Investigation of the Lambda(b) -> Lambda l(+)l(-) transition in universal extra dimension using form factors from full QCD

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JHEP01(2011)087

Published for SISSA by Springer

Received: November 29, 2010 Revised: December 29, 2010 Accepted: December 31, 2010 Published: January 20, 2011

Investigation of the Λ

b

→ Λℓ

+

ℓ

−

transition in universal

extra dimension using form factors from full QCD

K. Azizi and N. Katırcı

Physics Division, Faculty of Arts and Sciences, Do˘gu¸s University, Acıbadem-Kadık¨oy, 34722 Istanbul, Turkey

E-mail: [email protected],[email protected]

Abstract:_{Using the related form factors from full QCD which recently are available, we} provide a comprehensive analysis of the Λb→ Λℓ+ℓ−transition in universal extra dimension model in the presence of a single universal extra dimension called the Applequist-Cheng-Dobrescu model. In particular, we analyze some related observables like branching ratio, forward-backward asymmetry, double lepton polarization asymmetries and polarization of the Λ baryon in terms of compactification radius and corresponding form factors. We present the sensitivity of these observables to the compactification parameter, 1/R up to 1/R = 1000 GeV . We also compare the results with those obtained using the form factors from heavy quark effective theory as well as the SM predictions.

Keywords: _{Beyond Standard Model, Heavy Quark Physics, Standard Model}

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Contents

1 Introduction 1

2 Effective Hamiltonian, transition matrix elements and form factors

re-sponsible for Λb → Λℓ+ℓ− ₃

2.1 Effective Hamiltonian 3

2.2 Transition matrix elements and form factors 7

3 Some observables related to the Λb _{→ Λℓ}+_ℓ− ₈

3.1 Branching ratio 8

3.2 Forward backward asymmetry 10

3.3 Λ baryon polarizations 11

3.4 Double lepton polarization asymmetries 12

4 Conclusion 17

1 Introduction

The Standard Model (SM) of particle physics describes all known particles and their in-teractions except than gravity. The SM is the only minimal model which is in perfect consistency with all confirmed collider data despite it needs a missing ingredient, the Higgs boson or something else to give masses to the elementary particles. However, there are some problems such as origin of the matter in the universe, gauge and fermion mass hier-archy, number of generations, matter-antimatter asymmetry, unification, quantum gravity and so on, which can not addressed by the SM. Such problems show that the SM can not be the ultimate theory of nature and it can be considered as a low energy manifestation of some fundamental theories.

Models with extra dimensions (ED) [1–4] are among the most interesting candidates beyond the SM to overcome the aforementioned problems. A category of ED which allows the SM fields (both gauge bosons and fermions) propagate in the extra dimensions called the universal extra dimension (UED) model. The most simple example of the UED model also, where just a single universal extra dimension compactified on a circle of radius R is considered, is called the Appelquist, Cheng and Dobrescu (ACD) model [5]. The radius R is the extra parameter that causes the difference between SM and its beyond. The particles

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with momentum in extra dimension are called Kaluza-Klein (KK) particles. The mass of

KK particles and their interaction with themselves as well as with the SM particles are de-scribed in terms of compactification scale, 1/R. One of the important property of the model is the conservation of KK parity that guarantees the absence of tree level KK contributions to low energy processes occurring at scales very smaller than the compactification scale [6] (for more information about the model see also [7,8]). The flavor changing neutral current (FCNC) transition of Λb → Λℓ+ℓ−, which may be in the future program of the LHCb to study, lies in this scale. This transition is proceed via the FCNC transition of b → sℓ+_ℓ− at loop level in SM via electroweak penguin and weak box diagrams, which are sensitive to new physics contributions. Looking for SUSY particles [9], light dark matter [10], probable fourth generation of the quarks, and also KK modes (extra dimensions ) [6] is possible by investigating such loop level transitions.

The aim of the paper is to find the effects of the KK modes on various observables related to the Λb → Λℓ+ℓ− transition. These observables are total decay rate, branching ratio, forward-backward asymmetry, double lepton polarization asymmetries and polariza-tion of the Λ baryon. We analyze these observables in terms of the corresponding form factors as well as the compactification factor. From the electroweak precision tests, the lower limit for 1/R is obtained as 250 GeV if Mh≥ 250 GeV expressing larger KK contribu-tions to the low energy FCNC processes like, Λb → Λℓ+ℓ−, and 300 GeV if Mh ≤ 250 GeV , respectively [5,11]. We will consider the 1/R from 200 GeV up to 1000 GeV . We will use also the form factors obtained both using QCD sum rules in full theory, which they recently are available [12] and also those obtained in heavy quark effective theory (HQET) [13]. Us-ing the values of the form factors, we present the sensitivity of these observables to the compactification parameter, 1/R. Note that, using the form factors obtained in HQET, the transitions, Λb → Λν ¯ν and Λb → Λγ [14], Λb → Λl+l−[15], Λb → Λγ and Λb→ Λl+l−[16] have been analyzed in the same framework. The ACD model also has been applied to investigate some B and K mesons decays in [6–8,17–20].

The layout of the paper is as follows. In next section, we introduce responsible Hamil-tonian and present Wilson coefficients in UED model. We also present the transition matrix elements in terms of form factors responsible for Λb → Λℓ+ℓ− transition in this section. In section 3, we analyze the branching ratio, forward-backward asymmetry, double lepton polarization asymmetries and polarization of the Λ baryon in terms of the compactification factor, 1/R. In this section, using the form factors both from full theory and HQET, we also compare our results obtained both in the UED and SM models for each observable and discuss the results. Finally, we will present our concluding remark in section 4.

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2 Effective Hamiltonian, transition matrix elements and form factors

responsible for Λb _{→ Λℓ}+ℓ− 2.1 Effective Hamiltonian

The Λb → Λℓ+ℓ−transition is governed by the FCNC transition of the b → sl+l− at quark level and is described by the following effective Hamiltonian:

Heff = GFαemVtbVts∗ 2√2π C₉effsγ¯ _{µ(1 − γ5})b ¯ℓγµℓ + C10sγ¯ µ(1 − γ5)b ¯ℓγµγ5ℓ −2mbC₇eff 1 q2siσ¯ µνq ν_{(1 + γ} 5)b ¯ℓγµℓ , (2.1)

where αemis the fine structure constant at Z mass scale, GF is the Fermi constant, Vij are elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix and Ceff

7 , C9eff and C10are the Wilson coefficients. These coefficients are the main source of the deviation of the ACD model results for the observables from the SM models predictions. These coefficients are expressed in terms of some periodic functions, F (xt, 1/R) with xt= m

2 t

M2 W

and mtbeing the top quark mass. The mass of the KK particles are represented in terms of the zero modes corresponding to the ordinary SM particles and an extra part coming from the ACD model, i.e., m2_n= m2₀+_Rn22. Similar to the mass of the KK particles, the functions, F (x_t, 1/R) are

also described in terms of the corresponding SM functions, F0(xt) and functions in terms of the compactification factor, 1/R,

F (xt, 1/R) = F0(xt) + ∞ X n=1 Fn(xt, xn), (2.2) where xn= m2 n m2 W and mn= n

R. The Glashow-Illiopoulos-Maiani (GIM) mechanism under-takes the finiteness of the functions, F (xt, 1/R) and fulfills the condition, F (xt, 1/R) → F0(xt), when R → 0. As far as the compactification factor, 1/R is recorded in order of a few hundreds of GeV , these functions and as a result, the Wilson coefficients and consid-ered observables differ considerably from the SM predictions. In the following, we present the explicit expressions of the Wilson coefficients as well as their numerical values from 1/R = 200 GeV up to 1/R = 1000 GeV in ACD model ( see also [6–8]).

In ACD model with a single universal extra dimension, the C₇eff(1/R) in leading log approximation is written as (see also [21]):

C₇eff(µb, 1/R) = η 16 23C7(µW, 1/R)+8 3 η1423−η 16 23 C8(µW, 1/R) + C2(µW) 8 X i=1 hiηai , (2.3)

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where the first and second arguments show the scale and dependency on the

compactifica-tion parameters, 1/R, respectively and, C2(µW) = 1 , C7(µW, 1/R) = − 1 2D ′_(x t, 1/R) , C8(µW, 1/R) = − 1 2E ′_(x t, 1/R) . (2.4) The functions, D′_(x

t, 1/R) and E′(xt, 1/R) are given as: D′(xt, 1/R) = D′₀(xt) + ∞ X n=1 D_n′(xt, xn), E′(xt, 1/R) = E′₀(xt) + ∞ X n=1 E_n′(xt, xn) , (2.5) where, D₀′(x_{t) = −}(8x 3 t + 5x2t − 7xt) 12(1 − xt)3 + x2 t(2 − 3xt) 2(1 − xt)4 ln xt, (2.6) E₀′(x_{t) = −}xt(x 2 t − 5xt− 2) 4(1 − xt)3 + 3x2 t 2(1 − xt)4 ln xt, (2.7) and ∞ X n=1 D_n′(xt, xn) = x_{t[37 − xt}(44 + 17xt)] 72(xt− 1)3 +πmWR 12 " Z 1 0 dy (2y1/2+ 7y3/2+ 3y5/2) coth(πmWR√y) −xt(2 − 3xt_(x )(1 + 3xt) t− 1)4 J(R, −1/2) −_(x 1 t− 1)4{xt (1 + 3x_{t) + (2 − 3xt)[1 − (10 − xt})x_{t]}J(R, 1/2)} −_(x 1 t− 1)4[(2 − 3xt )(3 + x_{t) + 1 − (10 − xt})xt]J(R, 3/2) −_(x(3 + xt) t− 1)4 J(R, 5/2) # , (2.8) ∞ X n=1 E_n′(xt, xn) = x_{t[17 + (8 − xt})xt] 24(xt− 1)3 +πmWR 4 " Z 1 0 dy (y1/2+ 2y3/2_{− 3y}5/2) coth(πmWR√y) −xt(1 + 3xt)_(x t− 1)4 J(R, −1/2) + 1 (xt− 1)4 [xt(1 + 3xt_{) − 1 + (10 − x}t)xt]J(R, 1/2) − 1 (xt_{− 1)}4[(3 + xt) − 1 + (10 − xt)xt)]J(R, 3/2) +(3 + xt) (xt_{− 1)}4J(R, 5/2) # , (2.9)

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with, J(R, α) = Z 1 0 dy yαcoth(πmWR√_{y) − x}1+α_t coth(πmtR√y) . (2.10)

The remaining parameters in eq. (2.3) are defined as: η = αs(µW) αs(µb) , (2.11) αs(x) = αs(mZ) 1 − β0αs(mZ) 2π ln(m Z x ) , (2.12)

where in fifth dimension, αs(mZ) = 0.118 and β0= 23₃. The coefficients ai and hi are given as [22,23]:

ai= ( 14₂₃, 16₂₃, ₂₃6 _{, −}12₂₃, _{0.4086, −0.4230, −0.8994,} 0.1456), hi= (2.2996, −1.0880, −3₇, −₁₄1, −0.6494, −0.0380, −0.0186, −0.0057).

(2.13) The Wilson coefficient C10 is given as:

C_{10(1/R) = −}Y (xt, 1/R) sin2_θ

W

, (2.14)

where, sin2θW = 0.23 and,

Y (xt, 1/R) = Y0(xt) + ∞ X n=1 Cn(xt, xn) , (2.15) with, Y0(xt) = xt 8 xt− 4 xt− 1 + 3xt (xt− 1)2 ln xt , (2.16) and, ∞ X n=1 Cn(xt, xn) = x_t(7−xt) 16(x_t−1) − πmWRxt 16(x_t−1)2 [3(1+xt)J(R, −1/2) + (xt−7)J(R, 1/2)] . (2.17) Finally, we consider the Wilson coefficient, Ceff

9 . It can be written in leading log approxi-mation as [22,23]: C₉eff(ˆs′, 1/R) = C₉NDR(1/R)η(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) , (2.18) where, ˆs′ ₌ q2 m2 b with 4m2_l _{≤ q}2 _{≤ (mΛ}b− mΛ) 2 _and, C₉NDR(1/R) = P₀NDR+ Y (xt) sin2θW − 4Z(xt ) + PEE(xt), (2.19)

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here, NDR stands for the naive dimensional regularization scheme. We neglect the last term

in this equation since due to the order of PE, the contribution of this term is negligibly small. The P₀NDR_{= 2.60 ± 0.25 [}22,23] and the function, Z(xt, 1/R) is defined as:

Z(xt, 1/R) = Z0(xt) + ∞ X n=1 Cn(xt, xn) , (2.20) where, Z0(xt) = 18x4 t − 163x3t + 259x2t − 108xt 144(xt_{− 1)}3 + 32x4 t − 38x3t − 15x2t + 18xt 72(xt_{− 1)}4 − 1 9 ln xt. (2.21) In eq. (2.18), η(ˆs′_{) = 1 +}αs(µb) π ω(ˆs ′_), _(2.22) 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′₎, (2.23) and at µb scale, Cj = 8 X i=1 kjiηai (j = 1, . . . 6) (2.24)

where 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). (2.25)

The remaining functions in eq. (2.18) are also given as: h(y, ˆs′_{) = −}8 9ln mb µb − 8 9ln y + 8 27 + 4 9x (2.26) −2 9(2 + x)|1 − x| 1/2    ln √ 1−x+1 √ 1−x−1 − iπ , for x ≡ 4zˆs′2 < 1 2 arctan√1 x−1, for x ≡ 4z2 ˆ s′ > 1, (2.27)

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where y = 1 or y = z = mc mb and, h(0, ˆs′) = 8 27− 8 9ln mb µb − 4 9ln ˆs ′₊4 9iπ. (2.28)

Numerical results show that the Wilson coefficients in UED differ from their SM values, considerably. In particular, the C10is enhanced and the C₇eff is suppressed (for more details see [21–23]).

2.2 Transition matrix elements and form factors

The decay amplitude of the Λb→ Λℓ+ℓ−is obtained sandwiching the aforementioned effec-tive Hamiltonian between the initial and final baryonic states. As a result, the transition matrix elements, hΛ(p) | ¯sγµ(1 − γ5)b | Λb(p + q)i and hΛ(p) | ¯siσµνqν_{(1 + γ}

5)b | Λb(p + q)i are appeared. In full theory of QCD, they can be parametrized in terms of twelve form factors, fi, gi, fiT and gTi (i running from 1 to 3) in the following manner:

hΛ(p)| ¯sγµ(1 − γ5)b | Λb(p + q)i = ¯uΛ(p)hγµf1(q2) + iσµνqνf2(q2) + qµf3(q2) − γµγ5g1(q2) −iσµνγ5qνg2(q2) − qµγ5g3(q2) i uΛb(p + q) , (2.29) hΛ(p)| ¯siσµνqν(1+γ5)b |Λb(p+q)i=¯uΛ(p) h γµf1T(q2)+iσµνqνf2T(q2)+qµf3T(q2)+γµγ5g1T(q2) +iσµνγ5qνg2T(q2) + qµγ5g3T(q2) i uΛb(p + q) . (2.30)

These form factors have been recently calculated in [12] using light cone QCD sum rules in full theory.

On the other hand, the aforesaid transition matrix elements in HQET is defined in terms of only two form factors, F1 and F2 as [24,25]:

hΛ(p) | ¯sΓb | Λb(p + q)i = ¯uΛ(p)[F1(q2)+ 6vF2(q2)]ΓuΛb(p + q), (2.31)

where Γ denotes any Dirac matrices, 6 v = (6p + 6q)/mΛb and the form factors, F1(q

2_{) and} F2(q2) have been calculated in [13]. Comparing the definitions of the transition matrix elements both in full and HQET theories, one can easily find relations among the form factors mentioned above (see [12,26–29]).

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3 Some observables related to the Λb _{→ Λℓ}+_ℓ−

3.1 Branching ratio

Using the decay amplitude discussed in the previous section, the angular and 1/R depen-dent differential decay rate can be written as (see [15,30,31]):

dΓ dˆsdz(z, ˆs, 1/R) = G2_Fα2_emmΛb 16384π5 |VtbVts∗| 2_v√_λ " T0(ˆ_{s, 1/R) + T1}(ˆ_{s, 1/R)z + T2}(ˆs, 1/R)z2 # , (3.1)

where z = cos θ, θ being the angle between the momenta of Λb and ℓ−in the center of mass of leptons, λ = λ(1, r, ˆs) = 1 + r2+ ˆs2_{− 2r − 2ˆs − 2rˆs, r = m}2_Λ/m2_Λ_b and v = q 1 −4m 2 ℓ q2 .

The functions, T0(ˆ_{s, 1/R), T1}(ˆ_{s, 1/R) and T2}(ˆs, 1/R) are given as (see also [12]):

T0(ˆs, 1/R) = 32m2_ℓm4_Λ_b_{s(1 + r − ˆs)}ˆ _|D3|2_{+ |E3|}2 (3.2) +64m2_ℓm3_Λ_b_{(1 − r − ˆs) Re[D}₁∗E3+ D3E1∗] +64m2_Λ_b√r(6m2_ℓ _{− m}2_Λ_bˆs)Re[D₁∗E1] +64m2_ℓm3_Λ_b√r2mΛbsRe[Dˆ ∗ 3E3] + (1 − r + ˆs)Re[D1∗D3+ E∗1E3] +32m2_Λ_b(2m2_ℓ + m2_Λ_bs)ˆn_{(1 − r + ˆs)mΛ}b √ r Re[A∗₁A2+ B1∗B2] −mΛb(1 − r − ˆs) Re[A ∗ 1B2+ A∗2B1] − 2 √ rRe[A∗₁B1] + m2Λbˆs Re[A ∗ 2B2] o +8m2_Λ_bn4m2_ℓ_{(1 + r − ˆs) + m}2_Λ_bh_{(1 − r)}2_{− ˆs}2io _|A1|2_{+ |B1|}2 +8m4_Λ_bn4m2_ℓh_{λ + (1 + r − ˆs)ˆs}i+ m2_Λ_bˆsh_{(1 − r)}2_{− ˆs}2io _|A2|2_{+ |B2|}2 −8m2Λb n 4m2_ℓ_{(1 + r − ˆs) − m}2_Λ_bh_{(1 − r)}2_{− ˆs}2io _|D1|2+ |E1|2 +8m5_Λ_bsvˆ 2n_{− 8mΛ}bsˆ √ r Re[D₂∗E_{2] + 4(1 − r + ˆs)}√r Re[D₁∗D2+ E1∗E2] −4(1 − r − ˆs) Re[D1∗E2+ D∗2E1] + mΛb h (1 − r)2− ˆs2i |D2|2+ |E2|2 o , T1(ˆ_{s, 1/R) = −16m}4_Λ_bˆsv√λn2Re(A∗₁D1) − 2Re(B1∗E1) +2mΛbRe(B ∗ 1D2− B2∗D1+ A∗2E1− A∗1E2) o +32m5_Λ_bˆs v√λnmΛb(1 − r)Re(A∗2D2− B2∗E2) +√rRe(A∗₂D1+ A∗1D2− B2∗E1− B1∗E2) o , (3.3) T2(ˆ_{s, 1/R) = −8m}4_Λ_bv2λ_|A1|2_{+ |B1|}2_{+ |D1}_|2_{+ |E1|}2 +8m6_Λ_bsvˆ 2λ_|A2_|2_{+ |B2|}2_{+ |D2|}2_{+ |E2}_|2, (3.4)

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where, A1 = 1 q2 f T 1 + gT1 −2mbC₇eff(1/R)+ (f1− g1) C9eff(ˆs, 1/R) A2 = A1(1 → 2) , A3 = A1(1 → 3) , B1 = A1 g1 → −g1; gT1 → −g1T , B2 = B1(1 → 2) , B3 = B1(1 → 3) , D1 = (f1_{− g}1) C10(1/R) , D2 = D1_{(1 → 2) ,} (3.5) D3 = D1_{(1 → 3) ,} E1 = D1(g1 → −g1) , E2 = E1(1 → 2) , E3 = E1(1 → 3) , (3.6)

and the relation between the ˆs′ _{used in the previous section and ˆ}_{s in the present section} is: ˆs′ = s mˆ

2

Λ_b

m2 b

. Integrating out the angular dependent differential decay rate, the following dilepton mass spectrum is obtained:

dΓ dˆs(ˆs, 1/R) = G2_Fα2_emmΛb 8192π5 |VtbVts|∗ 2_v√_λ T0(ˆs, 1/R) +1 3T2(ˆs, 1/R) . (3.7)

Integrating also the above equation over ˆs in the allowed physical region, 4m

2 ℓ

m2

Λ_b ≤ ˆs ≤

(1−√r)2, one can obtain the 1/R dependent total decay width. Multiplying the total decay rate to the lifetime of the Λb baryon, we obtain the 1/R dependent branching ratio. Using the numerical values, mt = 167 GeV , mW = 80.4 GeV , mZ = 91 GeV , mb = 4.8 GeV , mc = 1.46 GeV , µb = 5 GeV , µW = 80.4 GeV , |VtbVts∗| = 0.041, GF = 1.17 × 10−5 GeV−2, αem = ₁₃₇1 , τΛb = 1.383 × 10−12 s, mΛ = 1.115 GeV , mΛb = 5.620 GeV [32], me =

0.51 M eV , mµ= 0.1056 GeV and mτ = 1.771 GeV , we present the dependence of branching ratios on compactification factor, 1/R in figure1. From this figure, we deduce the following results:

• There is considerable discrepancy between the predictions of the ACD and SM models for low values of the compactification factor, 1/R. As 1/R increases, this difference tends to diminish so that for higher values of 1/R (1/R ≃ 1000 GeV ), the predictions of ACD become very close to the results of SM . Such discrepancy at low values of 1/R can be considered as a signal for the existence of extra dimensions.

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SM SM,HQET UED UED,HQET 200 400 600 800 1000 4.0 4.5 5.0 5.5 6.0 1R@GeVD BR HL b ® L e +e -Lx10 6 SM SM,HQET UED UED,HQET 200 400 600 800 1000 3.0 3.5 4.0 4.5 5.0 5.5 1R@GeVD BR HL b ® LΜ +Μ -Lx10 6 SM SM, HQET UED UED, HQET 200 400 600 800 1000 4 6 8 10 12 1R@GeVD BR HL b ® LΤ +Τ -Lx10 7

Figure 1. The dependence of branching ratios on compactification factor, 1/R.

• As it is expected, an increase in the lepton mass ends up in a decrease in the branching ratio.

• The order of magnitude of the branching ratio shows a possibility to study such channels at the LHC.

• The Λb → Λℓ+ℓ− _{transition is more probable, specially for τ case, in full theory in} comparison with HQET.

3.2 Forward backward asymmetry

The lepton forward-backward asymmetry is one of the promising tools in looking for new physics beyond the SM such as extra dimensions. This asymmetry is defined as:

AF B = Nf − Nb Nf + Nb

(3.8) where Nf is the number of events that particle is moving ”forward” with respect to any chosen direction, while Nb is the number of events for particle motion in ”backward” direc-tion. The forward-backward asymmetry AF B(ˆs, 1/R) is defined in terms of the differential decay rate as:

AF B(ˆs, 1/R) = Z 1 0 dΓ dˆsdz(z, ˆs, 1/R) dz − Z 0 −1 dΓ dˆsdz(z, ˆs, 1/R) dz Z 1 0 dΓ dˆsdz(z, ˆs, 1/R) dz + Z 0 −1 dΓ dˆsdz(z, ˆs, 1/R) dz . (3.9)

We depict the dependence of AF B(ˆs, 1/R) asymmetry on 1/R for different leptons and at a fixed value of ˆs = 0.5 common for allowed physical regions of all leptons in figure 2. A quick glance at these figures leads to the following results:

• The AF B is approximately the same for e and µ and about 2-2.5 times greater than that of τ case.

• As far as HQET is considered, there is considerable discrepancy between the pre-dictions of the ACD and SM models for low values of 1/R. As 1/R increases, this

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SM SM, HQET UED UED, HQET 200 400 600 800 1000 -0.70 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.35 -0.30 1RHGeVL AFB HL b ® L e +e -L SM SM,HQET UED UED,HQET 200 400 600 800 1000 -0.70 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.35 -0.30 1RHGeVL AFB HL b ® LΜ +Μ -L SM SM,HQET UED UED,HQET 200 400 600 800 1000 -0.28 -0.26 -0.24 -0.22 -0.20 -0.18 -0.16 -0.14 1R@GeVD AFB HL b ® LΤ +Τ -L

Figure 2_{. The dependence of A}F B(ˆs, 1/R) asymmetry on compactification factor, 1/R for different

leptons at ˆs = 0.5.

difference starts to diminish and at 1/R ≃ 1000 GeV , the two models have approx-imately the same results. In full theory, two models have approxapprox-imately the same predictions for all leptons and all 1/R values.

• For all leptons, the forward-backward asymmetries show considerable differences be-tween the full theory and HQET predictions.

3.3 Λ baryon polarizations

The definitions for polarizations of Λ baryon in Λb → Λℓ+ℓ−channel are given in [33]. Using those definitions, the 1/R-dependent normal (PN), transversal (PT) and longitudinal (PL) polarizations of the Λ baryon in the massive lepton case are found as (for the general model independent case see [29,34]):

PN(ˆs, 1/R) = 8πm3_Λ_bv√sˆ ∆(ˆs, 1/R) − 2mΛb(1 − r + ˆs) √ r Re[A∗₁D1+ B1∗E1] +mΛb(1 − √ r)[(1 +√r)2_{− ˆs]}mℓRe[(A2− B2)∗F1] +mℓ[(1 +√r)2− ˆs] Re[A∗1F1] +4m2_Λ_bˆs√r Re[A∗₁E2+ A2∗E1+ B1∗D2+ B2∗D1] −2m3Λbˆs √ r(1 − r + ˆs) Re[A∗2D2+ B2∗E2∗] +2mΛb(1 − r − ˆs) Re[A∗₁E1+ B₁∗D1] + m2_Λ_bsRe[Aˆ ∗₂E2+ B₂∗D2] −m2Λb[(1 − r) 2 − ˆs2] Re[A∗₁D2+ A∗2D1+ B1∗E2+ B2∗E1] −mℓ[(1 +√r)2_{− ˆs] Re[B}∗₁F1] , (3.10)

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PT(ˆs, 1/R) = − 8πm3 Λbv √ ˆ sλ ∆(ˆs, 1/R) mℓ Im[(A1+ B1)∗F1] −mℓmΛb h (1 +√r) Im[(A2+ B2)∗F1] i +m2_Λ_b_{(1 − r + ˆs)}Im[A∗ 2D1− A∗1D2] − Im[B2∗E1− B∗1E2] +2mΛb Im[A∗₁E1− B1∗D1] − m2Λbs Im[Aˆ ∗ 2E2− B2∗D2] , (3.11) PL(ˆs, 1/R) = 16m2_Λ_b√λ ∆(ˆs, 1/R) 8m2_ℓmΛb Re[D₁∗E3− D∗3E1] +√rRe[D∗1D3− E1∗E3)] +2mℓmΛb(1 + √ r)Re[(D1− E1)∗F2] −2mℓm2_Λ_bsˆnRe[(D3− E3)∗F2] + 2mℓ(|D3|2− |E3|2) o −4mΛb(2m 2 ℓ+ m2Λbs) Re[Aˆ ∗ 1B2− A∗2B1] −4₃m3_Λ_bsvˆ 23Re[D∗₁E2− D∗2E1] +√rRe[D1∗D2− E1∗E2] −4₃mΛb √ r(6m2_ℓ+ m2_Λ_bˆsv2) Re[A∗₁A2− B1∗B2] +1 3 n 3[4m2_ℓ + m2_Λ_b_{(1 − r + ˆs)](|A}1|2− |B1|2) − 3[4m2ℓ − m2Λb(1 − r + ˆs)] ×(|D1|2− |E1|2) − m2Λb(1 − r − ˆs)v 2 (|A1|2− |B1|2+ |D1|2− |E1|2) o −1₃m2_Λ_b_{12m2_ℓ_{(1 − r) + m}2_Λ_bˆ_{s[3(1 − r + ˆs) + v}2_{(1 − r − ˆs)]}(|A2}_|2_{− |B2|}2) −2₃m4_Λ_b_{s(2 − 2r + ˆs)v}ˆ 2_(|D2_|2_{− |E2|}2) , (3.12) where, ∆(ˆ_{s, 1/R) = T0}(ˆs, 1/R) +1 3T2(ˆs, 1/R). (3.13)

For instance, we show the dependence of the PN and PT polarizations of the Λ baryon on compactification factor at a fixed value of ˆs = 0.5 in figures3 and 4 , respectively. From these figures, we infer the following information:

• In the case of PN and all leptons, we observe a (25-35)% HQET violations. This violation is very small for the transverse polarization of the Λ.

• The UED predictions deviate considerably from the SM results in the case of PT and small values of the compactification factor. This deviation is small for the PN compared to the PT. In the case of τ and HQET, two models have approximately the same predictions for the normal polarization.

3.4 Double lepton polarization asymmetries

The double lepton polarization asymmetries related to the Λb → Λℓ+ℓ− transition are defined in [35] for general model independent form of the effective Hamiltonian. In our

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SM SM,HQET UED UED,HQET 200 400 600 800 1000 -0.55 -0.50 -0.45 -0.40 1RHGeVL PN HL b ® L e +e -L SM SM,HQET UED UED,HQET 200 400 600 800 1000 -0.55 -0.50 -0.45 -0.40 1RHGeVL PN HL b ® LΜ +Μ -L SM SM,HQET UED UED,HQET 200 400 600 800 1000 -0.24 -0.23 -0.22 -0.21 -0.20 -0.19 -0.18 1R@GeVD PN HL b ® LΤ +Τ -L

Figure 3. The dependence of PN(ˆs, 1/R) on compactification factor, 1/R for different leptons at

ˆ s = 0.5. SM SM,HQET UED UED,HQET 200 400 600 800 1000 1.8 ´10-6 2. ´10-6 2.2 ´10-6 2.4 ´10-6 2.6 ´10-6 1RHGeVL PT HL b ® L e +e -L SM SM,HQET UED UED,HQET 200 400 600 800 1000 0.00030 0.00035 0.00040 0.00045 0.00050 0.00055 0.00060 1RHGeVL PT HL b ® LΜ +Μ -L SM SM,HQET UED UED,HQET 200 400 600 800 1000 0.0026 0.0028 0.0030 0.0032 0.0034 0.0036 0.0038 0.0040 1RHGeVL PT HL b ® LΤ +Τ -L

Figure 4. The dependence of PT(ˆs, 1/R) on compactification factor, 1/R for different leptons at

ˆ s = 0.5.

case, in the rest frame of ℓ±_{, the 1/R-dependent double longitudinal, transverse and normal} asymmetries are obtained as (see also [36, 37]):

PLN(ˆs, 1/R)= 16πm4_Λ_bmˆℓ √ λ ∆(ˆs, 1/R)√sˆIm (1−r)(A∗1D1+B1∗E1)+mΛbs(Aˆ ∗ 1E3−A∗2E1+B1∗D3−B2∗D1) +mΛb √ rˆs(A∗₁D3+A∗2D1+B1∗E3+ B2∗E1)−m2Λbˆs 2_B∗ 2E3+A∗2D3 , (3.14) PN L(ˆs, 1/R)=− 16πm4 Λbmˆℓ √ λ ∆√sˆ Im (1−ˆrΛ)(A∗1D1+B1∗E1)+mΛbs(Aˆ ∗ 1E3−A∗2E1+B∗1D3−B2∗D1) −mΛbp ˆrΛˆs(A ∗ 1D3+A∗2D1+B1∗E3+B2∗E1) − m2Λbˆs 2_B∗ 2E3+A∗2D3 , (3.16) PLT(ˆs, 1/R) = 16πm4_Λ_bmˆℓ √ λv ∆(ˆs, 1/R)√ˆs Re (1 − r)|D1|2+ |E1|2− ˆsA1D1∗− B1E1∗ −mΛbsˆ h

B1D∗₂+ (A2+ D2_{− D}3)E∗₁_{− A}1E₂∗_{− (B}2− E2+ E3)D1∗ i +mΛb √ rˆshA1D∗2+ (A2+ D2+ D3)D1∗− B1E2∗− (B2− E2− E3)E1∗ i +m2_Λ_b_{s(1 − r)(A2}ˆ D∗₂_{− B2}E₂∗_{) − m}_Λ2_bsˆ2(D2D3∗+ E2E3∗) , (3.16)

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PT L(ˆs, 1/R) = 16πm4_Λ_bmˆℓ √ λv ∆√ˆs Re (1 − ˆrΛ)_|D1|2_{+ |E1|}2+ ˆsA1D∗1− B1E1∗ +mΛbˆs h B1D∗₂+ (A2_{− D}2+ D3)E1∗− A1E2∗− (B2+ E2− E3)D1∗ i −mΛbp ˆrΛˆs h A1D2∗+ (A2− D2− D3)D1∗− B1E2∗− (B2+ E2+ E3)E1∗ i −m2Λbˆs(1 − ˆrΛ)(A2D ∗ 2− B2E2∗) − m2Λbsˆ 2_(D 2D∗3+ E2E∗3) , (3.17) PLL(ˆs, 1/R) = 16m4 Λb 3∆(ˆs, 1/R)Re −6mΛb √ r(1 − r + ˆs)hs(1 + vˆ 2)(A1A∗2+ B1B2) − 4 ˆ∗ m2ℓ(D1D∗3+ E1E3∗) i +6mΛb(1 − r − ˆs) h ˆ

s(1 + v2)(A1B₂∗+ A2B₁∗) + 4 ˆm2_ℓ(D1E₃∗+ D3E₁∗)i +12√rˆs(1 + v2)A1B1∗+ D1E1∗+ m2ΛbˆsA2B ∗ 2 +12m2_Λ_bmˆ2_ℓ_{s(1 + r − ˆs)}ˆ _|D3|2_{+ |E}₃∗_|2 −(1 + v2)h1 + r2_{− r(2 − ˆs) + ˆs(1 − 2ˆs)}i_|A1|2_{+ |B1|}2 −h(5v2_−3)(1−r)2+ 4 ˆm2_ℓ(1 + r)+ 2ˆs(1 + 8 ˆm2_ℓ _{+ r) − 4ˆs}2i_|D1|2_{+ |E1|}2 −m2Λb(1 + v 2_)ˆ_sh_{2 + 2r}2 − ˆs(1 + ˆs) − r(4 + ˆs)i |A2|2+ |B2|2 −2m2Λbˆsv 2h_{2(1 + r}2_{) − ˆs(1 + ˆs) − r(4 + ˆs)}i_|D2|2 + |E2|2 +12mΛbˆs(1 − r − ˆs)v 2_D1E∗ 2 + D2E1∗ −12mΛb √ rˆ_{s(1 − r + ˆs)v}2D1D∗2+ E1E2∗ +24m2_Λ_b√rˆsˆsv2D2E2∗+ 2 ˆm2ℓD3E3∗ , (3.18) PN T(ˆs, 1/R) = 64m4 Λbλv 3∆(ˆs, 1/R)Im (A1D1∗+ B1E1∗) + m2Λbs(Aˆ ∗ 2D2+ B2∗E2) , (3.19) PT N(ˆs, 1/R) = − 64m4_Λ_bλv 3∆ Im (A1D∗1+ B1E1∗) + m2Λbˆs(A ∗ 2D2+ B2∗E2) , (3.20) PN N(ˆs, 1/R)= 32m4 Λb 3ˆs∆(ˆs, 1/R)Re 24 ˆm2_ℓ√rˆs(A1B1∗+ D1E1∗) −12mΛbmˆ 2 ℓ √ rˆ_{s(1 − r + ˆs)(A1}A∗₂+ B1B2∗) +6mΛbmˆ 2 ℓsˆ h mΛbs(1+r−ˆs)ˆ |D3|2+|E3|2+ 2√_{r(1−r+ˆs)(D1}D₃∗+E1E3∗) i +12mΛbmˆ 2 ℓs(1 − r − ˆs)(A1ˆ B∗2+ A2B1∗+ D1E3∗+ D3E1∗) −[λˆs + 2 ˆm2_ℓ(1 + r2_{− 2r + rˆs + ˆs − 2ˆs}2)]_|A1|2_{+ |B1|}2_{− |D1|}2_{− |E1|}2 +24m2_Λ_bmˆ2_ℓ√rˆs2(A2B2∗+ D3E3∗) − m2Λbλˆs 2_v2_|D2|2 + |E2|2 +m2_Λ_bˆ_{s{λˆs − 2 ˆ}m2_ℓ[2(1+r2_{) − ˆs(1 + ˆs) − r(4 + ˆs)]}}_|A2|2_+|B2|2 , (3.21)

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SM SM,HQET UED UED,HQET 200 400 600 800 1000 0.000028 0.000029 0.00003 0.000031 0.000032 0.000033 0.000034 0.000035 1RHGeVL PLN HL b ® L e +e -L SM SM, HQET UED UED,HQET 200 400 600 800 1000 0.0058 0.0060 0.0062 0.0064 0.0066 0.0068 0.0070 0.0072 1RHGeVL PLN HL b ® LΜ +Μ -L SM SM,HQET UED UED,HQET 200 400 600 800 1000 0.090 0.095 0.100 0.105 0.110 0.115 0.120 1RHGeVL PLN HL b ® LΤ +Τ -L

Figure 5. The dependence of PLN(ˆs, 1/R) on compactification factor, 1/R for different leptons at

ˆ s = 0.5. PT T(ˆs, 1/R) = 32m4_Λ_b 3ˆs∆(ˆs, 1/R)Re − 24 ˆm2_ℓ√rˆs(A1B1∗+ D1E1∗) −12mΛbmˆ 2 ℓ √ rˆ_{s(1 − r + ˆs)(D1}D∗₃+ E1E3∗)−24m2Λbmˆ 2 ℓ √ rˆs2(A2B2∗+ D3E3∗) −6mΛbmˆ 2 ℓsˆ h mΛbs(1+r−ˆs)ˆ |D3|2+ |E3|2−2√r(1−r+ˆs)(A1A∗₂+B1B2∗) i −12mΛbmˆ 2 ℓˆs(1 − r − ˆs)(A1B2∗+ A2B1∗+ D1E3∗+ D3E1∗) −[λˆs − 2 ˆm2_ℓ(1 + r2_{− 2r + rˆs + ˆs − 2ˆs}2)]_|A1_|2_{+ |B1|}2 +m2_Λ_bˆ_{s{λˆs + ˆ}m2_ℓ_{[4(1 − r)}2_{− 2ˆs(1 + r) − 2ˆs}2_]}_|A2|2+ |B2|2 +{λˆs − 2 ˆm2_ℓ_{[5(1 − r)}2_{− 7ˆs(1 + r) + 2ˆs}2_]}_|D1|2_{+ |E1|}2 −m2Λbλˆs 2_v2_|D2|2 + |E2|2 ) , (3.22)

where, ˆml = _mm_Λbl . As examples, we depict the 1/R dependence of some double lepton polarization asymmetries at a fixed value of ˆs = 0.5 in figures5–9. From these figures, we obtain the following conclusions:

• In all cases, there are substantial differences between predictions of the ACD and SM models in low values of the compactification parameter, 1/R.

• We observe overall considerable differences between predictions of the full QCD and HQET for double lepton polarization asymmetries.

• All polarization asymmetries have the same sign for all leptons except the PT T, which predicts a different sign for τ compared to the e and µ. In the case of e and µ and HQET, the PN N changes its sign around 1/R = 600 GeV . In PN N, the full QCD predicts different sign for the ACD and SM models for these two leptons although the SM results are very small.

At the end of this section, we would like to compare the full theory and HQET pre-dictions on some observables considering the errors of form factors. In figures 1–9, we

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SM SM,HQET UED UED,HQET 200 400 600 800 1000 0.00014 0.00015 0.00016 0.00017 0.00018 0.00019 0.00020 1RHGeVL PLT HL b ® L e +e -L SM SM,HQET UED UED,HQET 200 400 600 800 1000 0.020 0.025 0.030 0.035 0.040 0.045 0.050 1RHGeVL PLT HL b ® LΜ +Μ -L SM SM,HQET UED UED,HQET 200 400 600 800 1000 0.20 0.22 0.24 0.26 0.28 0.30 0.32 1RHGeVL PLT HL b ® LΤ +Τ -L

Figure 6. The dependence of PLT(ˆs, 1/R) on compactification factor, 1/R for different leptons at

ˆ s = 0.5. SM SM,HQET UED UED,HQET 200 400 600 800 1000 0.041 0.042 0.043 0.044 0.045 0.046 0.047 0.048 1RHGeVL PTN HL b ® L e +e -L SM SM,HQET UED UED,HQET 200 400 600 800 1000 0.040 0.042 0.044 0.046 0.048 0.050 1RHGeVL PTN HL b ® LΜ +Μ -L SM SM,HQET UED UED,HQET 200 400 600 800 1000 0.0180 0.0185 0.0190 0.0195 0.0200 1RHGeVL PTN HL b ® LΤ +Τ -L

Figure 7. The dependence of PT N(ˆs, 1/R) on compactification factor, 1/R for different leptons at

ˆ s = 0.5. SM SM,HQET UED UED,HQET 200 400 600 800 1000 -0.02 0.00 0.02 0.04 0.06 1RHGeVL PNN HL b ® L e +e -L SM SM,HQET UED UED,HQET 200 400 600 800 1000 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 1RHGeVL PNN HL b ® LΜ +Μ -L SM SM,HQET UED UED,HQET 200 400 600 800 1000 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 1R@GeVD PNN HL b ® LΤ +Τ -L

Figure 8. The dependence of PN N(ˆs, 1/R) on compactification factor, 1/R for different leptons at

ˆ s = 0.5. SM SM,HQET UED UED,HQET 200 400 600 800 1000 0.03 0.04 0.05 0.06 0.07 0.08 0.09 1RHGeVL PTT HL b ® L e +e -L SM SM,HQET UED UED,HQET 200 400 600 800 1000 0.03 0.04 0.05 0.06 0.07 0.08 0.09 1RHGeVL PTT HL b ® LΜ +Μ -L SM SM,HQET UED UED,HQET 200 400 600 800 1000 -0.75 -0.70 -0.65 -0.60 -0.55 -0.50 -0.45 1R@GeVD PTT HL b ® LΤ +Τ -L

Figure 9. The dependence of PT T(ˆs, 1/R) on compactification factor, 1/R for different leptons at

ˆ s = 0.5.

compared the results of two theories when the central values of the form factors are used. Now, in figures 10–12, we depict the dependence of some considered observables on com-pactification factor, 1/R at a fixed value of ˆs = 0.5 and compare predictions of two theories when the uncertainties of the form factors are taken into account. The red bands in these

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200 400 600 800 1000 0 2 4 6 8 10 12 1R@GeVD BR HL b ® LΜ +Μ -Lx10 6 200 400 600 800 1000 4 6 8 10 12 14 16 18 1R@GeVD BR HL b ® LΤ +Τ -Lx10 7

Figure 10. The dependence of branching ratios on compactification factor, 1/R, when errors of the form factors are considered. The red and blue bands belong to the full QCD and HQET, respectively. 200 400 600 800 1000 -0.6 -0.4 -0.2 0.0 0.2 1RHGeVL AFB HL b ® LΜ +Μ -L 200 400 600 800 1000 -0.3 -0.2 -0.1 0.0 0.1 0.2 1RHGeVL AFB HL b ® LΤ +Τ -L

Figure 11. The same as figure10, but for AF B asymmetries.

figures belong to the full theory and they are obtained considering the errors of the form factors presented in [12], while the blue bands correspond to the HQET and they are ob-tained using the errors of the form factors presented in [13]. Here, we should stress that the reported errors of the form factors in HQET are small comparing those presented in full QCD, hence the HQET bands are narrow comparing to the full theory bands. From figure10, we see a significant difference between the predictions of two theories for τ case, while in the µ case, the HQET band lies inside the full QCD region. In the case of AF B in figure 11, we see also considerable difference between delimited regions of full and HQET theories for both leptons. In the case of µ and PT T and PT polarizations (see figures 12 and 13), predictions of the HQET lie inside the full theory bands, but in the case of τ and PT T, the band of HQET is out of the band of full theory but very close to it. In PT polarization and τ , two bands partly coincide with each other.

4 Conclusion

We analyzed the branching ratio, forward-backward asymmetry, double lepton polarization asymmetries and polarization of the Λ baryon for the channel, Λ_b _{→ Λℓ}+_ℓ−_{in the universal}

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200 400 600 800 1000 0.00 0.05 0.10 0.15 1RHGeVL PTT HL b ® LΜ +Μ -L 200 400 600 800 1000 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 1RHGeVL PTT HL b ® LΤ +Τ -L

Figure 12. The same as figure10, but for PT T polarization.

200 400 600 800 1000 0.0003 0.0004 0.0005 0.0006 1RHGeVL PT HL b ® LΜ +Μ -L 200 400 600 800 1000 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 1RHGeVL PT HL b ® LΤ +Τ -L

Figure 13. The same as figure10, but for PT polarization.

extra dimension scenario using the form factors obtained from both full QCD and HQET. For each case, we compared the obtained results with predictions of the SM. In lower values of the compactification factor, we see considerable discrepancy between the UED and SM models. However, when 1/R grows, the results of UED tend to diminish and at 1/R = 1000 GeV , two models have approximately the same predictions. The order of magnitude for branching ratios shows a possibility to study this channel at LHCb. The obtained results for the branching fractions show also that this transition is more probable in full QCD compared to the HQET. For other observables, we see also overall substantial differences between predictions of the full theory and HQET specially when the central values of the form factors from both theories are used. Any measurements on the considered physical quantities in this manuscript and their comparison with our predictions, can give useful information about existing of extra dimensions.

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