Comparative analysis of the Λb → Λℓ+ℓ- decay in the SM, SUSY, and RS model with custodial protection

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arXiv:1508.03980v2 [hep-ph] 21 Dec 2015

Comparative analysis of the

_Λ

_b

_{→ Λℓ}

+

ℓ

−

decay in the

SM, SUSY and RS model with custodial protection

K. Azizi1 ∗, A. T. Olgun2 †, Z. Tavuko˘glu2 ‡

1 _{Department of Physics, Do˘gu¸s University, Acıbadem-Kadık¨}_{oy, 34722 ˙Istanbul, Turkey}

2 _{Vocational School Kadık¨}_{oy Campus, Okan University, Hasanpa¸sa-Kadık¨}_{oy, 34722 ˙Istanbul, Turkey}

Abstract

We comparatively analyze the rare Λb → Λℓ+ℓ− channel in standard model,

su-persymmetry and Randall-Sundrum model with custodial protection (RSc). Using

the parametrization of the matrix elements entering the low energy effective Hamilto-nian in terms of form factors, we calculate the corresponding differential decay width and lepton forward-backward asymmetry in these models. We compare the results obtained with the most recent data from LHCb as well as lattice QCD results on the considered quantities. It is obtained that the standard model, with the form factors calculated in light-cone QCD sum rules, can not reproduce some experimental data on the physical quantities under consideration but the supersymmetry can do it. The

RSc model predictions are roughly the same as the standard model and there are no

considerable differences between the predictions of these two models. In the case of

differential decay rate, the data in the range 4 GeV2/c4 _{≤ q}2 _{≤ 6 GeV}2/c4 can not

be described by any of the considered models.

PACS number(s): 12.60.-i, 12.60.Jv, 13.30.-a, 13.30.Ce, 14.20.Mr

∗_{e-mail: kazizi@dogus.edu.tr} †_{e-mail: tugba.olgun@okan.edu.tr} ‡_{e-mail: zeynep.tavukoglu@okan.edu.tr}

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1 Introduction

The ATLAS and CMS Collaborations at CERN have independently reported their discovery of the Higgs boson with a mass of about 125 GeV using the samples of proton-proton collision data collected in 2011 and 2012, commonly referred to as the first LHC run [1–3]. Recently, a measurement of the Higgs boson mass based on the combined data samples of the ATLAS and CMS experiments has been presented as mH = 125.09 ±0.21(stat)±0.11(syst)

GeV in Refs [3–7]. At the same time, all LHC searches for signals of new physics above the TeV scale have given negative results. However, the LHC constraints on new physics effects can help theoreticians in the course of searching for these new effects and answering the questions that the standard model (SM) has not answered yet. We hope that the upcoming LHC run can bring unexpected surprises to observe signals of new physics in the experiment [8].

Although the SM could be valid up to some arbitrary high scale, new scenarios should exist because we are lacking a proper understanding of some important issues like origin of the matter, matter-antimatter asymmetry, dark matter and dark energy etc. [9]. In the baryonic sector, the loop-induced flavor changing neutral current (FCNC) decay of the Λb → Λℓ+ℓ− with ℓ = e, µ, τ , which is described by the b → sℓ+ℓ− transition at quark

level, is one of the important rare processes that can help us in the course of indirectly searching for new physics effects [10]. Recently, the differential branching fraction of the Λ0

b → Λµ+µ− decay channel has been measured as a function of the square of the di-muon

invariant mass (q2_{), corresponding to an integrated luminosity of 3.0 f b}−1 _using

proton-proton collision data collected by the LHCb experiment [11]. The measured result at 15 GeV2_/c4 _{≤ q}2 _{≤ 20 GeV}2_/c4 _{region for the differential branching fraction is dBr(Λ}0

b →

Λµ+_µ−_)/dq2 _{= (1.18} + 0.09

− 0.08 ± 0.03 ± 0.27) × 10−7 GeV2/c4. The LHCb Collaboration has

also reported the measurement on the forward-backward asymmetries of this transition at the µ channel. The measured result at the 15 GeV2_/c4 _{≤ q}2 _{≤ 20 GeV}2_/c4 _{region for the}

lepton forward-backward asymmetry is Aµ_{F B} _{= −0.05 ± 0.09(stat) ± 0.03(syst) [11]. In the} literature, there are a lot of studies on this decay channel via different approaches (for some recent studies see for instance Refs. [12–17]).

In the present work, we calculate the differential decay rate and lepton forward-backward asymmetry related to the FCNC Λb → Λℓ+ℓ− transition for all leptons in the SM,

super-symmetry (SUSY) and Randall-Sundrum scenario with custodial protection (RSc). We

compare the results with the experimental data provided by LHCb [11] as well as the ex-isting lattice QCD predictions [18]. Comparison of the LHCb results with the lattice QCD

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predictions shows that there are some deviations of data from the SM predictions. Such deviations can be attributed to the new physics effects that can contribute to such loop level processes. In this connection, we comparatively analyze the Λb → Λℓ+ℓ− decay channel in

SM and some new physics scenarios. In the calculations, we use the form factors calculated via the light-cone QCD sum rules in [19]. Hence, to get ride of any misleading, we will use SMLCSR instead of SM referring to the results that are obtained via using the from factors predicted by the light-cone sum rules in [19] when we speak about the predictions of different models. Note that there are many studies devoted to the calculations of the form factors defining the transition under consideration via different approaches (se for instance [20, 21]), but our aim here is to use those form factors that are obtained in the full theory of QCD in [19] without any approximation.

The outline of the paper is as follows. In the next section, we introduce a detailed discussion of the effective Hamiltonian responsible for the semileptonic Λb → Λℓ+ℓ− decay

channel and Wilson coefficients in SM, RSc and SUSY models. In this section, we also

present a basic introduction of the RSc scenario. In section 3, we calculate the differential

decay rate and lepton forward-backward asymmetry at different scenarios and compare the predictions of different models.

2 The semileptonic

_Λ

_b

_{→ Λℓ}

+

ℓ

−

transition in SM,

SUSY and RS

_c

models

2.1 The effective Hamiltonian and Wilson Coefficients

At quark level, the FCNC transition of Λb → Λℓ+ℓ−is governed by the b → sℓ+ℓ−transition

whose effective Hamiltonian in the SM can be written as

Hef fSM = GFαemVtbVts∗ 2√2π " C₉ef f,SMsγ¯ µ(1 − γ5)b ¯ℓγµℓ + C10SMsγ¯ µ(1 − γ5)b ¯ℓγµγ5ℓ − 2mbC7ef f,SM 1 q2¯siσµνq ν_{(1 + γ} 5)b ¯ℓγµℓ # , (2.1)

where Vtb and Vts∗ are elements of the Cabibbo-Kobayashi-Maskawa (CKM) mixing

ma-trix, αem is the fine structure constant at Z mass scale, GF is the Fermi weak coupling

constant, q2 _{is the transferred momentum squared; and the C}ef f,SM

9 , C10SM and C ef f,SM 7

are the Wilson coefficients representing different interactions. The explicit expressions of the Wilson coefficients entered to the above Hamiltonian are given in the following. The

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Wilson coefficient C₉ef f,SM which is a function of ˆs′ ₌ q2 m2

b with q

2 _{lies in the allowed region}

4m2 l ≤ q2 ≤ (mΛb − mΛ) 2 _{is given by [22, 23]} C₉ef f,SM(ˆs′) = C₉N DRη(ˆs′) + h(z, ˆs′) (3C1+ C2+ 3C3+ C4+ 3C5+ C6) −1₂h(1, ˆs′) (4C3 + 4C4+ 3C5+ C6) −1₂h(0, ˆs′) (C3+ 3C4) + 2 9(3C3+ C4+ 3C5+ C6) , (2.2) where the CN DR

9 in the naive dimensional regularization (NDR) scheme is expressed as

C₉N DR = P₀N DR+ Y

SM

sin2θW − 4Z

SM_{+ P}

EESM . (2.3)

The last term in the right hand side is neglected due to smallness of the order of PE. Here

PN DR

0 = 2.60 ± 0.25, YSM = 0.98, ZSM = 0.679 and sin2θW = 0.23 [22–24]. The parameter

η(ˆs′_{) in Eq.(2.2) is given as} η(ˆs′_{) = 1 +}αs(µb) π ω(ˆs ′_{) ,} _(2.4) with ω(ˆs′_{) = −}2 9π 2 −4₃Li2(ˆs′) − 2 3ln ˆs ′_{ln(1 − ˆs}′_{) −} 5 + 4ˆs′ 3(1 + 2ˆs′₎ln(1 − ˆs ′_{) −} 2ˆs′_{(1 + ˆ}_s′_{)(1 − 2ˆs}′₎ 3(1 − ˆs′₎2_{(1 + 2ˆ}_s′₎ ln ˆs ′ ₊ 5 + 9ˆs′ − 6ˆs′2 6(1 − ˆs′_{)(1 + 2ˆ}_s′₎ , (2.5) and αs(x) = αs(mZ) 1 − β0αs(m_2πZ)ln(m_xZ) . (2.6)

Here αs(mZ) = 0.118 and β0 = 23₃. The function h(y, ˆs′) in Eq.(2.2) is also defined by

h(y, ˆs′_{) = −}8 9ln mb µb − 8 9ln y + 8 27+ 4 9x (2.7) −2 9(2 + x)|1 − x| 1/2    ln √ 1−x+1 √ 1−x−1 − iπ , for x ≡ 4zsˆ′2 < 1 2 arctan_√1 x−1, for x ≡ 4z2 ˆ s′ > 1 , (2.8) where y = 1 or y = z = mc mb and, h(0, ˆs′_{) =} 8 27− 8 9ln mb µb − 4 9ln ˆs ′₊ 4 9iπ . (2.9)

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In Eq.(2.2), the remaining coefficients are given by [24] Cj = 8 X i=1 kjiηai (j = 1, ...6), (2.10)

where the kji are given as

k1i = ( 0, 0, 1₂, −1₂, 0, 0, 0, 0 ) , k2i = ( 0, 0, 1₂, 1₂, 0, 0, 0, 0 ) , k3i = ( 0, 0, −₁₄1, 1₆, 0.0510, −0.1403, −0.0113, 0.0054 ) , k4i = ( 0, 0, −₁₄1, −1₆, 0.0984, 0.1214, 0.0156, 0.0026 ) , k5i = ( 0, 0, 0, 0, −0.0397, 0.0117, −0.0025, 0.0304 ) , k6i = ( 0, 0, 0, 0, 0.0335, 0.0239, −0.0462, −0.0112 ) . (2.11)

The explicit expression for the Wilson coefficient CSM

10 is given as

C₁₀SM _{= −} Y

SM

sin2θW

. (2.12)

Finally, the Wilson coefficient C₇ef f,SM in the leading log approximation is defined by [22–25]

C₇ef f,SM(µb) = η 16 23C 7(µW) + 8 3 η1423 − η 16 23 C8(µW) + C2(µW) 8 X i=1 hiηai , (2.13) where η = αs(µW) αs(µb) , (2.14) and C7(µW) = − 1 2D ′ SM 0 (xt) , C8(µW) = − 1 2E ′ SM 0 (xt) , C2(µW) = 1 . (2.15) The functions D′ SM 0 (xt) and E0′ SM(xt) with xt = m 2 t m2 W are given by D₀′ SM(xt) = − (8x3 t + 5x2t − 7xt) 12(1 − xt)3 +x 2 t(2 − 3xt) 2(1 − xt)4 ln xt , (2.16) and E0′ SM(xt) = − xt(x2t − 5xt− 2) 4(1 − xt)3 + 3x 2 t 2(1 − xt)4 ln xt . (2.17)

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The coefficients hi and ai inside the C7ef f,SM are also given by [22, 23]

hi = ( 2.2996, −1.0880, −3₇, −₁₄1 , −0.6494, −0.0380, −0.0186, −0.0057 ),

ai = ( 14₂₃, 16₂₃, ₂₃6 , −12₂₃, 0.4086, −0.4230, −0.8994, 0.1456 ).

(2.18)

One of the most important new physics scenarios is SUSY. The different SUSY models involve the SUSY I, SUSY II, SUSY III and SUSY SO(10) scenarios according to the values of tanβ and an extra parameter µ with dimension of mass [26–29]. In SUSY I, the Wilson coefficient C₇ef f changes its sign, the µ takes a negative value and the contributions of the neutral Higgs bosons (NHBs) have been disregarded. In the SUSY II model, the value of the tanβ is large and the masses of the superparticles are relatively small. In SUSY III, the tanβ takes a large value and the masses of the superparticles are relatively large. In SUSY SO(10), the contributions of the NHBs are taken into account. The supersymmetric effective Hamiltonian in terms of the new operators coming from the NHBs exchanges diagrams and the corresponding Wilson coefficients is written as

Hef fSU SY = GFαemVtbVts∗ 2√2π " C₉ef f,SU SY¯sγµ(1 − γ5)b ¯ℓγµℓ + C9′ eff,SUSYsγ¯ µ(1 + γ5)b ¯ℓγµℓ + CSU SY 10 sγ¯ µ(1 − γ5)b ¯ℓγµγ5ℓ + C10′ SUSY¯sγµ(1 + γ5)b ¯ℓγµγ5ℓ − 2mbC7ef f,SU SY 1 q2siσ¯ µνq ν_{(1 + γ} 5)b ¯ℓγµℓ − 2mbC7′ eff,SUSY 1 q2siσ¯ µνq ν (1 − γ5)b ¯ℓγµℓ + CSU SY Q1 s(1 + γ¯ 5)b ¯ℓℓ + C ′ SUSY Q1 s(1 − γ¯ 5)b ¯ℓℓ + CQSU SY2 s(1 + γ¯ 5)b ¯ℓγ5ℓ + C ′ SUSY Q2 s(1 − γ¯ 5)b ¯ℓγ5ℓ # , (2.19)

where C₉ef f,SU SY, C₉′ eff,SUSY, CSU SY

10 , C10′ SUSY, C ef f,SU SY 7 , C7′ eff,SUSY, CQSU SY1 , C ′ SUSY Q1 , CSU SY Q2 and C ′ SUSY

Q2 are the new Wilson coefficients in the different SUSY models. The new

Wilson coefficients, C_Q(′)SUSY₁ and C_Q(′)SUSY₂ come from NHBs exchanging [29]. The primed Wilson coefficients only appear in SUSY SO(10) model. The values of Wilson coefficients in different supersymmetric models are presented in table 1 [27–30].

The last new physics scenario which we consider in this work is the Randall-Sundrum scenario proposed to solve the gauge hierarchy and the flavor problems in 1999 [31, 32]. It is a successful model, featuring one compact extra dimension with non-factorizable anti-de Sitter (AdS5) space-time [33]. This model describes the five-dimensional space-time

manifold with coordinates (x; y) and metric

ds2 = e−2kyηµνdxµdxν − dy2 ,

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Coefficient SUSY I SUSY II SUSY III SUSY SO(10)(A0= −1000) C₇ef f,SU SY +0.376 +0.376 _−0.376 _−0.219 C₉ef f,SU SY 4.767 4.767 4.767 4.275 CSU SY 10 −3.735 −3.735 −3.735 −4.732 CSU SY Q1 0 6.5(16.5) 1.2(4.5) 0.106 + 0i(1.775 + 0.002i) CSU SY Q2 0 −6.5(−16.5) −1.2(−4.5) −0.107 + 0i(−1.797 − 0.002i) C₇′ eff,SUSY 0 0 0 0.039 + 0.038i C₉′ eff,SUSY 0 0 0 0.011 + 0.072i C′ SUSY 10 0 0 0 −0.075 − 0.67i C′ SUSY Q1 0 0 0 −0.247 + 0.242i(−4.148 + 4.074i) C′ SUSY Q2 0 0 0 −0.25 + 0.246i(−4.202 + 4.128i)

Table 1: The Wilson coefficients in different SUSY models [27–30]. The values inside the parentheses are for the τ lepton.

The scale parameter k is defined as k ≃ O(MP lanck). We choose it as k = 1019 GeV. The

fifth coordinate y varies in a range between two branes 0 and L. y = 0 and y = L correspond to the so-called UV brane and IR brane, respectively. The simplest RS model with only the SM gauge group in the bulk has many important problems with the electroweak precision parameters [34]. In the present work, we consider the RS model with an enlarged custodial protection based on SU(3)c × SU(2)L× SU(2)R× U(1)×× PLR, where PLR interchanges

the two SU(2) groups and is responsible for the protection of the ZbLbL vertex (for more

information on the model see [33–41]).

The effective Hamiltonian for the b → sℓ+_ℓ− _{transition in the RS}

c model is given as Hef fRSc = GFαemVtbVts∗ 2√2π " Cef f,RSc 9 ¯sγµ(1 − γ5)b ¯ℓγµℓ + C′ eff,RS c 9 sγ¯ µ(1 + γ5)b ¯ℓγµℓ + CRSc 10 sγ¯ µ(1 − γ5)b ¯ℓγµγ5ℓ + C10′ RScsγ¯ µ(1 + γ5)b ¯ℓγµγ5ℓ − 2mbC7ef f,RSc 1 q2siσ¯ µνq ν_{(1 + γ} 5)b ¯ℓγµℓ − 2mbC7′ eff,RSc 1 q2siσ¯ µνq ν (1 − γ5)b ¯ℓγµℓ # , (2.21)

where the new Wilson coefficients are modified considering the new interactions. The new coefficients in terms of the SM coefficients are written as [33–41]

C(′)RSc

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where ∆C9 = " ∆Ys sin2_(θ w) − 4∆Z s # , ∆C′ 9 = " ∆Y′ s sin2_(θ w) − 4∆Z ′ s # , ∆C10 = − ∆Ys sin2_(θ w) , (2.23) and ∆C′ 10 = − ∆Y′ s sin2_(θ w) , (2.24) with ∆Ys = − 1 VtbVts∗ X X ∆ℓℓ L(X) − ∆ℓℓR(X) 4M2 Xg2SM ∆bs_L(X) , ∆Y_s′ _{= −} 1 VtbVts∗ X X ∆ℓℓ L(X) − ∆ℓℓR(X) 4M2 Xg2SM ∆bs_R(X) , ∆Zs = 1 VtbVts∗ X X ∆ℓℓ R(X) 8M2 XgSM2 sin2(θw) ∆bs L(X) , (2.25) and ∆Z′ s = 1 VtbVts∗ X X ∆ℓℓ R(X) 8M2 XgSM2 sin2(θw) ∆bs_R(X) . (2.26)

In the above equations, X = Z, ZH, Z′ and A(1), gSM2 = G√F₂2πsinα2_(θ

w) and θw is the Weinberg

angle. The functions inside ∆Ys, ∆Ys′, ∆Zs and ∆Zs′ are given in [33–41].

In the case of ∆C7(′), ∆C7(′)(µb) = 0.429∆C7(′)(MKK) + 0.128∆C8(′)(MKK) is used where

the following three contributions are included [35]:

(∆C7)1 = iQu r X F =u,c,t [A + 2m2_F(A′+ B′)]h_D_L†Yu(Yu)†Yd_DR i 23 (∆C7)2 = −iQd r 8 3(g 4D s )2 X F =d,s,b [I0+ A + B + 4m2F(I0′ + A′+ B′)] h DL†RLYdRRDR i 23 (∆C7)3 = iQd r 8 3(g 4D s )2 X F =d,s,b mF[I0+ A + B] nh D†LRLRLYdDR i 23 + mb ms h DL†YdRRRRDR i 23 o (2.27)

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(∆C′ 7)1 = iQu r X F =u,c,t [A + 2m2_F(A′_{+ B}′_)]h_D† R(Yd)†Yu(Yu)†DL i 23 (∆C₇′)2 = −iQd r 8 3(g 4D s )2 X F =d,s,b [I0+ A + B + 4m2F(I0′ + A′+ B′)] h DR†RR(Yd)†RLDL i 23 (∆C₇′)3 = iQd r 8 3(g 4D s )2 X F =d,s,b mF[I0+ A + B] nh D†RRRRR(Yd)†DL i 23 + mb ms h DR†(Yd)†RLRLDL i 23 o (2.28) (∆C8)1 = ir X F =u,c,t [A + 2m2_F(A′+ B′)]h_D_L†Yu(Yu)†Yd_DR i 23 (∆C8)2 = −ir 9 8(g 4D s )2 v2 mbmsT 3 X F =d,s,b [ ¯A + ¯B + 2m2_F( ¯A′_{+ ¯}_B′_)] h DL†YdRR(Yd)†RLYdDR i 23 (∆C8)3 = −ir 9 4(g 4D s )2T3 X F =d,s,b [ ¯A + ¯B + 2m2_F( ¯A′_{+ ¯}_B′_)] h DL†RLYdRRDR i 23 (2.29) (∆C′ 8)1 = ir X F =u,c,t [A + 2m2_F(A′_{+ B}′_)]h_D† R(Y d₎†_Yu_(Yu₎†_D L i 23 (∆C₈′)2 = −ir 9 8(g 4D s )2 v2 mbmsT3 X F =d,s,b [ ¯A + ¯B + 2m2_F( ¯A′+ ¯B′)] h DR†(Yd)†RLYdRR(Yd)†DL i 23 (∆C₈′)3 = −ir 9 4(g 4D s )2T3 X F =d,s,b [ ¯A + ¯B + 2m2_F( ¯A′+ ¯B′)] h DR†RR(Yd)†RLDL i 23 (2.30) where r = v GF 4π2VtbV ∗ tsmb and T3 = 1 L RL 0 dy[g(y)]

2_{. For the parameters inside the above}

equa-tions and the related diagrams see [33–41]. The Qu and Qd are representing the electric

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given as I0(t) = i (4π)2 1 M2 KK − 1 t − 1 + ln(t) (t − 1)2 ! I0′(t) = i (4π)2 1 M4 KK 1 + t 2t(t − 1)2 − ln(t) (t − 1)3 ! A(t) = B(t) = i (4π)2 1 4M2 KK t − 3 (t − 1)2 + 2ln(t) (t − 1)3 ! A′_{(t) = 2B}′_{(t) =} i (4π)2 1 M4 KK − t 2_{− 5t − 2} 6t(t − 1)3 − ln(t) (t − 1)4 ! ¯ A(t) = B(t) =¯ i (4π)2 1 4M2 KK − 3t − 1 (t − 1)2 + 2t2_ln(t) (t − 1)3 ! ¯ A′(t) = B¯′(t) = i (4π)2 1 4M4 KK 5t + 1 (t − 1)3 − 2t(2 + t)ln(t) (t − 1)4 ! , (2.31) with t = m2

F/MKK2 (for more information see [35]).

Fitting the parameters to the B → K∗_µ+_µ− _{channel, the modifications on Wilson}

coefficients in RSc model are found as table 2 [35].

∆C7 ∆C7′ ∆C9 ∆C9′ ∆C10 ∆C10′

Values 0.046 0.05 0.0023 0.038 0.030 0.50

Table 2: The values of modifications in Wilson coefficients in RScmodel used in the analysis

[35].

2.2 Transition amplitude and matrix elements

The amplitude of the transition under consideration is obtained by sandwiching the corre-sponding effective Hamiltonian between the initial and final baryonic states, i.e.,

MΛb→Λℓ+ℓ−

= hΛ(pΛ) | Hef f | Λb(pΛb)i , (2.32)

where pΛb and pΛare momenta of the initial and final baryons. To calculate the amplitude,

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form factors in full QCD: hΛ(pΛ) | ¯sγµ(1 − γ5)b | Λb(pΛb)i = ¯uΛ(pΛ) " γµf1(q2) + iσµνqνf2(q2) + qµf3(q2) − γµγ5g1(q2) − iσµνγ5qνg2(q2) − qµγ5g3(q2) # uΛb(pΛb) , (2.33) hΛ(pΛ) | ¯sγµ(1 + γ5)b | Λb(pΛb)i = ¯uΛ(pΛ) " γµf1(q2) + iσµνqνf2(q2) + qµf3(q2) + γµγ5g1(q2) + iσµνγ5qνg2(q2) + qµγ5g3(q2) # uΛb(pΛb) , (2.34) hΛ(pΛ) | ¯siσµνqν(1 + γ5)b | Λb(pΛb)i = ¯uΛ(pΛ) " γµf1T(q2) + iσµνqνf2T(q2) + qµf3T(q2) + γµγ5gT1(q2) + iσµνγ5qνg2T(q2) + qµγ5g3T(q2) # uΛb(pΛb) , (2.35) hΛ(pΛ) | ¯siσµνqν(1 − γ5)b | Λb(pΛb)i = ¯uΛ(pΛ) " γµf1T(q2) + iσµνqνf2T(q2) + qµf3T(q2) − γµγ5g1T(q2) − iσµνγ5qνg2T(q2) − qµγ5g3T(q2) # uΛb(pΛb) , (2.36) hΛ(pΛ) | ¯s(1 + γ5)b | Λb(pΛb)i = 1 mb ¯ uΛ(pΛ) " 6qf1(q2) + iqµσµνqνf2(q2) + q2f3(q2) − 6qγ5g1(q2) − iqµσµνγ5qνg2(q2) − q2γ5g3(q2) # uΛb(pΛb) , (2.37) and hΛ(pΛ) | ¯s(1 − γ5)b | Λb(pΛb)i = 1 mb ¯ uΛ(pΛ) " 6qf1(q2) + iqµσµνqνf2(q2) + q2f3(q2) + 6qγ5g1(q2) + iqµσµνγ5qνg2(q2) + q2γ5g3(q2) # uΛb(pΛb) , (2.38)

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where the f_i(T ) and g_i(T ) (i running from 1 to 3) are transition form factors and uΛb and uΛ

are spinors of the Λb and Λ baryons, respectively. We will use these form factors from [19]

that have been calculated using the light-cone QCD sum rules.

Using the above transition matrix elements in terms of form factors, we find the ampli-tude of the transition under consideration at different scenarios. In the SM, we find

MΛb→Λℓ+ℓ− SM = GFαemVtbVts∗ 2√2π ( h ¯ uΛ(pΛ)(γµ[ASM1 R + B1SML] + iσµνqν[ASM2 R + B2SML] + qµ_[ASM3 R + BSM3 L])uΛb(pΛb) i (¯ℓγµℓ) + hu¯Λ(pΛ)(γµ[DSM1 R + E1SML] + iσµνqν[D2SMR + E2SML] + qµ_[D₃SM_{R + E}₃SML])uΛb(pΛb) i (¯ℓγµγ5ℓ) ) . (2.39)

In the case of SUSY we get

MΛb→Λℓ+ℓ− SU SY = GFαemVtbVts∗ 2√2π ( h ¯ uΛ(pΛ)(γµ[ASU SY1 R + B1SU SYL] + iσµνqν[ASU SY2 R + B2SU SYL] + qµ_[ASU SY₃ _{R + B}₃SU SYL])uΛb(pΛb) i (¯ℓγµℓ) + hu¯Λ(pΛ)(γµ[DSU SY1 R + E1SU SYL] + iσµνqν[D2SU SYR + E2SU SYL] + qµ_[D₃SU SY_{R + E}₃SU SYL])uΛb(pΛb) i (¯ℓγµγ5ℓ) + hu¯Λ(pΛ)(6q[G1SU SYR + H1SU SYL] + iqµσµνqν[G2SU SYR + H2SU SYL] + q2_[G₃SU SY_{R + H}SU SY₃ L])uΛb(pΛb) i (¯ℓℓ) + hu¯Λ(pΛ)(6q[KSU SY1 R + S1SU SYL] + iqµσµνqν[KSU SY2 R + S2SU SYL] + q2_[KSU SY₃ _{R + S}₃SU SYL])uΛb(pΛb) i (¯ℓγ5ℓ) ) , (2.40)

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and for RSc we obtain MΛb→Λℓ+ℓ− RSc = GFαemVtbVts∗ 2√2π ( h ¯ uΛ(pΛ)(γµ[ARS1 cR + BRS c 1 L] + iσµνqν[ARS2 cR + BRS c 2 L] + qµ_[ARSc 3 R + BRS c 3 L])uΛb(pΛb) i (¯ℓγµℓ) + hu¯Λ(pΛ)(γµ[D1RScR + ERS c 1 L] + iσµνqν[D2RScR + ERS c 2 L] + qµ_[DRSc 3 R + ERS c 3 L])uΛb(pΛb) i (¯ℓγµγ5ℓ) ) , (2.41)

where R = (1 + γ5)/2 is the right-handed and L = (1 − γ5)/2 is the left-handed projectors.

In the above equations, the calligraphic coefficients are defined at different models as

A1 = f1C9ef f +− g1C9ef f −− 2mb 1 q2 h f₁TC₇ef f ++ g₁TC₇ef f −i_{, A}2 = A1(1 → 2), A3 = A1(1 → 3) , B1 = f1C9ef f ++ g1C9ef f −− 2mb 1 q2 h f₁TC₇ef f +_{− g}T₁C₇ef f −i_{, B}2 = B1(1 → 2) , B3 = B1(1 → 3) , D1 = f1C10+ − g1C10−, D2 = D1(1 → 2) , D3 = D1(1 → 3) , E1 = f1C10+ + g1C10−, E2 = E1(1 → 2) , E3 = E1(1 → 3) , G1 = 1 mb h f1CQ+1 − g1C − Q1 i , _G2 = G1(1 → 2) , G3 = G1(1 → 3) , H1 = 1 mb h f1CQ+1 + g1C − Q1 i , _H2 = H1(1 → 2) , H3 = H1(1 → 3) , K1 = 1 mb h f1C_Q+2 − g1C − Q2 i , _K2 = K1(1 → 2) , K3 = K1(1 → 3) , S1 = 1 mb h f1CQ+2+ g1C − Q2 i , _S2 = S1(1 → 2) , S3 = S1(1 → 3) , (2.42) with C₉ef f + = C₉ef f + C₉′ eff, C₉ef f −= C₉ef f _{− C}₉′ eff ,

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C₇ef f + = C₇ef f + C₇′ eff, C₇ef f −= C₇ef f _{− C}₇′ eff , C₁₀+ = C10 + C10′ , C10− = C10 − C10′ , C_Q+₁ = CQ1 + C ′ Q1, C − Q1 = CQ1 − C ′ Q1 , C+ Q2 = CQ2 + C ′ Q2, C − Q2 = CQ2 − C ′ Q2 . (2.43)

3 Physical Observables

3.1 The differential decay width

In the present subsection, we would like to calculate the differential decay width for the decay channel under consideration. Using the decay amplitude and the transition matrix elements in terms of form factors, the supersymmetric differential decay rate as the most comprehensive differential decay rate among the models under consideration is obtained as

d2_Γ SU SY dˆsdz (z, ˆs) = G2 Fα2emmΛb 16384π5 |VtbV ∗ ts|2vpλ(1, r, ˆs) " T0SU SY(ˆs) + T1SU SY(ˆs)z + T2SU SY(ˆs)z2 # , (3.44)

where z = cos θ with θ being the angle between the momenta of the lepton l+ _{and the}

Λb in the center of mass of leptons, v =

q

1 − 4m2ℓ

q2 is the lepton velocity, λ = λ(1, r, ˆs) =

(1 −r − ˆs)2−4rˆs is the usual triangle function, ˆs = q2/m2_Λb and r = m 2

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TSU SY

0 (ˆs), T1SU SY(ˆs) and T2SU SY(ˆs) are obtained as

TSU SY 0 (ˆs) = 32m2ℓm4Λbs(1 + r − ˆs)ˆ |D3|2+ |E3|2 + 64m2_ℓm3_Λb(1 − r − ˆs) Re h D1∗E3+ D3E1∗ i + 64m2_Λ_b√r(6m2_ℓ _{− m}2_Λ_bˆs)Reh_D∗ 1E1 i + 64m2_ℓm3_Λ_b√r ( 2mΛbsReˆ h D∗ 3E3 i + (1 − r + ˆs)RehD∗ 1D3+ E1∗E3 i ) + 32m2_Λ_b(2m2_ℓ + m2_Λ_bs)ˆ ( (1 − r + ˆs)mΛb √ r Reh_A∗₁_A2+ B1∗B2 i − mΛb(1 − r − ˆs) Re h A∗1B2+ A∗2B1 i − 2√rReh_A∗1B1 i + m2Λbs Reˆ h A∗2B2 i ) + 8m2_Λ_b ( 4m2_ℓ_{(1 + r − ˆs) + m}2_Λ_bh_{(1 − r)}2_{− ˆs}2i ) |A1|2+ |B1|2 + 8m4_Λ_b ( 4m2_ℓh_{λ + (1 + r − ˆs)ˆs}i+ m2_Λ_bˆsh_{(1 − r)}2_{− ˆs}2i ) |A2|2 + |B2|2 − 8m2Λb ( 4m2_ℓ_{(1 + r − ˆs) − m}2_Λ_bh_{(1 − r)}2_{− ˆs}2i ) |D1|2+ |E1|2 + 8m5Λbsvˆ 2 ( − 8mΛbsˆ √ r Reh_D∗2E2 i + 4(1 − r + ˆs)√r Reh_D∗1D2+ E1∗E2 i − 4(1 − r − ˆs) RehD∗ 1E2 + D2∗E1 i + mΛb h (1 − r)2_{− ˆs}2i_|D 2|2 + |E2|2 ) − 8m4Λb ( 4mℓ h (1 − r)2− ˆs(1 + r)iReh_D∗ 1K1+ E1∗S1 i + (4m2_ℓ _{− m}2_Λ_bs)ˆh_{(1 − r)}2 _{− ˆs(1 + r)}i _|G1|2+ |H1|2 + 4m2_Λ_b√rˆs2(4m_ℓ2_{− m}2_Λ_bs) Reˆ h_G∗ 3H3 i ) − 8m5Λbsˆ ( 2√r(4m2_ℓ _{− m}2_Λ_b_{s) (1 − r + ˆs) Re}ˆ h_G∗ 1G3+ H∗1H3 i + 4mℓ√r(1 − r + ˆs)Re h D∗ 1K3+ E1∗S3+ D3∗K1+ E3∗S1 i

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+ 4mℓ(1 − r − ˆs)Re h D∗ 1S3+ E1∗K3+ D∗3S1+ E3∗K1 i + 2(1 − r − ˆs)(4m2ℓ − m2Λbs) Reˆ h G∗ 1H3+ H∗1G3 i − mΛb h (1 − r)2− ˆs(1 + r)i|K1|2+ |S1|2 ) − 32m4Λb √ rˆs ( 2mℓRe h D∗ 1S1+ E1∗K1 i + (4m2_ℓ _{− m}2_Λ_bs) Reˆ h_G∗ 1H1 i ) + 8m6_Λ_bsˆ2 ( 4√r Reh_K∗ 1S1 i + 2mΛb √ r(1 − r + ˆs)RehK∗ 1K3+ S1∗S3 i + 2mΛb(1 − r − ˆs)Re h K∗ 1S3+ S1∗K3 i − (4m2ℓ − m2Λbs)(1 + r − ˆs)ˆ |G3|2+ |H3|2 − 4mℓ(1 + r − ˆs)Re h D∗ 3K3+ E3∗S3 i − 8mℓ√rRe h D∗ 3S3+ E3∗K3 i ) + 8m8_Λ_bsˆ3 ( (1 + r − ˆs)|K3|2+ |S3|2 + 4√rReh_K∗ 3S3 i ) , (3.45) T1SU SY(ˆs) = −32m4Λbmℓ √ λv(1 − r)ReA∗1G1+ B1∗H1 − 16m4Λbsvˆ √ λ ( 2Re_A∗₁_D1 − 2ReB∗1E1 + 2mΛbRe B∗ 1D2− B2∗D1+ A∗2E1− A∗1E2 + 2mΛbmℓRe A∗ 1H3+ B∗1G3− A∗2H1− B2∗G1 ) + 32m5_Λbs vˆ √ λ ( mΛb(1 − r)Re A∗2D2− B2∗E2 + √rRe_A∗₂_D1+ A∗1D2− B2∗E1− B1∗E2 − √rmℓRe A∗1G3+ B1∗H3+ A∗2G1+ B2∗H1 ) + 32m6_Λbmℓ √ λvˆs2Re_A∗₂_G3+ B∗2H3 , (3.46)

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and TSU SY 2 (ˆs) = −8m4Λbv 2_λ_|A 1|2 + |B1|2+ |D1|2+ |E1|2 + 8m6_Λbsvˆ 2_λ |A2|2+ |B2|2+ |D2|2+ |E2|2 . (3.47)

Integrating the Eq.(3.44) over z in the interval [−1, 1], we obtain the differential decay width only in terms of ˆs as

dΓSU SY dˆs (ˆs) = G2 Fα2emmΛb 8192π5 |VtbV ∗ ts|2v √ λ " T0SU SY(ˆs) + 1 3T SU SY 2 (ˆs) # . (3.48)

The differential decay rate of RSc is found from dΓSU SY_dˆ_s (ˆs) by replacing CQ1, C_Q′ 1, CQ2 and

C′

Q2 with zero. In the case of SM, dΓSM

dˆs (ˆs) is found from the supersymmetric differential

decay rate via setting C₇′ eff, C₉′ eff, C′

10, CQ1, CQ′ 1, CQ2 and CQ′2 to zero.

3.2 The differential branching ratio

In this subsection, we numerically analyze the differential branching ratio that depends on q2 _{for the Λ}

b → Λℓ+ℓ− decay in SMLCSR, SUSY and RSc scenarios. In order to discuss

the variation of the differential branching ratio with respect to q2_{, we shall present some}

values of input parameters in table 3 besides the form factors as the main inputs.

LHCb Collab. SMLCSR RSc Lattice 0 5 10 15 20 0.0 0.5 1.0 1.5 2.0 2.5 3.0 q2_@GeV2_c4_D dBr HL b ® L Μ +Μ -L dq 2x 10 7@H GeV 2c 4L -1D

Figure 1: The dependence of the differential branching ratio on q2 _{for the Λ}

b → Λµ+µ−

transition in the SMLCSR and RSc models. The lattice QCD [18] and recent experimental

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Some Input Parameters Values mµ 0.10565 GeV mτ 1.77682 GeV mc 1.275 ± 0.025 GeV mb 4.18 ± 0.03 GeV mt 173.21 ± 0.51 ± 0.71 GeV mW 80.385 ± 0.015 GeV mΛb 5.6195 ± 0.0004 GeV mΛ 1.11568 GeV τΛb (1.451 ± 0.013) × 10−12 s ~ _{6.582 × 10}−25 _{GeV s} GF 1.166 × 10−5 GeV−2 αem 1/137 |VtbVts∗| 0.040

Table 3: The values of some input parameters used in our calculations, taken generally from PDG [42]. SMLCSR RSc 13 14 15 16 17 18 19 20 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 q2_@GeV2_c4_D dBr HL b ® L Τ +Τ -L dq 2x 10 7@H GeV 2c 4L -1D

b → Λτ+τ−

transition in the SMLCSR and RSc models.

By using all these input parameters and the form factors with their uncertainties, we present the dependence of the differential branching ratio of the Λb → Λℓ+ℓ− on q2 in

SMLCSR, RSc and different SUSY models in figures 1-6. In these figures we also show the

experimental data provided by LHCb [11] as well as the existing lattice QCD predictions [18]. We do not present the results for e in the presentations since the predictions at e

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LHCb Collab. SMLCSR SUSY I Lattice 0 5 10 15 20 0 2 4 6 8 10 q2_@GeV2_c4_D dBr HL b ® L Μ +Μ -L dq 2x 10 7@H GeV 2c 4L -1D LHCb Collab. SMLCSR SUSY II Lattice 0 5 10 15 20 0 10 20 30 40 q2_@GeV2_c4_D dBr HL b ® L Μ +Μ -L dq 2x 10 7@H GeV 2c 4L -1D

b → Λµ+µ−

transition in SMLCSR and SUSY I and II models. The lattice QCD [18] and recent exper-imental data by LHCb [11] Collaboration are also included.

LHCb Collab. SMLCSR SUSY III Lattice 0 5 10 15 20 0 2 4 6 8 q2_@GeV2_c4_D dBr HL b ® L Μ +Μ -L dq 2x 10 7@H GeV 2c 4L -1D LHCb Collab. SMLCSR SUSY SOH10L Lattice 0 5 10 15 20 0 2 4 6 8 q2_@GeV2_c4_D dBr HL b ® L Μ +Μ -L dq 2x 10 7@H GeV 2c 4L -1D

b → Λµ+µ−

transition in SMLCSR and SUSY III and SO(10) models. The lattice QCD [18] and recent experimental data by LHCb [11] Collaboration are also included.

SMLCSR SUSY I 13 14 15 16 17 18 19 20 0 1 2 3 4 5 6 q2_@GeV2_c4_D dBr HL b ® L Τ +Τ -L dq 2x 10 7@H GeV 2c 4L -1D SMLCSR SUSY II 13 14 15 16 17 18 19 20 0 20 40 60 80 q2_@GeV2_c4_D dBr HL b ® L Τ +Τ -L dq 2x 10 7@H GeV 2c 4L -1D

b → Λτ+τ−

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SMLCSR SUSY III 13 14 15 16 17 18 19 20 0 1 2 3 4 5 6 q2_@GeV2_c4_D dBr HL b ® L Τ +Τ -L dq 2x 10 7@H GeV 2c 4L -1D SMLCSR SUSY SOH10L 13 14 15 16 17 18 19 20 0 2 4 6 8 10 12 14 q2_@GeV2_c4_D dBr HL b ® L Τ +Τ -L dq 2x 10 7@H GeV 2c 4L -1D

b → Λτ+τ−

transition in SMLCSR and SUSY III and SO(10) models.

channel are very close to those of µ. From figures 1-6 we see that

• for all lepton channels, the SMLCSR and RSc models have roughly the same

pre-dictions except for some values of q2 _{at which there are small differences between}

predictions of the SMLCSR and RSc models on the differential branching ratio.

• The areas swept by the SMLCSR are wider compared to those of lattice QCD [18] existing in the µ channel but they include those predictions.

• The experimental data in the intervals 4 GeV2_/c4 _{≤ q}2 _{≤ 6 GeV}2_/c4 _{and 18 GeV}2_/c4

≤ q2 _{≤ 20 GeV}2_/c4 _{cannot be described by the SMLCSR, lattice QCD or RS}

c

mod-els. In the remaining intervals the SMLCSR, lattice and RSc models reproduce the

experimental data, except for 6 GeV2_/c4 _{≤ q}2 _{≤ 8 GeV}2_/c4_{, for which the datum}

remains outside of the lattice predictions.

• In the τ channel, the bands of the SMLCSR and RSc scenarios intersect each other,

except for higher values of q2_{, for which the errors of the form factors do not kill the}

differences between the two model predictions.

• At all lepton channels, the SUSY models show overall considerable deviations from the SMLCSR, lattice QCD and experimental data although they include the predictions of these models for some values of q2_{. The maximum deviations of the SUSY predictions}

from the results of SMLCSR, lattice QCD and experiment belongs to the SUSY II such that the SMLCSR, lattice QCD and experimental results remain out of the regions swept by the SUSY II model at higher values of q2_.

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• In the µ channel, the experimental data in the interval 18 GeV2_/c4_{≤ q}2 _{≤ 20 GeV}2_/c4

are reproduced by SUSY I, III and SO(10) but not by SUSY II. Note that in this interval other models (SMLCSR, lattice QCD and RSc) also can not describe the

experimental data.

• Again in the µ channel, the experimental data in the interval 4 GeV2_/c4 _{≤ q}2 _{≤ 6}

GeV2_/c4 _{cannot be reproduced by any SUSY models like the SMLCSR, lattice QCD}

and RSc scenarios.

• In the case of τ as the final lepton, there are considerable differences between different SUSY models’ predictions and that of the SMLCSR and these cannot be completely killed by the errors of form factors. The maximum deviations of the SUSY results from the SMLCSR predictions belong to the SUSY II at higher q2 _values.

3.3 The lepton forward-backward asymmetry

In this subsection, we would like to present the results of the lepton forward-backward asymmetry obtained in different scenarios. The lepton AF B is defined as

AF B(ˆs) = Z 1 0 d2_Γ dˆsdz(z, ˆs) dz − Z 0 −1 d2_Γ dˆsdz(z, ˆs) dz Z 1 0 d2_Γ dˆsdz(z, ˆs) dz + Z 0 −1 d2_Γ dˆsdz(z, ˆs) dz . (3.49) LHCb Collab. SMLCSR RSc Lattice 0 5 10 15 20 -1.0 -0.5 0.0 0.5 1.0 1.5 q2@GeV2c4D AFB HL b ® L Μ +Μ -L

Figure 7: The dependence of the AF B on q2 for Λb → Λµ+µ−transition in SMLCSR, lattice

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SMLCSR RSc 13 14 15 16 17 18 19 20 -0.4 -0.2 0.0 0.2 0.4 q2_@GeV2_c4_D AFB HL b ® L Τ + Τ -L

Figure 8: The dependence of the AF B on q2 for Λb → Λτ+τ− transition in SMLCSR and

RSc models. LHCb Collab. SMLCSR SUSY I Lattice 0 5 10 15 20 -1.0 -0.5 0.0 0.5 1.0 1.5 q2_@GeV2_c4_D AFB HL b ® L Μ +Μ -L LHCb Collab. SMLCSR SUSY II Lattice 0 5 10 15 20 -1.0 -0.5 0.0 0.5 1.0 1.5 q2_@GeV2_c4_D AFB HL b ® L Μ +Μ -L

Figure 9: The dependence of the AF B on q2 for Λb → Λµ+µ−transition in SMLCSR, lattice

QCD [18] and SUSY I and II together with recent experimental data by LHCb [11].

LHCb Collab. SMLCSR SUSY III Lattice 0 5 10 15 20 -1.0 -0.5 0.0 0.5 1.0 1.5 q2_@GeV2_c4_D AFB HL b ® L Μ +Μ -L LHCb Collab. SMLCSR SUSY SOH10L Lattice 0 5 10 15 20 -1.0 -0.5 0.0 0.5 1.0 1.5 q2_@GeV2_c4_D AFB HL b ® L Μ +Μ -L

Figure 10: The dependence of the AF B on q2 for Λb → Λµ+µ− transition in SMLCSR,

lattice QCD [18] and SUSY III and SO(10) together with recent experimental data by LHCb [11].

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SMLCSR SUSY I 13 14 15 16 17 18 19 20 -0.4 -0.2 0.0 0.2 0.4 q2_@GeV2_c4_D AFB HL b ® L Τ +Τ -L SMLCSR SUSY II 13 14 15 16 17 18 19 20 -0.4 -0.2 0.0 0.2 0.4 q2_@GeV2_c4_D AFB HL b ® L Τ +Τ -L

SUSY I and II scenarios.

SMLCSR SUSY III 13 14 15 16 17 18 19 20 -0.4 -0.2 0.0 0.2 0.4 q2_@GeV2_c4_D AFB HL b ® L Τ +Τ -L SMLCSR SUSY SOH10L 13 14 15 16 17 18 19 20 -0.4 -0.2 0.0 0.2 0.4 q2_@GeV2_c4_D AFB HL b ® L Τ +Τ -L

SUSY III and SO(10) scenarios.

Considering the form factors with their uncertainties from [19], we plot the dependence of the lepton forward-backward asymmetry on q2 _{for the decay under consideration in both}

lepton channels in the SMLCSR, RSc and different SUSY models in figures 7-12. From

these figures, we obtain that

• in the µ channel, the SMLCSR, lattice QCD and RSc models predictions on AF B

coincide with each other. Except for the lattice QCD, they can only describe the experimental data existing in the 0 GeV2_/c4 _{≤ q}2 _{≤ 2 GeV}2_/c4 _{and 18 GeV}2_/c4

≤ q2 _{≤ 20 GeV}2_/c4 _{regions. The remaining data lie out of the swept regions by all}

these models.

• As far as the SUSY models are considered, in the µ channel, the SUSY models have predictions that deviate from the SMLCSR and lattice QCD predictions, considerably. All SUSY models reproduce the experimental data in the regions 0 GeV2_/c4 _{≤ q}2 _{≤ 2}

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regions swept by different SUSY models except for the SUSY II, which reproduces the experimental data also in the interval 15 GeV2_/c4 _{≤ q}2 _{≤ 18 GeV}2_/c4_.

• In the case of the τ lepton, the SMLCSR and RSc have roughly the same predictions

on AF B.

• In the τ lepton channel, the SMLCSR and SUSY I have roughly the same predictions for AF B; however, the remaining SUSY models’ predictions deviate from the SMLCSR

predictions considerably, although they intersect each other at some points.

4 Conclusion

In the present work, we have analyzed the semileptonic Λb → Λℓ+ℓ− decay mode in

SMLCSR, different SUSY models and the RSc scenario. Using the form factors calculated

in light cone QCD sum rules in the full theory [19], we evaluated the differential branching ratio and lepton forward-backward asymmetry for different leptons in those scenarios. We also compared the results obtained via SMLCSR, RSc and different SUSY scenarios with

the recent experimental data provided by LHCb [11] as well as the existing lattice QCD predictions [18] on the considered quantities. We observed that the regions swept by the SMLCSR model include the RSc predictions although they are somewhat wider compared

to those of RSc models for the considered physical quantities. The SMLCSR predictions on

the considered quantities in the present work are overall consistent with the lattice QCD predictions provided by Ref. [18].

The predictions of different SUSY models on the differential branching ratio deviate considerably from the SMLCSR and lattice predictions. The maximum deviations belong to the SUSY II model. In the case of AF B and the µ channel, the predictions of different

SUSY models have considerable deviations from the SMLCSR and lattice QCD predictions. For AF B and the τ channel, the SUSY I and SMLCSR have roughly the same predictions

but the other SUSY models have predictions different from that of the SMLCSR.

The experimental data on the differential branching ratio in the µ channel can be re-produced by SMLCSR, lattice QCD and RSc models except for the intervals 4 GeV2/c4

≤ q2 _{≤ 6 GeV}2_/c4 _{and 18 GeV}2_/c4 _{≤ q}2 _{≤ 20 GeV}2_/c4_{, which cannot be described by}

SMLCSR, lattice QCD or RSc models. As far as the SUSY models are considered,

differ-ent SUSY models also cannot reproduce the experimdiffer-ental data in the interval 4 GeV2_/c4

≤ q2 _{≤ 6 GeV}2_/c4_{. However, except for SUSY II, the remaining SUSY scenarios can explain}

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In the case of AF B and the µ channel, the SMLCSR, RSc and different SUSY models

can only describe the experimental data existing in the 0 GeV2_/c4 _{≤ q}2 _{≤ 2 GeV}2_/c4 _and

18 GeV2_/c4 _{≤ q}2 _{≤ 20 GeV}2_/c4 _{regions. The other existing data remain out of the swept}

areas by these models, except for SUSY II, which can also reproduce the experimental data in 15 GeV2_/c4 _{≤ q}2 _{≤ 18 GeV}2_/c4_.

More experimental data in the µ channel related to different physical quantities asso-ciated with the Λb → Λµ+µ− mode, the future experimental data in the τ channel and

comparison of the results with our predictions on the quantities considered in the present work may help us in the course of searching for new physics effects.

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