Radiative charmless B_{(s)}\to V γand B_{(s)}\to A γdecays in pQCD approach
aa r X i v : . [ h e p - ph ] N ov Radiative charmless B ( s ) → V γ and B ( s ) → Aγ decays in pQCD approach Wei Wang a , Run-Hui Li b,a and Cai-Dian L¨u a a Institute of High Energy Physics, P.O. Box 918(4) Beijing 100049, P.R. China b Department of Physics, Shandong University, Jinan 250100, P.R. China c Graduate University of Chinese Academy of Sciences, Beijing 100049, P.R. China
We study the radiative charmless B ( s ) → V ( A ) γ decays in perturbative QCD (pQCD)approach to the leading order in α s (here V and A denotes vector mesons and two kinds ofaxial-vector mesons: P and P states, respectively.). Our predictions of branching ratiosare consistent with the current available experimental data. We update all B ( s ) → V formfactors and give the predictions for B → A form factors using the recent hadronic inputs. Inaddition to the dominant factorizable spectator diagrams, which is form factor like, we alsocalculate the so-called “power suppressed” annihilation type diagrams, the gluonic penguin,charming penguin, and two photon diagrams. These diagrams give the main contributionsto direct CP asymmetries, mixing-induced CP asymmetry variables, the isospin asymmetryand U-spin asymmetry variables. Unlike the branching ratios, these ratios or observablespossess higher theoretical precision in our pQCD calculation, since they do not depend onthe input hadronic parameters too much. Most of the results still need experimental testsin the on-going and forthcoming experiments. I. INTRODUCTION
Present studies on B decays are mainly concentrated on the precise test of standard model (SM)and the search for signals of possible new physics. Radiative processes b → sγ and b → dγ areamong the most ideal probes and thus have received considerable efforts [1]. In the standard model,these processes are induced by the flavor-changing-neutral-current transitions which are purely loopeffects. Such decays are rare and the measurement of parameters in these channels, especially theCP asymmetry variables, may shed light on detailed information on the flavor structure of theelectroweak interactions. This predictive power relies on the accuracy of both the experimentalside and the theoretical side. Thanks to the technique of operator-product-expansion, remarkabletheoretical progress has been made in the SM to the next-to-next-to-leading order accuracy [2].Compared with experimental results [3], the theoretical prediction is consistent with but 1 . σ belowthe experimental data [4]. This consistence could certainly provide a rather stringent constrainton non-standard model scenarios.Compared with inclusive processes, the exclusive processes B → V γ is more tractable on theexperimental side, but more difficult on the theoretical side. Theoretical predictions are oftenhampered by our ability to calculate the decay amplitude h V γ | O i | B i , where O i is a magneticmoment or a four-quark operator. We have to use some non-perturbative hadronic quantities todescribe the bound state effects. In the heavy-quark limit, the non-perturbative contributions canbe organized in a universal and channel-independent manner. Factorization analysis, the separationof the short-distance and long-distance dynamics, can give many important predictions.The dominant contribution to the radiative decay amplitudes is proportional to the transitionform factor. Different treatments on the dynamics in form factor F B → V result in different explicitapproaches. There are many approaches such as Lattice QCD (LQCD) [5], light-front quark model(LFQM) [6, 7] and light-cone sum rules (LCSR) [8]. Recently three commonly-accepted approachesare developed to study exclusive B decays: QCD factorization (QCDF) [9], soft collinear effectivetheory (SCET) [10] and perturbative QCD approach [11]. In pQCD approach, the recoiling mesonmoves on the light-cone and a large momentum transfer is required. Keeping quarks’ transversemomentum, pQCD approach is free of endpoint divergence and the Sudakov formalism makes itmore self-consistent. It has been successfully applied to various decay channels and quite recentlythis approach is accessing to next-to-leading order accuracy [12]. A bigger advantage is that we canreally do the form factor calculation and the quantitative annihilation type diagram calculation inthis approach. The importance of annihilation diagrams are already tested in the predictions ofdirect CP asymmetries in B → π + π − , K + π − decays [11, 13] and in the explanation of B → φK ∗ polarization problem [14, 15]. These “power suppressed” contributions are the main source ofisospin symmetry breaking (or SU(3) breaking) effects in radiative decays B → V γ [16, 17].Some of charmless B → V γ decay channels have been studied in pQCD approach [16, 17, 18]separately. According to the transition at the quark level, all the 10 decay channels can be dividedinto three different groups: b → s , b → d and purely annihilation type. The two B → K ∗ γ modes( b → s process) have been well measured experimentally. The agreement of pQCD approach resultsand experimental data [3] is very encouraging. In this paper, we study all those channels includingthe corresponding B s decays in a comprehensive way. The input hadronic parameters will bechosen the same as that in Ref. [19]. We also updated all the B → V decay form factors in pQCDapproach [20]. Despite branching ratios and CP asymmetry parameters which heavily depend onthe input parameters, we also study some ratios characterizing the isospin and SU(3) breakingeffects, where most of the uncertainties cancel. Experimental measurements of these quantitieswill prove a good test of our theory, since different method gives different prediction. With theongoing B factories BaBar and Belle, the B -Physics program on CDF and the onset of the LHCexperiments, as well as the Super B-factories being contemplated for the future, we expect a wealthof data involving these decays.Although experimentalists have already measured one channel of the B → Aγ decays [21], thereare not many discussions on the theoretical side. The pQCD study on B → V γ can be straight-forward extended to radiative processes involving higher resonants such as K (1270), K (1400).In the quark model, the quantum numbers J P C for the orbitally excited axial-vector mesons are1 ++ or 1 + − , depending on different spins of the two quarks. In SU(3) limit, these mesons can notmix with each other; but since the s quark is heavier than u, d quarks, K (1270) and K (1400)are not purely 1 P or 1 P states. These two mesons are believed to be mixtures of K A and K B , where K A and K B are P and P states, respectively. In general, the mixing angle canbe determined by experimental data. But unfortunately, there is not too much data on the mixingof these mesons which leaves the mixing angle much free. Analogous to η and η ′ , the flavor-singletand flavor-octet axial-vector meson can also mix with each other. Using those hadronic parametersdetermined in B → V γ decays, the production of Aγ in B decays could provide a unique insightto these mysterious axial-vector mesons.This paper is organized as follows: In section II, we briefly review pQCD approach with theoperator basis used subsequently. Some input quantities which enter pQCD approach, wave func-tion of the B -meson, distribution amplitudes for light vector mesons, and for light axial-vectormesons and input values of the various mesonic decay constants, are also given here. In section III,we give the factorization formulae and the numerical results for B → V and B → A form fac-tors. Section IV contains the calculation of B → V γ decays, making explicit the contributionsfrom the electromagnetic diploe operator, the chromo-magnetic moment operator, some higher or-der ( O ( α s )) corrections from tree operators, the contribution from two-photon diagrams, the treeoperator annihilation diagrams and penguin operator annihilation diagrams. Numerical resultsfor the charge-conjugated averages of decay branching ratios are given in comparison with thecorresponding numerical results obtained in QCDF approach and SCET, as well as the availableexperimental data. We also give direct CP-asymmetries, time-dependent CP asymmetries S f andobservables H f (for B s system) in the time-dependent decay rates in this section. In section V, westudy B → Aγ decays by predicting branching ratios and direct CP asymmetries. Our summary isgiven in the last section. Some functions are relegated to the appendix: appendix A contains thevarious functions that enter the factorization formulae in the pQCD approach; appendix B and Cgive the analytic formulae for the B → V γ and B → Aγ decays used in numerical calculations,respectively. II. FORMALISM OF PQCD APPROACHA. Notations and conventions
We specify the weak effective Hamiltonian which describes b → D ( D = d, s ) transitions [22]: H eff = G F √ (cid:26) X q = u,c V qb V ∗ qD (cid:2) C ( µ ) O q ( µ ) + C ( µ ) O q ( µ ) i − V tb V ∗ tD h , γ, g X i =3 C i ( µ ) O i ( µ ) (cid:3)(cid:27) + H.c. , (1)where V qb ( D ) and V tb ( D ) are Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. Functions O i ( i = 1 , ..., , γ, g ) are local four-quark operators or the moment type operators: • current–current (tree) operators O q = (¯ q α b β ) V − A ( ¯ D β q α ) V − A , O q = (¯ q α b α ) V − A ( ¯ D β q β ) V − A , (2) • QCD penguin operators O = ( ¯ D α b α ) V − A X q ′ (¯ q ′ β q ′ β ) V − A , O = ( ¯ D β b α ) V − A X q ′ (¯ q ′ α q ′ β ) V − A , (3) O = ( ¯ D α b α ) V − A X q ′ (¯ q ′ β q ′ β ) V + A , O = ( ¯ D β b α ) V − A X q ′ (¯ q ′ α q ′ β ) V + A , (4) • electro-weak penguin operators O = 32 ( ¯ D α b α ) V − A X q ′ e q ′ (¯ q ′ β q ′ β ) V + A , O = 32 ( ¯ D β b α ) V − A X q ′ e q ′ (¯ q ′ α q ′ β ) V + A , (5) O = 32 ( ¯ D α b α ) V − A X q ′ e q ′ (¯ q ′ β q ′ β ) V − A , O = 32 ( ¯ D β b α ) V − A X q ′ e q ′ (¯ q ′ α q ′ β ) V − A , (6) • magnetic moment operators O γ = − e π ¯ D α σ µν ( m D P L + m b P R ) b α F µν , O g = − g π ¯ D α σ µν ( m D P L + m b P R ) T aαβ b β G aµν , (7)where α and β are color indices and q ′ are the active quarks at the scale m b , i.e. q ′ = ( u, d, s, c, b ).The left handed current is defined as (¯ q ′ α q ′ β ) V − A = ¯ q ′ α γ ν (1 − γ ) q ′ β and the right handed current(¯ q ′ α q ′ β ) V + A = ¯ q ′ α γ ν (1 + γ ) q ′ β . The projection operators are defined as P L = (1 − γ ) / P R = (1 + γ ) /
2. The combinations a i of Wilson coefficients are defined as usual [23]: a = C + C / , a = C + C / , a = C + C / , a = C + C / , a = C + C / ,a = C + C / , a = C + C / , a = C + C / , a = C + C / , a = C + C / . (8)For the explicit formulae, we will consider ¯ B meson decays and use light-cone coordinates todescribe the momentum: p = ( p + , p − , ~p T ) = (cid:16) p + p √ , p − p √ , ~p T (cid:17) . (9)where ~p T = ( p , p ). In the ¯ B meson rest frame, momenta of ¯ B meson, vector (axial-vector) mesonand the photon are chosen as: P = m B √ , ,~ T ) , P = m B √ , ,~ T ) , P γ = m B √ , ,~ T ) , (10)where the vector (axial-vector) is mainly moving on the minus direction n − and the photon ismoving on the plus direction n + . Longitudinal momenta fractions of the spectator anti-quarks in¯ B and final state meson are chosen as x = k +1 /P +1 and x = k − /P − . Including the transversecomponents, the momenta of these spectator antiquarks are expressed by: k = ( m B √ x , , ~k T ) , k = (0 , m B √ x , ~k T ) , (11) One should be cautious that in the discussion of light cone distribution amplitudes (LCDAs) for the vectors andaxial-vectors, x is defined as the momentum fraction of the positive quark which is different with our definitionhere. ¯ qDO γ O γ D ¯ q γγb ¯ q ¯ qb FIG. 1: Feynman diagrams of the electromagnetic penguin operator O γ then the b and D (= s, d ) quark momenta are p b = P − k and p D = P − k . For convenience, wecan define the following useful ratio variables: r b = m b m B , r D = m D m B , r V = m V m B , r A = m A m B , (12)where m D and m V ( A ) are masses for the d ( s ) quark and the vector (axial-vector) meson, respec-tively. In the calculation of decay amplitudes, we will only keep terms of leading order in r V but wewill consider the corrections together with kinematic corrections in the phase space as in Eq. (87).According to the Lorentz structure, decay amplitudes from various operators can be generallydecomposed into scalar and pseudoscalar components as: M = ( ǫ ∗ γ · ǫ ∗ V ) M S + iǫ µναβ ǫ ∗ µγ ǫ ∗ νV n α + n β − M P , (13)and where we adopt the convention ǫ = +1. In order to study mixing-induced CP asymmetries,it is convenient to separate different chiralities in the amplitudes. If the emitted photon is left-handed, the relationship between the scalar M S and pseudoscalar component M P is requiredas M S = −M P ≡ M L , (14)while the condition M S = M P ≡ M R , (15)is required for the right-handed photon. B. A brief review of pQCD approach
The basic idea of pQCD approach is that it takes into account the intrinsic transverse momentumof valence quarks. The decay amplitude, taking the first diagram in Fig. 1 as an example, can l H l H l H l H FIG. 2: O ( α s ) corrections to the hard scattering kernel H . be expressed as a convolution of wave functions φ B , φ V and hard scattering kernel T H by bothlongitudinal and transverse momenta: M = Z dx dx Z d ~k T (2 π ) d ~k T (2 π ) φ B ( x , ~k T , p , t ) T H ( x , x , ~k T , ~k T , t ) φ V ( x , ~k T , p , t ) . (16)Usually it is convenient to compute the amplitude in coordinate space. Through Fourier transfor-mation, the above equation can be expressed by: M = Z dx dx Z d ~b d ~b φ B ( x ,~b , p , t ) T H ( x , x ,~b ,~b , t ) φ V ( x ,~b , p , t ) . (17)This derivation is mainly concentrated on tree level diagrams, but actually we have to take intoaccount some loop effects which can give sizable corrections. The O ( α s ) radiative corrections tohard scattering process H are depicted in Fig. 2. In general, individual higher order diagrams maysuffer from two types of infrared divergences: soft and collinear. Soft divergence comes from theregion of a loop momentum where all it’s momentum components vanish: l µ = ( l + , l − ,~l T ) = (Λ , Λ , ~ Λ) , (18)where Λ is the typical scale for hadronization. Collinear divergence originates from the gluonmomentum region which is parallel to the massless quark momentum, l µ = ( l + , l − ,~l T ) ∼ ( m B , Λ /m B , ~ Λ) . (19)In both cases, the loop integration corresponds to R d l/l ∼ log Λ, thus logarithmic divergencesare generated. It has been shown order by order in perturbation theory that these divergencescan be separated from the hard kernel and absorbed into meson wave functions using eikonalapproximation [24]. But when soft and collinear momentum overlap, there will be double logarithmdivergences in the first two diagrams of Fig. 2. These large double logarithm can be resummedinto the Sudakov factor whose explicit form is given in Appendix A.Furthermore, there are also another type of double logarithm which comes from the loop correc-tion for the weak decay vertex correction. The left diagram in Fig. 1 gives an amplitude proportionalto 1 / ( x x ). In the threshold region with x → x ∼ O (Λ QCD /m B ))], additionalsoft divergences are associated with the internal quark at higher orders. The QCD loop correctionsto the electro-weak vertex can produce the double logarithm α s ln x and resummation of thistype of double logarithms lead to the Sudakov factor S t ( x ). Similarly, resummation of α s ln x due to loop corrections in the other diagram leads to the Sudakov factor S t ( x ). These doublelogarithm can also be factored out from the hard part and grouped into the quark jet function.Resummation of the double logarithms results in the threshold factor [25]. This factor decreasesfaster than any other power of x as x →
0, which modifies the behavior in the endpoint region tomake pQCD approach more self-consistent. For simplicity, this factor has been parameterized in aform which is independent on channels, twists and flavors [26].Combing all the elements together, we can get the typical factorization formulae in pQCDapproach: M = Z dx dx Z d ~b d ~b (2 π ) φ B ( x ,~b , p , t ) × T H ( x , x , Q,~b ,~b , t ) φ V ( x ,~b , p , t ) S t ( x ) exp[ − S B ( t ) − S ( t )] . (20) C. Wave functions of B mesons In order to calculate the analytic formulas of decay amplitudes, we will use light cone wavefunctions Φ
M,αβ decomposed in terms of the spin structure. In general, Φ
M,αβ with Dirac indices α, β can be decomposed into 16 independent components, 1 αβ , γ αβ , γ µαβ , ( γ µ γ ) αβ , σ µναβ . If theconsidered meson is the B meson, a heavy pseudo-scalar meson, the B meson light-cone matrixelement can be decomposed [27, 28, 29] by: Z d ze ik · z h | b β (0) ¯ D α ( z ) | ¯ B ( P B ) i = i √ N c (cid:26) ( P B + m B ) γ (cid:20) φ B ( k ) − 6 n − 6 v √ φ B ( k ) (cid:21)(cid:27) βα , (21)where N c = 3 is the color factor. n and v are two light-like vectors: n = v = 0. From equation(21), one can see that there are two Lorentz structures in the B meson distribution amplitudes.They obey the following normalization conditions: Z d k (2 π ) φ B ( k ) = f B √ N c , Z d k (2 π ) ¯ φ B ( k ) = 0 . (22)In general, one should consider these two Lorentz structures in the calculations of B mesondecays. However, it is found that the contribution of ¯ φ B is numerically small [20], thus its con-tribution can be safely neglected. With this approximation, we only retain the first term in thesquare bracket from the full Lorentz structure in Eq. (21):Φ B = i √ N c ( P B + m B ) γ φ B ( k ) . (23)In the following calculation, we will see that the hard part is always independent of one of the k +1 and/or k − . The B meson wave function is then a function of the variables k − (or k +1 ) and k only, φ B ( k − , ~k T ) = Z dk +1 π φ B ( k +1 , k − , ~k T ) . (24)In the b -space, the B meson’s wave function can be expressed byΦ B ( x, b ) = i √ N c [ P B γ + m B γ ] φ B ( x, b ) , (25)where b is the conjugate space coordinate of the transverse momentum k T .In this study, we use the following phenomenological wave function: φ B ( x, b ) = N B x (1 − x ) exp (cid:20) − m B x ω b −
12 ( ω b b ) (cid:21) , (26)with N B the normalization factor. In recent years, a lot of studies have been performed for B d and B ± decays in pQCD approach. The parameter ω b = (0 . ± .
05) GeV has been fixed using therich experimental data on B d and B ± meson decays. In the SU(3) symmetry limit, this parametershould be the same in B s decays. Considering a small SU(3) breaking, s quark momentum fractionshould be a little larger than that of u or d quark in the lighter B mesons, since s quark is heavierthan u or d quark. We will use ω b = (0 . ± .
05) GeV in this paper for the B s decays [19]. D. Light-cone distribution amplitudes of light vector mesons
Decay constants for vector mesons are defined by: h | ¯ q γ µ q | V ( p, ǫ ) i = f V m V ǫ µ , h | ¯ q σ µν q | V ( p, ǫ ) i = if TV ( ǫ µ p ν − ǫ ν p µ ) . (27)The longitudinal decay constants of charged vector mesons can be extracted from the data on τ − → ( ρ − , K ∗− ) ν τ [30]. Neutral vector meson’s longitudinal decay constant can be determined byits electronic decay width through V → e + e − and results are given in Table I. Transverse decayconstants are mainly explored by QCD sum rules [31] that are also collected in Table I. There is a recent study on extracting vectors’ decay constants from experimental data, which has taken intoaccount the effects of ρ - ω and ω - φ mixing [31]. Since we do not consider the mixing for the decay amplitudes inour calculation, we will not use those values for self-consistence. TABLE I: Input values of the decay constants for the vector mesons (in MeV) f ρ f Tρ f ω f Tω f K ∗ f TK ∗ f φ f Tφ ± ± ± ± ± ±
10 231 ± ± We choose the vector meson momentum P with P = m V , which is mainly on the plus direction.The polarization vectors ǫ , satisfying P · ǫ = 0, include one longitudinal polarization vector ǫ L andtwo transverse polarization vectors ǫ T . The vector meson distribution amplitudes up to twist-3 aredefined by: h V ( P, ǫ ∗ L ) | ¯ q β ( z ) q α (0) | i = 1 √ N c Z dxe ixP · z (cid:2) m V ǫ ∗ L φ V ( x )+ ǫ ∗ L P φ tV ( x ) + m V φ sV ( x ) (cid:3) αβ , h V ( P, ǫ ∗ T ) | ¯ q β ( z ) q α (0) | i = 1 √ N c Z dxe ixP · z (cid:2) m V ǫ ∗ T φ vV ( x )+ ǫ ∗ T P φ TV ( x )+ m V iǫ µνρσ γ γ µ ǫ ∗ νT n ρ v σ φ aV ( x )] αβ , (28)for the longitudinal polarization and transverse polarization, respectively. Here x is the momentumfraction associated with the q quark. n is the moving direction of the vector meson and v is theopposite direction. These distribution amplitudes can be related to the ones used in QCD sumrules by: φ V ( x ) = f V √ N c φ || ( x ) , φ tV ( x ) = f TV √ N c h ( t ) || ( x ) ,φ sV ( x ) = f TV √ N c ddx h ( s ) || ( x ) , φ TV ( x ) = f TV √ N c φ ⊥ ( x ) ,φ vV ( x ) = f V √ N c g ( v ) ⊥ ( x ) , φ aV ( x ) = f V √ N c ddx g ( a ) ⊥ ( x ) . (29)The twist-2 distribution amplitudes can be expanded in terms of Gegenbauer polynomials C / n with the coefficients called Gegenbauer moments a n : φ || , ⊥ ( x ) = 6 x (1 − x ) " ∞ X n =1 a || , ⊥ n C / n ( t ) , (30)where t = 2 x −
1. The Gegenbauer moments a || , ⊥ n are mainly determined by the technique of QCDsum rules. Here we quote the recent numerical results [32, 33, 34, 35] as a k ( K ∗ ) = 0 . ± . , a ⊥ ( K ∗ ) = 0 . ± . , (31) a k ( ρ ) = a k ( ω ) = 0 . ± . , a ⊥ ( ρ ) = a ⊥ ( ω ) = 0 . ± . , (32) a k ( K ∗ ) = 0 . ± . , a ⊥ ( K ∗ ) = 0 . ± . , (33) a k ( φ ) = 0 . ± . , a ⊥ ( φ ) = 0 . ± . , (34)1 TABLE II: Input values of the decay constants (absolute values) for the axial-vector mesons (in MeV). Thetransverse decays constants for P are evaluated at µ = 1 GeV. f a (1260) f f (1 P ) f f (1 P ) f K A f Tb (1235) f Th (1 P ) f Th (1 P ) f TK B ±
10 245 ±
13 239 ±
13 250 ±
13 180 ± ±
12 190 ±
10 190 ± where the values are taken at µ = 1 GeV.Using equation of motion, two-particle twist-3 distribution amplitudes are related to twist-2LCDAs and three-particle twist-3 distribution amplitudes. But in some B → V V decays, thereexists the so-called polarization problem. It has been suggested that using asymptotic LCDAs canresolve this problem in pQCD approach. Thus to be self-consistent, we should also use the sameform to calculate radiative decays. As in Ref. [19], we use the asymptotic form for twist-3 LCDAs: h ( t ) k ( x ) = 3 t , h ( s ) || ( x ) = 6 x (1 − x ) , (35) g ( a ) ⊥ ( x ) = 6 x (1 − x ) , g ( v ) ⊥ ( x ) = 34 (1 + t ) . (36) E. Light-cone distribution amplitudes of axial-vectors
Longitudinal and transverse decay constants for axial-vectors are defined by: h A ( P, ǫ ) | ¯ q γ µ γ q | i = if A m A ǫ ∗ µ , h A ( P, ǫ ) | ¯ q σ µν γ q | i = f TA ( ǫ ∗ µ P ν − ǫ ∗ ν P µ ) . (37)In SU(2) limit, due to G-parity, the longitudinal (transverse) decay constants vanish for thenon-strange P [ P ] states. This will affect the normalization for the corresponding distribu-tion amplitudes which will be discussed in the following. For convenience, we take f P ≡ f [ f T P ( µ = 1 GeV) ≡ f ] as the “normalization constant”. The numbers of axial vector meson decayconstants shown in table II are taken from Ref. [36, 37].Distribution amplitudes for axial-vectors with quantum numbers J P C = 1 ++ or 1 + − are definedby: h A ( P, ǫ ∗ L ) | ¯ q β ( z ) q α (0) | i = − i √ N c Z dxe ixp · z (cid:2) m A ǫ ∗ L γ φ A ( x ) − 6 ǫ ∗ L P γ φ tA ( x ) − m A γ φ sA ( x ) (cid:3) αβ , h A ( P, ǫ ∗ T ) | ¯ q β ( z ) q α (0) | i = − i √ N c Z dxe ixp · z (cid:2) m A ǫ ∗ T γ φ vA ( x ) − 6 ǫ ∗ T P γ φ TA ( x ) − m A iǫ µνρσ γ µ ǫ ∗ νT n ρ v σ φ aA ( x )] αβ . (38)Besides the factor − iγ from the right hand, axial-vector mesons’ distribution amplitudes can be2related to the vector ones by making the following replacement: φ V → φ A , φ tV → − φ tA , φ sV → − φ sA ,φ TV → − φ TA , φ vV → φ vA , φ aV → φ aA . (39)These distribution amplitudes can be related to the ones calculated in QCD sum rules by: φ A ( x ) = f √ N c φ || ( x ) , φ tA ( x ) = f √ N c h ( t ) || ( x ) ,φ sA ( x ) = f √ N c ddx h ( s ) k ( x ) , φ TA ( x ) = f √ N c φ ⊥ ( x ) ,φ vA ( x ) = f √ N c g ( v ) ⊥ ( x ) , φ aA ( x ) = f √ N c ddx g ( a ) ⊥ ( x ) , (40)where we use f as the “normalization” constant for both longitudinal polarized and transverselypolarized mesons.In SU(2) limit, due to G-parity, φ k , g ( a ) ⊥ and g ( v ) ⊥ are symmetric [antisymmetric] under thereplacement x ↔ − x for non-strange 1 P [1 P ] states, whereas φ ⊥ , h ( t ) || , and h ( s ) || are antisym-metric [symmetric]. In the above, we have taken f T P = f P = f [ f P = f T P ( µ = 1 GeV) = f ],thus we have h P ( P, ǫ ) | ¯ q σ µν γ q | i = f T P a ⊥ , P ( ǫ ∗ µ P ν − ǫ ∗ ν P µ ) , (41) h P ( P, ǫ ) | ¯ q γ µ γ q | i = if P a k , P m P ǫ ∗ µ , (42)where a ⊥ , P and a k , P are the Gegenbauer zeroth moments. That can give the following normal-ization for the distribution amplitudes: Z dxφ ⊥ ( x ) = a ⊥ " Z dxφ || ( x ) = a || , (43)for the 1 P [1 P ] states. The zeroth Gegenbauer moments a ⊥ , P and a k , P , characterizing thedegree of the flavor SU(3) symmetry breaking, are non-zero for only strange mesons. We normalizethe distribution amplitude φ k of the 1 P states as Z dxφ || ( x ) = 1 . (44)For convenience, we formally define a || = 1 for the 1 P states so that we can use Eq. (43) asthe normalization condition. Similarly, we also define a ⊥ = 1 for 1 P states so that φ ⊥ ( x ) has acorrect normalization.3 TABLE III: Gegenbauer moments of φ ⊥ and φ || for 1 P and 1 P mesons evaluated in Ref. [37], where thevalues are taken at µ = 1 GeV. a || ,a (1260)2 a || ,f P a || ,f P a || ,K A a || ,K A − . ± . − . ± . − . ± . − . ± .
03 0 . ± . a ⊥ ,a (1260)1 a ⊥ ,f P a ⊥ ,f P a ⊥ ,K A a ⊥ ,K A a ⊥ ,K A − . ± . − . ± . − . ± . − . ± .
48 0 . ± .
09 0 . ± . a || ,b (1235)1 a || ,h P a || ,h P a || ,K B a || ,K B a || ,K B − . ± . − . ± . − . ± . − . ± .
45 0 . ± .
15 0 . ± . a ⊥ ,b (1235)2 a ⊥ ,h P a ⊥ ,h P a ⊥ ,K B a ⊥ ,K B . ± .
19 0 . ± .
22 0 . ± . − . ± .
22 0 . ± . Up to conformal spin 6, twist-2 distribution amplitudes for axial-vector mesons can be expandedas: φ k ( x ) = 6 x ¯ x (cid:20) a k + 3 a k t + a k
32 (5 t − (cid:21) , (45) φ ⊥ ( x ) = 6 x ¯ x (cid:20) a ⊥ + 3 a ⊥ t + a ⊥
32 (5 t − (cid:21) , (46)wher the Gegenbauer moments are calculated in Refs. [36, 37] shown in table III. From theresults in table III, we can see that there are large uncertainties in Gegenbauer moments whichcan inevitably induce large uncertainties to branching ratios and CP asymmetries. We hope theuncertainties could be reduced in future studies in order to make more precise predictions.As for twist-3 LCDAs, we use the following form: g ( v ) ⊥ ( x ) = 34 a k (1 + t ) + 32 a k t , g ( a ) ⊥ ( x ) = 6 x ¯ x ( a k + a k t ) , (47) h ( t ) k ( x ) = 3 a ⊥ t + 32 a ⊥ t (3 t − , h ( s ) k ( x ) = 6 x ¯ x ( a ⊥ + a ⊥ t ) . (48)4 III. B → V AND B → A FORM FACTORS ¯ B → V form factors are defined under the conventional form as follows: h V ( P , ǫ ∗ ) | ¯ qγ µ b | ¯ B ( P ) i = − V ( q ) m B + m V ǫ µνρσ ǫ ∗ ν P ρ P σ , h V ( P , ǫ ∗ ) | ¯ qγ µ γ b | ¯ B ( P ) i = 2 im V A ( q ) ǫ ∗ · qq q µ + i ( m B + m V ) A ( q ) (cid:20) ǫ ∗ µ − ǫ ∗ · qq q µ (cid:21) − iA ( q ) ǫ ∗ · qm B + m V (cid:20) ( P + P ) µ − m B − m V q q µ (cid:21) , h V ( P , ǫ ∗ ) | ¯ qσ µν q ν b | ¯ B ( P ) i = − iT ( q ) ǫ µνρσ ǫ ∗ ν P ρ P σ , h V ( P , ǫ ∗ ) | ¯ qσ µν γ q ν b | ¯ B ( P ) i = T ( q ) (cid:2) ( m B − m V ) ǫ ∗ µ − ( ǫ ∗ · q )( P + P ) µ (cid:3) + T ( q )( ǫ ∗ · q ) (cid:20) q µ − q m B − m V ( P + P ) µ (cid:21) , (49)where q = P − P and the relation 2 m V A (0) = ( m B + m V ) A (0) − ( m B − m V ) A (0) is obtainedin order to cancel the pole at q = 0.In pQCD approach, the factorization formulae for these form factors at maximally recoiling( q = 0) are expressed by: V = 8 πC F m B ( m B + m V ) Z dx dx Z ∞ b db b db φ B ( x , b ) × (cid:26) E e ( t a ) h e ( x x , x , b , b ) h φ TV ( x ) + (2 + x ) r V φ aV ( x ) − r V x φ vV ( x ) i + E ′ e ( t ′ a ) h e ( x x , x , b , b ) r V h φ aV ( x ) + φ vV ( x ) i(cid:27) , (50) A = 8 πC F m B Z dx dx Z ∞ b db b db φ B ( x , b ) × (cid:26) E e ( t a ) h e ( x x , x , b , b ) h (1 + x ) φ V ( x ) + (1 − x ) r V ( φ sV ( x ) + φ tV ( x )) i +2 r V E ′ e ( t ′ a ) h e ( x x , x , b , b ) φ sV ( x ) (cid:27) , (51) A = 8 πC F m B ( m B − m V ) Z dx dx Z ∞ b db b db φ B ( x , b ) × (cid:26) E e ( t a ) h e ( x x , x , b , b ) h φ TV ( x ) + (2 + x ) r V φ vV ( x ) − r V x φ aV ( x ) i + E ′ e ( t ′ a ) h e ( x x , x , b , b ) r V h φ aV ( x ) + φ vV ( x ) i(cid:27) , (52) A = 1 m B − m V h ( m B + m V ) A − m V A i , (53)5 T = T = 8 πC F m B Z dx dx Z ∞ b db b db φ B ( x , b ) × (cid:26) E e ( t a ) h e ( x x , x , b , b ) h (1 + x ) φ TV ( x ) + (1 − x ) r V ( φ vV ( x ) + φ aV ( x )) i + E ′ e ( t ′ a ) h e ( x x , x , b , b ) r V ( φ vV ( x ) + φ aV ( x )) (cid:27) , (54) T = T − πC F m B r V Z dx dx Z ∞ b db b db φ B ( x , b ) × (cid:26) E e ( t a ) h e ( x x , x , b , b ) h φ V ( x ) + 2 r V φ tV ( x ) + r V x ( φ tV ( x ) − φ sV ( x )) i +2 r V E ′ e ( t ′ a ) h e ( x x , x , b , b ) φ sV ( x ) (cid:27) , (55)with C F = 4 /
3. The definitions of functions E i , the factorization scales t i and hard functions h i ,are given in Appendix A.With the above expressions for form factors, we obtain the numerical results that are collectedin table IV. When evaluating the form factor A , we used 2 m V A (0) = ( m B + m V ) A (0) − ( m B − m V ) A (0) and assume A and A are linearly correlated to estimate the uncertainties. The firsterror comes from decay constants of B ( s ) meson and shape parameters ω b ; while the second one isfrom hard scale t and Λ QCD . In the calculation, f B = (0 . ± .
02) GeV, ω B = (0 . ± .
05) GeV(for B ± and B d mesons) and f B s = (0 . ± .
02) GeV, ω B s = (0 . ± .
05) GeV (for B s meson) havebeen used. It is clear that these hadronic parameters give the dominant theoretical uncertainties.They quantify the SU(3)-symmetry breaking effects in the form factors in pQCD approach. Tomake a comparison, we also collect the results using other approaches [5, 6, 8, 39, 40]. Fromtable IV, we can see that most of our results are consistent with others within theoretical errors.Likewise, ¯ B → A form factors are defined by: h A ( P , ǫ ∗ ) | ¯ qγ µ γ b | ¯ B ( P ) i = − iA ( q ) m B − m A ǫ µνρσ ǫ ∗ ν P ρ P σ , h A ( P , ǫ ∗ ) | ¯ qγ µ b | ¯ B ( P ) i = − m V V ( q ) ǫ ∗ · qq q µ − ( m B − m A ) V ( q ) (cid:20) ǫ ∗ µ − ǫ ∗ · qq q µ (cid:21) + V ( q ) ǫ ∗ · qm B − m A (cid:20) ( P + P ) µ − m B − m A q q µ (cid:21) , h A ( P , ǫ ∗ ) | ¯ qσ µν γ q ν b | ¯ B ( P ) i = − T ( q ) ǫ µνρσ ǫ ∗ ν P ρ P σ , h A ( P , ǫ ∗ ) | ¯ qσ µν q ν b | ¯ B ( P ) i = − iT ( q ) (cid:2) ( m B − m A ) ǫ ∗ µ − ( ǫ ∗ · q )( P + P ) µ (cid:3) − iT ( q )( ǫ ∗ · q ) (cid:20) q µ − q m B − m A ( P + P ) µ (cid:21) , (56)with a factor − i different from B → V and the factor m B + m V ( m B − m V ) is replaced by m B − m A ( m B + m A ). Similar with B → V form factors, the relation 2 m A V = ( m B − m A ) V − ( m B + m A ) V TABLE IV: B → V form factors at maximally recoil, i.e. q = 0. The first error comes from decay constantsof B mesons and shape parameters ω b ; while the second one is from hard scale t and Λ QCD . B → ρ B → K ∗ B → ω B s → K ∗ B s → φ LFQM[6] V .
27 0 . A .
28 0 . A .
22 0 . A .
20 0 . V .
323 0 .
411 0 .
293 0 .
311 0 . A .
303 0 .
374 0 .
281 0 .
360 0 . A .
242 0 .
292 0 .
219 0 .
233 0 . A .
221 0 .
259 0 .
198 0 .
181 0 . T .
267 0 .
333 0 .
242 0 .
260 0 . V . A . A . A . T . V .
298 0 .
339 0 .
275 0 .
323 0 . A .
260 0 .
283 0 .
240 0 .
279 0 . A .
227 0 .
248 0 .
209 0 .
228 0 . A .
215 0 .
233 0 .
198 0 .
204 0 . T = T .
260 0 .
290 0 .
239 0 .
271 0 . T .
184 0 .
194 0 .
168 0 .
165 0 . V . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . A . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . A . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . A . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . T = T . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . T . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . is obtained at q = 0. In pQCD approach, B → A form factors’ formulas can be derived from thecorresponding B → V form factor formulas in eq.(50-55) using the replacement in Eq. (39) withthe proper change of the momentum fraction.In the following, we will use a to denote a (1260), b to denote b (1235). In Table V, we give the7 TABLE V: B → A form factors at maximally recoil, i.e. q = 0. Results in the first line of each formfactor are calculated using θ K = 45 ◦ , θ P = 10 ◦ or θ P = 38 ◦ , while the second line corresponds to theangle θ K = − ◦ , θ P = 45 ◦ or θ P = 50 ◦ . The errors are from: decay constants of B ( s ) meson and shapeparameter ω b ; Gegenbauer moments in axial-vectors’ LCDAs. B → K (1270) B → h (1170) B s → K (1270) B → h (1380) B s → h (1170) B s → h (1380) A − . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . V . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . V − . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . V − . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . T ( T ) − . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . T − . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . B → K (1400) B → f (1285) B s → K (1400) B → f (1420) B s → f (1420) B s → f (1285) A . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . V − . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . V . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . V . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . T ( T ) 0 . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . T . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . . +0 . . − . − . . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . − . +0 . . − . − . This work B → b B → a LFQM[6, 7] B → b ( K B ) B → a ( K A ) A . +0 . . − . − . . +0 . . − . − . A .
10 (0.11) 0 . V . +0 . . − . − . . +0 . . − . − . V . . V . +0 . . − . − . . +0 . . − . − . V . . V . +0 . . − . − . . +0 . . − . − . V − . − .
05) 0 . T ( T ) 0 . +0 . . − . − . . +0 . . − . − . T ( T ) –(0 .
13) –(0 . T . +0 . . − . − . . +0 . . − . − . T – ( − .
07) –(0 . TABLE VI: Distinct contributions to form factor T from various distribution amplitudes. B → ρ B → a (1260) B → b (1235) φ T .
086 0 .
142 0 . φ a .
047 0 .
086 0 . φ v .
063 0 .
115 0 . .
196 0 .
343 0 . numerical results for B → A form factors, in which we have used minus values for decays constantsof P mesons . The errors are from: decay constants of B ( s ) mesons and shape parameters ω b ;Gegenbauer moments in axial-vectors’ LCDAs. In the calculation, we use the mass of the twophysical states K (1270), K (1400) as that of two spin states K A , K B for simplicity and similarfor the branching ratios which are given in the following. That only involves a slight differenceto the form factors. As the quark contents (to be precise the mixing angles) of the axial-vectors K ( f , h ) have not been uniquely determined, we give two different kinds of results for form factorsas in Ref. [37]: the results in the first line are calculated using θ K = 45 ◦ while the second linecorresponds to the angle θ K = − ◦ . This is also done for the results involving the flavor-singletand flavor-octet mesons: the results in the first line are calculated using θ P = 10 ◦ , θ P = 38 ◦ ;while the second line corresponds to the angle θ P = 45 ◦ , θ P = 50 ◦ .A number of remarks on B → A form factors are in order.1. Form factors are strongly dependent on mixing angles. Many of them even are different byorder of magnitude, because the mixing angles describe directly the inner quark contentsof the meson. The large difference of form factors surely will induce large differences inbranching ratios, which we will see later.2. We give a comparison of the B → ρ , B → a (1260) and B → b (1235) form factors. Formfactors V , V , T for B → A transition are larger than the corresponding B → V ones. Itseems that the form factor A B → ( a ,b ) is somewhat equal to or even smaller than V B → ρ . Butactually that is artificial: as in Eq. (50), the pre-factor is m B + m V while for B → A formfactor A , the factor becomes m B − m A . We take T as an example to explain the reason Decay constants given in QCD sum rules [36, 37] are both positive for two kinds of axial-vectors and we find thatthis will give negative values for B → P form factors, like in [41]. For non-strange P mesons, this minus signwill not give any differences as it can not be observed experimentally. But we should point out the minus sign willaffect the mixing between K A and K B by changing the mixing angle θ to − θ . B → A form factors. In table VI, we give contributions from three kinds ofLCDAs: φ T , φ v and φ a . The contribution from φ T is larger for B → a , than the other twotransitions only because the axial-vector a decay constant is larger. Furthermore, largeraxial vector meson mass implies larger contribution from twist-3 distribution amplitudes φ v , φ a for both of T B → b and T B → a .3. In our calculation for form factors involving f mesons, we have used the mixing anglebetween the octet and singlet: θ = 38 ◦ (50 ◦ ) which is very close to the ideal mixing angle θ = 35 . ◦ . That implies that the lighter meson f (1285) is almost made up of ¯ uu + ¯ dd √ whilethe heavier meson f (1420) is dominated by the ¯ ss component. Thus B → f (1420) and B s → f (1285) form factors are suppressed by the flavor structure and are numerically small.The form factors involving h are similar if the mixing angle is taken as 45 ◦ .4. From the table V, we can see that the form factor T B → a is almost equal to T B → b . In theflavor SU(3) symmetry limit, B → K A and B → K B form factors are also almost equal witheach other. But the physical states K (1270) and K (1400) are mixtures of B → K A, B .With the mixing angle θ K = ± ◦ , the B d,s → K (1270)( K (1400)) form factors are eitherenhanced by a factor √ B → A form factors. Howeverthere are lots of studies using some non-perturbative methods: quark meson model [42], ISGW[43, 44], QCD sum rules and light-cone sum rules [38, 45, 46] and light-front quark model [6, 7].Results in LFQM are also given in table V to make a comparison. These two approaches arevery different in the treatment of dynamics of transition form factors, but at first we will analyzedifferences caused by non-perturbative inputs. For B → a and B → K A form factors, most ofour results (except V and T , ) are slightly larger than or almost equal with these of evaluated inLFQM, as slightly larger decay constants for a and K A are used: f a = 203 MeV and f K A = 186MeV. The form factor V is calculated by the relation 2 m A V = ( m B − m A ) V − ( m B + m A ) V .Small differences in V and V have induced a large difference in V , which could be reduced infuture studies using more precise hadronic inputs. We have found there are large differences in B → P transition form factors. As the decay constant of b is zero in isospin limit, thus in LFQMthe shape parameter ω can not be directly determined and Cheng and Chua used the same valuewith that of a [6]. It is also similar for K B : they used the same shape parameter with that of K A which predicts f K B = 11 MeV. Compared with the QCD sum rule results f K B = f TK B × a || given0in table II and III, we can see: although they are consistent within large theoretical errors, there arestill large differences in the central value. Thus our predictions for B → P form factors (centralvalues) are larger than those in LFQM. We have to confess that the differences in decays constantsare not responsible for all differences in form factors. That may arise from further differences inthe dynamics. Compared with the recent light-cone sum rules results [46]: V Ba (0) = 0 . +0 . − . , V BK A (0) = 0 . +0 . − . , (57) V Bb (0) = − . +0 . − . , V BK B (0) = − . +0 . − . , (58)where uncertainties are from Borel window and input parameters, we can see that they are well con-sistent with our calculations in pQCD approach. As mentioned in Ref. [41], the Babar measurementof ¯ B → a +1 π − [47] favors V B → a (0) ≃ .
30 and this is very close to our result: V B → a (0) = 0 . IV. CALCULATION OF RADIATIVE DECAY B → V γ
A. The factorization formulae for decay amplitude
For convenience, we define a common factor F which appears in many diagrams by: F = em B C F π . (59)As we have mentioned in the above section, we have to use the amplitudes with distinct chiralities.The explicit factorization formulae for the left-handed and right-handed photon from operator O γ depicted in Fig. 1 are given by: M L γ = 4 r b F Z dx dx Z ∞ b db b db φ B ( x , b ) (cid:26) C γ ( t a ) E e ( t a ) × h (1 + x ) φ TV ( x ) + r V (1 − x )( φ vV ( x ) + φ aV ( x )) i h e ( x x , x , b , b )+ r V h φ vV ( x ) + φ aV ( x ) i C γ ( t ′ a ) E ′ e ( t ′ a ) h e ( x x , x , b , b ) (cid:27) , (60) M R γ = − r D r b M L γ , (61)where the left-helicity amplitude is from the m b term in the effective Hamiltonian and the right-helicity amplitude is from the m D term which is obviously highly suppressed. This O γ contribution1 b ¯ q γ ¯ qDO g ¯ qD ¯ qb γ O g b ¯ q γ ¯ qDO g ¯ qb D ¯ qO g γ FIG. 3: Feynman diagrams of the chromo-magnetic penguin operator O g is the dominant one characterizing by the form factor T ( T ). The formulas in eq.(60,61) are thesame as that in eq.(54) times the Wilson coefficient C .In pQCD approach, a hard gluon is required to kick the soft spectator in the B meson toturn into an energetic collinear anti-quark. This gluon could be generated from QCD interactionHamiltonian or from the O g operator. In Fig. 3, we give the four diagrams from O g operatorgiven in the effective Hamiltonian. The factorization formulae for the first two diagrams in Fig. 3are M L ( a )8 g = 2 r b F Z dx dx Z ∞ b db b db φ B ( x , b ) (cid:26) Q D C g ( t b ) E e ( t b ) × h (2 x − x ) φ TV ( x ) − x r V ( φ vV ( x ) + φ aV ( x )) i h e ( x x , x − , b , b ) (62)+ h x φ TV ( x ) + x r V ( φ vV ( x ) + φ aV ( x )) i Q b C g ( t ′ b ) E ′ e ( t ′ b ) h e ( x x , x , b , b ) (cid:27) , M R ( a )8 g = − r D r b M L ( a )8 g . (63)If we consider the last two diagrams in Fig. 3, there will be more sources to generate the right-handed photon in addition to the m D term in the effective electro-weak Hamiltonian. The thirddiagram can give a small contribution which is from the higher twist component: M L ( b )8 g ( Q q ) = 2 r b Q q F Z dx dx Z ∞ b db b db φ B ( x , b ) (cid:26) C g ( t c ) E e ( t c ) × h (2 + x − x ) φ TV ( x ) + 3 x r V ( φ vV ( x ) + φ aV ( x )) i h e ( x − x , − x , b , b )+ h x r ( φ vV ( x ) + φ aV ( x )) − x φ TV ( x ) i C g ( t ′ c ) E ′ e ( t ′ c ) h e ( x − x , x , b , b ) (cid:27) − r D Q q F Z dx dx Z ∞ b db b db φ B ( x , b ) C g ( t c ) E e ( t c ) × h x r V ( φ vV ( x ) − φ aV ( x )) i h e ( x − x , − x , b , b ) , (64)2 ¯ qD ¯ qb γ O b ¯ q γ ¯ qDO b ¯ q γ ¯ qDO ¯ qb D ¯ qγO FIG. 4: Feynman diagrams in which the operator O is inserted in the loop with a photon emitted from theexternal quark line M R ( b )8 g ( Q q ) = − r D Q q F Z dx dx Z ∞ b db b db φ B ( x , b ) (cid:26) C g ( t c ) E e ( t c ) × h (2 + x − x ) φ TV ( x ) + 3 x r V ( φ vV ( x ) + φ aV ( x )) i h e ( x − x , − x , b , b )+ h x r V ( φ vV ( x ) + φ aV ( x )) − x φ TV ( x ) i C g ( t ′ c ) E ′ e ( t ′ c ) h e ( x − x , x , b , b ) (cid:27) +2 r b Q q F Z dx dx Z ∞ b db b db φ B ( x , b ) C g ( t c ) E e ( t c ) × h x r V ( φ vV ( x ) − φ aV ( x )) i h e ( x − x , − x , b , b ) . (65)Next we want to mention some higher order corrections as usual which may give importantcontributions: charm and up quark loop ( O ( α s )) contributions in Fig. 4, Fig. 5 and Fig. 6. Itshould be pointed out that these contributions are not related to next-to-leading order correctionsin pQCD approach, while next-to-leading order corrections to the exclusive processes πγ ∗ → γ inpQCD approach have been investigated in Ref. [12].We use the subtitle “quark line photon emission” to denote that a photon is emitted throughthe external quark lines as in Fig.4. We define the c and u loop function in order that the b → Dg vertex can be expressed as ¯ Dγ µ (1 − γ ) I aµν A aν b . It has the gauge invariant form [48] as follows: I aµν = gT a π ( k g µν − k µ k ν ) Z dxx (1 − x ) (cid:20) (cid:18) m i − x (1 − x ) k t (cid:19)(cid:21) = − gT a π ( k g µν − k µ k ν ) (cid:20) G ( m i , k , t ) − (cid:21) , (66)where k is the gluon momentum and m i is the loop internal quark mass. G is the function from3the loop integration: G ( m i , k , t ) = θ ( − k ) 23 "
53 + 4 m i k − ln m i t + (cid:18) m i k (cid:19) r − m i k ln p − m i /k − p − m i /k + 1 + θ ( k ) θ (4 m i − k ) 23 "
53 + 4 m i k − ln m i t − (cid:18) m i k (cid:19) r m i k − p m i /k − ! + θ ( k − m i ) 23 "
53 + 4 m i k − ln m i t + (cid:18) m i k (cid:19) r − m i k ln 1 − p − m i /k p − m i /k + iπ ! . (67) The loop function G has the dependence of gluon momentum square of k . But there is nosingularity when we take the limit of k →
0, so we can neglect k T components of k in the loopfunction G . Using this effective vertex, the factorization formulae for the first two diagrams ofFig.4 is calculated as: M L ( a )1 i = − Q D F Z dx dx Z b db b db φ B ( x , b ) C ( t b ) E e ( t b ) h G ( m i , − x x m B , t b ) − i × h x r V ( φ vV ( x ) + φ aV ( x )) + 3 x x φ TV ( x ) i h e ( x x , x − , b , b ) , (68) M R ( a )1 i = − Q b F Z dx dx Z b db b db φ B ( x , b ) C ( t ′ b ) E ′ e ( t ′ b ) h G ( m i , − x x m B , t ′ b ) − i × x x r V ( φ aV ( x ) − φ vV ( x )) h e ( x x , x , b , b ) , (69)where Q D = − . For the other two diagrams of Fig.4, the factorization formulas are M L ( b )1 i ( Q q ) = Q q F Z dx dx Z ∞ b db b db φ B ( x , b ) (cid:26) C ( t c ) E e ( t c ) × h x r V (1 + 2 x )[ φ vV ( x ) + φ aV ( x )] + 3( x − x ) φ TV ( x ) i × h G ( m i , − ( x − x ) m B , t c ) − i h e ( x − x , − x , b , b )+ h x r V ( φ vV ( x ) + φ aV ( x )) − x φ TV ( x ) ih G ( m i , − ( x − x ) m B , t ′ c ) − i × C ( t ′ c ) E ′ e ( t ′ c ) h e ( x − x , x , b , b ) (cid:27) , (70) M R ( b )1 i ( Q q ) = Q q F Z dx dx Z ∞ b db b db φ B ( x , b ) C ( t c ) E e ( t c ) h e ( x − x , − x , b , b ) × h G ( m i , − ( x − x ) m B , t c ) − i × (2 + x ) x r V h φ vV ( x ) − φ aV ( x ) i . (71)In Fig. 5, we give the diagrams in which a photon emitted from the internal loop quark line.The sum of the effective vertex b → Dγg ∗ in Fig.5 has been derived by [49, 50]: I = ¯ Dγ ρ (1 − γ ) T a bI µνρ A µ A aν , (72)4 b O b O ¯ qγ γ ¯ q ¯ qD ¯ qD FIG. 5: Feynman diagrams in which the operator O is inserted in the loop with both of the photon andthe virtual gluon emitted from the internal quark line. with the tensor structure given by I µνρ = A [( q · k ) ǫ µνρσ ( q − k ) σ + ǫ νρστ q σ k τ k µ − ǫ µρστ q σ k τ q ν ]+ A (cid:2) ǫ µρστ q σ k τ k ν − k ǫ µνρσ q σ (cid:3) , (73)and A = 4 ieg π Z dx Z − x dy xyx (1 − x ) k + 2 xyq · k − m i + iε , (74) A = − ieg π Z dx Z − x dy x (1 − x ) x (1 − x ) k + 2 xyq · k − m i + iε , (75)where q is the momentum of the photon q = P B − P V , and k is the momentum of the gluon. Thenthe amplitudes can be expressed as follows: M L i = − F Z dx Z − x dy Z dx dx Z b db φ B ( x , b ) C ( t d ) α s ( t d ) exp[ − S B ( t d )] × h ′ e xyx m B − m i × (cid:26) xyx h (1 − x ) r V ( φ vV ( x ) + φ aV ( x )) − (1 + 2 x ) φ TV ( x ) i + x (1 − x ) h x r V ( φ vV ( x ) + φ aV ( x )) + 3 x x φ TV ( x ) i(cid:27) , (76) M R i = 83 F Z dx Z − x dy Z dx dx Z b db φ B ( x , b ) C ( t d ) α s ( t d ) exp[ − S B ( t d )] × h ′ e xyx m B − m i × xyx r V h φ vV ( x ) − φ aV ( x ) i , (77)where the function h ′ e is defined by: h ′ e ≡ K ( √ x x m B b ) − h θ ( B ) K ( b p | B | ) + θ ( − B ) i π H ( b p | B | ) i , (78)with B = x x m B − y − x x m B + m i x (1 − x ) . (79)Diagrams in Fig. 6 in which the photon is emitted from the external loop and the gluon isemitted from QCD interaction Hamiltonian do not give any contribution to B → V γ . The5 b D ¯ q ¯ qγO D ¯ qb ¯ q γO FIG. 6: Feynman diagrams in which the operator O is inserted in the loop with a photon emitted from theinternal quark line b ¯ q b ¯ q b ¯ q γ b ¯ q γ γγO i O i O i O i q ¯ q q ¯ q q q ¯ q ¯ q FIG. 7: Feynman diagrams for annihilation topologies reason is as follows. Similar with b → Dg , the vertex for b → Dγ can also be expressed as¯ Dγ µ A ν (1 − γ )( k g µν − k µ k ν ) b only with a different coefficient. For an on-shell photon, the fol-lowing conditions are required: k = 0 and ǫ · k = 0, thus the contribution from diagrams in Fig. 6vanishes in b → s ( d ) γ decays. But it should be noted that these diagrams can give a non-zerocontribution to b → s ( d ) γ ∗ → s ( d ) l + l − .In annihilation diagrams, there are three different kinds of operators in the ⊗ depicted in Fig. 7.In the following, we use LL to denote the left-handed current between b and ¯ q quark and the left-handed current between the final state two quarks; LR denotes the left-handed current between b and ¯ q quark and the right-handed current between the final state two quarks; we use SP to denotethe ( S − P )( S + P ) current which is from the Fierz transformation of ( V − A )( V + A ) operators.6The factorization formulae for these diagrams are given by: M L ( a,LL ) ann ( a i , Q q ) = M L ( a,LR ) ann ( a i , Q q )= F √ Q q r V f V π m B Z dx Z ∞ b db a i ( t ′ e ) E a ( t ′ e ) φ B ( x , b ) K ( √ x m B b ) , (80) M R ( a,LL ) ann ( a i , Q q ) = M R ( a,LR ) ann ( a i , Q q ) (81)= − F √ Q b r V f V π m B Z dx Z ∞ b db a i ( t e ) E a ( t e ) φ B ( x , b ) K ( √ x m B b ) , M L ( b,LL ) ann ( a i , Q q , Q q ) = M R ( b,LR ) ann ( a i , Q q , Q q )= − F √ r V f B π m B Z dx Z b db h φ vV ( x ) + φ aV ( x ) i × (cid:26) Q q a i ( t f ) E ′ a ( t f ) i π H (1)0 ( √ − x m B b ) − x Q q a i ( t ′ f ) E ′ a ( t ′ f ) i π H (1)0 ( √ x m B b ) (cid:27) , (82) M R ( b,LL ) ann ( a i , Q q , Q q ) = M L ( b,LR ) ann ( a i , Q q , Q q )= − F √ r V f B π m B Z dx Z b db × (cid:26) (1 − x ) h φ vV ( x ) − φ aV ( x ) i Q q a i ( t f ) E ′ a ( t f ) i π H (1)0 ( √ − x m B b ) − h φ vV ( x ) − φ aV ( x ) i Q q a i ( t ′ f ) E ′ a ( t ′ f ) i π H (1)0 ( √ x m B b ) (cid:27) , (83) M L ( SP ) ann ( a i ) = F √ f B πm B Z dx Z b db φ TV ( x ) (84) × (cid:26) Q q a i ( t f ) E ′ a ( t f ) i π H (1)0 ( √ − x m B b ) + Q q a i ( t ′ f ) E ′ a ( t ′ f ) i π H (1)0 ( √ x m B b ) (cid:27) . Finally, there is another kind of contribution from O γ : the neutral vector meson ¯ qq is generatedby a photon as depicted in Fig. 8. Although these diagrams are suppressed by the electromagneticcoupling constant, the enhancement factor m B / Λ QCD can make it important in some cases [51].We include these diagrams in our calculation. The first two diagrams of Fig. 8 are equal to eachother, so are the last two diagrams. The factorization formulae are given by: M Len ( Q q ) = − F √ α em Q q Q b r b f V m B m V Z dx Z ∞ b db φ B ( x , b ) × (cid:26) C γ ( t e ) E a ( t e ) K ( √ x m B b ) + C γ ( t ′ e ) E a ( t ′ e ) K ( √ x m B b ) (cid:27) , (85) M Ren ( Q q ) = − r D r b M Len ( Q q ) . (86)7 O γ b ¯ D O γ b ¯ D O γ b ¯ D O γ b ¯ Dq qqq
FIG. 8: Feynman diagrams with double-photon contributions
B. Numerical results of Branching ratios
With those decay amplitude formulas for different Feynman diagrams in the last subsection, itis easy to get the total decay amplitude for each channel of B → V γ : B − → ρ − γ , B → ρ γ , B → ωγ , B − → K ∗− γ , B → φγ , B → K ∗ γ , B s → K ∗ γ , B s → ρ γ , B s → ωγ , and B s → φγ .The explicit expressions are shown in Appendix B. The CP-averaged decay width is thenΓ( B → V γ ) = |A ( ¯ B → V γ ) | + |A ( B → ¯ V γ ) | πm B (1 − r V ) , (87)where the summation on polarizations is implemented.For CKM matrix elements, we use the same values as in Ref. [19]: | V ud | = 0 . , | V us | = 0 . , | V ub | = (3 . +0 . − . ) × − , | V td | = (8 . +0 . − . ) × − , | V ts | = 40 . × − , | V tb | = 1 . ,α = (99 +4 − . ) ◦ , γ = (59 . +9 . − . ) ◦ , arg[ − V ts V ∗ tb ] = 1 . ◦ , (88)where we have adopted the updated results from [52] and drop the (small) errors on V ud , V us , V ts and V tb . The CKM factors mostly give an overall factor to branching ratios. However, the CKMangles do give large uncertainties to branching ratios of some decay modes and to all the non-zeroCP asymmetries which will be discussed in the following subsection. CP -averaged branching ratios of B → V γ decays are listed in table VII. The first error in theseentries arises from the input hadronic parameters, which is dominated by B ( B s )-meson decayconstants (taken as f B = (0 . ± .
02) GeV and f B s = (0 . ± .
02) GeV) and B ( B s ) mesonwave function shape parameters (taken as ω B = (0 . ± .
05) GeV and ω B s = (0 . ± .
05) GeV).The second error is from the hard scale t , defined in Eqs. (A1) – (A8) in Appendix A, which8 TABLE VII: CP -averaged branching ratios ( × − ) of B → V γ decays obtained in pQCD approach (Thiswork); the errors for these entries correspond to uncertainties in input hadronic quantities, from the scale-dependence, and CKM matrix elements, respectively. For comparison, we also listed the current experimentalmeasurements [3, 69, 70, 71] and theoretical estimates of branching ratios recently given in Ref. [31] (QCDF)and in Ref. [64] (SCET).Modes QCDF SCET This work Exp. B − → K ∗− γ . ± . ± . ± ± ± ± . +17 . . . − . − . − . . ± . B → ¯ K ∗ γ . ± . ± . ± ± ± ± . +17 . . . − . − . − . . ± . B s → φγ . ± . ± . ± ± ± ± . +13 . . . − . − . − . +18+12 − − [Belle]Modes QCDF This work Exp. B − → ρ − γ . ± . ± .
13 1 . +0 . . . − . − . − . . +0 . − . ± .
09 [BaBar] 0 . +0 . . − . − . [Belle]¯ B → ρ γ . ± . ± .
07 0 . +0 . . . − . − . − . . +0 . − . ± .
06 [BaBar] 1 . +0 . . − . − . [Belle]¯ B → ωγ . ± . ± .
05 0 . +0 . . . − . − . − . . +0 . − . ± .
05 [BaBar] 0 . +0 . . − . − . [Belle]¯ B s → K ∗ γ . ± . ± .
18 1 . +0 . . . − . − . − . —¯ B → φγ — (7 . +2 . . . − . − . − . ) × − < .
85 [HFAG]¯ B s → ρ γ — (1 . +0 . . . − . − . − . ) × − —¯ B s → ωγ — (1 . +0 . . . − . − . − . ) × − — we vary from 0 . t to 1 . t (not changing 1 /b i ), and from Λ (5) QCD = 0 . ± .
05 GeV. This scale-dependence characterize the size of next-to-leading order contributions in pQCD approach. A partof this perturbative improvement coming from next-to-leading order Wilson coefficients is alreadyavailable [22]. However, the complete next-to-leading order corrections to hard spectator kernelsare still missing. The third error is the combined uncertainties in CKM matrix elements and anglesof the unitarity triangle. It is clear that the largest uncertainty here is the first one from the inputhadronic parameters.These ten B → V γ decay channels can be divided into three different types: b → s transitions, b → d transitions and purely annihilation decays. The first type contains ¯ B → ¯ K ∗ γ , B − → K ∗− γ and ¯ B s → φγ . Among these decays, the dominant contribution from O γ , is proportional to V tb V ∗ ts ∼ λ . This contribution can be related to the form factor T B → V . In the flavor SU(3)symmetry limit, form factors for these three channels should be equal which could also relates thethree decays. We do obtain similar branching ratios for this kind decays only with small deviations.The penguin contribution in b → dγ processes is proportional to V tb V ∗ td ∼ λ , which is expected9to be suppressed by one order magnitude relative to the b → s transitions. In our calculation,branching ratio results for pure annihilation processes are mainly from two-photon diagrams inFig. 8. Thus BR ( B → φγ ) is surely consistent with Ref. [51]. This feature can also certainlyinterpret the large differences between ¯ B s → ρ γ and ¯ B s → ωγ . This contribution is proportionalto the charge of the constitute quark. This factor is − − for ρ while + − for ω . Thus thebranching ratio of B s → ρ γ is one order in magnitude larger than that of ( B s → ωγ ).In the literature, there are many studies concentrating on B → V γ [16, 17, 18, 53, 54, 55, 56, 57,58, 59, 60, 61, 62, 63, 64]. Recently, a comprehensive study [31] using QCDF method and QCD sumrules appears. In that paper, the authors used QCDF approach to calculate all B → V γ approachand included some power corrections: weak annihilation contributions, the soft-gluon emission fromquark (charm and light-quark) loops and long distance photon emission from the soft quarks. InTable VII, we quote them to make a comparison. Their uncertainties come from form factors, therenormalization scale, the soft-gluon terms, the CKM parameters, decay constants, Gegenbauermoments, the first inverse moments of B ( B s ) mesons and quark masses, etc. Their results forbranching ratios of B − → K ∗− γ and ¯ B → ¯ K ∗ γ are about (20 − T : they used T B → K ∗ = 0 . ± .
04 while ourcalculation gives a smaller T = 0 . +0 . − . . The smaller form factor is also preferred by recentLattice QCD result: T B → K ∗ = 0 . ± . +0 . − . . Although the difference is not too large, it canalready induce a sizable difference to branching ratios. b → d transitions and ¯ B s → φγ are wellconsistent with each other, as the effective form factors used in Ref. [31] are smaller and almostequal to our results. Very recently, the authors in Ref. [64] used the more theoretical approachSCET to investigate the three b → s decays channels: B − → K ∗− γ , ¯ B → ¯ K ∗ γ and ¯ B s → φγ .After integrating out the hard scale m b which results in SCET I , contributions to B → V γ decayamplitudes can be divided into two different groups: contributions from operators J A and J B .The B -type operator is power suppressed in SCET I but it can give leading power contributionswhen matching onto SCET II as A -type operator receives power suppressions. When performingthe matching from SCET I to SCET II , we have to be cautious about the A -type operator as thisterm suffers from the end-point singularities. Thus one has to leave it as a non-perturbative freeparameter determined from experiments or some non-perturbative QCD approaches, but recentstudies using zero-bin subtractions show that this term can also factorized in rapidity space [65]. InRef. [64], the authors calculated the two-loop corrections ( α s ) to short-distance coefficients of the A -type operator (called vertex functions) determined from QCD to SCET I matching and utilizedthe physical B → V form factor T to extract the soft form factor in SCET. While for the B -0type operator which can been factorized into convolutions of LCDAs and hard kernels, both ofthe Wilson coefficient and jet function have been calculated up to one loop order. Short-distanceWilson coefficients for the B -type operator are extracted at the scale m b and have been evolveddown the intermediate scale µ ∼ . α s correctionsfrom vertex corrections and hard spectator scattering can be as large as 10% and both of themalso provide imaginary amplitudes about 5% in magnitude; the order α s corrections are not toolarge. We quote their final results for branching ratios in table VII and they are also consistentwith ours.The three experimental collaborations, BaBar [66], Belle [67] and CLEO [68], have reported theirmeasurements on BR ( B → K ∗ γ ). Since all of these results are well consistent with each other,we quote the averaging results from Heavy Flavor Averaging Group (HFAG) [3] in table VII. Wealso include the very recent result on BR ( ¯ B s → φγ ) [69]. All of them agree with our calculations.On the b → d transition, branching ratios of some channels have been given by the BaBar [70]and Belle [71] collaboration. We find that our results agree well with BaBar’s results but notwith Belle’s central value results. In flavor SU(3) symmetry limit, the relation BR ( B − → ρ − γ ) =2 BR ( ¯ B → ρ γ ) = BR ( ¯ B → ωγ ) should be held. The small deviation in our calculation is causedby the SU(3) symmetry breaking effect. Since the electro-magnetic penguin operator O γ givesthe dominant contribution, it is difficult to understand the results from Belle collaboration: whyis the branching ratio of ¯ B → ρ γ larger than B − → ρ − γ . But before we conclude it is the signalfor non-standard model scenarios, it is necessary to re-examine this channel on the experimentalside. All other decay modes, including ¯ B s decays and annihilation type decays, have not beenexperimentally measured. C. CP asymmetry studies
The direct CP-asymmetry in ¯ B → V γ is defined by: A dir CP ≡ BR ( ¯ B → ¯ V γ ) − BR ( B → V γ ) BR ( ¯ B → ¯ V γ ) + BR ( B → V γ ) = |A ( ¯ B → ¯ V γ ) | − |A ( B → V γ ) | |A ( ¯ B → ¯ V γ ) | + |A ( B → V γ ) | . (89)In order to give a non-zero direct CP asymmetry, we need two kinds of contributions with differentstrong phases and different weak phases. The magnitude of the CP asymmetry also depends onrelative sizes of the two different amplitudes: if one amplitude is much larger than the other one,we can only get a small CP asymmetry. Since there is only penguin contribution in ¯ B → φγ B → V γ decays, there are contributions fromthe penguin operator and two kinds of tree operators (proportional to V cb V ∗ cd,cs and V ub V ∗ ud,us ).Taking these amplitudes into account, we obtain the numerical results for direct CP asymmetries(in %) in other B → V γ decays as: A dir CP ( B − → ρ − γ ) = 12 . +0 . . . − . − . − . , (90) A dir CP ( ¯ B → ρ γ ) = 12 . +0 . . . − . − . − . , (91) A dir CP ( B − → K ∗− γ ) = − . ± . ± . ± . , (92) A dir CP ( ¯ B → ¯ K ∗ γ ) = − . +0 . . . − . − . − . , (93) A dir CP ( ¯ B → ωγ ) = 12 . +0 . . . − . − . − . , (94) A dir CP ( ¯ B s → ρ γ ) = − . +0 . . . − . − . − . , (95) A dir CP ( ¯ B s → K ∗ γ ) = 12 . +0 . . . − . − . − . , (96) A dir CP ( ¯ B s → ωγ ) = − . +0 . . . − . − . − . , (97) A dir CP ( ¯ B s → φγ ) = − . +0 . . . − . − . − . , (98)where the three kinds of errors are given as that of the branching ratios case. It is easy to see thattheoretical uncertainties here are much smaller than branching ratios in table VII, especially thefirst one from hadronic input parameters, since they are mostly canceled in eq.(89). In the three b → s channels B − → K ∗− γ , ¯ B → ¯ K ∗ γ and ¯ B s → φγ , CKM matrix element for the magneticpenguin operator is V tb V ∗ ts ∼ λ , while the tree operator is either proportional to V cb V ∗ cs ∼ λ or V ub V ∗ us ∼ λ . The CKM matrix element in the first tree operator is almost parallel to thepenguin operator. This kind of contribution has a same weak phase with the penguin contribution.The second tree operator is small in magnitude. Thus we expect small CP asymmetries in thesechannels. Experimentally, both BaBar and Belle collaboration give their combined measurementsof the two B → K ∗ γ channels [72]: A dir CP ( B → K ∗ γ ) = − . ± . ± . , [Belle] , − . ± . ± . , [BaBar] . (99)Although the central value of CP asymmetry is larger than the one in our calculation in eq.(92,93),it is still consistent with zero.In annihilation-type decays B s → ρ ( ω ) γ , the tree amplitude is also suppressed by CKM matrixelements, thus the CP asymmetry is small too. In b → d transitions B − → ρ ∗− γ , ¯ B → ρ ( ω ) γ and ¯ B s → K ∗ γ , the CKM matrix element for the magnetic penguin operator is V tb V ∗ td ∼ λ , while2the tree operator is either proportional to V cb V ∗ cd ∼ λ or V ub V ∗ ud ∼ λ . Then tree contribution isnot suppressed and can be comparative with the penguin contribution. Thus we expect relativelylarge CP asymmetries in these four processes, which are also shown in the above.Restricting the final vector state V to have definite CP-parity, the time-dependent decay widthfor the B → f decay is:Γ( B ( t ) → f ) = e − Γ t Γ( B → f ) h cosh (cid:16) ∆Γ t (cid:17) + H f sinh (cid:16) ∆Γ t (cid:17) − A dir CP cos(∆ mt ) − S f sin(∆ mt ) i , (100)where ∆ m = m H − m L >
0, Γ is the average decay width, and ∆Γ = Γ H − Γ L is the differenceof decay widths for the heavier and lighter B mass eigenstates. The time dependent decay widthΓ( ¯ B ( t ) → f ) is obtained from the above expression by flipping the signs of the cos(∆ mt ) andsin(∆ mt ) terms. In the B d system, ∆Γ is small and can be neglected. In the B s system, we expecta much larger decay width difference (∆Γ / Γ) B s = − . ± .
024 [73] within the standard model,while experimentally (∆Γ / Γ) B s = − . +0 . − . [3], so that both S f and H f , can be extracted fromthe time dependent decays of B s mesons. The definition of the various quantities in the aboveequation are as follows: S f ( V γ ) = 2 Im (cid:16) qp ( A ∗ L ¯ A L + A ∗ R ¯ A R ) (cid:17) |A L | + |A R | + | ¯ A L | + | ¯ A R | , (101) H f ( V γ ) = 2 Re (cid:16) qp ( A ∗ L ¯ A L + A ∗ R ¯ A R ) (cid:17) |A L | + |A R | + | ¯ A L | + | ¯ A R | , (102)where ¯ A and A denote the amplitudes for the ¯ B and B meson decays. q/p is given in terms of the B q - ¯ B q mixing matrix M , qp = r M ∗ M = e iφ q (103)with φ d ≡ − arg[( V ∗ td V tb ) ] = − β , φ s ≡ − arg[( V ∗ ts V tb ) ] = 2 ǫ. (104)where the convention arg[ V cb ] = arg[ V cs ] = 0 is adopted.In b → Dγ ( D = d, s ) processes, the dominant contribution to decay amplitudes comes fromthe chiral-odd dipole operator O . As only left-handed quarks participate in the weak interaction,an effective operator of this type necessitates, a helicity flip on one of the external quark lines,which results in a factor m b (and a left-handed photon) in b R → D L γ L and a factor m D (and aright-handed photon) in b L → D R γ R . Hence, the emission of right-handed photon is suppressed3 TABLE VIII: Mixing-induced CP-asymmetry parameters (in percentage) of B → V γ decays obtained inpQCD approach. The errors are the same with table VII. The H f parameter in B d decays could hardly bemeasured as the decay width difference is small.Modes S f H f ¯ B → ¯ K ∗ γ − . +0 . . . − . − . − . . +0 . . . − . − . − . ¯ B s → φγ . +0 . . . − . − . − . . +0 . . . − . − . − . ¯ B → ρ γ . +0 . . . − . − . − . . +0 . . . − . − . − . ¯ B → ωγ . +0 . . . − . − . − . . +0 . . . − . − . − . ¯ B s → K ∗ γ . +0 . . . − . − . − . − . +0 . . . − . − . − . by a factor m D /m b . In the b → Dγ process, the emitted photon is predominantly left-handed,and right-handed in ¯ b decays. This leads to very small predictions of S f and H f . The mixing-induced CP asymmetry variables are calculated and summarized in table VIII. ¯ B → ¯ K ∗ γ hasbeen treated as an effective flavor eigenstate. Apparently, the numerical results agree with ourexpectations. On the experimental side, the mixing-induced CP asymmetries have been measuredby Belle and BaBar as follows [74, 75, 76]: S f ( B → K ∗ γ → K S π γ ) = − . +0 . − . ± . , [Belle] , − . ± . ± . , [BaBar] , (105) S f ( ¯ B → ρ γ ) = − . ± . ± . . (106)They are consistent with zero since there are large uncertainties in these results. The theoreticalresults agree with the experimental data taking the experimental uncertainty into account. But asthis parameter could be a good probe to detect the non-standard scenarios, more studies, includingboth of the precise experimental studies and the theoretical studies, are strongly deserving. D. Isospin asymmetry and U-spin asymmetry
Apart from branching ratios and CP asymmetry variables, we will also consider some ratiosof branching fractions defined below. In the evaluations for branching fractions, there are manyuncertainties, especially from hadronic input parameters, which can blur our predictions, but we canimprove the accuracy of our predictions by using ratios of branching fractions. Many uncertainties,such as those from decay constants, will cancel in these parameters. The most important ratios4are the parameters characterizing isospin asymmetries which are defined by: A ( ρ, ω ) = Γ( B → ωγ )Γ( B → ρ γ ) − , (107) A I ( ρ ) = 2Γ( ¯ B → ρ γ )Γ( ¯ B ± → ρ ± γ ) − , (108) A I ( K ∗ ) = Γ( ¯ B → K ∗ γ ) − Γ( B ± → K ∗± γ )Γ( ¯ B → K ∗ γ ) + Γ( B ± → K ∗± γ ) , (109)where the partial decay rates are CP-averaged.In the flavor SU(3) symmetry limit and if we neglect diagrams which are proportional to thequark charge, all of these three parameters should be equal to 0. The ω meson decay constantis smaller than that for ρ meson, the ¯ uu component contributes with a different sign and theelectomagnetic diagrams with two photons are different in the charge factor. These differencesmake A ( ρ, ω ) deviate from 0 (smaller than 0). The origins for deviations for A I ( ρ ) and A I ( K ∗ )from 0 are similar: the spectator quarks are different and annihilation diagrams are also different.Taking on all those power suppressed contributions, our predictions are A ( ρ, ω ) = − . +0 . . . − . − . − . , (110) A I ( ρ ) = 0 . +0 . . . − . − . − . , (111) A I ( K ∗ ) = 0 . +0 . . . − . − . − . . (112)As we expected, theoretical uncertainties due to the hadronic parameters are indeed smaller due tocancelations. Using experimental results listed in table VII, we can calculate the isospin asymmetryparameters from experiments A ( ρ, ω ) = − . +0 . − . [BaBar] , − . +0 . − . [Belle] , (113) A I ( ρ ) = − . +0 . − . [BaBar] , . +3 . − . [Belle] , (114) A I ( K ∗ ) = 0 . ± . , (115)where we have assumed all the uncertainties are not correlated by adding them quadratically.From the experimental results, except A I ( K ∗ ), we find that there are large differences between theresults from the two collaborations. Our results are consistent with them, since the error bars inthe experiments are too large. We wish a more precise measurement on these parameters in thefuture.5Apart from the isospin symmetry, U-spin symmetry is another kind symmetry which is wellheld in strong interactions. U-spin can connect two different kinds of weak decays [77, 78, 79]: b → s (∆ S = 1) and b → d by exchange of d ↔ s . The decay amplitudes of b → s process can beexpressed as: A ( B → f ) = V ∗ ub V us A u + V ∗ cb V cs A c , (116) A ( ¯ B → ¯ f ) = V ub V ∗ us A u + V cb V ∗ cs A c , (117)while the decay amplitudes of U B → U f are A ( U B → U f ) = V ∗ ub V ud U A u + V ∗ cb V cd U A c , (118) A ( U ¯ B → U ¯ f ) = V ub V ∗ ud U A u + V cb V ∗ cd U A c . (119)Using the relation A u = U A u and A c = U A c in U-spin symmetry limit and the CKM unitarityrelation I m ( V ∗ ub V us V cd V ∗ cs ) = − I m ( V ∗ ub V ud V cd V ∗ cd ) , (120)we obtain |A ( B → f ) | − |A ( ¯ B → ¯ f ) | = −|A ( U B → U f ) | + |A ( U ¯ B → U ¯ f ) | . (121)This equation relates the differences of partial decay widths. The following radiative B decayscan be related to each other by this symmetry: B − → ρ − γ and B − → K ∗− γ ; ¯ B → ¯ K ∗ γ and¯ B s → K ∗ γ . As an example, we define the following parameter to test U-spin symmetry breaking:∆ ≡ A CP ( B − → K ∗− γ ) − A CP ( B − → ρ − γ ) × BR ( B − → ρ − γ ) BR ( B − → K ∗− γ ) . (122)In our pQCD calculation, we find∆ = ( − . +0 . . . − . − . − . ) × − . (123)This result is close to 0. The U spin symmetry seems quite good here. But the most importantreason is the small CP asymmetry in B − → K ∗− γ and the small ratio BR ( B − → ρ − γ ) BR ( B − → K ∗− γ ) . The absolutevalue for the b → s channel’s CP asymmetry is small. This direct CP asymmetry may be dramati-cally enhanced by some new physics with a different weak phase. The precise measurement of CPasymmetries can at least give a constraint on the non-standard model scenario parameters.6 V. CALCULATION OF B → A (1 P ) γ AND A (1 P ) γ DECAYS
The factorization formulae for B → Aγ is more complicated than B → V γ because of themixing between different mesons: K A and K B ; f and f ; h and h . The real physical states K (1270) and K (1400) are mixtures of the K A and K B states with the mixing angle θ K : | K (1270) i = | K A i sin θ K + | K B i cos θ K , (124) | K (1400) i = | K A i cos θ K − | K B i sin θ K . (125)In flavor SU(3) symmetry limit, these mesons can not mix with each other; but since s quark isheavier than the u, d quarks, K (1270) and K (1400) are not purely 1 P or 1 P states. In general,the mixing angle can be determined by experimental data. The partial decay rate for τ − → K ν τ is given by: Γ( τ − → K ν τ ) = m τ π G F | V us | f A (cid:18) − m A m τ (cid:19) (cid:18) m A m τ (cid:19) , (126)with the measured results for branching fractions [30]: BR ( τ − → K (1270) ν τ ) = (4 . ± . × − , BR ( τ − → K (1400) ν τ ) = (1 . ± . × − . (127)We can straightforward obtain the longitudinal decay constant (in MeV): | f K (1270) | = 169 +19 − ; | f K (1400) | = 125 + 74 − . (128)In principle, one can combine the decay constants for K A , K B evaluated in QCD sum rules withthe above results to determine the mixing angle θ K . But since there are large uncertainties inEq. (128), the constraint on the mixing angle is expected to be rather smooth: − ◦ < θ K < − ◦ , or − ◦ < θ K < − ◦ , or 37 ◦ < θ K < ◦ , or 131 ◦ < θ K < ◦ , (129)where we have taken the uncertainties from the branching ratios in Eq.(127) and the first Gegen-bauer moment a K into account but neglected the mass differences as usual. For simplicity, we usetwo reference values in Ref. [37] θ K = ± ◦ . (130)Besides, the flavor-octet and the flavor-singlet can also mix with each other: | f (1285) i = | f i cos θ P + | f i sin θ P , | f (1420) i = −| f i sin θ P + | f i cos θ P , (131) | h (1170) i = | h i cos θ P + | h i sin θ P , | h (1380) i = −| h i sin θ P + | h i cos θ P . (132)7The references points are chosen as: θ P = 38 ◦ or θ P = 50 ◦ ; θ P = 10 ◦ or θ P = 45 ◦ [37]. Weshould point out that if the mixing angle is θ = 35 . ◦ , the mixing is ideal: f (1285) is made up of ¯ uu + ¯ dd √ while f (1420) is composed of ¯ ss . Thus some of the form factors are very small which willof course give small production rates of this meson.Apart from these differences, the expression for B → Aγ is different from B → V γ in moreaspects. Since the twist-2 LCDA φ || is normalized to a || , we should replace the decay constant f V by f A a || in the first two annihilation diagrams. As there is no overlap between an axial-vectormeson and a photon (wrong parity), there is no contribution from the two photon electromagneticoperator diagrams in Fig. 8. Regardless of these differences, the factorization formulae for B → Aγ decays can be obtained from the corresponding B → V γ ones using the replacement in Eq. (39) ifthe electroweak current is σ µν (1 + γ ) or γ µ (1 − γ ) type. If the current is σ µν (1 − γ ), we shouldadd an additional minus sign. In annihilation diagrams, if the electroweak current is LL or SP inthe lower two diagrams, we need replace the distribution amplitudes as in Eq. (39); while we adda minus sign if the current is LR . We show the formulas in Appendix C.Branching ratios (in unit of 10 − ) and direct CP asymmetries (in %) for B → ( a , b ) γ processesare calculated straightforward as follows: BR ( B − → a − (1260) γ ) = 3 . +1 . . . . − . − . − . − . , (133) BR ( ¯ B → a (1260) γ ) = 1 . +0 . . . . − . − . − . − . , (134) BR ( ¯ B s → a (1260) γ ) = (2 . +0 . . . . − . − . − . − . ) × − , (135) BR ( B − → b − (1235) γ ) = 2 . +1 . . . . − . − . − . − . , (136) BR ( ¯ B → b (1235) γ ) = 1 . +0 . . . . − . − . − . − . , (137) BR ( ¯ B s → b (1235) γ ) = (5 . +1 . . . . − . − . − . − . ) × − , (138) A dirCP ( B − → a − (1260) γ ) = 11 . +2 . . . . − . − . − . − . , (139) A dirCP ( ¯ B → a (1260) γ ) = 3 . +0 . . . . − . − . − . − . , (140) A dirCP ( ¯ B s → a (1260) γ ) = 0 . +0 . . . . − . − . − . − . , (141) A dirCP ( B − → b − (1235) γ ) = 16 . +1 . . . . − . − . − . − . , (142) A dirCP ( ¯ B → b (1235) γ ) = 11 . +0 . . . . − . − . − . − . , (143) A dirCP ( ¯ B s → b (1235) γ ) = − . +0 . . . . − . − . − . − . , (144)while we give branching ratios and CP asymmetries for B → K ( f , h ) γ in table IX and X, respec-tively. The errors for these entries correspond to uncertainties in the input hadronic quantities,8 TABLE IX: CP -averaged branching ratios ( × − ) of B → Aγ decays obtained in pQCD approach usingtwo different mixing angles; the errors for these entries correspond to uncertainties in the input hadronicquantities, from the scale-dependence, and CKM matrix elements, the Gegenbauer moments of the axial-vector mesons respectively.Modes θ K = 45 ◦ θ K = − ◦ Exp. B − → K − (1270) γ +68+21+4+41 − − − − . +1 . . . . − . − . − . − . . ± . ± . B → ¯ K (1270) γ +64+19+4+45 − − − − . +0 . . . . − . − . − . − . B − → K − (1400) γ . +1 . . . . − . − . − . − . +68+21+4+41 − − − − < . B → ¯ K (1400) γ . +0 . . . . − . − . − . − . +64+19+4+45 − − − − ¯ B s → K (1270) γ . +0 . . . . − . − . − . − . . +0 . . . . − . − . − . − . ¯ B s → K (1400) γ . +0 . . . . − . − . − . − . . +0 . . . . − . − . − . − . Modes θ P = 38 ◦ θ P = 50 ◦ ¯ B → f (1285) γ . +0 . . . . − . − . − . − . . +0 . . . . − . − . − . − . ¯ B → f (1420) γ (4 . +2 . . . . − . − . − . − . ) × − . +0 . . . . − . − . − . − . ¯ B s → f (1285) γ . +0 . . . . − . − . − . − . . +1 . . . . − . − . − . − . ¯ B s → f (1420) γ . +24 . . . . − . − . − . − . . +22 . . . . − . − . − . − . Modes θ P = 10 ◦ θ P = 45 ◦ ¯ B → h (1170) γ . +0 . . . . − . − . − . − . . +0 . . . . − . − . − . − . ¯ B → h (1380) γ . +0 . . . . − . − . − . − . (2 . +0 . . . . − . − . − . − . ) × − ¯ B s → h (1170) γ . +2 . . . . − . − . − . − . . +0 . . . . − . − . − . − . ¯ B s → h (1380) γ . +16 . . . . − . − . − . − . . +18 . . . . − . − . − . − . the scale-dependence, CKM matrix elements, and the Gegenbauer moments of the axial-vectormesons. It is noted that theoretical uncertainties for branching ratios are quite large. The branch-ing fractions of B → a (1260)( b (1235)) γ are larger than that of B → ργ , as we have shown that B → A form factors are larger. As we have mentioned in the above, there are some ambiguities inthe quark content of B → K ( f , h ) γ : these mesons are mixtures but the mixing angles are notuniquely determined. The reference points for the mixing angles are two-fold, thus we give twodifferent kinds of results collected in these two tables. The branching ratios and CP asymmetriesare very sensitive to the mixing angles, which is not quite constrained.Experimentalist gave results for B − → K − (1270) γ [21] shown also in table IX. Compared withit, our result for B − → K − (1270) γ , is about 3 times larger, when θ K = 45 ◦ ; or very smaller than theexperimental results, when θ K = − ◦ . In Fig. 9, we show the strong dependence of the branching9 TABLE X: Direct CP asymmetries of B → Aγ decays obtained in pQCD approach using two differentmixing angles; the errors for these entries correspond to uncertainties in the input hadronic quantities, fromthe scale-dependence, and CKM matrix elements, respectively.Modes θ K = 45 ◦ θ K = − ◦ B − → K − (1270) γ − . +0 . . . . − . − . − . − . − . +0 . . . . − . − . − . − . ¯ B → ¯ K (1270) γ − . ± . ± . ± . ± . . +0 . . . . − . − . − . − . B − → K − (1400) γ − . +0 . . . . − . − . − . − . − . +0 . . . . − . − . − . − . ¯ B → ¯ K (1400) γ . +0 . . . . − . − . − . − . − . ± . ± . ± . ± . B s → K (1270) γ − . +0 . . . . − . − . − . − . − . +3 . . . . − . − . − . − . ¯ B s → K (1400) γ − . +3 . . . . − . − . − . − . − . +0 . . . . − . − . − . − . Modes θ P = 38 ◦ θ P = 50 ◦ ¯ B → f (1285) γ . +0 . . . . − . − . − . − . . +0 . . . . − . − . − . − . ¯ B → f (1420) γ . +0 . . . . − . − . − . − . . +0 . . . . − . − . − . − . ¯ B s → f (1285) γ − . +0 . . . . − . − . − . − . − . +0 . . . . − . − . − . − . ¯ B s → f (1420) γ − . ± . ± . ± . ± . − . ± . ± . ± . ± . θ P = 10 ◦ θ P = 45 ◦ ¯ B → h (1170) γ . +0 . . . . − . − . − . − . . +0 . . . . − . − . − . − . ¯ B → h (1380) γ . +0 . . . . − . − . − . − . . +0 . . . . − . − . − . − . ¯ B s → h (1170) γ − . ± . ± . ± . ± . − . +0 . . . . − . − . − . − . ¯ B s → h (1380) γ − . ± . ± . ± . ± . − . ± . ± . ± . ± . ratio on the mixing angle. At θ K = 45 ◦ , the B − → K − (1270) γ receives almost a maximalbranching ratio. The current B − → K − (1270) γ experiment implies our chosen two referencepoints of the mixing angle are not favored. From this Fig. 9, we could read out the experimentalconstrained mixing angle value, which are also two fold. However large hadronic uncertainties andthe missing next-to-leading order corrections [80] plus the still large experimental error bars makethis constraint not very effective. Except for these two processes, other decay modes have notbeen measured. Here we refrain from a direct comparison with the previous studies on B → Aγ [7, 38, 81, 82, 83, 84], as the analysis is similar in B → A form factors which has been performedin section III.0 - - -
50 0 50 100 150 Θ K BR H B - ® K - H L Γ H X - L FIG. 9: The θ K dependence of the B − → K − (1270) γ branching ratio. The region between the two horizontallines are allowed by the experimental 1 σ bound, where we add the statistic and the systematic uncertaintieslinearly: 29 . < BR < . VI. SUMMARY pQCD approach is based on k T factorization where we keep the transverse momentum of valencequarks in the meson, to smear the endpoint singularity. k T resummation of double logarithmsresults in the Sudakov factor. Resummation of double logarithms from the threshold region leadsto the jet function. Sudakov factor and jet function can suppress the contribution from the large b region and small x region, respectively. This makes the pQCD approach self-consistent. Inspiredby the success of pQCD approach in non-leptonic B decays [85], we give a comprehensive study onthe charmless B ( s ) → V ( A ) γ decays in pQCD approach.Semi-leptonic and radiative decays are somewhat simpler than non-leptonic decays as onlyone hadronic meson involved in the final state. In this case, the dominant amplitude can beparameterized in form factors. In order to make precise prediction and extract CKM matrixelements, we have to know the behavior of form factors. In pQCD approach, the final statemeson moves nearly on the light-cone and a hard-gluon-exchange is required. Thus the dominantcontribution is from the hard region which can be factorized. In section III, we have used thesame input hadronic parameters with Ref. [19] and updated all the B → V decay form factors inpQCD approach. Compared with the results evaluated from other approaches, we find: despite ofa number of theoretical differences in different approaches, all the numerical results of the formfactors are surprisingly consistent with each other.The 10 B → V γ decay channels can be divided into three categories based on their dominantquark transition b → s , b → d and the annihilation topology. Our prediction on the first category1of decays BR ( B → K ∗ γ ) is consistent with the averaged value from experiments. On the b → d transition, branching ratios have been given by BaBar and Belle collaborations with still large errorbars. We find our results are well consistent with BaBar’s results but a little far from the Belle’scentral value results in some channels. We also give our predictions on the purely annihilation typedecays with very small branching ratios in SM. In three b → s transitions B − → K ∗− γ , ¯ B → ¯ K ∗ γ and ¯ B s → φγ , the direct CP asymmetry is small, since the tree contribution is suppressed by theCKM matrix element. In the b → d transitions B − → ρ − γ , ¯ B → ¯ ρ ( ω ) γ and ¯ B s → K ∗ γ ,the tree contribution can be comparable with the penguin contribution. Thus we obtain large CPasymmetries in these four processes. In SM, the two quantities S f and H f in time-dependent decayare expected to be rather small. This is due to the fact that the dominant contribution to decayamplitudes comes from the chiral-odd dipole operator O . Except for a few decays discussed in theabove, all other decay modes including ¯ B s decays and annihilation type decays, remain essentiallyunexplored. We wish a wealth of measurements at the B factories and other experiments in thefuture.In section III, we also study B → A form factors. As the quark contents (to be precise themixing angle) for the axial-vectors have not been uniquely determined, we give two different kindsof results for the form factors according to different mixing angles. For the axial-vector mesons f , we have used the mixing angle between the octet and singlet: θ = 38 ◦ (50 ◦ ) which is close tothe ideal mixing angle θ = 35 . ◦ . With this mixing angle, one can easily check that the lightermeson f (1285) is made almost up of ¯ uu + ¯ dd √ while the heavier meson f (1420) is composed of ¯ ss .Thus partial decay widths of B → f (1420) γ and B s → f (1285) γ are suppressed by the flavorstructure. In Fig. 9, we show the strong dependence of the B − → K − (1270) γ decay branchingratio on the mixing angle θ K . Our calculation can be used to constrain this mixing angle usingexperimental measurements provided with well understood hadronic inputs. The study of higherresonance production in B decays can help us to uncover the mysterious structure of these excitedstates. Acknowledgements
This work is partly supported by National Science Foundation of China under the GrantNo. 10475085 and 10625525. We would like to acknowledge S.-Y. Li, Y. Li, Y.-L. Shen, X.-X.Wang, Y.-M. Wang, K.-C. Yang, M.-Z. Yang and H. Zou for valuable discussions.2
APPENDIX A: PQCD FUNCTIONS
In this appendix, we group the functions which appear in the factorization formulae. The hardscales are chosen as t a = max {√ x m B , /b , /b } , t ′ a = max {√ x m B , /b , /b } , (A1) t b = max {√ x x m B , p (1 − x ) m B , /b , /b } , (A2) t ′ b = max {√ x x m B , √ x m B , /b , /b } , (A3) t c = max { p | x − x | m B , √ x m B , /b , /b } , (A4) t ′ c = max { p | x − x | m B , √ x m B , /b , /b } , (A5) t d = max {√ x x m B , p | B | , /b } , (A6) t e = max {√ x m B , /b } , t ′ e = max {√ x m B , /b } , (A7) t f = max {√ − x m B , /b } , t ′ f = max {√ x m B , /b } . (A8)The functions h i in decay amplitudes are from the propagators of virtual quark and gluon andare defined by: h e ( A, B, b , b ) = h θ ( A ) K ( √ Am B b ) + θ ( − A ) i π H ( √− Am B b ) i × (cid:26) θ ( b − b ) h θ ( B ) K ( √ Bm B b ) I ( √ Bm B b )+ θ ( − B ) i π H (1)0 ( √− Bm B b ) J ( √− Bm B b ) i + ( b ↔ b ) (cid:27) , (A9)where H (1)0 ( z ) = J ( z ) + i Y ( z ).The Sudakov factor from threshold resummation is universal, independent of flavors of internalquarks, twists, and the specific processes. To simplify the analysis, the following parametrizationhas been used [26]: S t ( x ) = 2 c Γ(3 / c ) √ π Γ(1 + c ) [ x (1 − x )] c , (A10)with c = 0 .
4. This parametrization, symmetric under the interchange of x and 1 − x , is convenientfor evaluation of the amplitudes. It is obvious that the threshold resummation modifies the end-point behavior of the meson distribution amplitudes, rendering them vanish faster at x → E ( ′ ) e and E ( ′ ) a are given by E e ( t ) = α s ( t ) S t ( x ) exp[ − S B ( t ) − S ( t )] , E ′ e ( t ) = α s ( t ) S t ( x ) exp[ − S B ( t ) − S ( t )] , (A11) E a ( t ) = S t ( x ) exp[ − S B ( t )] , E ′ a ( t ) = S t ( x ) exp[ − S ( t )] , (A12)3in which the Sudakov exponents are defined as S B ( t ) = s (cid:18) x m B √ , b (cid:19) + 53 Z t /b d ¯ µ ¯ µ γ q ( α s (¯ µ )) , (A13) S ( t ) = s (cid:18) x m B √ , b (cid:19) + s (cid:18) (1 − x ) m B √ , b (cid:19) + 2 Z t /b d ¯ µ ¯ µ γ q ( α s (¯ µ )) , (A14)with the quark anomalous dimension γ q = − α s /π . The explicit form for the function s ( Q, b ) is: s ( Q, b ) = A (1) β ˆ q ln (cid:18) ˆ q ˆ b (cid:19) − A (1) β (cid:16) ˆ q − ˆ b (cid:17) + A (2) β (cid:18) ˆ q ˆ b − (cid:19) − " A (2) β − A (1) β ln (cid:18) e γ E − (cid:19) ln (cid:18) ˆ q ˆ b (cid:19) + A (1) β β ˆ q " ln(2ˆ q ) + 1ˆ q − ln(2ˆ b ) + 1ˆ b + A (1) β β h ln (2ˆ q ) − ln (2ˆ b ) i , (A15)where the variables are defined byˆ q ≡ ln[ Q/ ( √ , ˆ b ≡ ln[1 / ( b Λ)] , (A16)and the coefficients A ( i ) and β i are β = 33 − n f , β = 153 − n f ,A (1) = 43 , A (2) = 679 − π − n f + 83 β ln( 12 e γ E ) , (A17) n f is the number of the quark flavors and γ E is the Euler constant. We will use the one-looprunning coupling constant, i.e. we pick up only the four terms in the first line of the expression forthe function s ( Q, b ). APPENDIX B: ANALYTIC FORMULAE FOR THE B → V γ
DECAY AMPLITUDES
The analytic formulae for B − → ρ − γ is: A i ( B − → ρ − γ ) = G F √ V ub V ∗ ud n M i ( a )1 u + M i ( b )1 u ( Q u ) + M i u + M i ( a,LL ) ann ( a , Q u ) + M i ( b,LL ) ann ( a , Q d , Q u ) o + G F √ V cb V ∗ cd n M i ( a )1 c + M i ( b )1 c ( Q u ) + M i c o − G F √ V tb V ∗ td n M i γ + M i ( a )8 g + M i ( b )8 g ( Q u ) + M i ( a,LL ) ann ( a + a , Q u )+ M i ( b,LL ) ann ( a + a , Q d , Q u ) + M i ( SP ) ann ( a + a , Q d , Q u ) o , (B1)4while the expression for B − → K ∗− γ is basically the same except with the only difference in theCKM matrix elements: V qd → V qs . The formulas for other channels are √ A i ( ¯ B → ρ γ ) = G F √ V ub V ∗ ud n M i ( a,LL ) ann ( a , Q d ) + M i ( b,LL ) ann ( a , Q u , Q u ) − M i ( a )1 u − M i ( b )1 u ( Q d ) − M i u o + G F √ V cb V ∗ cd n − M i ( a )1 c − M i ( b )1 c ( Q d ) − M i c o − G F √ V tb V ∗ td n − M i γ − M i ( a )8 g −M i ( b )8 g ( Q d ) + M i ( a,LL ) ann ( − a + 32 a + 32 a + 12 a , Q d )+ M i ( b,LL ) ann ( a + a , Q u , Q u ) + M i ( b,LR ) ann ( a + a , Q u , Q u )+ M i ( b,LL ) ann ( − a − a + 12 a + 12 a , Q d , Q d ) + M i ( b,LR ) ann ( − a + 12 a , Q d , Q d )+ M i ( SP ) ann ( − a + 12 a , Q d , Q d ) + M ien ( Q u − Q d ) o , (B2) √ A i ( ¯ B → ωγ ) = G F √ V ub V ∗ ud n M i ( a,LL ) ann ( a , Q d ) + M i ( b,LL ) ann ( a , Q u , Q u ) + M i ( a )1 u + M i ( b )1 u ( Q d ) + M i u o + G F √ V cb V ∗ cd n M i ( a )1 c + M i ( b )1 c ( Q d ) + M i c o − G F √ V tb V ∗ td n M i γ + M i ( a )8 g + M i ( b )8 g ( Q d ) + M i ( a,LL ) ann (2 a + a + 2 a + 12 a + 12 a − a , Q d )+ M i ( b,LL ) ann ( a + a , Q u , Q u ) + M i ( b,LR ) ann ( a + a , Q u , Q u )+ M i ( b,LL ) ann ( a + a − a − a , Q d , Q d ) + M i ( b,LR ) ann ( a − a , Q d , Q d )+ M i ( SP ) ann ( a − a , Q d , Q d ) + M ien ( Q u + Q d ) o , (B3) A i ( ¯ B → ¯ K ∗ γ ) = G F √ V ub V ∗ us n M i ( a )1 u + M i ( b )1 u ( Q d ) + M i u o + G F √ V cb V ∗ cs n M i ( a )1 c + M i ( b )1 c ( Q d ) + M i c o − G F √ V tb V ∗ ts n M i γ + M i ( a )8 g + M i ( b )8 g ( Q d ) + M i ( a,LL ) ann ( a − a , Q d )+ M i ( b,LL ) ann ( a − a , Q s , Q d ) + M i ( SP ) ann ( a − a , Q s , Q d ) o . (B4)The expression for ¯ B s → K ∗ γ can be obtained by replacing V qs by V qd from ¯ B → ¯ K ∗ γ .5The formulas for the B s → φγ decay are A i ( ¯ B s → φγ ) = G F √ V ub V ∗ us n M i ( a )1 u + M i ( b )1 u ( Q s ) + M i u o + G F √ V cb V ∗ cs n M i ( a )1 c + M i ( b )1 c ( Q s ) + M i c o − G F √ V tb V ∗ ts n M i γ + M i ( a )8 g + M i ( b )8 g ( Q s )+ M i ( a,LL ) ann ( a + a + a − a − a − a , Q s )+ M i ( b,LL ) ann ( a + a − a − a , Q s , Q s ) + M i ( b,LR ) ann ( a − a , Q s , Q s )+ M i ( SP ) ann ( a − a , Q s , Q s ) + M ien ( Q s ) o . (B5)For the annihilation type decays, we have A i ( ¯ B → φγ ) = − G F √ V tb V ∗ td n M i ( a,LL ) ann ( a + a − a − a , Q s ) + M ien ( Q s )+ M i ( b,LL ) ann ( a − a , Q s , Q s ) + M i ( b,LR ) ann ( a − a , Q s , Q s ) o , (B6) √ A i ( ¯ B s → ρ γ ) = G F √ V ub V ∗ us n M i ( a,LL ) ann ( a , Q s ) + M i ( b,LL ) ann ( a , Q u , Q u ) o − G F √ V tb V ∗ ts n M i ( a,LL ) ann ( 32 a + 32 a , Q s ) + M ien ( Q u − Q d )+ M i ( b,LL ) ann ( a + a , Q u , Q u ) + M i ( b,LR ) ann ( a + a , Q u , Q u )+ M i ( b,LL ) ann ( − a + 12 a , Q d , Q d ) + M i ( b,LR ) ann ( − a + 12 a , Q d , Q d ) o , (B7) √ A i ( ¯ B s → ωγ ) = G F √ V ub V ∗ us n M i ( a,LL ) ann ( a , Q s ) + M i ( b,LL ) ann ( a , Q u , Q u ) o − G F √ V tb V ∗ ts n M i ( a,LL ) ann (2 a + 2 a + 12 a + 12 a , Q s ) + M ien ( Q u + Q d )+ M i ( b,LL ) ann ( a + a , Q u , Q u ) + M i ( b,LR ) ann ( a + a , Q u , Q u )+ M i ( b,LL ) ann ( a − a , Q d , Q d ) + M i ( b,LR ) ann ( a − a , Q d , Q d ) o . (B8) APPENDIX C: ANALYTIC FORMULAE FOR THE B → Aγ DECAY AMPLITUDES
The expression for B → P γ is different from B → V γ in various aspects. The first twoannihilation diagrams vanish for neutral axial vector mesons (not including K B ). There is not anytwo-photon diagram contribution in B → Aγ decays. The explicit formula for the B − → b − (1235) γ A i ( B − → b − (1235) γ ) = G F √ V ub V ∗ ud n M i ( a )1 u + M i ( b )1 u ( Q u ) + M i u + M i ( b,LL ) ann ( a , Q d , Q u ) o + G F √ V cb V ∗ cd n M i ( a )1 c + M i ( b )1 c ( Q u ) + M i c o − G F √ V tb V ∗ td n M i γ + M i ( a )8 g + M i ( b )8 g ( Q u )+ M i ( b,LL ) ann ( a + a , Q d , Q u ) + M i ( SP ) ann ( a + a , Q d , Q u ) o , (C1)while the expression for U-spin related process B − → K − B γ is basically the same except with theonly difference in the CKMmatrix elements: V qd → V qs . The formulas for the neutral decays modesare √ A i ( ¯ B → b (1235) γ ) = G F √ V ub V ∗ ud n M i ( b,LL ) ann ( a , Q u , Q u ) − M i ( a )1 u − M i ( b )1 u ( Q d ) − M i u o + G F √ V cb V ∗ cd n − M i ( a )1 c − M i ( b )1 c ( Q d ) − M i c o − G F √ V tb V ∗ td n − M i γ −M i ( b )8 g ( Q d ) + M i ( b,LL ) ann ( a + a , Q u , Q u ) + M i ( b,LR ) ann ( a + a , Q u , Q u )+ M i ( b,LL ) ann ( − a − a + 12 a + 12 a , Q d , Q d ) − M i ( a )8 g + M i ( b,LR ) ann ( − a + 12 a , Q d , Q d ) + M i ( SP ) ann ( − a + 12 a , Q d , Q d ) o , (C2) A i ( ¯ B → ¯ K B γ ) = G F √ V ub V ∗ us n M i ( a )1 u + M i ( b )1 u ( Q d ) + M i u o + G F √ V cb V ∗ cs n M i ( a )1 c + M i ( b )1 c ( Q d ) + M i c o − G F √ V tb V ∗ ts n M i γ + M i ( a )8 g + M i ( b )8 g ( Q d ) + M i ( a,LL ) ann ( a − a , Q d )+ M i ( b,LL ) ann ( a − a , Q s , Q d ) + M i ( SP ) ann ( a − a , Q s , Q d ) o , (C3)and the expression for ¯ B s → K B γ can be obtained by replacing V qs by V qd .The decay amplitudes involving h (1170) and h (1380) can obtained as A i ( B → h (1170) γ ) = A i ( B → h γ )cos θ P + A i ( B → h γ )sin θ P , (C4) A i ( B → h (1380) γ ) = − A i ( B → h γ )sin θ P + A i ( B → h γ )cos θ P , (C5)7where B denotes ¯ B or ¯ B s with √ A i ( ¯ B → h γ ) = G F √ V ub V ∗ ud √ n M i ( b,LL ) ann ( a , Q u , Q u ) + M i ( a )1 u + M i ( b )1 u ( Q d ) + M i u o + G F √ V cb V ∗ cd √ n M i ( a )1 c + M i ( b )1 c ( Q d ) + M i c o − G F √ V tb V ∗ td √ n M i γ + M i ( a )8 g + M i ( SP ) ann ( a − a , Q d , Q d ) o + M i ( b )8 g ( Q d ) + M i ( b,LL ) ann ( a + a , Q u , Q u ) + M i ( b,LR ) ann ( a + a , Q u , Q u )+ M i ( b,LL ) ann ( a + a − a − a , Q d , Q d ) + M i ( b,LR ) ann ( a − a , Q d , Q d ) − M i ( b,LL ) ann ( a − a , Q s , Q s ) − M i ( b,LR ) ann ( a − a , Q s , Q s ) o , (C6) √ A i ( ¯ B → h γ ) = G F √ V ub V ∗ ud √ n M i ( b,LL ) ann ( a , Q u , Q u ) + M i ( a )1 u + M i ( b )1 u ( Q d ) + M i u o + G F √ V cb V ∗ cd √ n M i ( a )1 c + M i ( b )1 c ( Q d ) + M i c o − G F √ V tb V ∗ td √ n M i γ + M i ( a )8 g + M i ( SP ) ann ( a − a , Q d , Q d ) o + M i ( b )8 g ( Q d ) + M i ( b,LL ) ann ( a + a , Q u , Q u ) + M i ( b,LR ) ann ( a + a , Q u , Q u )+ M i ( b,LL ) ann ( a + a − a − a , Q d , Q d ) + M i ( b,LR ) ann ( a − a , Q d , Q d )+ M i ( b,LL ) ann ( a − a , Q s , Q s ) + M i ( b,LR ) ann ( a − a , Q s , Q s ) o , (C7) A i ( ¯ B s → h γ ) = G F √ V ub V ∗ us − √ n M i ( a )1 u + M i ( b )1 u ( Q s ) + M i u o + G F √ V cb V ∗ cs − √ n M i ( a )1 c + M i ( b )1 c ( Q s ) + M i c o − G F √ V tb V ∗ ts − √ n M i γ + M i ( a )8 g + M i ( b )8 g ( Q s ) + M i ( SP ) ann ( a − a , Q s , Q s )+ M i ( b,LL ) ann ( a + a − a − a , Q s , Q s ) + M i ( b,LR ) ann ( a − a , Q s , Q s ) o , + G F √ V ub V ∗ us √ n M i ( b,LL ) ann ( a , Q u , Q u ) o − G F √ V tb V ∗ ts √ n M i ( b,LL ) ann ( a + a , Q u , Q u ) + M i ( b,LR ) ann ( a + a , Q u , Q u )+ M i ( b,LL ) ann ( a − a , Q d , Q d ) + M i ( b,LR ) ann ( a − a , Q d , Q d ) o , (C8)8 A i ( ¯ B s → h γ ) = G F √ V ub V ∗ us √ n M i ( a )1 u + M i ( b )1 u ( Q s ) + M i u o + G F √ V cb V ∗ cs √ n M i ( a )1 c + M i ( b )1 c ( Q s ) + M i c o − G F √ V tb V ∗ ts √ n M i γ + M i ( a )8 g + M i ( b )8 g ( Q s ) + M i ( SP ) ann ( a − a , Q s , Q s )+ M i ( b,LL ) ann ( a + a − a − a , Q s , Q s ) + M i ( b,LR ) ann ( a − a , Q s , Q s ) o , + G F √ V ub V ∗ us √ n M i ( b,LL ) ann ( a , Q u , Q u ) o − G F √ V tb V ∗ ts √ n M i ( b,LL ) ann ( a + a , Q u , Q u ) + M i ( b,LR ) ann ( a + a , Q u , Q u )+ M i ( b,LL ) ann ( a − a , Q d , Q d ) + M i ( b,LR ) ann ( a − a , Q d , Q d ) o . (C9)The annihilation type decay amplitude is: √ A i ( ¯ B s → b (1235) γ ) = G F √ V ub V ∗ us n M i ( b,LL ) ann ( a , Q u , Q u ) o − G F √ V tb V ∗ ts n M i ( b,LL ) ann ( a + a , Q u , Q u ) + M i ( b,LR ) ann ( a + a , Q u , Q u )+ M i ( b,LL ) ann ( − a + 12 a , Q d , Q d ) + M i ( b,LR ) ann ( − a + 12 a , Q d , Q d ) o . (C10)In the following, we will give the analytic factorization formulae for B → P γ which is similarwith B → V γ except for some differences in the flavor structure and zero contribution from thetwo-photon diagrams, A i ( B − → a − (1235) γ ) = G F √ V ub V ∗ ud n M i ( a )1 u + M i ( b )1 u ( Q u ) + M i u + M i ( a,LL ) ann ( a , Q u )+ M i ( b,LL ) ann ( a , Q d , Q u ) o + G F √ V cb V ∗ cd n M i ( a )1 c + M i ( b )1 c ( Q u ) + M i c o − G F √ V tb V ∗ td n M i γ + M i ( a )8 g + M i ( b )8 g ( Q u ) + M i ( a,LL ) ann ( a + a , Q u )+ M i ( b,LL ) ann ( a + a , Q d , Q u ) + M i ( SP ) ann ( a + a , Q d , Q u ) o , (C11)while the expression for B − → K − A γ is the same except with the only difference in the CKM9matrix elements: V qd → V qs . The formulas for other channels are √ A i ( ¯ B → a (1235) γ ) = G F √ V ub V ∗ ud n M i ( a,LL ) ann ( a , Q d ) + M i ( b,LL ) ann ( a , Q u , Q u ) − M i ( a )1 u −M i ( b )1 u ( Q d ) − M i u o + G F √ V cb V ∗ cd n − M i ( a )1 c − M i ( b )1 c ( Q d ) − M i c o − G F √ V tb V ∗ td n − M i γ − M i ( a )8 g −M i ( b )8 g ( Q d ) + M i ( a,LL ) ann ( − a + 32 a + 32 a + 12 a , Q d )+ M i ( b,LL ) ann ( a + a , Q u , Q u ) + M i ( b,LR ) ann ( a + a , Q u , Q u )+ M i ( b,LL ) ann ( − a − a + 12 a + 12 a , Q d , Q d ) (C12)+ M i ( b,LR ) ann ( − a + 12 a , Q d , Q d ) + M i ( SP ) ann ( − a + 12 a , Q d , Q d ) o , A i ( ¯ B → ¯ K A γ ) = G F √ V ub V ∗ us n M i ( a )1 u + M i ( b )1 u ( Q d ) + M i u o + G F √ V cb V ∗ cs n M i ( a )1 c + M i ( b )1 c ( Q d ) + M i c o − G F √ V tb V ∗ ts n M i γ + M i ( a )8 g + M i ( b )8 g ( Q d ) + M i ( a,LL ) ann ( a − a , Q d )+ M i ( b,LL ) ann ( a − a , Q s , Q d ) + M i ( SP ) ann ( a − a , Q s , Q d ) o , (C13)and the expression for ¯ B s → K A γ can be obtained by replacing V qs by V qd .The decay amplitudes involving f (1285) and f (1420) can obtained as A i ( B → f (1285) γ ) = A i ( B → f γ )cos θ P + A i ( B → f γ )sin θ P , (C14) A i ( B → h (1420) γ ) = − A i ( B → f γ )sin θ P + A i ( B → f γ )cos θ P , (C15)where B denotes ¯ B or ¯ B s with √ A i ( ¯ B → f γ ) = G F √ V ub V ∗ ud √ n M i ( a,LL ) ann ( a , Q d ) + M i ( b,LL ) ann ( a , Q u , Q u ) + M i ( a )1 u + M i ( b )1 u ( Q d ) + M i u o + G F √ V cb V ∗ cd √ n M i ( a )1 c + M i ( b )1 c ( Q d ) + M i c o − G F √ V tb V ∗ td √ n M i γ + M i ( a )8 g + M i ( b )8 g ( Q d )+ M i ( a,LL ) ann (2 a + a + 2 a + 12 a + 12 a − a , Q d )+ M i ( b,LL ) ann ( a + a , Q u , Q u ) + M i ( b,LR ) ann ( a + a , Q u , Q u )+ M i ( b,LL ) ann ( a + a − a − a , Q d , Q d ) + M i ( b,LR ) ann ( a − a , Q d , Q d )+ M i ( SP ) ann ( a − a , Q d , Q d ) − M i ( a,LL ) ann ( a + a − a − a , Q s ) − M i ( b,LL ) ann ( a − a , Q s , Q s ) − M i ( b,LR ) ann ( a − a , Q s , Q s ) o , (C16)0 √ A i ( ¯ B → f γ ) = G F √ V ub V ∗ ud √ n M i ( a,LL ) ann ( a , Q d ) + M i ( b,LL ) ann ( a , Q u , Q u ) + M i ( a )1 u + M i ( b )1 u ( Q d ) + M i u o + G F √ V cb V ∗ cd √ n M i ( a )1 c + M i ( b )1 c ( Q d ) + M i c o − G F √ V tb V ∗ td √ n M i γ + M i ( a )8 g + M i ( b )8 g ( Q d ) + M i ( a,LL ) ann (2 a + a + 2 a + 12 a + 12 a − a , Q d )+ M i ( b,LL ) ann ( a + a , Q u , Q u ) + M i ( b,LR ) ann ( a + a , Q u , Q u )+ M i ( b,LL ) ann ( a + a − a − a , Q d , Q d ) + M i ( b,LR ) ann ( a − a , Q d , Q d )+ M i ( SP ) ann ( a − a , Q d , Q d ) + M i ( a,LL ) ann ( a + a − a − a , Q s )+ M i ( b,LL ) ann ( a − a , Q s , Q s ) + M i ( b,LR ) ann ( a − a , Q s , Q s ) o , (C17) √ A i ( ¯ B s → f γ ) = G F √ V ub V ∗ us √ n M i ( a,LL ) ann ( a , Q s ) + M i ( b,LL ) ann ( a , Q u , Q u ) o − G F √ V tb V ∗ ts √ n M i ( a,LL ) ann (2 a + 2 a + 12 a + 12 a , Q s )+ M i ( b,LL ) ann ( a + a , Q u , Q u ) + M i ( b,LR ) ann ( a + a , Q u , Q u )+ M i ( b,LL ) ann ( a − a , Q d , Q d ) + M i ( b,LR ) ann ( a − a , Q d , Q d ) o + G F √ V ub V ∗ us − √ n M i ( a )1 u + M i ( b )1 u ( Q s ) + M i u o + G F √ V cb V ∗ cs − √ n M i ( a )1 c + M i ( b )1 c ( Q s ) + M i c o − G F √ V tb V ∗ ts − √ n M i γ + M i ( a )8 g + M i ( b )8 g ( Q s ) + M i ( SP ) ann ( a − a , Q s , Q s )+ M i ( a,LL ) ann ( a + a + a − a − a − a , Q s ) (C18)+ M i ( b,LL ) ann ( a + a − a − a , Q s , Q s ) + M i ( b,LR ) ann ( a − a , Q s , Q s ) o , √ A i ( ¯ B s → f γ ) = G F √ V ub V ∗ us √ n M i ( a,LL ) ann ( a , Q s ) + M i ( b,LL ) ann ( a , Q u , Q u ) o − G F √ V tb V ∗ ts √ n M i ( a,LL ) ann (2 a + 2 a + 12 a + 12 a , Q s )+ M i ( b,LL ) ann ( a + a , Q u , Q u ) + M i ( b,LR ) ann ( a + a , Q u , Q u )+ M i ( b,LL ) ann ( a − a , Q d , Q d ) + M i ( b,LR ) ann ( a − a , Q d , Q d ) o + G F √ V ub V ∗ us √ n M i ( a )1 u + M i ( b )1 u ( Q s ) + M i u o + G F √ V cb V ∗ cs √ n M i ( a )1 c + M i ( b )1 c ( Q s ) + M i c o − G F √ V tb V ∗ ts √ n M i γ + M i ( a )8 g + M i ( b )8 g ( Q s ) + M i ( SP ) ann ( a − a , Q s , Q s )+ M i ( a,LL ) ann ( a + a + a − a − a − a , Q s ) (C19)+ M i ( b,LL ) ann ( a + a − a − a , Q s , Q s ) + M i ( b,LR ) ann ( a − a , Q s , Q s ) o . For annihilation type decays, we have √ A i ( ¯ B s → a (1235) γ ) = G F √ V ub V ∗ us n M i ( a,LL ) ann ( a , Q s ) + M i ( b,LL ) ann ( a , Q u , Q u ) o − G F √ V tb V ∗ ts n M i ( a,LL ) ann ( 32 a + 32 a , Q s )+ M i ( b,LL ) ann ( a + a , Q u , Q u ) + M i ( b,LR ) ann ( a + a , Q u , Q u ) (C20)+ M i ( b,LL ) ann ( − a + 12 a , Q d , Q d ) + M i ( b,LR ) ann ( − a + 12 a , Q d , Q d ) o . Decay amplitudes of A i ( B → K (1270) γ ) and A i ( B → K (1400) γ ) (here B denotes ¯ B u,d,s and K denotes K − ( ¯ K )) can be obtained by: A i ( B → K (1270) γ ) = sin( θ K ) A i ( B → K A γ ) + cos( θ K ) A i ( B → K B γ ) , (C21) A i ( B → K (1400) γ ) = − sin( θ K ) A i ( B → K B γ ) + cos( θ K ) A i ( B → K A γ ) . (C22) [1] For a review, see T. Hurth, Rev. Mod. Phys. , 1159 (2003) [arXiv:hep-ph/0212304].[2] M. Misiak et al. , Phys. Rev. Lett. , 022002 (2007) [arXiv:hep-ph/0609232].[3] E. Barberio et al. [Heavy Flavor Averaging Group (HFAG)], arXiv:hep-ex/0603003.[4] For a review on the present status, please see T. Hurth, Int. J. Mod. Phys. A , 1781 (2007)[arXiv:hep-ph/0703226] and references therein.[5] L. Del Debbio, J. M. Flynn, L. Lellouch and J. Nieves [UKQCD Collaboration], Phys. Lett. B , 392(1998) [arXiv:hep-lat/9708008]. [6] H. Y. Cheng, C. K. Chua and C. W. Hwang, Phys. Rev. D , 074025 (2004) [arXiv:hep-ph/0310359].[7] H. Y. Cheng and C. K. Chua, Phys. Rev. D , 094007 (2004) [arXiv:hep-ph/0401141].[8] P. Ball and R. Zwicky, Phys. Rev. D , 014029 (2005) [arXiv:hep-ph/0412079].[9] M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Phys. Rev. Lett. , 1914 (1999)[arXiv:hep-ph/9905312].[10] C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. Lett. , 201806 (2001) [arXiv:hep-ph/0107002];C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. D , 054022 (2002) [arXiv:hep-ph/0109045].[11] Y. Y. Keum, H. n. Li and A. I. Sanda, Phys. Lett. B , 6 (2001) [arXiv:hep-ph/0004004]; Phys. Rev.D , 054008 (2001) [arXiv:hep-ph/0004173];C. D. Lu, K. Ukai and M. Z. Yang, Phys. Rev. D , 074009 (2001) [arXiv:hep-ph/0004213].[12] S. Nandi and H. n. Li, Phys. Rev. D , 034008 (2007) [arXiv:0704.3790 [hep-ph]].[13] B. H. Hong and C. D. Lu, Sci. China G49 , 357 (2006) [arXiv:hep-ph/0505020].[14] H.-n Li, and S. Mishima, Phys. Rev. D , 054025 (2005) [hep-ph/0411146]; H.-n. Li, Phys. Lett. B ,63 (2005) [hep-ph/0411305].[15] A.V. Gritsan, Invited talk at 5th Flavor Physics and CP Violation Conference (FPCP 2007), Bled,Slovenia, 12-16 May 2007; arXiv:0706.2030 [hep-ex][16] Y. Y. Keum, M. Matsumori and A. I. Sanda, Phys. Rev. D , 014013 (2005) [arXiv:hep-ph/0406055].[17] C. D. Lu, M. Matsumori, A. I. Sanda and M. Z. Yang, Phys. Rev. D , 094005 (2005) [Erratum-ibid.D , 039902 (2006)] [arXiv:hep-ph/0508300].[18] Y. Li and C. D. Lu, Phys. Rev. D , 097502 (2006) [arXiv:hep-ph/0605220].[19] A. Ali, G. Kramer, Y. Li, C. D. Lu, Y. L. Shen, W. Wang and Y. M. Wang, Phys. Rev. D , 074018(2007) [arXiv:hep-ph/0703162].C.D. Lu, Talk given at 4th International Workshop on the CKM Unitarity Triangle (CKM 2006),Nagoya, Japan, 12-16 Dec 2006 and talk given at 42nd Rencontres de Moriond on QCD and HadronicInteractions, La Thuile, Italy, 17-24 Mar 2007, arXiv:0705.1782 [hep-ph][20] T. Kurimoto, H. n. Li and A. I. Sanda, Phys. Rev. D , 014007 (2002) [arXiv:hep-ph/0105003];C. D. Lu and M. Z. Yang, Eur. Phys. J. C , 515 (2003) [arXiv:hep-ph/0212373].[21] K. Abe et al. [BELLE Collaboration], arXiv:hep-ex/0408138.[22] For a review, see G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. , 1125 (1996)[arXiv:hep-ph/9512380].[23] A. Ali, G. Kramer and C. D. Lu, Phys. Rev. D , 094009 (1998) [arXiv:hep-ph/9804363].[24] H. n. Li and H. L. Yu, Phys. Rev. D , 2480 (1996) [arXiv:hep-ph/9411308].[25] H. n. Li, Phys. Rev. D , 094010 (2002) [arXiv:hep-ph/0102013].[26] H. n. Li and K. Ukai, Phys. Lett. B , 197 (2003) [arXiv:hep-ph/0211272].[27] A. G. Grozin and M. Neubert, Phys. Rev. D , 272 (1997) [arXiv:hep-ph/9607366].[28] M. Beneke and T. Feldmann, Nucl. Phys. B , 3 (2001) [arXiv:hep-ph/0008255].[29] H. Kawamura, J. Kodaira, C. F. Qiao and K. Tanaka, Phys. Lett. B , 111 (2001) [Erratum-ibid. B , 344 (2002)] [arXiv:hep-ph/0109181].[30] W. M. Yao et al. [Particle Data Group], J. Phys. G , 1 (2006).[31] P. Ball, G. W. Jones and R. Zwicky, Phys. Rev. D , 054004 (2007) [arXiv:hep-ph/0612081].[32] V. M. Braun and A. Lenz, Phys. Rev. D , 074020 (2004) [arXiv:hep-ph/0407282].[33] P. Ball and R. Zwicky, Phys. Lett. B , 289 (2006) [arXiv:hep-ph/0510338].[34] P. Ball and R. Zwicky, JHEP , 046 (2006) [arXiv:hep-ph/0603232].[35] P. Ball and G. W. Jones, JHEP , 069 (2007) [arXiv:hep-ph/0702100].[36] K. C. Yang, JHEP , 108 (2005) [arXiv:hep-ph/0509337].[37] K. C. Yang, Nucl. Phys. B , 187 (2007) [arXiv:0705.0692 [hep-ph]].[38] J. P. Lee, Phys. Rev. D , 074001 (2006) [arXiv:hep-ph/0608087].[39] D. Becirevic, V. Lubicz and F. Mescia, Nucl. Phys. B , 31 (2007) [arXiv:hep-ph/0611295].[40] C. D. Lu, W. Wang and Z. T. Wei, Phys. Rev. D , 014013 (2007) [arXiv:hep-ph/0701265].[41] H. Y. Cheng and K. C. Yang, arXiv:0709.0137 [hep-ph].[42] A. Deandrea, R. Gatto, G. Nardulli and A. D. Polosa, Phys. Rev. D , 074012 (1999)[arXiv:hep-ph/9811259].[43] D. Scora and N. Isgur, Phys. Rev. D , 2783 (1995) [arXiv:hep-ph/9503486].[44] N. Isgur, D. Scora, B. Grinstein and M. B. Wise, Phys. Rev. D , 799 (1989).[45] T. M. Aliev and M. Savci, Phys. Lett. B , 256 (1999) [arXiv:hep-ph/9901395].[46] K. C. Yang, in preparation.[47] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. , 181803 (2007) [arXiv:hep-ex/0612050].[48] M. Bander, D. Silverman and A. Soni, Phys. Rev. Lett. , 242 (1979).[49] J. Liu and Y. P. Yao, Phys. Rev. D , 1485 (1990).[50] H. Simma and D. Wyler, Nucl. Phys. B , 283 (1990).[51] C. D. Lu, Y. L. Shen and W. Wang, Chin. Phys. Lett. , 2684 (2006) [arXiv:hep-ph/0606092].[52] J. Charles et al. [CKMfitter Group], Eur. Phys. J. C , 1 (2005) [arXiv:hep-ph/0406184].[53] H. n. Li and G. L. Lin, Phys. Rev. D , 054001 (1999) [arXiv:hep-ph/9812508].[54] M. Beneke, T. Feldmann and D. Seidel, Eur. Phys. J. C , 173 (2005) [arXiv:hep-ph/0412400].[55] S. W. Bosch and G. Buchalla, Nucl. Phys. B , 459 (2002) [arXiv:hep-ph/0106081].[56] S. W. Bosch, arXiv:hep-ph/0208203.[57] T. Becher, R. J. Hill and M. Neubert, Phys. Rev. D , 094017 (2005) [arXiv:hep-ph/0503263].[58] J. g. Chay and C. Kim, Phys. Rev. D , 034013 (2003) [arXiv:hep-ph/0305033].[59] S. W. Bosch and G. Buchalla, JHEP , 035 (2005) [arXiv:hep-ph/0408231].[60] A. Ali and A. Y. Parkhomenko, Eur. Phys. J. C , 89 (2002) [arXiv:hep-ph/0105302].[61] X. q. Li, G. r. Lu, R. m. Wang and Y. D. Yang, Eur. Phys. J. C , 97 (2004) [arXiv:hep-ph/0305283].[62] A. Ali, E. Lunghi and A. Y. Parkhomenko, Phys. Lett. B , 323 (2004) [arXiv:hep-ph/0405075].[63] A. Ali and A. Parkhomenko, arXiv:hep-ph/0610149.[64] A. Ali, B. D. Pecjak and C. Greub, arXiv:0709.4422 [hep-ph]. [65] A. V. Manohar and I. W. Stewart, Phys. Rev. D , 074002 (2007) [arXiv:hep-ph/0605001].[66] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D , 112006 (2004) [arXiv:hep-ex/0407003].[67] M. Nakao et al. [BELLE Collaboration], Phys. Rev. D , 112001 (2004) [arXiv:hep-ex/0402042].[68] T. E. Coan et al. [CLEO Collaboration], Phys. Rev. Lett. , 5283 (2000) [arXiv:hep-ex/9912057].[69] A. Drutskoy, arXiv:0710.1647 [hep-ex].[70] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. , 151802 (2007) [arXiv:hep-ex/0612017].[71] K. Abe et al. , Phys. Rev. Lett. , 221601 (2006) [arXiv:hep-ex/0506079].[72] M. Nakao, et. al., Belle Collaboration, Phys. Rev. D69, 112001 (2004); B. Aubert, et al., BaBarcollaboration, Phys. Rev. D70, 112006 (2004)[73] M. Beneke, G. Buchalla, C. Greub, A. Lenz and U. Nierste, Phys. Lett. B , 631 (1999)[arXiv:hep-ph/9808385]; A. Lenz and U. Nierste, JHEP , 072 (2007) [arXiv:hep-ph/0612167].[74] Y. Ushiroda et al. , Phys. Rev. Lett. , 231601 (2005) [arXiv:hep-ex/0503008].[75] B. Aubert et al. [BABAR Collaboration], arXiv:0708.1614 [hep-ex].[76] Y. Ushiroda et al. [BELLE Collaboration], arXiv:0709.2769 [hep-ex].[77] J. M. Soares, Nucl. Phys. B , 575 (1991).[78] M. Gronau, Phys. Lett. B , 297 (2000) [arXiv:hep-ph/0008292].[79] T. Hurth and T. Mannel, Phys. Lett. B , 196 (2001) [arXiv:hep-ph/0103331].[80] H. n. Li and S. Mishima, Phys. Rev. D , 094020 (2006) [arXiv:hep-ph/0608277].[81] Y. J. Kwon and J. P. Lee, Phys. Rev. D , 014009 (2005) [arXiv:hep-ph/0409133].[82] M. Jamil Aslam and Riazuddin, Phys. Rev. D , 094019 (2005) [arXiv:hep-ph/0509082].[83] M. J. Aslam, Eur. Phys. J. C , 651 (2007) [arXiv:hep-ph/0604025].[84] M. Jamil Aslam and Riazuddin, Phys. Rev. D , 034004 (2007) [arXiv:hep-ph/0607114].[85] H. n. Li, Prog. Part. Nucl. Phys. , 85 (2003) [arXiv:hep-ph/0303116];C. D. Lu, Mod. Phys. Lett. A22