Perturbative QCD analysis of B ->phi K-* decays

Perturbative QCD analysis of B \

^K * decays

Chuan-Hung Chen*

Institute of Physics, Academia Sinica, Taipei, Taiwan 115, Republic of China Yong-Yeon Keum^†

Department of Physics, Graduate School of Science, Nagoya University, Nagoya 464-8602, Japan Hsiang-nan Li^‡

Institute of Physics, Academia Sinica, Taipei, Taiwan 115, Republic of China

and Department of Physics, National Cheng-Kung University, Tainan, Taiwan 701, Republic of China 共Received 17 April 2002; published 26 September 2002兲

We study the first observed charmless B→VV modes, the B→␾K*decays, in perturbative QCD formalism.

The obtained branching ratios B(B→␾K*)⬃15⫻10^⫺6 are larger than⬃9⫻10^⫺6 from QCD factorization.

The comparison of the predicted magnitudes and phases of the different helicity amplitudes, and branching ratios with experimental data can test the power counting rules, the evaluation of annihilation contributions, and the mechanism of dynamical penguin enhancement in perturbative QCD, respectively.

DOI: 10.1103/PhysRevD.66.054013 PACS number共s兲: 12.38.Bx, 11.10.Hi, 12.38.Qk, 13.25.Hw The branching ratios of the penguin-dominated B→K␲

decays, about 3– 4 times larger than those of the tree- dominated B→␲␲ decays, indicate that penguin contribu- tions must be enhanced. This enhancement can be achieved either by large Wilson coefficients C_4,6 associated with the penguin operators in perturbative QCD共PQCD兲 关1–3兴, or by a large chiral symmetry breaking scale m₀ associated with the kaon in QCD factorization 共QCDF兲 关4,5兴. The latter mechanism, called chiral enhancement, corresponds to a characteristic scale of O(m_b), at which we have m₀(m_b)

⬃3 GeV and the smaller Wilson coefficients C4,6(m_b). The former mechanism, called dynamical enhancement, corre- sponds to a characteristic scale of O(冑⌳¯ m_b), ⌳¯⫽MB⫺mb

being the B meson and b quark mass difference, at which we have m₀(冑⌳¯ m_b)⬃1.5 GeV and the larger Wilson coeffi- cients C_4,6(冑⌳¯ mb)⬃1.5C4,6(m_b). Recently, we have pro- posed the B→␾K decays as the appropriate modes to clarify the above issue关6,7兴. These modes are not chirally enhanced because ␾ is a vector meson, and they are insensitive to the variation of the unitarity angle ␾3 because they are pure penguin processes. If the data of the branching ratios B(B

→␾K) are settled down at values around 10⫻10^⫺6 关7,8兴 instead of 4⫻10^⫺6 关9,10兴, the dynamical enhancement of penguin contributions to charmless nonleptonic B meson de- cays will gain strong support.

Here we argue why the characteristic scale involved in two-body B meson decays must be of O(冑⌳¯ MB) in PQCD from two points of view. Consider a two-body nonleptonic decay, in which the two final-state light mesons move back- to-back with large momenta. The lowest-order diagram for its amplitude contains a hard gluon attaching the spectator quark. Intuitively, the spectator quark in the B meson, form-

ing a soft cloud around the heavy b quark, carries momentum of order ⌳¯ . The spectator quark on the light-meson side carries momentum of O( M_B) in order to form the fast- moving light meson with the u quark produced in the b quark decay. Note that the end-point singularities from the small spectator momentum on the light-meson side do not exist in a self-consistent PQCD formalism, because of Sudakov sup- pression from k_Tand threshold resummations关11,12兴. Based on the above argument, the hard gluon is off-shell by order of ⌳¯ M_B. This scale characterizes the corresponding quark- level hard amplitude, which involves the four-fermion decay vertex. Theoretically, the hard scale ⌳¯ M_B is essential for constructing a gauge invariant B meson wave function. This wave function, though being a nonlocal matrix element, is gauge invariant in the presence of the path-ordered Wilson line integral. A careful investigation 关13,14兴 shows that the O(␣s

2) diagram with the second gluon attaching the hard gluon contributes to this line integral. That is, this diagram contains the soft divergence, which is factorized into the B meson wave function. This is possible only when the hard gluon is off shell by the intermediate scale⌳¯ M_B rather than by M_B².

In this work we shall perform a PQCD analysis of the first observed charmless B→VV modes, the B→␾^K* decays, which are similar to B→␾K, also appropriate for distin- guishing the different penguin enhancing mechanism. Be- sides, the B→VV modes reveal dynamics of exclusive B meson decays more than the B→PP and VP modes through the measurement of the magnitudes and the phases of various helicity amplitudes. According to the power counting rules defined in 关7兴, the longitudinal amplitude is leading, and the other two amplitudes are down by a power of M_␾/ M_B or of M_K*/ M_B, M_␾and M_K*being the␾^{and K}*meson masses, respectively. Since the B→␾^K*decays are insensitive to the unitarity angle, the relative phases among the helicity ampli- tudes mainly arise from strong interaction. The annihilation contributions, which can be evaluated unambiguously in our

*Email address: [email protected]

†Email address: [email protected]

‡Email address: [email protected]

approach, generate the strong phases. Therefore, comparing the predicted magnitudes and relative phases among the dif- ferent helicity amplitudes, and the predicted branching ratios with experimental data, we test the power counting rules, the evaluation of annihilation contributions, and the mechanism of dynamical penguin enhancement in PQCD, respectively.

The idea of the PQCD factorization theorem for two-body nonleptonic B meson decays has been reviewed in关1,15,16兴, which is subject to corrections of O(␣s

2) and O(⌳¯ /M_B). In this formalism decay amplitudes are expressed as the convo- lutions of the corresponding hard parts with universal meson distribution amplitudes 关13,14兴, which are regarded as the nonperturbative inputs. Because of the Sudakov effects from k_T and threshold resummations, the end-point singularities do not exist as stated above. Therefore, PQCD involves in- puts less than in QCDF, for which form factors, meson dis- tribution amplitudes, and infrared cutoffs for regulating the end-point singularities are all independent parameters 关4,5兴.

Strictly speaking, the infrared cutoffs, signifying important soft contributions to the nonfactorizable and annihilation am- plitudes, imply that the factorization formulas in QCDF are not self-consistent.

We work in the frame with the B meson at rest, i.e., with the B meson momentum P₁⫽(MB/冑^2)(1,1,0T) in the light- cone coordinates. Assume that the ␾^(K*) meson moves in the plus共minus兲 z direction carrying the momentum P2( P₃) and the polarization vectors ⑀2(⑀3). The B→␾^K* decay rates are written as

⌫⫽ G_F²P_c 16␲^MB

2 _␴⫽L,T

兺

where P_c⬅兩P2z兩⫽兩P3z兩 is the momentum of either of the outgoing vector mesons, and the superscript ␴ denotes the helicity states of the two vector mesons with L(T) standing for the longitudinal 共transverse兲 component. The amplitude M^(␴)is decomposed into

M^(␴)⫽⑀2*␮_共␴兲⑀3*␯_共␴兲

冋

⫹i c

M_␾M_K*⑀^{␮␯␣␤P}2␣P_3␤

册

⬅MB

2ML⫹MB

2MN⑀2*_共␴⫽T兲•⑀3*_共␴⫽T兲

⫹iMT⑀^␣␤␥␳⑀2*␣_共␴兲⑀3*␤_共␴兲P2␥P_3␳, 共2兲 with the convention¹⑀⁰¹²³⫽1 and the definitions

M_B²ML⫽a⑀2*_共L兲•⑀3*_共L兲⫹ ^b

M_␾M_K*⑀2*_共L兲•P1⑀3*_共L兲•P1,

M_B²MN⫽a⑀2*_共T兲•⑀3*_{共T兲, 共3兲}

MT⫽ c M_␾M_K*.

We define the helicity amplitudes A₀⫽⫺␰^MB

2ML, A_储⫽␰冑^{2 M}B

2MN, 共4兲

A_⬜⫽␰^M_␾^MK*冑²共r²⫺1兲MT, with the normalization factor ␰⫽冑^GF

2P_c/(16␲^MB 2⌫) and the ratio r⫽P2•P3/( M_␾M_K*). These helicity amplitudes satisfy the relation

兩A0兩²⫹兩A_储兩²⫹兩A_⬜兩²⫽1, 共5兲 following the helicity summation in Eq. 共1兲. We also intro- duce another equivalent set of helicity amplitudes,

H₀⫽MB 2ML, H_⫾⫽MB

2MN⫿M_␾M_K*冑^r2⫺1MT, 共6兲

with the helicity summation,

兺

The B→␾^K* decays involve the emission and annihila- tion topologies, both of which are classified into factorizable diagrams, where hard gluons attach the valence quarks in the same meson, and nonfactorizable diagrams, where hard glu- ons attach the valence quarks in different mesons. The am- plitudes are written as

MH⫽ f_␾V_t*F_He^(s)⫹Vt*_MHe^(s)_{⫹ fB}V_t*F_Ha^(d)⫹Vt*_MHa^(d), 共8兲 MH⫽ f_␾V_t*F_He^(s)⫹Vt*_MHe^(s)_{⫹ fB}V_t*F_Ha^(u)⫹Vt*_MHa^(u)

⫺ fBV_u*F_Ha⫺Vu*_MHa, 共9兲 for the B_d⁰→␾^K*⁰ and B^⫹→␾^K*^⫹ modes, respectively, where the subscript H⫽L,N,T denotes the different helicity amplitudes, e(a) denotes the emission 共annihilation兲 topol- ogy, and V_q⫽Vqs*V_qb are the products of the Cabibbo- Kobayashi-Maskawa共CKM兲 matrix elements. The hard parts for the factorizable amplitudes F and for the nonfactorizable amplitudes M are derived by contracting the following structures to the lowest-order one-gluon-exchange diagrams:

1

冑^2Nc

共P”1⫹MB兲␥5⌽共x,b兲, 共10兲

1

冑^2Nc

关M␾⑀”²共L兲⌽␾共x兲⫹⑀”²共L兲P”²⌽_␾^t共x兲⫹M␾I⌽_␾^s共x兲兴, 共11兲

1This convention corresponds to tr(␥5a” b”c”d”)⫽

⫺4i⑀^␣␤␥␳a_␣b_␤c_␥d_␳.

1

冑^2Nc

冋

⫹ M_␾

P₂•n⫺i⑀␮␯␳␴␥5␥^␮⑀2␯共T兲P2␳n_⫺^␴⌽_␾^a共x兲

册

1

冑^2Nc

关MK*⑀”3共L兲⌽K*共x兲⫹⑀”3共L兲P”3⌽_K^t_*共x兲

⫹MK*I⌽_K^s_*共x兲兴, 共13兲

1

冑^2Nc

冋

⫹ M_K*

P₃•n⫹i⑀␮␯␳␴␥5␥^␮⑀3␯共T兲P3␳n_⫹^␴⌽_K^a_*共x兲

册

where n_⫹⫽(1,0,0T) and n_⫺⫽(0,1,0T) are dimensionless vectors on the light cone. Equations 共11兲 and 共12兲 are asso- ciated with the longitudinally and transversely polarized ␾ mesons, respectively. The structures associated with the K* meson are similar as shown above.

To extract the contributions to the helicity amplitudeML, the following parametrization for the longitudinal polariza- tion vectors is useful:

⑀2共L兲⫽ P₂

M_␾⫺ M_␾ P₂•n⫺n_⫺,

⑀3共L兲⫽ P₃

M_K*⫺ M_K*

P₃•n⫹n_⫹, 共15兲 which satisfy the normalization⑀2

2(L)⫽⑀3

2(L)⫽⫺1 and the orthogonality⑀2(L)•P2⫽⑀3(L)•P3⫽0 for the on-shell con- ditions P₂²⫽M_␾² and P₃²⫽M_K²_*. We first keep the full depen- dence on the light meson masses M_␾ and M_K* in the mo- menta P₂ and P₃. After deriving the factorization formulas, which are well defined in the limit M_␾, M_K*→0, we drop the terms proportional to r_␾²,r_K²_*⬃0.04, with the ratios r_␾

⫽M_␾/ M_B and r_K*⫽MK*/ M_B. Under this approximation, the expressions of the␾^{and K}*meson momenta are then as simple as

P₂⫽M_B

冑^{2共1,0,0T兲, P3⫽}

M_B

冑^{2共0,1,0T兲. 共16兲}

For the extraction of the helicity amplitudesMN andMT, Eq. 共16兲 and the transverse polarization vectors,

⑀2共T兲⫽共0,0,1T兲, ⑀3共T兲⫽共0,0,1T兲, 共17兲 can be adopted directly. The explicit factorization formulas are collected in the Appendix.

The power counting rules in PQCD 关7兴 tell that the fac- torizable amplitude F_Le 共corresponding to the B→K* tran- sition form factor兲 is leading, and the other factorizable am- plitudes are at least down by a power of r_␾ or r_K*. The

nonfactorizable amplitudesM are suppressed by a power of

⌳¯ /M_B. Hence, the formalism presented in this work is com- plete at O( M_␾,K*/ M_B), and subject to corrections of O(⌳¯ /MB). Equation共4兲 then implies that the helicity ampli- tude A₀ is leading in the heavy-quark limit, and A_储 and A_⬜ are next to leading. The factorizable annihilation amplitudes F_Ha, being suppressed only by M_␾,K*/ M_B and almost imaginary, are the major source of the strong phases in PQCD. Since the B→␾^K*decays are the pure penguin pro- cesses with a weak dependence on the unitarity angle ␾3, these strong phases determine the relative phases among the helicity amplitudes A₀,A_储 and A_⬜.

For the B meson wave function, we employ the model关1兴,

⌽B共x,b兲⫽NBx²共1⫺x兲²

⫻exp

冋

冉

冊

册

where the shape parameter ␻B⫽0.4 GeV has been adopted in all our previous analyses of exclusive B meson decays.

The normalization constant N_B⫽91.784 GeV is related to the decay constant f_B⫽190 MeV 共in the convention f_␲

⫽130 MeV). It is known that there are two B meson wave functions⌽B and⌽¯_B, which are related to the three-parton B meson wave functions through a set of equations of motion 关17–20兴. Because of the unknown three-parton wave func- tions, the equations of motion in fact do not impose any constraint on the functional form of⌽Band⌽¯

B. Our simple choice of the model wave functions corresponds to⌽Bin Eq.

共18兲 and ⌽¯

B⫽0. This choice is legitimate, since the contri- bution from⌽¯_Bis suppressed by a power of⌳¯ /MB关11兴, and negligible within the accuracy of the current formalism.

The ␾ ^{and K}*meson distribution amplitudes up to twist 3 are given by关21兴

⌽_␾共x兲⫽ 3 f_␾

冑^2Nc

x共1⫺x兲, 共19兲

⌽_␾^t共x兲⫽ f_␾^T

2冑^2Nc

再

⫹0.69

冋

册冎

⌽_␾^s共x兲⫽ f_␾^T

4冑^2Nc

冋

⫹1.38ln x

1⫺x

册

⌽_␾^T共x兲⫽ 3 f_␾^T

冑^2Nc

x共1⫺x兲关1⫹0.2C2

3/2共1⫺2x兲兴,

共22兲

⌽_␾^v共x兲⫽ f_␾

2冑^2Nc

再

⫻关3共1⫺2x兲²⫺1兴⫹0.96C4

1/2共1⫺2x兲

冎

⌽_␾^a共x兲⫽ 3 f_␾ 4冑^2Nc

共1⫺2x兲关1⫹0.93共10x²⫺10x⫹1兲兴, 共24兲

⌽K*共x兲⫽3 f_K*

冑^2Nc

x共1⫺x兲关1⫹0.57共1⫺2x兲

⫹0.07C2

3/2共1⫺2x兲兴, 共25兲

⌽_K^t_*共x兲⫽ f_K

*

T

2冑^2Nc

兵^0.3共1⫺2x兲关3共1⫺2x兲²

⫹10共1⫺2x兲⫺1兴⫹1.68C4

1/2共1⫺2x兲

⫹0.06共1⫺2x兲²关5共1⫺2x兲²⫺3兴

⫹0.36兵¹⫺2共1⫺2x兲关1⫹ln共1⫺x兲兴其其^{, 共26兲}

⌽_K^s_*共x兲⫽ f_K^T_* 2冑^2Nc

兵³共1⫺2x兲关1⫹0.2共1⫺2x兲

⫹0.6共10x²⫺10x⫹1兲兴⫺0.12x共1⫺x兲

⫹0.36关1⫺6x⫺2ln共1⫺x兲兴其^, 共27兲

⌽_K^T_*共x兲⫽3 f_K

*

T

冑^2Nc

x共1⫺x兲关1⫹0.6共1⫺2x兲⫹0.04C2 3/2

⫻共1⫺2x兲兴, 共28兲

⌽_K^v_*共x兲⫽ f_K*

2冑^2Nc

再

⫹0.4C2

1/2共1⫺2x兲⫹0.88C4

1/2共1⫺2x兲

⫹0.48关2x⫹ln共1⫺x兲兴

冎

⌽_K^a_*共x兲⫽ f_K* 4冑^2Nc

兵³共1⫺2x兲关1⫹0.19共1⫺2x兲

⫹0.81共10x²⫺10x⫹1兲兴⫺1.14x共1⫺x兲

⫹0.48关1⫺6x⫺2ln共1⫺x兲兴其^, 共30兲 with the Gegenbauer polynomials,

C₂^1/2共␰兲⫽1

2共3␰²⫺1兲,

C₄^1/2共␰兲⫽1

8共35␰⁴⫺30␰²⫹3兲,

C₂^3/2共␰兲⫽3

2共5␰²⫺1兲. 共31兲

We employ G_F⫽1.16639⫻10^⫺5 GeV^⫺2, the Wolfenstein parameters ␭⫽0.2196, A⫽0.819, and Rb⫽0.38, the unitar- ity angle ␾3⫽90°, the masses MB⫽5.28 GeV,M_␾

⫽1.02 GeV and MK*⫽0.89 GeV, the decay constants f_␾

⫽237 MeV, f_␾^T⫽220 MeV, fK*⫽200 MeV, and fK*

T

⫽160 MeV, and the Bd

0(B^⫹) meson lifetime ␶B⁰

⫽1.55 ps(␶B⫹⫽1.65 ps) 关22兴. We have confirmed that the above distribution amplitudes and decay constants lead to the B→K* transition form factors关23兴 in agreement with those from light-cone QCD sum rules 关24兴. We have also con- firmed that the averaged values of the running hard scales t defined by Eqs.共A20兲 and 共A21兲 in the Appendix are indeed about 冑⌳¯ MB⬃1.6 GeV. Note that the B→␾^K* branching ratios are insensitive to the variation of ␾3. The results for the helicity amplitudes A₀, A_储 and A_⬜, including their rela- tive phases␾储⬅Arg(A储/A₀) and␾_⬜⬅Arg(A_⬜/A₀), are dis- played in Table I. The contributions to the B→␾^K*branch- ing ratios mainly arise from the longitudinal polarizations A₀ because of the relation 兩A0兩²Ⰷ兩A储兩²⬃兩A_⬜兩², which is ex- pected from the power counting rules. It is easy to observe that the ratios 兩H_⫺/H₀兩² and兩H_⫹/H₀兩² obtained in PQCD are close to those in QCDF关25兴. The annihilation contribu- tions are the major source of the strong phases, and the non- factorizable contributions are the minor one. The values of

␾储 and␾_⬜ in the rows共I兲–共III兲 of Table II indicate that the phases from the former are about 4 –5 times those from the latter共but opposite in sign兲. Without these sources, we have

␾储⫽␾_⬜⫽␲. Note that the relative phases among the differ- ent helicity amplitudes cannot be predicted unambiguously

TABLE I. Helicity amplitudes and relative phases.

Mode BR(10^⫺6) 兩A0兩² 兩A_储兩² 兩A_⬜兩² ␾储(rad) ␾⬜(rad)

␾K*⁰ 14.86 0.750 0.135 0.115 2.55 2.54

␾K*^⫹ 15.96 0.748 0.133 0.111 2.55 2.54

TABLE II. Helicity amplitudes and relative phases:共I兲 without annihilation and nonfactorizable contributions,共II兲 without annihi- lation contributions, and共III兲 without nonfactorizable contributions.

Mode BR(10^⫺6) 兩A0兩² 兩A储兩² 兩A⬜兩² ␾储(rad) ␾⬜(rad)

␾K*⁰_共I兲 14.48 0.923 0.040 0.035 ␲ ␲ 共II兲 13.25 0.860 0.072 0.063 3.30 3.33

共III兲 16.80 0.833 0.089 0.078 2.37 2.34

␾K*^⫹_共I兲 15.45 0.923 0.040 0.035 ␲ ␲ 共II兲 14.17 0.860 0.072 0.063 3.30 3.33

共III兲 17.98 0.830 0.094 0.075 2.37 2.34

in QCDF due to the arbitrary complex cutoffs for the evalu- ation of the nonfactorizable and annihilation contributions.

We examine the theoretical uncertainty from the variation of the hard scales t, which are defined as the invariant masses of the internal particles and are required to be higher than the factorization scales 1/b, b being the transverse extents of the mesons. This examination estimates higher-order corrections to the hard amplitudes, which are the most important theo- retical uncertainty for penguin-dominated B meson decays.

The light meson distribution amplitudes have been deter- mined in QCD sum rules. The possible 30% variation of the coefficients of the Gegenbauer polynomials in these distribu- tion amplitudes lead only to little changes of our predictions.

We consider the hard scales t located between 0.75–1.25 times the invariant masses of the internal particles. The pre- dictions for the B→␾K branching ratios from the above range are consistent with the data with uncertainty 关7兴. We then obtain the B→␾^K*branching ratios,

B共Bd

0→␾^K*⁰_{兲⫽共14.86⫺3.36}^⫹4.88_兲⫻10^⫺6,

B共B^⫾→␾^K*^⫾_{兲⫽共15.96⫺3.61}^⫹5.24_兲⫻10^⫺6. 共32兲 The relative phases ␾储 and␾_⬜, and the magnitudes 兩A0兩², 兩A_储兩² and 兩A_⬜兩² of the helicity amplitudes are quite stable under the variation of the hard scales t. They change within 0.05 rad and within 0.01, respectively. There is another mi- nor source of theoretical uncertainty from the light meson decay constants f_␾^(T) and f_K

*

(T). If they reduce by 5%, the predicted branching ratios will decrease by 10%. The C P asymmetries of the B→␾^K*modes are, as of B→␾^{K, van-} ishingly small共less than 2%兲.

The above branching ratios are larger than those from QCDF关25兴,

B共Bd

0→␾^K*⁰_{兲⫽8.71⫻10}^⫺6,

B共B^⫾→␾^K*^⫾_{兲⫽9.30⫻10}^⫺6, 共33兲 due to the dynamical enhancement of penguin contributions.

We emphasize that the annihilation amplitudes, though not negligible, are not responsible for the large branching ratios in PQCD, since they are mainly imaginary. This is under- stood by comparing the branching ratios in Table I and in row 共II兲 of Table II. The nonfactorizable contributions are not shown either by the branching ratios in Table I or in the row共III兲 of Table II. However, the annihilation contributions, parametrized as being real, are important in QCDF in order to explain the large B→␾K branching ratios. With the al- most real annihilation contributions, the B→␾K branching ratios obtained in QCDF can increase from 4⫻10^⫺6 to 7

⫻10^⫺6关9兴. The values quoted in Eq. 共33兲 do not include the annihilation contributions. The current experimental data of B(B⁰→␾^K*⁰),

CLEO关26兴: 共11.5_⫺3.7⫺1.7^⫹4.5⫹1.8兲⫻10^⫺6, BELLE关27兴: 共15_⫺6^⫹8⫾3兲⫻10^⫺6,

BABAR关28兴: 共8.6_⫺2.4^⫹2.8⫾1.1兲⫻10^⫺6, 共34兲

and those of B(B^⫾→␾^K*^⫾),

CLEO关26兴: 共10.6_⫺4.9⫺1.6^⫹6.4⫹1.8兲⫻10^⫺6,

BELLE关27兴: ⬍36⫻10^⫺6,

BABAR关28兴: 共9.7_⫺3.4^⫹4.2⫾1.7兲⫻10^⫺6, 共35兲

are not yet precise enough to distinguish the two different approaches.

In this paper we have studied the first observed B→VV modes, the B→␾^K* decays, using the PQCD formalism. It has been stressed that two-body heavy meson decays are characterized by a scale of O(⌳¯ M_B) in PQCD, for which penguin contributions are dynamically enhanced. This en- hancement makes penguin-dominated decay modes acquire branching ratios larger than those in QCDF, even when the final-state particles are vector mesons. We have proposed the B→␾^K(*⁾ decays as the ideal modes to test the significance of this mechanism. If their branching ratios are as large as 10⫻10^⫺6(15⫻10^⫺6) 共independent of the unitarity angle

␾3), dynamical enhancement will be convincing. We have also emphasized that the relative importance and the relative strong phases among the different helicity amplitudes in the B→VV modes can be predicted unambiguously in PQCD, which are determined by the power counting rules and by the annihilation contributions, respectively. These predictions are insensitive to the variation of the hard scales. Therefore, the comparison of the results presented here with future ex- perimental data will provide a stringent confrontation of the PQCD approach.

ACKNOWLEDGMENTS

We thank H.Y. Cheng, K.C. Yang and the members in the PQCD Collaboration for helpful discussions. The work was supported in part by Grant-in Aid for Special Project Re- search 共Physics of CP Violation兲, and by Grant-in Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan. The work of H.N.L. was supported in part by the National Science Council of R.O.C. under the Grant No. NSC-90-2112-M-001-077 and by National Center for Theoretical Sciences of R.O.C.

APPENDIX: FACTORIZATION FORMULAS In this appendix we present the explicit expressions of the factorizable and nonfactorizable amplitudes in Eq. 共9兲. The effective Hamiltonian for the flavor-changing b→s transition is given by

H_eff⫽G_F

冑² _q

兺

冋

⫹i

兺

C_i共␮兲Oi共␮兲

册

with the CKM matrix elements V_q⫽Vqs*V_qb and the opera- tors

O₁^(q)⫽共s¯iq_j兲V⫺A共q¯jb_i兲V⫺A, O₂^(q)⫽共s¯iq_i兲V⫺A共q¯jb_j兲V⫺A,

O₃⫽共s¯ib_i兲V⫺A

兺

O₄⫽共s¯ib_j兲V⫺A

兺

O₅⫽共s¯ib_i兲V⫺A

兺

O₆⫽共s¯ib_j兲V⫺A

兺

O₇⫽3

2共s¯ib_i兲V⫺A

兺

O₈⫽3

2共s¯ib_j兲V⫺A

兺

e_q共q¯jq_i兲V⫹A,

O₉⫽3

2共s¯ib_i兲V⫺A

兺

O₁₀⫽3

2共s¯ib_j兲V⫺A

兺

i and j being the color indices. Using the unitarity condition, the CKM matrix elements for the penguin operators O₃–O₁₀ can also be expressed as V_u⫹Vc⫽⫺Vt. The unitarity angle

␾3 is defined via

V_ub⫽兩Vub兩exp共⫺i␾3兲. 共A3兲

Here we adopt the Wolfenstein parametrization for the CKM matrix up to O(␭³),

冉

冊

冉

冊

with the parameters关29兴,

␭⫽0.2196⫾0.0023, A⫽0.819⫾0.035,

R_b⬅冑␳²⫹␩²⫽0.41⫾0.07. 共A5兲

The factorizable amplitudes F_He^(q) and F_Ha^(q)⫽FHa4 (q) ⫹FHa6

(q) are written as

F_Le^(q)⫽8␲^CFM_B²

冕

dx₁dx₃

冕

⬁

b₁db₁b₃db₃⌽B共x1,b₁兲兵关共1⫹x3兲⌽K*共x3兲⫹rK*共1⫺2x3兲„⌽K*

t 共x3兲

⫹⌽_K^s_*共x3兲…兴Ee (q)共te

(1)兲he共x1,x₃,b₁,b₃兲⫹2rK*⌽_K^s_*共x3兲Ee (q)共te

(2)兲he共x3,x₁,b₃,b₁兲其^{, 共A6兲}

F_Ne^(q)⫽8␲^CFM_B²

冕

冕

⫺⌽_K^a_*共x3兲…兴Ee (q)共te

(1)兲he共x1,x₃,b₁,b₃兲⫹rK*关⌽_K^v_*共x3兲⫹⌽_K^a_*共x3兲兴Ee (q)共te

(2)兲he共x3,x₁,b₃,b₁兲其^, 共A7兲

F_Te^(q)⫽16␲^CFM_B²

冕

dx₁dx₃

冕

⬁

b₁db₁b₃db₃⌽B共x1,b₁兲r_␾兵关⌽K*

T 共x3兲⫹2rK*⌽_K^a_*共x3兲⫺rK*x₃„⌽K*

v 共x3兲

⫺⌽_K^a_*共x3兲…兴Ee (q)共te

(1)兲he共x1,x₃,b₁,b₃兲⫹rK*关⌽K*

v 共x3兲⫹⌽K*

a 共x3兲兴Ee (q)共te

(2)兲he共x3,x₁,b₃,b₁兲其^{, 共A8兲}