (A1D∗1+ B1E∗1)

+M_Λ2bs  −s (1 + r − s) + 2(1 − r)2− 2s2Re (A2D∗2+ B2E∗2) −6s√rhRe (A1E∗1+ B1D∗1) + M2ΛbsRe (A2E ∗ 2+ B2D∗2) i +3sMΛ(1 − r + s) [Re (A1D∗2+ B1E2∗) + Re (A2D∗1 + B2E∗1)] −3sMΛb(1 − r − s) [Re (A1E ∗ 2+ B1D∗2) − + e (A2E∗1+ B2D∗1)] , (75) PT = 3 4π ˆml v u u t1 − 4m 2 l q2 q sφ (s) 1 RΛb(s) {−Im (A1D∗1− B1E∗1) +MΛ[Im (A1D∗2− B1E2∗) + Im (A2D∗1− B2E∗1)] +MΛb[Im (A1E ∗ 2− B1D∗2) − Im (A2E∗1− B2D∗1)] +MΛ2b(1 − r) Im (A2D ∗ 2− B2E∗2) . (76)

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Figure Captions

Figure 1: BRs as a function of q2_/M2

Λb for (a) Λb → Λµ

+_µ− _{and (b) Λ}

b → Λτ+τ−.

The curves with and without resonant shapes represent including and no LD contributions, respectively. The dashed, solid and dash-dotted curves stand for ω = 0.15, 0, and −0.15, respectively.

Figure 2: Same as Figure 1 but for the FBAs.

Figure 3: Same as Figure 1 but for the longitudinal polarization asymmetries. Figure 4: Same as Figure 1 but for the transverse polarization asymmetries. Figure 5: FBAs in the generic SUSY model as a function of q2_/M2

Λb for (a) Λb → Λµ

+_µ−

and (b) Λb → Λτ+τ−. The solid and dashed curves stand for the SM and

SUSY model, respectively.

Figure 6: Same as Figure 5 but for the longitudinal polarization asymmetries. Figure 7: Same as Figure 5 but for the transverse polarization asymmetries. Figure 8: Same as Figure 5 but for ∆Γ.

Figure 9: Same as Figure 5 but for ∆F BA.

Figure 10: Same as Figure 5 but for ∆PL.

0 2 4 6 5 10 15 20 q2(GeV2)

dB(

Λ

→Λµ

µ

)/dq

×

10

(GeV

)

dB(

Λ

→Λτ

τ

)/dq

×

10

(GeV

)

curves with and without resonant shapes represent including and no LD contributions, respectively. The dashed, solid and dash-dotted curves stand for ω = 0.15, 0, and −0.15, respectively.

-0.75 -0.5 -0.25 0 0.25 0.5 0.75 0 0.2 0.4 0.6 q2/M_Λ2_b

A

(

Λ

→Λµ

µ

)

A

(

Λ

→Λτ

τ

)

-1 -0.5 0 0.5 1 0 0.2 0.4 0.6 q2/M_Λ2_b

P

(

Λ

→Λµ

µ

)

P

(

Λ

→Λµ

µ

)

0 0.2 0.4 0.6 0.8 0.2 0.3 0.4 0.5 0.6 q2/M_Λ2_b

P

(

Λ

→Λµ

µ

)

×

10

P

(

Λ

→Λτ

τ

)

×

10

-0.5 -0.25 0 0.25 0.5 0 0.2 0.4 0.6 q2/M_Λ2_b

A

(

Λ

→Λµ

µ

)

A

(

Λ

→Λτ

τ

)

Figure 5: FBAs in the generic SUSY model as a function of q2_/M2

Λb for (a) Λb → Λµ

+_µ−

and (b) Λb → Λτ+τ−. The solid and dashed curves stand for the SM and SUSY model,

-1 -0.5 0 0.5 1 0 0.2 0.4 0.6 q2/M_Λ2_b

P

(

Λ

→Λµ

µ

)

P

(

Λ

→Λτ

τ

)

-2 -1 0 1 2 0 0.2 0.4 0.6 q2/M_Λ2_b

P

(

Λ

→Λµ

µ

)

×

10

P

(

Λ

→Λτ

τ

)

×

10

0 2 4 6 8 0 0.2 0.4 0.6 q2/M_Λ2_b

∆

(

Λ

→Λµ

µ

)

×

10

∆

(

Λ

→Λτ

τ

)

×

10

-15 -10 -5 0 0 0.2 0.4 0.6 q2/M_Λ2_b

∆

(

Λ

→Λµ

µ

)

×

10

∆

(

Λ

→Λτ

τ

)

×

10

0 0.1 0.2 0.3 0 0.2 0.4 0.6 q2/M_Λ2_b

∆

(

Λ

→Λµ

µ

)

∆

(

Λ

→Λτ

τ

)

-2 0 2 0 0.2 0.4 0.6 q2/M_Λ2_b

∆

(

Λ

→Λµ

µ

)

×

10

∆

(

Λ

→Λτ

τ

)

×

10