Pseudoscalar and Scalar Operators of Higgs-Penguins in the MSSM and B to phi K^*, K eta(') Decays
aa r X i v : . [ h e p - ph ] D ec arXiv:0711.3086 [hep-ph] Pseudoscalar and Scalar Operators of Higgs-Penguins in theMSSM and B → φK ∗ , K η ( ′ ) Decays
Hisaki Hatanaka and Kwei-Chou Yang
Department of Physics, Chung-Yuan Christian University, Chungli, Taiwan 320, R.O.C. (Dated: December 28, 2007)
Abstract
We study the effect of b → s ¯ ss scalar/pseudoscalar operators in B → Kη ( ′ ) , φK ∗ decays. Inthe minimal supersymmetric standard model (MSSM), such scalar/pseudoscalar operators canbe induced by the penguin diagrams of neutral Higgs bosons. These operators can be Fierz-transformed into tensor operators, and the resultant tensor operators could affect the transversepolarization amplitudes in B → φK ∗ decays. A combined analysis of the decays B → φK ∗ and B → Kη ( ′ ) , including b → s ¯ ss scalar/pseudoscalar operators and their Fierz-transformed tensoroperators originated from the MSSM, is performed. Our study is based on the followings: (1)Assuming that weak annihilations in B → φK ∗ is negligible and the polarization puzzle is resolvedby Fierz-transformed tensor operators, it results in too large coefficients of scalar/pseudoscalaroperators, such that the resulting B → Kη ( ′ ) branching fractions are much larger than observations.(2) When we take the weak annihilations in B → φK ∗ into account, the polarization puzzle canbe resolved. In this case, new physics effects are strongly suppressed and no more relevant to theenhancement of the transverse modes in B → φK ∗ decays. . INTRODUCTION Recent experimental results for polarization fractions in ¯ B , − → φ (1020) ¯ K ∗ (892) , − are f L ( ¯ B → φ ¯ K ∗ ) = . ± . ± .
015 BaBar [1]0 . ± . ± .
02 Belle [2]0 . ± . ± .
05 CDF [3] ,f ⊥ ( ¯ B → φ ¯ K ∗ ) = . ± . ± .
013 BaBar [1]0 . +0 . − . ± .
02 Belle [2]0 . ± . ± .
05 CDF [3] ,f L ( B − → φK ∗− ) = . ± . ± .
03 BaBar [4]0 . ± . ± .
03 Belle [2] ,f ⊥ ( B − → φK ∗− ) = . ± . ± .
02 BaBar [4]0 . ± . ± .
02 Belle [2] . (1)Here, the polarization fractions f λ ( λ = L, k , ⊥ ) are given by f λ = | A λ | / P σ = L, k , ⊥ | A σ | , withpolarization amplitudes A L ≡ A , A k and A ⊥ being longitudinal, parallel and perpendicularmodes in the transversity basis, respectively. Experimental results show that f L ∼ . f ⊥ ∼ f k . On the other hand, the power-counting estimate in the standard model (SM) tellsthat the longitudinal mode is dominant [5]. In the SM, the QCD factorization (QCDF)calculation yields [5] f L : f k : f ⊥ = 1 − O (1 /m b ) : O (1 /m b ) : O (1 /m b ). The experimentalresults largely deviate from the intuition in the SM. Similar discrepancies have been observedin penguin-dominated B ± , → ρ ± , K ∗ decays [6, 7]. These discrepancies are referred as thepolarization puzzle/anomaly in B → V V (where V denotes a vector meson) decays.Solutions to the puzzle have been discussed within or beyond the standard model [5, 8, 9].The recipe of fine-tuning form factors is proposed in [10]. Effects of final-state interactionsare discussed in [11, 12]. Sizable annihilation effects are considered in [13–15]. As discussedin [14], the magnitude of annihilation correction is of O (cid:2) /m b log m b / Λ h (cid:3) . Furthermore,the effect is destructive to longitudinal, and constructive to transverse modes. Thus we mayresolve the polarization puzzles by introducing annihilation effects. We note that, however,the perturbative QCD (pQCD) yields f L & .
75 even with annihilation effects [8].The b → s g (where g denotes a gluon) operator, which enhances the transverse compo-2ents, was discussed in [16]. However, it was found [13, 17] that the contribution due to theoperator mainly affects the longitudinal mode.As for the solutions of the puzzle, the effects of NP-induced tensor operators are discussedin [17] and right-handed currents ¯ sγ µ (1 + γ ) b ¯ qγ µ (1 ± γ ) q are in [18–20]. Because theright-handed currents may decrease the magnitude of | A | and increase | A ⊥ | , it can explainthe ratio | A ⊥ /A | . However, the resulting | A k | ≪ | A ⊥ | [18] is in contrast with the data | A k | ∼ | A ⊥ | .New physics (NP) contributions to B → φK ∗ decays due to b → s ¯ ss tensor operators,first mentioned in [13], are systematically discussed in [17], and later the idea is applied to B → ρK ∗ by considering the 4-quark tensor operators related to the processes b → s ¯ dd and s ¯ uu [21]. In the helicity basis, four-quark tensor operators have leading effects to H −− (or H ++ ), but sub-leading to H . The possibility of solving B → φK ∗ polarization puzzle byusing four-quark tensor operators is extensively studied in [14, 17] and further investigatedin [22–26].In [17] the general approach of resolving the polarization anomaly of B → φK ∗ by usingfour-quark NP operators is studied. There are two types of NP operators which are relevantto solve the polarization anomaly. They are tensor operators with σ µν (1 ± γ ) ⊗ σ µν (1 ± γ )structure. The tensor operator σ µν (1 + γ ) ⊗ σ µν (1 + γ ) results in H : H −− : H ++ = O (1 /m b ) : O (1) : O (1 /m b ), while σ µν (1 − γ ) ⊗ σ µν (1 − γ ) leads to H : H −− : H ++ = O (1 /m b ) : O (1 /m b ) : O (1).The decays B → P P (where P denotes a pseudoscalar meson) are sensitive toscalar/pseudoscalar 4-quark operators whereas B → V V are sensitive to tensor operators.Furthermore, it is known that scalar/pseudoscalar and tensor operators are not independent;scalar/pseudoscalar operators can be Fierz-transformed into tensor operators and vice versa.Therefore, the combined analysis of scalar/pseudoscalar and tensor operators for B → P P and B → V V modes will give more severe constraints about NP scalar/pseudoscalar andtensor operators.In this paper we focus on the b → s ¯ ss decay processes. We consider thescalar/pseudoscalar operators induced by Higgs penguin diagrams of the MSSM neu- Amplitudes in the helicity basis and the transversity basis are related by A = H , A k = ( H ++ + H −− ) / √ A ⊥ = − ( H ++ − H −− ) / √ B → φK ∗ decays, whereas originalscalar/pseudoscalar operators can affect B → Kη ( ′ ) decays. We study the consistency ofboth modes to see the validity of the scenario. One should note that this NP effect is furthersuppressed by m q /m s in the b → s ¯ qq channel (with q ≡ u or d ) as compared with b → s ¯ ss .Although the recent observations of the sizable transverse fraction in B ± , → ρ ± , K ∗ decaysmay hint at large annihilation effects, the present study for decays B → φK ∗ and B → Kη ( ′ ) can offer more severe constraints on the NP.The organization of the present article is as follows: In Sec. II, we summarize the for-mulation of MSSM-NHB scalar/pseudoscalar operators and its contributions to B → Kη ( ′ ) and B → φK ∗ decays. In Sec. III, we numerically analyze the decays for B → Kη ( ′ ) and B → φK ∗ . Sec. IV is devoted to summary and discussions. II. FORMULATIONA. SM and NP operators
In the SM the effective Hamiltonian relevant to B → Kη ( ′ ) and B → φK ∗ decays is givenby H SMeff = G F √ " X q = u,c V qb V ∗ qs ( c ( µ ) O q ( µ ) + c ( µ ) O q ( µ )) − V tb V ∗ ts X i =3 c i ( µ ) O i ( µ ) + c γ ( µ ) O γ ( µ ) + c g ( µ ) O g ( µ ) + h.c. , (2)where the operators O i =1 ,..., are four-quark operators. O γ and O g are electromagneticand chromomagnetic dipole operators, respectively. µ is the renormalization scale. V qb and V qs ( q = u, c, t ) are elements of Cabibbo-Kobayashi-Maskawa (CKM) matrix. The b → s ¯ ss four-quark NP effective Hamiltonian is given by H NPeff = − G F √ V tb V ∗ ts ) X i =11 c i ( µ ) O i ( µ ) + h.c. , (3)4here O i and c i ( i = 11 , . . . ,
26) are four-quark NP operators introduced in [17], and corre-sponding Wilson coefficients , respectively. Explicit forms of O i ( i = 11 , . . . ,
26) are shownin the following:(i) right-handed current operators O = ¯ s α γ µ (1 + γ ) b α ¯ s β γ µ (1 + γ ) s β , O = ¯ s α γ µ (1 + γ ) b β ¯ s β γ µ (1 + γ ) s α ,O = ¯ s α γ µ (1 + γ ) b α ¯ s β γ µ (1 − γ ) s β , O = ¯ s α γ µ (1 + γ ) b β ¯ s β γ µ (1 − γ ) s α , (4)(ii) scalar/pseudoscalar operators O = ¯ s α (1 + γ ) b α ¯ s β (1 + γ ) s β , O = ¯ s α (1 + γ ) b β ¯ s β (1 + γ ) s α ,O = ¯ s α (1 − γ ) b α ¯ s β (1 − γ ) s β , O = ¯ s α (1 − γ ) b β ¯ s β (1 − γ ) s α ,O = ¯ s α (1 + γ ) b α ¯ s β (1 − γ ) s β , O = ¯ s α (1 + γ ) b β ¯ s β (1 − γ ) s α ,O = ¯ s α (1 − γ ) b α ¯ s β (1 + γ ) s β , O = ¯ s α (1 − γ ) b β ¯ s β (1 + γ ) s α , (5)(iii) tensor/axial-tensor operators O = ¯ s α σ µν (1 + γ ) b α ¯ s β σ µν (1 + γ ) s β , O = ¯ s α σ µν (1 + γ ) b β ¯ s β σ µν (1 + γ ) s α ,O = ¯ s α σ µν (1 − γ ) b α ¯ s β σ µν (1 − γ ) s β , O = ¯ s α σ µν (1 − γ ) b β ¯ s β σ µν (1 − γ ) s α . (6)Since B → P P ( B → V V ) decays are not sensitive to the factorized tensor(scalar/pseudoscalar) b → s ¯ ss operators, it is a good approximation to use O i with i = 1 , . . . ,
22 ( i = 1 , . . . , , , . . . ,
26) for B → P P ( B → V V ). Some of these NP operatorsare not independent, and can be related with each other by the Fierz transformation: O = − O , O = − O , O = − O , O = − O ,O = − O − O , O = − O − O ,O = − O − O , O = − O − O . (7)Due to the Fierz transformation, we can introduce modified Wilson coefficients ¯ c i , which aredefined by ¯ c i = c i − c j with ( i, j ) = (5 , , (6 , , (13 , , (14 , , (8) In [17], CKM factors and Wilson coefficients are not separated. σ µν (1 ± γ ) σ µν (1 ∓ γ ) type operators O i ( i = 27 , . . . ,
30) in [17] are found to vanish. ¯ c ¯ c ¯ c ¯ c = M c c c c , ¯ c ¯ c ¯ c ¯ c = M c c c c with M = − −
80 1 − − − −
16 112 . (9)Thus we can replace the Wilson coefficients by the effective ones: { c i , ¯ c j } i = 1 , . . . , , , . . . , , , . . . , , j = 5 , , , , (10)for B → P P decays, and { c i , ¯ c j } i = 1 , . . . , , , . . . , , j = 5 , , , , , . . . , , (11)for B → V V decays, so that the decay amplitudes can be simplified. B. B → Kη ( ′ ) Decay Amplitudes
In the SM, B → Kη ( ′ ) decay amplitudes are given by [31] A ( ¯ B → ¯ Kη ( ′ ) ) = X p = u,c V pb V ∗ ps T p ¯ Kη ( ′ ) , (12)where √ T pB → K − η ( ′ ) = A ¯ Kη ( ′ ) q (cid:2) δ pu α + 2 α p + α p ,EW + 2 β pS (cid:3) + √ A ¯ Kη ( ′ ) s (cid:2) δ pu β + α p + α p − α p ,EW − α p ,EW + β p + β p ,EW + β pS (cid:3) + √ A ¯ Kη ( ′ ) c [ δ pc α + α p ]+ A η ( ′ ) q ¯ K (cid:2) δ pu ( α + β ) + α p + α p ,EW + β p + β p ,EW (cid:3) , (13) √ T p ¯ B → ¯ K η ( ′ ) = A ¯ Kη ( ′ ) q (cid:2) δ pu α + 2 α p + α p ,EW + 2 β pS (cid:3) + √ A ¯ Kη ( ′ ) s (cid:2) α p + α p − α p ,EW − α p ,EW + β p − β p ,EW + β pS (cid:3) + √ A ¯ Kη ( ′ ) c [ δ pc α + α p ]+ A η ( ′ ) q ¯ K (cid:2) α p − α p ,EW + β p − β p ,EW (cid:3) . (14)For the B → P P decays α , , , , EW, EW are defined as α , = a , , α p = a p − a p , α p = a p + r M χ a p ,α p ,EW = a p − a p , α p ,EW = a p + r M χ a p , (15)6ith r Kχ = 2 m K /m b ( m q + m s ) and r η ( ′ ) s χ ≡ h sη ( ′ ) / ( f sη ( ′ ) m b m s ). A M M are given by A M M = i G F √ · m B F B → M (0) f M . (16)Contributions from annihilation diagram are represented by β Q ( Q = 3 , , EW, EW, S β Q ≡ b Q · B M M /A M M with B Kη ( ′ ) r = i G F √ f B f K f rη ( ′ ) , B η ( ′ ) q K = i G F √ f B f K f qη ( ′ ) , (17)(where r = q or s ) respectively. b , , EW, EW are the coefficients due to weak annihilation ofpenguin operators. b S is originated from the singlet penguin contribution which is intro-duced in [31, 32]. Note that following the approximation adopted in [31], we have neglectedsingle weak annihilations β S , β S , β S ,EW , and only keep β S .In the above results, we adopt a pi =1 ,..., and b pQ , given by the QCD factorization (QCDF)calculation in [31]. The NP effects due to scalar/pseudoscalar operators can be includedby replacing c and c with the effective ones c eff5(6) ≡ c + ∆ c in the ¯ B → ¯ Kη s decayamplitudes in the following way:∆ c = ( − ¯ c + ¯ c + c − c ) , ∆ c = ( − ¯ c + ¯ c + c − c ) , for α p , β p , (18)and ∆ c = ( c − c ) , ∆ c = ( c − c ) , for α p , β p , β S . (19)Here ¯ c , ¯ c , ¯ c , and ¯ c are defined in (9). η and η ′ mesons states can be regarded asmixed states of | η q i ≡ √ ( | ¯ uu i + | ¯ dd i ) and | η s i ≡ | ¯ ss i with a mixing angle φ η [32]: | η i| η ′ i = cos φ η − sin φ η sin φ η cos φ η | η q i| η s i . (20)Decay constants f q,sη ( ′ ) , pseudoscalar densities h q,sη ( ′ ) are defined by h η ( ′ ) ( p ) | ¯ qγ µ γ q | i = − i √ f qη ( ′ ) p µ , h η ( ′ ) ( p ) | ¯ sγ µ γ s | i = − if sη ( ′ ) p µ , m q h η ( ′ ) ( p ) | ¯ qγ q | i = − i √ h qη ( ′ ) , m s h η ( ′ ) ( p ) | ¯ qγ q | i = − ih sη ( ′ ) , (21)with m q = ( m u + m d ) /
2. As for explicit forms of f q,sη ( ′ ) , h q,sη ( ′ ) and form factors F B → η ( ′ ) ( q ), wesummarize in Appendix A. 7 . B → φK ∗ Decay Amplitudes
Decay amplitudes of ¯ B → φ ¯ K ∗ can be decomposed as A ( ¯ B → φ ¯ K ∗ ) = X h =0 , ± H hh , (22)where H hh = X p = u,c V pb V ∗ ps (cid:16) T p,hφK ∗ ,A + T p,hφK ∗ ,B (cid:17) , ( h = 0 , ± ) , (23)is the amplitudes in the helicity basis. Amplitudes for the emission topology (the T A part)is given by X p = u,c V pb V ∗ ps T p,hφK ∗ ,A = ( − V tb V ∗ ts ) n A h ( ¯ B ¯ K ∗ ,φ ) − (cid:2) a h + a h + a h − r φχ a h − (cid:0) a h − r φχ a h + a h + a h (cid:1)(cid:3) + A h ( ¯ B ¯ K ∗ ,φ )+ (cid:2) a h + a h + a h − r φχ a h (cid:3) + A h ( ¯ B ¯ K ∗ ,φ ) T + (cid:2) a h + a h (cid:3) + A h ( ¯ B ¯ K ∗ ,φ ) T − (cid:2) a h + a h (cid:3)o , (27)with r φχ given by r φχ = 2 m φ m b ( µ ) f Tφ ( µ ) f φ . (28) The coefficient “1/2” in front of a h , can be realized as follows. Take O as an example. Because, underthe Fierz transform, O can be written by O = (1 / s α σ µν (1 + γ ) b β ¯ s β σ µν (1 + γ ) s α − s α (1 + γ ) b β ¯ s β (1 + γ ) s α , (24)therefore, in the factorization limit, we obtain h φ ¯ K ∗ | O | ¯ B i = h φ | ¯ sσ µν (1 + γ ) s | ih ¯ K ∗ | ¯ sσ µν (1 + γ ) b | ¯ B i + h φ | ¯ s β σ µν (1 + γ ) s α | ih ¯ K ∗ | ¯ s α σ µν (1 + γ ) b β | ¯ B i = N c ! h φ | ¯ sσ µν (1 + γ ) s | ih ¯ K ∗ | ¯ sσ µν (1 + γ ) b | ¯ B i . (25)Note that the second term in the right hand side of (24) gives no contribution since the local scalar currentcannot couple to φ . Similarly, in the factorization limit we have h φ ¯ K ∗ | O | ¯ B i = N c + 12 ! h φ | ¯ sσ µν (1 + γ ) s | ih ¯ K ∗ | ¯ sσ µν (1 + γ ) b | ¯ B i . (26)The same procedure can be applied to the matrix elements containing O and O . a hi ( i = 3 , . . . ,
10) have been calculated in QCDF [15, 31]. However instead of c and c , ¯ c and ¯ c should be used in the calculation of a and a (see (11)). a hi = (cid:18) ¯ c i + ¯ c i ± N c (cid:19) + O ( α s ) , with i = 23 , , , , (29)where the radiative corrections are negligible. We summarized the explicit form of a hi for i = 11 , . . . ,
14 due to the right-handed four-quark operators in Appendix C.In (27), coefficients A h ( BV ,V ) ± and A h ( BV ,V ) ,T ± are given by A h ( ¯ B ¯ K ∗ ,φ ) ∓ ≡ G F √ h φ ( q, ε ( h )) | ¯ sγ µ (1 − γ ) s | ih ¯ K ∗ ( p ′ , ε ( h )) | ¯ sγ µ (1 ∓ γ ) b | ¯ B ( p ) i = G F √ { if φ m φ (cid:20) − im B + m K ∗ ǫ µναβ ε µ ∗ ε ν ∗ p α p ′ β V ( q ) (cid:21) ∓ if φ m φ (cid:20) ( m B + m K ∗ )( ε ∗ · ε ∗ ) A ( q ) − ( ε ∗ · p )( ε ∗ · p ) 2 A ( q ) m B + m K ∗ (cid:21) } , (30) A h ( ¯ B ¯ K ∗ ,φ ) ,T ± ≡ G F √ h φ ( q, ε ( h )) | ¯ sσ µν s (1 ± γ ) | ih ¯ K ∗ ( p ′ , ε ( h )) | ¯ sσ µν (1 ± γ ) b | ¯ B ( p ) i = G F √ f Tφ { ǫ µνρσ ε µ ∗ ε ν p ρ p ′ σ T ( q ) ∓ iT ( q ) (cid:2) ( ε ∗ · ε ∗ )( m B − m K ∗ ) − ε ∗ · p )( ε ∗ · p ) (cid:3) ± iT ( q )( ε ∗ · p )( ε ∗ · p ) m φ m B − m K ∗ } , (31)or in the explicit forms A
0( ¯ B ¯ K ∗ ,φ ) ∓ = ∓ G F √ if φ m φ )( m B + m K ∗ )[ aA ( m φ ) − bA ( m φ )] ,A ± ( ¯ B ¯ K ∗ ,φ ) − = G F √ if φ m φ ) (cid:20) ( m B + m K ∗ ) A ( m φ ) ∓ m B p c m B + m K ∗ V ( m φ ) (cid:21) ,A ± ( ¯ B ¯ K ∗ ,φ )+ = G F √ if φ m φ ) (cid:20) − ( m B + m K ∗ ) A ( m φ ) ∓ m B p c m B + m K ∗ V ( m φ ) (cid:21) ,A
0( ¯ B ¯ K ∗ ,φ ) T ± = ∓ G F √ if Tφ ) m B [ h T ( m φ ) − h T ( m φ )] ,A ± ( ¯ B ¯ K ∗ ,φ ) T + = − G F √ if Tφ ) m B [ ± f T ( m φ ) − f T ( m φ )] ,A ± ( ¯ B ¯ K ∗ ,φ ) T − = − G F √ if Tφ ) m B [ ± f T ( m φ ) + f T ( m φ )] , (32)with a = ( m B − m φ − m K ∗ ) / (2 m φ m K ∗ ), b = (2 m B p c ) / [ m φ m K ∗ ( m B + m K ∗ )] and f = 2 p c /m B , f = ( m B − m K ∗ ) /m B ,h = 12 m K ∗ m φ (cid:20) ( m B − m φ − m K ∗ )( m B − m K ∗ ) m B − p c (cid:21) ,h = 12 m K ∗ m φ (cid:18) p c m φ m B − m K ∗ (cid:19) . (33)9ere we have used decay constants and form factors defined in Appendix B.Weak annihilation contributions (the T B -part) to ¯ B → φ ¯ K ∗ and B − → φK ∗− decayamplitudes in helicity basis can be given by X p = u,c V pb V ∗ ps T p,hφK ∗ ,B = B φK ∗ ( − V tb V ∗ ts ) (cid:2) b h − b h EW + b h (cid:3) , X p = u,c V pb V ∗ ps T p,hφK ∗− ,B = B φK ∗ (cid:8) ( − V tb V ∗ ts ) (cid:2) b h + b h EW + b h (cid:3) + V ub V ∗ us · b h (cid:9) , (34)where B φK ∗ ≡ i G F √ f B f K ∗ f φ , (35)and b h = C F N c h c A i,h + ¯ c ( A i,h + A f,h ) + N c ¯ c A f,h i ,b h EW = C F N c h c A i,h + c ( A i,h + A f,h ) + N c c A f,h i ,b h = C F N c c A i,h ,b h = − C F N c h c A i,h + ¯ c ( A i,h + A f,h ) + N c ¯ c A f,h i , (36)with h = 0 , − , +. Building blocks A i ( f ) ,h , can be found in Appendix of [15]. D. Scalar and Pseudoscalar operators in the MSSM
In the MSSM, scalar and pseudoscalar operators can be induced by neutral-Higgs boson(NHB) penguin diagrams. We refer such operator as the MSSM-NHB scalar/pseudoscalaroperators. In [28], b → s ¯ ℓℓ (where ℓ denotes a charged lepton) scalar/pseudoscalar op-erators Q ( ′ )1 , induced by the MSSM-NHB penguin diagrams are considered. b → s ¯ qq scalar/pseudoscalar operators can be obtained by replacing Higgs-¯ ℓ - ℓ vertex with Higgs-¯ q - q vertex [29]. They are O = ¯ s (1 + γ ) b X q m q m b ¯ q (1 + γ ) q, O = ¯ s i (1 + γ ) b j X q m q m b ¯ q j (1 + γ ) q i , O = ¯ s (1 − γ ) b X q m q m b ¯ q (1 − γ ) q, O = ¯ s i (1 − γ ) b j X q m q m b ¯ q j (1 − γ ) q i , O = ¯ s (1 + γ ) b X q m q m b ¯ q (1 − γ ) q, O = ¯ s i (1 + γ ) b j X q m q m b ¯ q j (1 − γ ) q i , O = ¯ s (1 − γ ) b X q m q m b ¯ q (1 + γ ) q, O = ¯ s i (1 − γ ) b j X q m q m b ¯ q j (1 + γ ) q i , (37)10here q = u, d, s, c . The Wilson coefficients C i ( µ ) of O i with i = 15 , . . . ,
22 at µ = m W aregiven by [28, 29]: C ( m W ) = e π ( C S + C P ) , C ( m W ) = e π ( C ′ S − C ′ P ) , C ( m W ) = e π ( C S − C P ) , C ( m W ) = e π ( C ′ S + C ′ P ) , (38) C i = 0 ( i = 16 , , , , and four-quark tensor operators are not directly induced. Here C ( ′ ) S = 43 λ t g s g sin θ W m b m H cos α + r s sin α cos β m ˜ g m b f ′ b ( x ) δ dLL ( RR )23 δ dLR ( LR ∗ )33 ,C ( ′ ) P = ∓ λ t g s g sin θ W m b m A ( r p + tan β ) m ˜ g m b f ′ b ( x ) δ dLL ( RR )23 δ dLR ( LR ∗ )33 , (39)with r s = m H /m h , r p = m A /m Z , x = m q /m g and λ t ≡ V tb V ∗ ts . g and g s are gaugecouplings for the weak and strong interactions, respectively. m h , m H and m A are massesof neutral Higgs bosons h , H , A , respectively. α is the neutral Higgs mixing angle andtan β is the ratio of the two Higgs vacuum expectation values, m ˜ g and m ˜ q are the gluinomass and common squark mass, respectively. Factors δ dLL , δ dRR and δ dLR are down-typeleft-light second-third, right-right second-third and left-right third generation squark mixingparameters, respectively. The loop function f ′ b ( x ) is defined as f ′ b ( x ) ≡ ( x / ∂ f b ( x ) /∂x and f b ( x ) is defined in [29]. f ′ b ( x ) is given by f ′ b ( x ) = − x ( − x − x log x )2( x − . (40)Because O i ( i = 15 , ...,
22) and O i are related by m s m b O i ⊆ O i , (41)the Wilson coefficients c i ( µ ) for O i ( µ ) with i = 15 , . . . ,
26 at µ = m W are given by c ( m W ) = m s m b C = D ( A − B ) ξ, c ( m W ) = m s m b C = D ( A − B ) ξ ′ ,c ( m W ) = m s m b C = D ( A + B ) ξ, c ( m W ) = m s m b C = D ( A + B ) ξ ′ , (42) c i = 0 for i = 16 , , , , , , , , Precisely speaking, in the two doublet Higgs model, couplings of the light neutral Higgs h to up-typequarks are suppressed by tan β compared with the down-type quarks. Therefore we can neglect contribu-tions of up-type quarks. D ≡ π λ t e g s g sin θ W f ′ b ( m q /m g ) m s m ˜ g ,A ≡ m H (cid:18) cos α + ( m H /m h ) sin α cos β (cid:19) , B ≡ m A (cid:18) m A m Z + tan β (cid:19) ,ξ ≡ δ dLL δ dLR , ξ ′ ≡ δ dRR δ dLR ∗ . (43)Since the Wilson coefficients for (¯ sb )( ¯ dd ) scalar and pseudoscalar operators are suppressedby m d /m s , we neglect contributions for operators with q = d and we consider only b → s ¯ ss type operators. In (43), we note that D is almost positive and real because f ′ b ( x ) < λ t < λ t is negligibly small.There are three enhancement factors: tan β , 1 /m A and 1 /m H in Eq. (43). Therefore,if δ dLR is sizable and neutral Higgses, A and H , are sufficiently light and tan β is large,then the Wilson coefficients for scalar/pseudoscalar operators can be large enough. In thepresent paper, for simplicity, we neglect the mixing between scalar/pseudoscalar and tensoroperators through renormalization-group equations (RGEs) since such a mixing effect issmall. Thus we assume c i ( m b ) ∝ c i ( m W ) and we use same symbols A , B , D , ξ and ξ ′ at µ ∼ m b .Under the existence of the MSSM-NHB scalar/pseudoscalar operators (43), ∆ c , , givenin (18), (19), in B → Kη s decays are rewritten by∆ c = DB ( | ξ | e ± iφ − | ξ ′ | e ± iφ ′ ) , for α , β , DB (cid:18) − B − AB (cid:19) ( | ξ | e ± iφ − | ξ ′ | e ± iφ ′ ) , for α , β , β S , ∆ c = 0 , (44)where ξ ( ′ ) = | ξ ( ′ ) | exp( iφ ( ′ ) ). As for the Wilson coefficients in B → φK ∗ decays, we use thereplacements (11). They are given by¯ c − c = − D ( A + B ) ξ ′ = 12 (cid:18) B − AB − (cid:19) DB | ξ ′ | e iφ ′ , ¯ c = − D ( A + B ) ξ = 12 (cid:18) B − AB − (cid:19) DB | ξ | e iφ , ¯ c = c , c = c = ¯ c = 0 , ¯ c = D ( A − B ) ξ, ¯ c = − D ( A − B ) ξ, ¯ c = D ( A − B ) ξ ′ , ¯ c = − D ( A − B ) ξ ′ . (45)12 IG. 1: ( B − A ) /B for various tan β and m A . Β H B - A L(cid:144) B mA = = = = As for a − , we parametrized the following coefficients that appear in (27). a + 12 a ≈ N c B − AB DB | ξ | e i ( δ ± φ ) ,a + 12 a ≈ N c B − AB DB | ξ ′ | e i ( δ ′ ± φ ′ ) , (46)where the α s corrections are negligible. However, we still parametrize the strong phases δ and δ ′ here. Actually the strong phases consistent with zero in the fit. Here and below thehelicity labels are omitted for a − since these coefficients very weakly depend on theirhelicities.The factor ( B − A ) /B depends on the details of neutral Higgs sector. In FIG. 1, wehave plotted the value of ( B − A ) /B for various m A and tan β in the MSSM. In the MSSM,( B − A ) /B is always smaller than one and − . . ( B − A ) /B . . β &
1. When( B − A ) /B ∼ .
2, the ratio | ( a + a ) / ∆ c | ≈ . B − A ) /B ≈ O (10 − ). III. NUMERICAL ANALYSISA. Numerical Inputs
We summarize input parameters in Table I. As for B → K ∗ vector and tensor formfactors, we follow the light-cone sum-rule (LCSR) results [33], defined as F ( q ) = F (0) exp( c q /m B + c q /m B ) , (47)13 ABLE I: Input ParametersDecay constants [5, 31] f π = 131 MeV f K = 160 MeV f B = 210 MeV f q = (1 . ± . f π f s = (1 . ± . f π f K ∗ = 218 MeV f TK ∗ = 175 MeV f φ = 221 MeV f Tφ = 175 MeV B meson parameter [31] λ B = 200 +250 − MeV η − η ′ mixing angle [31] φ η = 39 . ◦ ± . ◦ CKM parameters [35] A = 0 . ± . λ = 0 . ± . | V ub /V cb | = 0 . +0 . − . φ ( γ ) = (67 . +2 . − . ) ◦ sin 2 φ (sin 2 β ) = 0 . +0 . − . B meson lifetimes [36] τ ¯ B = 1 .
530 ps τ B − = 1 .
638 ps B → P form factors [31] F B → π (0) = 0 . F B → K (0) = 0 . for F ≡ A , A , A , V, T , T , T , where c and c are listed in Table III of [17]. We use m B ± ≈ m B = 5 .
279 GeV. We parametrize λ B to be m B /λ B ≡ R dy Φ B ( y ) /y , whereΦ B ( y ) is one of the two B meson light cone distribution amplitudes with y being themomentum fraction carried by the light spectator quark in the B meson. As for the renor-malization scale we use µ = m b /
2. In the calculation of the hard spectator and the weakannihilation, we adopt µ h = √ Λ h · µ ∼ h = 0 . X MH , X MA ( M = K, η ( ′ ) , K ∗ ), and X K ∗ L . For simplicity, in the fit, we assume that X KA,H = X η ( ′ ) A,H ≡ X ( Kη ( ′ ) ) A,H in B → Kη ( ′ ) decays, and parameterize them as [34] X ( Kη ( ′ ) ) A,H = [1 + ρ A,H exp( iφ A,H )] log (cid:18) m B Λ h (cid:19) , ≤ ρ A,H ≤ . (48)In B → φK ∗ decays we have fixed X L = m B / Λ h and X H = log( m B / Λ h ) (i.e., ρ H = ρ L = 0),because in these decays the branching ratios are very insensitive to them.As for NP effects, we use DB | ξ | , DB | ξ ′ | , φ , φ ′ , δ , δ ′ and ( B − A ) /B as the independentparameters. We have constrained weak and strong phases to be | φ ( ′ ) | ≤ π and | δ ( ′ ) | ≤ π/ ξ ′ = 0 and (ii) NP-(B) for which ξ = 0. 14 ABLE II: World averages of observables for B → Kη ( ′ ) are shown in the second column [36–46]. Upper limits are at 90% CL. In the third and fourth columns we have shown best fit valuesfor combined fit with corrections received from B → φK ∗ annihilation. Corresponding best fitparameters are shown in Table IV and in FIG. 2. Best fit values of B → φK ∗ are shown inTable III.Observable Experiment Combined Fit with φK ∗ ann.NP-(A) NP-(B) B ( B + → η ′ K + ) × . +2 . − . . ± . . ± . B ( B → η ′ K ) × . ± . . ± . . ± . B ( B + → ηK + ) × . ± . . ± . . ± . B ( B → ηK ) × ( < .
9) 1 . ± . . ± . A CP ( B + → η ′ K + ) 0 . ± .
021 0 . ± .
021 0 . ± . A CP ( B → η ′ K ) 0 . ± .
06 0 . ± .
01 0 . ± . A CP ( B + → ηK + ) − . ± . − . ± . − . ± . A CP ( B → ηK ) (N.A.) − . ± . − . ± . − η CP S K η ′ ( B → η ′ K ) 0 . ± .
07 0 . ± .
03 0 . ± . − η CP S K η ( B → ηK ) (N.A.) 0 . ± .
17 0 . ± . B. Experimental Data
In the fit for ¯ B → ¯ Kη ( ′ ) decays, we use 7 observables including 3 averaged branchingfractions, 3 direct CP violations and the ¯ B → ¯ K η ′ indirect CP violation − η CP S Kη ′ . Here S Kη ( ′ ) is defined by S Kη ( ′ ) = 2 Im( λ f )1 + | λ f | , λ f = qp A ( ¯ B → K S,L η ( ′ ) ) A ( B → K S,L η ( ′ ) ) , (49)with q/p ≃ e − iφ for B d , and η CP is the CP eigenvalue of | K S,L η ( ′ ) i . The value of − η CP S Kη ( ′ ) should be close to sin 2 φ in the SM. The experimental data for B → Kη ( ′ ) are listed inTable II.For the B → φK ∗ decays we have 20 observables, which include ¯ B , − → φ ¯ K ∗ , − branchingfractions ( B ), polarization fractions ( f L , f ⊥ ), and CP asymmetries ( A tot CP , A CP , A ⊥ CP ), phasesof polarized modes ( φ k , φ ⊥ ), and phase differences (∆ φ k , ∆ φ ⊥ ), where A tot CP = ( P λ | ¯ A λ | − λ | A λ | ) / ( P λ | ¯ A λ | + P λ | A λ | ), A λCP = ( | ¯ A λ | − | A λ | ) / ( | ¯ A λ | + | A λ | ), φ λ = arg( A λ /A ),∆ φ λ = arg( ¯ A λ / ¯ A · A /A λ ) with λ = 0 , k , ⊥ . The experimental data for the B → φK ∗ decays are shown in Table III. C. Combined Fits
If we ignore the annihilation effects in B → φK ∗ decays, the resulting χ &
170 istoo huge; i.e. we cannot have a reliable fitting result. This is the fact that if the B → φK ∗ polarization anomaly was mainly due to the tensor operators induced by the Fierztransformation, then the NP effects would lead to too large B → Kη ( ′ ) branching ratios ascompared with the data.Once the B → φK ∗ annihilation effects are included, we can see that the χ is drasti-cally small. In Table IV, we have summarized the best fit values of the χ and parameters. The resulting δ and δ ′ are consistent with zero, which are also consistent with the fact thatthe α s -corrections to the tensor operators are negligible. The fitted results for B → Kη ( ′ ) and B → φK ∗ are collected in Tables II and III.The results obtained in [17], where the weak annihilation effects are not included, aregiven by | ˜ a , DY | = 4 . +0 . − . × − , | ˜ a , DY | = 5 . +0 . − . × − , (50)to be compared with our present upper bounds, | ˜ a | ≤ . × − , | ˜ a | ≤ . × − , (51)which are extracted from FIG. 2 and (46), and are much smaller than the values in (50). Thisindicates that the contributions of tensor operators induced from the scalar/pseudoscalaroperators in the MSSM-Higgs are too small to explain the polarization puzzle, while thepuzzle can be accommodated by the weak annihilations effects. In BaBar measurements [49], instead of A λCP , the asymmetries of f +1 λ and f − λ are defined. f +1 λ and f − λ are the polarization fractions measured in ¯ B and B decays, respectively. The errors of parameters in Table IV are obtained from the error matrix (covariance matrix) at the globalminimum of χ . The error matrix is the inverse matrix of the curvature matrix of chi-square functionwith respect to its free parameters. The errors of best-fit values in Tables II and III are estimated fromthe same error matrices for each NP scenario. Here ˜ a and ˜ a are defined by ˜ a ≡ a + a and ˜ a ≡ a + a , respectively [17]. We have factoredout the CKM factor. ABLE III: World averages and best fit values of observables for ¯ B → φ ¯ K ∗ (upper) and B − → φK ∗− (lower) [1–4, 36, 37]. In the third and fourth column, best fit values for combined fit of B → Kη ( ′ ) and B → φK ∗ for NP-(A) and NP-(B) scenarios, including the contributions from B → φK ∗ annihilations, are shown. Corresponding best fit parameters are show in FIG. 2 andTable IV. Best fit values for B → Kη ( ′ ) are shown in Table II.Observable Experiment Combined Fit with φK ∗ ann.NP-(A) NP-(B) B tot × . ± . . ± . . ± . . ± . . ± . . ± . f L . ± .
032 0 . ± .
025 0 . ± . . ± .
05 0 . ± .
028 0 . ± . f ⊥ . ± .
031 0 . ± .
013 0 . ± . . ± .
05 0 . ± .
010 0 . ± . A tot CP − . ± . − . ± .
05 0 . ± . − . ± .
08 0 . ± .
03 0 . ± . A CP . ± . − . ± .
03 0 . ± . . ± .
11 0 . ± .
07 0 . ± . A ⊥ CP − . ± .
12 0 . ± .
03 0 . ± . . ± .
25 0 . ± .
07 0 . ± . φ k . +0 . − . . ± .
09 2 . ± . . ± .
17 2 . ± .
09 2 . ± . φ ⊥ . ± .
14 2 . ± .
06 2 . ± . . ± .
17 2 . ± .
06 2 . ± . φ k . ± .
14 0 . ± .
06 0 . ± . . ± .
21 0 . ± . − . ± . φ ⊥ . ± .
14 0 . ± .
07 0 . ± . . ± .
21 0 . ± . − . ± . IG. 2: Contour plots for ∆ χ ≡ χ − χ in Re( DBξ ) v.s. Im(
DBξ ) [or Re(
DBξ ′ ) v.s.Im( DBξ ′ )] for the NP scenario-A [or NP scenario-B]. Allowed regions of ∆ χ <
1, 1 < ∆ χ < < ∆ χ < × ”symbol indicates the location of the global minimum, χ . The origin corresponds to the SM.The circle at the origin indicates the allowed upper-limit from the B s → µ + µ − data. - - Re H DB Ξ L - - I m H D B Ξ L - - - - NP - H A L - - Re H DB Ξ ' L - - I m H D B Ξ ' L - - - - NP - H B L ABLE IV: Best fit parameters obtained in NP-(A) and NP-(B) with considering B → φK ∗ anni-hilations. Numbers with ( ∗ ) indicates that they reach the upper or lower bound in the parameterspace. Best fit values of B → Kη ( ′ ) and B → φK ∗ are shown in Tables II and III, respectively.Weak phased NP parameters ( DBξ , DBξ ′ ) are plotted in FIG. 2.Combined fit with φK ∗ annihilation effectsScenario (A) Scenario (B) χ / d.o.f. 9 . /
17 15 . / B − A ) /B . ± .
76 (*) 0 . ± .
19 (*) δ , δ ′ δ = +30 ± ◦ δ ′ = − ± ◦ ρ A [ Kη ( ′ ) ] 1 . ± .
33 (*) 0 . ± .
41 (*) φ A [ Kη ( ′ ) ] 117 ± ◦ ± ◦ ρ H [ Kη ( ′ ) ] 0 . ± .
67 (*) 1 . ± .
59 (*) φ H [ Kη ( ′ ) ] 69 ± ◦ − ± ◦ ρ A [ φK ∗ ] 0 . ± .
04 0 . ± . φ A [ φK ∗ ] − ± ◦ − ± ◦ D. Consistency with SM and B s → µ + µ − The current upper-bound for the branching fraction of B s → µ + µ − [37, 47] at 90% CL is B ( B s → µ + µ − ) ≤ . × − . (52)The branching fraction of b → s ¯ ℓℓ due to operators O ( ℓ ) i (with i = 7 , , , , ,
21) for ℓ = e, µ, τ is given by B ( B s → µ + µ − )= τ B s G F m B s π f B s (cid:18) m B s m b + m s (cid:19) | V tb V ∗ ts | √ − m × (cid:26) (1 − m ) (cid:12)(cid:12)(cid:12) c ( ℓ )15 − c ( ℓ )17 + c ( ℓ )19 − c ( ℓ )21 (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) c ( ℓ )15 + c ( ℓ )17 − c ( ℓ )19 − c ( ℓ )21 + 2 ˆ m ( c ( ℓ )7 − c ( ℓ )9 ) (cid:12)(cid:12)(cid:12) (cid:27) , (53) We have defined O ( ℓ ) i as O ( ′ )7(9) = ¯ s (1 − γ ) b ¯ ℓ (1 ± γ ) ℓ , and O ( ℓ ) i ( i = 15 , . . . ,
21) with replacements:¯ s (1 ± γ ) s → ¯ ℓ (1 ± γ ) ℓ and ¯ sγ µ (1 + γ ) s → ¯ ℓγ µ (1 + γ ) s . m ≡ m µ /m B s , c ( ℓ ) i are the Wilson coefficients of O ( ℓ ) i at µ = m b , and we have used h | ¯ sγ b | ¯ B s i = − if B s m B s m b + m s . (54)Here we note that if RGE effects are not large, c ( ℓ ) i ∼ ( m ℓ /m s ) c i and( c − c + c − c ) = 2 DA ( ξ − ξ ′ ) = 2 (1 − ( B − A ) /B ) DB ( ξ − ξ ′ ) , ( c + c − c − c ) = − DB ( ξ + ξ ′ ) . (55)Using B s ’s lifetime τ B s = 1 .
437 ps, mass m B s = 5 .
366 GeV, decay constant f B s = 215 ±
25 MeV, quark masses m b = 4 . ± . m s = 145 ±
25 MeV, we obtain upperbounds of
DBξ ( ′ ) as | DBξ ( ′ ) | . . × − . (56)In FIG. 2 we have shown the SM ( DBξ = DBξ ′ = 0) and the upper bound of the NPeffect constrained by the B s → µ + µ − decay as the origin and a small circle at the origin,respectively. In the figures we have also drawn the contours of ∆ χ = 1 , χ ≡ χ − χ . The B s → µ + µ − allowed region shares partly the ∆ χ ≤ σ ) regionin NP-(B), and is just outside of the ∆ χ ≤ σ ) region in the scenario NP-(A). Since inboth cases the B s → µ + µ − data and the SM are located within contours where χ / d.o.f. issufficiently small, we can safely conclude that our two scenarios are consistent with the datafor B s → µ + µ − decay and with the SM.In [23], the authors discuss the scalar/pseudoscalar operators induced by R-parity vio-lating interactions in the supersymmetric standard models. Because they did not take intoaccount the weak annihilation effects and possible constraints from B → Kη ( ′ ) , large con-tributions due to tensor operators to explain the B → φK ∗ polarization puzzle are requiredand therefore the estimated magnitudes of the effects of scalar/pseudoscalar operators aremuch larger than the upper bound of B s → µ + µ − . IV. SUMMARY
We have studied the scalar/pseudoscalar operators, and tensor operators where the latterare obtained from scalar/pseudoscalar operators by the Fierz transformation, in B → φK ∗ and B → Kη ( ′ ) decays. We have considered the scalar/pseudoscalar operators induced bypenguin diagrams of MSSM neutral Higgs bosons.20ithout the weak annihilations in B → φK ∗ , we cannot obtain any reasonable solutionto explain both B → φK ∗ and B → Kη ( ′ ) decays simultaneously in the NP region ( − . ≤ ( B − A ) /B ≤
1) of the MSSM induced by the neutral Higgs bosons. Taking into accountweak annihilation effects in B → φK ∗ , we obtain best fit results in good agreement with the B → Kη ( ′ ) and B → φK ∗ data. From the fitted parameters we estimate the magnitudesof the contributions due to NP tensor operators. They are, however, much smaller thanthe results of [17], which are introduced to explain the B → φK ∗ polarization puzzle.The polarization puzzle can be explained mostly by weak annihilation effect, as pointedout in [13–15]. The contributions of NP operators are constrained mainly by the fit of B → Kη ( ′ ) data. While our results may allow non-vanishing NP effects, the data for thedecays B → Kη ( ′ ) , B → φK ∗ are consistent with the B s → µ + µ − data as well as the SMprediction.Finally, we remark on the recently observed large longitudinal polarization fraction f L in B → φK ∗ (1430) [1]. If tensor operators play an significant role in B → V T (where T denotes a tensor meson) decays, f L may significantly deviate from unity. The current B → φK ∗ (1430) experiment seems to be consistent with our conclusion since in our anal-ysis the effect due to tensor operators is found to be very small. However, in the presentstudy we cannot exclude the possibility that sizable NP effects contribute directly to ten-sor operators, instead of scalar/pseudiscalar operators, and, moreover, a cancelation maytake place between weak annihilations and contributions due to NP tensor operators in the B → φK ∗ (1430) decay. For the point of view of the new physics, B → φK ∗ (1430) maybe sensitive to the B → K ∗ tensor form factor which can be further explored from the B → K ∗ (1430) γ decay. Acknowledgments
We thank Andrei Gritsan for many helpful comments on the manuscript. This work ispartly supported by National Science Council (NSC) of Republic of China under GrantsNSC 96-2811-M-033-004 and NSC 96-2112-M-033-MY3.21
PPENDIX A: DECAY CONSTANTS AND FORM FACTORS FOR η AND η ′ MESONS
The | η i and | η ′ i meson states are defined as the mixed states of | η q i and | η s i , as statedin (20) [32]. In this section we summarize the notations in [32]. Decay constants f q,sη ( ′ ) aregiven by f qη = f q cos φ η , f sη = − f s sin φ η ,f qη ′ = f q sin φ η , f sη ′ = f s cos φ η , (A1)and in the same way, pseudoscalar densities h q,sη ( ′ ) are defined as h qη = h q cos φ η , h sη = − h s sin φ η ,h qη ′ = h q sin φ η , h sη ′ = h s cos φ η , (A2)where h q,s are defined by h q = f q ( m η cos φ η + m η ′ sin φ η ) − √ f s ( m η ′ − m η ) sin φ η cos φ η ,h s = f s ( m η ′ cos φ η + m η sin φ η ) − √ f q ( m η ′ − m η ) sin φ η cos φ η . (A3) B → η ( ′ ) form factors are defined as F B → η ( ′ ) = F f qη ( ′ ) f π + F √ f qη ( ′ ) + f sη ( ′ ) √ f π (A4)and in the present paper we take F = F B → π (0) and F = 0. f π is the decay constant of thepion. APPENDIX B: DECAY CONSTANTS AND FORM FACTORS IN B → V V
DE-CAYS
We have used h φ ( q, ε φ ) | V µ | i = f φ m φ ε µ ∗ φ , (B1) h ¯ K ∗ ( p K ∗ , ε K ∗ ) | V µ | ¯ B ( p B ) i = 2 m B + m K ∗ ǫ µναβ ε ν ∗ K ∗ p αB p βK ∗ V ( q ) , (B2) h ¯ K ∗ ( p K ∗ , ε K ∗ ) | A µ | ¯ B ( p B ) i = i (cid:20) ( m B + m K ∗ ) ε ∗ K ∗ µ A ( q ) − ( ε ∗ K ∗ · p B )( p B + p K ∗ ) µ A ( q ) m B + m K ∗ (cid:21) − im K ∗ ε K ∗ · p B q q µ (cid:2) A ( q ) − A ( q ) (cid:3) , (B3)22or current operators, and h φ ( q, ε ) | ¯ sσ µν s | i = − if Tφ ( ε µ ∗ q ν − ε ν ∗ q µ ) , (B4) h ¯ K ∗ ( p ′ , ε ) | ¯ sσ µν (1 + γ ) b | ¯ B ( p ) i = iǫ µνρσ ε ν ∗ p α p ′ β T ( q )+ { ε ∗ µ ( m B − m K ∗ ) − ( ε ∗ · p )( p + p ′ ) µ } T ( q )+( ε ∗ · p B ) (cid:20) q µ − q m B m K ∗ ( p + p ′ ) µ (cid:21) T ( q ) , (B5)for tensor operators. In (B3) and (B5), A (0) = A (0), T (0) = T (0) and A ( q ) = m B + m K ∗ m K ∗ A ( q ) − m B − m K ∗ m K ∗ A ( q ) . (B6) APPENDIX C: THE COEFFICIENTS a p,hi CORRESPONDING TO RIGHT-HANDED 4-QUARK OPERATORS
In (27), the expressions for effective parameters a p,h corresponding to right-handed4-quark operators are a p,hi ( V V ) = "(cid:18) c i + c i ± N c (cid:19) N i ( V )+ c i ± N c C F α s π (cid:16) V hi ( V ) + 4 π N c H hi ( V V ) (cid:17) + P p,hi ( V ) , (C1)with N i ( V ) = 1 for i = 11 ,
12. For a p,h , one should replace c i by ¯ c i , and have N ( V ) = 1, N ( V ) = 0. V hi ( V ) account for vertex corrections, H hi ( V V ) for hard spectator interactionswith a hard gluon exchange between the emitted meson and the spectator quark of the B meson and P i ( V ) for penguin contractions. The vertex corrections read V i ( V ) = Z dx Φ V ( x ) h
12 ln m b µ −
18 + g T ( x ) i , ( i = 11,12) , Z dx Φ V ( x ) h −
12 ln m b µ + 6 − g (1 − x ) i , ( i = 13) , Z dx Φ v ( x ) h − h ( x ) i , ( i = 14) , (C2)23nd V + i ( V ) = Z dx Φ b ( x ) h
12 ln m b µ −
18 + g T ( x ) i , ( i = 11,12) , Z dx Φ a ( x ) h −
12 ln m b µ + 6 − g T (1 − x ) i , ( i = 13)0 , ( i = 14) , (C3)where Φ V ( x ) , Φ v ( x ) , Φ a ( x ) , Φ b ( x ), g ( x ) , h ( x ) and g T ( x ) are defined in [31] and [15]. H hi ( V V ) have the expressions: H ( V V ) = H ( V V ) = f B f V f V X ( BV ,V )0 Z dρ Φ B ( ρ ) ρ × Z dv Z du Φ V ( v )Φ V ( u )¯ u ¯ v + r V χ Φ v ( v )Φ V ( u ) u ¯ v ! , (C4) H ( V V ) = − f B f V f V X ( BV ,V )0 Z dρ Φ B ( ρ ) ρ × Z dv Z du Φ V ( v )Φ V ( u ) u ¯ v + r V χ Φ v ( v )Φ V ( u )¯ u ¯ v ! , (C5) H ( V V ) = 0 and H +11 ( V V ) = H +12 ( V V ) = − f B f ⊥ V f V X ( BV ,V )+ Z dρ Φ B ( ρ ) ρ Z dv Z du Φ ⊥ V ( v )Φ b ( u ) u ¯ v ,H +13 ( V V ) = f B f ⊥ V f V X ( BV ,V )+ Z dρ Φ B ( ρ ) ρ Z dv Z du Φ ⊥ V ( v )Φ a ( u )¯ u ¯ v ,H +14 ( V V ) = − f B f V f V X ( BV ,V )+ Z dρ Φ B ( ρ ) ρ Z dv Z du Φ a ( v )Φ ⊥ V ( u ) u ¯ u ¯ v , (C6)where X ( B V ,V )0 = f V m V " ( m B − m V − m V )( m B + m V ) A BV ( m V ) − m B p c m B + m V A BV ( m V ) ,X ( B V ,V )+ = − f V m V " ( m B + m V ) A BV ( m V ) ∓ m B p c m B + m V V BV ( m V ) , (C7)with q = p B − p V ≡ p V . Here Φ B ( ρ ) is one of the two light-cone distribution amplitudes of24he B meson [48]. P h,pi are strong penguin contractions. We obtain P ,p ( V ) = C F α s πN c ( ( c + c ) b X i = u (cid:20) n f m b µ − ( n f − G V (0) − G V ( s c ) − G V (1) − (cid:21) + c b X i = u (cid:20)
83 ln m b µ − G V (0) − G V (1) + 43 (cid:21)) , (C8) P ,p ( V ) = − C F α s πN c ( ( c + c ) b X i = u (cid:20) ( n f −
2) ˆ G V (0) + ˆ G V ( s c ) + ˆ G V (1) (cid:21) + c b X i = u (cid:20) ˆ G V (0) + ˆ G V (1) (cid:21)) , (C9) P h,p = P h,p = P + ,p = P + ,p , where s i = m i /m b and the functions G M ( s ) and ˆ G M ( s ) aregiven by G M ( s ) = − Z du Φ V ( u ) (cid:20) Z dx x ¯ x ln( s − ¯ ux ¯ x − iǫ ) (cid:21) , ˆ G V ( s ) = − Z du Φ v ( u ) (cid:20) Z dx x ¯ x ln( s − ¯ ux ¯ x − iǫ ) (cid:21) . (C10) [1] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. , 051801 (2007) [arXiv:hep-ex/0610073].[2] K. F. Chen et al. [BELLE Collaboration], Phys. Rev. Lett. , 221804 (2005) [arXiv:hep-ex/0503013].[3] P. Bussey for the CDF Collaboration, ICHEP 2006.[4] BABAR group, arXiv:0705.1798 [hep-ex]; A. V. Gritsan, In the Proceedings of 5th FlavorPhysics and CP Violation Conference (FPCP 2007), Bled, Slovenia, 12-16 May 2007, pp 001 [arXiv:0706.2030 [hep-ex]].[5] H. Y. Cheng and K. C. Yang, Phys. Lett. B , 40 (2001) [arXiv:hep-ph/0104090].[6] K. Abe et al. [BELLE-Collaboration], Phys. Rev. Lett. , 141801 (2005) [arXiv:hep-ex/0408102].[7] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. , 201801 (2006) [arXiv:hep-ex/0607057].[8] H. n. Li and S. Mishima, Phys. Rev. D , 054025 (2005) [arXiv:hep-ph/0411146].[9] X. Q. Li, G. r. Lu and Y. D. Yang, Phys. Rev. D , 114015 (2003) [Erratum-ibid. D ,019902 (2005)] [arXiv:hep-ph/0309136].
10] H. n. Li, Phys. Lett. B , 63 (2005) [arXiv:hep-ph/0411305].[11] H. Y. Cheng, C. K. Chua and A. Soni, Phys. Rev. D , 014030 (2005) [arXiv:hep-ph/0409317].[12] M. Ladisa, V. Laporta, G. Nardulli and P. Santorelli, Phys. Rev. D , 114025 (2004)[arXiv:hep-ph/0409286].[13] A. L. Kagan, Phys. Lett. B , 151 (2004) [arXiv:hep-ph/0405134].[14] K. C. Yang, Phys. Rev. D , 034009 (2005) [Erratum-ibid. D , 059901 (2005)] [arXiv:hep-ph/0506040].[15] M. Beneke, J. Rohrer and D. Yang, Nucl. Phys. B , 64 (2007) [arXiv:hep-ph/0612290].[16] W. S. Hou and M. Nagashima, arXiv:hep-ph/0408007.[17] P. K. Das and K. C. Yang, Phys. Rev. D , 094002 (2005) [arXiv:hep-ph/0412313].[18] A. L. Kagan, arXiv:hep-ph/0407076.[19] E. Alvarez, L. N. Epele, D. G. Dumm and A. Szynkman, Phys. Rev. D , 115014 (2004)[arXiv:hep-ph/0410096].[20] C. H. Chen and H. Hatanaka, Phys. Rev. D , 075003 (2006) [arXiv:hep-ph/0602140].[21] S. Baek, A. Datta, P. Hamel, O. F. Hernandez and D. London, Phys. Rev. D , 094008(2005) [arXiv:hep-ph/0508149].[22] Y. D. Yang, R. M. Wang and G. R. Lu, Phys. Rev. D , 015009 (2005) [arXiv:hep-ph/0411211].[23] A. Faessler, T. Gutsche, J. C. Helo, S. Kovalenko and V. E. Lyubovitskij, Phys. Rev. D ,074029 (2007) [arXiv:hep-ph/0702020].[24] C. H. Chen and C. Q. Geng, Phys. Rev. D , 115004 (2005) [arXiv:hep-ph/0504145].[25] Q. Chang, X. Q. Li and Y. D. Yang, JHEP , 038 (2007) [arXiv:hep-ph/0610280].[26] C. S. Huang, P. Ko, X. H. Wu and Y. D. Yang, Phys. Rev. D , 034026 (2006) [arXiv:hep-ph/0511129].[27] C. S. Huang and Q. S. Yan, Phys. Lett. B , 209 (1998) [arXiv:hep-ph/9803366];C. S. Huang, W. Liao and Q. S. Yan, Phys. Rev. D , 011701 (1999) [arXiv:hep-ph/9803460];C. S. Huang, W. Liao, Q. S. Yan and S. H. Zhu, Phys. Rev. D , 114021 (2001) [Erratum-ibid. D , 059902 (2001)] [arXiv:hep-ph/0006250]; K. S. Babu and C. F. Kolda, Phys. Rev.Lett. , 228 (2000) [arXiv:hep-ph/9909476]; S. R. Choudhury and N. Gaur, Phys. Lett. B , 86 (1999) [arXiv:hep-ph/9810307].[28] C. S. Huang and X. H. Wu, Nucl. Phys. B , 304 (2003) [arXiv:hep-ph/0212220].
29] J. F. Cheng, C. S. Huang and X. H. Wu, Nucl. Phys. B , 54 (2004) [arXiv:hep-ph/0404055].[30] F. Borzumati, C. Greub, T. Hurth and D. Wyler, Phys. Rev. D , 075005 (2000) [arXiv:hep-ph/9911245].[31] M. Beneke and M. Neubert, Nucl. Phys. B , 333 (2003) [arXiv:hep-ph/0308039].[32] M. Beneke and M. Neubert, Nucl. Phys. B , 225 (2003) [arXiv:hep-ph/0210085].[33] A. Ali and A. S. Safir, Eur. Phys. J. C , 583 (2002) [arXiv:hep-ph/0205254]; A. Ali, P. Ball,L. T. Handoko and G. Hiller, Phys. Rev. D , 074024 (2000) [arXiv:hep-ph/9910221].[34] M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Nucl. Phys. B , 313 (2000)[arXiv:hep-ph/0006124].[35] CKMfitter group http://ckmfitter.in2p3.fr , Results as of Summer 2007.[36] W. M. Yao et al. [Particle Data Group], J. Phys. G (2006) 1.[37] E. Barberio et al. [Heavy Flavor Averaging Group (HFAG) Collaboration], arXiv:0704.3575[hep-ex].[38] B. Aubert et al. [BaBar Collaboration], Phys. Rev. Lett. , 191802 (2005) [arXiv:hep-ex/0502017].[39] J. Schumann et al. [Belle Collaboration], Phys. Rev. Lett. , 061802 (2006) [arXiv:hep-ex/0603001].[40] S. J. Richichi et al. [CLEO Collaboration], Phys. Rev. Lett. , 520 (2000) [arXiv:hep-ex/9912059].[41] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. , 131803 (2005) [arXiv:hep-ex/0503035].[42] K. Abe et al. [Belle Collaboration], arXiv:hep-ex/0608033.[43] B. Aubert et al. [BaBar Collaboration], Phys. Rev. D , 051106 (2006) [arXiv:hep-ex/0607063].[44] S. Chen et al. [CLEO Collaboration], Phys. Rev. Lett. , 525 (2000) [arXiv:hep-ex/0001009].[45] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. , 031801 (2007) [arXiv:hep-ex/0609052].[46] K. F. Chen et al. [Belle Collaboration], Phys. Rev. Lett. , 031802 (2007) [arXiv:hep-ex/0608039].[47] D. Tonelli [CDF Collaboration], In the Proceedings of 4th Flavor Physics and CP ViolationConference (FPCP 2006), Vancouver, British Columbia, Canada, 9-12 Apr 2006, pp 001 arXiv:hep-ex/0605038]; DO Collaboration, (V. Abazov etal), DO Note 5344-CONF (2007);[48] M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Phys. Rev. Lett. , 1914 (1999)[arXiv:hep-ph/9905312].[49] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. , 231804 (2004) [arXiv:hep-ex/0408017]., 231804 (2004) [arXiv:hep-ex/0408017].