Nonperturbative Charming Penguin Contributions to Isospin Asymmetries in Radiative B decays
aa r X i v : . [ h e p - ph ] J un Nonperturbative Charming Penguin Contributions to IsospinAsymmetries in Radiative B decays Chul Kim, ∗ Adam K. Leibovich, † and Thomas Mehen ‡ Department of Physics, Duke University, Durham NC 27708, USA Department of Physics and Astronomy, University of Pittsburgh, PA 15260, USA
Abstract
Recent experimental data on the radiative decays B → V γ , where V is a light vector meson,find small isospin violation in B → K ∗ γ while isospin asymmetries in B → ργ are of order 20%,with large uncertainties. Using Soft-Collinear Effective Theory, we calculate isospin asymmetries inthese radiative B decays up to O (1 /m b ), also including O ( vα s ) contributions from nonperturbativecharming penguins (NPCP). In the absence of NPCP contributions, the theoretical predictions forthe asymmetries are a few percent or less. Including the NPCP can significantly increase the isospinasymmetries for both B → V γ modes. We also consider the effect of the NPCP on the branchingratio and CP asymmetries in B ± → V ± γ . ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] B decays, B → V γ , where V is a light vector meson, are important inheavy flavor physics because the dominant processes are due to the flavor changing neutralcurrent. Isospin asymmetries are interesting observables for testing the Standard Model(SM) and investigating new physics in the flavor sector. The isospin asymmetries for B → K ∗ γ and B → ργ are defined to be∆ K ∗ − = Γ( B → K ∗ γ ) − Γ( B − → K ∗− γ )Γ( B → K ∗ γ ) + Γ( B − → K ∗− γ ) , ∆ ρ − = 2Γ( B → ρ γ ) − Γ( B − → ρ − γ )2Γ( B → ρ γ ) + Γ( B − → ρ − γ ) . (1)In these asymmetries, the decay rates are averaged over charge conjugate modes. Recentexperimental measurements find [1]∆ K ∗ − = 0 . ± . , ∆ ρ − = 0 . ± . , (2)where the average values for the decay rates for B → K ∗ ( ρ ) γ are taken from the HeavyFlavor Averaging Group [2]. The isospin asymmetry for B → K ∗ γ is consistent with zerowithin an error of a few percent. The data suggests that the asymmetry in B → ργ is significantly larger, but because of large uncertainties it is not yet possible to draw adefinitive conclusion. The work in this paper is motivated by the question of whether ananomalously large isospin asymmetry in B → ργ can be understood within the SM. Inparticular, we calculate subleading contributions to the leading QCD factorization theoremsfor B → V γ to see if they can explain the observed asymmetries.In the heavy quark limit, the leading amplitudes are factorizable [3, 4, 5, 6, 7]. However,isospin violating asymmetries come from O (1 /m b ) suppressed power corrections for which thefactorization is necessarily more complicated. In this paper we calculate O (1 /m b ) correctionsto the asymmetries using Soft-Collinear Effective Theory (SCET) [8], which provides asystematic power counting. In addition, possible endpoint divergences in these higher ordercorrections can be regulated without imposing an arbitrary infrared cutoff by including thezero-bin subtraction of Ref. [9]. Previous QCD analyses of isospin asymmetries in radiative B xp xp (b)(a) bb γ − ¯ xp − ¯ xp FIG. 1: Nonperturbative charming penguin (NPCP) contributions for (a) B → V γ and (b) B → M M arise when ¯ x = 1 − x ≈ m c /m b , in which case the charm quark pair is in the thresholdregion. B decays [13, 14, 15]. (For analternative point of view, see Ref. [16]).The NPCP contributions to B → V γ are depicted in Fig. 1-(a). In certain kinematicregimes, the invariant mass of the charm quark pair in the loop in Fig. 1-(a) is near thethreshold, 2 m c , in which case the charm quark pair is described by nonrelativistic QCD(NRQCD) and additional interactions need to be taken into account. As pointed out inRef. [15], contributions from this regime are suppressed by only vα s (2 m c ) compared to theleading contribution. Here v is the relative velocity of the charm quarks in the thresholdregion. Therefore, the NPCP contribution to the isospin asymmetry could dominate overother 1 /m b suppressed contributions. In this paper, we calculate the isospin asymmetriesincluding the NPCP along with 1 /m b suppressed contributions. We also calculate the NPCPcontributions to the branching ratio and CP asymmetries for B ± → V ± γ .In the absence of NPCP contributions, the theoretical predictions for ∆ K ∗ − and ∆ ρ − areno larger than a few percent. Including the NPCP contributions can significantly modifythe theoretical predictions for the isospin asymmetries. We will see below that the NPCPcontribution can be factorized using SCET, and the result expressed in terms of nonper-turbative matrix elements. The NPCP matrix elements are fitted to available data on theisospin asymmetries, ∆ K ∗ − and ∆ ρ − , the CP asymmetry for B ± → ρ ± γ (recently measuredby Belle [17]), and the branching ratio for B + → ρ + γ [2]. The predictions for ∆ K ∗ − and ∆ ρ − are of order 10%, with uncertainties large enough that both predictions are consistent withexperiment. However, the NPCP does not predict a large difference between ∆ K ∗ − and ∆ ρ − ,as suggested by the central values in Eq. (2).The isospin asymmetry in B → V γ can arise from either the mass difference of thespectator quark in the B meson or the electric charge difference when the spectator quarkemits the photon in the final state. However the isospin asymmetry due to the mass differenceis negligible because it is O (( m u − m d ) / Λ QCD ) and therefore of order 1% or smaller. So thedominant piece comes from electromagnetic (EM) interactions with the spectator quark.In order to describe the isospin breaking corrections to B → V γ , we need the followingeffective weak Hamiltonian H W = G F √ " X p = u,c λ ( q ) p (cid:16) C O p + C O p (cid:17) − λ ( q ) t (cid:16) X i =3 C i O i + C g O g + C γ O γ (cid:17) , (3)where the operators are O p = ( pb ) V − A ( qp ) V − A , O p = ( p β b α ) V − A ( q α p β ) V − A ,O , = ( qb ) V − A X q ′ = u,d,s,c,b ( q ′ q ′ ) V ∓ A , O , = ( q β b α ) V − A X q ′ = u,d,s,c,b ( q ′ α q ′ β ) V ∓ A , (4) O γ = − em b π qσ µν F µν (1 + γ ) b, O g = − gm b π qσ µν G aµν T a (1 + γ ) b. Here q is the d or s quark, the CKM factor is λ ( q ) p = V pb V ∗ pq , and V ± A = γ µ (1 ± γ ).3 a) (b) (c) b q ′ = u, d q = d, sq ′ O c O , · · · , O O g FIG. 2: Various isospin breaking contributions in full QCD. Here crosses denote another possiblephoton emissions from the spectator quark. If we do not consider the long distance contributionin the diagram (c), all the contributions are power-suppressed by O (Λ /m b ). For the asymmetric contributions, the photon radiates from either the initial or finalspectator quark as shown in Fig. 2. If the photon is radiated from the initial spectatorquark, we need SCET operators of the type O (0 , q ) = χ n Γ b v ξ n Γ ξ n where ξ n ( χ n ) is n ( n )-collinear field and n and n are light-cone vectors satisfying n = n = 0 , n · n = 2.The analysis of factorization for these operators appears in Refs. [19, 20] and the Wilsoncoefficients at next-to-leading order (NLO) in α s have been calculated in Refs. [18, 20].The leading operator in SCET only contributes to longitudinally polarized vector mesonproduction, but in B → V γ the vector meson must be transversely polarized. Transverselypolarized vector mesons can be produced from subleading operators that are higher orderin the SCET expansion parameter, λ . The relevant effective weak Hamiltonian in SCET is H (1 , q ) W, SCET = G F √ X p λ ( q ) p X i =1 Z dx B pi ( x, µ ) O (1 , q ) i ( x, µ ) , (5)where O (1 , q ) i are O (1 , q )1 = χ un W n γ ⊥ µ (1 − γ ) Y † n b v h ξ qn W n γ µ ⊥ (1 − γ ) W † n ξ un + ξ qn W n γ µ ⊥ (1 − γ ) W † n ξ un i x , O (1 , q )2 , = χ qn W n γ ⊥ µ (1 − γ ) Y † n b v h ξ un W n γ µ ⊥ (1 ∓ γ ) W † n ξ un + ξ un W n γ µ ⊥ (1 ∓ γ ) W † n ξ un i x , (6) O (1 , q )4 = X q ′ = u,d,s χ q ′ n W n γ ⊥ µ (1 − γ ) Y † n b v h ξ qn W n γ µ ⊥ (1 − γ ) W † n ξ q ′ n + ξ qn W n γ µ ⊥ (1 − γ ) W † n ξ q ′ n i x , O (1 , q )5 , = X q ′ = u,d,s χ qn W n γ ⊥ µ (1 − γ ) Y † n b v h ξ q ′ n W n γ µ ⊥ (1 ∓ γ ) W † n ξ q ′ n + ξ q ′ n W n γ µ ⊥ (1 ∓ γ ) W † n ξ q ′ n i x . Here the superscript ‘1’ denotes suppression by one power of λ compared to the leadingoperator, W n ( n ) is a collinear Wilson line in n ( n )-direction, and Y n is a ultrasoft (usoft)Wilson line. The subscript outside the square brackets denotes that a delta function whichfixes the momentum fraction, x , is included in the bilinear operator: h ξ ¯ n W n Γ W † n ξ n i x ≡ ξ ¯ n W n δ (cid:16) x − P † E V (cid:17) Γ W † n ξ n , (7) Our conventions are the same as Ref. [18] except that we have exchanged n ↔ n compared to that paper. P = n · P is a derivative operator taking the largest momentum component and E V is the energy of the produced vector meson. Here, ξ n is the power-suppressed component inthe spin projection of q n = ( n/n// ξ n + ( n/n// ξ n , where q n is the collinear quark field. Usingthe equation of the motion, ξ n can be expressed in terms of ξ n as ξ n = W n P W † n iD/ ⊥ n/ ξ n . (8)The Wilson coefficients B pi in Eq. (5) are the same as the leading Wilson coefficients C pi of Ref. [18] due to reparameterization invariance [21]. Because the isospin-breakingcontributions from H (1 , q ) W, SCET are suppressed by Λ
QCD /m b compared to the leading decayamplitude, we will suppress EM penguins with C , , , at tree level and keep only C , , g atone loop order in B pi for our phenomenological analysis.The four-quark operators in the effective weak SCET Hamiltonian, Eq. (5), contributeto the radiative weak decay when the photon is emitted from the initial spectator quark,as shown in Fig. 3-(a). To calculate their contribution, we need to take the time-orderedproducts of O (1 , q ) i with the electromagnetic interaction term L (1)EM , L (1)EM = e q q us Y n A / ⊥ W † n χ qn + h . c ., (9)where A µ is a photon field and e q is the electric charge of the quark. The time-orderedproducts are performed in SCET I with possible offshellness p ∼ m b Λ and then matchedonto SCET II , which describes dynamics with fluctuations of p ∼ Λ . In the matching, n -collinear fields having large offshellness of O ( m b Λ) must be integrated out, giving a jetfunction h | T { W † n χ n ( z ) , χ n W n (0) }| i = i n/ δ ( z − ) δ ( z ⊥ ) Z dk − π e − ik − z + / J n · p γ ( k − ) , (10)where k + = n · k , k − = n · k , and the momentum of the photon, p µγ = n · p γ n µ / m b n µ / α s , the jet function is simply J n · p γ = 1 /k − .The n - and n -collinear degrees of freedom are decoupled at leading order in SCET, andby using a field redefinition it is possible to decouple usoft degrees of freedom from both. (a) (b) q ( u ) s ξ n b v ξ n O (1 , qγ ) O (1 , q ) FIG. 3: SCET diagrams for the isospin breaking corrections. Each diagram represent the electro-magnetic interactions from initial and final spectator quarks, respectively. n -collinear piece of the matrix element describing the production of the lightvector meson is decoupled from the n -collinear and usoft parts. Thus, we can compute B to γ from H (1 , q ) W, SCET , which only depends on usoft and n -collinear physics, independently ofthe light-meson production process, which depends on n -collinear physics. After a briefcalculation, we findˆ T µ q = i Z d z h γ ( ǫ ∗⊥ ) | T { χ qn W n γ ⊥ µ (1 − γ ) Y † n b v (0) , L (1)EM ( z ) }| B i , (11)= − i e q f B m B ( ǫ ∗ µ ⊥ + iε µν ⊥ ǫ ∗⊥ ν ) Z dl − J n · p γ ( − l − ) φ + B ( l − ) , where ε µν ⊥ = ε µνρσ n ρ n σ / ε = − φ + B is a light-cone distribution amplitude(LCDA) of B meson [22]. We use the convention of Ref. [23] with n and n interchanged.The light meson production is described by the n -collinear part of the matrix elements.The matrix elements for production of transversely polarized vector mesons in SCET are h V ⊥ ( η ∗⊥ ) | h ξ n W n γ µ ⊥ W † n ξ n + ξ n W n γ µ ⊥ W † n ξ n i x | i = − if V m V η ∗ µ ⊥ g ( v ) ⊥ ( x ) , (12) h V ⊥ ( η ∗⊥ ) | h ξ n W n γ µ ⊥ γ W † n ξ n + ξ n W n γ µ ⊥ γ W † n ξ n i x | i = − f V m V ε µν ⊥ η ∗⊥ ν ∂∂x g ( a ) ⊥ ( x ) , (13)where g ( v,a ) ⊥ are chiral-even, twist-3 LCDAs [24] whose asymptotic forms are g ( v ) ⊥ ( x ) = 3( x + x ) / g ( a ) ⊥ ( x ) = 6 xx , where x = 1 − x .Combining Eqs. (11), (12), and (13), the matrix element of O (1 , q )1 for B − → ρ − γ , forexample, is h ρ − γ |O (1 , q )1 ( x, µ ) | B − i = − e u f B f ρ m B m ρ A L ( ǫ ∗⊥ , η ∗⊥ ) Z dl − J n · p γ ( − l − , µ, µ ) φ + B ( l − , µ ) × h g ( v ) ⊥ ( x, µ ) − ∂ ∂x g ( a ) ⊥ ( x, µ ) i , (14)where the renormalization scales are roughly given by µ ∼ p m b Λ QCD and µ ∼ Λ QCD , and A L = ǫ ∗⊥ · η ∗⊥ − iε µν ⊥ ǫ ∗⊥ µ η ∗⊥ ν is the polarization factor for the left-handed ρ − and γ . In the caseof B → ργ , it is necessary to include one-loop corrections to the jet function, see Ref. [25]for details.When the photon radiates from a final state quark (i.e., from the crosses in Fig. 2), theintermediate quark line is hard with offshellness of order m b . Matching onto SCET weobtain localized four-quark operators with photons, shown in Fig. 3-(b). In this case, theeffective weak Hamiltonian that contributes to the decay amplitudes is H (1 , qγ ) W, SCET = G F √ X p λ ( q ) p X i =1 Z dx A pi ( x, µ ) O (1 , qγ ) i ( x, µ ) , (15)where the five-particle operators O (1 , qγ ) i are O (1 , qγ ) { , } ( x ) = X q ′ = u,d,s e q ′ q ′ Y n { n/, n/ } (1 + γ ) Y † n b v h ξ qn W n n/ / ⊥ (1 + γ ) 1 P W † n ξ q ′ n i x . (16)6n Eq. (15), the Wilson coefficients A pi at NLO are A p ( x, µ ) = C + C N + α s π C F N ( C h m B µ − G ( s p , x ) i − C g m b ¯ xm B ) , (17) A p ( x, µ ) = C + C N + α s π C F N ( C h m B µ − G ( s p , x ) i − C g m b m B ) , where G ( s p , x ) is G ( s p , x ) = − Z dzz ¯ z ln( s p − z ¯ z ¯ x − iǫ ) , (18)and s p ≡ m p /m B . Similar to B pi , we neglect EM penguins at tree level and only keep C , , g at one loop. The sum of the terms in Eq. (17) proportional to C differs by a factor of 3/4from Ref. [10].Again, the B → γ piece of the matrix element factors from the light-meson productionpiece. The n -collinear part in Eq. (16), describing the vector meson production, gives aleading twist LCDA, φ ⊥ ( x ), whose asymptotic form is 6 x ¯ x . It can be obtained from thefollowing projection: D V ⊥ ( η ∗⊥ ) (cid:12)(cid:12)(cid:12)h ξ n W n δ (cid:16) x − P † E V (cid:17)i αa h W † n ξ n i βb (cid:12)(cid:12)(cid:12) E twist − = − i f ⊥ V E V (cid:16) η/ ∗⊥ n/ (cid:17) ba δ βα N φ ⊥ ( x ) . (19)As an example, the B → K ∗ γ matrix elements from O (1 , qγ ) i are h K ∗ γ |O (1 , qγ )1 | B i = h K ∗ γ |O (1 , qγ )2 | B i = − e d f B f ⊥ K ∗ m B A L ( ǫ ∗⊥ , η ∗⊥ ) φ ⊥ ( x, µ )¯ x . (20)Next we turn to the contributions from NPCP, shown in Fig. 1-(a). The size of thiscontribution is O ( vα s (2 m c )) [15] and therefore is suppressed only logarithmically in the large m c limit, compared to the power suppression of previously considered O (1 /m b ) contributions.Numerically, Λ QCD /m b and vα s (2 m c ) are roughly the same size, so a priori it is sensibleto include them at the same order. In fact we will see below that the NPCP gives thedominant contribution to isospin violation in B → ργ . When ¯ x is close to 4 s c , long-distanceinteractions govern the charm quark pair in the loop and hence it cannot be separatedfrom the B meson. However, the n -collinear piece of V ⊥ can still be decoupled, with thedominant part obtained from the leading twist projection of Eq. (19). The factorizationprocess is similar to the treatment in Ref. [18] and we refer the reader to that paper fordetails. The NPCP contribution to the decay amplitude is M c ¯ c = G F √ λ ( q ) c h V ⊥ γ | C O c | B i NPCP (21)= i G F N √ λ ( q ) c f ⊥ V m B η ∗⊥ µ Z dxδ (¯ x − s c ) φ ⊥ ( x ) H c ¯ c ( x, m B ) h γ |O µc ¯ c | B i , where H c ¯ c = 4 C πα s / (¯ xm B ) at lowest order and the six-quark operator O µc ¯ c , includingnonrelativistic charm quark fields, c ± v , is defined as O µc ¯ c = i Z d y c − v Y n γ ν ⊥ T a Y † n c +v ( y ) χ q ′ n W n ( y ) γ ⊥ ν γ µ ⊥ n/ γ ρ (1 − γ ) T a Y † n c − v × c +v γ ρ (1 − γ ) b v (0) . (22)7n order to integrate out n -collinear fields with fluctuations greater than Λ QCD , we considertime-ordered products of O µc ¯ c with L (1)EM ,ˆ T c ¯ c = iη ∗⊥ µ Z d z h γ ( ǫ ∗⊥ ) | T {O µc ¯ c (0) , L (1)EM ( z ) }| B i (23)= ie q ′ Z d y dz + dk − π e − ik − ( z + − y + ) / J n · p γ ( k − ) h |O c ¯ cq ′ b ( ǫ ∗⊥ , η ∗⊥ , z + , y ) | B i , where we employed Eq. (10) to obtain the second line of Eq. (23) and O c ¯ cq ′ b is O c ¯ cq ′ b ( ǫ ∗⊥ , η ∗⊥ , z + , y ) = q ′ us Y n ( z + , y ⊥ , y − n/ ǫ/ ∗⊥ γ ν ⊥ η/ ∗⊥ n/ γ ρ (1 − γ ) T a Y n c − v (0) (24) × c − v Y n γ ⊥ ν T a Y † n c +v ( y ) c +v γ ρ (1 − γ ) b v (0) . The matrix element of O c ¯ cq ′ b in Eq. (23) is purely nonperturbative. It can be decomposedinto left- and right-handed polarized contributions, Z d y h |O c ¯ cq ′ b ( ǫ ∗⊥ , η ∗⊥ , z + , y ) | B i e ik − y + / = f B m B Z dl − e − il − z + / (25) × [ A L ( ǫ ∗⊥ , η ∗⊥ ) F Lc ¯ c ( k − , l − ) + A R ( ǫ ∗⊥ , η ∗⊥ ) F Rc ¯ c ( k − , l − )] , where A R = ǫ ∗⊥ · η ∗⊥ + iε µν ⊥ ǫ ∗⊥ µ η ∗⊥ ν is the polarization factor for decay into right-handed finalstates. An important point is that the NPCP can give a right-handed polarized contributionwhich is not O (1 /m b ) suppressed. As pointed out in Ref. [26], any other right-handedpolarized contributions to the decay amplitude should be suppressed by 1 /m b . ThereforeNPCP could give the dominant contribution to the right-handed polarized decay amplitudes.Since the right-handed polarized contribution from NPCP cannot interfere with the leadingorder amplitude which produces only left-handed final states, the right-handed contributiondoes not enter into the asymmetries until higher orders. Therefore, we neglect any possibleright-handed contribution from NPCP in our calculations of the asymmetries.Combining Eqs.(21), (23), and (25), we obtain M c ¯ c = − G F √ λ ( q ) c e q ′ f B f ⊥ V m B A L ( ǫ ∗⊥ , η ∗⊥ ) πα s N Λ c ¯ c Z dx φ ⊥ ( x )¯ x δ (¯ x − s c ) ˆ H c ¯ c (¯ x ) , (26)where q = d or s , q ′ is the B meson spectator quark, and ˆ H c ¯ c = ¯ xm B H c ¯ c / (4 πα s ) = C + · · · .Here we have defined Λ − c ¯ c to beΛ − c ¯ c = − Z dl − dz + dk − π e − i ( k − + l − ) z + / J n · p γ ( k − ) F Lc ¯ c ( k − , l − ) (27)= − Z dl − J n · p γ ( − l − ) F Lc ¯ c ( − l − , l − ) ∼ Z dl − F Lc ¯ c ( − l − , l − ) l − . Following Ref. [15], we can power count the size of this correction. The NPCP contri-bution is suppressed relative to the leading order term by vα s (2 m c ), and thus of order m b vα s (2 m c ) / Λ QCD relative to the other isospin breaking terms considered. Based on thispower counting, we expect that m B Λ c ¯ c φ ⊥ (4 s c )4 s c ∼ v m b Λ QCD . (28)8he factor φ ⊥ (4 s c ) / (4 s c ) is formally O (1) in the power counting of Ref. [15], but numerically φ ⊥ (4 s c ) / (4 s c ) ≈ .
3, so we keep this factor in estimating m B / Λ c ¯ c . Taking v m b / Λ QCD ∼ m B / Λ c ¯ c ∼ .
7. The extracted values of m B / Λ c ¯ c are consistent with this naive estimatebut smaller. For these values of m B / Λ c ¯ c , the NPCP gives significant contributions to theisospin asymmetries.Finally, there is another interesting isospin-breaking source, a double photon contributionwith the EM penguin O γ . It is only available for the decay with an unflavored vector meson,i.e., B → ρ γ . The largest contributions are depicted in Fig. 4. Concentrating first on Fig. 4-(a), the off-shell photon coming from O γ produces the vector meson and then an additionalphoton is emitted from the B meson spectator quark. Integrating out the hard photon, wecan match onto the SCET I operator C γ O γ −→ ee q ′ π m b m B m V Z dx C γγ ( x ) χ qn W n n/ γ µ ⊥ (1 + γ ) Y † n h v (29) × h ξ q ′ n W n γ ⊥ µ W † n ξ q ′ n + ξ q ′ n W n γ ⊥ µ W † n ξ q ′ n i x , where C γγ is equal to C γ at tree level. Next we integrate out the n -collinear fields in thetime-ordered product with L (1)EM and match onto SCET II . Applying Eqs. (10), (11), and(12), we find h V ⊥ γ | C γ O γ | B i γ (a) = ee q e q ′ π f B f V m b m B m V C γ A L ( ǫ ∗⊥ , η ∗⊥ ) Z dl − J n · p γ ( − l − ) φ + B ( l − ) , (30)and, in case of B → ρ γ , h ρ ⊥ γ | C γ O γ | B i γ (a) = Q d ( Q u − Q d ) √ eα π f B f ρ m b m B m V C γ A L ( ǫ ∗⊥ , η ∗⊥ ) Z dl − J n · p γ ( − l − ) φ + B ( l − ) , (31)where Q q = e q /e with Q u = 2 / Q d = − / α s , the contribution of Fig. 4-(b) is the same as Fig. 4-(a) with n and ¯ n exchanged. So the result can be written as h ρ ⊥ γ | C γ O γ | B i γ (b) = Q d ( Q u − Q d ) √ eα π f B f ρ m b m B m V C γ A L ( ǫ ∗⊥ , η ∗⊥ ) Z dl + J n · p γ ( − l + ) φ + B ( l + ) . (32)The double-photon contribution is suppressed by α , but enhanced by a factor of m b /m V due to the virtual photon. Compared to the other isospin-breaking breaking contributions,such as those in Eqs. (14) and (20), this contribution is rather small.The isospin asymmetries in Eq. (1) are given by∆ V − = Re( b Vd − b Vu ) + R Re(¯ b Vd − ¯ b Vu )1 + R , (33)where b Vd = A V c V L V , b Vu = A V − L V , (34)9 us ξ q ′ n b v ξ q ′ n (b)(a) FIG. 4: Leading double photon contribution to the isospin asymmetry, where ⊗ represents O γ .They are suppressed by αm b /m V compared to other usual isospin breaking contributions. A V , − are the leading isospin breaking corrections to the decay amplitude, L V are the leadingisospin symmetric decay amplitudes, and c V = 1 for K ∗ , c V = − / √ ρ . In Eq. (33),¯ b Vu,d are the corresponding ratios for the charge conjugate modes, and R = | L V | / | L V | . L V can be written as L K ∗ = G F √ e π m b m B A L ( ǫ ∗⊥ , η ∗⊥ ) λ ( s ) c a c ,K ∗ ζ K ∗ ⊥ , (35) L ρ = G F √ e π m b m B A L ( ǫ ∗⊥ , η ∗⊥ ) X p = u,c λ ( d ) p a p ,ρ ζ ρ ⊥ , where the transition form factor for B → V , ζ V ⊥ , is defined as h V ( η ∗⊥ ) | ξ n W n γ µ ⊥ (1 − γ ) Y † n b v | B i = m B ( iε µν ⊥ η ∗⊥ ν − η ∗ µ ⊥ ) ζ V ⊥ , (36)and we will use ζ K ∗ ⊥ = 0 . ± .
07 and ζ ρ ⊥ = 0 . ± .
05 [11] for the numerical analysis. TheWilson coefficients a p ,V in Eq. (35) are a p ,V = C γ A (0)7 + α s C F π [ C G ( s p ) + C g G ( s p )] (37)+ πα s C F N f B f ⊥ V m B m b Z dl + φ + B ( l + ) l + Z dx φ ⊥ ( x )¯ x h C γ + C H ( x, s p ) + C g i ζ V ⊥ , where the hard functions A (0)7 , G , , and H are available in Refs. [3, 5, 27], and we followedthe conventions of Ref. [5].Finally we obtain b K ∗ q = Q q π f B m b a c ,K ∗ ζ K ∗ ⊥ h f K ∗ ⊥ m b K K ∗ + f K ∗ m K ∗ λ B m b K K ∗ q i , (38) b ρq = Q q π f B m b P p = u,c λ ( d ) p a p ,ρ ζ ρ ⊥ h f ρ ⊥ m b K ρ + f ρ m ρ λ B m b K ρ q i . (39)Here λ − B = R dlφ + B ( l ) /l and we use the following model for φ + B ( l ) [28] φ + B ( l, µ ) = 4 µlπλ B ( l + µ ) h µ l + µ − σ B − π ln lµ i , (40)10here the parameters λ B and σ B are λ B = 460 ±
110 MeV, σ B = 1 . ± . µ = 1 GeV. K , can be written as K K ∗ = Z dx φ ⊥ ( x )¯ x n − h A c ( x ) + A c ( x ) i − C πα s N m B Λ c ¯ c δ (¯ x − s c ) o (41) K K ∗ q = Z dx h g ( v ) ⊥ − ∂ ∂x g ( a ) ⊥ i ( x ) n λ ( s ) u λ ( s ) c (cid:16) C + C N (cid:17) δ qu + B c ( x, m b ) o , (42) K ρ = X p = u,c λ ( d ) p Z dx φ ⊥ ( x )¯ x n − h A p ( x ) + A p ( x ) i − δ pc C πα s N m B Λ c ¯ c δ (¯ x − s c ) o , (43) K ρ q = X p = u,c λ ( d ) p (Z dx h g ( v ) ⊥ − ∂ ∂x g ( a ) ⊥ i ( x ) h − λ B Z dl − φ + B ( l − ) J p γ ( − l − ) (44) × (cid:16) δ qu B p ( x ) − δ qd B p ( x ) (cid:17) + B p ( x ) i + 2 δ qd C γ απ m b m B m ρ ) . Here we include only the tree-level contributions to Cabibbo-suppressed terms with λ ( s ) u inEq. (42) because it is numerically comparable to the other term with B c .In the convolutions of A and φ ⊥ ( x ) / ¯ x in Eqs. (41,43) and B and g ( v ) ⊥ in Eqs. (42,44),there are endpoint divergences, which can be eliminated with the zero-bin subtractions [9] Z dx φ ⊥ ( x )¯ x −→ Z dx φ ⊥ ( x ) + ¯ xφ ′⊥ (1)¯ x , (45) Z dx g ( v ) ⊥ ( x )¯ x −→ Z dx g ( v ) ⊥ ( x ) − g ( v ) ⊥ (1)¯ x . The zero-bin subtraction removes infrared divergences from the x integrals. We have droppedall finite terms including logarithms associated with rapidity scale dependence. We estimatethe uncertainty associated with this procedure to be 50%.For numerical estimates of b K ∗ ,ρq in Eqs. (38) and (39), we use the following set ofparameters: { m b , m c , m B , m K ∗ , m ρ , f B , f K ∗ , f ρ , f K ∗ ⊥ , f ρ ⊥ } = { . , . , . , . , . , . ± . , . , . , . ± . , . ± . } GeV. The CKM parameters are ρ =0 . ± .
064 and η = 0 . ± . µ = m b , we do not include any renormalization group evolution, and we use theasymptotic forms for the vector meson wave function, φ ⊥ and g ( v ) , ( a ) ⊥ . Our estimates for theisospin asymmetries in the absence of NPCP contributions are∆ K ∗ − = 0 . ± . , ∆ ρ − = 0 . ± . , (46)where the dominant errors come from λ B , ζ V ⊥ , and CKM factors. These estimates for ∆ V − are comparable to previous theoretical results [10, 11, 29]. Comparing to Eq. (2), we seethat ∆ K ∗ − is consistent, but ∆ ρ − disagrees by about 1.7 σ .Next we include the NPCP contribution in our calculation. In addition to the isospinasymmetries, the NPCP can contribute to the CP violating asymmetry, ∆ ρ + − [17], and tothe branching ratio, Br[ B + → ρ + γ ] [2]. In order to obtain values of m B / Λ c ¯ c that are not11 xp. w/o NPCP w/ NPCP∆ K ∗ − . ± .
04 0 . ± .
02 0 . ± . ρ − . ± .
14 0 . ± .
02 0 . ± . ρ + − . ± .
33 0 . ± .
02 0 . ± . B + → ρ + γ ] × . ± .
24 1 . ± .
69 1 . ± . inconsistent with measurements of these quantities, we perform a least squares fit to all fourobservables. In our calculations of ∆ ρ + − and Br[ B + → ρ + γ ], we include only the leadingorder and NPCP contributions, without any O (1 /m b ) corrections. An analysis that includesthe O (1 /m b ) corrections to all four observables is clearly required but is beyond the scopeof this paper.The results of the fits along with experimental results are shown in Table I. The firstcolumn lists the observables considered and the second gives their measured values includingerrors. In the third column, we show the theoretical prediction in the absence of the NPCPcontribution for the values of the parameters given earlier. The last column gives the resultsof the fit with the NPCP included. We extractRe (cid:20) m B Λ c ¯ c (cid:21) = − . ± . , Im (cid:20) m B Λ c ¯ c (cid:21) = 0 . ± . . (47)The χ of the predictions in column three of Table I is 15.2. Including the NPCP, the χ is 12.1, so the overall agreement between experiment and theory is slightly improved.Note that after including the NPCP the theoretical prediction for ∆ ρ − increases so that the1 σ error band of the experimental result and the theoretical result now overlap. However,the prediction for ∆ K ∗ − is now significantly increased. The trend suggested by the centralvalues of the experimental data, namely a large value of ∆ ρ − and small value of ∆ K ∗ − , doesnot seem to be naturally accommodated by including NPCP contribution. However, oncetheoretical and experimental uncertainties are taken into account, the theoretical predictionsare consistent with both isospin asymmetries. For the range of values of m B / Λ c ¯ c obtainedin our fit, the NPCP does not have significant impact on the theoretical predictions for∆ ρ + − and Br[ B + → ρ + γ ]. Finally, inclusion of NPCP contributions substantially increasesthe uncertainty in all theoretical predictions because the parameter m B / Λ c ¯ c is not wellconstrained.These results indicate that the NPCP can increase the isospin violating asymmetry ∆ ρ − to bring theoretical predictions closer to current data, while maintaining consistency withthe other observed asymmetries. It is not possible to obtain predictions for the two isospinasymmetries that are in agreement with the central values of both ∆ K ∗ − and ∆ ρ − . How-ever, the uncertainties in the current measurements of all asymmetries are large and bettermeasurements are need to determine whether the NPCP is an important contribution to B → ργ isospin and CP violating asymmetries.12o summarize, we have used SCET to calculate the isospin asymmetries in B → V γ decays, including all O (1 /m b ) contributions as well as the O ( vα s ) NPCP contribution. Asin nonleptonic B decays [30], our analysis allows for large NPCP contributions, which couldaccount for the large isospin asymmetries measured (with large errors) in B → ργ . If theisospin asymmetries are large and the NPCP is the source of these asymmetries, then wealso expect the NPCP to contribute to CP asymmetries in B → ργ and give larger thanexpected contributions to the right-handed polarized decay rates in B → V γ . We speculatethat the NPCP could also be responsible for the recently measured enhanced transverselypolarized decay amplitude for B → V V [31]. More precise experiments will be needed todetermine the exact size of NPCP and confirm these predictions.We are thankful to Jure Zupan for initial discussions of this work. We thank Dan Pir-jol and Zoltan Ligeti for useful comments on an earlier version of this paper. C. K. andT. M. were supported in part by the Department of Energy under grant numbers DE-FG02-05ER41368 and DE-FG02-05ER41376. A. K. 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