Predicting B−>ργ and B s −>ργ using holographic AdS/QCD Distribution Amplitudes for the ρ meson
PPredicting ¯ B ◦ → ρ ◦ γ and ¯ B ◦ s → ρ ◦ γ using holographic AdS/QCDDistribution Amplitudes for the ρ meson M. Ahmady
Department of Physics, Mount Allison University, Sackville, N-B. E46 1E6, Canada ∗ R. Sandapen
D´epartement de Physique et d’Astronomie,Universit´e de Moncton, Moncton, N-B. E1A 3E9, Canada &Department of Physics, Mount Allison University, Sackville, N-B. E46 1E6, Canada † We derive holographic AdS/QCD Distribution Amplitudes for the transverselypolarised ρ meson and we use them to predict the branching ratio for the decays¯ B ◦ → ρ ◦ γ and ¯ B ◦ s → ρ ◦ γ beyond leading power accuracy in the heavy quark limit.For ¯ B ◦ → ρ ◦ γ , our predictions agree with those generated using Sum Rules (SR)Distribution Amplitudes and with the data from the BaBar and Belle collaborations.In computing the weak annihilation amplitude which is power-supressed in ¯ B ◦ → ρ ◦ γ but is the leading contribution in ¯ B ◦ s → ρ ◦ γ , we find that, in its present form, theAdS/QCD DA avoids the end-point divergences encountered with the SR DA . INTRODUCTION
Flavor Changing Neutral Currents (FCNC) are excellent probes to the Standard Modeland beyond. In particular, the b → ( s, d ) γ transitions are most important for the extractionof the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements as well as for the search ofNew Physics (NP) signals. The experimental measurements for the b → dγ transistion arecurrently available for exclusive radiative B decay to a ρ meson, i.e. B → ργ . For a recentreview of radiative B decays, we refer to [1].The theory of exclusive decays is complicated by their sensitivity to non-perturbativephysics. The standard[40] theoretical framework is QCD factorization (QCDF)[2, 3] whichstates that, to leading power accuracy in the heavy quark limit, the decay amplitude factor-izes into perturbatively computable kernels and non-perturbative objects namely the B → V transition form factor, the meson couplings and the leading twist Distribution Amplitudes a r X i v : . [ h e p - ph ] F e b (DAs) of the mesons. Traditionally the DAs for the vector meson are obtained from QCDSum Rules[4–6]. The numerical values of the transition form factor and the tensor couplingof the vector meson are obtained from light-cone Sum Rules or lattice QCD. The predic-tive power of QCDF is limited by the uncertainties associated with these non-perturbativequantities and also by power corrections to the leading amplitude [7]. The computation ofthe power corrections is often problematic due to the appearance of end-point divergencesin convolution integrals that contribute to the decay amplitude [7, 8].Our goal in this paper is to use new holographic AdS/QCD DAs for the ρ meson tocompute the branching ratios for two exclusive radiative decays, namely ¯ B ◦ → ρ ◦ γ and ¯ B ◦ s → ρ ◦ γ , beyond leading power accuracy. We derive the AdS/QCD DAs using a holographicAdS/QCD light-front wavefunction for the ρ meson [9] which was recently shown to generatepredictions for the cross-sections in diffractive ρ meson production that are in agreementwith the data collected at the HERA electron-proton collider [10]. Reference [10] also showsthat the second moment of the AdS/QCD twist-2 DA of the longitudinally polarised ρ meson is in agreement with both Sum Rules and lattice predictions. Here we shall extendthe comparison between AdS/QCD and Sum Rules for the twist-2 and twist-3 DAs of thetransversely polarized ρ meson since both of these non-perturbative quantities are requiredto compute the decay amplitudes for ¯ B ◦ → ρ ◦ γ and ¯ B ◦ s → ρ ◦ γ beyond leading poweraccuracy.Theoretical predictions for the branching ratio of ¯ B ◦ → ρ ◦ γ , using the leading twist-2 SRDA for the ρ meson, can be found in references [2, 11]. In both references, the leading powercorrection ( O (Λ QCD /m b )) due to annihilation is taken into account within QCDF and thesubleading ( O (Λ QCD /m b ) ) annihilation contributions are neglected. On the other hand, inreference [11], two other classes of power corrections due to long-distance photon emissionand soft gluon emission are also taken into account. Here we shall investigate the numericalimportance of three additional subleading annihilation contributions to ¯ B ◦ → ρ ◦ γ , two ofwhich turn out to be sensitive to the higher twist-3 DA of the ρ meson. In fact, the raredecay ¯ B ◦ s → ρ ◦ γ proceeds mainly via these four annihilation processes and a theoreticalprediction for its branching ratio using the SR twist-3 DA is available in reference [12].However this prediction suffers from a large uncertainty because of end-point divergencesencountered when computing those annihilation contributions sensitive to the twist-3 DAfor the ρ meson. We shall update this prediction using the twist-3 AdS/QCD DA which, aswe shall see, avoids the end-point divergence problem.On the experimental side, the branching ratio for ¯ B ◦ → ρ ◦ γ has been measured withincreasing precision by the BaBar and Belle collaborations [13]. On the other hand, the raredecay ¯ B ◦ s → ρ ◦ γ has not been measured experimentally but it is an interesting process toinvestigate at LHCb because of its sensitivity to NP especially those which allow FCNC attree level [12].We start by an outline of the computation of the amplitudes for both decays followingreferences [2, 12], devoting attention to the quantities which are dependent on the DAs ofthe ρ meson. DECAY AMPLITUDES
The effective weak Hamiltonian for the underlying b → dγ transition is given by H eff = G F √ (cid:88) p = u,c V ∗ pq V pb [ C Q p + C Q p + (cid:88) i =3 C i Q i ] (1)where q = d for ¯ B ◦ → ρ ◦ γ and q = s for ¯ B ◦ s → ρ ◦ γ . Q p and Q p are the current-currentoperators, Q i =3 .. are the QCD penguin operators and Q and Q are the electromagneticand chromomagnetic operators. The coeffecients C i are the perturbatively known Wilsoncoeffecients and V ij are the CKM matrix elements. In this paper, we shall use the NLOWilson coeffecients given in [11, 14][41] and we use the numerical values of CKM matrixelements given in reference [15].We start with the amplitude for the decay ¯ B ◦ → ρ ◦ γ . At leading power accuracy in theheavy quark limit and to all orders in the strong coupling α s , the matrix elements of theseoperators factorize as[2]: (cid:104) ρ ( P, e T ) γ ( q, (cid:15) ) | Q i | ¯ B (cid:105) = [ F B → ρ T Ii + (cid:90) d ζ d z Φ B ( ζ ) T IIi ( ζ, z ) φ ⊥ ρ ( z )] · (cid:15) (2)i.e. into perturbatively computable hard-scattering kernels T Ii and T IIi and three non-perturbative quantities namely the transition form factor F B → ρ , the leading twist DA ofthe B meson, Φ B ( ζ ), and the twist-2 DA of ρ meson, φ ⊥ ρ ( z ). In equation (2), P and e T arethe 4-momentum and polarization vector of the ρ meson while q and (cid:15) are the 4-momentumand polarization vector of the photon. The form factor F B → ρ is obtained from light-coneQCD Sum Rules [11]. The second term in equation (2) describes mechanisms involvingthe spectator quark, hence its dependence on the DAs of the mesons. In what follows, weshall only need the first inverse moment of Φ B ( ζ ) which is parametrized as M B /λ B where λ B = O (Λ QCD ) [2].At zeroth order in α s , the leading power amplitude is given by A Leading ¯ B ◦ → ρ ◦ γ = G F √ (cid:88) p = u,c V ∗ pd V pb C (cid:104) Q (cid:105) (3)where (cid:104) Q (cid:105) is the matrix element of the operator Q , i.e. (cid:104) ρ ( P, e ) γ ( q, (cid:15) ) | Q | ¯ B ◦ (cid:105) = − em b ( µ ) F B → ρ √ π (cid:2) ε µναβ (cid:15) µ e νT P α q β + i (( (cid:15) · e T )( P · q ) − ( (cid:15) · P )( q · e T )) (cid:3) (4)where e = √ πα em and m b ( µ ) is the running mass of the b quark evaluated at the hard scale µ = m b .At order α s , the leading power amplitude becomes[2] A Leading ¯ B ◦ → ρ ◦ γ = G F √ (cid:88) p = u,c V ∗ pd V pb a p (cid:104) Q (cid:105) (5)with a p = C + α s ( µ ) C F π [ C ( µ ) G ( s p ) + C ( µ ) G ]+ α s ( µ h ) C F π [ C ( µ h ) H ( s p , µ h ) + C ( µ h ) H ( µ h )] . (6)In equation (6), the strong coupling is evaluated at two different scales: µ = m b and ahadronic scale µ h = (cid:112) Λ QCD µ . The functions G and H depend on s p = ( m p /m b ) where m p is the quark mass in loops contributing at next-to-leading order accuracy in α s . Thehard scattering functions G ( s p ) and G are given explicitly in reference [2]. Here, we focuson the functions H ( µ h ) and H ( s p , µ h ) which depend on the twist-2 DA of the ρ meson.The function H ( s p , µ ) is given by [2] H ( s p , µ ) = − (cid:32) π f B f ⊥ ρ ( µ )3 N c M B (cid:33) (cid:18) M B λ B (cid:19) I tw2 ( s p , µ ) (7)where f B is the decay constant of the B meson which is obtained from lattice QCD [16, 17]. I tw2 ( s p , µ ) is a convolution of the twist-2 DA with a hard scattering kernel, i.e. I tw2 ( s p , µ ) = (cid:90) d z h ( s p , ¯ z ) φ ⊥ ρ ( z, µ ) (8)where the hard scattering kernel is given by h ( s p , ¯ z ) = s p ¯ z L − (cid:113) ¯ z − s p + i(cid:15) ¯ z + L
21 + (cid:113) ¯ z − s p + i(cid:15) ¯ z − z (9)with L being the dilogarithmic function and ¯ z = 1 − z . The function H ( µ ) is given by H ( µ ) = (cid:32) π f B f ⊥ ρ ( µ )3 N c F B → ρ M B (cid:33) (cid:18) M B λ B (cid:19) I tw2 ( µ ) (10)where I tw2 ( µ ) is the first inverse moment of the twist-2 DA, i.e. I tw2 ( µ ) = (cid:90) d z φ ⊥ ρ ( z, µ ) z . (11)We note that if m b (cid:29) m p , then I tw2 ( s p , µ ) ≈ I tw2 (0 , µ ) = − I tw2 ( µ ) (12)so that both H and H become simply proportional to the first inverse moment of thetwist-2 DA of the ρ meson. In practice, this approximation is not justified for a charm loop,i.e. when p = c and we do not make it here. In what follows, we shall take m u,d = 0 .
14 GeV[10], m c = 1 . m b = 4 . B ◦ → ρ ◦ γ , all annihilation topologies are suppressed by at least one power of Λ QCD /m b [2]. The leading annihilation contribution is given by [2] A leadingannihilation ( ¯ B ◦ → ρ ◦ γ ) = G F √ V ∗ ud V ub (cid:18) C + 1 N c C (cid:19) b d (cid:104) Q (cid:105) (13)with b d = 2 πf B f ρ M ρ F B → ρ M B λ B . (14)This leading contribution can be taken into account by adding an extra term to the coeffe-cient a u in the leading power amplitude given by equation (5): a u → a u + b d (cid:18) C + 1 N c C (cid:19) (15)where b d is given by equation (14). The leading annihilation contribution corresponds to theannihilation diagram in which the photon is radiated off the spectator quark of the B meson,i.e. the third diagram in figure 1. Here we wish to investigate the numerical importanceof the three other subleading annihilation contributions shown in figure 1. In fact, the fourannihilation diagrams of figure 1 are the dominant contributions to the decay ¯ B ◦ s → ρ ◦ γ [12]. The total annihilation amplitude is given by[12] A annihilation d ( s ) = eG F √ V ∗ td ( s ) V tb f B ( s ) f ρ M ρ ( A d ( s ) + A d ( s ) + A d ( s ) + A d ( s ) ) (16)where to zeroth order in α s , A d ( s ) + A d ( s ) = 2 C [ I tw3 s ) ( µ ) − I tw3 s ) ( µ )] ε µναβ (cid:15) µ e νT P α q β (17)with I tw3 s ) ( µ ) = (cid:90) d z g ⊥ ( v ) ρ ( z, µ ) zM B ( s ) + z ¯ zM ρ − m f (18)and I tw3 s ) ( µ ) = (cid:90) d z zg ⊥ ( v ) ρ ( z, µ ) zM B ( s ) + z ¯ zM ρ − m f (19)while A d ( s ) = C (cid:32) E γ λ B ( s ) (cid:33) (cid:8) ε µναβ (cid:15) µ e νT P α q β + i [( (cid:15) · e T )( P · q ) − ( (cid:15) · P )( q · e T )] (20)+ i λ B ( s ) M B ( s ) (cid:34) (cid:15) · P )( q · e T ) − M B ( s ) (cid:32) m d ( s ) M B ( s ) (cid:33) ( (cid:15) · e T ) (cid:35)(cid:41) and A d ( s ) = C (cid:32) M B ( s ) E γ (cid:33) (cid:8) ε µναβ (cid:15) µ e νT P α q β − i [( (cid:15) · e T )( P · q ) − ( (cid:15) · P )( q · e T )] (21) − i (cid:34) (cid:15) · P )( q · e T ) − M B ( s ) (cid:32) m b M B ( s ) (cid:33) ( (cid:15) · e T ) (cid:35)(cid:41) In the above equations, C and C are the combinations of the Wilson coefficients[42][12]: C = 1 √ (cid:34) (cid:18) C + C (cid:19) V ub V ∗ ud ( s ) V tb V ∗ td ( s ) + (cid:18) C + C − C − C (cid:19)(cid:35) , (22) C = 1 √ (cid:34) (cid:18) C + C (cid:19) V ub V ∗ ud ( s ) V tb V ∗ td ( s ) (cid:35) (23)and E γ ( s ) is the energy of the photon in the B ( s ) meson rest frame, i.e. E γ ( s ) = M B ( s ) (cid:32) − M ρ M B ( s ) (cid:33) . (24) bb bbq ¯ qq ¯ q qq ¯ q ¯ q ¯ s, ¯ d ¯ s, ¯ d ¯ s, ¯ d ¯ s, ¯ d (2)(1)(3) (4) FIG. 1: Annihilation processes contributing to the decay ¯ B ◦ → ρ ◦ γ and ¯ B ◦ s → ρ ◦ γ . The diagrams(1),(2) and (4) are power-suppressed compared to the diagram (3). The contributions (1) and (2)are sensitive to the twist-3 DA of the ρ meson. The quantity λ B s is analogous to λ B , i.e. it parametrizes the first inverse moment of the¯ B ◦ s meson DA. As expected the amplitudes A , d ( s ) , corresponding to annihilation topologiesin which the photon is radiated off the light quark or antiquark of the ρ , are sensitive tothe twist-3 DA of the ρ meson. Note that the annihilation amplitude A d , evaluated toleading power in the heavy quark limit, coincides with equation (13) which is the leadingannihilation contribution given in reference [2].The total decay amplitude for ¯ B ◦ → ρ ◦ γ is then A ( ¯ B ◦ → ρ ◦ γ ) = A Leading ¯ B ◦ → ρ ◦ γ + A Annihilation d (25)where A Leading ¯ B ◦ → ρ ◦ γ and A Annihilation d are given by equation (5) and (16) respectively. On the otherhand, the total decay amplitude for ¯ B ◦ s → ρ ◦ γ is given by A ( ¯ B ◦ s → ρ ◦ γ ) = A Annihilation s (26)where A Annihilation s is given by equation (16).In order to compute the decay amplitudes given by equations (25) and (26), we mustspecify the twist-2 DA φ ⊥ ρ in equations (8) and (11) as well as the twist-3 DA g ⊥ ( v ) ρ appearingin equations (18) and (19). We also need to specify the numerical value of the tensor coupling f ⊥ ρ which appears in equations (7) and (10). We shall do this in the next section using aholographic AdS/QCD light-front wavefunction for the ρ meson. The numerical values ofthe decay constants f B ( s ) , the parameters λ ( s ) and the form factor F B → ρ are shown in tableI. Parameter Numerical value λ B ( s ) . ± .
12 (0 . ± .
20) GeV f B ( s ) . ±
47 (227 . ± .
0) MeV F B → ρ . ± . HOLOGRAPHIC ADS/QCD DAS AND COUPLINGS OF THE ρ MESON
Distribution Amplitudes parameterize the operator product expansion of vacuum-to-meson transition matrix elements of quark-antiquark non-local gauge invariant operatorsat light-like separations. At equal light-front time x + = 0 and in the light-front gauge A + = 0, we have [4, 5] (cid:104) | ¯ q (0) γ µ q ( x − ) | ρ ( P, λ ) (cid:105) = f ρ M ρ e λ · xP + x − P µ (cid:90) d u e − iuP + x − φ (cid:107) ρ ( u, µ )+ f ρ M ρ (cid:16) e µλ − P µ e λ · xP + x − (cid:17) (cid:90) d u e − iuP + x − g ⊥ ( v ) ρ ( u, µ ) , (27) (cid:104) | ¯ q (0)[ γ µ , γ ν ] q ( x − ) | ρ ( P, λ ) (cid:105) = 2 f ⊥ ρ ( e µλ P ν − e νλ P µ ) (cid:90) d u e − iuP + x − φ ⊥ ρ ( u, µ ) (28)and (cid:104) | ¯ q (0) γ µ γ q ( x − ) | ρ ( P, λ ) (cid:105) = − (cid:15) µνρσ e νλ P ρ x σ f ρ M ρ (cid:90) d u e − iuP + x − g ⊥ ( a ) ρ ( u, µ ) (29)for the vector, tensor and axial-vector current respectively. The polarization vectors e λ arechosen as e L = (cid:18) P + M ρ , − M ρ P + , ⊥ (cid:19) and e T ( ± ) = 1 √ , , , ± i ) (30)where P + is the “plus” component of the 4-momentum of the meson given by P µ = (cid:18) P + , M ρ P + , ⊥ (cid:19) . (31)All four DAs satisfy the normalization condition (cid:90) d z ϕ ( z, µ ) = 1 (32)where ϕ = { φ (cid:107) , ⊥ ρ , g ⊥ ( v,a ) ρ } so that for a vanishing light-front distance x − = 0, the definitionsof the vector coupling f ρ and tensor coupling f ⊥ ρ are recovered, i.e. (cid:104) | ¯ q (0) γ µ q (0) | ρ ( P, λ ) (cid:105) = f ρ M ρ e µλ (33)and (cid:104) | ¯ q (0)[ γ µ , γ ν ] q (0) | ρ ( P, λ ) (cid:105) = 2 f ⊥ ρ ( e µλ P ν − e νλ P µ ) . (34)The vector coupling f ρ is accessible experimentally via the leptonic decay width of the ρ meson [11] f ρ = (cid:18) e + e − M ρ πα (cid:19) / (35)where Γ e + e − = 7 . ± .
06 keV [15]. On the other hand, the tensor coupling f ⊥ ρ is notmeasured experimentally but is predicted theoretically by QCD Sum Rules and lattice QCD.It follows from equations (27), (28) and (29) that the twist-2 DAs are given by f ρ φ (cid:107) ρ ( z, µ ) = (cid:90) d x − e izP + x − (cid:104) | ¯ q (0) γ + q ( x − ) | ρ ( P, L ) (cid:105) (36)and f ⊥ ρ φ ⊥ ρ ( z, µ ) = 12 (cid:90) d x − e izP + x − (cid:104) | ¯ q (0)[ e ∗ T ( ± ) .γ, γ + ] q ( x − ) | ρ ( P, T ( ± )) (cid:105) (37)while the twist-3 DAs are given by f ρ g ⊥ ( v ) ρ ( z, µ ) = P + M ρ (cid:90) d x − e izP + x − (cid:104) | ¯ q (0) e ∗ T ( ± ) .γq ( x − ) | ρ ( P, T ( ± )) (cid:105) (38)and f ρ d g ⊥ ( a ) ρ d z ( z, µ ) = ∓ P + M ρ (cid:90) d x − e izP + x − (cid:104) | ¯ q (0) e ∗ T ( ± ) .γγ q ( x − ) | ρ ( P, T ( ± )) (cid:105) . (39)To relate the DAs to the light-front wavefunctions of the ρ meson, we use the relation[20] P + (cid:90) d x − e ix − zP + (cid:104) | ¯ q (0)Γ q ( x − ) | ρ ( P, λ ) (cid:105) = N c π (cid:88) h, ¯ h (cid:90) | k | <µ d k (2 π ) S ρ,λh, ¯ h ( z, k ) φ λ ( z, k ) × (cid:40) ¯ v ¯ h ((1 − z ) P + , − k ) (cid:112) (1 − z ) Γ u h ( zP + , k ) √ z (cid:41) (40)0where we have identified the renormalization scale µ as a cut-off on the transverse momentumof the quark [20] and φ λ ( z, k ) is the meson light-front wavefunction in momentum space.A two-dimensional Fourier transform of φ λ ( z, k ) gives the light-front wavefunction, φ λ ( z, r ),in configuration space. The light-front wavefunctions can be modelled [21–23] or extractedfrom data [20]. Here we use the AdS/QCD holographic wavefunction predicted in [24, 25]and which can be written as [26] φ λ ( z, ζ ) = N λ κ √ π (cid:112) z (1 − z ) exp (cid:18) − κ ζ (cid:19) exp (cid:18) − m f κ z (1 − z ) (cid:19) (41)where ζ = (cid:112) z (1 − z ) r is the transverse distance between the quark and antiquark at equallight-front time[43] and is the variable that maps onto the fifth dimension of AdS space[9, 27, 28]. The AdS/QCD wavefunction given by equation (41) is obtained using thesoft-wall model [29] to simulate confinement and in that case the parameter κ = M ρ / √ M ρ is the mass of the ρ meson. This AdS/QCD wavefunction has recently beenused within the dipole model to generate parameter-free[44] predictions for diffractive ρ meson electroproduction that are in agreement with the HERA data [10]. As discussed inreference [10], the normalization N λ of the AdS/QCD wavefunction is allowed to depend onthe polarisation of the meson λ = L, T .Going back to equation (40), the spinor wavefunctions S ρ,λh, ¯ h ( z, k ) are given by [20] S ρ,Lh, ¯ h ( z, k ) = (cid:20) M ρ + m f + k z (1 − z ) M ρ (cid:21) δ h, − ¯ h (42)and S ρ,T ( ± ) h, ¯ h ( z, k ) = √ z (1 − z ) { [(1 − z ) δ h ∓ , ¯ h ± − zδ h ± , ¯ h ∓ ] ke ± iθ k ∓ m f δ h ± , ¯ h ± } (43)while Γ stands for γ + , [ e ∗ T ( ± ) .γ, γ + ], e ∗ T ( ± ) .γ or e ∗ T ( ± ) .γγ . The matrix element in curlybrackets of equation (40) can then be evaluated explicitly for each case using the light-frontspinors of reference [30]: ¯ v ¯ h (cid:112) (1 − z ) γ + u h √ z = 2 P + δ h, − ¯ h , (44)¯ v ¯ h (cid:112) (1 − z ) [ e ∗ T ( ± ) .γ, γ + ] u h √ z = ∓ √ P + δ h ± , ¯ h ± , (45)¯ v ¯ h (cid:112) (1 − z ) e ∗ T ( ± ) .γ u h √ z = √ z (1 − z ) { [(1 − z ) δ h ∓ , ¯ h ± − zδ h ± , ¯ h ∓ ] ke ∓ iθ k ∓ m f δ h ± , ¯ h ± } (46)1and ¯ v ¯ h (cid:112) (1 − z ) e ∗ T ( ± ) .γγ u h √ z = √ z (1 − z ) {∓ [ zδ h ± , ¯ h ∓ + (1 − z ) δ h ∓ , ¯ h ± ] ke ∓ iθ k + (1 − z ) m f δ h ± , ¯ h ± } . (47)We are then able to deduce that φ (cid:107) ρ ( z, µ ) = N c πf ρ M ρ (cid:90) d r µJ ( µr )[ M ρ z (1 − z ) + m f − ∇ r ] φ L ( r, z ) z (1 − z ) , (48) φ ⊥ ρ ( z, µ ) = N c m f πf ⊥ ρ (cid:90) d r µJ ( µr ) φ T ( r, z ) z (1 − z ) , (49) g ⊥ ( v ) ρ ( z, µ ) = N c πf ρ M ρ (cid:90) d r µJ ( µr ) (cid:2) m f − ( z + (1 − z ) ) ∇ r (cid:3) φ T ( r, z ) z (1 − z ) (50)and d g ⊥ ( a ) ρ d z ( z, µ ) = √ N c πf ρ M ρ (cid:90) d r µJ ( µr )(1 − z )[ m f − ∇ r ] φ T ( r, z ) z (1 − z ) . (51)Equations (48) and (50) were derived in reference [20] where the light-front wavefunctions φ λ ( r, z ) were extracted from data. Equations (49) and (51) are new results which showhow the twist-2 and twist-3 DAs of the transversely polarised ρ meson are related to itslight-front wavefunction.We are also able to express the vector and tensor couplings f ρ and f ⊥ ρ in terms of thelight-front wavefunctions. From the definitions (33) and (34), it follows that (cid:104) | ¯ q (0) e ∗ L · γq (0) | ρ ( P, L (cid:105) = f ρ M ρ (52)and (cid:104) | ¯ q (0)[ e ∗ T ( ± ) · γ, γ + ] q (0) | ρ ( P, T ) (cid:105) = 2 f ⊥ ρ P + . (53)After expanding the left-hand-sides of equations (52) and (53), we obtain the decay widthconstraint [22] f ρ = N c M ρ π (cid:90) d z [ z (1 − z ) M ρ + m f − ∇ r ] φ L ( r, z ) z (1 − z ) (cid:12)(cid:12)(cid:12)(cid:12) r =0 (54)and f ⊥ ρ ( µ ) = m f N c π (cid:90) d z (cid:90) d r µJ ( µr ) φ T ( r, z ) z (1 − z ) (55)respectively. Note that equations (54) and (55) can also be obtained by inserting equations(48) and (49) into the normalization conditions on the twist-2 DAs, i.e. into (cid:90) d z φ (cid:107) ρ ( z, ∞ ) = 1 (56)2and (cid:90) d z φ ⊥ ρ ( z, µ ) = 1 (57)respectively. COMPARISON TO DAS AND COUPLINGS FROM SUM RULES
Inserting equation (41) in equations (54) and (55), we can compute the AdS/QCD pre-dictions for the vector and tensor couplings of the ρ meson. We compare our predictionsto experiment, Sum Rules and the lattice in table II. As can be seen, there is reasonableagreement between the AdS/QCD prediction for the vector coupling f ρ and experiment.We note that our prediction for f ⊥ ρ ( µ ) hardly depends on µ for µ ≥ µ ∼ .
14 GeV which is also the value used in [31–33].
Couplings of the ρ meson Reference Approach Scale µ f ρ [MeV] f ⊥ ρ ( µ ) [MeV] f ⊥ ρ ( µ ) /f ρ [15] Experiment 220 ± ∼ . ± ± . ± . . ± . . ± . ρ meson compared toSum Rules predictions, lattice predictions and experiment. We use m f = 0 .
14 GeV to make thesepredictions.
The twist-2 DAs can be expanded as [4, 5] φ || , ⊥ ρ ( z, µ ) = 6 z (1 − z ) (cid:34) (cid:88) j =2 , ,... a || j ( µ ) C / j ( ξ ) (cid:35) , (58)3where C / j ( ξ ) are the Gegenbauer polynomials and ξ = 2 z −
1. Standard Sum Rulespredictions are usually available only for a (cid:107) , ⊥ . The twist-2 DAs are thus approximated as φ (cid:107) , ⊥ ρ ( z, µ ) = 6 z (1 − z ) (cid:20) a (cid:107) , ⊥ ( µ ) 32 (5 ξ − (cid:21) (59)i.e. by keeping only the first term in equation (58). We use here the Sum Rules estimatesgiven in reference [36]: a (cid:107) = 0 . ± .
05 and a ⊥ = 0 . ± .
05 . Reference [36] also givesexplicit expressions for the twist-3 DAs: g ⊥ ( v ) ρ ( z, µ ) = 34 (1 + ξ ) + (cid:18) a (cid:107) ( µ ) + 5 ζ ( µ ) (cid:19) (cid:0) ξ − (cid:1) + (cid:20) a (cid:107) ( µ ) + 1532 ω (cid:107) ( µ ) − ω (cid:107) ( µ ) (cid:21) (cid:0) − ξ + 35 ξ (cid:1) . (60)and g ⊥ ( a ) ρ ( z, µ ) = 6 z (1 − z ) (cid:20) (cid:18) a (cid:107) ( µ ) + 109 ζ (cid:107) ( µ ) + 512 ω (cid:107) ( µ ) −
524 ˜ ω (cid:107) ( µ ) (cid:19) C / ( ξ ) (cid:21) . (61)The Sum Rules estimates are ζ (cid:107) (2 GeV) = 0 . ± . ω (cid:107) (2 GeV) = 0 . ± .
03 and˜ ω (cid:107) (2 GeV) = − . ± .
02 [36].In figure 2, we compare the AdS/QCD twist-2 DAs to the SR twist-2 DAs at a scale µ = 2 GeV. We note that, as is the case for the AdS/QCD tensor coupling, the AdS/QCDDAs hardly depend on µ for µ ≥ µ ∼ µ = 2GeV. The agreement between AdS/QCD and SR is quite good for both the axial vectorDA but we note a difference between SR and AdS/QCD vector DA at the end-points: theAdS/QCD, unlike the SR DA, falls to zero at the end-points.4 z f || r ( z , m ) (a) Twist-2 DA for the longitudinally polarized ρ meson z fr^ ( z , m ) (b) Twist-2 DA for the transversely polarized ρ meson FIG. 2: Twist-2 DAs for the ρ meson. Solid Red: AdS/QCD DA at µ ∼ µ = 2 GeV. BRANCHING RATIOS
We are now in a position to compute the branching ratios given by BR ( ¯ B ◦ ( s ) → ρ ◦ γ ) = τ B ( s ) πM B ( s ) − (cid:32) M ρ M B ( s ) (cid:33) |A ( ¯ B ◦ ( s ) → ρ ◦ γ ) | , (62)5 z g r^ ( a ) ( z , m ) (a) Axial-vector twist-3 DA for the transversely polarized ρ meson z g r^ ( v ) ( z , m ) (b) Vector twist-3 DA for the transversely polarized ρ meson FIG. 3: Twist-3 DAs for the ρ meson. Solid Red: AdS/QCD DA at µ ∼ µ = 2 GeV. where the amplitude A ( ¯ B ◦ ( s ) → ρ ◦ γ ) is given by either equation (25) for ¯ B ◦ → ρ ◦ γ orequation (26) for ¯ B ◦ s → ρ ◦ γ and τ B ( s ) is the measured lifetime of the B ( s ) meson [15] .Before presenting our predictions for the branching ratios, it is instructive to compare theAdS/QCD and SR predictions for the integrals I tw2 ( s p , µ ) and I tw2 ( µ ) given by equations (8)6and (11) respectively. Our results are shown in table III. We note that the integrals are notvery sensitive to the precise shape of the twist-2 DA. Integral SR AdS/QCD I tw2 ( s c , µ ) 1 .
902 + 2 . i .
590 + 2 . iI tw2 ( s u , µ ) − .
561 + 0 . i − .
866 + 0 . iI tw2 (0 , µ ) − . − . I tw2 .
330 4 . I tw2 and I tw2 given by equations (8)and (11) respectively. The SR predictions are at a scale µ = 2 GeV and the AdS/QCD predictionare at a scale µ ∼ We next compare the Sum Rules and the AdS/QCD predictions for the integrals I tw3 and I tw3 given by equations (18) and (19) respectively. Our results are shown in table IV. Inthis case, the AdS/QCD and SR predictions are drastically different. The SR DA yieldsdivergent integrals for both I tw3 and I tw3 unlike the AdS/QCD DA which leads to finiteresults in both cases. The divergent SR integrals could be estimated by introducing an IRcut-off but this procedure leads to a large uncertainty in the prediction for the annihilationamplitude [12]. Integral SR AdS/QCD I tw3 s ) ( µ ) ∞ . . I tw3 s ) ( µ ) ∞ . . I tw3 and I tw3 given by equations(18) and (19) respectively. The SR predictions are at a scale µ = 2 GeV and the AdS/QCDprediction are at a scale µ ∼ It is instructive to investigate the influence of perturbative QCD scale evolution on theinfrared divergence encountered with the SR DA. As shown in figure 4, we evolve the SR DAfrom µ = 1 GeV to µ = 2 , µ → ∞ . As can be seen, the SR7 z g r^ ( v ) ( z , m ) FIG. 4: Evolution of the twist-3 SR DA. Blue: SR DA at µ = 1 (Dot-dashed), 2 (Long-dashed), 3(Dot-dot-dashed) and 5 (Short-dashed) GeV; Dotted Black: Asymptotic DA; Solid Red: AdS/QCDDA at µ ∼ DAs do not vanish at the end-points and we find that the divergence problem persists atscales other than 2 GeV. Also shown in figure 4 is the AdS/QCD DA which, unlike the SRDA, vanishes at the end-points and avoids the end-point divergences. On the other hand,the AdS/QCD DA lacks the perturbative evolution with the scale µ and must be viewedto be a parametrization of the DA at some low scale µ ∼ µ = 2 GeV relevant to the decays wecompute here although we cannot make a strong case that it will still avoid the end-pointdivergences if its perturbative QCD evolution with the scale µ is taken into account.Our predictions for the branching ratio of ¯ B ◦ → ρ ◦ γ are shown in table V. In this table,we show how the predictions vary with the degree of accuracy of the calculation. The pre-dicted branching ratio computed using the leading power amplitude at zeroth order in α s (i.e. equation (3)) is clearly lower than the measured value. At this level of accuracy, theamplitude does not depend on the DAs. The leading power amplitude becomes sensitive tothe twist-2 DA at first order in α s and at this level of accuracy, we find that the AdS/QCDand SR predictions agree with each other and with experiment. We confirm that all fourpower-suppressed annihilation contributions in ¯ B ◦ → ρ ◦ γ are numerically small. Neverthe-8less, the AdS/QCD DA allows us to compute the annihilation contributions beyond leadingpower accuracy without the ambiguity due to end-point divergences encountered with theSR DA. At the same time, the AdS/QCD DA allows us to provide a more reliable theoreticalestimate for the branching ratio of the decay ¯ B ◦ s → ρ ◦ γ which proceeds mainly via annihi-lation and cannot be reliably predicted using the SR DA due to end-point divergences[12] .Using the AdS/QCD DAs, we predict a branching ratio of 5 . × − for this decay. Thisrare decay can be enhanced by NP [12] and it would be interesting to investigate it at theLHCb. Branching ratio ( × − ) for ¯ B ◦ → ρ ◦ γ DA Accuracy SR AdS/QCD PDG Belle BaBartw2 + tw3 Lead.( α s ) + Anni.[ α s , (1 /m b ) ] 7 .
67 8 . ± . . ± . . ± . . . ± . . ± . . tw2 Lead.( α s ) + Anni.[ α s , (1 /m b )] 7 .
86 7 . α s ) 7 .
87 7 . α s ) 4 .
76 4 . × − ) of ¯ B ◦ → ρ ◦ γ using AdS/QCD or Sum Rules compared to the measurements from Belle [13], BaBar [37] and theaverage value from PDG [15]. CONCLUSIONS
We have used new holographic AdS/QCD DAs for the transversely polarised ρ mesonin order to compute the branching ratios for the decays ¯ B ◦ → ρ ◦ γ and B s → ρ ◦ γ beyondleading power accuracy. The AdS/QCD prediction for the branching ratio of ¯ B ◦ → ρ ◦ γ agrees with experiment and we provide a theoretical estimate for the branching ratio ofthe rare decay ¯ B ◦ s → ρ ◦ γ . We find that the AdS/QCD DAs are complementary to thestandard SR DAs: they agree with the SR predictions to leading power accuracy and avoidthe end-point divergence ambiguity when computing some power corrections. However, inits present form, the AdS/QCD DA lacks the perturbative QCD evolution and it remainsto be seen if our conclusion remains valid if this evolution is taken into account.9 ACKNOWLEDGEMENTS
This research is supported by the Natural Sciences and Engineering Research Council ofCanada (NSERC). ∗ Electronic address: [email protected] † Electronic address: [email protected] [1] T. Hurth and M. Nakao, Ann.Rev.Nucl.Part.Sci. , 645 (2010), 1005.1224.[2] S. W. Bosch and G. Buchalla, Nucl.Phys. B621 , 459 (2002), hep-ph/0106081.[3] M. Beneke, T. Feldmann, and D. Seidel, Nucl.Phys.
B612 , 25 (2001), hep-ph/0106067.[4] P. Ball and V. M. Braun, Phys. Rev.
D54 , 2182 (1996), hep-ph/9602323.[5] P. Ball and V. M. Braun (1998), hep-ph/9808229.[6] P. Ball and V. M. Braun, Nucl. Phys.
B543 , 201 (1999), hep-ph/9810475.[7] M. Antonelli, D. M. Asner, D. A. Bauer, T. G. Becher, M. Beneke, et al., Phys.Rept. ,197 (2010), 0907.5386.[8] B. D. Pecjak (2008), 0806.4846.[9] G. F. de Teramond and S. J. Brodsky, Phys.Rev.Lett. , 081601 (2009), 0809.4899.[10] J. R. Forshaw and R. Sandapen, Phys.Rev.Lett. , 081601 (2012), 1203.6088.[11] P. Ball, G. W. Jones, and R. Zwicky, Phys.Rev.
D75 , 054004 (2007), hep-ph/0612081.[12] M. Ahmady and F. Mahmoudi, Mod.Phys.Lett.
A24 , 3173 (2009), 0706.1427.[13] N. Taniguchi et al. (Belle), Phys.Rev.Lett. , 111801 (2008), 0804.4770.[14] G. Buchalla, A. J. Buras, and M. E. Lautenbacher, Rev.Mod.Phys. , 1125 (1996), hep-ph/9512380.[15] J. Beringer et al. (Particle Data Group), Phys.Rev. D86 , 010001 (2012).[16] H. Na, C. J. Monahan, C. T. Davies, R. Horgan, G. P. Lepage, et al., Phys.Rev.
D86 , 034506(2012), 1202.4914.[17] A. Bazavov et al. (Fermilab Lattice and MILC), Phys.Rev.
D85 , 114506 (2012), 1112.3051.[18] C. McNeile, C. Davies, E. Follana, K. Hornbostel, and G. Lepage, Phys.Rev.
D85 , 031503(2012), 1110.4510.[19] J. Laiho, E. Lunghi, and R. S. Van de Water, Phys.Rev.
D81 , 034503 (2010), 0910.2928. [20] J. R. Forshaw and R. Sandapen, JHEP , 093 (2011), 1104.4753.[21] G. Kulzinger, H. G. Dosch, and H. J. Pirner, Eur. Phys. J. C7 , 73 (1999), hep-ph/9806352.[22] J. R. Forshaw, R. Sandapen, and G. Shaw, Phys. Rev. D69 , 094013 (2004), hep-ph/0312172.[23] J. Nemchik, N. N. Nikolaev, E. Predazzi, and B. G. Zakharov, Z. Phys.
C75 , 71 (1997),hep-ph/9605231.[24] S. J. Brodsky and G. F. de Teramond, Phys.Rev.
D77 , 056007 (2008), 0707.3859.[25] S. J. Brodsky and G. F. de Teramond (2008), 0802.0514.[26] A. Vega, I. Schmidt, T. Branz, T. Gutsche, and V. E. Lyubovitskij, Phys.Rev.
D80 , 055014(2009), 0906.1220.[27] S. J. Brodsky and G. de Teramond (2012), 1208.3020.[28] G. F. de Teramond and S. J. Brodsky, PoS
QNP2012 , 120 (2012), 1206.4365.[29] A. Karch, E. Katz, D. T. Son, and M. A. Stephanov, Phys.Rev.
D74 , 015005 (2006), hep-ph/0602229.[30] G. P. Lepage and S. J. Brodsky, Phys. Rev.
D22 , 2157 (1980).[31] G. Soyez, Phys. Lett.
B655 , 32 (2007), 0705.3672.[32] J. R. Forshaw, R. Sandapen, and G. Shaw, JHEP , 025 (2006), hep-ph/0608161.[33] J. R. Forshaw and G. Shaw, JHEP , 052 (2004), hep-ph/0411337.[34] D. Becirevic, V. Lubicz, F. Mescia, and C. Tarantino, JHEP , 007 (2003), hep-lat/0301020.[35] V. Braun, T. Burch, C. Gattringer, M. Gockeler, G. Lacagnina, et al., Phys.Rev. D68 , 054501(2003), hep-lat/0306006.[36] P. Ball, V. M. Braun, and A. Lenz, JHEP , 090 (2007), 0707.1201.[37] B. Aubert et al. (BABAR), Phys.Rev. D78 , 112001 (2008), 0808.1379.[38] C.-D. Lu, M. Matsumori, A. Sanda, and M.-Z. Yang, Phys.Rev.
D72 , 094005 (2005), hep-ph/0508300.[39] A. Ali and A. Parkhomenko, Eur.Phys.J.
C23 , 89 (2002), hep-ph/0105302.[40] Alternative frameworks can be found in reference [38, 39].[41] Note that C , (here)= C , (reference [11]).[42] For notational simplicity, we suppress the dependence of the Wilson coeffecients on the scale µ = m b .[43] The transverse separation between the quark and antiquark at equal ordinary time is r . [44] The quark mass m f is chosen as 0 .
14 GeV which is the value used in the dipole fits to thestructure function F2