Expectations on B ->(K-0(*)(1430),K-2(*)(1430))phi decays

Expectations on B !
K

^₀

1430; K

₂^

1430
decays

Chuan-Hung Chen

^1,2,*

and Chao-Qiang Geng

^3,4,†

1Department of Physics, National Cheng-Kung University, Tainan 701, Taiwan

2National Center for Theoretical Sciences, Taiwan

3Department of Physics, National Tsing-Hua University, Hsinchu 300, Taiwan

4Theory Group, TRIUMF, 4004 Wesbrook Mall, Vancouver, B.C. V6T 2A3, Canada (Received 9 January 2007; published 13 March 2007)

As the annihilation contributions play important roles in solving the puzzle of the small longitudinal polarizations in B ! K^
decays, we examine similar effects in the decays of B ! K^_0;2
1430
. For the calculations on the annihilated contributions, we adopt that the form factors in B ! K
^
decays are parameters determined by the observed branching ratios (BRs), polarization fractions, and relative angles in experiments, and we connect the parameters between B ! K^_0
2
and B ! K
^
. We find that the BR of B_d ! K^0₀
1430
is
3:69  0:47  10^6. We show that the BR of B_d! K₂^0
1430
is
1:70  0:80  10^6in the 2nd version of the Isgur-Scora-Grinstein-Wise (ISGW2) model, whereas it can be a broad range of values in the light-front quark model. In terms of the recent BABAR observations on BRs and polarization fractions in B_d ! K₂^0
1430
, the results in the light-front quark model are found to be more favorable. In addition, due to the annihilation contributions to B ! K₂^
and B ! K^
being opposite in sign, we demonstrate that the longitudinal polarization of B_d! K^0₂
1430
is always O
1

with or without including the annihilation contributions.

DOI:10.1103/PhysRevD.75.054010 PACS numbers: 13.25.Hw

I. INTRODUCTION

Since the transverse decay amplitudes of vector meson productions in B decays are associated with their masses, by naive estimations, the longitudinal polarization (LP) of B decaying into two light vector mesons is close to unity.

The expectation is confirmed by BELLE [1] and BABAR [2,3] in B ! 
! decays, in which the longitudinal parts occupy over 88%. Furthermore, the LPs could be small if the final states include heavy vector mesons. The conjec- ture is verified in B ! J=
K

^

decays [4,5], in which the longitudinal contribution is only about 60%. However, the rule for the small LPs seems to be violated in B !
K

^

decays. From the measurements of BELLE [6] and BABAR [2,7], it is quite clear that the LPs in B ! K

^

are only around 50%. To solve the unexpected observations, many mechanisms have been proposed, such as those with new QCD effects [8–11] as well as extensions of the standard model (SM) [12,13].

Recently, the BABAR Collaboration has observed the branching ratios (BRs) and polarization fractions (PFs) for the decays of B

_d

! K

^0_0;2

1430
[14], given by BR
B

_d

! K

₂^0

1430
 
7:8  1:1  0:6  10

^6

;

R

_L

B

_d

! K

^0₂

1430  0:853

^0:061_0:069

 0:036;

R

?

B

_d

! K

^0₂

1430  0:045

^0:049_0:040

 0:013;

BR
B

_d

! K

₀^0

1430
 
4:6  0:7  0:6  10

^6

: (1)

By the observations, it seems that the LP of the p-wave

tensor-meson production is much larger than those of the s-wave vector mesons in B decays. To find out whether the data are just the statistical fluctuation or the correct ten- dency for the behavior of the p-wave productions in B decays, it is important to study the phenomena from a theoretical viewpoint.

It is known that the annihilation contributions play important roles in the PFs of B ! K

^

decays [8,10,11].

As the corresponding timelike form factors are more un- certain than those of the transition form factors, we first adopt that the form factors of the annihilation contributions on B ! K

^

are parameters fixed by the data in B ! K

^

and then connect them to those in B

_d

! K

^0₂

1430
. To have more illustrative examples, we also examine the decays of B ! K

^₀

1430
simultaneously.

The paper is organized as follows. In Sec. II, we carry out a general study on the decay amplitudes and hadronic matrix elements. We present our numerical analysis in Sec. III. Our conclusions are presented in Sec. IV.

II. DECAY AMPLITUDES AND HADRONIC MATRIX ELEMENTS

It is known that the effective interactions for the decays of B ! K

_n^

1430
(n  0, 2) are described by b ! sq q, which are the same as B ! K

^

, given by [15]

H

_eff

 G

_F

2

p X

qu;c

V

_q



C

₁

O

^q₁

  C

₂

O

^q₂



 X

¹⁰

i3

C

_i

O

_i





; (2)

where V

_q

 V

_qs^

V

_qb

are the Cabibbo-Kobayashi-Maskawa

*Email address: [email protected]

†Email address: [email protected]

PHYSICAL REVIEW D 75, 054010 (2007)

(CKM) matrix elements [16] and the operators O

₁

–O

₁₀

are defined as

O

^q₁


s

_

q

_



_VA

q

_

b

_



_VA

; O

^q₂


s

_

q

_



_VA

q

_

b

_



_VA

;

O

₃


s

_

b

_



_VA

X

q

q

_

q

_



_VA

; O

₄


s

_

b

_



_VA

X

q

q

_

q

_



_VA

; O

₅


s

_

b

_



_VA

X

q

q

_

q

_



_VA

; O

₆


s

_

b

_



_VA

X

q

q

_

q

_



_VA

;

O

₇

 3

2
s

_

b

_



_VA

X

q

e

_q

q

_

q

_



_VA

;

O

₈

 3

2
s

_

b

_



_VA

X

q

e

_q

q

_

q

_



_VA

;

O

₉

 3

2
s

_

b

_



_VA

X

q

e

_q

q

_

q

_



_VA

;

O

₁₀

 3

2
s

_

b

_



_VA

X

q

e

_q

 q

_

q

_



_VA

;

(3)

with  and  being the color indices and C

₁

–C

₁₀

the corresponding Wilson coefficients (WCs). In Eq. (2), O

₁

–O

₂

are from the tree level of weak interactions, O

₃

–O

₆

are the so-called gluon penguin operators, and O

₇

–O

₁₀

are the electroweak penguin operators. Using the unitarity condition, the CKM matrix elements for the penguin operators O

₃

–O

₁₀

can also be expressed by V

_u

 V

_c

 V

_t

. Besides the weak effective interactions, to study exclusive two-body decays, we should know how to calculate the transition matrix elements such as hM

₁

M

₂

jH

_eff

jBi, where nonperturbative effects dominate the uncertainties. By taking the heavy quark limit, we consider that the productions of light mesons satisfy the assumption of color transparency [17], i.e., the final state interactions are negligible subleading effects. Hence, the decays of B ! K

_n^

1430
could be treated as short- distance dominant processes. As the wave functions of p-wave states are quite uncertain, unlike those of s-wave states which are known at least in the leading twist-2 and twist-3 [18], in our calculations we will employ the gen- eralized factorization assumption [19,20], in which the factorized parts are regarded as the leading effects and the nonfactorized effects are lumped and characterized by the effective number of colors, denoted by N

^eff_c

[21].

Based on the effective interactions of Eq. (2), the matrix elements hK

^_n

1430
jH

eff

jBi could be classified by vari- ous flavor diagrams displayed in Fig.

1, where (a) and (b)

denote the penguin emission topologies, while (c) [(d)] is the penguin [tree] annihilation topology. Furthermore, in

terms of the flavor diagrams, we can group the effects of Eq. (2) for the transition matrix elements to be

X

^BKn;


 h
j
ss

_VA

j0ihK

_n^

j
 bs

_VA

jBi;

Z

^B;K₁ ^n


 hK

_n^

j
qs 

_VA

j0ih0j
 bq

_VA

jBi;

Z

^B;K₂ ^n


 hK

_n^

j
qs 

_SP

j0ih0j
 bq

_SP

jBi;

(4)

where X

^BKn;


represent the factorized parts of the emis- sion topology and Z

^B;K_1;2 ^n


stand for the factorized parts of the annihilation topology. Note that the currents associated with
S  P
S  P in Eq. (4) are from the Fierz trans- formations of
V  A
V  A. From Eqs. (2) –(4), the decay amplitudes for B ! K

_n^

1430
can be written as

A
B

_d

! K

^0_n

1430
  G

_F

2

p fV

_tb

V

_ts^

~ a

^s

X

^BKn;


 a

^s₄

Z

^B;K₁ ^n


 2a

^s₆

Z

^B;K₂ ^n


g;

A
B

^_u

! K

^_n

1430
  G

_F

2

p fV

_us

V

_ub^

a

₁

Z

^B;K₁ ^n


 V

_tb

V

^_ts

~ a

^s

X

^BKn;


 a

^u₄

Z

^B;K₁ ^n


 2a

^u₆

Z

^B;K₂ ^n


g;

(5)

with ~ a

^s

 a

^s₃

 a

^s₄

 a

^s₅

. To be more convenient for our analysis, we have redefined the useful WCs by combin- ing the gluon and electroweak penguin contributions to be

b

s s

s

(a)

b s

s

(b) s

s

b q

q

(c) V − A V A

V ±

±

± V − A A

V − A V A

V − A

V − A

s s s u u

b

(d)

FIG. 1. Flavor diagrams for B !
K₀^
1430; K^₂
1430
de- cays, where (a) and (b) denote the penguin emission topologies, while (c) and (d) are the annihilation topologies for penguin and tree contributions, respectively.

a

₁

 C

^eff₂

 C

^eff₁

N

_c^eff

; a

₂

 C

^eff₁

 C

^eff₂

N

_c^eff

; a

^q₃

 C

^eff₃

 C

^eff₄

N

_c^eff

 3 2 e

_q



C

^eff₉

 C

^eff₁₀

N

_c^eff



; a

^q₄

 C

^eff₄

 C

^eff₃

N

_c^eff

 3 2 e

_q



C

^eff₁₀

 C

^eff₉

N

_c^eff



;

a

^q₅

 C

^eff₅

 C

^eff₆

N

_c^eff

 3

2 e

_q



C

^eff₇

 C

^eff₈

N

_c^eff



;

a

^q₆

 C

^eff₆

 C

^eff₅

N

_c^eff

 3

2 e

_q



C

^eff₈

 C

^eff₇

N

_c^eff



;

(6)

where the effective WCs of C

^eff_i

contain vertex corrections for smearing the -scale dependences in the transition matrix elements [20] and the effective color number of N

^eff_c

is a variable, which may not be equal to 3 [19–21].

The hadronic matrix elements defined in Eq. (4) are the essential quantities that we have to deal with in the two- body exclusive B decays. In the following discussions, we analyze the quantities X

^BKn;

, Z

^B;K₁ ^n

, and Z

^B;K₂ ^n

individu- ally. Since the degrees of freedom of K

^₀

are less than those of K

₂^

, we start with B ! K

₀^

1430
. As usual, we define the various normal hadronic matrix elements as follows [22]:

h0j  b

^



₅

qjB
p

_B

i  if

_B

p

^_B

; h

q; hj s

^

sj0i  m

f

"

^

h;

hK

₀^

pjV

_

 A

_

jB
p

_B

i  i



P

_

 m

²_B

 m

²_K 0

q

²

q

_

 F

^BK

 0

1

q

²

  m

²_B

 m

²_K 0

q

²

q

_

F

^BK

 0

0

q

²





;

(7)

where
V

_

; A

_

   b


_

; 

_



₅

s, m

_B;
;K^

0

are the meson masses, P  p

_B

 p, q  p

_B

 p and P  q  m

²_B

 m

²_K n

. Similarly, the timelike form factors for hK

^₀

jV

_

 A

_

j0i could be defined by

hK

^₀

p

q; "

jV

_

 A

_

j0i  i V

^K0

q

²

 m

 m

_K^

0



_

"

^

Q

^

s

; 

 2m

A

^K

 0

0

q

²

 "

^

 Q Q

²

Q

_


m

 m

_K^

0

A

^K₁^0

q

²





"

^
_

 "

^

 Q Q

²

Q

_



 A

^K₂^0

q

²

 "

^

 Q m

 m

_K^

0



s

_

 s  Q Q

²

Q

_



(8)

with Q

_


p  q

_

 p

_B

and s

_


p  q

_

. In terms of form factors in Eqs. (7) and (8), Eq. (4) could be rewritten as

X

^BK0;


 i2m

f

F

^BK

 0

1

m

²

"

^

 p

_B

; Z

^B;K

 0


1

 i2m

f

_B

A

^K

 0

0

m

²_B

"

^

 p

_B

; Z

^B;K

 0


2

 i2m

f

_B

m

²_B

m

_b

 m

_q


m

_s

 m

_q

 A

^K

 0

0

m

²_B

"

^

 p

_B

:

(9)

To compare with the s-wave states, we also give the hadronic matrix elements in B ! K
as

X

^BK;


 2m

f

F

^BK₁

m

²

"

^

 p

_B

; Z

^B;K
₁

 2m

f

_B

A

^K
₀

m

²_B

"

^

 p

_B

; Z

^B;K
₂

 2m

f

_B

m

²_B

m

_b

 m

_q


m

_s

 m

_q

 A

^K
₀

m

²_B

"

^

 p

_B

:

(10)

We note that, except the factor of i associated with the p-wave states [22], the definitions of the form factors for hKjV

_

 A

_

jBi and hK
jV

_

 A

_

j0i are similar to those for hK

₀^

jV

_

 A

_

jBi and hK

^₀

jV

_

 A

_

j0i, respectively. We will discuss the behaviors of A

^K

 0

0

and A

^K
₀

later on.

We now investigate the decays of B ! K

₂^

1430
, which are similar to B ! K

^

. The analogy of Eq. (4) for B ! K

^

can be presented by

EXPECTATIONS ON B !
K^₀
1430; K^₂
1430
. . . PHYSICAL REVIEW D 75, 054010 (2007)

X

^BK;


 im

f



m

_B

 m

_K^

A

^BK₁ ^

q

²

"

^

 "

^_K

 2A

^BK₂ ^

q

²



m

_B

 m

_K^

"

^

 p

_B

"

^_K

 p

_B

 i 2V

^BK

q

²



m

_B

 m

_K^



_

"

^

"

^_K

q

^

p



; Z

^B;K₁ ^


 if

_B

fm

²_B

V

₁^K

Q

²

"

^

 "

^_T

 V

₂^K

Q

²

"

^

 Q"

^_K

 Q  i2A

^K

Q

²



_

"

^

"

^_T

q

^

p

g;

Z

^B;K₂ ^


 i m

²_B

f

_B

m

_b

 m

_q

 m

²_B

V

₁^K

Q

²



m

_s

 m

_q

"

^

 "

^_K

 V

₂^K

Q

²



m

_s

 m

_q

"

^

 Q"

^_K

 Q  i2 A

^K

Q

²



m

_s

 m

_q



_

"

^

"

^_K

q

^

p



;

(11) where we have used the form factors in the transition of B ! K

^

and hK

^

jV

_

A

_

j0i, defined by [22]

hK

^

p; "

_K^

jV

_

jB
p

_B

i   V

^BK

q

²



m

_B

 m

_K^



_

"

^_K^

P

^

q

; hK

^

p; "

_K^

jA

_

jB
p

_B

i  i



2m

_V

A

^BK₀ ^

q

²

 "

^_K

 q

q

²

q

_


m

_B

 m

_V

A

^BK₁ ^

q

²





"

^_V

 "

^_K

 q q

²

q

_



 A

^BK₂ ^

q

²

 "

^_K

 q m

_B

 m

_K^



P

_

 P  q q

²

q

_



;

(12)

and [23]

hK

^

j q 

_



₅

sQ

^

j0i  iA

^K

q

²



_

"

^

"

^_K

Q

^

s

; hK

^

j q 

_

sQ

^

j0i  m

²_B

V

₁^K

q

²

"

^

 "

^_K

 V

₂^K

q

²

"

^

 Q"

^_K

 Q;

(13)

respectively. Here, "

_K^

denotes the polarization vector of the K

^

meson. To study the production of a tensor meson such as K

^₂

1430 in B decays, we need to introduce the properties of polarization vectors for the tensor meson. It is known that the polarization tensor ~ 

^

of a tensor meson satisfies

~



^

p; h  ~ 

^

p; h;

 ~

^

p; hp

_

 ~ 

^

p; hp

_

 0; g

_

 ~

^

 0; (14) where h is the meson helicity. The states of a massive spin- 2 particle can be constructed by using two spin-1 states. To analyze PFs in the production of the tensor mesonic B decays, we explicitly express ~ 

^

p; h to be [24]

~



^

2  e

^

e

^

;

~



^

1  1

2

p e

^

e

^

0  e

^

0e

^

;

~



^

0  1

6

p e

^

e

^

  e

^

e

^





2 3 s

e

^

0e

^

0;

(15)

where e

^

0;  denote the polarization vectors of a mas- sive vector state and their representations are chosen as

e

^

0  1

m

_T

j ~ pj; 0; 0; E

_T

;

e

^

  1

2

p
0; 1; i; 0

(16)

with m

_T

j ~ pj being the mass (momentum) of the particle.

Since the B meson is a spinless particle, the helicities carried by decaying particles in the two-body B decay should be the same. Moreover, although the tensor meson contains 2 degrees of freedom, only h  0 and 1 give the contributions. Hence, it will be useful to redefine the new polarization vector of a tensor meson to be "

_T

h 

~



_

p; hp

^_B

, where

"

^_T

2  0; "

^_T

1  1

2

p e
0  p

_B

e

^

;

"

^_T

0 

2 3 s

e
0  p

_B

e

^

0;

(17)

with e
0  p

_B

 m

_B

j ~ pj=m

_T

. Based on the new polariza- tion vector "

^_T

, the transition form factors for B ! K

₂^

could be defined by [22]

hK

₂^

p; "

_T

jV

_

jB
p

_B

i  h
q

²



_

"

^_T

P

^

q

; hK

₂^

p; "

_T

jA

_

jB
p

_B

i  ik
q

²

"

^_T

 "

^_T

 P
P

_

b



q

²



 q

_

b



q

²

;

(18)

and the timelike form factors for hK

₂^

jV

_

 A

_

j0i could

be parametrized as

hK

^₂

p; "

_T



q; "

j
V

_

 A

_

Q

^

j0i

 iA

^K2

Q

²



_

"

^

"

^_T

Q

^

s

 V

₁^K2

Q

²

"

^

 "

^_T

 V

₂^K2

Q

²

"

^

 Q"

^_T

 Q: (19)

Consequently, the analogy of Eq. (4) for B ! K

^₂

1430

could be explicitly expressed by

X

^BK2;


 im

f

fk
q

²

"

^

 "

^_T

 2b



q

²

"

^

 p

_B

"

^_T

 p

_B

 i2h
q

²



_

"

^

"

^_T

q

^

p

g;

Z

^B;K

 2


1

 if

_B

fV

₁^K2

Q

²

"

^

 "

^_T

 V

₂^K2

Q

²

"

^

 Q"

^_T

 Q  i2A

^K2

Q

²



_

"

^

"

^_T

q

^

p

g;

Z

^B;K

 2


2

 i m

²_B

f

_B

m

_b

 m

_q

 V

^K

 2
1

Q

²



m

_s

 m

_q

"

^

 "

^_T

 V

^K

 2
2

Q

²

 m

_s

 m

_q

 "

^

 Q"

^_T

 Q  i2 A

^K2

Q

²



m

_s

 m

_q



_

"

^

 "

^_T

q

^

p



: (20)

Since the final sates in the decays B ! VV and B ! TV carry spin degrees of freedom, the decay amplitudes in terms of helicities can be generally described by [25]

M

^h

 "

^_1

h"

^_2

h



ag

^

 b

m

₁

m

₂

p

^_B

p

^_B

 i c

m

₁

m

₂



^

p

_1

p

_2



;

which can be decomposed in terms of

H

₀₀

 ax  b
x

²

 1; H



 a  c

x

²

 1

p ;

(21)

and

H

₀₀

 

2 3 s

e
0  p

_B

ax  b
x

²

 1; H

_

 1

2

p e
0  p

_B

a  c

x

²

 1

p 

(22)

for B ! K

^

and B ! K

^₂

, respectively, with x 

m

²_B

 m

²₁

 m

²₂

=
2m

₁

m

₂

. In addition, we can also write the amplitudes in terms of polarizations as

A

_L

 H

₀₀

; A

k

 1

2

p
H



 H



;

A

_?

 1

2

p
H



 H

_

:

(23)

As a result, the BRs are given by

BR
B ! M
  j ~ pj

8m

²_B

jA

_L

j

²

 jA

_k

j

²

 jA

²_?

j (24) where M 
K

^

; K

₂^

1430 and j ~ pj is the magnitude of the outgoing momentum, and the corresponding PFs can be defined to be

R

_i

 jA

_i

j

²

jA

_L

j

²

 jA

k

j

²

 jA

²_?

j ;
i  L; k; ?; (25) representing longitudinal, transverse parallel, and trans- verse perpendicular components, respectively. Note that P

i

R

_i

 1.

III. NUMERICAL ANALYSIS

In Tables

I

and

II, we display the meson decay constants

and the transition form factors, respectively. Since the numerical values of B ! K

^₂

in the LFQM are different from those in the 2nd version of the Isgur-Scora-Grinstein- Wise approach (ISGW2) [26,27], in Table

II

we list both results. In Table

III, we show the results without the

annihilated topologies with various effective N

^eff_c

, where the  scale for the effective Wilson coefficients is fixed to be   2:5 GeV which is usually adopted in the literature.

For an explicit example of the effective WCs, we have that

~

a

^s

  2:5 GeV 
584  97i; 418  73i; 284  55i; 84  27i  10

^4

for N

_c^eff


2; 3; 5; 1, respectively.

In our naive estimations, we see that the BRs for B

_d

!

K

^0

; K

_n^0

1430
decays are close to the world average values when N

_c^eff

 3. This could indicate that the non- factorized contributions in the processes are small.

However, from Table

III, the longitudinal and transverse

PFs for B

_d

! K

^0

are inconsistent with the measure-

TABLE II. Transition form factors by the LFQM and ISGW2.

F
m²
 F^BK₁ V^BK A^BK₁ ^ A^BK₂ ^ F^BK

 0

1 k b h

LFQM 0.37 0.33 0.27 0.26 0.275 0.013 0.0065 0.0087

ISGW2 0.217 0.0045 0.0045

TABLE I. Meson decay constants (in units of GeV).

f_K f_B f_K^ f^T_K f
f^T

0.16 0.19 0.20 0.16 0.23 0.20

EXPECTATIONS ON B !
K^₀
1430; K^₂
1430
. . . PHYSICAL REVIEW D 75, 054010 (2007)