Nonuniversal Z '(') couplings in B decays

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(1)PHYSICAL REVIEW D 73, 075003 (2006). Nonuniversal Z0 couplings in B decays Chuan-Hung Chen1,2,* and Hisaki Hatanaka3,† 1. Department of Physics, National Cheng-Kung University, Tainan 701, Taiwan 2 National Center for Theoretical Sciences, Taiwan 3 Department of Physics, National Tsing-Hua University, Hsin-Chu 300, Taiwan (Received 16 January 2006; revised manuscript received 21 February 2006; published 4 April 2006) We study the impacts of the nonuniversal Z0 model, providing flavor-changing neutral current at tree level, on the branching ratios (BRs), CP asymmetries (CPAs), and polarization fractions of B decays. We find that, for satisfying the current data, the new left- and right-handed couplings have to be included at the same time. The new introduced effective interactions not only could effectively explain the puzzle of small longitudinal polarization in B ! K decays, but also provide a solution to the small CPA of B ! 0 K . We also find that the favorable CPA of B ! 0 K is opposite in sign to the standard model; meanwhile, the CPA of Bd ! 0 K has to be smaller than 10%. In addition, by using the values of parameters which are constrained by B ! K, we find that the favorable ranges of BRs, CPAs, longitudinal polarizations, and perpendicular transverse polarizations for B ! K ; Bd ! K are 17:1 3:9; 10:0 2:0 106 , 3 5; 21 7%, 0:66 0:10; 0:44 0:08, and 0:14 0:10; 0:25 0:09, respectively. DOI: 10.1103/PhysRevD.73.075003. PACS numbers: 12.60.Cn, 13.20.He. I. INTRODUCTION Some puzzles have been found in B meson decaying processes, such as (a) the large branching ratios (BRs) of B ! K0 [1,2], (b) the small longitudinal polarizations of B ! K decays [3–6], and (c) the unmatched CP asymmetries (CPAs) and BRs in B ! K decays [7–11]. The interesting thing is that the mazy problems are all related to the flavor-changing neutral current (FCNC) b ! s decays. The resultants push us not only to consider more precise QCD effects but also to speculate the existence of new physics. It is known that the FCNC processes usually arise from the loop corrections in the standard model (SM) like models. The preference of loop correction originates from the strict constraints of K meson oscillation. However, the constraints of the K system are only on the first two generations; the FCNC at tree level in the third generation has no significant limit yet. Especially, the direct constraint from Bs B s mixing, dictated by b ! s interactions, is a lower bound in experiments [1]. Hence, it will be interesting to investigate the models in which FCNC occurs at tree level, and they could provide the solutions to the puzzles in b ! s decays. One of the simple extensions of the SM for the effects of FCNC is the unconventional Z0 model, in which FCNC is arisen from the family of nonuniversal couplings [12]; i.e. the couplings of Z0 to different families are not the same. One of the possible ways to get the family nonuniversal couplings is to include an additional U10 gauge symmetry; and then by the requirement of anomaly free, the gauge charges for different families are different [13]. Other *Electronic address: [email protected] † Electronic address: [email protected]. 1550-7998= 2006=73(7)=075003(14)$23.00. models giving the family nonuniversal Z0 interactions could be referred to Ref. [12]. The detailed phenomenological analyses for various low energy physics could be found in Ref. [14]. Especially, the implications on timedependent CPA of B ! Ks and on the BRs of B ! 0 K have been studied by Ref. [15]. Moreover, the solution to the B ! K puzzle by the nonuniversal Z0 couplings is also discussed by the authors of Ref. [16]. To further pursue the effects on B decaying processes, in this paper we will take all the measurements of B ! K and B ! K into account to constrain the free parameters. By the constrained parameters, we investigate the implications of the Z0 model on BRs, CPAs, and polarization fractions (PFs) for the decays B ! K , B ! K, and B ! K . For two-body color-allowed processes of B decays, it is known that the dominant hadronic effects are the factorized parts which could be simply expressed as the multiplication of effective Wilson coefficients (WCs), the decay constant, and the transition form factors, e.g. A / CfM1 FB!M2 q2 m2M1 . Furthermore, by the observed CPA of O(10%) in Bd ! K , we know that the large strong phases have to be introduced. In order to selfconsistently calculate the hadronic effects, we employ the perturbative QCD (PQCD) approach [17,18] to evaluate the hadronic matrix elements, where the large strong phases could be generated by the annihilation effects of the effective operator V A V A. It is known that in the B system which is composed of a heavy quark and a light quark, the residual momentum of the light quark is typically k OmB mb with mBb being the mass of the B-meson (b-quark). In B decays, if we regard that the processes are dominated by short-distant interactions, for catching up the energetic quark from the b-quark decay to form a energetic meson with the typical energy being OmB , the light quark inside the B meson has. 075003-1. © 2006 The American Physical Society.

(2) CHUAN-HUNG CHEN AND HISAKI HATANAKA. PHYSICAL REVIEW D 73, 075003 (2006). to obtain a large momentum from the b-quark via the gluon exchange. Hence, the momentum transfer carried by the hard gluon could be estimated to be k1 k2 , where k1 and k2 denote the momenta of spectator quarks inside the B meson and produced meson, respectively. In terms of lightcone coordinates, the large components of ki i 1; 2 p x m = 2 with xi being the could be defined by k. i B i momentum fractions. Hence, the squared momentum of the exchanged hard gluon is q2 x1 x2 m2B . As known that the residual momentum of light quark in the B meson is OmB mb , x1 is roughly OmB mb =mB . Since the produced light meson is energetic, the momenta of valence p quarks should be OmB = 2, i.e. x2 O1. By taking x1 0:16, x2 0:5, and mBb 5:284:4 GeV, we get p q2 1:5 GeV. Since the value reflects the typical reacting scale of B decays in the framework of the PQCD, for the SM contributions, in our calculations the values of weak WCs are estimated at the scale 1:5 GeV. The paper is organized as follows: In Sec. II, we introduce the nonuniversal Z0 effects for b ! s transition. In Sec. III, based on the flavor diagrams, we explicitly write out the factorizable amplitudes associated with the new physics for the decays Bd ! K , B ! K, and B ! K . In addition, we also define direct CPA and PFs. Then by setting the values of parameters, in Sec. IV we give the calculated values for hadronic effects, present various current experimental data for constraining the unknown parameters, display the SM predictions, and discuss the results of the Z0 model. Finally, we give a summary. II. FCNC FOR b ! s TRANSITION IN THE Z0 MODEL In this section, we will introduce the neutral current interactions in the SM and its extension with an extra Z0 boson. Since we will study the nonleptonic decays, in following discussions we only concentrate on the quark sector. Although we concentrate on the study of new physics, the used notation for new interacting operators will be similar to those presented in the SM. Therefore, it is useful to introduce the effective operators of the SM. Thus, we describe the effective Hamiltonian for b ! sqq decays as [19] G X q Heff pF Vq C1 Oq 1 C2 O2 2 qu;c 10 X. Ci Oi ; (1). b VA ; Oq 1 s q VA q b VA ; Oq 2 s q VA q X O3 s b VA q q VA ; q. O4 s b VA. q q VA ;. q. O5 s b VA. X. q q V A ;. q. O6 s b VA. X. q q V A ;. (2). q. X 3 O7 s b VA eq q q V A ; 2 q X 3 O8 s b VA eq q q V A ; 2 q X 3 O9 s b VA eq q q VA ; 2 q X 3 O10 s b VA eq q q VA ; 2 q with and being the color indices. In Eq. (1), O1 –O2 are from the tree level of weak interactions, O3 –O6 are the socalled gluon penguin operators, and O7 –O10 are the electroweak penguin operators, while C1 –C10 are the corresponding WCs. Using the unitarity condition, the CKM matrix elements for the penguin operators O3 –O10 can also be expressed as Vu Vc Vt . For studying the Z0 model, as usual we describe the Lagrangian for the neutral current interactions in terms of weak eigenstates as [14,21] LNC g1 J1 Z01 g2 J2 Z02 ; J1 q i

(3) Lq PL Rq PR qi ; J2. (3). q i

(4) ~ LqL ij PL ~ RqR ij PR qi ;. where the subscript of qi denotes the flavor index of quark, PL;R 1 5 =2, Z01 and Z02 are the neutral gauge bosons, corresponding to SU2 U1 and the extended Abelian gauge symmetries, and g1 and g2 are the associare ated gauge couplings, respectively. We note that L;R q universal couplings of the SM while ~L;R q are 3 3 matrices and denote the effects of nonuniversal couplings. In general, the Z01 and Z02 bosons will mix each other so that the physical states of Z bosons could be parametrized by . i3. V where Vq Vqs qb are the Cabibbo-Kobayashi-Maskawa (CKM) [20] matrix elements and the operators O1 –O10 are defined as. X. cos. Z sin. Z0. sin. cos. . Z01 ; Z02. (4). where denotes the Z Z0 mixing angle. In addition, the physical states of quarks could be related to the weak. 075003-2.

(5) NONUNIVERSAL Z0 COUPLINGS IN B DECAYS p ULR. w. DpLR. PHYSICAL REVIEW D 73, 075003 (2006) w. T. eigenstates by VULR U and VDLR D in which U u; c; t, DT d; s; b, and the VULR and VDLR are the unitary matrices for diagonalizing weak states to physical eigenstates. The CKM matrix is defined by y VCKM VUL VDL . As a result, in terms of physical states the effective interactions for b ! s decays could be written as X X 8GF g 2 1 P 1 q H:c:; H Z p sin cos 2

(6) Bsb q b P 2 sq g1 1 ; 2 q 2 (5) 8G g m 2 X X 2 1 P 1 q H:c: H Z0 pF cos2 2 Z

(7) Bsb Bqq b P 2 sq g1 mZ0 1 ; 2 q 2 where H Z and H Z0 express the effects of Z Z0 mixing and Z0 , respectively, the q could be u, d, s, and c quark, i1;2 L and R, and Lq T3q Qq sin2 W ;. B DD V D ~ D V Dy ;. Rq Qq sin2 W ;. B UU V U ~ U V Uy :. (6). Here, the capital UU and DD in the subscript of the B parameter could be the flavors u; c and d; s; b, respectively, and the W is the Weinberg’s angle. By current experimental data, it is known that the mixing angle is limited to be less than O103 [15,16]. If the mass of Z0 is in the range of a few hundred GeV to 1 TeV, the dominant effects only come from the Z0 exchange. Therefore, under this assumption, we will neglect the contributions of Z Z0 mixing. According to the interactions of Eq. (5), the new effective Hamiltonian for b ! sqq decays could be written as Z0 Heff. with. C9

(8) 7 C09

(9) 7. 2. 2 g2 M Z 1 BL BL

(10) R 2BL

(11) R DD ; 3 g1 MZ0 Vtb Vts sb UU 2 g2 M Z 2 1 BR BR

(12) L 2BR

(13) L DD ; 3 g1 MZ0 Vtb Vts sb UU (8) 4 g2 M Z 2 1 L

(14) R L L

(15) R V Bsb BUU BDD ; 3 g1 MZ0 Vtb ts 4 g2 M Z 2 1 BR BR

(16) L BR

(17) L DD : 3 g1 MZ0 Vtb Vts sb UU. C3

(18) 5 C03

(19) 5. . GF X 3 3 VA C5 C7 eq qq V A p Vtb Vts bsVA C3 C9 eq qq 2 2 2 q 3 0 0 3 0 0 V A C5 C7 eq qq VA. bsV A C3 C9 eq qq 2 2. The expressions have been written as the four-fermion operators of the SM, shown in Eq. (2). The operators associated with the new unprimed WCs C3;5;7;9 are the same as SM. However, the operators associated with primed coefficients C03;5;7;9 have different chirality from those in the SM for b s couplings. That is, the flavorchanging (FC) Z0 model provides different chiral flavor structures for FCNC processes. It has been found that the new effective WCs of Eq. (8) could be simplified if one assumes B UU ’ 2B DD [16]. Although in general the assumption is unnecessary, for simplicity we still impose the condition in our case. Hence, we get C0 3

(20) 5 0 and 2 gM 1 C9

(21) 7 4 2 Z BL BL

(22) R ; g1 MZ0 Vtb Vts sb DD (9) g2 M Z 2 1 R

(23) L 0 R C9

(24) 7 4 B B : g1 MZ0 Vtb Vts sb DD. (7). Although the hadronic matrix elements, describing the B decaying to two final mesons through the effective Hamiltonian, depend on the chiral and color structures of four-fermion operators, we find that the associated effective WCs could be classified and reexpressed to be a more useful form by C1 C ; a2 C1 2 ; Nc Nc C4 3 C10 NP C3. e C. ; Nc 2 q 9 Nc CNP C 3 C4 3 eq C10 9 ; Nc 2 Nc C6 3 C8 NP C5. e C. ; Nc 2 q 7 Nc C 3 CNP C6 5 eq C8 7 ; Nc 2 Nc. a1 C2. aq3 aq4 aq5 aq6. 3 eq C09 ; a0qNP 3 2. 3 C09 e a0qNP ; 4 2 q Nc. 3 a0qNP eq C07 ; 5 2. 3 C07 a0qNP eq ; 6 2 Nc. (10). with CNP 97 C97 C97 . The superscript q of Eq. (10) denotes the corresponding flavor and eq is its charge. Since the considering Z0 model has the flavor structures which are the same as SM, to more clearly understand the influ-. 075003-3.

(25) CHUAN-HUNG CHEN AND HISAKI HATANAKA. PHYSICAL REVIEW D 73, 075003 (2006). of Fig. 1, we can investigate the decay amplitudes for B ! K, B ! K, B ! K , and B ! K .. ence of new physics, we rewrite Eq. (10) to be 3. eq C9 ; aq3 aqSM 3 2 3 aq5 aqSM. eq C7 ; 5 2. 3 C9 aq4 aqSM. eq ; 4 2 Nc 3 C7 aq6 aqSM. eq : 6 2 Nc (11). III. DECAY AMPLITUDES FOR B ! K AND B ! K DECAYS To describe the amplitudes for B decays, we have to know not only the relevant effective weak interactions but also all possible topologies for the specific process. We display the general involving flavor diagrams for b ! sqq in Fig. 1, where (a) and (b) denote the emission topologies while (c) is the annihilation topology. The flavor q in Figs. 1(a) and 1(b) is produced by gauge bosons and could be u, or d or s quark if the final states are the light mesons; however, q0 stands for the spectator quark and could only be u or d quark, depending on the B meson being the charged or the neutral one. However, the role of q and q0 in Fig. 1(c) is reversed so that q u, or d or s is the spectator quark and q0 u or d is dictated by gauge interactions. We note that the presented flavor diagrams are based on the penguin operators of the SM. Except the different type of interactions at vertices, the flavor diagrams induced by new physics should be similar to those generated by the SM. In addition, since the matrix elements obtained by the Fierz transformation of O3;4 are the same as those of O1;2 , we do not further consider the matrix elements of tree operators. Hence, in terms of the effective interactions of the SM and those shown in Eq. (7), the expressions of Eqs. (10) and (11), and the flavor diagrams. A. Bd ! K0 Although there are charged and neutral modes in B ! K decays, since the differences of flavor diagrams in charged and neutral modes are only the parts of small tree annihilation, we concentrate only on the decay Bd ! K 0 . At quark level, the process is controlled by the decay b ! sss, therefore, q s and q0 d in Figs. 1(a) and 1(b), but they are reversed in Fig. 1(c). Hence, the decay amplitude for Bd ! K 0 is written as 0. V

(26) f

(27) Fe f

(28) Fa a M ZK Vtb ts 1 1K B 2 1K

(29) 3 F2K . (12) and

(30) 1 as a0sNP ;.

(31) 2

(32) 3 ad4

(33) 6 a0dNP 4

(34) 6 ;. (13). a0sNP. with as as3 as4 as5 and a0sNP a0sNP 3 4 0sNP a5 . The f means the decay constant of the meson and is defined by h0js sji f m , in which m and express the mass and polarization vector of the e is from the meson. The hadronic matrix element F1K a a diagrams 1(a) and 1(b). However, the F1K and F2K come from the annihilation topology diagram 1(c). The detailed expressions of the hadronic matrix elements are given in Appendix A 3. The decay rate for B ! PV is written as . G2F m2B Pc jMj2 ; 16. (14). where Pc m21 m22 r2 1=m2B with r P1 P2 =m1 m2 is the momentum of the outgoing vector mesons. By the decay width, we can also define the direct CPA to be ACP ; . (15). where is the decay rate of antiparticle. B. B ! K There are four specific modes in B ! K decays. Since all BRs and CPA of Bd ! K are observed well in experiments, we have to analyze all of them in detail. We begin the analysis from the decay B ! K 0 . According to the flavor diagrams of Figs. 1(a) and 1(c), the emission and annihilation topologies of the decay are described by q d and q0 u. Hence, the decay amplitude for the B ! K 0 decay could be expressed by 0. e e MZ K0 Vtb Vts

(35) fK 1d F1. 2d rK F2 . (a) and (b) stand for FIG. 1. The flavor diagrams for b ! sqq: the emission topologies while (c) is annihilation topology.. a a. 2u F2K ;. fB 1u F1K. (16). where fK is the decay constant of kaon and defined by. 075003-4.

(36) NONUNIVERSAL Z0 COUPLINGS IN B DECAYS. h0js 5 djK sociated with. 0. i fK p , rK m0K =mB with h0js5 djK 0 i fK m0K , and. ; 1q aq4 a0qNP 4. m0K. PHYSICAL REVIEW D 73, 075003 (2006). being as-. 2q aq6 a0qNP : 6. once one determines the first three decays, the decay B ! 0 K is also fixed.. (17). e , Fa , and Fa The hadronic matrix elements F1 1K 2K are 0 similar to those for Bd ! K and the detailed expressions are given in Appendix A 2. In addition, we have a new e contribution F2K which arises from the emission topologies of O6;8 . Similar to B ! K 0 , we can obtain the decay amplitude of Bd ! K easily by using qq0 ud instead of qq0 du. Thus, the decay amplitude for Bd ! K decay is written as 0. e e MZ K Vtb Vts

(37) fK 1u F1. 2u rK F2 a a e. 2d F2K Vub Vus fK a1 F1 ;. fB 1d F1K. (18) where we have included the tree contributions. As mentioned before, except the CKM matrix elements and effective WCs, the hadronic effects of tree are the same as the penguin operators O3;4 . Hence, the hadronic effects encountered in Bd ! K decay are the same as those in B ! K 0 . Next, we analyze the situation of Bd ! 0 K 0 decay. One can easily find that, besides the involving flavor diagrams appeared in the decays B ! K 0 and Bd ! K , the diagram of Fig. 1(b) corresponding to the electroweak penguin contributions of the SM should also be included. Taking proper flavors for q and q0 , the decay amplitude for Bd ! 0 K 0 decay is given by p 0 V

(38) f d Fe d r Fe f Fe 2MZ0 K0 Vtb ts K 1 1 1K 2 K 2 a a V V f a Fe ; fB 1d F1K. 2d F2K ub us 2 1K. C. Bd ! K0 For the production of two vector mesons in B decays, since both vector mesons carry spin degrees of freedom, the decay amplitudes are related to not only the longitudinal parts but also transverse parts. In terms of the notation of Ref. [22], the amplitude Mh could be expressed by M h m2B ML m2B MN 1 t 2 t. iMT 1 t 2 tP1 P2 ;. with the convention 0123 1, where the superscript h is the helicity, the subscript L stands for the h 0 component while N and T express another two h 1 components, and 1 t 2 t 1 with t 1. Hence, each helicity amplitude could be written as [22] H0 m2B ML ;. p H m2B MN mV1 mV2 r2 1MT :. ad3. au5. ad5. a0uNP 3. AL H0. jAi j2 ; jAL j2 jAk j2 jA2? j. Ri . (23). i L; k; ?:. (24). Consequently, the decay rate for B ! V2 V1 is given by . G2F Pc

(39) jAL j2 jAk j2 jA? j2 ; 16m2B. (25). where Pc jP1z j jP2z j is the momentum of either of the outgoing vector mesons. From Fig. 1, it is easy to see that the associated flavor diagrams for B ! K are the same as those for B ! K decays. Furthermore, as the results for the neutral and the charged modes are expected to be similar by neglecting the small annihilation contributions from tree operators Ou1;2 appearing in the charged mode, which will be discussed in Sec. IVA. We will concentrate on the neutral B decay. In terms of the distribution amplitudes of vector mesons, defined in Appendix A 1, the decay amplitudes with various helicities defined by Eq. (21) are given by. a a . fB 1u F1K. 2u F2K V f a Fe f a Fe : Vub us K 1 1 2 1K. 1 Ak? p H H : 2. Accordingly, the PFs can be defined as. a0dNP 3. . with 0dNP e a0uNP a . We note that the new term f F , corre 1K 5 5 sponding to Fig. 1(b), is opposite in sign to other terms. The reason comespfrom the flavor wave function of 0 dd= being uu 2. Figure 1(b) picks both components component. Since the tree while others only take the dd contributions are color suppressed, the corresponding WC is a2 . After introducing the decay amplitudes of B ! K 0 , Bd ! K , and Bd ! 0 K 0 , the amplitude for B ! 0 K decay could be immediately obtained as p 0 V

(40) f u Fe u r Fe f Fe 2MZ0 K Vtb ts K 1 1 2 K 2 1K. (22). In addition, we can also write the amplitudes in terms of polarizations as. (19) au3. (21). 0. a M ZK H Vtb Vts

(41) f

(42) 1H FKe H fB

(43) 2H F1K H. (20). Clearly, the amplitudes shown in the first three decay modes all appear in the decay B ! 0 K . That is,. a. fB

(44) 3H F2K H ;. where H L; N; T and. 075003-5. (26).

(45) CHUAN-HUNG CHEN AND HISAKI HATANAKA. PHYSICAL REVIEW D 73, 075003 (2006).

(46) 1L

(47) 1k as a0sNP ;. M. Z0 K H.

(48) 1? as a0sNP ;

(49) 2

(50) 3 ? . ad4

(51) 6. d a e. fB 3H F2K H Vub Vus a1 fK FH :. (27).

(52) 2

(53) 3 L

(54) 2

(55) 3 k ad4

(56) 6 a0dNP 4

(57) 6 ; a0dNP 4

(58) 6 :. V

(59) f u Fe f d Fa Vtb ts K 1H H B 2H 1K H. (30) q 12. The definitions of as and a0sNP are the same as those for Bd ! K 0 decay. The explicit expressions for fFe ; Fa g could be referred to Appendix A 4.. The definitions of are the same as those for B ! K, expressed by Eq. (17). IV. NUMERICAL ANALYSIS A. Theoretical inputs. D. B ! K Since the quark compositions of and K mesons are the same as those of and K, respectively, the flavor diagrams for B ! K and B ! K decays should be 0 j0i / V p p 0 the same. However, due to hVpjqq in which the scalar vertex is arisen from the Fierz transformation of V A V A, the emitted factorizable contributions of four-fermion operators O6;8 are vanished, i.e. aq6 have no contributions. Consequently, it could be expected that BRs of B ! K are smaller than those of B ! K in the SM. Although there are four possible modes in the B ! K decays, we only concentrate on the decays B ! K 0 and B0 ! K that have larger BRs. Following the definition of Eq. (21), the various helicity amplitudes for B ! K 0 decay would be written as M. Z0 K 0 H. u a fB 3H F2K H :. therefore, the values of SM shown in Eq. (10) are estimated to be a1 1:07; auSM 3. q 2HL;k. . 1q ;. q 3HL;k. q q 2? aq4 a0qNP ; 1? 4. auSM 0:036; 4. auSM 0:008; 5. auSM 0:055; 6. adSM 3 adSM 5. . 2q ;. q 3? aq6 a0qNP : 6. (29). Similarly, the decay amplitude for B0 ! K decay is written as. a2 0:028;. 0:002;. (28). The associated effective WCs are given by . fKT fT 200160 MeV s. The lifetimes of charged and neutral B mesons are chosen as B 1:67 1012 s and Bd 1:56 1012 s, respectively. Since we use the PQCD approach to calculate the hadronic matrix q elements, B 1:5 GeV; we set the scale of weak WCs at m. V

(60) f d Fe f u Fa Vtb ts K 1H H B 2H 1K H. q 1HL;k. To obtain numerical estimations, the values of theoretical parameters in the SM related to the weak interactions are taken as follows: GF 1:166 105 GeV2 , Vus 0:224, Vts 0:041, and Vub 3:5 103 ei3 with 3 720 . The decay constants of mesons are set to be T f 130, fK 160, fB 190, f 237170, and. 0:012;. adSM 4. 0:036;. 0:008;. adSM 6. 0:056;. (31). and asSM adSM . In addition, in Table I we present the values of the hadronic effects, which are displayed in Secs. III A, III B, III C, and III D and calculated by the PQCD approach. From the table, we clearly see that the annihilation contributions from V A V A opa a erators which correspond to F1PP and F1VV are negligible. To more clearly understand the results, we use B ! PP. TABLE I. The values of factorizable amplitudes. e F1K 0.37. a F1K 9:88 i7:54104. a F2K 0:047 i0:14. e F1 0.24. e F2 0.50. a F1K 0:39 i8:16104. a F2K 1:99 i3:36102. FKe L 0.36. FKe k 0.06. FKe ? 0.11. a F1K L 1:4 i1:0103. a F1K k 6:6 i6:5104. a F1K ? 1:2 i6:4103. a F2K L 0:03 i0:14. a F2K k 0:03 i0:02. e FL 0.31. e Fk 0.04. e F? 0.08. a F1K L 2:3 i5:4103. a F1K ? 1:9 i2:9103. a F2K L 0:03 i0:16. a F2K k 0:65 i8:3102. a F2K ? 0:01 i0:17. a F2K ? 0:06 i0:11 a F1K k 3:1 i0:9104. 075003-6.

(61) NONUNIVERSAL Z0 COUPLINGS IN B DECAYS. PHYSICAL REVIEW D 73, 075003 (2006). decays to illustrate the property. For B ! PP decays, the factorized amplitude associated with the V A V A interaction for annihilated topology can be expressed as [23] a ifB m21 m22 F0P1 P2 m2B ; hP1 P2 jq 1 1 5 q2 q 3 1 5 bjBi. (32). where m12 are the masses of outgoing particles and F0P1 P2 m2B corresponds to the timelike form factor, defined by B i ifB p 5 bjBp h0jq B; m21 m22 m21 m22 P1 P2 2 Q Q . Q F0P1 P2 Q2 ; F hP1 p1 P2 p2 jq 1 q2 j0i q 1 2 2 Q Q respectively, with q p1 p2 and Q p1 p2 . From Eq. (32), it is clear that, if m1 m2 , the factorized effects of annihilation topology vanish. However, the cancellation factor will be removed when the interactions correspond to V A V A operators [23]. B. Experimental inputs and predictions of the SM As mentioned before, the accuracies of some experimental data on BRs and CPAs are quite well, thus we could utilize these observed values to constrain the new parameters of the Z0 model. To more clearly know what the experimental inputs and the predictions are, in the following we definitely display the ranges of current experimental data for the inputs. Hence, taking the world averages with 2 errors presented in Ref. [24], the inputs of BRs are Bd ! K 0 , B ! K 0 , and all B ! K decays, and their limits are taken to be. jACP B ! 0 K j < 10%. Because there is no any significant information on the CPA of Bd ! 0 K 0 , we leave the value as our prediction. In addition, we also take the longitudinal polarization RL of Bd ! K 0 as the input and the limit is chosen to be 44% < RL Bd ! K 0 < 57%. Before further discussing the contributions of the Z0 model, it is worth knowing the SM results which are based on the taken values in Sec. IVA. Hence, the SM predictions on BRs are BRBd ! K 8:98 106 ; BRBd ! K 12:9 106 ; BRB ! K 15:3 106 ; BRBd ! K 13:4 106 ; BRB ! K 22:2 106 ; . . (36). 6. BRBd ! K 19:0 10 ;. 7:3 < BRBd ! K106 < 9:5;. BRBd ! 0 K 7:87 106 ;. 8:6 < BRBd ! K 106 < 10:4;. BRB ! 0 K 12:5 106 ;. 21:5 < BRB ! K106 < 26:7; 16:6 < BRBd ! K 106 < 19:8;. (33). (34). 9:5 < BRBd ! 0 K106 < 13:5;. SM the ratios of BRs are estimated by RSM 1 0:92, Rc SM 1:18, and Rn 1:22; and the predictions on CPAs are. ACP Bd ! K 22:0%;. 10:5 < BRB ! 0 K 106 < 13:7:. ACP Bd ! K 12:1%; Moreover, we also take into account the ratios of BRs, defined by. 2BRB ! 0 K ; BRB ! K. Rn . (37). ACP B ! 0 K 8:3%;. BRBd ! K R1 B ; Bd BRB ! K Rc . ACP Bd ! 0 K 1:35%;. BRBd ! K ; 2BRBd ! 0 K (35). where the vanished CPAs are not shown. Moreover, the estimations of the various PFs for VV modes are also given to be. as 0:76 < R1 < 0:88, 0:91 < Rc < 1:09, and 0:74 < Rn < 0:88 [11]. Since there are no measurements on the CPAs of B ! K , we artificially set the limits as 0 < jACP B ! K j < 0:05 in which the CPAs vanish in the SM. Other limits from data are taken as 0 < jACP B ! K 0 j < 5%, 7:1% < jACP Bd ! K j < 14:7%, and 0 <. 075003-7. RL Bd ! K 0 0:71; R? Bd ! K 0 0:15; RL B ! K 0 0:72; R? B ! K 0 0:13; RL Bd ! K 0:52; R? Bd ! K 0:22:. (38).

(62) CHUAN-HUNG CHEN AND HISAKI HATANAKA 20 ∗. 5 0. 5. 0.2 0.16 0. 5. C. Results of the Z0 model on B ! K , B ! K, and B ! K decays. 0.56 0. 5. 10 20 15 ∗ −6 BR(Bd→Κ φ)10 (d). 0.24 0.2 0.16. 10 20 15 ∗ −6 BR(Bd→Κ φ)10. 0.56. 0.8 0.64 0.72 ∗ RL(Bd→Κ φ). FIG. 2 (color online). The SM predictions for Bd ! K decays, where (a), (b), (c), and (d) represent the correlations between BRs of K and K modes, RL? and BRBd ! K , and R? and RL , respectively. The circle, square, diamond, and triangle-up symbols stand for the results of 1:3, 1.5, 2.0, and 4:0 GeV, respectively. The error bars are the world averages with 2 errors.. 3 0. 10. 3 (c). 2 1. ±. 0 -1 -2. 2. 4 8 10 6 −6 BR(Bd → π° Κ)10. -8 (b). ±. 15 20 25 30 ± ± −6 BR(B → π Κ)10 ±. -6. ACP(B → π° Κ )%. ACP(Bd → π° Κ)%. -3. ACP(Bd → π Κ )%. (a). ±. ±. ACP(B → π Κ)%. According to our estimations, we see that, compared to the data, Bd ! K Bd ! 0 K has a larger (smaller) BR, the ratios R1;c;n do not fit the data well, and RL of Bd ! K 0 is much larger than observations. We also find RL Bd ! K could be around 50%. For displaying the influence of different scales, in Fig. 2, we present the correlations between BRs in K and K modes, RL;? and BRBd ! K , and RL and R? , where the circle, square, diamond, and triangle-up symbols stand for the results of 1:3, 1.5, 2.0, and 4:0 GeV, respectively. The error bars presented in the figures are the world averages with 2 errors. Similarly, we also show the SM. 6. predictions on the CPAs of B ! K and the corresponding BRs in Fig. 3.. 0.64. 10 20 15 ∗ −6 BR(Bd→ Κ φ)10 (c). 0.24. (b). 0.72. ∗. ∗. R ⊥(Bd→ Κ φ). RL(Bd→ Κ φ). 10. R⊥(Bd→Κ φ). BR(Bd→Κφ)10. −6. (a). 15. 0. PHYSICAL REVIEW D 73, 075003 (2006). 0.8. -12 -16 -20 -24 0 4 0. 5. 10 15 20 25 30 ± −6 BR(Bd → π Κ )10 (d). -4 -8 -12 -16 0. 10 20 5 15 ± ± −6 BR(B → π° Κ )10. FIG. 3 (color online). The SM predictions for the CPAs (in units of 102 ) versus the corresponding BRs (in units of 106 ). Legend is the same as Fig. 2.. Before performing the numerical calculations, we first discuss the allowed regions of new effects which are from C09

(63) 7 . According to the results of Refs. [15,16], it is known that the unknown parameters, defined by

(64) LX X g2 mZ =g1 mZ0 2 BL sb Bdd =Vtb Vts with X L; R, have been LX limited to be j

(65) j 0:02 in which m0Z is at TeV scale. L

(66) R That is, if we assume that BLsb BRsb and BR

(67) L DD BDD , consequently we obtain jC0 9

(68) 7 j 0:08. In the following analysis, we will take this value as the upper bound of new effects. Since C0 9

(69) 7 in general are complex, we totally have eight parameters for each b ! sqq with q u; d and b ! sss decays. In principle, the eight free parameters for b ! sqq could be fixed by the eight chosen measurements such as four BRs and four CPAs in B ! K decays. Then, using the constrained parameters we can make predictions on B ! K . Although there are not eight measurements related to b ! sss directly, however, due to the new effects in Eq. (13) for B ! K being different from that in Eq. (27) for

(70) 12L of B ! K , we find that when the data of K and K are considered simultaneously, the unknowns have been strictly constrained. For convenience, we parametrize the unknowns to be C9 LL eiLL , C7 LR eiLR , C09 RL eiRL , C07 RR eiRR , so that jXY j 0:08 and 0 XY 2 with X and Y each being L or R. Now, we could investigate the contributions of the Z0 model to the considering processes. At first, we study the decays governed by b ! sss. It has been known that, in terms of the flavor diagrams of Fig. 1, all effects contributing to Bd ! K will also influence Bd ! K . It could be expected that, in SM-like models, to reduce the longitudinal polarization RL of Bd ! K will also lower the BR of B ! K. We find that, based on the hadronic values of Table I, if we tune C09

(71) 7 0, there are no solutions for the C9

(72) 7 to satisfy the data of BRBd ! K and RL Bd ! K at the same time. Also, if we set C9

(73) 7 C09

(74) 7 or C9

(75) 7 C07

(76) 9 etc., no possible solutions are found. That is, in order to fit the current experimental data, XY XY cannot have a simple relationship for different X and Y. Hence, by taking each XY 0:08 and each XY

(77) ; and including the limits of Eq. (34) and 44% < RL Bd ! K 0 < 57%, we present the possible solutions in Fig. 4. By Fig. 4(a), we could see the correlation of BR between K and K . From Figs. 4(b) and 4(c), we see clearly how the changes of RL? are associated with the BR of K . We also present the correlation of RL and R? in Fig. 4(d). According to these results, it could be concluded that the Z0 model which provides the left- and right-. 075003-8.

(78) NONUNIVERSAL Z0 COUPLINGS IN B DECAYS. 0.3 0.25. 0.35 0.3 0.25. 7.6. 8. 0.48. 8.4 8.8 9.2 9.6 ∗ −6 BR(Bd→Κ φ)10. 0.51. handed couplings could solve the anomalies of small RL B ! K . In addition, the Z0 model also provides the room for large R? B ! K , say above 25%, in which R? of the SM is around 16%. As comparisons, we also show the results of 1:3 GeV in Fig. 5. Clearly, more solutions are allowed. We note that no solution can be found when > 1:5 GeV. Concerning the processes dictated by the decays b ! we also find that if we tune C09

(79) 7 0, or C9

(80) 7 sqq, C09

(81) 7 , or C9

(82) 7 C07

(83) 9 etc., no possible solutions for simultaneously matching the data are found. Hence, all unknowns should be regarded as independent of parameters. Following the formulas introduced in Sec. III B, the constraints of Eqs. (34) and (35), as well as the bounds of CPA, we display the results in Fig. 6. From the figure, we. 0.57. 8 7 8. 7.6. 8. ∗. R⊥(Bd→Κ φ). 0.35 0.3 0.25. (d). 0.35 0.3 0.25. 7.6. 8. 8.4 8.8 9.2 9.6 ∗ −6 BR(Bd→Κ φ)10. FIG. 5 (color online). 1:3 GeV.. 0.48. 0.51. -20. 10 10.5 11 11.5 −6 BR(Bd → π° Κ)10. (d). 5 0 -5 10.5. 11 11.5 12 12.5 ± ± −6 BR(B → π° Κ )10. ±. ACP(Bd → π Κ )%. could see that the CPA of B ! K could be as large as 4% while it vanishes in the SM. To satisfy all current experimental data, the CPA of Bd ! 0 K should be smaller than 10%. Since the CPA of K has a good accurate measurement, if one could further confirm that the magnitude of CPA for B ! K is small, say less than 4%, we could conclude that the large CPA of Bd ! 0 K could be very good evidence to display the existence of new physics, where the SM prediction is only around 3%. Furthermore, by Fig. 6(d), we also see that the CPA of B ! 0 K could be much smaller than that of Bd ! K , in which they should have similar values in the SM. Also, the results show that the CPAs of K and 0 K favor being opposite in sign but the SM predicts the same sign. As mentioned in the end of Sec. III B, when the decay amplitudes for B ! K, Bd ! K , and Bd ! 0 K decays are determined, those for B ! 0 K . 0.54 0.57 ∗ RL(Bd→Κ φ). Legend is the same as Fig. 4 but for . 10. FIG. 6 (color online). The CPAs (in units of 102 ) versus the corresponding BRs (in units of 106 ). The world averages with 2 errors are included.. 8.4 8.8 9.2 9.6 ∗ −6 BR(Bd→Κ φ)10. 0.4 (c). -15. 9.5. ±. 0.51. 8.4 8.8 9.2 9.6 ∗ −6 BR(Bd→ Κ φ)10. 0.4 ∗. 0.54. 0.48 7.6. -10. 6 (a). 3 0. ±. 9. 6. R ⊥(Bd→ Κ φ). (b). ∗. RL(Bd→ Κ φ). −6. BR(Bd→Κφ)10. (a). 10. (c). -5. 0.54 0.57 ∗ RL(Bd→Κ φ). FIG. 4 (color online). (a), (b), (c), and (d) denote the correlations between BRs of K and K modes, RL? and BRBd ! K , and R? and RL , respectively. The world averages with 2 errors are presented.. 11. 0. 17 17.5 18 18.5 ± −6 BR(Bd → π Κ )10. 16.5. 22 23 24 25 ± ± −6 BR(B → π Κ)10 ±. (d). -14. ±. ∗. 0.35. R⊥(Bd→Κ φ). 0.4 (c). -12. ACP(B → π° Κ )%. 8.4 8.8 9.2 9.6 ∗ −6 BR(Bd→Κ φ)10. (b). -10. -3 -6 24 0. (c). -10 -15. 10.5 11 11.5 12 −6 BR(Bd → π° Κ)10. FIG. 7 (color online). 1:3 GeV.. 075003-9. (b). -10 -12 -14. 18. 27 26 25 ± ± −6 BR(B → π Κ)10. -5. -20 10. -8. ±. 8. -6 21. -8. ACP(Bd → π Κ )%. 7.6. -3. ±. 8.4 8.8 9.2 9.6 ∗ −6 BR(Bd→ Κ φ)10. 0. ±. 8. (a). 3. ACP(B → π° Κ )%. 7.6. 6. ±. 0.48. 0.4 ∗. 0.51. ACP(Bd → π° Κ)%. 7. 0.54. ACP(B → π Κ)%. 8. ±. ∗. 9. 6. R ⊥(Bd→ Κ φ). (b). ACP(B → π Κ)%. 0.57. PHYSICAL REVIEW D 73, 075003 (2006). ACP(Bd → π° Κ)%. (a). 10. RL(Bd→ Κ φ). BR(Bd→Κφ)10. −6. 11. 19 20 ± −6 BR(Bd → π Κ )10. 10. (d). 5 0 -5. 11. 12 13 14 ± ± −6 BR(B → π° Κ )10. Legend is the same as Fig. 6 but for .

(84) CHUAN-HUNG CHEN AND HISAKI HATANAKA. 5. ±. 22 18 21 15 ± ± ∗ −6 BR(B → ρ Κ )10. 15. 18 ±. 21 ±. ∗. −6. (d). 0.3. ±. ±. ∗. R⊥(B →ρ Κ ). ± ±. 0.7. -5 12. 0.4 (c). 0.8. 0. BR(B →ρ Κ )10. 0.9 ∗. (b). 10. ±. ∗. ACP(B →ρ Κ )%. −6. 23. ±. ±. BR(B →π Κ)10. (a). 12. RL(B → ρ Κ ). PHYSICAL REVIEW D 73, 075003 (2006). 15. 24. 0.6 0.5 12. 15. 0.2 0.1 0 12. 18 21 ± ∗ −6 BR(B →ρ Κ )10. 15. ±. 18 ±. 21 ±. ∗. −6. BR(B →ρ Κ )10. FIG. 8. (a) correlation of BR (in units of 106 ) between K and K; (b), (c), and (d) denote the correlations between (CPA, Rk , R? ) and the BR, respectively.. decays are also fixed. Therefore, the sign difference could also be clear evidence that new physics exists. In Fig. 7, we also presented the results with 1:3 GeV. Since the data of B ! K have better accuracy, we find that no possible solution appears when the scale is smaller (larger) than 1:31:5 GeV. Finally, we discuss the contributions of the Z0 model to the decays B ! K . By the analysis in Sec. III D, we know that, except the transverse parts, the weak WCs for longitudinal polarization RL of B ! K should be the same for the decays B ! K, i.e. they have the same weak effective WCs 1;2 , as shown in Eqs. (17) and (29). This is because the final states in both processes have the same parity properties. However, the case encountered in the decays B ! K is different because the parity properties of the final state K are different. Hence, the values constrained by B ! K could directly make predictions. 40 ∗. 22. 10. 12 14 ± ∗ −6 BR(Bd→ρ Κ )10 (d). 0.4. ±. ∗. (c). ∗ ±. 10 0. 12 14 ± ± ∗ −6 BR(B →ρ Κ )10. 0.5 0.4. FIG. 9.. 20. 0.5. 0.6. 0.3. (b). 30. ±. 23. 10. RL(Bd→ ρ Κ ). ACP(Bd→ρ Κ )%. (a). R⊥(Bd→ρ Κ ). ±. BR(Bd→π Κ)10. −6. 24. 10. 12 14 ± ∗ −6 BR(Bd→ρ Κ )10. 0.3. . on B ! K . We present the results of B ! K and Bd ! K in Figs. 8 and 9, respectively. Since the observed BR of Bd ! K has reached a good accuracy, in Figs. 8(a) and 9(a) we show how the BRs are associated with BRB ! K. Moreover, we display the CPA, RL , and R? versus the corresponding BR in (b), (c), and (d) diagrams of both figures, respectively. We note that the observed BR (RL ) of B ! K by BABAR and BELLE are not consistent with each other. The former observes 17:0 2:9 2:0 0:79 0:08 0:04 [25] while the latter is 8:9 1:7 1:0 0:43 0:11 0:05 0:02 [26]. By Fig. 8, we could see clearly that (i) B ! K can have sizable CPA in which it vanishes in the SM; (ii) RL could be less than 0.60 while the corresponding BR is above 15 106 ; (iii) the solutions of small R? exist, i.e. Rk R? where the prediction of SM is Rk R? . As for the results of Bd ! K shown in Fig. 9, due to just like the case of Bd ! K in which the results with new effects are similar to the SM, we expect that the derivations from the SM are not too much. Hence, we could summarize the favorable ranges of BRs, CPAs, RL , and R? for B ! K ; Bd ! K are 17:1 3:9; 10:0 2:0 106 , 3 5; 21 7%, 0:66 0:10; 0:44 0:08, and 0:14 0:10; 0:25 0:09, respectively. V. SUMMARY We have studied the effects of the nonuniversal Z0 model on the processes dictated by the b ! sqq decays with q u, d, and s. By using the PQCD approach, we calculate the needed hadronic matrix elements. For B ! K decays, we find that their BRs and the RL of Bd ! K have provided strict constraints on the new parameters. After marching the currents data, we find the R? of Bd ! K 0 favors to be larger than 25%. For B ! K decays, by requiring that the magnitude of ACP B ! K is less than 5% and all BRs satisfy the current observations, we find that the magnitude of CPA of Bd ! 0 K should be larger than 10% but the sign is the same as the SM. Meanwhile, the CPA of B ! 0 K could be as low as a few percent which is indicated by the current experiments. Moreover, we also obtain that the CPA of B ! 0 K is opposite in sign to the SM. In sum, to satisfy current data, the new left- and righthanded couplings have to be included simultaneously. It is clear that the FC Z0 model provides the needed couplings naturally. With more physical observations and accurate data by B factories, we could further examine the effects of the nonuniversal Z0 model. ACKNOWLEDGMENTS. 0.2 0.1. 10. 12 14 ± ∗ −6 BR(Bd→ρ Κ )10. The legend is the same as Fig. 8 but for Bd ! K .. We thank Professor Chao Qiang Geng and Professor Cheng-Wei Chiang for useful discussions. This work is supported in part by the National Science Council of R. O. C. under Grant No. NSC-94-2112-M-006-009.. 075003-10.

(85) NONUNIVERSAL Z0 COUPLINGS IN B DECAYS. PHYSICAL REVIEW D 73, 075003 (2006). APPENDIX A: DISTRIBUTION AMPLITUDES AND DECAY AMPLITUDES 1. Distribution amplitudes We describe the spin structures of the meson to be i Z1 dxeixpz f5 p 6 P x 5 m0P pP x m0P 5 n 6 6n 1P g hPpjq 2 zq1 0j0i p 2Nc 0. (A1). for pseudoscalar, where p p ; 0; 0? , n 1; 0; 0? , and n 0; 1; 0? ; and 1 Z1 dxeixpz

(86) mV 6 L V x 6 L p 6 tV x mV sV x ; hVp; L jq 2 zq1 0j0i p 2Nc 0 1 Z1 mV hVp; T jq 2 zq1 0j0i p dxeixpz 6 T

(87) p 6 TV x mVo vV x . i T 5 p n aV x p n 2Nc 0 (A2) p; t;s;v;a for vector meson. The notations P and T stand for the twist-3 V denote the twist-2 wave functions while P and V wave functions of the pseudoscalar and the vector meson, respectively. Their explicit expressions could be found in Ref. [27].. 2. Hard functions for B ! P2 P1 decays In terms of the spin structures of mesons defined by Appendix A 1, we write the factorizable amplitudes for the B ! P transition form factors and the B ! PP annihilations as e 8CF MB2 F1P. Z1 0. dx1 dx3. Z1 0. b1 db1 b3 db3 B x1 ; b1 f

(88) 1 x3 P x3 rP 1 2x3 pP x3 . p 2. P x3 Ee t1 e he x1 ; x3 ; b1 ; b3 2rP P x3 Ee te he x3 ; x1 ; b3 ; b1 g;. e 16C M 2 r F2P F B P. Z1 0. dx1 dx3. Z1 0. b1 db1 b3 db3 B x1 ; b1 f

(89) P x3 rP 2 x3 pP x3 . p 2 x2 P x3 Ee t1 e he x1 ; x3 ; b1 ; b3 2rP P x3 Ee te he x3 ; x1 ; b3 ; b1 g;. a 8CF MB2 F1P 2 P1. 1 . Z1 0. dx2 dx3. x3 P2 1. Z1 0. (A3). (A4). b2 db2 b3 db3 f

(90) x3 P1 x2 P2 1 x3 2rP1 rP2 pP1 x2 1 x3 pP2 1 x3 . x3 Ea t1 a ha x2 ; x3 ; b2 ; b3

(91) x2 P1 x2 P2 1 x3 . 2rP1 rP2 pP2 1 x3 1 x2 pP2 x2 1 x2 P1 x2 Ea t2 a ha x3 ; x2 ; b3 ; b2 g; a 16CF MB2 F2P 2 P1. Z1 0. dx2 dx3. Z1 0. (A5). b2 db2 b3 db3 f

(92) rP2 x3 P1 x2 pP2 1 x3 P2 1 x3 . p . 2rP1 pP1 x2 P2 1 x3 Ea t1 a ha x2 ; x3 ; b2 ; b3

(93) x2 rP1 P1 x2 P1 x2 P2 1 x3 . 2rP2 P1 x2 pP2 1 x3 Ea t2 a ha x3 ; x2 ; b3 ; b2 g; with mPi =mB , where the hard functions hea are given by p p p he x1 ; x3 ; b1 ; b3 K0 x1 x3 mB b1 St x3

(94) b1 b3 K0 x3 mB b1 I0 x3 mB b3 p p. b3 b1 K0 x3 mB b3 I0 x3 mB b1 ; 2 i p p p H01 x2 x3 mB b2 St x3

(95) b2 b3 H01 x3 mB b2 J0 x3 mB b3 ha x2 ; x3 ; b2 ; b3 2 p p. b3 b2 H01 x3 mB b3 J0 x3 mB b2 :. 075003-11. (A6). (A7). (A8).

(96) CHUAN-HUNG CHEN AND HISAKI HATANAKA. PHYSICAL REVIEW D 73, 075003 (2006). The evolution factors Eea are defined as Ee t s tSB tSP t;. Ea t s tSP1 tSP2 t;. (A9). where SM t denote the Sudakov factor of the M-meson; the explicit expressions could be found in Ref. [22] and the references therein. 3. Hard functions for B ! PV decays Similarly, the factorizable amplitudes for B ! PV modes are given to be a F1PV 8CF MB2. Z1. dx2 dx3. 0. Z1. b2 db2 b3 db3 f

(97) x3 V x2 P 1 x3 2rP rV sV x2 1 x3 pP 1 x3 . 0. s. 1 x3 P 1 x3 Ea t1 a ha x2 ; x3 ; b2 ; b3

(98) x2 V x2 P 1 x3 2rP rV 1 x2 V x2 . 1 x2 tV x2 pP 1 x3 Ea ta2 ha x3 ; x2 ; b3 ; b2 g; a 16CF MB2 F2PV. Z1 0. dx2 dx3. Z1. (A10). b2 db2 b3 db3 f

(99) rP x3 V x2 pP 1 x3 P 1 x3 2rV sV x2 P 1 x3 . 0. p s t Ea t1 a ha x2 ; x3 ; b2 ; b3

(100) 2rP V x2 P 1 x3 x2 rV V x2 V x2 P 1 x3 . Ea t2 a ha x3 ; x2 ; b3 ; b2 g;. (A11). with rV mV =mB . 4. Hard functions for B ! V2 V1 decays The needed factorizable amplitudes for VV modes are given by FVe 2 L 8CF MB2. Z1 0. dx1 dx3. Z1 0. b1 db1 b3 db3 B x1 ; b1 f

(101) 1 x3 V2 x3 rV2 1 2x3 tV2 x3 . s e 2. sV2 x3 Ee t1 e he x1 ; x3 ; b1 ; b3 2rV2 V2 x3 E te he x3 ; x1 ; b3 ; b1 g;. Z1. FVe 2 N 8CF MB2. 0. Z1. dx1 dx3. 0. b1 db1 b3 db3 B x1 ; b1 rV1 f

(102) TV2 x3 2rV2 vV2 x3 rV2 x3 vV2 x3 . v a e 2 aV2 x3 Ee t1 e he x1 ; x3 ; b1 ; b3 rV2

(103) V2 x3 V2 x3 E te he x3 ; x1 ; b3 ; b1 g;. FVe 2 T 16CF MB2. Z1 0. dx1 dx3. Z1 0. Z1 0. dx2 dx3. Z1 0. (A13). b1 db1 b3 db3 B x1 ; b1 rV1 f

(104) TV2 x3 2rV2 aV2 x3 rV2 x3 vV2 x3 . v a e 2 aV2 x3 Ee t1 e he x1 ; x3 ; b1 ; b3 rV2

(105) V2 x3 V2 x3 E te he x3 ; x1 ; b3 ; b1 g;. a 8CF MB2 F1V 2 V1 L. (A12). (A14). b2 db2 b3 db3 f

(106) x3 V1 x2 V2 1 x3 2rV1 rV2 sV1 x2 1 x3 tV2 1 x3 . 1 x3 sV2 1 x3 Ea t1 a ha x2 ; x3 ; b2 ; b3

(107) x2 V1 x2 V2 1 x3 . 2rV1 rV2 sV2 1 x3 1 x2 tV1 x2 1 x2 sV1 x2 Ea t2 a ha x3 ; x2 ; b3 ; b2 g; a 8CF MB2 F1V 2 V1 N. Z1 0. dx2 dx3. Z1 0. (A15). b2 db2 b3 db3 rV1 rV2 f

(108) 1 x3 vV1 x2 vV2 1 x3 aV1 x2 aV2 1 x3 . 1 x3 vV1 x2 aV2 1 x3 aV1 x2 vV2 1 x3 Ea t1 a ha x2 ; x3 ; b2 ; b3

(109) 1 x2 vV1 x2 vV2 1 x3 aV1 x2 aV2 1 x3 1 x2 vV1 x2 aV2 1 x3 . aV1 x2 vV2 1 x3 Ea t2 a ha x3 ; x2 ; b3 ; b2 g;. 075003-12. (A16).

(110) NONUNIVERSAL Z0 COUPLINGS IN B DECAYS a F1V 16CF MB2 2 V1 T. Z1. dx2 dx3. 0. Z1 0. PHYSICAL REVIEW D 73, 075003 (2006). b2 db2 b3 db3 rV1 rV2 f

(111) 1 x3 vV1 x2 vV2 1 x3 aV1 x2 aV2 1 x3 . 1 x3 vV1 x2 aV2 1 x3 aV1 x2 vV2 1 x3 Ea t1 a ha x2 ; x3 ; b2 ; b3 .

(112) 1 x2 vV1 x2 vV2 1 x3 aV1 x2 aV2 1 x3 1 x2 vV1 x2 aV2 1 x3 . aV1 x2 vV2 1 x3 Ea t2 a ha x3 ; x2 ; b3 ; b2 g; a F2V 16CF MB2 2 V1 L. Z1 0. dx2 dx3. Z1 0. (A17). b2 db2 b3 db3 f

(113) rV2 x3 V1 x2 tV2 1 x3 sV2 1 x3 . s t 2rV1 sV1 x2 V2 1 x3 Ea t1 a ha x2 ; x3 ; b2 ; b3

(114) rV1 x2 V1 x2 V1 x2 V2 1 x3 . 2rV2 V1 x2 sV2 1 x3 Ea t2 a ha x3 ; x2 ; b3 ; b2 g; a F2V 16CF MB2 2 V1 N. Z1 0. dx2 dx3. rV2 TV1 x2 vV2 1. a F2V 32CF MB2 2 V1 T. Z1 0. Z1 0. b2 db2 b3 db3 frV1 vV1 x2 aV1 x2 TV2 1 x3 Ea ta1 ha x2 ; x3 ; b2 ; b3 . x3 aV1 1 x3 Ea t1 a ha x3 ; x2 ; b3 ; b2 g;. dx2 dx3. Z1 0. (A18). (A19). b2 db2 b3 db3 frV1 vV1 x2 aV1 x2 TV2 1 x3 Ea ta1 ha x2 ; x3 ; b2 ; b3 . rV2 TV1 x2 vV2 1 x3 aV2 1 x3 Ea t2 a ha x3 ; x2 ; b3 ; b2 g:. (A20). We define rVi mVi =mB .. [1] S. Eidelman et al. (Particle Data Group), Phys. Lett. B 592, 1 (2004). [2] E. Kou and A. I. Sanda, Phys. Lett. B 525, 240 (2002); C. W. Chiang et al., Phys. Rev. D 68, 074012 (2003). [3] K. F. Chen et al. (BELLE Collaboration), Phys. Rev. Lett. 94, 221804 (2005); B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett. 91, 171802 (2003); B. Aubert et al., Phys. Rev. Lett. 93, 231804 (2004); A. Gritsan, hep-ex/0409059. [4] A. Kagan, Phys. Lett. B 601, 151 (2004); W. S. Hou and M. Nagashima, hep-ph/0408007; P. Colangelo, F. De Fazio, and T. N. Pham, Phys. Lett. B 597, 291 (2004); M. Ladisa et al., Phys. Rev. D 70, 114025 (2004); H. Y. Cheng, C. K. Chua, and A. Soni, Phys. Rev. D 71, 014030 (2005); H. N. Li, Phys. Lett. B 622, 63 (2005). [5] A. Kagan, hep-ph/0407076; E. Alvarez et al., Phys. Rev. D 70, 115014 (2004); Y. D. Yang, R. M. Wang, and G. R. Lu, Phys. Rev. D 72, 015009 (2005); A. K. Giri and R. Mohanta, hep-ph/0412107; P. K. Das and K. C. Yang, Phys. Rev. D 71, 094002 (2005); C. S. Kim and Y. D. Yang, hep-ph/0412364; C. S. Hung et al., hep-ph/ 0511129; S. Nandi and A. Kundu, hep-ph/0510245; S. Baek et al., Phys. Rev. D 72, 094008 (2005). [6] C. H. Chen and C. Q. Geng, Phys. Rev. D 71, 115004 (2005). [7] S. Chen et al. (CLEO Collaboration), Phys. Rev. Lett. 85, 525 (2000); A. Bornheim et al., Phys. Rev. D 68, 052002 (2003).. [8] Y. Chao et al. (BELLE Collaboration), Phys. Rev. D 69, 111102 (2004); Y. Chao and P. Chang, Phys. Rev. D 71, 031502 (2005); Y. Chao et al., Phys. Rev. Lett. 93, 191802 (2004); K. Abe et al., hep-ex/0409049. [9] B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett. 89, 281802 (2002); hep-ex/0408062; hep-ex/0408080; hep-ex/0408081; B. Aubert et al., Phys. Rev. Lett. 93, 131801 (2004). [10] H. N. Li et al., Phys. Rev. D 72, 114005 (2005); X. Q. Li and Y. D. Yang, Phys. Rev. D 72, 074007 (2005); R. Arnowitt et al., Phys. Lett. B 633, 748 (2006); D. Chang et al., hep-ph/0510328. [11] C. S. Kim, S. Oh, and C. Yu, Phys. Rev. D 72, 074005 (2005). [12] S. Chaudhuri et al., Nucl. Phys. B456, 89 (1995); G. Cleaver et al., Nucl. Phys. B525, 3 (1998); G. Cleaver et al., Phys. Rev. D 59, 055005 (1999); 59, 115003 (1999). [13] D. A. Demir, G. L. Kane, and Ting T. Wang, Phys. Rev. D 72, 015012 (2005). [14] P. Langacker and M. Plu¨macher, Phys. Rev. D 62, 013006 (2000). [15] V. Barger et al., Phys. Lett. B 580, 186 (2004). [16] V. Barger et al., Phys. Lett. B 598, 218 (2004). [17] G. P. Lepage and S. J. Brodsky, Phys. Lett. 87B, 359 (1979); Phys. Rev. D 22, 2157 (1980). [18] H. N. Li and G. Sterman, Nucl. Phys. B381, 129 (1992); G. Sterman, Phys. Lett. B 179, 281 (1986); Nucl. Phys. B281, 310 (1987); S. Catani and L. Trentadue, Nucl. Phys.. 075003-13.

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