Systematic analysis of the B-s -> f(0)l(+)l(-) in the universal extra dimension

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JHEP02(2012)021

Published for SISSA by Springer

Received: November 7, 2011 Revised: December 20, 2011 Accepted: January 18, 2012 Published: February 8, 2012

Systematic analysis of the B

s

→

f

0

ℓ

+

ℓ

−

in the

universal extra dimension

V. Bashirya and K. Azizib

a

Cyprus International University, Faculty of Engineering, Department of Computer Engineering, Nicosia, Northern Cyprus, Mersin 10, Turkey

b

Department of Physics, Do˘gu¸s University, Acıbadem-Kadık¨oy, 34722 Istanbul, Turkey

E-mail: bashiry@ciu.edu.tr,kazizi@dogus.edu.tr

Abstract: _{Using form factors enrolled to the transition matrix elements and calculated}

via light-cone QCD sum rules including next-to-leading order corrections in the strong

coupling constant, we provide a systematic analysis of the Bs → f0(980)ℓ+ℓ− both in the

standard and universal extra dimension models. In particular, we discuss sensitivity of the differential branching ratio and various double lepton polarization asymmetries on the compactification factor of extra dimension and show how the results of the extra dimension model deviate from the standard model predictions. The order of branching ratio makes this decay mode possible to be checked at LHCb in near future.

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Contents

1 Introduction 1

2 Transition matrix elements and observables related to the Bs→f0(980)

ℓ+_ℓ− channel 3

3 Numerical results 6

1 Introduction

It is well known that the decays of the Bs meson are very promising tools to constrain

the standard model (SM) parameters, serve to determine the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, enable us to understand the origin of the CP violation and help us search for new physics (NP) effects beyond the SM. One of the possible decay

modes of the Bs meson is the semileptonic Bs → f0(980)ℓ+ℓ−. This channel proceeds via

flavor-changing neutral currents (FCNC) transition of b → sℓ+_ℓ− _{at quark level, which is}

induced at loop level in the SM and is therefore sensitive to NP effects. The extra dimen-sions as NP effects can contribute to such loop level transitions and enhance the branching ratio. It has been shown that, in the presence of a single universal extra dimension (UED)

compactified on a circle with radius R, the branching ratios of the Bs→ η(

′₎

ℓ+ℓ−_{/ν ¯}_{ν and}

Bs → φν ¯ν which are also based on b → sℓ+ℓ−/ν ¯ν, increase significantly at lower values of

the compactification factor, 1/R [1,2].

In the present work, we investigate the effect of the UED on some physical observables

related to the semileptonic Bs → f0(980)ℓ+ℓ−. The UED with a single extra dimension

called the Appelquist, Cheng and Dobrescu (ACD) model [3], is a kind of extra dimension

(ED) [4–6] which allows the SM fields (both gauge bosons and fermions) to propagate in the

extra dimensions (for more details see for instance [7]). The ACD model has been previously

applied to many decay channels. For some of them see [8,9,9–21] and references therein.

In the SM, the effective Hamiltonian describing the Bs → f0(980)ℓ+ℓ− transition at

quark level can be written as:

Heff = GFαemVtbV ∗ ts 2√2π C₉eff¯sγµ(1 − γ5)b ¯ℓγµℓ + C10sγ¯ µ(1 − γ5)b ¯ℓγµγ5ℓ −2mbC7eff 1 q2siσ¯ µνq ν_{(1 + γ} 5)b ¯ℓγµℓ . (1.1)

where Vij are elements of the CKM matrix, αem is the fine structure constant, GF is

the Fermi constant, and C₇eff, C₉eff and C10 are Wilson coefficients. In the ACD model,

the form of the effective Hamiltonian remains unchanged but due to the interaction of the Kaluza-Klein (KK) particles with the usual SM particles and also with themselves, the

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1/R [GeV] C₇eff C10 C9eff

200 _{−0.195212 −5.61658 4.83239 + 3.59874i} 400 _{−0.266419 −4.65118} 4.7538 + 3.54366i 600 _{−0.283593 −4.43995} 4.7366 + 3.53161i 800 _−0.29003 _{−4.36279 4.73032 + 3.52721i} 1000 _{−0.293092 −4.32646 4.72736 + 3.52514i} SM _{−0.298672 −4.26087 4.72202 + 3.52139i}

Table 1. Numerical values for Ceff

7 , C10 and values of C9eff at transferred momentum square, q2_{= 14 for different values of 1/R as well as the SM}

Wilson coefficients are modified. This modification is done in [22–26] in leading logarithmic

approximation in a way that each Wilson coefficient is written in terms of some periodic functions as: F (xt, 1/R) = F0(xt) + ∞ X n=1 Fn(xt, xn), (1.2)

where, F0(xt) is ordinary SM part and the other part can be written in terms of the

compactification factor 1/R, by means of the following definitions:

xt= m2t/MW2 , xn= m2n/m2W, mn= n/R, (1.3)

where, mt is the mass of the top quark, MW is the mass of the W boson, mn is the mass

of the KK particles and n = 0 corresponds to the ordinary SM particles. Few comments about the lower bound of the compactification factor are in order. From the electroweak

precision tests, the lower limit for 1/R had been previously obtained as 250 GeV in [3,14]

if Mh ≥ 250 GeV expressing larger KK contributions to the low energy FCNC processes,

and 300 GeV if Mh ≤ 250 GeV. Analysis of the B → Xsγ transition and also anomalous

magnetic moment had shown also that the experimental data are in a good agreement

with the ACD model if 1/R ≥ 300 GeV [27, 28]. Taking into account the leading order

(LO) contributions due to the exchange of KK modes as well as the available

next-to-next-to-leading order (NNLO) corrections to also B(B → Xsγ) in the SM, the authors of [21]

have obtained a lower bound on the inverse compactification radius 600 GeV. Using the electroweak precision measurements and also some cosmological constraints, the authors

of [29] and [30] have found that the lower limit on compactification factor is in or above

the 500 GeV range. We will plot the physical observables under consideration in the range 1/R ∈ [200 − 1000]GeV just to clearly show how the results of the UED deviate from those of the SM and grow decreasing the 1/R .

The numerical values for different Wilson coefficients both in the SM and ACD model

in the range 1/R ∈ [200 − 1000]GeV are presented in table1. From this table, we see that

in the ACD model, the C10is enhanced and C7eff is suppressed considerably in comparison

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Here we should mention that, besides the aforementioned contributions, the Wilson

coefficient C₉eff receives also long distance contributions from J/ψ family parameterized

using Breit-Wigner ansatz [31], i.e.,

YLD= 3π α2 em C(0) X Vi=ψ(1s)···ψ(6s) æi Γ(Vi→ ℓ+ℓ−)mVi m2 Vi− q2− imViΓVi ,

where, C(0)= 0.362. As the phenomenological factors, æihave not known for the transition

under consideration, we will choose the values of the q2 _{which do not lie on the J/ψ family}

resonances. This is possible in the case of the differential branching ratio and double-lepton polarization asymmetries under consideration in this work, however, to calculate the total branching ratio as well as the average double-lepton polarization asymmetries,

which require integration over q2, one should take also into account such contributions.

For more details about the long distance contributions see for instance [32,33].

The layout of the paper is as follows. In the next section, we present the transition matrix elements expressed in terms of form factors and the formula for decay rate as well as various double lepton polarization asymmetries. In the last section, we numerically analyze the observables in terms of compactification factor, 1/R of the ACD model and discuss the results.

2 Transition matrix elements and observables related to the Bs → f0(980)ℓ+ℓ− channel

To find the amplitude of the decay channel in question, we need to sandwich the aforemen-tioned effective Hamiltonian between the initial and final states. As a result, we obtain

the following transition matrix elements defined in terms of form factors F0(q2), F1(q2)

and FT(q2): hf0(pf0)|¯sγµγ5b|Bs(pBs)i = −i F1(q2) Pµ− m2_Bs_{− m}2_f0 q2 qµ + F0(q2) m2_Bs_{− m}2_f0 q2 qµ , (2.1) hf0(pf0)|¯sσµνγ5qνb|Bs(pBs)i = − FT(q2) mBs+ mf0 h q2Pµ− (m2Bs− m2f0)qµ i , (2.2)

where P = pBs+ pf0 and q = pBs− pf0. For simplicity in some parts of calculations, it is

convenient to introduce the auxiliary form factors f+ and f−,

hf0(pf0)|¯sγµγ5b|Bs(pBs)i = −if+(q2)Pµ+ f−(q2)qµ (2.3) such that F1(q2) = f+(q2) , F0(q2) = f+(q2) + q 2 m2 Bs − m2f0 f−(q2) . (2.4)

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Fi(q2 = 0) ai bi

F1 0.238 ± 0.036 1.50+0.13−0.09 0.58+0.09−0.07

F0 0.238 ± 0.036 0.53+0.14−0.10 −0.36+0.09−0.08

FT 0.308 ± 0.049 1.46+0.14_−0.10 0.58+0.09_−0.07

Table 2. Bs→ f₀(980) transition form factors including next-to-leading order corrections [34].

The form factors, F1, F0 and FT are calculated via light-cone QCD sum rules both at the

leading order and the next-to-leading order corrections in [34]. We use the latter to analyze

the considered physical observables. The fit parametrization of the form factors including

next-to-leading order corrections in αs is given as [34]:

Fi(q2) =

Fi(0)

1 − aiq2/m2_Bs+ bi(q2/m2_Bs)2

, (2.5)

where Fi denotes any function among F1,0,T. The parameters ai and bi as well as Fi(0) are

given in table 2.

Now we proceed to calculate some observables such as differential decay rate and double lepton polarization asymmetries. With the matrix elements in terms of form factors one can easily obtain the 1/R-dependent differential decay rate as:

dΓ( ¯Bs→ f0ℓ+ℓ−) dq2 (q 2_{, 1/R)} = G 2 Fα2em|Vtb|2|Vts∗|2 √ λ 512m3 Bsπ5 v 3q2 6m2_ℓ_|C10(1/R)|2(m2Bs − m2f0)2F02(q2) +(q2+ 2m2_ℓ)λ C₉eff(q2, 1/R)F1(q2) + 2Ceff 7 (1/R)(mb− ms)FT(q2) mBs+ mf0 2 +|C10(1/R)|2(q2− 4m2ℓ)λF12(q2) , (2.6) with v = q 1 −4m 2 ℓ

q2 , λ = λ(m2_Bs, m2_f0, q2) with λ(a, b, c) = (a − b − c)2− 4bc and m_ℓ is the

lepton’s mass.

To calculate the double-polarization asymmetries, we consider the polarizations of both lepton and anti-lepton, simultaneously and introduce the following spin projection

operators for the lepton ℓ− _{and the anti-lepton ℓ}+ _{(see also [}₃₅_–₃₇_]):

Λ1= 1 2(1 + γ56s − i ) , Λ2=1 2(1 + γ56s + i ) , (2.7)

where i = L, N and T correspond to the longitudinal, normal and transversal polarizations,

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and anti-lepton: s−µ_L = 0, ~e_L− = 0, ~p− |~p−| , s−µ_N = 0, ~e_N− = 0, ~pf0 × ~p− |~pf0 × ~p−| , s−µ_T = 0, ~e_T− = 0, ~e− N × ~e − L , s+µ_L = 0, ~e_L+ = 0, ~p+ |~p+| , s+µ_N = 0, ~e_N+ = 0, ~pf0 × ~p+ |~pf0 × ~p+| , s+µ_T = 0, ~e_T+ = 0, ~e+ N × ~e + L , (2.8)

where ~p∓ are the three-momenta of the leptons ℓ∓ and ~pf0 is three-momentum of the final

f0 meson in the center of mass (CM) frame of ℓ−ℓ+. By Lorenz transformations, the

longitudinal unit vectors are boosted to the CM frame of ℓ−_ℓ+_,

s−µ_L CM = |~p−| mℓ , E~p− mℓ|~p−| , s+µ_L CM = |~p−| mℓ , − E~p− mℓ|~p−| , (2.9)

while the other two vectors are kept the same. Now, we define the double-lepton

polariza-tion asymmetries as [35–37]: Pij(ˆs) = dΓ dˆs(~s − i , ~s + j ) − dΓ dˆs(−~s − i , ~s + j ) ! − dΓ_dˆ_s(~s−_i _{, −~s}+_j_{) −}dΓ dˆs(−~s − i , −~s + j ) ! dΓ dˆs(~s − i , ~s+j ) + dΓ dˆs(−~s − i , ~s+j ) ! + dΓ dˆs(~s − i , −~s+j) + dΓ dˆs(−~s − i , −~s+j ) ! , (2.10)

where the subindex j also stands for the L, N or T polarization. The subindexses, i and j correspond to the lepton and anti-lepton, respectively. Using the above definitions, the various 1/R-dependent double lepton polarization asymmetries are obtained in the following way: PLL(ˆs, 1/R) = −4m 2 Bs 3∆(ˆs, 1/R)Re[−24m 2 Bsmˆ2l(1 − ˆrf0)C∗D + λ ′ m2Bs(1 + v2)|A|2 −12m2Bsmˆ2ls|D|ˆ 2+ m2Bs|C|2(2λ ′ − (1 − v2)(2λ′_{+ 3(1 − ˆr}f0)2))], (2.11) PLN(ˆs, 1/R) = −4πm 3 Bs √ λ′ ˆ s ˆ s∆(ˆs, 1/R) Im[−mBsmˆlsAˆ ∗ D − mBsmˆl(1 − ˆrf0)A∗C], (2.12) PN L(ˆs, 1/R) = −PLN(ˆs, 1/R), (2.13) PLT(ˆs, 1/R) = 4πm3_Bs√λ′ ˆ s ˆ s∆(ˆs, 1/R) Re[mBsmˆlv(1 − ˆrf0)|C| 2_{+ m} BsmˆlvˆsC∗D], (2.14)

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PT L(ˆs, 1/R) = PLT(ˆs, 1/R), (2.15) PN T(ˆs, 1/R) = − 8m2_Bsv 3∆(ˆs, 1/R)Im[2λ ′ m2_BsA∗C], (2.16) PT N(ˆs, 1/R) = −PN T(ˆs, 1/R), (2.17) PT T(ˆs, 1/R) = 4m2_Bs 3∆(ˆs, 1/R)Re[−24m 2 Bsmˆ2l(1 − ˆrf0)C∗D − λ ′ m2_Bs(1 + v2_)|A|2 −12m2Bsmˆ2ls|D|ˆ 2+ m2Bs|C|2{2λ ′ − (1 − v2)(2λ′_{+ 3(1 − ˆr}f0)2)}], (2.18) PN N(ˆs, 1/R) = 4m2_Bs 3∆(ˆs, 1/R)Re[24m 2 Bsmˆ2l(1 − ˆrf0)C∗D − λ ′ m2_Bs_{(3 − v}2_)|A|2 +12m2_Bsmˆ2_l_s|D|ˆ 2+ m2_Bs_|C|2_{2λ′ _{− (1 − v}2)(2λ′_{− 3(1 − ˆr}f0)2)}], (2.19) where, ˆs = _mq22 Bs , ˆrf0 = m2 f0 m2 Bs , ˆml= _mml Bs, λ ′ = λ(1, ˆrf0, ˆs) and ∆(ˆs, 1/R) = 4m 2 Bs 3 Re[24m 2 Bsmˆ2l(1 − ˆrf0)D∗C + λ ′ m2_Bs_{(3 − v}2_)|A|2+ 12m2_Bsmˆ2_lˆ_s|D|2 +m2_Bs_|C|2_{2λ′_{− (1 − v}2)(2λ′ _{− 3(1 − ˆr}f0)2)}], (2.20) with

A = A(ˆs, 1/R) = 2C₉eff(ˆs, 1/R)f+(ˆs) − 4C7eff(1/R)(mb+ ms)

FT(ˆs) mBs + mf0 , B = B(ˆs, 1/R) = 2C₉eff(ˆs, 1/R)f−(ˆs) +4C₇eff(1/R)(mb+ ms) FT(ˆs) (mBs+ mf0)ˆsm2Bs (m2_Bs_{− m}2_f0), C = C(ˆs, 1/R) = 2C10(1/R)f+(ˆs), D = D(ˆs, 1/R) = 2C10(1/R)f−(ˆs) . (2.21) 3 Numerical results

In this section, we numerically analyze the physical observables and discuss their sensitivity to the compactification factor of extra dimension. The main input parameters are form factors in the matrix elements whose fit parametrization are presented in the previous section. To proceed in numerical calculations, we also need to know the numerical values

of the other input parameters. We use the values: mt = 167 GeV, mW = 80.4 GeV,

mb = 4.8 GeV, ms = 0.14 GeV, mµ = 0.105 GeV, mτ = 1.778, |VtbVts∗| = 0.041, GF =

1.166 × 10−5 _GeV−2_{, α}

em = ₁₃₇1 , τBs = 1.42 × 10−12 s, mf0 = 0.980 GeV and mBs =

5.36 GeV.

Considering the central values of the form factors, we plot the dependence of the differential branching ratio and various double lepton polarization asymmetries for the

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200 400 600 800 1000 0.15 0.20 0.25 0.30 0.35 1R@GeVD 10 6´ d BR ß _ds HB s ® f0 Μ +Μ -L

Figure 1. Dependence of the branching ratio on the 1/R for muon channel at three fixed values of the ˆs. The blue, green and red lines belong to the values ˆs = 0.2, ˆs = 0.3 and ˆs = 0.5, respectively. The straight line shows the result of SM and the curve depicts the ACD model prediction in each case. 200 400 600 800 1000 0.12 0.13 0.14 0.15 0.16 0.17 1R@GeVD 10 6´ d BR ß _ds HBs ® f0 Τ +Τ -L

Figure 2. Dependence of the branching ratio on the 1/R for tau channel at two fixed values of the ˆ

s. The blue and green lines belong to the values ˆs = 0.5 and ˆs = 0.6, respectively. The straight line shows the result of SM and the curve depicts the ACD model prediction in each case.

Bs → f0ℓ+ℓ− decay channel on the compactification factor (1/R) of the extra dimension

in figures 1-8. As the results of the electron channel are very close to those of the µ, we will depict only the results of the µ and τ channels. As it is evident from the formulas in

the previous section that the observables depend on ˆs, we will present our results at three

fixed values of this parameter for the µ and two fixed values for τ channel in the allowed

kinematical region (4 ˆm2_ℓ _{≤ ˆs ≤ (1 −}p ˆrf0)2). Note that in each figure we see graphs of the

lines with the same colors. The straight line in each case depicts the result of the SM and the curve line stands for the ACD model prediction.

From these figures, we obtain the following results:

• There are considerable discrepancies between the results of the UED and SM pre-dictions at lower values of the compactification factor for all observables and both lepton channels. When 1/R is increased, the differences between the predictions of two models become small so that two models have approximately the same predictions at 1/R = 1000 GeV.

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200 400 600 800 1000 -0.982 -0.981 -0.980 -0.979 -0.978 -0.977 -0.976 1R@GeVD PLL HB s ® f0 Μ + Μ -L 200 400 600 800 1000 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 1R@GeVD PNN HB s ® f0 Μ +Μ -L

Figure 3. Dependence of the PLLand PN N on the 1/R for muon channel at three fixed values of the ˆ

s. The blue, green and red lines belong to the values ˆs = 0.2, ˆs = 0.3 and ˆs = 0.5, respectively. The straight line shows the result of SM and the curve depicts the ACD model prediction in each case.

200 400 600 800 1000 0.80 0.82 0.84 0.86 0.88 1R@GeVD PLL HB s ® f0 Τ +Τ -L 200 400 600 800 1000 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 1R@GeVD PNN HB s ® f0 Τ +Τ -L

Figure 4. Dependence of the PLL and PN N on the 1/R for tau channel at two fixed values of the ˆ

s. The green and red lines belong to the values ˆs = 0.5 and ˆs = 0.6, respectively. The straight line shows the result of SM and the curve depicts the ACD model prediction in each case.

200 400 600 800 1000 0.55 0.60 0.65 0.70 0.75 0.80 0.85 1R@GeVD PTT HB s ® f0 Μ + Μ -L 200 400 600 800 1000 0.000 0.005 0.010 0.015 0.020 1R@GeVD PLN HB s ® f0 Μ +Μ -L

Figure 5. The same as figure 3 but for PT T and PLN.

• An increase in the value of ˆs ends up in a decrease in the value of the differential branching ratio.

• The deviations of the UED results from those of the SM on double lepton polarization asymmetries are small in comparison with the deviation of the differential branching ratios from corresponding SM values. However these can not be overlooked.

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200 400 600 800 1000 -0.900 -0.895 -0.890 -0.885 -0.880 -0.875 1R@GeVD PTT HB s ® f0 Τ +Τ -L 200 400 600 800 1000 0.07 0.08 0.09 0.10 0.11 1R@GeVD PLN HB s ® f0 Τ +Τ -L

200 400 600 800 1000 0.15 0.16 0.17 0.18 0.19 0.20 0.21 1R@GeVD PLT HB s ® f0 Μ + Μ -L 200 400 600 800 1000 0.00 0.05 0.10 0.15 1R@GeVD PTN HB s ® f0 Μ +Μ -L

Figure 7. The same as figure 3 but for PLT and PT N.

200 400 600 800 1000 0.32 0.33 0.34 0.35 0.36 0.37 0.38 1R@GeVD PLT HB s ® f0 Τ +Τ -L 200 400 600 800 1000 0.016 0.018 0.020 0.022 0.024 1R@GeVD PTN HB s ® f0 Τ +Τ -L

• The contributions of KK modes enhance the absolute values of the PLL for τ as well

as the PN N, PT T and PLT for both lepton channels, but they decrease the absolute

values of the PLL for µ and PLN and PT N for both leptons.

• The PLL for µ and PT T for τ have negative signs but the rest of double lepton

polarization asymmetries have positive signs.

Now, we would like to discuss how the uncertainties of the form factors affect the physical quantities under consideration. For this aim, we plot the aforementioned physical observables on the compactification factor in figures 9-16 when the uncertainties of the

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200 400 600 800 1000 0.28 0.30 0.32 0.34 0.36 0.38 0.40 1R@GeVD 10 6´ d BR ß _ds HB s ® f0 Μ +Μ -L

Figure 9. Dependence of the branching ratio on the 1/R for muon channel at ˆs = 0.2 when errors of the form factors are taken into account. The straight band shows result of the SM and the curve band refers to the ACD model prediction.

200 400 600 800 1000 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 1R@GeVD 10 6´ d BR ß _ds HBs ® f0 Τ +Τ -L

Figure 10. Dependence of the branching ratio on the 1/R for tau channel at ˆs = 0.6 when errors of the form factors are taken into account. The straight band shows result of the SM and the curve band refers to the ACD model prediction.

200 400 600 800 1000 -0.9770 -0.9765 -0.9760 -0.9755 -0.9750 1R@GeVD PLL HB s ® f0 Μ + Μ -L 200 400 600 800 1000 0.65 0.70 0.75 0.80 0.85 1R@GeVD PNN HB s ® f0 Μ +Μ -L

Figure 11. Dependence of the PLLand PN N on the 1/R for muon channel at ˆs = 0.2 when errors of the form factors are taken into account. The straight bands show results of the SM and the curve bands refer to the the ACD model predictions.

form factors are taken into account. These figures are plotted at ˆs = 0.2 and ˆs = 0.6 for

the µ and τ channels, respectively. From figures 9 and 10 for the branching fractions, it is clear that the difference between the UED and SM models predictions exist and can not be

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200 400 600 800 1000 0.84 0.85 0.86 0.87 0.88 0.89 1R@GeVD PLL HB s ® f0 Τ +Τ -L 200 400 600 800 1000 0.90 0.91 0.92 0.93 0.94 0.95 0.96 1R@GeVD PNN HB s ® f0 Τ +Τ -L

Figure 12. Dependence of the PLL and PN N on the 1/R for tau channel at ˆs = 0.6 when errors of the form factors are taken into account. The straight bands show results of SM and the curve bands refer to the ACD model predictions.

200 400 600 800 1000 0.65 0.70 0.75 0.80 0.85 1R@GeVD PTT HB s ® f0 Μ +Μ -L 200 400 600 800 1000 0.0014 0.0015 0.0016 0.0017 0.0018 0.0019 1R@GeVD PLN HB s ® f0 Μ + Μ -L

200 400 600 800 1000 -0.910 -0.905 -0.900 -0.895 -0.890 -0.885 1R@GeVD PTT HB s ® f0 Τ +Τ -L 200 400 600 800 1000 0.070 0.075 0.080 0.085 0.090 0.095 0.100 1R@GeVD PLN HB s ® f0 Τ +Τ -L

killed by uncertainties of the form factors especially at lower values of the compactification factor. The figures 11-16 for double lepton polarization asymmetries also depict that except

the PT T and PT N in τ channel and PLT in both lepton channels, there are discrepancies

between two model predictions at lower values of the 1/R even if the errors of the form factors are encountered.

In conclusion, making use of the related form factors calculated via light-cone QCD

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200 400 600 800 1000 0.175 0.180 0.185 0.190 0.195 0.200 0.205 0.210 1R@GeVD PLT HB s ® f0 Μ +Μ -L 200 400 600 800 1000 0.013 0.014 0.015 0.016 0.017 1R@GeVD PTN HB s ® f0 Μ + Μ -L

200 400 600 800 1000 0.370 0.375 0.380 0.385 1R@GeVD PLT HB s ® f0 Τ +Τ -L 200 400 600 800 1000 0.014 0.016 0.018 0.020 0.022 0.024 1R@GeVD PTN HB s ® f0 Τ +Τ -L

differential branching ratio and various double lepton polarization asymmetries on the

com-pactification factor of extra dimension for the Bs → f0(980)ℓ+ℓ−transition. Our numerical

calculations depict considerable deviations of the extra dimension model results from the SM predictions. These differences can not be killed by errors of the form factors in the allowed regions of the compactification parameter previously discussed. Such discrepancies can be interpreted as signals for existing extra dimensions in nature which can be searched for at hadron colliders. As a final note it is worth to estimate the accessibility to measure the branching ratio and lepton polarization asymmetries. An observation of a 3 σ signal

for asymmetry of the order of the 1% needs about ∼ 1012 BB pairs. This allow us to¯

measure the branching ratio and the polarization asymmetries shown in the figures 1–16

in principle. The order of branching ratios show that the Bs→ f0(980)ℓ+ℓ−decay channel

both for µ and τ leptons can be detected at LHC. However, as experimentalists say, there are some technical difficulties to measure the lepton polarizations. In the case of µ, this lepton should be stopped in order to measure its polarizations which is not yet possible experimentally. For τ lepton, we should reconstruct then analyze the decay products of this lepton. In this case, we face with the problem of the efficiency of the reconstruction. If these technical difficulties over come, by measuring the considered double-lepton po-larization asymmetries, we can get valuable information about the nature of interactions included in the effective Hamiltonian because as the large parts of the uncertainties are

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canceled out, the ratio of physical observables such as CP, forward-backward asymmetry and single or double-lepton polarization asymmetries less suffer from the uncertainty of the form factors compared to the branching ratio.

Acknowledgments

We would like to thank T. M. Aliev for his useful discussions.

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