B→ K 1 γ Decays in the Light-Cone QCD Sum Rules
aa r X i v : . [ h e p - ph ] S e p B → K γ Decays in the Light-Cone QCD Sum Rules
Hisaki Hatanaka and Kwei-Chou Yang
Department of Physics, Chung Yuan Christian University, Chung-Li 320, Taiwan
Abstract
We present a detailed study of B → K (1270) γ and B → K (1400) γ decays. Using the light-conesum rule technique, we calculate the B → K A (1 P ) and B → K B (1 P ) tensor form factors, T K A (0) and T K B (0), where the contributions are included up to the first order in m K /m b .We resolve the sign ambiguity of the K (1270)– K (1400) mixing angle θ K by defining the signsof decay constants, f K A and f ⊥ K B . From the comparison of the theoretical calculation and thedata for decays B → K γ and τ − → K − (1270) ν τ , we find that θ K = − (34 ± ◦ is favored.In contrast to B → K ∗ γ , the hard-spectator contribution suppresses the B → K (1270) γ and B → K (1400) γ branching ratios slightly. The predicted branching ratios are in agreement withthe Belle measurement within the errors. We point out that a more precise measurement for theratio R K = B ( B → K (1400) γ ) / B ( B → K (1270) γ ) can offer a better determination for the θ K and consequently the theoretical uncertainties can be reduced. . INTRODUCTION b → sγ decays contain rich phenomenologies relevant to the standard model and new physics.Radiative B decays involving a vector meson have been observed by CLEO, Belle, and BaBar[1, 2, 3]. Recently, the Belle Collaboration has measured the B → K γ decays for the first time [4]: B ( B − → K − (1270) γ ) = (43 ± ± × − , (1.1) B ( B − → K − (1400) γ ) < × − , (1.2) B ( ¯ B → ¯ K (1270) γ ) < × − , (1.3) B ( ¯ B → ¯ K (1400) γ ) < × − , (1.4)where K is the orbitally excited (P-wave) axial-vector meson. The data indicate that B ( B → K (1270) γ ) ∼ B ( B → K ∗ γ ) and B ( B → K (1270) γ ) ≫ B ( B → K (1400) γ ). It is quite hard toexplain the above-mentioned measurements using the existing theoretical calculations [5, 6, 7, 8,9, 10]. Therefore, these measurements represent a challenge for theory. The production of theaxial-vector mesons has been seen in the two-body hadronic D decays and in charmful B decays[11]. As for charmless hadronic B decays, B → a ± (1260) π ∓ are the first modes measured by B factories [12, 13]. The BaBar collaboration has recently reported the observation of the decays ¯ B → b ± π ∓ , b +1 K − , B − → b π − , b K − , a π − , a − π [14, 15], and ¯ B → K − (1270) π + , K − (1400) π + , a +1 K − , B − → a − ¯ K , f (1285) K − , f (1420) K − [16]. The related phenomenologies have been studied in theliterature [17, 18, 19, 20, 21, 22, 23].In this paper, we will focus on the study of the B → K γ decays. The physical states K (1270)and K (1400) are the mixtures of 1 P ( K A ) and 1 P ( K B ) states. K A and K B are not masseigenstates and they can be mixed together due to the strange and nonstrange light quark massdifference. Following the convention given in Ref. [24], their relations can be written as | ¯ K (1270) i = | ¯ K A i sin θ K + | ¯ K B i cos θ K , | ¯ K (1400) i = | ¯ K A i cos θ K − | ¯ K B i sin θ K . (1.5)In Ref. [24], two possible solutions with two-fold ambiguity | θ K | ≈ ◦ and 57 ◦ were obtained. Asimilar constraint 35 ◦ . | θ K | . ◦ was found in Ref. [25]. From the data of τ → K (1270) ν τ and K (1400) ν τ decays, the mixing angle is extracted to be ± ◦ and ± ◦ in [26]. The signambiguity for θ K is due to the fact that one can add arbitrary phases to | ¯ K A i and | ¯ K B i . Thissign ambiguity can be removed by fixing the signs for f K A and f ⊥ K B , which do not vanish in theSU(3) limit and are defined by h | ¯ ψγ µ γ s | ¯ K A ( P, λ ) i = − i f K A m K A ǫ ( λ ) µ , (1.6) h | ¯ ψσ µν s | ¯ K B ( P, λ ) i = if ⊥ K B ǫ µναβ ǫ α ( λ ) P β , (1.7)(with ψ ≡ u or d ) in the present paper. Following Ref. [27], we adopt the convention: f K A > f ⊥ K B > ǫ = −
1. Thus, the signs of the ¯ B → ¯ K A,B tensor form factors also depend onthe definition mentioned above. See also the discussions after Eq. (5.2).In the quark model calculation, it was argued that the radiative B decay involving the K B whichis the pure 1 P octet state is forbidden because the effective operator O is a spin-flip operator P meson is represented as aconstituent quark-antiquark pair with total spin S = 0 and angular momentum L = 1, a real hadronin QCD language should be described in terms of a set of Fock states, for which each state with thesame quantum number as the hadron can be represented using light-cone distribution amplitudes(LCDAs). In terms of LCDAs, the leading twist LCDAs of the ¯ K B do not vanish, so that ¯ B → ¯ K B tensor form factors are not zero. As a matter of fact, due to the G-parity, the leading-twist LCDAΦ K A ⊥ (Φ K B k ) of the ¯ K A ( ¯ K B ) meson defined by the nonlocal tensor current (nonlocal axial-vector current) is antisymmetric under the exchange of quark and anti - quark momentum fractionsin the SU(3) limit, whereas the Φ K A k (Φ K B ⊥ ) is symmetric [27, 28]. The above properties were notwell-recognized in the previous light-cone (LC) sum rule calculation [7, 29]. In Ref. [7], the authorused only the “symmetrically” asymptotic form for leading-twist distribution amplitudes of the realstates K (1270) and K (1400): Φ K (1270) ⊥ ( u ) = Φ K (1400) ⊥ ( u ) = 6 u ¯ u , in the LC sum rule calculation.In Ref. [29], only the ¯ B → ¯ K B tensor form factor T K B (0) (see Eq. (3.1) for the definition) iscomputed. The correct forms of LCDAs for the axial-vector mesons have been studied in details inRef. [27]. Using the LCDAs in Ref. [27], B → K γ decays have recently been investigated in theperturbative QCD (PQCD) approach [30].In this paper, making use of the LCDAs for the ¯ K A and ¯ K B in Ref. [27, 28], we study the B → K γ decays. We compute the relevant ¯ B → ¯ K A and ¯ K B tensor form factors in the LC sumrule approach. The method of LC sum rules has been widely used in the studies of nonperturbativeprocesses, including weak baryon decays [31], heavy meson decays [32], and heavy to light transitionform factors [33, 34, 35]. We find that the B → K γ data favor a negative θ K . The more preciseestimate can be made through the analysis for the τ − → K − (1270) ν τ data. The predicted branchingratios for B → K (1270) γ, K (1400) γ are in agreement with the data within errors.This paper is organized as follows. In Sec. II, the relevant effective Hamiltonian is given. InSec. III, we provide the definition of ¯ B → ¯ K tensor form factors and then gives the formula forthe B → K γ branching ratios. In Sec. IV we derive the LC sum rules for the relevant tensorform factors, T K A and T K B . The numerical results and detailed analyses are given in Sec. V. Weconclude in Sec. VI. The relevant expressions for two-parton and three-parton LCDAs are collectedin Appendixes A and B, respectively. II. THE EFFECTIVE HAMILTONIAN
Neglecting doubly Cabibbo-suppressed contributions, the weak effective Hamiltonian relevantto b → sγ is given by H eff ( b → sγ ) = G F √ ( V cb V ∗ cs ( c ( µ ) O c ( µ ) + c ( µ ) O c ( µ )) − V tb V ∗ ts X i =3 c i ( µ ) O i ( µ ) ) , (2.1)where O c = ( cb ) V − A ( sc ) V − A , O c = ( c α b β ) V − A ( s β c α ) V − A ,O = ( sb ) V − A X q ( qq ) V − A , O = ( s α b β ) V − A X q ( q β q α ) V − A , = ( sb ) V − A X q ( qq ) V + A , O = ( s α b β ) V − A X q ( q β q α ) V + A ,O = em b π ¯ s α σ µν (1 + γ ) b α F µν ,O = g s m b π ¯ s α σ µν (1 + γ ) T aαβ b β G aµν . (2.2)Here α, β are the SU (3) color indices, V ± A correspond to γ µ (1 ± γ ), and we have neglected correc-tions due to the s -quark mass. We will adopt the next-to-leading order (NLO) Wilson coefficientscomputed in Ref. [36]. III. THE FORMULA FOR THE B → K γ BRANCHING RATIO
The penguin form factors for B → K are defined as follows: (cid:10) ¯ K ( p, λ ) | ¯ s σ µν γ q ν b | ¯ B ( p B ) (cid:11) = 2 T K ( q ) ǫ µνρσ ǫ ∗ ν ( λ ) p ρB p σ , (3.1) (cid:10) ¯ K ( p, λ ) | ¯ s σ µν q ν b | ¯ B ( p B ) (cid:11) = − iT K ( q ) [( m B − m K ) ǫ ∗ µ ( λ ) − ( ǫ ∗ ( λ ) q ) ( p + p B ) µ ] (3.2) − iT K ( q ) ( ǫ ∗ ( λ ) q ) " q µ − q m B − m K ( p + p B ) µ , with T K (0) = T K (0) . (3.3)where ¯ K can be ¯ K A or ¯ K B (or ¯ K (1270), ¯ K (1400)).At the next-to-leading order of α s , the branching ratio can be expressed as [9, 37, 38]: B ( B → K γ ) = τ B Γ( B → K γ ) (3.4)= τ B G F α | V tb V ∗ ts | π m b,pole m B (cid:16) T K (0) (cid:17) − m K m B ! (cid:12)(cid:12)(cid:12) c (0)eff7 + A (1) (cid:12)(cid:12)(cid:12) , where m b,pole is the pole mass of the b quark, and α is the electromagnetic fine structure constant.The effective coefficient c (0)eff7 in the naive dimensional regularization (NDR) scheme is defined by c (0)eff7 = c − c − c . A (1) can be decomposed as A (1) ( µ ) = A (1) C ( µ ) + A (1)ver ( µ ) + A (1) K sp ( µ sp ) , (3.5)where A (1) c , A (1)ver , which are the NLO corrections due to the Wilson coefficient c (0)eff7 and in the b → sγ vertex, respectively, and A (1) K sp , which is the hard-spectator correction, are given by A (1) c ( µ ) = α s ( µ )4 π c (1)eff7 ( µ ) , (3.6) A (1)ver ( µ ) = α s ( µ )4 π (cid:26) h c (0)1 ( µ ) − c (0)eff8 ( µ ) i ln m b µ (3.7)+ 427 (cid:0) − π + 6 πi (cid:1) c (0)eff8 ( µ ) − c (0)eff7 + r ( z ) c (0)1 ( µ ) (cid:27) ,A (1) K sp ( µ sp ) = πα s ( µ sp ) C F N c f B f ⊥ K λ − B m B T K (0) ( c (0)eff8 ( µ sp ) h u − i ( K ) ⊥ − c (0)1 ( µ sp ) h ∆ i ( z ( c )0 , , u i ⊥ ) . (3.8)Here c eff8 = c + c , m B /λ B describes the first negative moment of the B -meson distributionamplitude Φ B [38, 39], and h u − i ( K ) ⊥ ≡ Z du Φ K ⊥ ( u ) u , (3.9) h ∆ i ( z ( c )0 , , u i ⊥ ≡ Z du ∆ i ( z ( c )0 , , u Φ K ⊥ ( u ) , (3.10)with z = ( m c /m b ) and z ( c )0 ≃ m B ¯ u/m c , where m c ≡ m c ( m c ) and m b ≡ m b ( m b ) are the MS c - and b - quark masses, respectively. The detailed definitions of the functions r ( z ) and ∆ i ( z ( c )0 , ,
0) canbe found in Refs. [36, 37]. In the numerical calculation, we set the scale for the vertex correctionsto be µ = m b and scale for the spectator interactions to be µ sp = √ Λ h m b , where Λ h ≃ IV. THE LIGHT-CONE SUM RULE FOR T K To calculate the form factor T K , we consider the two-point correlation function, which issandwiched between the vacuum and transverse polarized K meson, i Z d xe iqx h ¯ K ( P, ⊥ ) | T [¯ s ( x ) σ µν b ( x ) j † B (0)] | i = − i A ( p B , q ) { ǫ ∗ ( ⊥ ) µ (2 P + q ) ν − ǫ ∗ ( ⊥ ) ν (2 P + q ) µ } + i B ( p B , q ) { ǫ ∗ ( ⊥ ) µ q ν − ǫ ∗ ( ⊥ ) ν q µ } + 2 i C ( p B , q ) ǫ ∗ ( ⊥ ) qm B − m K { P µ q ν − q µ P ν } , (4.1)where j B = i ¯ ψγ b (with ψ ≡ u or d ) is the interpolating current for the B meson, p B = ( P + q ) ,and P the momentum of the K meson. Note that in this section K ≡ K A or K B . A is the onlyrelevant term in the present study, and at the hadron level can be written in the form A ( p B , q ) = T K ( q ) · m B − p B · m B f B m b + · · · , (4.2)where the dots denote contributions that have poles p B = m B ∗ with m B ∗ being the masses ofthe higher resonance B ∗ -mesons. To obtain the result for A , we have taken into account here thetransverse polarized K , instead of its longitudinal component, because for the longitudinal K , A mixes with B and C for an energetic K .In a region of sufficiently large virtualities: m b − p B ≫ Λ QCD m b , with a small q ≥
0, theoperator product expansion is applicable in Eq. (4.1), so that in QCD for an energetic K mesonthe correlation function in Eq. (4.1) can be represented in terms of the LCDAs of the K meson: i Z d xe iqx h ¯ K ( P, ⊥ ) | T [¯ s ( x ) σ µν b ( x ) j † B (0)] | i Z − i ( q + k ) − m b Tr h σ µν ( q + k + m b ) γ M K ⊥ i(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k = uEn − du + 14 Z dv Z Dα vE ( n − q ) (cid:16) f A K A ( α ) + f V K V ( α ) (cid:17) Tr( σ µν ǫ ∗ ( ⊥ ) n − ) n m b − [ q + ( α + α g v ) En − ] o + O (cid:18) m K E (cid:19) , (4.3)where f A K ∼ O ( f K m K ), f V K ∼ O ( f K m K ), E = | ~P | , P µ = En µ − + m K n µ + / (4 E ) ≃ En µ − withtwo light-like vectors satisfying n − n + = 2 and n − = n = 0. Here E ∼ m b and we have assignedthe momentum of the s -quark in the K meson to be k µ = uEn µ − + k µ ⊥ + k ⊥ uE n µ + , (4.4)where k ⊥ is of order Λ QCD . In Eq. (4.3), in calculating contributions due to the two-parton LCDAsof the ¯ K in the momentum space, we have used the following substitution for the Fourier transformof h ¯ K ( P, ⊥ ) | ¯ s α ( x ) ψ δ (0) | i , x µ → − i ∂∂k µ ≃ − i n µ + E ∂∂u + ∂∂k ⊥ µ ! , (4.5)where the term of order k ⊥ is omitted. Thus, we can obtain the light-cone transverse projectionoperator M K ⊥ of the ¯ K meson in the momentum space: M K ⊥ = i f ⊥ K E ( ǫ ∗ ( λ ) ⊥ n − γ Φ ⊥ ( u ) − f K f ⊥ K m K E " ǫ ∗ ( λ ) ⊥ γ g ( a ) ⊥ ( u ) − E Z u dv Φ a ( v ) n − γ ǫ ∗ ( λ ) ⊥ µ ∂∂k ⊥ µ + iε µνρσ γ µ ǫ ∗ ( λ ) ν ⊥ n ρ − n σ + g ( v ) ′⊥ ( u )8 − E g ( v ) ⊥ ( u )4 ∂∂k ⊥ σ ! k = up + O (cid:18) m K E (cid:19)) , (4.6)where Φ a ≡ Φ k − g ( a ) ⊥ and the detailed definitions for the relevant two-parton LCDAs are collectedin Appendix A. A similar discussion for the vector meson projection operators can be found inRef. [40]. From the expansion of the transverse projection operator, one can find that contributionsarising from Φ a , g ( v ) ′⊥ , and g ( v ) ⊥ are suppressed by m K /E as compared with that from Φ ⊥ . Note thatin Eq. (4.3) the derivative with respect to the transverse momentum acts on the hard scatteringamplitude before the collinear approximation is taken. The three-parton chiral-even distributionamplitudes of twist-3, A ( α ) and V ( α ), together with their decay constants, f A K and f V K , aredefined by h ¯ K ( P, λ ) | ¯ s ( x ) γ α γ g s G µν ( vx ) ψ (0) | i = p α [ p ν ǫ ∗ ( λ ) ⊥ µ − p µ ǫ ∗ ( λ ) ⊥ ν ] f A K A ( v, − px )+ · · · , (4.7) h ¯ K ( P, λ ) | ¯ s ( x ) γ α g s e G µν ( vx ) ψ (0) | i = ip α [ p µ ǫ ∗ ( λ ) ⊥ ν − p ν ǫ ∗ ( λ ) ⊥ µ ] f V K V ( v, − px )+ · · · , (4.8) here we have set p µ = P µ − m K ¯ z µ / (2 P ¯ z ) with¯ z µ = x µ − P µ m K ( xP − h ( xP ) − x m K i / ) . Here the ellipses stand for terms of twist higher than three, the following shorthand notations areused: A ( v, − px ) ≡ Z D α e ipx ( α + vα g ) A ( α ) , (4.9)etc., and the integration measure is defined as Z D α ≡ Z dα Z dα Z dα g δ (1 − X α i ) , (4.10)with α , α , α g being the momentum fractions carried by the s quark, ¯ ψ ( ≡ ¯ u or ¯ d ) quark, and gluon,respectively. At the quark-gluon level, after performing the integration of Eq. (4.3), the result for A QCD reads (with ¯ u = 1 − u ) A QCD = − m b f ⊥ K Z du ( m b − up B − ¯ uq × " Φ ⊥ ( u ) − m K f K m b f ⊥ K ug ( a ) ⊥ ( u ) + Φ a ( u ) + g ( v ) ⊥ ( u )4 − g ( v ) ′⊥ ( u )4 p B + q p B − q ! − m K f K m b f ⊥ K ( m b + q )( m b − up B − ¯ uq ) g ( v ) ⊥ ( u ) ) − Z vdv Z Dα f A K A ( α ) + f V K V ( α )2( α + vα g ) (cid:20) m b − ( α + vα g )( p B − q ) − q − m b − q [ m b − ( α + vα g )( p B − q ) − q ] (cid:21) . (4.11)We have given the results of A from the hadron and quark-gluon points of view, respectively. Thus,the contribution due to the lowest-lying K meson can be further approximated with the help ofquark-hadron duality: T K ( q ) · m B − p B · m B f B m b = 1 π Z s m b Im A QCD ( s, q ) s − p B ds , (4.12)where s is the excited state threshold. After applying the Borel transform p B → M to the aboveequation, we obtain T K ( q ) = m b m B f B e − m B /M π Z s m b e s/M Im A QCD ( s, q ) ds . (4.13)Finally, the light-cone sum rule for T K reads T K ( q ) = − m b f ⊥ K m B f B e ( m B − m b ) /M Z du ( u e ¯ u ( q − m b ) / ( uM ) θ [ c ( u, s )] " Φ ⊥ ( u ) − m K f K m b f ⊥ K (cid:16) ug ( a ) ⊥ ( u ) + Φ a ( u ) + g ( v ) ⊥ ( u )4 − g ( v ) ′⊥ ( u )4 m b + ( u − ¯ u ) q m b − q (cid:17) u e ¯ u ( q − m b ) / ( uM ) m K f K m b f ⊥ K ( m b + q ) g ( v ) ⊥ ( u ) θ [ c ( u, s )] uM + δ [ c ( u, s )] ! − m K f K m b f ⊥ K g ( v ) ′⊥ ( u )2 q m b − q e ( m b − q ) /M ) − m b m B f B e ( m B − m b ) /M Z vdv Z Dα f A K A ( α ) + f V K V ( α )( α + vα g ) × e (1 − α − vα g )( q − m b ) / [( α + vα g ) M ] ( θ [ c ( α + vα g , s )] − ( m b − q ) θ [ c ( α + vα g , s )]( α + vα g ) M + δ [ c ( α + vα g , s )] !) , (4.14)where c ( u, s ) = us − m b + (1 − u ) q and θ [ · · · ] is the step function. Note that here f ⊥ K A is chosento be f K A , while f K B is adopted to be f ⊥ K B (1 GeV). (See Eq. (A4) and related discussions.) V. RESULTSA. T K A and T K B LCSR results and B → K γ branching ratios Parameters relevant to the present study are collected in Table I. We first analyze the T (0)sum rules numerically. The pole b quark mass is adopted in the LC sum rule. The f ⊥ K andparameters appearing in the distribution amplitudes are evaluated at the factorization scale µ f = q m B − m b,pole . On the other hand, the form factor T (0) depends on the renormalization scaleof the effective Hamiltonian, for which the scale is set to be m b ( m b ). The working Borel windowis 7 . < M < . , where the correction originating from higher resonance statesamounts to 15% to 35%. We do not include the contributions of the twist-4 LCDAs and 3-partontwist-3 chiral-even LCDAs in the light-cone sum rule since these corrections to light-cone expansionseries is of order ( m K /m b ) and might be negligible. The excited state threshold s can bedetermined when the most stable plateau of the LC sum rule result is obtained within the Borelwindow. We find that the corresponding threshold s lies in the interval 32 ∼
36 GeV .Two remarks are in order. First, we have consistently used f B = 190 ±
10 MeV in all numericalanalysis. In the literature, it was assumed that the theoretical errors due to the radiative correctionsin the form factor sum rules can be canceled if one adopts the f B sum rule result with the sameorder of α s -corrections in the calculation [34, 35]. Nevertheless, the resulting sum rule result for T BK ∗ (0) seems to be significantly larger than the estimate extracted from the data [37], althoughthe sum rule result can be improved by including α s -corrections [35]. We have checked that using thephysical value of f B , that we adopt here, in the T BK ∗ (0) LC sum rule with the same order in α s and m K /m b , we get T BK ∗ (0) ≈ . +0 . − . which is in good agreement with the result constrained by thedata [37, 41]. Extracting from the data, the current estimation is T BK ∗ (0) = 0 . ± .
018 [41]. Thelattice QCD result is T BK ∗ (0) = 0 . ± . +0 . − . [42]. Therefore, although the radiative correctionscan be important in the form factor sum rule calculations, its effects are significantly reduced unning quark masses (GeV), pole b -quark mass (GeV), and couplings m c ( m c ) m s (2 GeV) m b ( m b ) m b,pole α s ( m Z ) α . ± .
10 0 . ± .
01 4 . ± .
15 4 . ± .
05 0 . B distribution amplitude | V cs | | V cb | λ B . ± .
095 (41 . ± . × − (0 . ± .
15) GeVMasses (GeV) and decay constants (MeV) for mesons m K A m K B f K A f ⊥ K B (1 GeV) f B . ± .
06 1 . ± .
08 250 ±
13 190 ±
10 190 ± K A meson at scales 1 GeV and 2 . a k ,K A a k ,K A a ⊥ ,K A a ⊥ ,K A a ⊥ ,K A − . +0 . − . − . ± .
03 0 . +0 . − . − . ± .
48 0 . ± . − . +0 . − . ) ( − . ± .
02) (0 . +0 . − . ) ( − . ± .
37) (0 . ± . K B meson at scales 1 GeV and 2 . a k ,K B a k ,K B a k ,K B a ⊥ ,K B a ⊥ ,K B − . ± . − . ± .
45 0 . +0 . − . . +0 . − . − . ± . − . ± .
15) ( − . ± .
36) (0 . +0 . − . ) (0 . +0 . − . ) ( − . ± . K A meson at the scale 2.2 GeV f V ,K A (in GeV ) ω VK A σ VK A f A ,K A (in GeV ) λ AK A σ AK A . ± . − . ± . − . ± .
16 0 . ± . . ± .
46 2 . ± . K B meson at the scale 2.2 GeV f V ,K B (in GeV ) λ VK B σ VK B f A ,K B (in GeV ) ω AK B σ AK B . ± . . ± .
24 0 . ± . − . ± . − . ± . − . ± . a k ,K A , a ⊥ ,K A , a ⊥ ,K A , a k ,K B , a k ,K B , and a ⊥ ,K B are G-parity violating Gegenbaur moments, which vanish in the SU(3) limit. Using the QCDsum rules, the relation a ⊥ ,K A + (0 . ± . a k ,K B = 0 . ± .
11 was obtained, instead of theirindividual values [27]. It will be seen later that due to the data for B ( B → K (1270) γ ) ≫ B ( B → K (1400) γ ) and for τ − → K − (1270) ν τ , θ K and a k ,K B should be negative. Here we further makereasonable assumptions that | a k ,K B f K B | ≤ × f ⊥ K B and | a ⊥ ,K A f ⊥ K A | (1 GeV) ≤ × f K A to account for the possible SU(3) breaking effect, i.e., we assume G-parity correction is roughly lessthan 30%. (See Eqs. (5.3)-(5.6) for the detailed definitions of parameters.) Finally, we arrive at a k ,K B = − . ± .
15 and a ⊥ ,K A = 0 . +0 . − . . As shown in Table I, once these two parameters aredetermined, the remaining G-parity violating Gegenbaur moments are thus updated according tothe relations given in Eq. (141) in Ref. [27].To illustrate the qualities and uncertainties of the sum rules, we plot the results for T K A (0) . 8. 9. 10. 11. 12. 13. M (cid:43) GeV (cid:47) T K (cid:3) A (cid:43) (cid:47)
7. 8. 9. 10. 11. 12. 13. M (cid:43) GeV (cid:47) (cid:16) (cid:16) (cid:16) (cid:16) T K (cid:3) B (cid:43) (cid:47) FIG. 1: T K A (0) and T K B (0) as functions of the Borel mass squared, where thecentral values of input parameters have been used in the solid curve. The dashed(dot-dashed) curves are for variation of the m b,pole (parameters for LCDAs) with thecentral values of the remaining theoretical parameters.and T K B (0) as functions of M in Fig. 1. We obtain T K A (0) = 0 . +0 . . . − . − . − . ,T K B (0) = − (cid:0) . +0 . . . − . − . − . (cid:1) , (5.1)where the first, second, and third error bars come from the variations of m b,pole , f B , and theremaining parameters, respectively. The third errors are mainly due to the G-parity violatingGegenbaur moments of the leading twist LCDAs. Corrections arising from the three-parton LCDAsare less than 3%.In calculating the B → K (1270) γ and K (1400) γ branching ratios, B → K tensor form factorshave the expressions T K (1270)1 (0) = T K A (0) sin θ K + T K B (0) cos θ K ,T K (1400)1 (0) = T K A (0) cos θ K − T K B (0) sin θ K . (5.2)From Eq. (4.14), we know that T K A and T K B depend on the definition of the signs of f K A and f ⊥ K B , so that the resultant θ K also depends on the signs of f K A and f ⊥ K B .As for the relevant physical properties of ¯ K mesons, we have h | ¯ ψγ µ γ s | ¯ K (1270)( P, λ ) i = − i f K (1270) m K (1270) ǫ ( λ ) µ = − i ( f K A m K A sin θ K + f K B m K B a k ,K B cos θ K ) ǫ ( λ ) µ , (5.3) h | ¯ ψγ µ γ s | ¯ K (1400)( P, λ ) i = − i f K (1400) m K (1400) ǫ ( λ ) µ = − i ( f K A m K A cos θ K − f K B m K B a k ,K B sin θ K ) ǫ ( λ ) µ , (5.4) h | ¯ ψσ µν s | ¯ K (1270)( P, λ ) i = if ⊥ K (1270) ǫ µναβ ǫ α ( λ ) P β = i ( f ⊥ K A a ⊥ ,K A sin θ K + f ⊥ K B cos θ K ) ǫ µναβ ǫ α ( λ ) P β , (5.5) (cid:16) (cid:16) (cid:16) (cid:16)
10 10 30 50 70 90 (cid:84) K (cid:43) degree (cid:47) r B (cid:43) B (cid:16) (cid:145) K (cid:16) (cid:74) (cid:47) (cid:151) (cid:91) (cid:16) (cid:16) (cid:16) (cid:16) (cid:16)
10 10 30 50 70 90 (cid:84) K (cid:43) degree (cid:47) r B (cid:43) B (cid:16) (cid:145) K (cid:16) (cid:74) (cid:47) (cid:151) FIG. 2: Branching ratios as functions of the mixing angle θ K . The upper five (red)curves at θ K = − ◦ are for the K (1270) γ mode, and the lower five (blue) curvesfor the K (1400) γ mode. The solid curves correspond to central values of the inputparameters. The dot-dashed and dashed curves denote the theoretical uncertaintiesdue to the parameters of LCDAs and m b,pole , respectively. The horizontal line isthe experimental limit on B → K (1400) γ , and the horizontal band shows theexperimental result for the K (1270) γ mode with its 1 σ error.and h | ¯ ψσ µν s | ¯ K (1400)( P, λ ) i = if ⊥ K (1400) ǫ µναβ ǫ α ( λ ) P β = i ( f ⊥ K A a ⊥ ,K A cos θ K − f ⊥ K B sin θ K ) ǫ µναβ ǫ α ( λ ) P β , (5.6)where the values of f K A , f ⊥ K B , m K A , m K B , a k ,K B and a ⊥ ,K A are given in Table I, and use of f K B = f ⊥ K B (1 GeV) and f ⊥ K A = f k K A is made in the present study. Following this definition, a k ,K B and a ⊥ ,K A vanish in the SU(3) limit, and we have the relationsΦ K (1270) ⊥ ( u ) = f ⊥ K A f ⊥ K (1270) Φ K A ⊥ ( u ) sin θ K + f ⊥ K B f ⊥ K (1270) Φ K B ⊥ ( u ) cos θ K , (5.7)Φ K (1400) ⊥ ( u ) = f ⊥ K A f ⊥ K (1400) Φ K A ⊥ ( u ) cos θ K − f ⊥ K B f ⊥ K (1400) Φ K B ⊥ ( u ) sin θ K . (5.8)In Fig. 2 we plot the branching ratios of B − → K − (1270) γ and B − → K − (1400) γ as functions of θ K . The mixing angle dependence of the K − (1270) γ mode is opposite to that of the K − (1400) γ mode. To satisfy the observable B ( B → K (1270) γ ) ≫ B ( B → K (1400) γ ), we find that the sign of θ K should be negative. The further constraint for θ K can be obtained from the τ − → K − (1270) ν τ analysis. B. The constraint for θ K from the τ − → K − (1270) ν τ data The decay constant f K (1270) can be extracted from the measurement τ − → K − (1270) ν τ byALEPH [43]: B ( τ − → K − (1270) ν τ ) = (4 . ± . × − , where the formula for the decay rate is iven by Γ( τ → K ν τ ) = G F π | V us | f K ( m τ + 2 m K )( m τ − m K ) m τ . (5.9)It was obtained in Refs. [26, 30] that (cid:12)(cid:12) f K (1270) (cid:12)(cid:12) = 169 +19 − MeV . (5.10)As obtained in the previous subsection, θ K should be negative to account for the observable B ( B → K (1270) γ ) ≫ B ( B → K (1400) γ ). Using the values for f K A and f K B as given in Table I,the result for f K (1270) in Eq. (5.10) and the relation in Eq. (5.3), we find that a k ,K B should benegative. Further substituting a k ,K B = − . ± .
15 into Eq. (5.3), we obtain that θ K lies in theinterval − ◦ ∼ − ◦ . We can use the obtained angle to predict the decay constants f K (1270) and f K (1400) : f K (1270) = − (cid:0) +25+49 − − (cid:1) MeV , (5.11) f K (1400) = 179 +13+30 − − MeV , (5.12)for θ K = ( − ± ◦ , where the first error is due to the uncertainties of decay constants and a k ,K B , and the second due to the variation of θ K . The first error is dominated by the variation of a k ,K B . The predicted θ K = ( − ± ◦ is also consistent with the result given in Ref. [24], where | θ K | ≈ ◦ or 57 ◦ . We thus predict B ( τ − → K − (1400) ν τ ) = (3 . +0 . . − . − . ) × − , (5.13)to be compared with the current data B ( τ − → K − (1400) ν τ ) = (1 . ± . × − [11] which has largeexperimental error. If a more precise measurement for B ( τ − → K − (1400) ν τ ) can also be achieved,we can extract directly the values of θ K and a k ,K B . Consequently, we can have more precisepredictions for the B ( B → K (1270) γ ) and B ( B → K (1400) γ ) branching ratios and B → K transition form factors. C. B → K γ branching ratios Using m c /m b = 1 .
25 GeV / .
25 GeV, one finds B ( B → K γ ) = τ B G F α | V tb V ∗ ts | π m b,pole m B − m K m B ! (cid:16) T K (0) (cid:17) × (cid:12)(cid:12)(cid:12) ( − . − i . A (1) K sp ( µ h ) (cid:12)(cid:12)(cid:12) , (5.14)where T K (1270)1 (0) and T K (1400)1 (0), as given in Eq. (5.2), are θ K -dependent. For θ K = − (34 ± ◦ , we have T K (1270)1 (0) = − (cid:0) . +0 . . . − . − . − . (cid:1) ,T K (1400)1 (0) = 0 . +0 . . . − . − . − . , (5.15) ( B − → K − (1270) γ ) B ( B − → K − (1400) γ )Expt . This work 43 ± +21+30+2+ 6 − − − − < . +4 . . . . − . − . − . − . B ( ¯ B → ¯ K (1270) γ ) B ( ¯ B → ¯ K (1400) γ )Expt . This work < +19+28+2+ 5 − − − − < . +3 . . . . − . − . − . − . TABLE II: Branching ratios for the radiative decays B → K (1270) γ, K (1400) γ (in units of 10 − )in this work and the experiment [4]. The branching ratios correspond to θ K = − (34 ◦ ± ◦ ) inour work, where the first error comes from the variation of m b,pole and f B , the second from theparameters of LCDAs, the third from λ B , and the forth from θ K . The annihilation amplitudes arenot included in the neutral B decay modes.where the first uncertainty comes from the variation of m b,pole and f B in the sum rules, the secondfrom the parameters of LCDAs, and the third from θ K . To illustrate the contribution due tothe hard-spectator correction, it is interesting to note that, using λ B = 0 .
35 GeV, θ K = − ◦ , T K A (0) = 0 . T K B (0) = − .
25, and the center values of the remaining input parameters, weobtain A (1) K (1270)sp ( µ h ) = 0 .
016 + i . ,A (1) K (1400)sp ( µ h ) = 0 . − i . , (5.16)which suppress the decay rates slightly by about 8%, in contrast to the B → K ∗ γ decay where theinterference between the hard-spectator correction A (1) K ∗ sp ( µ h ) = − . − i .
011 and the remainderis constructive [37].In Table II, we present a comparison of the resulting branching ratios in this work with thedata. Our results are consistent with the Belle measurement [4] within errors. A much moreprecise determination of θ K can be made by the measurement R K = B ( B → K (1400) γ ) B ( B → K (1270) γ ) . (5.17)The current upper bound of this ratio is R K < .
5. It can be seen from Fig. 3 that R K weaklydepends on the theoretical uncertainty. Thus, R K is a suitable quantity for measuring the mixingangle θ K . In the light-cone sum rule calculation, the physical quantities, including the branchingratios and transition form factors, receive large errors from the uncertainties of G-parity violatingGegenbaur moments. A more precise value for θ K can be used to extract a better result of a k ,K B from the data for B ( τ − → K − (1270) ν τ ); the remaining G-parity violating Gegenbaur momentscan thus be determined using Eq. (141) in Ref. [27]. On the other hand, we can also obtain goodestimates for θ K and a k ,K B from the data B ( τ − → K − (1270) ν τ ) and B ( τ − → K − (1400) ν τ ) if we (cid:16) (cid:16) (cid:16) (cid:16)
10 10 30 50 70 90 (cid:84) K (cid:43) degree (cid:47) R K (cid:91) FIG. 3: Same as Fig. 2 except for the ratio R K = B ( B → K (1400) γ ) / B ( B → K (1270) γ ) as a function of the mixing angle θ K .can improve the measurement for B ( τ − → K − (1400) ν τ ). Consequently, theoretical uncertaintiesdue to G-parity violating Gegenbaur moments and θ K can be reduced in the form factors andbranching ratios calculations. VI. CONCLUSIONS
We have presented a detailed study of B → K (1270) γ and B → K (1400) γ decays. Our mainresults are as follows. • Using the light-cone sum rule technique, we have evaluated the B → K A , K B ten-sor form factors, T K A (0) and T K B (0), where the contributions have been included upto the first order in m K /m b . We obtain T K A (0) = 0 . +0 . . . − . − . − . and T K B (0) = − (0 . +0 . . . − . − . − . ). • The sign ambiguity of the K (1270)– K (1400) mixing angle θ K can be resolved by defining f K A and f ⊥ K B to be positive. Combining the analysis for the decays B → K γ and τ − → K − (1270) ν τ , we find that the mixing angle θ K should be negative, and its value lies in theinterval − (34 ± ◦ . We obtain f K (1270) = − (cid:0) +25+49 − − (cid:1) MeV and f K (1400) = 179 +13+30 − − MeV, and predict B ( τ − → K − (1400) ν τ ) = (3 . +0 . . − . − . ) × − . • We find T K (1270)1 (0) = − (0 . +0 . . . − . − . − . ) , T K (1400)1 (0) = 0 . +0 . . . − . − . − . . The hard-spectator contribution suppresses the B → K (1270) γ and B → K (1400) γ decay ratesslightly by about 8%, in contrast with the situation for B → K ∗ γ . The predicted branchingratios for the decays B → K (1270) γ and B → K (1400) γ agree with the data within theerrors. • We point out that better determinations of the θ K and G-parity violating Gegenbaur mo-ments of leading-twist light-cone distribution amplitudes can be obtained from a more precise easurement for the ratio R K = B ( B → K (1400) γ ) / B ( B → K (1270) γ ) or from an im-proved measurement for B ( τ − → K − (1400) ν τ ) together with the B ( τ − → K − (1270) ν τ ) data.Thus, the theoretical uncertainties can be further reduced. Acknowledgments
This research was supported in part by the National Science Council of R.O.C. under Grant No.NSC96-2112-M-033-004-MY3 and No. NSC96-2811-M-033-004.
APPENDIX A: TWO-PARTON DISTRIBUTION AMPLITUDES
In the calculation, the LCDAs of the axial-meson appear in the following way h ¯ K ( P, λ ) | ¯ s α ( y ) ψ δ ( x ) | i = − i Z du e i ( u P y +¯ uP x ) ( f K m K " P γ ǫ ∗ ( λ ) zP z Φ k ( u )+ ǫ ∗ − 6 P ǫ ∗ ( λ ) zP z ! γ g ( a ) ⊥ ( u ) − 6 zγ ǫ ∗ ( λ ) z P z ) m K ¯ g ( u ) + ǫ µνρσ ǫ ∗ ( λ ) ν p ρ z σ γ µ g ( v ) ⊥ ( u )4 + f ⊥ K " (cid:18) P ǫ ∗ ( λ ) − 6 ǫ ∗ ( λ ) P (cid:19) γ Φ ⊥ ( u ) − (cid:18) P z − 6 z P (cid:19) γ ǫ ∗ ( λ ) z ( P z ) m K ¯ h ( t ) k ( u ) − (cid:18) ǫ ∗ ( λ ) z − 6 z ǫ ∗ ( λ ) (cid:19) γ m K P z ¯ h ( u ) + i ( ǫ ∗ ( λ ) z ) m K γ h ( p ) k ( u )2 δα + O (cid:16) ( x − y ) (cid:17) , (A1)where ¯ g ( u ) = g ( u ) + Φ k − g ( a ) ⊥ ( u ) , ¯ h ( t ) k ( u ) = h ( t ) k ( u ) −
12 Φ ⊥ ( u ) − h ( u ) , ¯ h ( u ) = h ( u ) − Φ ⊥ ( u ) , (A2) z = ( y − x ) = 0, and P = m K . The detailed LCDAs are defined in Ref. [27]. Here Φ k , Φ ⊥ are of twist-2, g ( a ) ⊥ , g ( v ) ⊥ , h ( t ) k , h ( p ) k of twist-3, and g , h of twist-4. In SU(3) limit, due to G-parity,Φ k , g ( a ) ⊥ , g ( v ) ⊥ , and g are symmetric (antisymmetric) under the replacement u ↔ − u for the 1 P (1 P ) states, whereas Φ ⊥ , h ( t ) k , h ( p ) k , and h are antisymmetric (symmetric). For convenience, wenormalize the distribution amplitudes of the 1 P and 1 P states to be subject to Z du Φ k ( u ) = 1 , Z du Φ ⊥ ( u ) = 1 . (A3)We take f ⊥ P = f P and f P = f ⊥ P ( µ = 1 GeV) in the study, such that we define h ¯ K A ( P, λ ) | ¯ s (0) σ µν γ ψ (0) | i = f ⊥ K A a ⊥ ,K A ( ǫ ∗ ( λ ) µ P ν − ǫ ∗ ( λ ) ν P µ ) , h ¯ K B ( P, λ ) | ¯ s (0) γ µ γ ψ (0) | i = if K B a k ,K B m K B ǫ ∗ ( λ ) µ , (A4) here a ⊥ ,K A and a k ,K B are the Gegenbauer zeroth moments, which vanish in the SU(3) limit.We take into account the approximate forms of twist-2 distributions for the ¯ K A meson to be[27] Φ k ( u ) = 6 u ¯ u (cid:20) a k ξ + a k
32 (5 ξ − (cid:21) , (A5)Φ ⊥ ( u ) = 6 u ¯ u (cid:20) a ⊥ + 3 a ⊥ ξ + a ⊥
32 (5 ξ − (cid:21) , (A6)and for the ¯ K B meson to beΦ k ( u ) = 6 u ¯ u (cid:20) a k + 3 a k ξ + a k
32 (5 ξ − (cid:21) , (A7)Φ ⊥ ( u ) = 6 u ¯ u (cid:20) a ⊥ ξ + a ⊥
32 (5 ξ − (cid:21) , (A8)where ξ = 2 u − / O ( m s ) [27]: g ( a ) ⊥ ( u ) = 34 (1 + ξ ) + 32 a k ξ + (cid:18) a k + 5 ζ V ,K A (cid:19) (cid:0) ξ − (cid:1) + (cid:18) a k + 10516 ζ A ,K A − ζ V ,K A ω VK A (cid:19) (cid:0) ξ − ξ + 3 (cid:1) +5 " ζ V ,K A σ VK A + ζ A ,K A (cid:18) λ AK A − σ AK A ! ξ (5 ξ − −
92 ¯ a ⊥ e δ + (cid:18)
32 + 32 ξ + ln u + ln ¯ u (cid:19) −
92 ¯ a ⊥ e δ − (3 ξ + ln ¯ u − ln u ) , (A9) g ( v ) ⊥ ( u ) = 6 u ¯ u ( a k + 203 ζ A ,K A λ AK A ! ξ + " a k + 53 ζ V ,K A (cid:18) − ω VK A (cid:19) + 354 ζ A ,K A (5 ξ − ζ V ,K A σ VK A − ζ A ,K A σ AK A ! ξ (7 ξ − ) − a ⊥ e δ + (3 u ¯ u + ¯ u ln ¯ u + u ln u ) − a ⊥ e δ − ( u ¯ uξ + ¯ u ln ¯ u − u ln u ) , (A10)for the ¯ K A state, and g ( a ) ⊥ ( u ) = 34 a k (1 + ξ ) + 32 a k ξ + 5 " ζ V ,K B + ζ A ,K B − ω AK B ! ξ (cid:0) ξ − (cid:1) + 316 a k (cid:0) ξ − ξ − (cid:1) + 5 ζ V ,K B λ VK B (cid:0) ξ − (cid:1) + 10516 (cid:18) ζ A ,K B σ AK B − ζ VK B σ VK B (cid:19) (cid:0) ξ − ξ + 3 (cid:1) − a ⊥ (cid:20)e δ + ξ + 12 e δ − (3 ξ − (cid:21) (cid:20)e δ + (2 ξ + ln ¯ u − ln u ) + e δ − (2 + ln u + ln ¯ u ) (cid:21) (1 + 6 a ⊥ ) , (A11) g ( v ) ⊥ ( u ) = 6 u ¯ u ( a k + a k ξ + " a k + 53 ζ V ,K B λ VK B − σ VK B ! + 354 ζ A ,K B σ AK B (5 ξ − ξ " ζ A ,K B + 2116 ζ V ,K B − ζ A ,K B ω AK B ! (7 ξ − − a ⊥ [2 e δ + ξ + e δ − (1 + ξ )] ) − (cid:20) e δ + (¯ u ln ¯ u − u ln u ) + e δ − (2 u ¯ u + ¯ u ln ¯ u + u ln u ) (cid:21) (1 + 6 a ⊥ ) , (A12)for the ¯ K B state, where e δ ± = ± f ⊥ K f K m s m K , ζ V,A ,K = f V,A K f K m K . (A13) APPENDIX B: THREE-PARTON CHIRAL-EVEN DISTRIBUTION AMPLI-TUDES OF TWIST-3
Taking into account the contributions up to terms of conformal spin 9 / m s , the twist-3 three-parton chiral-even distribution amplitudes, defined inEqs. (4.7) and (4.8), can be approximately written as [27] A ( α ) = 5040( α s − α ψ ) α s α ψ α g + 360 α s α ψ α g h λ AK A + σ AK A
12 (7 α g − i , (B1) V ( α ) = 360 α s α ψ α g h ω VK A
12 (7 α g − i + 5040( α s − α ψ ) α s α ψ α g σ VK A , (B2)for the ¯ K A state, and A ( α ) = 360 α s α ψ α g h ω AK B
12 (7 α g − i + 5040( α s − α ψ ) α s α ψ α g σ AK B , (B3) V ( α ) = 5040( α s − α ψ ) α s α ψ α g + 360 α s α ψ α g h λ VK B + σ VK B
12 (7 α g − i , (B4)for the ¯ K B state, where λ ’s correspond to conformal spin 7/2, while ω ’s and σ ’s are parameterswith conformal spin 9/2. Note that as the SU(3)-symmetry (and G-parity) is restored, we have λ ’s= σ ’s=0.[1] T. E. Coan et al. [CLEO Collaboration], Phys. Rev. Lett. , 5283 (2000)[arXiv:hep-ex/9912057].[2] M. Nakao et al. [BELLE Collaboration], Phys. Rev. D , 112001 (2004)[arXiv:hep-ex/0402042].[3] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D , 112006 (2004)[arXiv:hep-ex/0407003].
4] K. Abe et al. [BELLE Collaboration], arXiv:hep-ex/0408138; H. Yang et al. , Phys. Rev. Lett. , 111802 (2005) [arXiv:hep-ex/0412039]; Heavy Flavor Averaging Group, E. Barberio et al., , 677 (1988).[6] S. Veseli and M. G. Olsson, Phys. Lett. B , 309 (1996) [arXiv:hep-ph/9508255].[7] A. S. Safir, Eur. Phys. J. directC (2001) 1 [arXiv:hep-ph/0109232].[8] H. Y. Cheng and C. K. Chua, Phys. Rev. D , 094007 (2004) [arXiv:hep-ph/0401141].[9] J. P. Lee, Phys. Rev. D , 114007 (2004) [arXiv:hep-ph/0403034].[10] Y. J. Kwon and J. P. Lee, Phys. Rev. D , 014009 (2005) [arXiv:hep-ph/0409133].[11] Particle Data Group, Y.M. Yao et al., J. Phys. G , 1 (2006).[12] B. Aubert et al. (BaBar Collaboration), Phys. Rev. Lett. , 051802 (2006).[13] K. Abe et al. (Belle Collaboration), arXiv:0706.3279 [hep-ex].[14] B. Aubert et al. [The BABAR Collaboration], Phys. Rev. Lett. , 241803 (2007)[arXiv:0707.4561 [hep-ex]].[15] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. , 261801 (2007) [arXiv:0708.0050[hep-ex]].[16] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. , 051803 (2008) [arXiv:0709.4165[hep-ex]].[17] K.C. Yang, Phys. Rev. D , 034009 (2005); D , (E)059901 (2005).[18] C.H. Chen, C.Q. Geng, Y.K. Hsiao, and Z.T. Wei, Phys. Rev. D , 054011 (2005).[19] G. Nardulli and T.N. Pham, Phys. Lett. B , 65 (2005).[20] V. Laporta, G. Nardulli and T. N. Pham, Phys. Rev. D , 054035 (2006) [Erratum-ibid. D , 079903 (2007)] [arXiv:hep-ph/0602243].[21] G. Calderon, J. H. Munoz and C. E. Vera, Phys. Rev. D , 094019 (2007) [arXiv:0705.1181[hep-ph]].[22] K. C. Yang, Phys. Rev. D , 094002 (2007) [arXiv:0705.4029 [hep-ph]].[23] H. Y. Cheng and K. C. Yang, Phys. Rev. D , 114020 (2007) [arXiv:0709.0137 [hep-ph]].[24] M. Suzuki, Phys. Rev. D (1993) 1252.[25] L. Burakovsky and J. T. Goldman, Phys. Rev. D , 2879 (1998) [arXiv:hep-ph/9703271].[26] H.Y. Cheng, Phys. Rev. D , 094007 (2003).[27] K. C. Yang, Nucl. Phys. B , 187 (2007) [arXiv:0705.0692 [hep-ph]].[28] K. C. Yang, JHEP , 108 (2005) [arXiv:hep-ph/0509337].[29] J. P. Lee, Phys. Rev. D , 074001 (2006) [arXiv:hep-ph/0608087].[30] W. Wang, R. H. Li and C. D. Lu, arXiv:0711.0432 [hep-ph].[31] I. I. Balitsky, V. M. Braun and A. V. Kolesnichenko, Nucl. Phys. B , 509 (1989).[32] V. L. Chernyak and I. R. Zhitnitsky, Nucl. Phys. B (1990) 137.[33] V. M. Belyaev, A. Khodjamirian and R. Ruckl, Z. Phys. C , 349 (1993)[arXiv:hep-ph/9305348].[34] P. Ball and V. M. Braun, Phys. Rev. D , 5561 (1997) [arXiv:hep-ph/9701238].[35] See, for example, P. Ball and R. Zwicky, Phys. Rev. D , 014029 (2005)[arXiv:hep-ph/0412079], and references therein.
36] C. Greub, T. Hurth and D. Wyler, Phys. Rev. D , 3350 (1996) [arXiv:hep-ph/9603404].[37] M. Beneke, T. Feldmann and D. Seidel, Nucl. Phys. B (2001) 25 [arXiv:hep-ph/0106067];A. Ali and A. Y. Parkhomenko, Eur. Phys. J. C , 89 (2002) [arXiv:hep-ph/0105302].[38] S. W. Bosch and G. Buchalla, Nucl. Phys. B , 459 (2002) [arXiv:hep-ph/0106081].[39] M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Nucl. Phys. B , 313 (2000)[arXiv:hep-ph/0006124].[40] M. Beneke and T. Feldmann, Nucl. Phys. B , 3 (2001) [arXiv:hep-ph/0008255].[41] P. Ball, G. W. Jones and R. Zwicky, Phys. Rev. D , 054004 (2007) [arXiv:hep-ph/0612081].[42] D. Becirevic, V. Lubicz and F. Mescia, Nucl. Phys. B , 31 (2007) [arXiv:hep-ph/0611295].[43] R. Barate et al. (ALEPH Collaboration), Eur. J. Phys. C , 599 (1999)., 599 (1999).