meson and search for new physics signals

PHYSICAL REVIEW D 72, 034009 (2005)

Annihilation in factorization-suppressed B decays involving a 11P1 mesonand search for new physics signals

Kwei-Chou YangDepartment of Physics, Chung Yuan Christian University, Chung-Li, Taiwan 320, Republic of China

(Received 4 June 2005; published 11 August 2005; publisher error corrected 31 August 2005)

1550-7998=20

The recent measurements of large transverse fractions in B! �K� decays represent a challenge fortheory. It was shown that the QCD penguin annihilation could contribute to the transverse amplitudes suchthat the observation could be accounted for in the standard model. However, if one resorts to new physicsfor resolutions, then the relevant 4-quark operators should be tensorlike or scalarlike. We show that thesame annihilation effects could remarkably enhance the longitudinal rates of factorization-suppressedB0 ! h1�1380�K�0; b1�1235�K�0 modes to be �12:0�4:1

�3:0� � 10�6 and �7:0� 3:5� � 10�6, respectively, butattend to be canceled in the transverse components. Nevertheless, the transverse fractions of B!h1�1380�K� can become sizable due to new physics contributions. The measurements of these decayscan thus help us to realize the role of annihilation topologies in charmless B decays and offer atheoretically clean window to search for the evidence of new physics.

DOI: 10.1103/PhysRevD.72.034009 PACS numbers: 13.25.Hw, 12.38.Bx, 14.40.Cs

I. INTRODUCTION

The BABAR and Belle collaborations have recentlymeasured B! �K� decays with large transverse fractions[1–5] which are suppressed by �1=mb�

2 in the standardmodel (SM) perturbative picture. Within the SM, it hasbeen given in Ref. [6] that the annihilation graphs, whichare formally suppressed by 1=m2

b but logarithmically en-hanced, could account for the observations with a moderatevalue of the BBNS parameter �A [7]. However, the pertur-bative QCD (PQCD) analysis [8] yielded the longitudinalfraction fL * 0:75 even with considering the annihilation,in contrast with the observation of fL � 0:5. In the largeenergy limit [9], since the three SM helicity amplitudes ofthe �K� modes in the transversity basis are, respectively,proportional to:

A SM0 / f�m

2B�k; ASM

k / ��2pf�m�mB�?;

ASM? / �

��2pf�m�mB�?;

(1.1)

where

mB

mB �mK�V

mB �mK�

2EA1 �?; (1.2)

A0 EmK�

�mB �mK�

2EA1 �

mB �mK�

mBA2

� �k; (1.3)

with A0;1;2 and V being the axial-vector and vector currentform factors, respectively, and the PQCD results for theB! �K� branching ratios (BRs) are about 1.5 timeslarger than the data, it was thus suggested in Ref. [10]that choosing a smaller A0 could resolve the anomaly forobserving the large transverse fractions in the �K� modes.

05=72(3)=034009(14)$23.00 034009

On the other hand, it was argued that [11,12] the anomalymay be resolved if the long distance final-state interactionsvia charmed meson intermediate states exist in the B!�K� decays.

Some possible new physics (NP) solutions have beenproposed. If the mechanism is due to the right-handedcurrents, which could contribute constructively to A? butdestructively to A0;k, then one may have larger jA?=A0j

2 toaccount for the data fL � 0:5. Nevertheless, the resultingjAkj

2 jA?j2 will be in contrast to the recent observations

with f?�perpendicular fraction� � fk�parallel fraction�[2,4]. (See further explanations in Refs. [6,13].) NP inbsg chromomagnetic dipole operator was used to explainthe large transverse fractions of �K� modes in Ref. [14].However, since, in large mb limit, the strong interactionconserves the helicity of a produced light quark pair,helicity conservation requires that the outgoing s and �sarising from s� �s� n gluons vertex have opposite helic-ities. The contribution of the chromomagnetic dipole op-erator to the transverse polarization amplitudes should besuppressed as H00:H��:H�� O�1�:O�1=mb�:O�1=m

2b�

[13] which does not help us to understand the observation.Furthermore, it has been shown in Refs. [6,13] that ifconsidering only the two-parton scenario for the finalmesons, the contributions of the chromomagnetic dipoleoperator to the transverse polarization amplitudes are ac-tually equal to zero. An additional longitudinal gluon isnecessary for having nonvanishing transverse amplitudes.

A general discussion for searching possible NP solutionshas been given in [13] that only two classes of NP four-quark operators are relevant in resolving the transverseanomaly in the �K� modes. The first class of operatorswith structures ��1� �5� � ��1� �5� and �1� �5� ��1� �5� contributes to helicity amplitudes, which referto the NP scenario 1, as H00:H��:H�� O�1=mb�:O�1=m2

b�:O�1�. The second class of operators

-1 © 2005 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.72.034009

KWEI-CHOU YANG PHYSICAL REVIEW D 72, 034009 (2005)

with structures ��1� �5� � ��1� �5� and �1� �5� ��1� �5�, which refer to the NP scenario 2, results inH00:H��:H�� O�1=mb�:O�1=m2

b�:O�1�.1 It was found

that these two classes can separately satisfy the two pos-sible phase solutions for polarization data, owing to thephase ambiguity in the measurement, and resolve theanomaly for large transverse fractions in the �K� modes.(Some discussions due to the tensor operator ��1� �5� ��1� �5�, can be found in [6,16].) A model applicationcan be found in Ref. [17].

In this paper, we shall devote to the study forfactorization-suppressed B! VA decays, where A�V� isan axial-vector (vector) meson with quantum numberN2S�1Lj 11P1�1

3S1� in the quark model scenario.Some B decays involving 1P1 mesons were discussed inRef. [18]. In particular, we focus on h1�1380�K� modes,where h1�1380� is a 11P1 meson2 and its properties are notwell-established experimentally [19]. The quark content ofh1�1380� was suggested as �ss in the QCD sum rule calcu-lation [20]. Because of the G-parity, the distribution am-plitudes of a 11P1 meson defined by the nonlocal vectorand axial-vector current are antisymmetric under the ex-change of quark and antiquark momentum fractions in theSU(3) limit. We shall show that in the SM, while thetransverse components of h1�1380�K� modes are negli-gible, the longitudinal fraction receiving large QCD cor-rections is further enhanced by the annihilation topologiesalthough it vanishes in the factorization limit. Interestingly,the local tensor operator can couple mainly to a trans-versely polarized h1�1380� meson. This means that if thelarge transverse fractions of B! �K� decays are owing tothe NP 4-quark tensor operators, which contribute to theb! s�ss processes, then we expect transverse branchingratios: BRT�h1�1380�K�� ’ BRT��K�� which would bestriking evidence for physics beyond SM. We will alsoshow that the remarkable enhancement of the longitudinalpolarization due to the annihilation topologies could beobserved in b1�1235�K� modes.3

This paper is organized as follows. In Sec. II, we beginwith the summary of light-cone distribution amplitudes(LCDAs) and introduce a light-cone projection operatorin the momentum space that our QCD factorization resultsrely on. We then calculate the QCD factorization decayamplitudes and take B! h1�1380�K� as an example. InSec. III, we give a detailed NP calculation for B!h1�1380�K�, compared with B! �K� results. Sec. IVcontains numerical results for several decay modes, along

1For the b! s�ss processes, the tensor operators can be trans-formed as the scalar operators by Fierz transformation and viceversa [13]. However, it is not true for b! s �uu and b! s �dd.The tensor operators can be induced by box diagrams with theexchange of two gluinos [15].

2h1�1380� with IG�JPC� ?��1�� was denoted as H0 in oldclassification. Its isospin may be 0, but is not confirmed yet.

3b1�1235� was denoted as B�1235� in old classification.

034009

with a detailed estimation of theoretical uncertainties fromvarious sources. Finally, we conclude in Sec. V.

A. Brief summary of results

Since Sec. II contains the mathematical expressions forLCDAs of the 11P1 mesons and QCD factorization decayamplitudes, and Sec. III for new physics amplitudes, theycan be read independently. The reader, who is not familiarwith the theoretical framework, may temporarily omitthese two sections but should consult Sec. IV about thenumerical results, for which we summarize the mainbranching ratios as follows. If large transverse fractionsinB! �K� decays are owing to the annihilation topology,we predict

B Rtot�h1�1380�K�0� �12:0�4:1�3:0� � 10�6;

BRtot�b1�1235�K�0� �7:0� 3:5� � 10�6;(1.4)

which are predominated by the longitudinal component.Accordingly, large BR�h1�1380�K� and BR�b�1 �1235�K��could be observed. See also Sec. V for discussions. On theother hand, if large transverse components of the �K�

modes originate from the NP, we can also observe thatlarge transverse fractions in B! h1�1380�K�, such that

BRtot�h1�1380�K�0� �14:5� 4:0� � 10�6;

BRk�h1�1380�K�0� �3:2� 1:5� � 10�6;

BR?�h1�1380�K�0� �2:0� 1:0� � 10�6 (1.5)

in the NP scenario 1, and

BRtot�h1�1380�K�0� �8:5� 2:0� � 10�6;

BRk�h1�1380�K�0� �2:0� 0:5� � 10�6;

BR?�h1�1380�K�0� �1:8� 0:5� � 10�6 (1.6)

in the NP scenario 2. The detailed results and discussionscan be found in Sec. IV. We discuss possible NP effects for�K� modes in Sec. V.

II. B! h1�1380�K� STANDARD MODEL DECAYAMPLITUDES

Within the framework of QCD factorization, the SMeffective Hamiltonian matrix elements are written in theform of

hh1K�jH effjBi

GF��2p

Xpu;c

�phh1K�jTA

h;p �T Bh;pjBi;

(2.1)

where �p � VpbV�ps, and the superscript h denotes thefinal-state meson helicity. TA accounts for the topologiesof the form-factor and spectator scattering, while T B

contains annihilation topology amplitudes.

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ANNIHILATION IN FACTORIZATION-SUPPRESSED B . . . PHYSICAL REVIEW D 72, 034009 (2005)

A. Two-parton distribution amplitudes of 11P1axial-vector mesons

We consider B! VA processes where the 11P1 axialmeson A, which is made of q1 and �q2, is emitted from theweak decay vertex in the factorization amplitudes.4 In the

4If the 1P1 particle is made of �qq, then its charge conjugate Cis �1, i.e., its quantum number is JPC 1��.

034009

naive factorization, B! VA processes are highly sup-pressed since G-parity does not match between the Ameson and the axial-vector current �q1��5q2 in theSU(3) limit. In the QCD factorization, the QCD radiativecorrections can turn the local operators �q1��1 �5�q2

into a series of nonlocal operators as

hA�P0; ��j �q1�y�q2�x�j0i �i4

Z 1

0duei�up

0�y� �up0�x��fAmA

�6p0�5

�� zp0 � z

�k�u�� 6 �?�5g�v�? �u��

��p0�z��g�a�? �u�

4

�

� f?A

��6 �? 6p

0�5�?�u�� im2A� � z

�p0 � z�2��5p

0�z�h�t�k�u�� im2

A�� z

h�s�k�u�

2

��; (2.2)

where the chiral-even LCDAs are given by

hA�P0; ��j �q1�y��5q2�x�j0i ifAmA

Z 1

0duei�up

0�y� �up0�x�

�

�p0�

�� zp0 � z

�k�u�

� ��?�g�v�? �u�

�; (2.3)

hA�P0; ��j �q1�y��q2�x�j0i �ifAmA��p0�z�

�Z 1

0duei�up

0�y� �up0�x� g�a�? �u�

4;

(2.4)

with the matrix elements involving an odd number of �matrices and �u � 1� u, and the chiral-odd LCDAs aregiven by

hA�P0; ��j �q1�y��5q2�x�j0i

f?AZ 1

0duei�up

0�y� �up0�x��?�p

0� � ��?�p

0��?�u�

�m2A�� z

�p0 � z�2�p0�z� � p0�z��h

�t�k�u��; (2.5)

hA�P0;��j �q1�y��5q2�x�j0if?A m2A

�� zZ 1

0duei�up

0�y� �up0�x�

�h�s�k�u�

2; (2.6)

with the matrix elements containing an even number of �matrices. Here, throughout the present discussion, we de-fine z y� x with z2 0, and introduce the lightlikevector p0� P0� �m

2Az�=�2P

0 � z� with the meson’s mo-mentum satisfying P02 m2

A. Moreover, the meson polar-ization vector �� has been decomposed into longitudinal

and transverse projections defined as

��k��

�� zP0 �z

�P0��

m2A

P0 �zz�

�; ��?��

��

�k�; (2.7)

respectively. The LCDAs �k;�? are of twist-2, andg�v�? ; g

�a�? ; h

�t�? ; h

�s�k

of twist-3. Because of G-parity, �k; g�v�?

and g�a�? are antisymmetric with the replacement u! 1�u, whereas �?; h

�t�k

and h�s�k

are symmetric in the SU(3)limit. We restrict ourselves to two-parton LCDAs withtwist-3 accuracy.

To perform the calculation in the momentum space, wefirst represent Eq. (2.2) in terms of z-independent variables,P0 and ��. For simplicity, we introduce two lightlike vec-tors n�� 1; 0; 0;�1�; n�� 1; 0; 0; 1�. If neglecting themeson mass squared, we have p0� En�� where E is theenergy of the A meson in the B rest frame. Choosing themomentum of the quark q1 in the A meson as

k�1 uEn�� k�? �

k2?

4uEn��; (2.8)

we apply the following substitution in the calculation

z� ! �i@

@k1�’ �i

�n��2E

@@u�

@@k?�

�; (2.9)

where the term of order k2? is omitted. Note that all the

components of the coordinate z should be taken into ac-count in the calculation before the collinear approximationis applied. Then, the light-cone projection operator of an Ameson in the momentum space reads

MA MA

k �MA?; (2.10)

with the longitudinal part

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PHYSICAL REVIEW D 72, 034009 (2005)

MAk �i

fA4

mA�� n��2

6n��5�k�u� �if?A mA

4

�mA��

� � n��2E

�i2��5n��n��h

�t�k�u�

� iEZ u

0dv��?�v� � h

�t�k�v��5n��

@@k?�

� �5

h0�s�k�u�

2

��kup0; (2.11)

and the transverse part

MA? i

f?A4E�6 �?6n��5�?�u� � i

fAmA

4

��6 �?�5g

�v�? �u�

� EZ u

0dv��k�v� � g

�v�? �v��6n��5��?�

@@k?�

� i��? �

�n��

�n��

1

8

dg�a�? �u�

du

� Eg�a�? �u�

4

@@k?�

��kup0; (2.12)

where the transverse polarization vectors of the axial-vector meson are

��? � �� n�

2n��

� � n�2

n��; (2.13)

which is, instead of that in Eq. (2.7), independent of thecoordinate. In the present study, we choose the coordinatesystems in the Jackson convention [13], which is adoptedby BABAR and Belle measurements. In other words, in theB rest frame, if the z axis of the coordinate system is alongthe direction of the flight of the V meson, we can have

KWEI-CHOU YANG

034009

��V �0� �pc; 0; 0; EV�=mV;

��A �0� �pc; 0; 0;�EA�=mA;

��V ��1� 1��2p �0; 1;�i; 0�;

��A ��1� 1��2p �0; 1;�i; 0�; (2.14)

where pc is the center mass momentum of the final-statemeson. In the large energy limit, we have ��A�� n� 2EA=mA�;0 and ��A�� n� 0. Note that if the coordi-nate systems are chosen in the Jacob-Wick convention[13], the transverse polarization vectors of the A mesonbecome ��A ��1� �0;�1;�i; 0�=

��2p

. In general, the QCDfactorization amplitudes can be reduced to the form ofR

10 dutr�MA . . .�.In the following, we will give a brief discussion for

LCDAs of V and A mesons. The detailed information forLCDAs of the vector mesons can be found in [13,21]. Theasymptotic twist-2 distribution amplitudes are

�Vk�u� �V

?�u� �A?�u� 6u �u; (2.15)

but �Ak�u� 0 in SU(3) due to G-parity. �A

k�u� can be

expanded in Gegenbauer polynomials with only odd terms:

�Ak�u� 6u �u

� Xi1;3;5;...

aA;ki C3=2i �2u� 1�

�; (2.16)

where we have neglected the even terms due to possiblemq1

� mq2. Note that since the product fAa

A;k1 always

appears together, we simply take fA f?A in the presentstudy, while aA;k1 is determined in Ref. [22]. If neglectingthe three-parton distributions and terms proportional to thelight quark masses, the twist-3 distribution amplitudes forboth V and A mesons can be related to the twist-2 ones byWandzura-Wilczek relations [22,23]:

h0�s�k�v� �2

�Z v

0

�?�u��u

du�Z 1

v

�?�u�u

du�; h0�t�

k�v� �2u� 1�

�Z v

0

�?�u��u

du�Z 1

v

�?�u�u

du�;

Z v

0du��?�u��h

�t�k�u�� v �v

�Z v

0

�?�u��u

du�Z 1

v

�?�u�u

du�;

Z v

0du��k�u��g

�v�? �u��

1

2

��vZ v

0

�k�u��u

du�vZ 1

v

�k�u�u

du�; g�a�? �v� 2

��vZ v

0

�k�u��u

du�vZ 1

v

�k�u�u

du�;

g0�a�? �v�

4�g�v�? �v�

Z 1

v

�k�u�u

du;g0�a�? �v�

4�g�v�? �v� �

Z v

0

�k�u��u

du: (2.17)

B. B! h1�1380�K� amplitudes with topologies of the form-factor and spectator scattering

TAh;p describes contributions from naive factorization, vertex corrections, penguin contractions, and spectator

scattering. However, for B! h1�1380�K� processes, the naive factorization amplitudes are forbidden due to the mismatchof the G-parity between the h1�1380� meson and the local axial-vector current �s��5s. The resultant amplitude reads

-4


GF��2p

Xpu;c

�phh1K�jT A

h;pjBi GF��

2p ��VtbV

�ts�

�ah3 � a

h4 � a

h5 � r

h1� ah6 �

1

2��ah7 � r

h1� ah8 � a

h9 � a

h10�

�X�BK

�;h1�h ; (2.18)

where5

X�BK�;h1�

0 �fh1

2mK�

��m2

B �m2K� �m

2h1��mB �mK� �ABK

�

1 �m2h1� �

4m2Bp

2c

mB �mK�ABK

�

2 �m2h1�

�;

X�BK�;h1�

� fh1mh1

��mB �mK� �ABK

�

1 �m2h1�

2mBpcmB �mK�

VBK��m2

h1�

�;

(2.19)

with q pB � pK� � ph1, pc being the center mass momentum of the final-state mesons in the B rest frame and

rh1�

2mh1

mb��

f?h1��

fh1

: (2.20)

Here the form factors are defined as

hK��pK� ;�K� �jV�jB�pB�i2

mB�mK�� K�p

Bp

K�V�q

2�;

hK��pK� ;�K� �jA�jB�pB�i i��mB�mK� ��

�K��A1�q

2��K� �pB��pB�pK� ��A2�q2�

mB�mK�

�

�2imK��K� �pBq2 q��A3�q

2��A0�q2��; (2.21)

where A3�0� A0�0� and

A3�q2� mB �mK�

2mK�A1�q2� �

mB �mK�

2mK�A2�q2�: (2.22)

In general, for B! VA processes with the A meson emitted from the weak decay vertex, ahi ’s are given by

ah1s4�

CFNcc2�f

h;0I �f

h;0II �; ah2

s4�

CFNcc1�f

h;0I �f

h;0II �; ah3

s4�

CFNcc4�f

h;0I �f

h;0II �;

ah4s4�

CFNcfc3�f

h;0I �f

h;0II ��G

h�ss��Gh�sb��c1

��u�tGh�su��

�c�tGh�sc�

��c4�c6�

Xbiu

Gh�si��3

2�c8�c10�

Xbiu

eiGh�si�

�3

2c9

�eq0G

h�sq0 ��1

3Gh�sb�

��cgG

hg

�;

ah5�s4�

CFNcc6�f

h;1I �f

h;1II �;

ah6s4�

CFNc

�c5

~fhI �c1

��u�tGh�su��

�c�tGh�sc�

��c3�G

h�ss��Gh�sb��c4�c6�

Xbiu

Gh�si��;

ah7�s4�

CFNcc8�f

h;1I �f

h;1II �; ah8

s4�

CFNcc7

~fhI �

9�Che; ah9

s4�

CFNcc10�f

h;0I �f

h;0II �;

ah10s4�

CFNcc9�f

h;0I �f

h;0II ��

9�Che; (2.23)

where ci are the Wilson coefficients, CF �N2c � 1�=�2Nc�, si m2

i =m2b, and �q VqbV�qq0 , with q0 d; s. In Eq. (2.23),

the vertex corrections are given by

5There is a relative sign difference between X�BK�;h1�

0 and X�BK� ;h1�

� due to the adoption of the Jackson convention for the coordinatesystems.

034009-5


f0;iI

Z 1

0dx�A

k�x� �gi�x�;

f�;iI Z 1

0dx�gA�v�? �x� � ��1�i

1

4

dgA�a�? �x�

dx

��gi�x�;

~f0I

Z 1

0dx�h0�s�A

k�x�

2�2Li2�x� � ln2x� �1� 2i�� lnx� �x$ 1� x��;

~f�I 0; (2.24)

with

�g0�x� 3�1� 2x1� x

�lnx�

�2Li2�x� � ln2x�

2 lnx1� x

� �3� 2i�� lnx� �x$ 1� x��;

�g1�x� �g0�1� x�: (2.25)

Here fh;iII , arising from the hard spectator interactions with a hard gluon exchange between the emitted h1�1380�meson andthe spectator quark of the B meson, have the expressions:

f0;iII

4�2

Nc

fBfAfV

2X�B0V;A�

0

Z 1

0d�

�B1 ��

Z 1

0dv�V

k�v�

Z 1

0du��1�i

�1

u �v�

1

�u �v

��Ak�u��rA�

h0�s�Ak�u�

2

�

�

�1

u �v�

1

�u �v

��Ak�u��rh1

�h0�s�Ak�u�

2

��;

f�;iII �4�2

Nc

2fBfAf?V mA

mBX�B0V;A��

�1 1�Z 1

0d�

�B1 ��

Z 1

0dv

�V?�v�

�v2

Z 1

0du��gA�u�? �u��1�i

1

4

dgA�a�? �u�

du

��

�1

u�

1

�u

�

�Z u

0dx��A

k�x��gA�v�? �x��

��

4�2

Nc

fBfAfVmAmV

m2BX�B0V;A��

Z 1

0d�

�B1 ��

Z 1

0dv�gV�v�? �v��

1

4

dgV�a�? �v�

dv�

�Z 1

0du��1�i

�u� �v

u �v2 ��u� �v

�u �v2

��

�u� �v

u �v2 ��u� �v

�u �v2

��gA�v�? �u��1�i

1

4

dgA�a�? �u�

du

��

��1�i

�1

u �v2�1

�u �v2

�

�

�1

u �v2�1

�u �v2

��Z u

0dx��A

k�x��gA�v�? �x��

�; (2.26)

with �B1 �� being one of the two light-cone distribution amplitudes of the Bmeson [21,24].Gh; Gh; Che , and Che , originating

from QCD and electroweak contractions, respectively, are given by

G0�s� 4Z 1

0du�A

k�u�

Z 1

0dxx �x ln�s� �ux �x� i��;

G��s� 2Z 1

0du�gA�v�? �u� �

1

4

dgA�a�? �u�

du

�Z 1

0dxx �x ln�s� �ux �x� i��;

G0�s� 4Z 1

0du�h0�s�A

k�u�

2

Z 1

0dxx �x ln�s� �ux �x� i��; G��s� 0;

Che�s� ��u�tGh�su� �

�c�tGh�sc�

��c2 �

c1

Nc

�;

(2.27)

and Che can be obtained from Che with the replacement Gh ! Gh, where small electroweak corrections from c7�10 areneglected in Che and Che . The dipole operator Og gives

G0g �2

Z 1

0du

�Ak�u�

�u; G�g

Z 1

0

du�u

�Z u

0��Ak�v� � gA�v�? �v��dv� �ugA�v�? �u�

�u4

dgA�a�? �u�

du�gA�a�? �u�

4

�: (2.28)

Using Eq. (2.17), G�g can be further reduced to

034009-6


G�g Z 1

0du�Z u

0

�Ak�v�

�vdv�

Z 1

u

�Ak�v�

vdv� 0; G�g 0; (2.29)

where we have taken the approximation �Ak�u� 6u �uaA;k1 C3=2

1 �2u� 1�. Obviously, considering only two-parton distri-bution amplitudes, the dipole operator does not contribute to transverse amplitudes at O�s�. The result is consistent withthe fact that in large mb limit the transverse amplitudes are suppressed since the outgoing s and �s arising from s� �s�n gluons couplings have opposite helicities. Note that the linear infrared divergence, originating from twist-3� twist-3final-state LCDAs,6 is present in f�II and it may exist a mechanism in analogy to the heavy-light transition form factorswhere the linear divergences are consistently absorbed into the form factors [21]. However we will introduce an infraredcutoff �QCD=mb to regulate the linear divergence. The numerical results are very insensitive to the cutoff, and, moreover,the transverse contributions are already suppressed. On the other hand, we shall parametrize the logarithmic divergence,appearing in fhII, as

Z 1

0

dx�x! XhH ln

�mB

�h

��1� �hHe

i�hH �; (2.30)

with �hH � 1 and �h � 0:5 GeV.

C. B! h1�1380�K� amplitudes with topologies of annihilation

We shall see that B! h1�1380�K� helicity amplitudes in the SM may be governed by the annihilation topologies. Theweak annihilation contributions to B0 ! h1�1380�K�0 read

GF��2p

Xpu;c

�phh1K�0jT B

h;pjB0i ’GF��

2p

Xpu;c

VpbV�ps

�fBfK�fh1

�bh3 �

1

2bh3;EW

��; (2.31)

where

bh3 CFN2c�c3A

i;h1 � c5A

i;h3 � �c5 � Ncc6�A

f;h3 �; bh3EW

CFN2c�c9A

i;h1 � c7A

i;h3 � �c7 � Ncc8�A

f;h3 �: (2.32)

For B� ! h1�1380�K�� an additional term pubh2 is needed in the square bracket in Eq. (2.31) with bh2 c2A

i;h1 CF=N

2c .

The annihilation amplitudes originating from operators � �q1b�V�A� �q2q3�V�A and �2� �q1b�S�P� �q2q3�S�P are denoted asAi;f�h�1 and Ai;f�h�3 , respectively, where the superscript i�f� indicates gluon emission from the initial (final) state quarks in theweak vertex. In Eq. (2.31), we will neglect the terms proportional to Ai;h1�3� due partially to c3;5;9 �c5 � Ncc6� or Cabibbo-Kobayashi-Maskawa quark-mixing matrix (CKM) suppression (for B�). Although Ai;�1 contains a linear divergencearising from the twist-3� twist-3 final-state distribution amplitudes, it was argued in Ref. [6] that the divergence should becanceled by the twist-4� twist-2 ones. Moreover, Ai;�1 is still relatively small compared to Af;�3 [6]. Af;h3 , for which one ofthe final-state mesons arises with the twist-3 distribution amplitude, while the other is of twist-2, are given by

Af;03 �h1K�� s

Z 1

0dudv

�2mh1f?h1

mbfh1

�K�k�v�

�hh10�s�k�u�

2�v�u

Z u

0dx��h1

? �x� � hh1�t�k�x��

�2

v2 �u�

2mK�f?K�

mbfK��h1

k�u�

�

�hK�0�s�k�v�

2�

�uv

Z v

0dx��K�

? �x� � hK��t�k�x��

�2

�u2v

�; (2.33)

Af;�3 �h1K��s

Z 1

0dudv

�2mh1

f?K�mbfK�

�K�? �v�

�gh1�v�? �u��

gh10�a�? �u�

4

�2

v2 �u�

2mK�f?h1

mbfh1

�h1

? �u��gK

��v�? �v��

gK�0�a�? �v�

4

�2

�u2v

�;

(2.34)

where the detailed definitions of the distribution amplitudes of the h1 meson have been collected in Sec. II A.Using the asymptotic distribution amplitudes of �K�

k;?�u� and �h1

?�u�, and the approximation for �h1

k�u�

6u �uah11 C

3=21 �2u� 1� given in Eqs. (2.15) and (2.16), respectively, we obtain the annihilation amplitudes

6We have checked that the linear divergence is not canceled by twist-4� twist-2 ones.

034009-7


Af;03 �h1K�� 18�s�X

0A � 2�

�2mh1f?h1

mbfh1

�2X0A � 1� �

2mK�ah1;k1 f?K�

mbfK��6X0

A � 11��; (2.35)

Af;�3 �h1K�� 18�s�X�A � 1�

�2mK�f?h1

mbfh1

�2X�A � 3� �2mh1

ah1;k1 f?K�

mbfK�

�2X�A �

17

3

��; (2.36)

where the logarithmic divergences are parametrized asXhA �1� �

hAe

i’hA� ln�mB=�h� with �hA & 1. Comparedwith ASM

0 given in Eq. (1.1), the annihilation amplitudesthus contribute to the longitudinal and negative polarizedstates as O�1=m2

bln2�mb=�h��, where we use that fB �1=m�1=2

b [25] and �k � 1=m�3=2b [9]. We obtain

f?h1�1 GeV� ’ 0:2 GeV and fh1

ah1;k1 �1 GeV� �

�0:45 GeV from the QCD sum rule calculation [22],where the numerical values will be listed in Sec. IV. InEq. (2.35), the second term in Af;03 is numerically dominantsuch that annihilation corrections are constructive to thelongitudinal amplitude, while in Eq. (2.36), two terms inAf;�3 tend to cancel each other such that annihilation effectsare negligible for transverse fractions. For comparison, welist Af;0��3 for the B! �K� decays as follows:

Af;03 ��K��18i�s

�2m�f?�mbf�

�2mK�f

?K�

mbfK�

��X0

A�2�

��2X0A�1�;

Af;�3 ��K��18i�s

�2mK�f?�mbf�

�2m�f?K�mbfK�

��2X�A �3�

��X�A �1�: (2.37)

III. NEW PHYSICS EFFECTS: B! h1�1380�K� VSB! �K�

In addition to annihilation contributions, the other pos-sibility for explaining the polarization puzzle in B! �K�

is to introduce NP scalar- and/or tensorlike operators asdiscussed in Ref. [13]. In the present paper, we will explorethe existing evidence in B! h1�1380�K� channel. Therelevant NP effective Hamiltonian H NP, following thedefinition in Ref. [13], is given by

H NP GF��

2p

Xi15�18;23�26

ci��Oi�� H:c:; (3.1)

where the scalar-type operators are

O15 s�1� �5�bs�1� �5�s;

O16 s�1� �5�b s �1� �5�s;

O17 s�1� �5�bs�1� �5�s;

O18 s�1� �5�b ss �1� �5�s;

(3.2)

and the tensor-type operators are

034009

O23 s��1� �5�bs��1� �5�s;

O24 s��1� �5�b s ��1� �

5�s;

O25 s��1� �5�bs��1� �5�s;

O26 s��1� �5�b s ��1� �

5�s;

(3.3)

with ; being the color indices.We now calculate the decay amplitudes for B0 !

h1�1380�K�0 due to O15�18 and O23�26 operators definedin Eqs. (3.2) and (3.3). The CP-conjugate amplitudes forB0 ! h1�1380�K�0 can be obtained by CP-transformation.By the Fierz transformation, O15;16 and O17;18 operatorscan be expressed in terms of linear combination of O23;24

and O25;26 operators, respectively, i.e.,

O151

12O23�

1

6O24; O16

1

12O24�

1

6O23;

O171

12O25�

1

6O26; O18

1

12O26�

1

6O25:

(3.4)

In the computation, the matrix elements for tensor opera-tors O23;25 can be recast into

hh1�1380��q; �h1�;

K��pK� ; �K� �j �s��1� �5�s�s��1� �5�bjB�p�i

�1�

1

2Nc

�8f?h1

��i��

��h1��K�p

�Bp

�K�T1�m2

h1�

�

�T2�m

2h1� �

m2h1

m2B �m

2K�T3�m

2h1�

��h1� pB�

��K� � pB� �1

2T3�m2

h1��h1

� ��K� ��; (3.5)

under factorization, where the tensor decay constant f?h1

and the form factors are defined as [5,22,26,27]

hh1�1380��q; �h1�js��5sj0i f?h1

��h1q� � ��h1

q��;

(3.6)

-8

TABLE II. Form factors at zero momentum transfer for B!V transitions evaluated in the light-cone sum rule analysis. Set 1contains the new results with some improvements in Ref. [29]and set 2 is the original analysis in Ref. [26].

A1(0) A2(0) A0(0) V(0) T1(0) T2(0) T3(0)

B! � (set 1) 0.242 0.221 0.303 0.323 0.267 0.267 0.176B! K� (set 1) 0.292 0.259 0.374 0.411 0.333 0.333 0.202

TABLE I. Values of decay constants from QCD sum rule calculations [22,27,29,30]

� K� � b1 h1�1380�

fV�A� [MeV] 205� 9 217� 5 231� 4 f?b1�1 GeV� f?h1

�1 GeV�f?V�A�� 1 GeV� [MeV] 160� 10 170� 10 200� 10 180� 10 200� 20f?V�A�� 2:2 GeV� [MeV] 147� 10 156� 10 183� 10 165� 9 183� 18


hK��pK� ; �K� �j �s��5bjB�p�i

T1�q2��K� �pB � pK� �� K� �pB � pK� �

��

� �T1�q2� � T2�q

2��m2B �m

2K�

q2 ��K�q� � ��K�q

��

� 2�T3�q2� � �T1�q2� � T2�q2��

m2B �m

2K�

q2

��p�K�q

�

� p�K�q��;

(3.7)

with

T1�0� T2�0�: (3.8)

The helicity amplitudes, in units of GF=��2p

, for the B0

decay due to the NP operators are given by

HNP00 4f?h1

m2B�~a23 � ~a25��h2T2�m

2h1� � h3T3�m

2h1��;

HNP�� 4f?h1

m2B

��m2B �m

2K�

m2B

��~a23 � ~a25�T2�m2

h1�

2pc�~a23 � ~a25�T1�m2h1�

�; (3.9)

or in terms of the transversity basis,

ANP0 4f?h1

m2B�~a23 � ~a25��h2T2�m

2h1� � h3T3�m

2h1��;

ANPk �4

��2pf?h1�m2

B �m2K� ��~a23 � ~a25�T2�m2

h1�;

ANP? �8

��2pf?h1mBpc�~a23 � ~a25�T1�m

2h1�; (3.10)

where

h2 1

2mK�mh1

��m2B �m

2h1�m2

K� ��m2B �m

2K� �

m2B

� 4p2c

�;

h3 1

2mK�mh1

� 4p2cm

2h1

m2B �m

2K�

�; (3.11)

and

~a 23a23�a24

2�a16

8; ~a25a25�

a26

2�a18

8; (3.12)

are defined as ~a23 �j~a23jei23ei�23 ; ~a25 j~a25jei25ei�25

with 23;25 and �23;25 being the corresponding the strongphases and NP weak phases, respectively.7 Here we define

7We have restrained j23;25j<�=2.

034009

a2i c2i �c2i�1

Nc� nonfactorizable corrections;

a2i�1 c2i�1 �c2i

Nc� nonfactorizable corrections; (3.13)

with i 2 integer number.Since the �K� data showed that jA?j� ��H��

H��=��2p� ’ jAkj� �H�� H��=

��2p�, there exist two

possible solutions with H�� H�� and H�� H��,respectively, for which the former, referring to the NPscenario 2, corresponds to j~a23j ’ 1:5� 10�4; ~a25 � 0,while the latter, referring to the NP scenario 1, accordswith ~a23 � 0; j~a25j ’ 2:0� 10�4.

Comparing the NP results for B! h1�1380�K� with thatfor B! �K� [13] (in units of GF=

��2p

):

ANP0 �4if?�m

2B�~a23 � ~a25��h2T2�m

2�� h3T3�m

2��;

ANPk 4i

��2pf?� �m

2B �m

2K� ��~a23 � ~a25�T2�m2

��;

ANP? 8i

��2pf?�mBpc�~a23 � ~a25�T1�m

2��; (3.14)

we have jANP0 �h1�1380�K��=ANP

0 ��K��j ’ jANP

k �

�h1�1380�K��=ANPk ��K

��j ’ jANP? �h1�1380�K��=ANP

? �

��K��j ’ f?h1=f?� � 1 for the case of H�� H��

or H�� H��. Neglecting the annihilation contribu-tions, we conclude that in the NP scenariosBRL�h1�1380�K�� is dominated by the SM amplitudeswhich may contain sizable QCD corrections, whileBRT�h1�1380�K�� 0:6� 1:1�BRT��K�� due to the NPeffects together with the SM contributions in the �K�

modes. The detailed analysis will be given in the nextsection.

B! � (set 2) 0.261 0.223 0.372 0.338 0.285 0.285 0.202B! K� (set 2) 0.337 0.283 0.470 0.450 0.379 0.379 0.261

-9

TABLE III. The CP-averaged branching ratio (in units of 10�6) for B! �K�0. The annihilation corrections and possible newphysics effects are not included. The longitudinal fractions are larger than 85%. The results refer to �H 0, and the central values ofdecay constants and form factors given in Tables I and II. �B is in units of MeV.

Form factors: Set 1 Form factors: Set 2�B 200 �B 350 �B 500 �B 200 �B 350 �B 500

�v mb 2.53 2.93 3.26 4.14 4.64 5.06�H mb=2�v mb=2 6.19 6.82 7.34 9.81 10.6 11.2

�H mb=2�v mb=2 2.84 4.27 5.65 5.38 7.33 9.12

�H 1 GeV


IV. NUMERICAL RESULTS

A. Input parameters

To proceed the numerical analysis, we adopt next-to-leading order (NLO) Wilson coefficients in the naive di-mensional regularization (NDR) scheme given in [28]. ForCKM matrix elements, we adopt the Wolfenstein parame-trization with A 0:801, � 0:2265, � ��1��2=2� 0:189, and � ��1� �2=2� 0:58 [19]. Totake into account the possible uncertainty of form factorson our results, in the numerical analysis we combine twopossible sets of form factors, coming from the light-conesum rule calculation. The form factors at zero momentumtransfer are cataloged in Table II, for which we allow 15%uncertainties in values in the present analysis, and theirq2-dependence can be found in [26,29]. The decay con-stants [27,29,30] used in the numerical analysis are col-lected in Table II. In analogy with the QCD sum rulecalculation for f?b1

, one can obtain f?h1[22]. We will simply

take fb1�h1� f?b1�h1�

�1 GeV� in the study since only the

products of fb1�h1�ab1�h1�;k

1 are relevant, where ab1�h1�;k1 is the

first Gegenbauer moment of �b1�h1�

kdefined in Eq. (2.16).

Using the QCD sum rule technique, we have studiedab1�h1�;k

1 in Ref. [22], where the results are given by

ab1;k1 �� 1 GeV� �1:70� 0:45;

ah1;k1 �� 1 GeV� �1:75� 0:20;

ab1;k1 �� 2:2 GeV� �1:41� 0:37;

ah1;k1 �� 2:2 GeV� �1:45� 0:17: (4.1)

8It was argued in Ref. [7] that �H;�A � ��h��1=2 with � 2

�mb=2; mb�. However, here we have taken larger ranges of�H;�A into account.

The magnitudes of ab1�h1�;k1 have a large impact on the

longitudinal fraction of the penguin-dominated B! VAdecay rates. We use the LCDAs of mesons given inEqs. (2.15), (2.16), and (2.17). It turns out that our predic-tions are insensitive to the 2nd nonzero Gegenbauer mo-ments of LCDAs. The integral of the B meson wavefunction is parametrized as

034009

Z 1

0

d��

�B1 ��

mB

�B; (4.2)

with �B �350� 150� MeV [7]. For simplicity, the loga-rithmic divergences, XH; XA are taken to be independent ofthe helicities of the final states, with �H; �A � 1 and�H;�A 2 �0; 2��. As will be discussed below, the valuesof �A and �A are further constrained by the �K� data.There are three independent renormalization scales fordescribing the decay amplitudes: (i) �v for loop diagramsand penguin topologies, contributing to the hard-scatteringkernels, (ii) �H for the hard spectator scattering, and(iii) �A for the annihilation. We take �v 2 �mb=2; mb�and �H;A 2 �1 GeV; mb=2�.8 The working scales �B andvalues of form factors give a large impact on our results. Toreduce these theoretical uncertainties in predictions, weconstrain the parameters by means of B0 ! �K�0 data.The relevant QCDF formulas for the �K�0 mode can befound in Refs. [6,13]. Without the annihilation effects, weillustrate the B0 ! �K�0 branching ratio corresponding toseveral typical choices of parameters in Table III, wheresince XH gives corrections to H and H00 suppressed by1=m�2�b and r�� , respectively, the results are insensitive tothe magnitude of �H. Four remarks are in order. First, thelongitudinal fractions are * 85% in Table III. Second, weseparately consider the annihilation and new physics ef-fects. Third, the results in Ref. [6] indicate that if theannihilation corrections construct to the negative polariza-tion component (for B decays), they become destructive tothe longitudinal fraction with the same order of magnitude.Since the data give B�B0 ! �K�0� �0:95� 0:9� �10�6 and the longitudinal fraction fL 0:48� 0:04 [5],it seems to be favored to have a larger value ( * 0:8�10�6) of BR before adding the annihilation effects, as somechoices in Table III; otherwise the resulting branching ratiowill be too small. (Thus form factors of set 2 seem to bepreferable.) If further considering the �K� phase measure-

-10

TABLE V. CP-averaged branching ratios (in units of 10�6) for B! h1�1380�K� without/with annihilation contributions denoted asBRwo=BRw.

BRwotot BRwo

kBRwo? BRw

tot BRwk

BRw? arg�Ak=A0� arg�A?=A0�

B� ! h1�1380�K�� 3:4�1:5�1:2 & 0:2 & 0:2 13:1�4:1

�3:0 & 1:0 & 1:0 �0:72� 0:13 �0:71� 0:13B0 ! h1�1380�K�0 3:2�1:5

�1:2 & 0:2 & 0:2 12:0�4:1�3:0 & 1:0 & 1:0 �0:67� 0:13 �0:67� 0:13

TABLE IV. Effective coefficients ah�h0�

i for B! VA�B! VA� helicity amplitudes, where A�� 11P1� is formed by the emittedquarks from the weak vertex, obtained in the QCD factorization calculations. The results are given at �v �H mb=2.

<�a01� 0:07� 0:02 a��1 0:01� 0:01 a��1 0:02� 0:01

<�a01� �0:04� 0:02 a��1 �0:03� 0:02 a��1 �0:03� 0:02

<�a02� �0:27� 0:05 a��2 �0:04� 0:02 a��2 �0:10� 0:03

=�a02� 0:16� 0:04 a��2 0:11� 0:03 a��2 0:11� 0:03

<�a03� �0:012� 0:004 a��3 0:002� 0:001 a��3 0:004� 0:002

=�a03� �0:007� 0:003 a��3 �0:005� 0:002 a��3 �0:005� 0:002

<�a04� �0:020� 0:005 <�a��4 � �0:004� 0:002 <�a��4 � �0:0001� 0:0001

=�a04� �0:012� 0:004 =�a��4 � �0:001� 0:001 =�a��4 � �0:0003� 0:0001

<�a05� 0:014� 0:003 a��5 �0:007� 0:002 a��5 �0:004� 0:002

=�a05� �0:009� 0:003 a��5 0:006� 0:003 a��5 0:006� 0:003

<�a06� 0:003� 0:001 <�a��6 � 0 <�a��6 � 0

=�a06� 0:013� 0:005 =�a��6 � 0 =�a��6 � 0

<�a07� �0:0002� 0:0001 <�a��7 � 0:000 08� 0:000 03 <�a��7 � 0:000 04� 0:000 02

=�a07� 0:000 10� 0:000 04 =�a��7 � �0:000 07� 0:000 03 =�a��7 � �0:000 07� 0:000 05

<�a08� �0:000 02� 0:000 01 <�a��8 � 0 <�a��8 � 0

=�a08� 0:000 06� 0:000 02 =�a��8 � 0 =�a��8 � 0

<�a09� �0:0006� 0:0002 <�a��9 � �0:0001� 0:0001 <�a��9 � �0:0002� 0:0001

=�a09� 0:0003� 0:0002 =�a��9 � 0:0002� 0:0001 =�a��9 � 0:0002� 0:0001

<�a010� 0:0020� 0:0005 <�a��10 � 0:0003� 0:0001 <�a��10 � 0:0008� 0:0003

=�a010� �0:0014� 0:0004 =�a��10 � �0:0009� 0:0003 =�a��10 � �0:0009� 0:0004


ments with 1� errors, arg�Ak=A0� 2:36�0:18�0:16 and

arg�A?=A0� 2:49� 0:18, we obtain �45� & �A &

10�. Fourth, the new physics gives constructive correctionsto A0, of order 1=mb, and to Ak;?, of order 1. Thus, tojustify the measurements, in the SM (without annihilationcorrections), the �K� BR should be & 4:5� 10�6 beforeincluding new physics effects.

B. B! h1�1380�K�

To illustrate the nonfactorizable effects forfactorization-suppressed B! VA helicity amplitudes,where A�� 11P1� is formed by the emitted quarks fromthe weak vertex, we give the numerical results for effectivecoefficients ahi in Table IV. The results are evaluated at�v �H mb=2, including all theoretical uncertainties.

In the SM, with parameters constrained by the B! �K�

measurements and including the annihilation effects, wehave computed the branching ratios, together with relativephases among the amplitudes, of h1�1380�K� modes,which are summarized in Table V. The QCD correctionsturn the local operators �s��5s into a series of nonlocaloperators and the resultant magnitudes of the decay am-plitudes depend on the first Gegenbauer moment ah1;k

1 of

034009

�h1

k. Unlike the case of �K� modes, the two terms in the

square brackets of Af;�3 �h1K�� given in Eq. (2.36) are

mutually destructive such that the transverse (longitudinaland perpendicular) BRs are less than 1� 10�6 (seeTable V). Nevertheless, the second term in the squarebracket of Af;03 �h1K

�� given in Eq. (2.35) is much largerthan the first one such that annihilations contribute con-structively to the longitudinal amplitude which is thusremarkably enhanced. With annihilation BRL�h1K

�0�could be �9:0� 16:1� � 10�6.

Alternatively, if the large transverse component of theB! �K� branching ratio is due to the new physics, thenwe expect that sizable transverse fractions can be observedin h1�1380�K� modes. Without annihilations, we haveemployed the experimental information on polarization�K� decays [5] to determine the NP parameter, a23 ora25, which characterizes our NP scenario. Thus, in the NPscenario 1, we have ~a25 j~a25je

i25ei�25 with

j~a25j �2:0� 0:3� � 10�4; 25 1:00� 0:30;

�25 �0:02� 0:06: (4.3)

On the other hand, in the NP scenario 2, we obtain ~a23 �j~a23jei23ei�23 with

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TABLE VI. New physics predictions for B! h1�1380�K� modes, where BRs are given in units of 10�6, and phases in radians. Theinput parameters are used with constraints by the B! �K� data. ~a25 and ~a23 are given in Eqs. (4.3) and (4.4), respectively.

New physics Process BRtot BRk BR? arg�AkA0� arg�

AkA0� arg�A?A0

� arg�A?A0�

Scenario 1: B� ! h1�1380�K�� 15:3� 4:0 3:4� 1:5 2:1� 1:0 �2:32� 0:25 0:48� 0:15 �2:21� 0:25 0:49� 0:15~a25 B0 ! h1�1380�K�0 14:5� 4:0 3:2� 1:5 2:0� 1:0 �2:23� 0:25 0:49� 0:15 �2:22� 0:25 0:50� 0:15Scenario 2: B� ! h1�1380�K�� 9:1� 2:0 2:1� 0:5 2:0� 0:5 �0:87� 0:20�0:97� 0:20 �0:87� 0:20�0:97� 0:20~a23 B0 ! h1�1380�K�0 8:5� 2:0 2:0� 0:5 1:8� 0:5 �0:86� 0:20�0:97� 0:20 �0:87� 0:20�0:97� 0:20

TABLE VII. CP-averaged branching ratios (in units of 10�6) for B! b�1 �1235�K��; ��K�� without/with annihilation contribu-tions denoted as BRwo=BRw.

Process BRwotot BRwo

kBRwo? BRw

tot BRwk

BRw? arg�Ak=A0� arg�A?=A0�

B0 ! b�1 �1235�K�� 1:7� 1:3 & 0:01 & 0:01 7:0� 3:5 & 0:3 & 0:3 �0:66� 0:15 �0:66� 0:15B0 ! ��K�� 6:3� 2:0 0:2� 0:1 0:2� 0:1 6:2� 2:0 1:2� 0:7 1:2� 0:7 2:18� 0:18 2:17� 0:17

9ah6 and ah8 in Ref. [13] should be corrected as in Eq. (2.23) ofthe present paper.

10ABb11 is expected to be much smaller than AB�1 since the local

axial-vector current does not couple to b1 in SU(2) limit, and thelocal tensor current mainly couples to the transverse states of b1.


j~a23j �1:5� 0:3� � 10�4; 23 �0:47� 0:20;

�23 �0:07� 0:06: (4.4)

Consequently, using the above ~a23 and ~a25 in theh1�1380�K� modes, respectively, we show the resultsin Table VI. Because the transverse branching ratiosare enhanced by NP operators, we therefore obtainsizable transverse components: BRT�h1�1380�K�� 0:6� 1:1�BRT�B! �K��. It should be stressed that, un-like the case of �K� modes, the two possible NP solutionscan be distinguished in the h1�1380�K� modes since thereis no phase ambiguity existing between the two NPscenarios.

C. B0 ! b�1 �1235�K��vs:B0 ! ��K��

The SM decay amplitudes for B0 !b�1 �1235�K��; ��K�� read

AhB0!b�1 K

�� GF��

2p

�VubV

�usa

h1X�Bb1;K

��h �VtbV

�ts

��ah4�r

b1� ah6

�rb1� ah8�a

h10�X

�BK�;b1�h �fBfK�f�

�

�bh3�

1

2bh3;EW

��;

AhB0!��K��

GF��

2p

�VubV

�usa

h1X�B�;K��h �VtbV

�ts

��ah4�r

b1� ah6

�rb1� ah8�a

h10�X

�BK�;��h �fBfK�f�

��bh3�1

2bh3;EW�

��; (4.5)

respectively, where ahi ; bhi for B0 ! b�1 �1235�K�� are

034009

given in Eqs. (2.23), (2.32), (2.35), and (2.36), while thoseforB0 ! ��K�� can be found in Ref. [13]9 and Eqs. (2.32)and (2.37). b�1 �1235�K�� and ��K�� modes are penguin-dominant processes. The former, only receiving the tinyeffect from the CKM suppressed tree amplitude for whichthe longitudinal fraction is further suppressed by the B!b1 transition form factor,10 is highly factorization-suppressed. In the numerical study, we thus neglect thetree part of the B0 ! b�1 �1235�K�� amplitude.Considering the possible annihilation effects and usingthe input parameters also constrained by the B! �K�

data, we have collected the results for B0 !b�1 �1235�K�� as well as for B0 ! ��K�� in Table VII.The annihilation effects are negligible in BRT�b�1 K

��, butcould give a significant enhancement of BRL�b�1 K

��.Nevertheless, for B0 ! ��K�� the annihilation effectscontribute constructively to the transverse fractions witha large ratio of BRT��

�K��=BRtot��K�� 0:39�0:07

�0:16,but destructively to the longitudinal component. It shouldbe noted that the contributions of NP tensor-type andscalar-type four-quark operators to the transverse fractionsof ��K�� and b�1 K

�� modes are different from the casesof �K� and h1K

� modes. Since so far there is no data onconstraining these NP parameters, we do not further dis-cuss this possibility.

D. B� ! b�1 �0 and B0 ! b�1 �

�

The SM decay amplitudes with annihilation correctionsfor B� ! b�1 �

0 and B0 ! b�1 �� read

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AhB�!b�1 �

0 ’GF��

2p

�VubV�ud�a

h1X�B��0;b�1 �h � ah2X

�B�b�1 ;�0�

h � � VtbV�td

�3

2��ah7 � a

h9 � r

b1� ah8 � a

h10�X

�B��0;b�1 �h

��;

AhB0!b�1 �

� ’GF��

2p

�VubV

�uda

h1X�B��;b�1 �h � VtbV

�td

��ah4 � r

b1� ah6 � r

b1� ah8 � a

h10�X

�B��;b�1 �h � fBfb1

f�

�bh3 �

1

2bh3;EW

��: (4.6)

Since X�B�b�1 ;�

0�

h is negligible as explained in footnote 10,B0 ! b�1 �

0 can be roughly related to the tree dominatedB�B0 ! ��0� decay as

B�B0 ! b�1 �0�

B�B0 ! ��0��

��fb1a0

1�b�1 �

0�

f�a01��

��0�

��2� 0:12: (4.7)

Because the interference between the tree and penguin(including annihilation) amplitudes is constructive inB0 !b�1 �

�, we thus have a larger branching ratio for this mode.Including annihilation contributions in the b�1 �

� mode, weobtain

B�B� ! b�1 �0� �0:3� 0:2� � 10�6;

B�B0 ! b�1 �� 0:5� 0:3� � 10�6; (4.8)

which are predominated by the longitudinal fraction.

V. CONCLUSIONS

We have studied the factorization-suppressed B decaysinvolving a 11P1 meson in the final state. For the penguin-dominated B! VV decays, the annihilation could giverise to logarithmic divergent contributions�O��1=m2

b�ln2�mb=�h�� to the helicity amplitudes A0

and Ak;?. Moreover, the annihilation corrections interferedestructively and constructively in the former and latteramplitudes, respectively. The branching ratios for B0 !b�1 �

� and B0 ! b�1 �� are & 10�6. We show that if the

large transverse fractions of�K� mainly originate from theannihilation topologies, then the large enhancement shouldbe observed only in the longitudinal component of h1K�

and b1K� modes such that the resulting fL�h1�1380�K��and fL�b�1 �1235�K�� could be even larger than fL��K��and fL��K��, respectively. Consequently, it is interest-ing to note that BR�h1�1380�K� and BR�b�1 �1235�K��could be much larger than BR�h1�1380�K�� andBR�b�1 �1235�K��, respectively [31], since the annihila-tion effects are further enhanced by the ‘‘chirally-enhanced’’ factor. Roughly speaking, we obtain

034009

BR�h1�1380�K�BR�h1�1380�K��

�BR�b�1 �1235�K��BR�b�1 �1235�K��

� 2:

On the other hand, if the large transverse fractions of�K� arise from the new physics, the same order of magni-tudes of BRT�h1K

�� should be measured. Nonsmall strongphases of ~a23;25, as given in Eqs. (4.3) and (4.4), may hint atthe (SM or NP) inelastic annihilation topologies in decays.Although we do not simultaneously take the annihilationand NP into account, the two effects should be distinguish-able in h1K�0;� and b�1 K

�� modes.According to the annihilation scenario, it is found that

fT��K�� ’ 0:23� 0:46. Analogously, one can expect

2BRT��0K�� BRT��

�K�0� � BRT��K��. Any ob-

vious deviation of the above relation from the experimentsmay imply the new physics. It should be stressed that the�0;�K�� and ��K�0 modes are relevant for exploring NPfour-quark operators in b! s �uu and b! s �dd channels,respectively.

In analogy to the helicity discussion for B! �K� givenin Ref. [13], the helicity structures of two-body baryonic Bdecays were systematically studied in Ref. [32] based onthe perturbative argument. In the SM (even with consider-ing the possible annihilation effects), the dominant helicityamplitude is H�1=2�1=2 for the B decay. In particular, it isinteresting to note that H�1=2�1=2 could be remarkablyenhanced in the NP scenario 1, although H�1=2�1=2 isdominant in the NP scenario 2.

In summary, the measurements of B! h1�1380�K�0;�

and b�1 �1235�K�� can offer a crucial test of our NP sce-narios and annihilation contributions.

ACKNOWLEDGMENTS

This work was supported in part by the National ScienceCouncil of R. O. C. under Grant No. NSC93-2112-M-033-004. I am grateful to Alex Kagan and Andrei Gritsan foruseful discussions.

[1] B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett.91, 171802 (2003).

[2] B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett.93, 231804 (2004); A. Gritsan, hep-ex/0409059.

[3] K. F. Chen et al. (Belle Collaboration), Phys. Rev. Lett. 91,201801 (2003).

[4] K. Abe et al. (Belle Collaboration), hep-ex/0408141;Phys. Rev. Lett. 94, 221804 (2005).

[5] Heavy Flavor Averaging Group, http://www.slac.stanford.edu/xorg/hfag/.

[6] A. L. Kagan, Phys. Lett. B 601, 151 (2004); hep-ph/0407076.

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[7] M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda,Nucl. Phys. B606, 245 (2001).

[8] H. n. Li and S. Mishima, Phys. Rev. D 71, 054025 (2005).[9] J. Charles, A. Le Yaouanc, L. Oliver, O. Pene, and J. C.

Raynal, Phys. Rev. D 60, 014001 (1999).[10] H. n. Li, hep-ph/0411305.[11] P. Colangelo, F. De Fazio, and T. N. Pham, Phys. Lett. B

597, 291 (2004).[12] H. Y. Cheng, C. K. Chua, and A. Soni, Phys. Rev. D 71,

014030 (2005).[13] P. K. Das and K. C. Yang, Phys. Rev. D 71, 094002 (2005).[14] W. S. Hou and M. Nagashima, hep-ph/0408007.[15] F. Borzumati, C. Greub, T. Hurth, and D. Wyler, Phys.

Rev. D 62, 075005 (2000).[16] C. S. Kim and Y. D. Yang, hep-ph/0412364.[17] C. H. Chen and C. Q. Geng, Phys. Rev. D 71, 115004

(2005).[18] M. Suzuki, Phys. Rev. D 64, 117503 (2001).

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[19] S. Eidelman et al. (Particle Data Group), Phys. Lett. B592, 1 (2004).

[20] J. Govaerts, L. J. Reinders, F. De Viron, and J. Weyers,Nucl. Phys. B283, 706 (1987).

[21] M. Beneke and T. Feldmann, Nucl. Phys. B592, 3 (2001).[22] K. C. Yang (to be published).[23] P. Ball, V. M. Braun, Y. Koike, and K. Tanaka, Nucl. Phys.

B529, 323 (1998).[24] H. Y. Cheng and K. C. Yang, Phys. Lett. B 511, 40 (2001).[25] M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda,

Nucl. Phys. B591, 313 (2000) .[26] P. Ball and V. M. Braun, Phys. Rev. D 58, 094016 (1998).[27] P. Ball and V. M. Braun, Phys. Rev. D 54, 2182 (1996).[28] M. Beneke and M. Neubert, Nucl. Phys. B675, 333 (2003).[29] P. Ball and R. Zwicky, Phys. Rev. D 71, 014029 (2005).[30] P. Ball and M. Boglione, Phys. Rev. D 68, 094006 (2003).[31] A. Gritsan (private communication).[32] M. Suzuki, J. Phys. G 31, 755 (2005).

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