Lepton polarization and CP-violating effects in B→K∗τ+τ− decay in standard and two Higgs doublet models

25 May 2000

Physics Letters B 481 Ž2000. 275–286

in B

™

Lepton polarization and CP-violating effects K )tqty decay in standard and two Higgs doublet models T.M. Aliev 1, M. Savcı

2

Physics Department, Middle East Technical UniÕersity, 06531 Ankara, Turkey Received 29 March 2000; accepted 6 April 2000 Editor: R. Gatto

Abstract The most general model independent expressions for the CP-violating asymmetry, longitudinal, transversal and normal polarizations of leptons are derived. Application of these general results to the concrete models such as Standard model and three different types of two Higgs doublet model is discussed. q 2000 Elsevier Science B.V. All rights reserved. PACS: 12.60.-i; 13.20.-v; 13.25.Gv

1. Introduction

™

Rare B meson decays, induced by flavor-changing neutral current ŽFCNC. b s transition, is one of the most promising research area in high energy physics. Theoretical interest to the rare B decays lies in their role as a potential precision testing ground for the standard model ŽSM. at loop level and experimentally these decays will provide quantitative information about the Cabibbo–Kobayashi–Maskawa ŽCKM. matrix elements Vt d , Vt s and Vt b . Besides, these rare decays have the potential of establishing new physics beyond SM, such as two Higgs doublet model Ž2HDM., minimal supersymmetric extension of the SM ŽMSSM., left–right models, etc. At present, the main interest is focused on the rare B meson decays, for which the SM predicts ‘large’ branching ratios and that can be potentially measur1 2

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

™

able at working B factories and LHC. The rare B K ) l q l y Ž l s e, m , t . decays are such ones. At the same time this decay constitutes a suitable tool in looking for new physics beyond the SM. At quark level the process is described by the b s l ql y transition. This transition in framework of the SM and its various extensions have been extensively investigated w1–15x. One efficient way in establishing new physics is the measurement of lepton polarization. This problem is widely discussed in literature for the b s l q l y decay w16–19x. Note that all previous studies for the lepton polarization, except the work w19x, have been limited to SM and its minimal extensions. In w19x the analysis of the t lepton polarization for the b stqty decay was performed in a model independent way. In this work twelve Žten local and two nonlocal. four-Fermi operator interactions were introduced in a model independent way instead of three independent structures which are present in SM.

™

™

™

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 4 5 4 - 8

T.M. AlieÕ, M. SaÕcır Physics Letters B 481 (2000) 275–286

276

It is well known that the theoretical analysis of the inclusive decays is rather easy but their experimental detection is quite difficult. However for exclusive decays the situation is contrary to this case, i.e., their experimental study is easy but theoretical investigation is difficult. This is due to the fact that the description of the exclusive decays requires form factors, i.e., the matrix elements of the effective Hamiltonian between initial B and final meson states. This problem is related to to the nonperturbative sector of the QCD and it can be solved only by means of a nonperturbative approach. These matrix elements have been investigated in framework of different approaches such as chiral theory w20x, three point QCD sum rules method w21x, relativistic quark model by the light-front formalism w22x, effective heavy quark theory w23x and light cone QCD sum rules w24,25x. The aim of the present paper is to perform a comprehensive study the lepton polarizations and CP-violating asymmetry in the exclusive B K ) l q l y Ž l s m , t . decay in the SM and three versions of the 2HDM. Note that this decay in framework of 2HDM Žmodel I and model II. were investigated in w15x Žsecond reference., where PL and PT and A F B were studied. Here in this work we extend these considerations by studying in model III, including normal polarization of t lepton and CPviolating asymmetry, using the current limits for the parameters of 2HDM coming from low energy experiments like B 0 –B 0 mixing, r-parameter analysis and b sg decay. The paper is organized as follows. In Section 2, starting from a general form of four-Fermi interactions we derive model independent expressions for the longitudinal, transversal and normal polarizations of leptons. In Section 3, we apply the above-mentioned general results of the lepton polarizations to SM and to three types of 2HDM Žso called models I, II and III.. A brief summary of our results is presented in this section.

™

™

2. Lepton polarizations In this section we present the expressions for the longitudinal, transversal and normal polarizations of

the t lepton in a model independent way. For this aim we follow w19x where the matrix element for b stqty transition is given in terms of twelve most general independent four-Fermi interactions:

™

Ga Ms

'2 p

½

Vt bVt s) CS L si smn

qCB R si smn

qn q2

qn q2

Ž m s L . b l gm l

Ž m b R . b l gm l

qCL L sL gm bL l L g m l L qCL R sL gm bL l R g m l R qCR L sR gm bR l L g m l L qCR R sR gm bR l R g m l R qCL R L R sL bR l L l R q CR L L R sR bL l L l R qCL R R L sL bR l R l L qCR L R L sR bL l R l L q CT ssmn b l s mn l

5

qiCT E e mn a b ssmn b l sa b l ,

Ž 1.

where C X X are the coefficient of the four-Fermi interactions. Among them, there are two non-local Fermi interactions denoted by CS L and CB R , which correspond to y2C 7eff in the SM. Two Ž CL L , CL R . of the four vector type interactions Ž CL L , CL R , CR L and CR R . are also present in the SM Žin the forms of CL L s C9eff y C10 and CL R s C9eff q C10 .. CL R L R , CR L L R , CL R R L and CR L R L are the scalar type interactions and the last two terms with coefficients CT and CT E are the tensor type interactions. For simplicity we will take the mass of the strange quark to be zero and neglect tensor type interactions, since it was indicated in w26x that the physical observables are not sensitive to the presence of the tensor type interactions. Having established the matrix element for the b stqty transition, our next

™

T.M. AlieÕ, M. SaÕcır Physics Letters B 481 (2000) 275–286

problem is calculation of the matrix elements, K ) sgm Ž 1 " g 5 . b B , K ) si smn q n Ž 1 q g 5 . b B and ² K ) s Ž 1 " g 5 . b B :, in order to be able to calculate the physically measurable quantities at hadronic level. These matrix elements can be written in terms of the form factors in the following way:

¦

;¦

¦K ) Ž pK

)

;

mB q mK ) 2 mK )

A1 Ž q 2 . y

mB y mK ) 2 mK )

A2 Ž q 2 . .

Ž 5.

mB q mK )

² K ) Ž pK

)

, ´ . s Ž 1 " g 5 . b B Ž pB . :

2

Ž m B q m K ) . A1 Ž q .

. i Ž pB q pK ) . m Ž ´ q . . iqm

Using this relation we obtain

2V Ž q 2 .

2 mK ) q2

1 s

A2 Ž q 2 .

)

2

2

Ž ´ q . A 3 Ž q . y A 0 Ž q . , Ž 2. )

pKr ) q s T1

™

From Eqs. Ž1. – Ž3. and Ž6. we get the following expression for the matrix element of the B K ) l ql y decay Ga Ms

2

Žq .

q 2 i ´m) Ž m2B y m2K ) .

½

Vt bVt s) l gm Ž 1 y g 5 . l

qi Ž ´ ) q . Ž pB q pK ) . m B2 q iqm Ž ´ ) q . B3

q2 i Ž ´ ) q . qm y Ž pB q pK ) . m

m 2B y m2K )

4'2 p

= yemnrs ´ ) n pKr ) q s Ž 2 A1 . y i ´m) B1

y Ž p B q p K ) . m Ž ´ ) q . T2 Ž q 2 .

q2

Ž 6.

mB q mK )

;

)n

 .2 im K Ž ´ ) q . A 0 Ž q 2 . 4 . )

mb

, ´ . si smn q n Ž 1 q g 5 . b B Ž pB .

s 4emnrs ´

=

A3 Ž q 2 . s

;

" i ´m)

)

Using the equation of motion, the form factor A 3 can be expressed as a linear combination of the form factors A1 and A 2 Žsee w21x.

, ´ . sgm Ž 1 " g 5 . b B Ž pB .

s yemnrs ´ ) n pKr ) q s

¦K ) Ž pK

277

ql gm Ž 1 q g 5 . l

T3 Ž q 2 . ,

Ž 3.

= yemnrs ´ ) n pKr ) q s Ž 2C1 . y i ´m) D 1 qi Ž ´ ) q . Ž pB q pK ) . m D 2 q iqm Ž ´ ) q . D 3

where ´ is the polarization vector of K meson and q s pB y pK ) is the momentum transfer. In order to ensure finiteness of Ž2. at q 2 s 0, we assume that A 3 Ž q 2 s 0. s A 0 Ž q 2 s 0.. To calculate the matrix element ² K ) s Ž 1 " g 5 . b B :, we multiply both sides of Eq. Ž2. by qm and use the equation of motion. Neglecting the mass of the strange quark, we get )

² K ) Ž pK 1 s mb

)

, ´ . s Ž 1 " g 5 . b B Ž pB . :

 .i Ž ´ ) q . Ž m B q m K

)

ql Ž 1 y g 5 . l i Ž ´ ) q . B4

5

ql Ž 1 q g 5 . l i Ž ´ ) q . B5 , where A1 s Ž C L L q C R L .

. A1 Ž q 2 .

V Ž q2 . mB q mK )

y 2CB R

B1 s Ž CL L y CR L . Ž m B q m K ) . A1

"i Ž m B y m K ) . Ž ´ q . A 2 Ž q )

"2 im K ) Ž ´ q . A 3 Ž q )

2

2

. 2

. y A0 Ž q . 4 .

Ž 4.

Ž 7.

y 2CB R

mb q2

Ž m2B y m2K . T2 , )

mb q2

T1 ,

T.M. AlieÕ, M. SaÕcır Physics Letters B 481 (2000) 275–286

278

CL L B2 s

yCR L mB q mK ) y 2CB R

A2

mb

T2 q

q2

D 1 s B1 Ž CL L

™C ™C ™C ™C

m 2B y m2K )

2 mK )

B3 s Ž C L L y C R L . C1 s A1 Ž C L L

q2

q

LR

2

, CR L

L R , CR L

D 2 s B2 Ž C L L D 3 s B3 Ž C L L

Ž A 3 y A 0 . y 2CB R

™C ™C ™C ™C

RR

, CR L

B4 s y Ž CL R R L y CR L R L .

B4 s y Ž CL R L R y CR L L R .

mb q2

T3 ,

.,

dG

ž /

RR. ,

L R , CR L LR

T3 ,

ž ž

dq 2

RR

2 mK ) mb 2 mK )

eL s

< p1 <

,

eN s

pK ) = p 1 < pK ) = p 1 <

mb

G 2a 2 s

.,

/ /

A0 .

dq

2

1 s 2

dG

ž / dq 2

1 3

½

< Vt bVt s) < 2l1r2 Õ 32 l m 4B

Ž m2B s y m2l . Ž < A1 < 2 q < C1 < 2 .

q2 m 2l Re Ž A1C1) .

Ž 8.

q96m2l Re Ž B1 D 1) . 4 y m2B m l l Re Ž B1 y D 1 . Ž B4) y B5) . r 8 q m2B m2l l  Re r qRe

Ž B3) q D 2) y D 3) . B1

Ž B2) y B3) q D 3) . D1 y Re Ž B4 B5) . 4

4 q m4B m l Ž 1 y r . r =l  Re Ž B2 y D 2 . Ž B4) y B5) .

,

e T s eL = eN , Ž 9.

1 q Ž PL e L q PN e N 0

qPT e T . P n ,

2 14p 5 m B

=

A0 ,

where p 1 and pK ) are the three-momenta of the lepton l yand K ) meson, respectively, in the center of mass of final leptons. The differential decay rate for any spin direction n of the l y lepton, where n is a unit vector in the l y rest frame, can be expressed in the following form d G Ž n.

0

RR. ,

At this point we would like to make the following comment. The main difference from the SM case is, we have six different structures Žafter setting m s s 0 and neglecting tensor interactions. in the inclusive channel, while under the same conditions we have only two new structures, namely scalar type interactions proportional to B4 and B5 . On the other hand there appears no any new structure in the 2HDM. Having established this matrix element, let us now consider the final lepton polarization. We define the following three orthogonal unit vectors: p1

where the subscript ‘0’ corresponds to the unpolarized differential decay rate whose explicit form will be presented below. PL , PN and PT are recognized as the longitudinal, normal and transversal polarizations, respectively. It follows from the definition of unit vectors e i that PT obviously lies in the decay plane whose orientation is determined by the vectors p 1 and pK ) and PN is perpendicular to this plane. The expression for the unpolarized differential decay rate in Eq. Ž10. can be written as:

Ž 10 .

4

8 q m4B m2l Ž 1 y r . r =l  Re y Ž B2 y D 2 . Ž B3) y D 3) .

4

8 y m4B m2l l Ž 2 q 2 r y s . Re Ž B2 D 2) . r 4 y m4B m l s l Re Ž B3 y D 3 . Ž B4) y B5) . r 4 y m4B m2l s l < B3 < 2 q < D 3 < 2 y 2 Re Ž B3 D 3) . r

T.M. AlieÕ, M. SaÕcır Physics Letters B 481 (2000) 275–286

2 q m2B Ž m 2B y 2 m 2l . l < B4 < 2 q < B5 < 2 r 8

8 y 3r

3rs

4

qm2B s

q

Ž1 yr ys.

3r

= Re Ž B1 B2) . q Re Ž D 1 D 2) . 4 q rs

PT s

2 m2l Ž l y 6 rs . q m2B s Ž l q 12 rs .

= < B1 < 2 q < D1 < 2 q

4 3rs

= < B2 < 2 q < D 2 < 2

5

1

D

1 2r

4

dq 2 Pi Ž q . s dG

Ž n s ei . y

2

Ž 11 .

dq 2

Ž n s ei . q

dq 2 dG dq 2

1

D

Õ

½

4 3r

.

Ž 12 .

2

r's 1

Ž n s ye i .

r's

mB m l Ž 1 y r y s .

2

B1 y D 1

1 q 2r

m3B m l Ž 1 y r y s . 's

=Re 2 Ž B1 q D 1 . Ž B3) y D 3) . 1 q

r's

m3B m 2l l

2

= y2Re Ž B2 B5) . q 2Re Ž D 2 B4) . 1

4 y l m4B m l Ž 1 y r . r

q

=Re Ž B2 y D 2 . Ž B4) q B5) .

=Re y2 Ž B2 q D 2 . Ž B3) y D 3) .

2

2 2 y l m 4B s B4 y B5 r

2

m B m2l Ž 1 y r y s .

q

A1 y C1

,

y8m 3B m l 's

4 q l m2B m l Re Ž B1 y D 1 . Ž B4) q B5) . r

q 323 l m6B s

5

= 2Re Ž B1 B5) . y 2Re Ž D 1 B4) .

Ž n s ye i .

l2 m6B B2 y D 2

2

m3B m l Ž 1 q 3r q s . 's

1 q

q

After lengthy calculations we get the following general expressions for the longitudinal, transversal and normal polarizations of the l y lepton Žfor m s s 0 and neglecting the tensor interaction. PL s

½

2

B1 y D 1

= 2Re Ž B1 D 2) . y 2Re Ž B2 D 1) .

.

dG

'l p

q

The polarizations PL , PN and PT are defined as: dG

l m2B Ž l q 12 rs .

=Re Ž A1 q C1 . Ž B1) q D 1) .

m4B l  m2B s l

qm2l 2 l q 3s Ž 2 q 2 r y s .

l m4B Ž 1 y r y s .

= Re Ž B1 B2) . y Re Ž D1 D 2) .

m 2B l m2l Ž 2 y 2 r q s .

y

279

2

r's 1 2r

m5B m l Ž 1 y r . l B2 y D 2

m5B m l l's

1

2

q

2 r's

m3B m l Ž 1 y r y s . Ž 1 y r . q l

2

4 q l m4B m l s Re Ž B3 y D 3 . Ž B4) q B5) . r

= y2Re Ž B1 B2) . q 2Re Ž D 1 D 2) . 1 q

2

2 r's

mB Ž 1 y r y s .

T.M. AlieÕ, M. SaÕcır Physics Letters B 481 (2000) 275–286

280

= Ž y2 m 2l q m 2B s . = 2Re Ž D1 B5) . y 2Re Ž B1 B4) . 1 q

2 r's

m3B l Ž y2 m2l q m2B s .

5

= y2Re Ž D 2 B5) . q 2Re Ž B2 B4) .

PN s

1

D

,

½

p Õm3B'l 's 8 m l Im Ž B1) C1 q A1) D 1 .

1 q m 2B l Im Ž m l B3 y m l D 3 y B4 . B2) r qIm Ž yB5 y B3 q D 3 . D 2) 1 q m l Ž 1 q 3r y s . r = Im Ž D 1 q B1 . Ž B2) y D 2) . 1 q r

Ž1 yr ys.

™

= Im Ž B4 y m l B3 q m l D 3 . B1) qIm Ž m l B3 y B5 y m l D 3 . D 1)

5

,

Ž 13 .

where D is the expression within the curly parenthesis of the unpolarized differential decay rate in Eq. Ž11.. These expressions for the longitudinal, transversal and normal polarizations are general and model independent Žif the tensor interaction is neglected.. It follows from the expressions of PT and PN that they are proportional to the lepton mass and therefore they are nonvanishing only for the t lepton. In this work we also analyze the CP-violating asymmetry, which is defined as dG

A CP Ž q 2 . s

dG

ž / ž / ž / ž / dq 2

y

0

dG

dq 2

dq 2

0

dq 2

™

CL L s C9eff Ž m b . y C10 Ž m b . , CR L s 0 ,

CB R s y2C 7eff Ž m b . ,

CL R s C9eff Ž m b . q C10 Ž m b . ,

CR R s 0 ,

CL R R L s CR L L R s CL R L R s CR L R L s 0 . 0

dG

q

unpolarized differential decay rate for the antiparticle channel. Note that in SM, CP-violating asymmetry is equal to zero Žor suppressed very strongly., since all form factors are real Žsee below., Wilson coefficients C 7eff and C10 are real and only C9eff contains a strong phase. But this strong phase can not lead to CP-violation itself. Using these general expressions we can study the sensitivity of the t-lepton polarizations on the new Wilson coefficients. Furthermore one can investigate how strongly these polarizations deviate from the SM predictions and for which Wilson coefficient this departure is more essential. But in the present work we will apply these general results to concrete models, namely to the SM and three type of 2HDM, i.e., models I, II Žabout models I and II, see for example w27x and III. Note that in models I and II, the flavor changing neutral currents which appear at tree level are avoided by imposing ad hoc symmetry w28x. The phenomenological consequence of the 2HDM without this discrete symmetry has been investigated in w29x Žsee also w30–39x.. One novel feature of model III is existence of new weak phase which appears in Yukawa interaction of fermions with Higgs fields Žsee below.. Existence of this new weak phase can lead to sizeable CP violation in B K ) l ql y decay. Therefore if in future experiments sizeable CP violation in the B K ) l ql y decay is discovered, it is an unambiguous indication of the existence of new physics beyond SM, since in the SM the CP asymmetry suppressed very strongly. Making the following replacements in the expressions given in Eq. Ž8., the explicit forms of A i , Bi , Ci and Di can be obtained in SM and 2HDM easily. 1. SM

,

0

where Ž d Grdq 2 . 0 is the unpolarized differential decay rate given by Eq. Ž11. and Ž d Grdq 2 . 0 is the

Ž 14 .

2. 2HDM 2HDM CL L s C9eff 2HDM Ž m b . y C10 Ž mb . ,

CB R s y2C 7eff 2HDM Ž m b . ,

CR L s 0 ,

T.M. AlieÕ, M. SaÕcır Physics Letters B 481 (2000) 275–286 2HDM CL R s C9eff 2HDM Ž m b . q C10 Ž mb . ,

C L R R L s CQ 1 ,

C R L R L s CQ 2 ,

CR R s 0 ,

Ž 15 .

Ž B Ž mW . y C Ž mW . .

sin2u W

1

2

q lt t

The coefficients Ci2HDM Ž mW . Ž i s 7,9 and 10. to the leading order are given by Žsee for example w40,41x. C 72HDM

1

2HDM C10 Ž mW . s

C L R L R s CQ 1 ,

CR L L R s yCQ 2 .

281

xy

2

sin u W 8 1

ž

= y

1 q

yy1

Ž y y 1.

2

/

ln y ,

Ž mW .

Ž 18 . 2

sx

Ž7y5 xy8 x . 24 Ž x y 1 . 2

q lt t

3

2

x Ž 3 x y 2.

q

4 Ž x y 1.

4

ln x

mb m l

CQ 1Ž mW . s

y Ž7 y 5 y y 8 y 2 .

ž

72 Ž y y 1 .

1

m2h 0

lt t

1 2

x

2

sin u W 4

~°

3

= Ž sin2a q h cos 2a . f 1 Ž x , y .

¢

2

q

y Ž 3 y y 2. 12 Ž y y 1 .

ql t t l b b

ž

ln y

4

/

yŽ3y5 y. 12 Ž y y 1 .

2

q

y Ž 3 y y 2. 6 Ž y y 1.

3

/

ln y ,

q

sin2u W

2

sin u W

C Ž mW . y

18 Ž x y 1 .

=

ž

q lt t

yy

36 Ž x y 1 .

4

sin2u W 1

y

ž

2 m2H 0

Ž y y 1.

3

CQ 2 Ž m W . s

2

108 Ž y y 1 .

3 y3 y6 yq4 18 Ž y y 1 .

4

/

xs

m 2t 2 mW

,

BŽ x. sy

/

,

lt t

2

½

f 1Ž x , y .

m2H " y m2A 0

5

f2 Ž x , y . ,

2 mW

Ž 20 .

where

3

ln y

1

m 2H "

q 1q

8

ln y

mb m l

ln x

ys

47 y 2 y 79 y q 38

y

Ž 19 .

x Ž 25 y 19 x .

1 y 4sin2u W xy

2

1 yy1

2

0

2

y

9

0

¶• ß

y3 x 4 q 30 x 3 y 54 x 2 q 32 x y 8

q

Ž m2h q m2H .

=f3 Ž y . ,

1 y 4sin u W

4

m2h 0 y

2 m2H "

B Ž mW . 2

q

q Ž sin2a q h cos 2a . Ž 1 y z .

sin2 2 a

C92HDM Ž mW . 1

2 mW

=f 2 Ž x , y .

Ž 16 .

sy

m 2h 0

q

Ž 17 .

CŽ x. s

x 4

m2t

,

m 2H "

zs

x

x y

,

x q

4 Ž x y 1.

ž

hs

4 Ž x y 1.

xy6

2

2 Ž x y 1.

2 Ž x y 1.

m2H 0

ln x ,

3 xq2 q

m 2h 0

2

/

ln x ,

,

T.M. AlieÕ, M. SaÕcır Physics Letters B 481 (2000) 275–286

282

f 1Ž x , y . s

f2 Ž x , y . s

f3 Ž y . s

x ln x

y ln y

xy1

solve the corresponding renormalization group equation. Hence, including the NLO QCD corrections, C9eff Ž m b . can be written as:

,

y yy1 x ln y

ln z q

Ž z y x . Ž x y 1.

1 y y q y ln y

Ž y y 1.

2

Ž z y 1. Ž x y 1.

,

,

asŽ m.

C9eff Ž m . s C92HDM Ž m . 1 q

p

v Ž sˆ.

qg Ž m ˆ c , sˆ. 3C1 Ž m . q C2 Ž m .

Ž 21 .

q3C3 Ž m . q C4 Ž m . q 3C5 Ž m . 2

0

0

0

sin u W s 0.23 is the Weinberg angle, h , H and A are two scalar and pseudoscalar Higgs fields, respectively. The coefficients l t t and l b b for model I and model II of the 2HDM are:

qC6 Ž m . y 12 g Ž 0, sˆ. Ž C3 Ž m . q 3C4 Ž m . . y 12 g Ž 1, sˆ. Ž 4C3 q 4C4 q 3C5 q C6 .

l t t s cot b ,

l b b s ycot b , for model I ,

l t t s cot b ,

l b b s qtan b , for model II ,

y 12 g Ž 0, sˆ. Ž C3 q 3C4 .

Ž 22 .

while in model III l t t or l b b is complex, i.e.,

q 29 Ž 3C3 q C4 q 3C5 q C6 . , where m ˆ c s m crm b , sˆ s p 2rm 2b , and

v Ž sˆ. s y 29 p 2 y 43 Li 2 Ž sˆ. y 23 ln Ž sˆ. ln Ž 1 y sˆ.

lt t lb b ' lt t lb b e i f .

y

From Eqs. Ž16. – Ž20. we observe that the SM results for the Wilson coefficients C 7SM Ž mW ., C9SM Ž mW . SM Ž amd C10 mW . Žand correspondingly at m s m b scale. can all be obtained from 2HDM results by making the following replacements CQ 1

™0

,

Ž 23 .

™0 , Ž y ™ 0. , Ž y ™ 0. , Ž y ™ 0. ,

3 Ž 1 q 2 sˆ. y

s

C 72HDM

C9SM Ž mW .

s

C92HDM

SM C10 Ž mW .

s

2HDM C10

The evolution of the Wilson coefficients from the higher scale m s mW down to the low energy scale m s m b is described by the renormalization group equation. The coefficients C 7eff Ž m ., C9eff Ž m ., C10 Ž m . at the scale O Ž m s m b . are calculated in w42,43x and CQ 1 and CQ 2 at the same scale to leading order are calculated in w41x. The Wilson coefficient C10 is not modified as we move from m s mW to m s m b 2HDM Ž scale, i.e., C10 Ž m b . ' C10 mW .. In order to calcu2HDM late C9 at m b scale, it is enough to make the replacement C9SM Ž mW . C92HDM Ž mW . and then

™

ln Ž 1 y sˆ.

2 sˆŽ 1 q sˆ. Ž 1 y 2 sˆ. 2

3 Ž 1 y sˆ. Ž 1 q 2 sˆ. q

CQ 2

C 7SM Ž mW .

5 q 4 sˆ

ln Ž sˆ.

5 q 9 sˆ y 6 sˆ2

Ž 24 .

3 Ž 1 y sˆ. Ž 1 q 2 sˆ.

represents the O Ž a s . correction from the one gluon exchange in the matrix element of O 9 , while the function g Ž m ˆ c , sˆ. arises from one loop contributions of the four-quark operators O1 –O6 , whose form is g Ž yi , sˆ. s y 89 ln Ž m ˆ i . q 278 q 94 yi y 92 Ž 2 q yi . 1 y yi

(

½

ž

= Q Ž 1 y yi . ln

( 1y( 1yy

1 q 1 y yi

qQ Ž yi y 1 . 2arctan

1

(y y 1 i

5

yi p

i

,

/ Ž 25 .

T.M. AlieÕ, M. SaÕcır Physics Letters B 481 (2000) 275–286

283

where yi s 4 m ˆ 2i rpˆ 2 . The Wilson coefficient C9eff receives also long distance contributions, which have their origin in the real cc intermediate states, i.e., Jrc , c X , . . . . The Jrc family is introduced by the Breit–Wigner distribution for the resonances through the replacement w3,6x gŽ m ˆ c , sˆ.

™ gŽ mˆ , sˆ. y a3p k c

=

2 em

Ý

VisJr c i , c X , . . .

m V i G Ž Vi

™l

q y

l .

Ž p 2 y m2V . q imV G V i

i

, i

Ž 26 . where the phenomenological parameter k s 2.3 is chosen in order to reproduce correctly the experimental value of the branching ratio Žsee for example w44x.

3. Numerical analysis In this section we would like to present our numerical results. The main free parameters l t t , l b b of the 2HDM are restricted from B X sg decay, B 0 –B 0 mixing, r parameter and neutron electric-dipole moment w36x, that yields l b b s 50, l t t F 0.03. Throughout the numerical analysis for the mass of the Higgs bosons we have used m h 0 s 80 GeV, m H "s 250 GeV, m A 0 s 250 GeV and m H 0 s 150 GeV. For the values of the form factors, we have used the results of w25x, where the radiative corrections to

™

Fig. 1. The dependence of the CP-violating asymmetry A CP of the t lepton on q 2 and on the weak phase f in model III.

the leading twist contribution and SUŽ3. breaking effects are also taken into account. The q 2 dependence of the form factors can be represented in terms of three parameters as F Ž 0.

F Ž q2 . s 1 y aF

q2 m2B

q bF

q2

2

,

ž / m2B

™

where, the values of parameters F Ž0., a F and bF for the B K ) l ql y decay are listed in Table 1. In Fig. 1 we present the dependence of the CPviolating asymmetry on q 2 and on the weak phase f

Table 1 B meson decay form factors in a three-parameter fit, where the radiative corrections to the leading twist contribution and SUŽ3. breaking effects are taken into account w25x

™K ™K V ™K T1B ™ K T2B ™ K T3B ™ K A1B A 2B B

) ) ) ) ) )

F Ž0.

aF

bF

0.34"0.05 0.28"0.04 0.46"0.07 0.19"0.03 0.19"0.03 0.13"0.02

0.60 1.18 1.55 1.59 0.49 1.20

y0.023 0.281 0.575 0.615 y0.241 0.098

Fig. 2. The dependence of the averaged CP asymmetry ² A CP : of the t lepton on the weak phase f in model III, taking into account short and long distance contributions.

T.M. AlieÕ, M. SaÕcır Physics Letters B 481 (2000) 275–286

284

Fig. 3. The dependence of the longitudinal polarization PL of the t lepton on q 2 and on the weak phase f in model III, taking into account only the short distance contribution in C9eff .

™

Fig. 5. The dependence of the transversal polarization PT of the t lepton on q 2 and on the weak phase f in model III, taking into account only the short distance contribution in C9eff .

for the B K )tqty decay in model III, since we have already noted that in SM and in models I and II the CP-violating asymmetry is practically zero. We observe that CP asymmetry differs from zero in the region 0 - f - 2p , except at f s 0, p and 2p , and its value in the region 0 - f - p Žp - f - 2p . is negative Žpositive.. Fig. 2 depicts the dependence of the averaged CP asymmetry ² A CP : Žhere and in all of the following discussions, by the averaged values of the physical quantities we mean integration over q 2 in the region 14 GeV 2 F q 2 F Ž m B y m K ) . 2 . on the weak phase angle f in model III, taking into account short and long distance contributions. It follows from this figure that ² A CP : varies in the range Ž-0.04, 0.04. which is different from zero and it definitely is an

indication of the existence of new physics beyond SM, since ² A CP : is practically equal to zero in the SM. In Fig. 3, the dependence of PL on q 2 and the weak phase angle f without long distance effects in model III is presented. From this figure one can see that, for q 2 ) 14 GeV 2 , PL varies in the range Žy0.65, y0.8. which is larger than the SM prediction. This is due to the fact that the ‘new’ contribution which comes from the charged Higgs boson gives constructive interference to the SM results. In Fig. 4 we present the averaged longitudinal polarization ² PL : on the weak phase angle f , taking into account short and long distance contributions.

Fig. 4. The dependence of the averaged longitudinal polarization lepton on the weak phase f , taking into account short and long distance contributions.

Fig. 6. The dependence of the normal polarization PN of the t lepton on q 2 and on the weak phase f in model III, taking into account only the short distance contribution in C9eff .

² PL : of t

T.M. AlieÕ, M. SaÕcır Physics Letters B 481 (2000) 275–286

For completeness the predictions of SM, model I and model II on ² PL : are also presented. It is observed from this figure that ² PL : in model III as modulo, is larger than the ones predicted by SM, model I and model II. Therefore an observation of ² PL : G 0.65 is another conclusive confirmation of the existence of new physics beyond SM. In Figs. 5 and 6 we present the dependence of the transversal and normal polarizations of the t lepton on q 2 and on the weak phase angle f , respectively, without long distance effects in model III. The dependence of the averaged transversal and normal polarizations on the weak phase angle, taking into account short and long distance contributions, are depicted in Figs. 7 and 8, respectively. For sake of completeness we presented also the predictions of SM, model I and model II of the same physical quantity in both figures. Fig. 7 clearly depicts that, the prediction of model III on ² PT : as modulo, is approximately five times smaller than the ones predicted by SM, model I and model II. However the situation is totally different for ² PN :, having a range of values in the region Ž0.10, 0.15. in model III, it is approximately two or three times larger than the ones predicted by SM, model I and model II. Here we would like to make the following remark. It follows from Eq. Ž13. that PN is defined as the imaginary part of the form factors and of the corresponding Wilson coefficients C 7eff , C9eff , C10 , CQ1 and CQ 2 . In SM C 7eff and C10 are real and only C9eff has imaginary part. On the other side all theoretical methods predict these form factors to be real quantities. For this reason, if in future experi-

Fig. 7. The dependence of the averaged transversal polarization the t lepton on the weak phase f , taking into account short and long distance contributions

² PT : of

285

Fig. 8. The dependence of the averaged normal polarization ² PN : of the t lepton on the weak phase f , taking into account short and long distance contributions.

ments a different value for PN were observed compared to the SM prediction, it is an indication of unambiguous information about the existence of the above-mentioned CP-violating phase in theory. In conclusion, we have investigated the exclusive B K )tqty decay in the SM and in three different versions of the 2HDM. From the results we have obtained we conclude that the combined analysis of the CP-violating asymmetry and t lepton polarization effects are very useful tools in looking for new physics beyond SM.

™

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