Rare semileptonic B-S decays to eta and eta ' mesons in QCD

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Rare semileptonic B

s

decays to and 0 mesons in QCD

K. Azizi,1,*R. Khosravi,2,†and F. Falahati2,‡

1_{Physics Division, Faculty of Arts and Sciences, Dog˘us¸ University, Acbadem-Kadko¨y, 34722 Istanbul, Turkey}

2_{Physics Department, Shiraz University, Shiraz 71454, Iran}

(Received 18 August 2010; published 1 December 2010)

We analyze the rare semileptonic Bs! ð; 0Þlþl, ðl ¼ e; ; Þ, and Bs! ð; 0Þ transitions

probing the ss content of the and 0mesons via three-point QCD sum rules. We calculate responsible

form factors for these transitions in full theory. Using the obtained form factors, we also estimate the related branching fractions and longitudinal lepton polarization asymmetries. Our results are in a good consistency with the predictions of the other existing nonperturbative approaches.

DOI:10.1103/PhysRevD.82.116001 PACS numbers: 11.55.Hx, 13.20.He

I. INTRODUCTION

Among B mesons, the Bs has been received special

attention, since experimentally it is expected that an abun-dant number of Bs will be produced at LHCb. This will

provide the possibility to study the properties of this meson and its various decay channels. The first evidence for Bs

production at the ð5SÞ peak was found by the CLEO Collaboration [1,2]. Recently, the Belle Collaboration measured the branching ratios of the Bs! J=c

transi-tion as well as the Bs! J=c decay via the ! and

! þ0 channels to reconstruct the meson [3]. Semileptonic decays of the Bsto the and 0, induced

by the rare flavor changing neutral current transition of b ! slþland b ! s are a crucial framework to restrict the standard model (SM) parameters. They can provide the possibility of extracting the elements of the Cabbibo-Kobayashi-Maskawa matrix and search for origin of the CP and T violations. As these transitions occur at the lowest order through one-loop penguin diagrams, they are a good context to search for new physics effects beyond the SM. Looking for supersymmetric particles [4], light dark matter [5] and fourth generation of quarks is possible via these transitions. These transitions are also useful in studying the structures of the and 0mesons.

In the present work, we analyze the semileptonic Bs!

ð; 0_Þlþ

l= decays considering also the ss content of the and 0 mesons in the framework of the three-point QCD sum rules. Here, we consider also the mixing between the and 0 states with a single mixing angle [6,7] as

ji ¼ cos’jqi sin’jsi

j0_{i ¼ sin’j}

qi þ cos’jsi;

(1)

where in the quark favor (QF) basis (for more details see for instance [8–10]),

jqi ¼

1 ffiffiffi 2

p ðj uui þ j ddiÞ; jsi ¼ jssi: (2)

The decay constants of qq and ss parts are defined in terms of the pion decay constant as [6]

fq¼ ð1:02 0:02Þf; fs¼ ð1:34 0:06Þf: (3)

We will use the mixing angle ’ ¼ ð41:5 0:3stat

0:7syst 0:6thÞ [11], which has recently been obtained

by the KLOE Collaboration in the QF basis via measur-ing the ratio ð!_ð!Þ0Þ. In the QF basis with the single mixing angle, the form factors of Bs! ð0Þ transitions

are defined in terms of the form factors Bs! s as

fBs!ð0Þ

i ¼ sin’ðcos’Þf

Bs!s

i ; (4)

and their branching fractions are also related to the branching ratio of Bs! s as follows:

BRfBs! ð0Þlþlg ¼ sin2’ðcos2’Þ

BRfBs! slþlg: (5)

The paper is organized as follows: sum rules for form factors responsible for considered transitions are obtained in Sec.II. SectionIIIis devoted to the numerical analysis of the form factors, branching ratios, and longitudinal lepton polarization asymmetries as well as our discussions. In this section, we also compare the obtained results with the existing predictions of the other nonperturbative approaches.

II. QCD SUM RULES FOR TRANSITION FORM FACTORS

As we previously mentioned, to calculate the form fac-tors responsible for the rare semileptonic Bs!

ð; 0_Þlþ

l, ðl ¼ e; ; Þ, and Bs! ð; 0Þ decays,

we need to calculate the form factors of Bs!

slþl= . To this aim, we start with the following

three-point correlation function, which is constructed from the vacuum expectation value of time ordered product

*kazizi@dogus.edu.tr

†_{khosravi.reza@gmail.com}

‡_{falahati@shirazu.ac.ir}

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T of interpolating fields of initial and final mesons and transition currents, JV _{and J}T_{, as follows:}

V;T ¼ i2 Z d4xd4yeipxeip0y_{h0jT fJ}s 5ðyÞJV;Tð0ÞJ y BsðxÞgj0i; (6) where p and p0 are the initial and final momentums, respectively, JBsðxÞ ¼ sðxÞ5bðxÞ and J

s

5ðyÞ ¼ sðyÞ5sðyÞ,

are the interpolating currents of the Bs and s states and

JV

ð0Þ ¼ sð0Þbð0Þ and JTð0Þ ¼ sð0Þqbð0Þ are the

vector and tensor transition currents extracted from the effective Hamiltonian responsible for the Bs!

slþl= decays. At quark level, these transitions are

governed by b ! slþland b ! s via penguin and box diagrams (see Fig. 1). The corresponding effective Hamiltonian is presented in terms of the Wilson coeffi-cients, Ceff7 ; Ceff9 and C10as

Heff ¼ GF 2pffiffiffi2VtbV ts Ceff9 sð1 5Þb ‘‘ þ C10sð1 5Þb ‘5‘ 2Ceff 7 mb q2 siq _{ð1 þ}₅_{Þb ‘} ‘ ; (7)

where GF is the Fermi constant, is the fine structure

constant at Z mass scale, and Vij are elements of the

Cabbibo- Kobayashi-Maskawa matrix. For the case, only the term containing C10 is considered. It should

be mentioned that because of the parity conservations, the axial vector and pseudotensor currents do not contri-bute to the pseudoscalar-pseudoscalar hadronic matrix element, i.e., hPðp0_{Þ j J}AV ¼ s5b j BsðpÞi ¼ 0; hPðp0_{Þ j J}PT ¼ siq5b j BsðpÞi ¼ 0; (8) where P stands for the ð0Þ meson.

From the general aspect of the QCD sum rules, we calculate the aforementioned correlation function in two different ways. First, in the hadronic representation, it is calculated in the timelike region in terms of hadronic parameters called the phenomenological or physical side. Second, it is calculated in the spacelike region in terms of QCD degrees of freedom called the QCD or theoretical

side. The sum rules for the form factors can be obtained equating the coefficient of the selected structures from these two representations of the same correlation function through the dispersion relation and applying the double Borel transformation with respect to the momentums of the initial and final states to suppress the contributions coming from the higher states and continuum.

In order to obtain the phenomenological representation of the correlation function given in Eq. (6), two complete sets of intermediate states with the same quantum numbers as the interpolating currents Js and JBs are inserted to sufficient places. As a result of this procedure, we obtain V;T ðp2; p02; q2Þ ¼h0 j Js5j Pðp 0_ÞihPðp0_{Þ j J}V;T j BsðpÞihBsðpÞ j J y Bs j 0i ðp02_m2 PÞðp2 m2BsÞ þ ; (9)

where represents the contributions coming from the higher states and continuum. The following matrix ele-ments h0jJBsjPi and h0jJ

s

5jPi are defined in terms of the

leptonic decay constant and four parameters hs P as h0jJBsjBsi ¼ i fBsm 2 Bs mbþ ms ; h0jJs 5jPi ¼ i h s P 2ms ; (10) where correlating the hs

P to fs and fq, the values hs¼

0:053 GeV3 _{and h}s

0 ¼ 0:065 GeV3 are obtained (for

details see [6]). From the Lorentz invariance and parity considerations, the remaining matrix element, i.e., transi-tion matrix element in Eq. (9) is parameterized in terms of form factors in the following way:

hPðp0_{Þ j J}V j BsðpÞi ¼ Pfþðq2Þ þ qfðq2Þ; hPðp0_{Þ j J}T j BsðpÞi ¼ fTðq2Þ mBs þ mP ½Pq2 qðm2Bs m 2 PÞ; (11)

where fþðq2Þ, fðq2Þ and fTðq2Þ are the transition form

factors, which only depend on the momentum transfer squared q2,P ¼ ðp þ p0Þ and q ¼ ðp p0Þ.

Using Eqs. (10) and (11) in Eq. (9), we obtain V ðp2;p02;q2Þ ¼ fBsm 2 Bs 2msðmbþmsÞ hsP ðm2 Pp02Þðm2Bsp 2_Þ ½fþðq2ÞPþ fðq2Þq; T ðp2;p02;q2Þ ¼ fBsm 2 Bs 2msðmbþmsÞ hsP ðm2 Pp02Þðm2Bsp 2_Þ fTðq2Þ ðmBsþmPÞ ðq2_P ðm2Bsm 2 PÞqÞ : (12) B_s(p) b s s l− l+ 5 5 (Z0 ) B_s(p) 5 s 5 b s l− l+ u, c, t u, c, t u, c, t W W (a) (b) ( ) ) ( ( ) ( )− W l (−)

FIG. 1. Diagrams responsible for the Bs! ð; 0Þlþl=

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For extracting the sum rules for form factors fþðq2Þ and

fðq2Þ, we choose the coefficients of the structures Pand

qfrom Vðp2; p02; q2Þ, respectively, and the structure q

from T

ðp2; p02; q2Þ is considered to calculate the form

factor fTðq2Þ. Therefore, the correlation functions are

writ-ten in terms of the selected structures as V

ðp2; p02; q2Þ ¼ þPþ q;

T

ðp2; p02; q2Þ ¼ Tq:

(13) Now, we focus our attention to calculating the QCD side of the correlation function. This side is calculated at deep Euclidean space, where p2 ! 1 and p02! 1 via the operator product expansion (OPE). To this aim, we write each _ifunction (coefficient of each structure) in terms of the perturbative and nonperturbative parts as

_i¼ per_i þ non_i -per_; (14) where i stands for þ, , and T. The perturbative part is written in terms of the double dispersion integral as

per_i ¼ 1 ð2Þ2 Z ds0Z ds per i ðs; s 0 ; q2Þ ðs p2_Þðs0_p02_Þ þ subtraction terms; (15)

where the per_i ðs; s0; q2Þ are called spectral densities. To get the spectral densities, we need to evaluate the bare loop diagrams in Fig. 1. Calculating these diagrams via the usual Feynman integrals with the help of the Cutkosky rules, i.e., _p2_m1 2! 2ðp2 m2Þ, which implies that all quarks are real, leads to the following spectral densities:

perþ ðs; s0; q2Þ ¼ I0Ncf þ s0 2m2sþ 2mbmsþ ðE1þ E2Þug;

per ðs; s0; q2Þ ¼ I0Ncf þ s0þ 2m2s 2mbmsþ ðE1 E2Þug;

per_T ðs; s0; q2Þ ¼ I0Ncfðmb msÞ þ s0ðms mbÞ þ 2mss

þ 2½mbðE1 E2Þ þ msðE2 E1 1Þs0þ ðE1 E2Þðms mbÞug; (16)

where I0ðs;s0; q2Þ ¼ 1 41=2_ðs;s0 ; q2Þ; ðs; s0; q2Þ ¼ s2þ s02þ q4 2sq2 2s0q2 2ss0; E1¼ 1 ðs; s0; q2Þ½2s 0_s0 u; E2¼ 1 ðs; s0; q2Þ½2ss 0_u; u ¼ s þ s0 q2; ¼ s þ m2 s m2b; (17)

and Nc ¼ 3 is the color factor.

For the calculation of the nonperturbative contributions in the QCD side, the condensate terms of OPE are consi-dered. The condensate term of dimension three is related to contribution of quark condensate. Figure 2shows quark-quark condensate diagrams of dimension three. It should be reminded that the quark condensate are considered only for light quarks, and the heavy quark condensate is sup-pressed by inverse powers of the heavy quark mass. The contribution of diagram (c) in Fig.2is zero since applying the double Borel transformation with respect to both var-iables p2 and p02 kills its contribution, because only one variable appears in the denominator in this case. Therefore as dimension three, we consider only diagram (d) in Fig.2. The dimension four operator in OPE is the gluon-gluon condensate. Our calculations show that in this case, the gluon-gluon condensate contributions are very small in comparison with the quark-quark and quark-gluon con-densates contributions, and we can easily ignore their

contributions. The next operator is dimension five quark-gluon condensate. The diagrams corresponding to the quark-gluon condensate are presented in Fig. 3. Contributions of the diagrams (e) and (f ) vanish for the same reason as diagram (c) in Fig. 2. Therefore, only diagrams (g) and (h) contribute to the nonperturbative part of dimension five. In the QCD sum rule approach, the OPE is truncated at some finite order such that Borel transformations play an important role in this cutting. Mainly, the proper regions of the Borel parameters are adopted by demanding that in the truncated OPE, the condensate term with the highest dimension constitutes a small fraction of the total dispersion integral. In the next section, we will explain how these proper regions are obtained. Hence, we will not consider the condensates with d 6 that play a minor role in our calculations.

The explicit expressions of non_i -per, are given in the Appendix. b s s 5 ( ) 5 5 5 s 5 b s ( ) q ( ₅) ₅ q ( ₅) s s (c) (d)

;

FIG. 2. Quark-quark condensate diagrams.

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The next step is to apply the double Borel transforma-tions with respect to the p2ðp2! M12Þ and p02ðp02! M22Þ

on the phenomenological as well as the perturbative and nonperturbative parts of the QCD side and equate the two representations. As a result, the following sum rules for the form factors are derived:

f_i0ðq2Þ ¼ðmbþ msÞð2msÞ fBsm 2 Bsh s P em2Bs=M21_em2_P=M22 1 42 Zs00 2m2 s ds0Zs0 sL dsperi ðs; s 0 ; q2Þes=M12_es0=M22 þ ~Bnoni -perðp2; p02; q2Þ ; (18) where f0þðq2Þ ¼ fþðq2Þ, f0 ðq2Þ ¼ fðq2Þ and f0_Tðq2Þ ¼ fTðq2ÞðmBs mPÞ. The s0 and s 0

0 are the continuum

thresholds in initial and final channels, respectively, and sLis the lower limit of the integral over s. It is obtained as

sL¼ ðm2 sþ q2 m2b s0Þðm2bs0 q2m2sÞ ðm2 b q2Þðm2s s0Þ : (19)

Also the operator ~B in Eq. (18) is defined as ~

B ¼ Bp2ðM21ÞBp02ðM22Þ; (20)

where M12and M22are Borel mass parameters. It should be

also noted that to subtract the contributions of the higher states and the continuum the quark-hadron duality assump-tion is also used,

higherstatesðs; s0Þ ¼ OPEðs; s0Þ ðs s0Þ ðs0 s00Þ: (21)

III. NUMERICAL ANALYSIS

We are now ready to present our numerical analysis of the form factors fþðq2Þ, fðq2Þ, and fTðq2Þ and calculate

branching fractions and longitudinal lepton polarization asymmetries. In our numerical calculations, we use the following values for input parameters: ms¼ 0:13 GeV,

mb¼ 4:8 GeV, m¼ ð547:51 0:18Þ MeV, m0 ¼ ð957:78 0:14Þ MeV, mBs ¼ ð5366:3 0:6Þ MeV [12], jVtbVtsj ¼ 0:0385, Ceff7 ¼ 0:313, C9¼ 4:344, C10¼ 4:669 [13], fBs ¼ ð209 38Þ MeV [14], m 2 0¼ ð0:8

0:2Þ GeV2_, _{hssi ¼ ð0:8 0:2Þhu ui,} _and _{hu ui ¼}

ð0:240 0:010Þ3 _GeV3_.

The sum rules for the form factors contain also four auxiliary parameters, namely, Borel mass squares, M21and

M22 and continuum thresholds, s0 and s0. These are not

physical quantities, so our results should be independent of them. The parameters s0 and s00are not totally arbitrary

but they are related to the energy of the first excited states with the same quantum numbers as the interpolating cur-rents. They are determined from the conditions that guar-antee the sum rules to have the best stability in the allowed M12 and M22 regions. The value of continuum threshold s0

calculated from the two-point QCD sum rules are taken to be s0 ¼ ð34:2 2Þ GeV2 [15]. We use also the range,

ðmPþ 0:3Þ2 s00 ðmPþ 0:5Þ2 GeV2 in the P ¼ ð0Þ

channel. The working regions for M12 and M22 are

deter-mined demanding that not only the contributions of the higher states and continuum are effectively suppressed, but contributions of the higher dimensional operators are also small. Both conditions are satisfied in the regions 12 GeV2 M₁2 22 GeV2 and 4 GeV2 M2₂ 10 GeV2_.

The dependence of the form factors fþ, f and fT on

M12 and M22 for Bs! s transition when mP¼ m are

shown in Fig. 4. Figure 5, also depicts the dependence of the same form factors on Borel mass parameters for Bs! sdecay when mP¼ m0. These figures show good

stability of the form factors with respect to the Borel mass parameters in the working regions. Using these regions for M12 and M22, our numerical analysis shows that the

contri-bution of the nonperturbative part to the QCD side is about 21% of the total, and the main contribution comes from the perturbative part.

Now, we proceed to present the q2 dependency of the form factors. Since the form factors fðq2Þ and fTðq2Þ are

calculated in the spacelike (q2< 0) region, we should analytically continue them to the timelike (q2> 0) or physical region. Hence, we should change q2 to q2. As we previously mentioned, the form factors are truncated at approximately 1 GeV below the perturbative cut. Therefore, to extend our results to the full physical region, we look for parametrization of the form factors in such a way that in the reliable region the results of the parame-trization coincide with the sum rules predictions. Our b s s 5 ( ) 5 5 5 s 5 b s ( ) q ₅ 5 q 5 ( ) ( ) s s s s s s s s s 5 5 5 5 5 q 5 5 q 5 ( ) ( ) ( ) ( ) b b

;

G G G G (e) (f) ) h ( ) g (

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numerical calculations show that the sufficient parametri-zation of the form factors with respect to q2 is

fiðq2Þ ¼

fið0Þ

1 þ ^q þ ^q2; (22)

where ^q ¼ q2=m2Bs. The values of the parameters fið0Þ, , and are given in the Table Itaking M21¼ 12 GeV2 and

M22¼ 5 GeV2. TableIalso contains the predictions of the

light-front quark model (LFQM). The form factors of Bs! and Bs! 0 are obtained using the values in

TableIand also Eq. (4).

FIG. 4. The dependence of the form factors on M21and M22for Bs! sdecay when mP¼ m. The solid, dashed, and dashed-dotted

lines correspond to the fþ, f, and fT, respectively.

FIG. 5. The dependence of the form factors on M21and M22for Bs! sdecay when mP¼ m0. The solid, dashed, and dashed-dotted

lines correspond to the fþ, f, and fT, respectively.

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The values of the form factors at q2 ¼ 0 are also com-pared with the predictions of the other nonperturbative approaches such as, the LFQM and constituent quark model (CQM) in TableII.

Now, we would like to evaluate the branching ratios for the considered decays. Using the parametrization of these transitions in terms of the form factors, we get [17] d dq2ðBs! P Þ ¼ AG2FjVtsVtbj2m3Bs 2 285 jDðxtÞj2 sin4 W 3=2ð1; ^r; ^sÞjfþðq2Þj2; d dq2ðBs! Pl þ lÞ ¼AG 2 FjVtsVtbj2m3Bs 2 3 295 v1=2ð1; ^r; ^sÞ 1 þ2^l ^s ð1; ^r; ^sÞ 1þ 12^l1 ; (23)

where A ¼ sin2’ for Bs! and A ¼ cos2’ for Bs! 0transitions. The ^r, ^s, ^l, xtand ^mband the functions v, ð1; ^r; ^sÞ,

DðxtÞ, 1and 1 are defined as

^r ¼ m2P m2_B_s; ^s ¼ q2 m2_B_s; ^l ¼ m 2 l m2_B_s; xt¼ m 2 t m2_W; ^mb¼ m b mBs ; v ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4^l ^s s ; ð1; ^r; ^sÞ ¼ 1 þ ^r2þ^s2 2^r 2^s 2^r ^s; DðxtÞ ¼ x₈t2 þ xt xt 1 þ 3xt 6 ðxt 1Þ2 lnxt ; 1¼ Ceff9 fþðq2Þ þ 2 ^_m_b_Ceff 7 fTðq2Þ 1 þpffiffiffi^r 2 þjC10fþðq2Þj2; 1¼ jC10j2 1 þ ^r ₂^sjfþðq2Þj2þ ð1 ^rÞReðfþðq2Þfðq2ÞÞ þ1₂^sjfðq2Þj2 : (24)

Integrating Eq. (23) over q2 in the whole physical region and using the total mean lifetime Bs ¼ ð1:466 0:059Þ ps [12], the branching ratios of the Bs!

ð; 0_Þlþ

l= are obtained as presented in Table III. In Table III, we show only the values obtained

consid-ering the short-distance (SD) effects contributing to the Wilson coefficient Ceff9 for the charged lepton case. The

effective Wilson coefficient Ceff9 , including both the SD

and long distance (LD) effects, is [13]

Ceff9 ðsÞ ¼ C9þ YSDðsÞ þ YLDðsÞ: (25)

TABLE I. Parameters appearing in the fit function for form factors of Bs! s in two

approaches.

Bs! sðP ¼ Þ Bs! sðP ¼ 0Þ

Parameters This work LFQM [8] Parameters This work LFQM [8]

fþð0Þ 0:4 0:1 0.291 fþð0Þ 0:3 0:1 0.291 0:3 0:1 1:574 0:5 0:2 1:575 0:7 0:2 0.751 0:8 0:3 0.770 fð0Þ 0:2 0:1 0:231 fð0Þ 0:2 0:1 0:225 0:8 0:3 1:582 1:0 0:3 1:570 0:2 0:1 0.825 0:05 0:02 0.835 fTð0Þ 0:4 0:1 0:280 fTð0Þ 0:4 0:1 0:300 0:5 0:1 1:561 0:6 0:2 1:561 0:4 0:1 0.782 0:4 0:1 0.802

TABLE II. The form factors of the Bs! sdecay for M21¼ 12 GeV2and M22¼ 5 GeV2at

q2¼ 0 in different approaches: this work (3PSR), the LFQM, and CQM.

Mode Form factors This work LFQM [8] LFQM [16] CQM [16]

fþð0Þ 0:4 0:1 0.291 0.354 0.357 Bs! sðP ¼ Þ fð0Þ 0:2 0:1 0:231 0:360 0:304 fTð0Þ 0:4 0:1 0:280 0:369 0:365 fþð0Þ 0:3 0:1 0.291 0.354 0.357 Bs! sðP ¼ 0Þ fð0Þ 0:2 0:1 0:225 0:324 0:304 fTð0Þ 0:4 0:1 0:300 0:404 0:390

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The LD effect contributions are due to the J=c family. The explicit expressions of the YSDðsÞ and YLDðsÞ can

be found in [13] (see also [18]). Table III also includes a comparison between our results and the predictions of the other approaches, including the LFQM, CQM, and other methods [9]. Note that the results of [9] are not the results directly obtained by analysis of the Bs!

ð0Þ, but they have been found relating the form factors of Bs! s to the form factors of B ! K using

the quark flavor scheme (see [9]). Hence, the compari-son of our results with the predictions of [9] is an approximate and for the exact comparison, the form factors should be directly available. In Table III, the set A refers to the values computed using short-distance QCD sum rules, set B shows the results obtained by light-cone QCD sum rules, and set C corresponds to the results calculated via light-cone QCD sum rules within the soft collinear effective theory. From Table III, we see good consistency in the order of magnitude between our results and the predictions of the other nonpertur-bative approaches. Here, we should also stress that the results obtained for the electron are very close to the results of the muon and for this reason, we only present the branching ratios for muon in our Tables.

In this section, we would like to present the branching ratios, including the LD effects. We introduce some cuts around the resonances of J=c andc0and study the follow-ing three regions for muon:

I : ffiffiffiffiffiffiffiffiffi_q2 min q qffiffiffiffiffi_q2 _M J=c 0:20; II: MJ=c þ 0:04 ffiffiffiffiffi q2 q Mc0 0:10; III: M_c0þ 0:02 ffiffiffiffiffi q2 q mBs mP: (26)

and for tau

I : ffiffiffiffiffiffiffiffiffiq2min q ffiffiffiffiffiq2 q Mc0 0:02; II: Mc0 þ 0:02 ffiffiffiffiffi q2 q mBs mP; (27)

TABLE III. The branching ratios in different models corresponding to ’ ¼ 41:5. The values in parentheses related to ’ ¼ 39:3.

Mode This work LFQM [8] LFQM [16] CQM [16] _{Set A [}9] Set B [9] Set C [9]

BrðBs ! Þ 106 1:4 0:6 1.54 2.56(2.34) 2.38(2.17) 0:95 0:2 2:2 0:7 2:9 1:5 BrðBs ! 0 Þ 106 1:3 0:6 1.47 2.36(2.52) 2.23(2.38) 0:9 0:2 1:9 0:5 2:4 1:3 BrðBs ! þÞ 107 2:3 1:0 2.09 3.75(3.42) 3.42(3.12) 1:2 0:3 2:6 0:7 3:4 1:8 BrðBs ! 0þÞ 107 2:2 1:0 1.98 3.40(3.63) 3.19(3.41) 1:1 0:3 2:2 0:6 2:8 1:5 BrðBs ! þÞ 108 3:7 1:6 5.14 7.33(6.70) 7.33(6.70) 3 0:5 8 1:5 10 5:5 BrðBs ! 0þÞ 108 2:8 1:2 2.86 4.66(5.00) 4.04(4.30) 1:55 0:3 3:85 0:75 4:7 2:5

TABLE IV. The branching ratios of the semileptonic Bs! ð; 0Þþ decays, including

the LD effects.

Mode I II III

BrðBs! þÞ ð1:8 0:7Þ 107 ð2:2 0:9Þ 108 ð2:3 0:9Þ 108

BrðBs! 0þÞ ð1:8 0:7Þ 107 ð2:2 0:9Þ 108 ð1:3 0:5Þ 108

TABLE V. The branching ratios of the semileptonic Bs!

ð; 0_Þþ _{decays, including the LD effects.}

Mode I II

BrðBs ! þÞ ð0:4 0:2Þ 109 ð3:2 1:3Þ 108

BrðBs ! 0þÞ ð0:4 0:2Þ 109 ð2:3 0:9Þ 108

FIG. 6. The dependence of the differential branching fraction

of the Bs! þ decay with and without the LD effects on

q2. The solid and dotted lines show the results without and with

the LD effects, respectively.

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where ffiffiffiffiffiffiffiffiffi q2min

q

¼ 2ml. In Tables IVand V, we present the

branching ratios for muon and tau obtained using the regions shown in Eqs. (26) and (27), respectively. The errors presented in TablesIII,IV, andVare due to uncer-tainties in the determination of the auxiliary parameters, errors in input parameters, systematic errors in QCD sum

rules as well as the errors associated to the following approximations used in the present work: (a) the form factors are calculated in the low q2region and extrapolated to high q2 using the fit parametrization in Eq. (22), (b) the hadronic operators in the considered Hamiltonian can receive sizable nonfactorizable corrections, and the

FIG. 7. The same as Fig.6but for the Bs! þ.

FIG. 8. The same as Fig.6but for the Bs! 0þ.

FIG. 9. The same as Fig.6but for the Bs ! 0þ.

FIG. 10. The dependence of the differential branching fraction

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corresponding matrix elements may also be sensitive to the isosinglet content of the and 0 mesons. We show the dependency of the differential branching ratios on q2(with and without LD effects for the charged lepton case) in Figs.6–11.

Finally, we want to calculate the longitudinal lepton polarization asymmetry for the considered decays. It is given as [17] PL¼ 2v ð1þ2^l ^sÞð1; ^r; ^sÞ 1þ12^l1 Reð1; ^r; ^sÞðCeff9 fþðq2Þ2C7fTðq 2_Þ 1þpffiffiffi^r ðC10fþðq2ÞÞ ; (28) where v, ^l, ^r, ^s, ð1; ^r; ^sÞ, 1, and 1 were defined before.

The dependence of the longitudinal lepton polarization asymmetries for the Bs! ð; 0Þlþldecays on the

trans-ferred momentum square q2 with and without LD effects are plotted in Figs.12and13.

As a result, the order of the obtained values for branch-ing ratios as well as the longitudinal lepton polarization asymmetries show the possibility of studying the consid-ered transitions at LHC. Any experimental measurements on the presented quantities and those comparisons with the obtained results can give valuable information about the nature of the and 0 mesons and strong interactions inside them.

ACKNOWLEDGMENTS

The partial support of the Shiraz University Research Council is appreciated.

FIG. 11. The same as Fig.10but for the Bs! 0 .

FIG. 12. The dependence of the longitudinal lepton polarization asymmetry on q2. The left figure belongs to the Bs! þ

decay and the right figure corresponds to the Bs! þ. The solid lines and dotted lines show the results without and with the LD

effects, respectively.

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APPENDIX

In this appendix, the explicit expressions of the non_i -per are given, nonþ -perðp2; p02; q2Þ ¼ hssi ms 2rr0þ 4m2 0ms 2m20mbþ 3m3s 3m2smb 12rr02 þ m 2 0m3b m20m3sþ 3m4s 3m3bm2s 2m20m2bmsþ 2m20mbm2s 12r2_r02 þ m20msq2 m02mbq2þ 3m2bm3s 3mbm3sþ 3mbm2sq2 3m3sq2 12r2_r02 þ 2m2 0ms 4m20mbþ 3msm2b 3m2smb 12r2_r0 þ m20m3b 2m3bm2sþ 2m2bm3s m20m2bms 4r3_r0 þ 2m5 s m20m3s 2mbm4sþ m20mbm2s 4rr03 ; non-per ðp2; p02; q2Þ ¼ hssi2m 2 0mb 9m3sþ 3m2smb 12rr02 þ 3m5 s m20m3b m20m3sþ 3m3bm2sþ m20mbq2þ 12m20msq2 12r2_r02 þ3m2bm3sþ 3mbm4s 3mbm2sq2 3m3sq2 12r2_r02 þ 2m2 0ms 3msm2bþ 6m2smb 3m2smb 12r2_r0 þ2m3bm2s m20m3bþ 2m2bm3s m20m2bms 4r3 r0 þ 2m5 s m20m3sþ 2mbm4s m20mbm2s 4rr03 ; non_T -perðp2_{; p}02_{; q}2_{Þ ¼ hssi}2m3sþ 2msmbþ m20

4rr0 þ3m 5 s m20m2bþ 3m2sq2 m20q2þ 3m4s 6m2sm2b msm20mb 6rr02 þ m20m4b m20m4sþ 3m6s 3m4bm2s m20m3bmsþ m20mbm3s 6r2_r02 þ m 2 0m2sq2 m20m2bq2þ 3m2bm2sq2 3m4sq2 6r2_r02 þ m20m2sþ 3msm3bþ m20q2 3m2sm2bþ 3m4s 3m2sq2 6r2_r0 þ m 2 0mbms 3m3smb 6r2_r0 þ m20m4b 2m4bm2sþ 2m2bm4s m20m2bm2s 2r3_r0 þ 2m6 s m20m4s 2m2bm4sþ m20m2bm2s 2rr03 ; where r ¼ p2 m2band r0¼ p02 m2s.

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