: A signal for new physics?

PHYSICAL REVIEW D 71, 094002 (2005)

Data for polarization in charmless B! �K�: A signal for new physics?

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

(Received 22 December 2004; revised manuscript received 28 February 2005; published 5 May 2005)

1550-7998=20

The recent observations of sizable transverse fractions of B! �K� may hint for the existence of newphysics. We analyze all possible new-physics four-quark operators and find that two classes of new-physics operators could offer resolutions to the B! �K� polarization anomaly. The operators in the firstclass have structures �1� �5� � �1� �5�, ��1� �5� � ��1� �5�, and in the second class �1� �5� ��1� �5�, ��1� �5� � ��1� �5�. For each class, the new-physics effects can be lumped into a singleparameter. Two possible experimental results of polarization phases, arg�A?� � arg�Ak� � or 0,originating from the phase ambiguity in data, could be separately accounted for by our two new-physicsscenarios: the first (second) scenario with the first (second) class new-physics operators. The consistencybetween the data and our new-physics analysis suggests a small new-physics weak phase, together with alarge(r) strong phase. We obtain sizable transverse fractions �k k ��?? �00, in accordance with theobservations. We find �k k ’ 0:8�?? in the first scenario but �k k * �?? in the second scenario. Wediscuss the impact of the new-physics weak phase on observations.

DOI: 10.1103/PhysRevD.71.094002 PACS numbers: 13.25.Hw

I. INTRODUCTION

The studies for two-body charmless B decays haveraised a lot of interest among the particle physics commun-ity. Recently the BABAR and BELLE Collaborations pre-sented important results for the B meson decaying to a pairof light vector mesons (with V � �, , or K�) [1–6]. Thisimmediately surges a considerable amount of theoreticalattention to study nonperturbative features or to look forthe possibility of having new physics (NP) in order toexplain several discrepancies between the data and thestandard model (SM) based calculations [7–13].

From the existing SM calculations for the charmlessB! VV modes, it is known that the amplitude, H00

[ O�1�], with two vector mesons in the longitudinalpolarization state is much greater than those in transversepolarization states, since the latter are found to be H��

O�1=mb�, H�� O�1=m2b� [or Ak ’ A? O�1=mb� inthe transversity basis] [14]. For the B meson decays, therelation for different helicity amplitudes is modified asH00:H��:H�� O�1�:O�1=mb�:O�1=m2b�. Nevertheless,recently BABAR [1,3] first observed sizable transversefractions in the B! �K� decays, where the transversepolarization amplitudes are comparable to the longitudinalone. This result was confirmed later by BELLE [5,6]. Inother words, in terms of helicity amplitudes the data showthat jH�� H��j

2 jH00j2 (or 2jAkj2 2jA?j2

jA0j2 in the transversity basis). Such an anomaly in trans-verse fractions is rather unexpected within the SM frame-work. Efforts have already been made for finding apossible explanation in the SM or NP scenario. In theSM, according to Kagan [7], the non-factorizable contributions due to the annihilation couldgive rise to the following logarithmic divergent contribu-tions to the helicity amplitudes: H00; H�� O��1=m2b� �

05=71(9)=094002(16)$23.00 094002

ln2�mb=�h��, H�� O��1=m4b�ln2�mb=�h��, where �h is

the typical hadronic scale. This in turn may enhance thetransverse amplitudes required to explain the anomaly.However, in the perturbative QCD (PQCD) framework,Li and Mishima [8] have shown that the annihilations arestill not sufficient to enhance transverse fractions. Anotherpossibility for explaining the polarization anomaly advo-cated by Colangelo et al. [9] is the existence of largecharming penguin and final state interaction (FSI) effects.However they got jA0j

2�B! K��< jA0j2�B! �K��,

in contrast to the observations [1–6], where the norm-alization

PijAij

2 � 1 is adopted. With the similar FSIscenario, Cheng et al. [10] obtained jA0j2:jAkj2:jA?j2 �0:43:0:54:0:03, which is also in contrast to the recent data[3,6]. Now the question is as follows: Is it possible toexplain this anomaly by the NP? If yes, what types of NPoperators should one consider? Some NP related modelshave been proposed [12], where the so-called right-handedcurrents �s��1� �5�b �s��1� �5�s were emphasized [7].If the right-handed currents contribute constructively to A?but destructively to A0;k, then one may have largerjA?=A0j

2 to account for the data. However the resultingjAkj

2 � jA?j2 [7] will be in contrast to the recent obser-

vations [3,6]. See also the detailed discussions in Sec. II.In the present study, we consider general cases of four-

quark operators. Taking into account all possible color andLorentz structures, totally there are 20 NP four-quarkoperators which do not appear in the SM effectiveHamiltonian [see Eqs. (30) and (31)]. After analyzing thehelicity properties of quarks arising from various four-quark operators, we find that only two classes of four-quarkoperators are relevant in resolving the transverse anomaly.The first class is made of operators with structures ��1��5� � ��1� �5� and �1� �5� � �1� �5�, which contrib-

-1 2005 The American Physical Society

PRASANTA KUMAR DAS AND KWEI-CHOU YANG PHYSICAL REVIEW D 71, 094002 (2005)

ute to different helicity amplitudes as H00:H��:H�� O�1=mb�:O�1=m

2b�:O�1�. The second class consists of op-

erators with structures ��1� �5� � ��1� �5� and �1��5� � �1� �5�, from which the resulting amplitudes readas H00:H��:H�� O�1=mb�:O�1=m2b�:O�1�. Moreover,the above (pseudo)scalar operators can be written in termsof their companions, the (axial-)tensor operators, by Fierztransformation. Finally, there is only one effective coeffi-cient relevant for each class. We find that these two classescan separately satisfy the two possible solutions for polar-ization phase data, which is due to the phase ambiguity inthe measurement, and the anomaly for large transversefractions can thus be resolved. The tensor operator effectswere first noticed by Kagan [7] (see Sec. II for furtherdiscussions).

The organization of the paper is as follows. In Sec. II, wefirst introduce the SM results for the polarization ampli-tudes in the B0 ! �K�0 decay within the QCD factoriza-tion (QCDF) framework. After that we give a detaileddiscussion about how the NP can play a crucial role inresolving the large transverse polarization anomaly asobserved by BELLE and BABAR. The reason for choosingthe two classes of operators with structures (i) ��1� �5� ��1� �5�, �1� �5� � �1� �5� and (ii) ��1� �5� ��1� �5�, �1� �5� � �1� �5� is explained and the rele-vant calculations arising from these operators are per-formed. We discuss the possibility for the existence ofright-handed currents �s��1� �5�b�s��1� �5�s whichwas emphasized in [7]. From the point of view of helicityconservation in the strong interactions, we discuss variouscontributions originating from the chromomagnetic dipoleoperator, charming penguin mechanism, and annihilations.Some observables relevant in our numerical analysis aredefined in this section. In Sec. III, we summarize inputparameters e.g. Kobayashi-Maskawa (KM) elements, formfactors, meson decay constants, required for our study.Section IV is fully devoted to the numerical analysis. Wediscuss in detail two scenarios, which are separately con-sistent with the two possible polarization phase solutions indata due to the phase ambiguity. We obtain the best fitvalues for the NP parameters which can resolve the polar-ization anomaly. Numerical results for observables arecollected in this section. Finally, in Sec. V, we summarizeour results and make our conclusion.

II. FRAMEWORK

A. The standard model results in theQCD factorization approach

The best starting point for describing nonleptoniccharmless B decays is to write down first the effectiveHamiltonian describing the processes. The processes ofour concern are the B! �K� decays which are penguindominated. In the SM, the relevant effective weakHamiltonian H eff for the above �B � 1 transitions is

094002

H eff�GF��2

p

"VubV

�us�c1O

u1�c2O

u2��VcbV

�cs�c1O

c1�c2O

c2�

�VtbV�ts

X10i�3

ciOi

!�cgOg

#�H:c: (1)

Here ci’s are the Wilson coefficients and the four-quarkscurrent-current, penguin, and chromomagnetic dipole op-erators are defined by

(i) c

-2

urrent-current operators:

Ou1 � �ub�V�A�su�V�A;

Ou2 � �u�b��V�A�s�u��V�A;

Oc1 � �cb�V�A�sc�V�A;

Oc2 � �c�b��V�A�s�c��V�A;

(2)

(ii) Q
CD-penguin operators:
O3 � �sb�V�AXq

�qq�V�A;

O4 � �s�b��V�AXq

�q�q��V�A;

O5 � �sb�V�AXq

�qq�V�A;

O6 � �s�b��V�AXq

�q�q��V�A;

(3)

(iii) e
lectroweak-penguin operators:
O7 �3

2�sb�V�A

Xq

eq�qq�V�A;

O8 �3

2�s�b��V�A

Xq

eq�q�q��V�A;

O9 �3

2�sb�V�A

Xq

eq�qq�V�A;

O10 �3

2�s�b��V�A

Xq

eq�q�q��V�A;

(4)

(iv) c
hromomagnetic dipole operator:
O8g �gs8�2

mbs�� 1� �5�T

abGa� ; (5)

where �, � are the SU(3) color indices, V � A correspondto ��1� �5�, the Wilson coefficients ci’s are evaluated atthe scale �, e, and g are, respectively, QED and QCDcoupling constants and Ta’s are SU(3) color matrices. Forthe penguin operators, O3; . . . ; O10, the sum over q runsover different quark flavors, active at � ’ mb, i.e.q#fu; d; s; c; bg.

DATA FOR POLARIZATION IN CHARMLESS B! �K�: . . . PHYSICAL REVIEW D 71, 094002 (2005)

In the present work, we will embark on the study of B!�K� decays in the approach of the QCDF. The B0 ! �K�0

decay amplitude with the � meson being factorized [14]reads

A�B0 ! �K�0�SM �GF��2

p ��VtbV�ts�

�a3 � a4 � a5

�1

2�a7 � a9 � a10�

X�B

0K�0;��; (6)

which is penguin dominated. The annihilation contributionwhich is power suppressed is neglected here [14]. TheB0 ! �K�0 decay amplitude can be obtained by consid-ering CP transformation. As far as the charged B mesondecay is concerned, the dominant contribution also comesfrom the penguin operators, while the contribution due toO1, O2 is color and KM suppressed. In the scenario, where

1We choose the coordinate systems in the Jackson convention,consistent with what BABAR and Belle did [15]. In the B restframe, if the z axis of the coordinate system is along the directionof the flight of the � meson and the transverse polarizationvectors of � are chosen to be #��1� � �0;�1;�i; 0�=

��2

p, then

the transverse polarization vectors of K� are given by #�K� ��1� ��0;�1;�i; 0�=

��2

pin the Jackson convention, but become

#�K� ��1� � �0;�1;�i; 0�=��2

pin the Jacob-Wick convention.

Therefore in the NF, arg�Ak;?=A0� equals to � in the Jacksonconvention, but is zero in the Jacob-Wick convention. Here theamplitudes satisfy A � A0 � Ak � A?, A � A0 � Ak � A?,where the kinematic factors are not shown.

094002

� is factorized, the decay amplitudes for B0, B0, B�, B�

are almost the same. The factor X�B0K�0;�� in Eq. (6) is equal

to

X�B0K�0;�� h��q; #1�j�ss�V�Aj0i

� hK�0�p0; #2�j�sb�V�AjB0�p�i

� if�m�

��2i

mB �mK�

#� ��#��1 #� 2 p

�p0�V�q2�

� if�m�

��mB �mK� �#�1 � #

�2A1�q

2�

� �#�1 � p��#�2 � p�

2A2�q2�

mB �mK�

; (7)

where the decay constants and form factors are defined by

h��q; #1�jV�j0i � f�m�#

��1 ;

hK�0�p0; #2�jV�jB0�p�i �2

mB �mK�

#� ��#�2 p�p0�V�q

2�;

hK�0�p0; #2�jA�jB0�p�i � i

��mB �mK� �#��2 A1�q2� � �#�2 � p��p� p0��

A2�q2�

mB �mK�

� 2imK�

#�2 � p

q2q��A3�q

2� � A0�q2��;

(8)

with mB and mK� being the masses of B0 and K�0 mesons,respectively, q � p� p0, A3�0� � A0�0�, and

A3�q2� �

mB �mK�

2mK�

A1�q2� �

mB �mK�

2mK�

A2�q2�: (9)

It is straightforward to write down the decay width,

��B0 ! �K�0� �pc

8�m2B�jH00j

2 � jH��j2 � jH��j

2�;

(10)

where pc is the center mass momentum of the � or K�0

meson in the B rest frame. H00, H��, and H�� are thedecay amplitudes in the helicity basis and in QCDF,1 they

are given by

H00 �GF��2

p �VtbV�ts�a0SM�if�m��mB �mK� �

� �aA1�m2�� bA2�m2��;

H�� GF��2

p �VtbV�ts�a�SM�if�m��

��mB �mK� �A1�m2��

�2mBpc

mB �mK�

V�m2��;

(11)

with the constants a � �m2B �m2� �m2K� �=�2m�mK� �,b � �2m2Bp

2c�=�m�mK� �mB �mK� �2�. Here ahSM �

ah3 � ah4 � ah5 �12 �a

h7 � ah9 � ah10�. The superscript h in

ahi ’s denotes the polarization of � and K�0 mesons; h �0 is for the helicity 00 state and h � � for helicity ��states. Note that the weak phase effect is tiny in ahSM and isthus neglected in the study. Such helicity dependent effec-tive coefficients ahSM do arise in the QCDF, however in thenaive factorization (NF), they turn out to be the same, i.e.a0SM � a�SM � a�SM � aSM. In the NF, one can rewrite theabove amplitudes in the transversity basis as

-3


ASM0 �GF��2

p �VtbV�ts�aSM�if�m��mB �mK� ��aA1�m2�� bA2�m2��;

ASMk � �GF��2

p �VtbV�ts�aSM�i

��2

pf�m��mB �mK� �A1�m

2��;

ASM? � �GF��2

p �VtbV�ts�aSM�i

��2

pf�m��

2pcmB

�mB �mK� �V�m2��:

(12)

In the QCDF, ahi ’s are given by

ah1 � c1 �c2Nc

��s

4�CF

Ncc2�Fh � fhII�;

ah2 � c2 �c1Nc

��s

4�CF

Ncc1�F

h � fhII�;

ah3 � c3 �c4Nc

��s

4�CF

Ncc4�F

h � fhII�;

ah4 � c4 �c3Nc

��s

4�CF

Nc

(c3��F

h � fhII� �Gh�sq� �Gh�sb�� c1

�-u-tGh�su� �

-c-tGh�sc�

�� c4 � c6�

Xbi�u

�Gh�si� �

2

3

�

�3

2�c8 � c10�

Xbi�u

ei

�Gh�si� �

2

3

��3

2c9

�eqG

h�sq� �1

3Gh�sb�

� cgG

hg

);

ah5 � c5 �c6Nc

��s

4�CF

Ncc6� ~F

h � fhII � 12�;

ah6 � c6 �c5Nc

;

ah7 � c7 �c8Nc

��s

4�CF

Ncc8� ~F

h � fhII � 12� ��9�

NcChe;

ah8 � c8 �c7Nc

;

ah9 � c9 �c10Nc

��s

4�CF

Ncc10�F

h � fhII� ��9�

NcChe;

ah10 � c10 �c9Nc

��s

4�CF

Ncc9�Fh � fhII� �

�9�

Che;

(13)

where CF � �N2c � 1�=�2Nc�, si � m2i =m2b, -q � VqbV�

qq0 , and q0 � d; s. Note that we have given the expressions for ah1 ,ah2 , which may be relevant for charged B decays, arising due to O1 and O2 in Eq. (1). There are QCD and electroweak-penguin-type diagrams induced by the four-quark operators Oi for i � 1, 3, 4, 6, 8, 9, 10. These corrections are describedby the penguin-loop function Gh�s� given by

G0�s� �2

3�4

3ln�mb

� 4Z 1

0du�V

k�u�

Z 1

0dxx�1� x� ln�s� �ux�1� x��;

G��s� �2

3�2

3ln�mb

� 2Z 1

0du�g�v�? �u� �

1

4

dg�a�? �u�

du

�Z 1

0dxx�1� x� ln�s� �ux�1� x��:

(14)

In Eq. (13) we have also included the leading electroweak-penguin-type diagrams induced by the operators O1 and O2,

Che �

�-u-tGh�su� �

-c-tGh�sc�

��c2 �

c1Nc

�: (15)

The dipole operator O8g will give a tree-level contribution proportional to

094002-4


G0g � �2Z 1

0du�Vk�u�

1� u;

G�g �

Z 1

0

du�u

"Z u

0��V

k�v� � g�v�? �v��dv� �ug�v�? �u� �

�u4

dg�a�? �u�

du�g�a�? �u�

4

#:

(16)

In Eq. (13), the vertex correction is given by

Fh � �12 ln�mb

� 18� fhI ; (17)

where we have used the naive dimensional regularization scheme [16],

�� -�1� �5� � �� -�1� �5� � 4�4� "��1� �5� � ��1� �5�; (18)

�� -�1� �5� � �-�1� �5�� 4�1� 2"��1� �5� � ��1� �5�; (19)

�� -�1� �5� � �� -�1� �5� � 4�1� "��1� �5� � ��1� �5�; (20)

�� -�1� �5� � �-�1� �5�� 4�4� 4"��1� �5� � ��1� �5�; (21)

with D � 4� 2", and have adopted the MS subtraction. An explicit calculation for fhI , arising from vertex corrections,yields

f0I �Z 1

0dx��

k�x��31� 2x1� x

lnx� 3i��; f�I �

Z 1

0dx�g��v�? �x� �

1

4

dg��a�? �x�

dx

��31� 2x1� x

lnx� 3i��: (22)

The hard kernels fhII for hard spectator interactions, arising from the hard spectator interactions with a hard gluon exchangebetween the emitted vector meson and the spectator quark of the B meson, have the expressions:

f0II �4�2

Nc

2fBfK�mK�

h0

Z 1

0d�B1 ��

Z 1

0d2�K�

k�2�

�2

Z 1

0d3��k�3�

3;

f�II � �4�2

Nc

fBfTK�

mBh��1� 1�

Z 1

0d�B1 ��

Z 1

0d2�K�

? �2�

�22Z 1

0d3

"2�g��v�? �3� �

1

4

dg��a�? �3�

d3

��

�1

3�1�3

�Z 3

0dv��

k�v�

� g��v�? �v��

#�4�2

Nc

2fBfK�mK�

m2Bh�

Z 1

0d�B1 ��

Z 1

0d2

�gK

��v�? �2� �

1

4

dgK��a�

? �2�

d2

�

�Z 1

0d3

(�2� �3

�22 �3

�g��v�? �3� �

1

4

dg��a�? �3�

d3

��

1

�223

Z 3

0dv��

k�v� � g��v�? �v��

); (23)

where

h0 � �m2B �m2K� �m2��mB �mK� �ABK�

1 �m2��

�4m2Bp

2c

mB �mK�

ABK�

2 �m2��;

h� � �mB �mK� �ABK�

1 �m2�� 2mBpc

mB �mK�

VBK��m2��:

(24)

Note that ~Fh can be obtained from Fh in Eq. (17) with thereplacement of g0��a�? ! �g0��a�? . We will introduce a cut-off of order�QCD=mb to regulate the infrared divergence infII. Note also that we have corrected2 the QCDF results in

2We especially thank Kagan for pointing out that some termsin Ch

g may be missed in Ref. [14]. We were therefore motivatedto recalculate the QCDF decay amplitudes.

094002

Ref. [14] which were done by Cheng and one of us(K. C. Y.). The key point for the calculation is that oneneeds to consider correctly the projection operator in themomentum space, as discussed in the Appendix, whichmay explain the difference with Ref. [17].3 In the calcu-lation, we take the asymptotic light-cone distribution am-plitudes (LCDAs) for the light vector mesons, and aGaussian form for the B meson wave function [14]. Wenow make a SM estimate for various helicity amplitudesfrom a power counting point of view. For B0 ! �K�0, thehelicity amplitude H00 arising from the �V � A� � �V � A�operators is O�1�, since each of � and K�0 mesons, withthe quark and antiquark being left- and right-handed hel-

3Our G�g does not agree with that obtained by Yang et al.

-5

TABLE I. Possible NP operators and their candidacy in satisfying the anomaly resolutioncriteria. We have adopted the convention �1 � �2 � s�1bs�2s.

Model Operators H00 H�� H�� Choice

SM ��1� �5� � ��1� �5� O�1� O�1=mb� O�1=m2b�NP ��1� �5� � ��1� �5� O�1� O�1=m2b� O�1=mb� NNP ��1� �5� � ��1� �5� O�1� O�1=m2b� O�1=mb� NNP �1� �5� � �1� �5� O�1=mb� O�1� O�1=m2b� YNP �1� �5� � �1� �5� O�1=mb� O�1=m2b� O�1� YNP �1� �5� � �1� �5� O�1� O�1=m2b� O�1=mb� NNP �1� �5� � �1� �5� O�1� O�1=mb� O�1=m2b� NNP �� 1� �5� � �� 1� �5� O�1=mb� O�1� O�1=m2b� YNP �� 1� �5� � �� 1� �5� O�1=mb� O�1=m2b� O�1� YNP �� 1� �5� � �� 1� �5� O�1� O�1=m2b� O�1=mb� NNP �� 1� �5� � �� 1� �5� O�1� O�1=mb� O�1=m2b� N


icities, respectively, requires no helicity flip. For H��, thehelicity flip for the �s quark in the � meson is required,resulting in m�=mb suppression for the amplitude. Finallyin H�� two helicity flips for s quarks are required, one inthe � meson and the other in the B0 ! K�0 form factortransition, which cause a suppression by �m�=mb��

��=mb�,where �� mB �mb. In a nutshell, the three helicityamplitudes in the SM can be approximated asH00:H��:H�� O�1�:O�1=mb�:O�1=m2b�. One shouldnote that for the CP-conjugated B0 ! �K�0, the aboveresult is modified to be H00:H��:H�� O�1�:O�1=mb�:O�1=m

2b�. The results extended to various

FIG. 1 (color online). The main helicity directions of quarksand antiquarks arising from various four-quark operators duringthe B decay, where the solid circles denote the b quark, q1; q3 arethe s quarks, and �q2 is the �s quark. (q1; �q2) and (q3; �q4) form �and K�, respectively. �q4 is the spectator light quark which has nopreferable direction. The short arrows denote the helicities ofquarks and antiquarks. See the text for the detailed discussions.

094002

possible NP operators together with the SM operators willbe shown later in Table I and also illustrated in Fig. 1.

B. New physics: hints from the BABARand BELLE observations

The large transverse B! �K� fractions as have beenobserved by BELLE and BABAR [2,3,5,6] may hint adeparture from the SM expectation for the longitudinalone. Within the SM, the QCDF calculation [14] yields

1� R0 � O�1=m2b� � RT; (25)

where R0 � jA0j2=jAtotj

2 and RT � Rk � R? � �jAkj2 �

jA?j2�=jAtotj

2. The observation of large RT , as large as50%, may be possible to be accounted for in the SM, buthere we are considering the new physics alternatively.4 Inthe transversity basis the recent experiments [3,6] haveshown that

jA0j2�� jH00j

2� jATj2�� jAkj

2 � jA?j2�; (26)

where Ak � �H�� H��=��2

pand A? � ��H��

H��=��2

p. One may need NP to explain such a large RT

( R0). A set of NP operators contributing to the differenthelicity amplitudes like

H 00:H��:H�� O�1=mb�:O�1�:O�1=m2b�; (27)

or

H 00:H��:H�� O�1=mb�:O�1=m2b�:O�1�; (28)

could resolve such a polarization anomaly. Note that H��

(in the former case) and H�� (in the latter case) are of

4In the PQCD approach, annihilation contributions appear tobe too small to resolve the puzzle [8], but Li [18] recently arguedthat a decrease in one of the B! K� form factors could behelpful. Nevertheless, using the QCDF, Kagan [7] showed thatthe suppressed annihilations could account for the observationswith modest values for the BBNS [19] parameter A. See alsothe discussion after Eq. (35).

-6


O�1�, while H00 is always of O�1=mb�, in contrast to theSM expectation. The detailed reason will be seen below.

1. New-physics operators

Now Eqs. (27) and (28) can serve as guideline in select-ing NP operators. To begin with, consider the followingeffective Hamiltonian H NP:

H NP �GF��2

pX30i�11

�ci��Oi�� H:c:; (29)

which may be generated from some NP sources and con-tains the following general NP four-operators:

(i) f
our-quark operators with vector and axial-vectorstructures:
O11 � s��1� �5�bs��1� �5�s;

O12 � s��1� �5�b�s��1� �5�s�;

O13 � s��1� �5�bs��1� �5�s;

O14 � s��1� �5�b�s��1� �5�s�;

(30)

(ii) f
our-quark operators with scalar and pseudoscalarstructures:
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�;

O19 � s�1� �5�bs�1� �5�s;

O20 � s��1� �5�b�s��1� �5�s�;

O21 � s�1� �5�bs�1� �5�s;

O22 � s��1� �5�b�s��1� �5�s�;

(31)

(iii) f

5However, the contributions arising from the �s�� 1�

our-quark operators with tensor and axial-tensorstructures:

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�;

O27 � s�� 1� �5�bs�� 1� �5�s;

O28 � s�� 1� �5�b�s�� 1� �5�s�;

O29 � s�� 1� �5�bs�� 1� �5�s;

O30 � s�� 1� �5�b�s�� 1� �5�s�:

(32)

�5�b �s�� 1� �5�s operator to different polarization amplitudesshould be H00:H��:H�� O�1�:O�1=m2b�:O�1=mb�, not asmentioned in [7].

Here ci with i � 11; . . . ; 30 are the Wilson coefficients ofthe corresponding NP operators and � the renormalization

094002

scale, chosen to be mb here. Now we give an estimation ofseveral types of NP operators, contributing to variousB0 !�K� helicity amplitudes. In Fig. 1, we draw the diagramsin the B rest frame, where q1, q3 are the s quarks, and �q2 isthe �s quark. (q1; �q2) and (q3; �q4) form � and K�, respec-tively. q4 is the spectator light quark which has not anypreferable direction. q1, �q2, and q3 originated from thefollowing operators: �q3�1b �q1�2q2 for O3–6, O11–14,O23–30 or �q1�1b �q3�2q2 for O15–22. If the helicity for q1or �q2 is flipped, then the amplitude is suppressed by afactor of m�=mb. On the other hand, if the helicity of q3is further flipped, the amplitude will be suppressed by�m�=mb��

��=mb�, with �� mB �mb. The results aresummarized in Table I.

From Table I, we see that both (pseudo)scalar operatorsO15–18 and (axial-)tensor operators O23–26 satisfy theanomaly resolution criteria as given by Eqs. (27) and(28), while the rest are not. However, through the Fierztransformation, it can be shown that O15;16 and O17;18operators can be expressed as a linear combination ofO23;24 and O25;26 operators, respectively, i.e.,

O15 �1

12O23 �

1

6O24; O16 �

1

12O24 �

1

6O23;

O17 �1

12O25 �

1

6O26; O18 �

1

12O26 �

1

6O25:

(33)

Before we continue the study, five remarks are in order. (i)The �s�� 1� �5�b �s�� 1� �5�s operator, which couldmaintain jA?j

2 jAkj2, was first mentioned by Kagan

[7].5 (ii) We do not consider NP of left-handed currents,�s��1� �5�b �s�

��1� �5�s, which give corrections to SMWilson coefficients, c1–10, since they give no help forunderstanding large polarized amplitudes and are stronglyconstrained by other B! PP, VP observations [20]. (iii)O11–14 are the so-called right-handed currents, emphasizedrecently by Kagan [7]. These operators give corrections toamplitudes as

A0;k / ��VtbV�ts�aSM � �a11 � a12 � a13�;

A? / ��VtbV�ts�aSM � �a11 � a12 � a13�;

(34)

where

a11 � c11 �c12Nc

� nonfact:;

a12 � c12 �c11Nc

� nonfact:;

a13 � c13 �c14Nc

� nonfact:;

(35)

with ‘‘nonfact.’’ � nonfactorizable corrections. Note that

-7


a11;12;13 enter the A0;k amplitudes with a ‘‘minus’’ sign dueto the relative sign changed for A1; A2 form factors ascompared to the SM amplitudes in Eq. (11). If the right-handed currents contribute constructively to A? but de-structively to A0;k, then one may have larger jA?=A0j2 toaccount for the data. According to the SM resultjA?=A0j

2 ’ 0:02 in Eq. (55), we need to have ja11 � a12 �a13j=j � VtbV�

tsaSMj 1:5 such that jA?=A0j2 ’ 0:25.However the resulting jAkj

2 � jA?j2 will be in contrast

to the recent observations [3,6]. (iv) Since, in the large mblimit, the strong interaction conserves the helicity of aproduced light quark pair, helicity conservation requiresthat the outgoing s and �s arising from s� �s� n gluonsvertex have opposite helicities. The contribution of thechromomagnetic dipole operator to the transversely polar-ized amplitudes should be suppressed as H00:H��:H�� O�1�:O�1=mb�:O�1=m2b�; otherwise the results will violatethe angular momentum conservation. Actually if only con-sidering the two parton scenario for the meson, the con-tributions of the chromomagnetic dipole operator to thetransversely polarized amplitudes equal to zero [7] [seeEq. (16) and the Appendix for the detailed discussions].Similarly, the s and �s quark pairs generated from c; �cannihilation in the charming penguin always have oppositehelicities due also to the helicity conservation. Hence, thatcontributions to the transversely polarized amplitudes arerelatively suppressed, in contrast with the results inRefs. [9,10]. (v) With the same reason as the above dis-cussion, in the SM, the transversely polarized amplitudesoriginating from annihilations are subjected to helicitysuppression. A suggestion pointed out by Kagan [7] forthe polarization anomaly is the annihilation via the �S�P� � �S� P� operator, which contributes to helicity am-plitudes as H00; H�� O��1=m2b�ln

2�mb=�h��, H�� O��1=m4b�ln

2�mb=�h��. However this contribution toH�� is already of order 1=m2b although it is logarithmicdivergent.

We now calculate the decay amplitudes for B0 ! �K�0

due to O15–18 and O23–26 operators in Eqs. (31) and (32).The amplitudes for B0 ! �K�0 can be obtained by CPtransformation. The matrix elements for (axial-)tensor op-erators O23;25 can be recast into

h��q; #1�; K��p0; #2�j �s�� 1� �5�s�s�� 1� �5�bjB�p�i

�

�1�

1

2Nc

�fT�

�8#� �#

��1 #� 2 p

p0�T1�q2�

� 4iT2�q2��#�1 � #

�2��m

2B �m2K� � � 2�#�1 � p��#

�2 � p��

� 8iT3�q2��#�1 � p��#�2 � p�

m2�m2B �m2K�

�; (36)

under factorization, where the tensor decay constant fT� isdefined by [21–23]

h��q; #1�js�� sj0i � �ifT��#��1 q � # �1 q

��; (37)

094002

and

hK��p0; #2�j�s�� q �1� �5�bjB�p�i

� i#� �#� pp0�2T1�q

2� � T2�q2�f#�2;��m

2B �m2K� �

� �#� � p��p� p0��g � T3�q2��#�2 � pB�

�

�q� �

q2

m2B �m2K�

�p� p0��

�; (38)

with

T1�0� � T2�0�: (39)

The helicity amplitudes for the B0 decay due to the NPoperators are (in units of GF=

��2

p) given by

HNP00 � �4ifT�m

2B�~a23 � ~a25��h2T2�m

2�� h3T3�m

2��;

HNP�� 4ifT�m

2Bf~a23��f1T1�m

2�� f2T2�m2��

� ~a25��f1T1�m2�� f2T2�m

2��g;

(40)

and in the transversity basis, the amplitudes become (inunits of GF=

��2

p)

ANP0 � �4ifT�m2B�~a23 � ~a25��h2T2�m

2�� h3T3�m

2��;

ANPk � 4i��2

pfT�m

2B�~a23 � ~a25�f2T2�m2��;

ANP? � 4i��2

pfT�m

2B�~a23 � ~a25�f1T1�m

2��;

(41)

where

f1 �2pcmB

;

f2 �m2B �m2K�

m2B;

h2 �1

2mK�m�

��m2B �m2� �m2K� ��m2B �m2K� �

m2B� 4p2c

�;

h3 �1

2mK�m�

� 4p2cm2�

m2B �m2K�

�;

(42)

and

~a23 ��1�

1

2Nc

��c23 �

1

12c15 �

1

6c16

��

�1

Nc�1

2

�

�

�c24 �

1

12c16 �

1

6c15

�� nonfact:;

~a25 ��1�

1

2Nc

��c25 �

1

12c17 �

1

6c18

��

�1

Nc�1

2

�

�

�c26 �

1

12c18 �

1

6c17

�� nonfact:;

(43)

are NP effective coefficients defined by ~a23 �j~a23je

i723ei�23 , ~a25 � j~a25jei725ei�25 with �23, �25 being

the corresponding NP weak phases, while 723, 725 thestrong phases. Note that here we do not distinguish effec-tive coefficients for different helicity amplitudes since

-8


those differences are relatively tiny compared with thehierarchy results in Eqs. (27) and (28). A further modelcalculation for ~a23, ~a25 will be published elsewhere [24].Note that if applying the equation of motion to four-quarkoperators in deriving the matrix in Eq. (41), we can obtainthe following relations: T1 ’ VmB=�mB �mK� �, T2 ’A1mB=�mB �mK� �, T3 ’ A2, consistent with results bythe light-cone sum rule (LCSR) calculation [21–23]. TheB0 ! �K�0 polarization amplitudes can be obtained fromthe results of the B0 ! �K�0 decay by performing therelevant changes under CP transformation. The total SMand NP contributions for the B0 and B0 decays can bewritten as

A�B0 ! �K�0� � A�B0 ! �K�0�SM � A�B0 ! �K�0�NP;

A�B0 ! �K�0� � A�B0 ! �K�0�SM � A�B0 ! �K�0�NP:

(44)

With these decay amplitudes in the transversity basis, wecan evaluate physical observables: jA0j2; jAkj2; jA?j2,jA0j2; jAkj2; jA?j2,�00;�kk;�??,�?0;�?k;�k0,%00;%kk;%??,-and the triple products A0T; A

kT , A0T; A

kT [25]. The observ-

ables �hh and %hh are defined as

�hh �1

2�jAhj

2 � jAhj2�;

%hh �1

2�jAhj2 � jAhj2�;

�?i � �Im�A?A�i � A?A

�i �;

�k0 � Re�AkA�0 � AkA

�0�;

%?i � �Im�A?A�i � A?A

�i �;

%k0 � Re�AkA�0 � AkA�0�;

(45)

with h � 0; k;? and i � 0; k . Here we adopt the normal-ization conditions

PijAij

2 � 1 andP

ijAij2 � 1. The two

triple products A0T and AkT are defined as

A0T �Im�A?A

�0�

jA0j2 � jAkj

2 � jA?j2 ;

AkT �Im�A?A

�k�

jA0j2 � jAkj

2 � jA?j2 :

(46)

In our numerical analysis, we will focus on the studies ofthese quantities. The CP-conjugated A0T and AkT can beobtained by replacing A0; Ak; A? by their CP-transformedforms A0; Ak; A?. Observables like %--;%k0;�?i (with- � 0; k;? and i � 0; k ) are sensitive to the NP [25],which, in absence of the NP, strictly equal to zero. Thetriple product A0T or AkT can exhibit the relative phase

094002

between A? and A0 or between A? and Ak. The differencesbetween AiT and their CP-conjugated parts, i.e. �?i �

AiT � AiT (with i � 0; k ), are CP-violating (and also Tviolating following from the CPT invariance theorem)quantities. Therefore, any nonzero prediction of%--;%k0;�?i resembles the evidence of a new source ofCP violation. Moreover, since CP-violated effects areexpected to be negligible within the SM, sizable �?0 or�?k may also imply the existence of the NP. We will lookfor these possibilities from a detailed numerical study.

III. INPUT PARAMETERS

The decay amplitudes depend on the effective coeffi-cients ai’s, KM matrix elements, several form factors, anddecay constants.

A. KM matrix elements

We will adopt the Wolfenstein parametrization, withparameters A, -, , and 2, of the KM matrix as below

VKM �

Vud Vus VubVcd Vcs VcbVtd Vts Vtb

0@ 1A

�1� 1

2-2 - A-3�� i2�

�- 1� 12-

2 A-2

A-3�1� � i2� �A-2 1

0B@1CA:

We employ A and - � sin8c at the values of 0.815 and0.2205, respectively, in our analysis. The other parametersare found to be � �1� -2=2� � 0:20� 0:09 and 2 �2�1� -2=2� � 0:33� 0:05 [26].

B. Effective coefficients ai, form factors,and decay constants

The numerical values for the effective coefficients ahiwith i � 1; 2; . . . ; 10, which are obtained in the QCDFanalysis [14], are cataloged in Table II. The effectivecoefficients a0i are the same both for B and B, but not sofor a��;��i . In the third and fifth columns of Table II, the ai’swith the superscript being bracketed are for the B! �K�

process, otherwise for the B! �K� process.For the decay constants, we use [14] f� � fT� �

237 MeV, fK� � fTK� � 160 MeV, and fB � 190 MeV.For the B! K� transition form factors, we adopt theLCSR results in [21] with the parametrization

F�q2� � F�0� exp�c1�q2=m2B� � c2�q

2=m2B�2�; (47)

which were rescaled to account for the B! K�� data. Thevalues of the relevant form factors and parameters aregiven in Table III. The reason for choosing this set ofform factors is because the T1�0� value extracted from

-9

TABLE II. Effective coefficients for B! �K��B! �K�� obtained in the QCD factorization analysis [14], where a�;��2;3 , sensitiveto the nonfactorizable contribution f�II which is opposite in sign to f0;�II , are obviously different from a02;3; a

�;��2;3 .

a01 1:0370� 0:0135i a��1 1:0900� 0:0187i a��1 1:0180� 0:0135ia02 0:0764� 0:0793i a��2 �0:2351� 0:1096i a��2 0:1898� 0:0793ia03 0:0055� 0:0026i a��3 0:0156� 0:0035i a��3 0:0019� 0:0026ia04 �0:0347� 0:0068i a��4 �0:0366� 0:001i a��4 �0:0310� 0:0036ia05 �0:0046� 0:0030i a��5 0:0023� 0:0019i a��5 �0:0077� 0:0030ia07 �0:0001� 0:0001i a��7 0:000 01� 0:0001i a��7 0:000 09� 0:000 05ia09 �0:0092� 0:0003i a��9 �0:0096� 0:0002i a��9 �0:0088� 0:0002ia010 �0:0004� 0:0006i a��10 0:0023� 0:0010i a��10 �0:0014� 0:0007i


the B! K�� and B! Xs� data seems to prefer a smallerone [22,27,28].

TABLE III. The values for the parametrization of the B! K�

form factors in Eq. (47) [21]. The renormalization scale forT1; T2; T3 is � � mb.

IV. NUMERICAL ANALYSIS

We estimate the NP parameters which may resolve thepolarization anomaly in B�B� ! �K��K�� decays [3,6].An enhancement in transversely polarized amplitudes by50% can therefore take place in our NP scenario since theSM polarization amplitudes, ASM0 :ASMk :ASM?

O�1�:O�1=mb�:O�1=mb�, are modified to beANP0 :A

NPk :A

NP? O�1=mb�:O�1�:O�1�, as given in Eq. (41)

which allows us to find solutions in the NP parameter space(j~a23j; 723; �23; j~a25j; 725; �25) for explaining the B!�K� polarization anomaly.

Choosing the normalization conditionsP

ijAij2 �P

ijAij2 � 1, and setting arg�A0� � arg�A0� � 0, one can

measure the magnitudes and relative phases of the sixA0;k;?; A0;k;? polarization amplitudes,6 giving 8 measure-ments, and then extract 12 observables in Eq. (45) as wellas the triple products in Eq. (46). We take the average of theBABAR and BELLE data in our 92 analyses for estimatingNP parameters and consequently obtain the predictions forobservables. For simplicity, we neglect the correlationsamong the data. The 92i of any observable Oi with themeasurement Oi�expt� � �1�i�expt is defined as

92i ��Oi�expt� �Oi�theory�

�1�i�expt

2: (48)

For N different observables, the total 92 equals to 92 �PNi�1 9

2i . In the 92 best fit analysis, we consider the follow-

ing 8 observables:

jA0j2; jA?j2; jA0j2; jA?j2;arg�Ak�;arg�A?�;arg�Ak�;arg�A?�;

(49)

as our inputs. For the purpose of performing the numericalanalysis easily, we have converted the BABAR measure-

6For simplicity, in the present study we have chosen theconvention arg�A0� � arg�A0� � 0, i.e. we do not considerhere the physics arising from the difference between arg�A0�and arg�A0�.

094002

ments into the above quantities, as shown in Tables IV andV.

Since the interference terms in the angular distributionanalysis [3,6] are limited to Re�AkA

�0�, Im�A?A

�0�, and

Im�A?A�k�, there exists a phase ambiguity:

arg�Ak� ! � arg�Ak�; arg�A?� ! �� arg�A?�;

arg�A?� � arg�Ak� ! �� arg�A?� � arg�Ak��:

(50)

Therefore, the world averages for arg�Ak� and arg�A?�,given in Tables IV and V, can be

arg�Ak� � �2:33� 0:22; arg�A?� � 0:59� 0:24;

(51)

or, following from Eq. (50),

arg�Ak� � 2:33� 0:22; arg�A?� � 2:55� 0:24:

(52)

From Eq. (51), the phase difference for A? and Ak reads

arg�A?� � arg�Ak� �; (53)

but, on the other hand, from Eq. (52), becomes

arg�A?� � arg�Ak� 0: (54)

The resultant implications in Eqs. (53) and (54) are dis-cussed below. Numerically, SM and NP amplitudes in thetransversity basis for B0 ! �K�0 are given by

A1�0� A2�0� A0�0� V�0� T1�0� T2�0� T3�0�

F�0� 0.294 0.246 0.412 0.399 0.334 0.334 0.234c1 0.656 1.237 1.543 1.537 1.575 0.562 1.230c2 0.456 0.822 0.954 1.123 1.140 0.481 1.089

-10

TABLE IV. Comparison between the first NP scenario predictions and the average for BABAR and BELLE data [3,6] with the phasesgiven in Eq. (51). The 92min=d:o:f: for 8 inputs is 4:15=5.

NP parameters NP results

j~a25j �2:10�0:19�0:12� � 10�4

725 1:15� 0:09�25 �0:12� 0:09

Observables BABAR BELLE Average NP results

arg�Ak� �2:61� 0:31 �2:05� 0:31 �2:33� 0:22 �2:60� 0:14arg�Ak� �2:07� 0:31 �2:29� 0:37 �2:16� 0:24 �2:40� 0:14arg�A?� 0:31� 0:36 0:81� 0:32 0:59� 0:24 0:87� 0:12arg�A?� 1:03� 0:36 0:74� 0:33 0:87� 0:24 1:10� 0:13

jA0j2; �jA0j2� 0:49� 0:07 �0:55� 0:08� 0:59� 0:1 �0:41� 0:10� 0:52� 0:06 �0:50� 0:06� 0:52� 0:04 �0:53� 0:04�jAkj2; �jAkj2� 0:20� 0:02 �0:22� 0:02�jA?j

2; �jA?j2� 0:20� 0:07 �0:24� 0:08� 0:26� 0:09 �0:24� 0:10� 0:22� 0:06 �0:24� 0:07� 0:28� 0:03 �0:25� 0:02�

A0T; �A0T� 0:28� 0:10 �0:21� 0:09� 0:19� 0:04 �0:24� 0:03�

AkT; �AkT� 0:06� 0:06 �0:04� 0:08� 0:09� 0:01 �0:10� 0:00�


ASM0 � �0:003 07� 0:000 74i;

ASMk � �0:000 48� 0:000 064i;

ASM? � �0:000 40� 0:000 073i;(55)

and

ANP0 ’ 2:2�j~a23jei�723��23� � j~a25je

i�725��25��;

ANPk ’ �11:9�j~a23jei�723��23� � j~a25je

i�725��25��;

ANP? ’ �10:9�j~a23jei�723��23� � j~a25je

i�725��25��;(56)

respectively, in units of �iGF=��2

p. From Eq. (55), we find

that jASMk j=jASM0 j and jASM? j=jASM0 j are 0.17 and 0.14, re-

spectively. In other words, ASMk;? are O�1=mb� suppressed,

compared to ASM0 . On the other hand, the measurements forjAkj2 jA?j2

12 jA0j

2, as cataloged in Table IV, mean

TABLE V. Comparison between the second NP scenario predictiophases given in Eq. (52). The 92min=d:o:f: for 8 inputs is 0:56=5.

NP parameters

�j~a23j~723 � 723 � �

�23Observables BABAR BELLE

arg�Ak� 2:61� 0:31 2:05� 0:31arg�Ak� 2:07� 0:31 2:29� 0:37arg�A?� 2:83� 0:36 2:33� 0:32arg�A?� 2:11� 0:36 2:40� 0:33

jA0j2; �jA0j

2� 0:49� 0:07 �0:55� 0:08� 0:59� 0:1 �0:41�jAkj2; �jAkj2�jA?j2; �jA?j2� 0:20� 0:07 �0:24� 0:08� 0:26� 0:09 �0:24�

A0T; �A0T� 0:28� 0:10 �0:21�

AkT; �AkT� 0:06� 0:06 �0:04�

094002

that Ak and A? are dominated by ANPk and ANP? , respec-tively. We therefore find that the data for the amplitudephases in Eq. (51) prefer the ~a25 terms in ANPk and ANP?given in Eq. (56), since there is a phase difference of �between two ~a25 terms. Consequently, from Table I, we getH�� H�� if O17;18, O25;26 NP operators are dominant.On the other hand, for the data of the amplitude phases inEq. (52), we find that the ~a23 terms in ANPk and ANP? inEq. (56) are instead favored, since they have the same sign.Accordingly, also from Table I, as only O15;16, O23;24 NPoperators are considered we obtain H�� H��, which isconsistent with the SM expectation [29,30]. Thereforebecause of the phase ambiguity, the data prefer two differ-ent types of NP scenarios: (i) the first scenario, where theNP is characterized by O17;18;25;26 operators, while theoperators O15;16;23;24 are absent, (ii) the second scenario,

ns and the average for BABAR and BELLE data [3,6] with the

NP results

��1:70�0:11�0:07� � 10�4

�0:78� 0:100:14� 0:09

Average NP results

2:33� 0:22 2:42� 0:172:16� 0:24 2:21� 0:182:55� 0:24 2:44� 0:172:27� 0:24 2:24� 0:17

0:10� 0:52� 0:06 �0:50� 0:06� 0:51� 0:04 �0:52� 0:04�0:26� 0:02 �0:26� 0:02�

0:10� 0:22� 0:06 �0:24� 0:07� 0:23� 0:02 �0:23� 0:02�

0:09� 0:22� 0:04 �0:27� 0:04�0:08� 0:01�0:00�0:01 �0:01

�0:00�0:01�

-11


where the NP is dominated by O15;16;23;24 operators, whileO17;18;25;26 operators are absent.

A. The first scenario with O15;16;23;24 absent

In this scenario, the NP effects characterized byO17;18;25;26 operators are lumped into the single effectivecoefficient ~a25 � j~a25je

i�25ei725 , where �25 and 725 are theNP weak and strong phases associated with it. Therefore,in our 92 analysis, we have three fitted parameters, j~a25j,�25, and 725. The 92min=d:o:f: for this scenario is 4:15=5,where d:o:f: � degrees of freedom in the fit. Our best fitresults together with the data are cataloged in Tables IVandVI. For illustration, we obtain theoretical errors by scan-ning the 92 � 92min � 1 parameter space. The branchingratios (BRs) are sensitive only to the form factors, whilethe rest results depend very weakly on the theoretical inputparameters and the cutoff that regulates the hard spectatoreffects in the SM calculation. To estimate the errors forBRs, arising from the input parameters, we allow 10%variation in form factors and decay constants which maybe underestimated, and the resulting errors are displayed inTable VI. The NP parameters are given by

j~a25j � �2:10�0:19�0:12� � 10�4; 725 � 1:15� 0:09;

�25 � �0:12� 0:09;(57)

with the phases in radians. Note that the nonsmall 725 mayimply that the strong phase due to the annihilation mecha-

TABLE VI. Comparison between the NP predictions and data [3,6errors for BRs come from the uncertainties of input parameters, and tworld average for the total BR is �9:5� 0:9� � 10�6 [20].

BABAR BELLE

�00 0:50� 0:07�k k 0:25� 0:07�?? 0:25� 0:07�k0 �0:39� 0:14%?0 �0:49� 0:14%?k �0:09� 0:10�?0�� 0:22� 0:10 0:07� 0:12�?k�� 0:04� 0:08 0:02� 0:10%00�� 0:09� 0:06%k k�� 0:10� 0:06%??�� 0:01� 0:06%k0�� 0:11� 0:14

BR�B0 ! �K�0�

BR�B0 ! �K�0�0BR�B0 ! �K�0�kBR�B0 ! �K�0�?ACP�B! �K�� 0:01� 0:09A0CP � �%00=�00 �0:06� 0:10AkCP � �%k k=�k k

A?CP � �%??=�?? �0:10� 0:25ATCP � �

%k k�%? ?

�k k��? ?

094002

nism in the SM cannot be negligible. In Tables IV and VI,we obtain results in good agreement with the data. TheBABAR and BELLE [3,6] data show that

�00 ’ �k k ��??; �k k ’ �??; (58)

which can be realized as follows. The transverse ampli-tudes are given by

A k � ANPk � ASMk ; A? � ANP? � ASM? ; (59)

where ANPk �1:1ANP? in this scenario. In Ak of Eq. (59),

the interference of ANPk and ASMk is destructive, while in A?the interference of ANP? and ASM? becomes constructive. Wethus find �?? >�k k and accordingly �k k ’ 0:8�??.Interestingly, because 725 ��25 is much closer to 0 ascompared to 725 ��25, the above interference effects thusresult in jAkj2=jA?j2 < jAkj2=jA?j2. In other words, alarger j�25j yields larger magnitudes of AkCP; A

?CP. To get

the first relation of Eq. (58), we first take the squares of theAk and A? of Eq. (59), and then add them up together withtheir CP-conjugated parts. The interference terms are mu-tually canceled and one thus finds�k k ��?? �00, dueto jANPk;?j

2 � jASMk;?j2.

We obtain �k0 � �0:33� 0:04 and %?0 � �0:44�0:05, as compared with the BELLE data: �k0 � �0:39�0:14 and %?0 � �0:49� 0:14. Within the SM, �k0 ’0:30 and %?0 ’ 0:10 are in contrast to the data.

]. The NP related observables are denoted by ‘‘��.’’ The secondhe first ones are obtained with the constraint 92 � 92min � 1. The

The 1st scenario The 2nd scenario

0:53� 0:04 0:51� 0:040:21� 0:02 0:26� 0:020:26� 0:02 0:23� 0:02

�0:33� 0:04 �0:49� 0:07�0:44� 0:05 �0:49� 0:07�0:19� 0:01 �0:01� 0:00�0:05� 0:04 �0:05� 0:06�0:01�0:00�0:01 �0:00�0:00�0:010:00� 0:00 0:00� 0:000:01�0:00�0:01 �0:00� 0:00

�0:01� 0:01 �0:00� 0:000:06� 0:06 0:05� 0:07

�13:3�0:5�0:6 � 1:9� � 10�6 �12:2�0:5�0:6 � 1:8� � 10

�6

�6:5� 0:1� 1:0� � 10�6 �6:2� 0:1� 0:9� � 10�6

�3:8� 0:3� 0:4� � 10�6 �3:3� 0:3� 0:5� � 10�6

�3:0� 0:3� 0:5� � 10�6 �2:7� 0:3� 0:5� � 10�6

�0:01� 0:01 �0:01� 0:02�0:02� 0:02 �0:02� 0:02�0:05� 0:05 �0:01� 0:020:03� 0:03 �0:01� 0:01

�0:01� 0:01 �0:01� 0:02

-12


The consistency between data and this NP scenariorequires the presence of a large strong phase 725 and a(small) weak phase �25. Our numerical predictions for therest NP related observables are �?k;%00;%kk;%?? ��1–2�%, which are marginally sensitive to �25. Sinceour analysis yields AkT ’ 0:09 and AkT ’ 0:10, we thereforeobtain%?k � �AkT � AkT ’ �0:19 and�?k � AkT � AkT ’

�0:01. We observe that �?0;?k;%??; ACP�B!

�K��; A0;kCP have the same sign as �25, whereas%00;kk;k0; A?CP are of the opposite sign. For a small �25 ��0:12�� 7 �, we get NP related quantities �?0

�%k0 AkCP �0:05 which may become visible in thefuture once the experimental errors go down. It is interest-ing to note that the existence of the nonzero NP weak phase�25 may be hinted by the BABAR measurements ofarg�A? � A?� � 0 and arg�Ak � Ak� � 0. If taking alonethe BABAR data as inputs, we obtain �25 � �16� 7� ,which could cause sizable effects in observations: %k0��

Re�AkA�0 � AkA�0�� 0:15� 0:07, A0T�� 0:26� 0:03� �

A0T�� 0:11� 0:06), %?0�� A0T � A0T� � �0:38� 0:08,�?0�� A0T � A0T� � �0:15� 0:07, and AkCP ’ �2A

?CP ’

3A0CP ’ ��15� 6�%. As for the branching ratio, we obtainBR�B0 ! �K�0� ’ �1:33� 0:25� � 10�6, in good agree-ment with the world average �9:5� 0:9� � 10�6 [20],while without NP corrections the result becomes a muchsmaller value of 5:8� 10�6.

B. The second scenario with O17;18;25;26 absent

In the second scenario, the NP is characterized byO15;16;23;24 operators and the only relevant NP parameteris ~a23 � j~a23jei�23ei723 with �23 and 723 being the NPweak and strong phases, respectively. Following the sameway as in the first scenario, we show the results in Tables Vand VI, where 92min=d:o:f: is 0:56=5. The NP parameters inthis scenario are given by7

j~a23j � �1:70�0:11�0:07� � 10�4; 723 � 2:36� 0:10;

�23 � 0:14� 0:09;(60)

with phases in radians. ~a23 produces sizable contributionsto the transverse amplitudes. �k k ��?? �00 can beunderstood by following the analysis given in the firstscenario. In this scenario, because the two terms in bothamplitudes Ak and A? in Eq. (59) contribute constructively,we find �k k=�?? �ANPk =ANP? �2 � 1:1.

As for �23 � 0:14� 0:09�� 8� 5� �, we obtain A0T �0:27� 0:04, A0T � 0:22� 0:04, and accordingly %?0 ��0:49� 0:07, �?0 � �0:05� 0:06. Since the numerical

7It may be better to rewrite as ~a23 � �j~a23jei�23ei~723 , where

the redefined strong phase is ~723 � 723 � � � �0:78� 0:10��45� 6� �. The reason is that it is hard to have a large strongphase in the perturbation calculation.

094002

analysis gives the triple products AkT , AkT ’ 0:00 0:01, wetherefore obtain %?k � �AkT � AkT ’ �0:01 and �?k �

AkT � AkT ’ 0. Note that �?0;?k are CP-violating observ-ables. We get �k0 � �0:49� 0:07, while the SM result is�k0 ’ 0:30. For NP related observables, we obtain %k0 ��?0 � 0:05� 0:07 but �?k %-- 0 which arerather small. Larger magnitudes of �?0 and %k0 are im-plied for a larger j�23j. The BABAR results, displayingarg�A? � A?� � 0 and arg�Ak � Ak� � 0, may hint at theexistence of the NP weak phase; consequently, if takingalone the BABAR data, the numerical analysis yields�23 �0:23�0:15�0:12 such that A0T�� 0:29� 0:04� � A0T�� 0:14�0:10�0:07),which can be rewritten as %?0�� A0T � A0T� ��0:43�0:08�0:11 and �?0�� A0T � A0T� � �0:16�0:12�0:09, and%k0�� Re�AkA

�0 � AkA

�0�� 0:15� 0:09. Finally, we get

BR�B0 ! �K�0� ’ �1:22� 0:24� � 10�6 which is in goodagreement with the world average �9:5� 0:9� � 10�6

[20].

V. SUMMARY AND CONCLUSION

The large transverse polarization anomaly in the B!�K� decays has been observed by BABAR and BELLE. Weresort to the new physics for seeking the possible resolu-tions. We have analyzed all possible new-physics four-quark operators. Following the analysis for the helicitiesof quarks arising from various four-quark operators in theB decays, we have found that there are two classes ofoperators which could offer resolutions to the B! �K�

polarization anomaly. The first class is made of O17;18 andO25;26 operators with structures �1� �5� � �1� �5� and��1� �5� � ��1� �5�, respectively. These operatorscontribute to different helicity amplitudes asH00:H��:H�� O�1=mb�:O�1=m

2b�:O�1�. The second

class consists of O15;16 and O23;24 operators with structures�1� �5� � �1� �5� and ��1� �5� � ��1� �5�, respec-tively, and the resulting amplitudes are given asH00:H��:H�� O�1=mb�:O�1=m2b�:O�1�. Moreover,we have shown in Eq. (33) that by Fierz transformationO17;18 can be rewritten in terms of O25;26, and O15;16 interms of O23;24. For each class of new physics, we havefound that all new-physics effects can be lumped into a soleparameter: ~a25 (or ~a23) in the first (or second) class. Ourconclusions are as follows:

(1) T

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wo possible experimental results of polarizationphases, arg�A?� � arg�Ak� � or 0, originatingfrom the phase ambiguity in data, could be sepa-rately accounted for by our two new-physics scenar-ios with the presence of a large(r) strong phase, 725(or 723�, and a small weak phase, �25 (or �23). Inthe first scenario only the effective coefficient ~a25 isrelevant, which is related to O17;18;25;26 operatorssuch that H�� H��, while in the second sce-nario only the effective coefficient ~a23 is relevant,which is associated with O15;16;23;24 operators such


that H�� H��. Note that if simultaneously con-sidering the six parameters j~a25j, 725, �25, j~a23j,723, and �23 in the fit, the final results still convergeto the above two scenarios.

(2) W
e obtain �k k ’ 0:8�?? in the first scenario, but�k k * �?? in the second scenario.
(3) O
ur numerical analysis yields AkT; AkT 0:10 and
%?k �0:19 in the first scenario, but givesAkT; A

kT ’ 0:01 and %?k ’ �0:01 in the second sce-

nario. These two scenarios can thus be distin-guished. Furthermore, a larger magnitude of theweak phase, �25 or �23, can result in sizable�?0;%k0. As displayed in Table VI, we obtain�?0 ’ �%k0 ’ �0:05 for �25;�23� � �0:11�0:14�.

(4) T
he NP related observations %00;k k;??;�?k areonly marginally affected by weak phases �25;23.
(5) W
e obtain BR�B! �K�� ’ �1:3� 0:3� � 10�6 intwo scenarios. Note that we have used the rescaled
094002-14

LCSR form factors in Refs. [21,23], where smallervalues for form factors were used in explaining B!K��; Xs� data [22].

ACKNOWLEDGMENTS

We are grateful to Hai-Yang Cheng and Kai-Feng Chenfor useful discussions. We thank Andrei Gritsan and AlexKagan for many helpful comments on the manuscript. Thiswork was supported in part by the National ScienceCouncil of R.O.C. under Grants No. NSC92-2112-M-033-014, No. NSC93-2112-M-033-004, and No. NSC93-2811-M-033-004.

APPENDIX

The LCDAs of the vector meson relevant for the presentstudy are given by [31]

hV�P0; #�j �q1�y��q2�x�j0i � fVmV

Z 1

0duei�up

0�y� �up0�x��p0�

#� � zp0 � z

�k�u� � #�?�g�v�? �u�

�; (A1)

hV�P0; #�j �q1�y��5q2�x�j0i � �fV

�1�

fTVfV

mq1 �mq2

mV

�mV#� �#� p0z�

Z 1

0duei�up

0�y� �up0�x� g�a�? �u�

4; (A2)

hV�P0; #�j �q1�y�� q2�x�j0i � �ifTVZ 1

0duei�up

0�y� �up0�x��#�?�p0 � #�? p

0��?�u�; (A3)

where z � y� x with z2 � 0, and we have introduced the lightlike vector p0� � P0� �m2Vz�=�2P0 � z� with the meson’s

momentum P02 � m2V . Here the longitudinal and transverse projections of the polarization vectors are defined as

#�k� �

#� � zP0 � z

�P0� �

m2VP0 � z

z�

�; #�?� � #�� #�

k�: (A4)

Note that these are not exactly the polarization vectors of the vector meson. In the QCDF calculation, the LCDAs of themeson appear in the following way:

hV�P0; #�j �q1��y�q27�x�j0i �1

4

Z 1

0duei�up

0�y� �up0�x��fVmV

�p6 0#� � zp0 � z

�k�u� � #6 �?g�v�? �u� � #� �#��p0z��5

g�a�? �u�

4

�

� fTV#6�?p6

0�?�u��7�: (A5)

Note that to perform the calculation in the momentum space, we first represent the above equation in terms ofz-independent variables, P0 and #�. Then, the light-cone projection operator of a light vector meson in the momentumspace reads

MV7� � MV

7�k �MV7�?; (A6)

with the longitudinal projector

MVk�fV4

mV�#� � n��2

n6 ��k�u��k�up0

; (A7)

and the transverse projector

PHYSICAL REVIEW D 71, 094002 (2005)

MV? �

fTV4#6 �?p6

0�?�u� �fVmV

4

(#6 �?g

�v�? �u�

�Z u

0dv��k�v� � g�v�? �v��p6 0#�?�

@@k?�

� i#� �#� ? ��5

�n�n��

1

8

dg�a�? �u�

du

� p0g�a�? �u�

4

@@k?�

)��k�up0; (A8)

where n�� 1; 0; 0;�1�, n�� 1; 0; 0; 1�, k? is the trans-verse momentum of the q1 quark in the vector meson, andthe polarization vectors of the vector meson are

#�? � #� �# � n�2

n�� # � n�2

n��: (A9)

In the present study, we consider only the leading contri-bution in �QCD=mb for MV

k. In Eqs. (A1)–(A3), �k;�?

are twist-2 LCDAs, while g�v�? ; g�a�? are twist-3 ones.Applying the equations of motion to LCDAs, one canobtain the following Wandzura-Wilczek relations:

g�v�? �u� �1

2

"Z u

0

�k�v��v

dv�Z 1

u

�k�v�v

dv

#� � � � ;

(A10)

DATA FOR POLARIZATION IN CHARMLESS B! �K�: . . .

094002

g�a�? �u� � 2

"�uZ u

0

�k�v��v

dv� uZ 1

u

�k�v�v

dv

#� � � � ;

(A11)

where the ellipses in Eqs. (A10) and (A11) denote addi-tional contributions from three-particle distribution ampli-tudes containing gluons and terms proportional to lightquark masses, which we do not consider here.Equations (A10) and (A11) further give

1

4

dg�a�? �u�

du� g�v�? �u� �

Z 1

u

�k�v�v

dv� � � � ; (A12)

Z u

0��k�v� � g�v�? �v��dv �

1

2

"�uZ u

0

�k�v��v

dv

� uZ 1

u

�k�v�v

dv

#� � � � :

(A13)

After considering Eqs. (A10)–(A13), G�g in Eq. (16) are

actually equal to zero.

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: A signal for new physics?