Understanding the penguin amplitude in B φk decays

Transcription

1 Understanding the penguin amplitude in B φk decays S. Mishima Department of hysics, Nagoya University, Nagoya , Japan (July, 1) DNU-1-15 We calculate the branching ratios for pure penguin decay modes, the B φk decays using perturbative QCD approach. Our results of branching ratios are consistent with the experimental data and larger than those obtained from the naive factorization assumption and the QCD-improved factorization approach. This is due to the penguin enhancement in perturbative QCD approach. ACS index : 13.5.Hw, 11.1.Hi, 1.38.Bx I. INTRODUCTION Recently the branching ratios of the B φk decays have been measured by the BaBar [1], BELLE [] and CLEO [3] collaborations. These decay modes are important to understand penguin dynamics, because the B φk modes are pure penguin processes. There is a following problem related the penguin contribution for amplitude [4]. A naive estimate of the loop diagram leads to /T α s /(1π)log(m t /m c ) O(.1) where is a penguin amplitude and T is a tree amplitude. But experimental data for Br(B Kπ)andBr(B ππ) leadsto/t O(.1). The penguin contribution is enhanced dynamically. This enhancement can not be explained by the factorization assumption for nonleptonic two-body B meson decays [5]. This problem is explained by Keum, Li and Sanda using perturbative QCD (QCD) approach [6]. QCD method for inclusive decays was developed by many researcher over many years, and this formalism has been successful. Recently, QCD has been applied to the exclusive B meson decays, B Kπ [6], ππ [7], πρ, πω [8], KK [9] and Kη ( ) [1]. QCD approach is based on the three scale factorization theorem [11], [1] in which the matrix element can be written as the convolution of the hard part H with the hard scale t O(M B ), the meson wave function Φ with the soft scale 1/b Λ QCD and the Wilson coefficient C whose evolves from the W boson mass M W down to the scale t. b is the conjugate variable of the parton transverse momenta k T For instance, the B K transition form factor is written as F BK [dx][db]c(t)φ K (x,b )H(t)Φ B (x 1,b 1 ) [ exp i=1, t 1/b i d µ µ γ φ(α s ( µ)) ], (1) where x is the momentum fraction of parton and γ φ is the anomalous dimension of the mesons. The hard part H can be calculated perturbatively. It is important that the scale of the Wilson coefficient is determined dynamically, since the scale depends on x and b. Then in QCD, the scale of the Wilson coefficient can reach lower scale than M B or M B / which are usually taken in the factorization assumption case. The Wilson coefficient for the penguin operator is enhanced as the scale evolves to lower scale. Therefore the penguin amplitudes are larger than those in the naive factorization assumption. In this method, we can calculate not only factorizable amplitudes but also nonfactorizable and annihilation amplitudes which can not be calculated in the factorization assumption. We can calculate the hard part perturbatively, and we use the meson wave functions which are formed in the light-cone QCD sum rules. The hard part depends on processes, but the wave functions are independent of those. We can mishima@eken.phys.nagoya-u.ac.jp 1

2 determine the parameters in the wave functions, since there are many modes in the B meson two-body decays, So, we can predict quantitative values for the other decays. In this paper, we introduce the branching ratios for the B φk modes using QCD approach. The detail is discussed in Ref. [15]. The B φk modes depend only on the penguin amplitudes, and the branching ratio for this modes is proportional to. Thus the B φk modes are useful to seeing the size of the penguin amplitudes. We predict the branching ratios for the B φk decays, and our prediction agree with the current experimental data and are larger than the values obtained from the naive factorization assumption and the QCD-improved factorization [13], [14]. II. B φk AMLITUDES We consider that the B meson is at rest. In the light-cone coordinate, the B meson momentum 1, the K meson momentum and the φ meson momentum 3 are taken to be 1 = M B (1, 1, T ), = M B (1 r φ,, T ), 3 = M B (r φ, 1, T ), () where r φ = M φ /M B,andtheK meson mass is neglected. The momenta of the light quark in the B meson is written as k 1. We can integrate over k 1 + and choose k+ 1 =, since the hard part is independent of k+ 1. k 1 has the minus component k1 and small transverse components k 1T. k1 is given as k 1 = x 11,wherex 1 is the momentum fraction. The quarks in the K meson have plus components x + and (1 x ) +,and the small transverse components k T and k T, respectively. The quarks in the φ meson have the minus components x 3 3 and (1 x 3 )3, and the small transverse components k 3T and k 3T, respectively. The φ meson longitudinal polarization vector ɛ φ and two transverse polarization vector ɛ φt are given by ɛ φ =(1/ r φ )( rφ, 1, T )andɛ φt =(,, 1 T ). The B meson wave function for incoming state and the K and φ meson wave functions for outgoing state with up to twist-3 terms are then written as Φ (in) B,αβ,ij = iδ ij dx 1 d k 1T e i(x1 1 z+ 1 k1t z1t ) [( 1 + M B )γ 5 φ B (x 1, k 1T )] αβ, (3) Nc Φ (out) K,αβ,ij = iδ 1 ij dx e ix z γ 5 [ φ A K (x )+m K φ K (x )+m K ( v n 1)φ T K (x ) ], (4) Nc αβ Φ (out) φ,αβ,ij = δ 1 ij Nc dx 3 e ix33 z3 [ M φ ɛ φ φ φ (x 3 )+ ɛ φ 3 φ t φ(x 3 )+M φ φ s φ(x 3 ) ] αβ, (5) where i and j is the color indices and α and β are the Dirac indices. m K is related to the chiral symmetry breaking scale, m K = MK /(m d +m s ). v and n are defined as v µ = µ / + and nµ = z µ /z =(, 1, T ). We neglect the wave functions which are proportional to the transverse polarization vector ɛ T φ, because these terms do not appear in our calculations. It must be noted that the expression of Φ (out) φ depends on how to define the sign of ɛ φ. The explicit form of these wave functions will be shown in Sec. III. The decay rates of the B φk have the expressions Γ= G F 3πM B A. (6) The decay amplitudes A and Ā corresponding to B φk and B φ K respectively, are written as A = f φ V ts Vtb F e Ā = f φ Vts V tbfe + V tsvtb M e + f BV ts Vtb F a + V ts V tbm e + f BVts V tbfa + V tsvtb a, (7) + V ts tbm a. (8) F e is the amplitude for the factorizable diagrams which is considered in the factorization assumption. F a and M are the annihilation factorizable and the nonfactorizable diagrams which are neglected in the factorization assumption. The indices e and a denote the tree topology and annihilation topology respectively. The index denotes the contribution from the diagram with a penguin operator. The decay amplitudes A + and A corresponding to B + φk + and B φk respectively, are written as

3 A + = f φ V ts VtbF e A = f φ VtsV tb Fe + V ts VtbM e + f B V ts VtbF a + VtsV tb M e + f B VtsV tb Fa + V ts VtbM a f B V us VubF a T V us VubM T a, (9) + VtsV tb M a f B VusV ub Fa T VusV ub M T a, (1) where the index T denotes tree contributions. Since the CKM matrix elements for the tree amplitudes are much smaller than for the penguin amplitudes, then the tree contributions are very small. FIG. 1. Feynman diagrams contributing to factorizable amplitudes for B φk The factorizable diagrams are given as Fig. 1. The factorizable penguin amplitude Fe from Fig. 1(a) and Fig. 1(b) is written as which comes F e 1 =8πC F MB 4 dx 1 dx b 1 db 1 b db φ B (x 1,b 1 ) { [(1 + x )φ A K(x )+r K (1 x ) ( φ K(x )+φ T K(x ) )] E e (t (1) e )N t {x (1 x )} c h e (x 1,x,b 1,b ) } +r K φ K (x )E e (t () e )N t{x 1 (1 x 1 )} c h e (x,x 1,b,b 1 ), (11) where N t {x(1 x)} c is the factor for the threshold resummation [16]. We use N t =1.775 and c =.3 [17]. The evolution factors are defined by E e (t) =α s (t)a e (t)exp[ S B (t) S K (t)] where exp[ S i (t)] is the factor for the k T resummation [18], [19]. The explicit forms of the factor S i (t) are given, for example, in Ref. [6]. t (1) e =max( x M B, 1/b 1, 1/b )andt () e The hard scales t, which are the typical scales in hard process, are given by =max( x 1 M B, 1/b 1, 1/b ). The Wilson coefficient is given by ( C 7 + C 8 + C 9 + C 1 + C 1 + C ) 9. (1) N c N c N c a e (t) =C 3 + C 4 N c + C 4 + C 3 N c + C 5 + C 6 N c 1 The hard functions, which are the Fourier transformation of the virtual quark propagator and the hard gluon propagator, are given by h e (x 1,x,b 1,b )=K ( x 1 x M B b 1 )[θ(b 1 b )K ( x M B b 1 ) I ( x M B b )+(b 1 b )]. (13) The factorizable annihilation diagrams as Fig. 1(c) and Fig. 1(d), and the nonfactorizable diagrams as Fig. (a)-(d) can be also calculated in the same way as F e [15]. FIG.. Feynman diagrams contributing to nonfactorizable amplitudes for B φk 3

4 III. NUMERICAL RESULTS We use the model of the B meson wave function written as [ φ B (x, b) =N B x (1 x) exp 1 ( ) ] xmb ω B b ω B, (14) where ω B =.4 GeV []. N B is determined by normalization condition given by 1 dxφ B (x, b =)= f B. (15) N c The K meson wave functions are given as φ A K (x) = f [ K 6x(1 x) 1+3a 1 (1 x)+ 3 ] N c a (5(1 x) 1), (16) φ K(x) = f [ K 1+ 1 ( 3η 3 5 ) {3(1 N c ρ K x) 1 } 1 ( 3η 3 ω ρ K + 81 ) {3 1 ρ K a 3(1 x) + 35(1 x) 4} ], (17) φ T K (x) = f [ ( K (1 x) 1+6 5η 3 1 N c η 3ω 3 7 ρ K 3 ) ] 5 ρ K a (1 1x +1x ), (18) where ρ K = (M d + m s )/M K [1], []. The parameters of these wave functions are given as a 1 =.17, a =., η 3 =.15, ω 3 = 3. where the renormalization scale is 1 GeV. We fix the parameters as above values. The φ meson wave functions are given as f φ φ φ (x) = 6x(1 x), (19) N c φ t φ(x) = f φ T [ 3(1 x) + 35 N c 4 ζt 3 {3 3(1 x) + 35(1 x) 4 } + 3 { δ + 1 (1 x)log 1 x }], () x φ s φ (x) = f φ T [ 4 (1 x) ( ] 6+9δ ζ3 T N 14ζT 3 x + 14ζT 3 x) x +3δ + log, (1) c 1 x where ζ3 T =.4, δ + =.46 [3]. We fix the parameters as above values, since the numerical results is not sensitive to these parameters. We use the Wolfenstein parameters for the CKM matrix elements A =.819, λ =.196, R b ρ + η =.38 [4], and choose the angle φ 3 = π/ [6]. For the values of the meson masses, we use M B =5.8 GeV, M K =.49 GeV and M φ =1. GeV. In addition, for the values of the meson decay constants, we use f B = 19 MeV, f K = 16 MeV, f φ = 37 MeV and fφ T = 15 MeV. The B meson life times are given as τ B = sec and τ B ± = sec. And we use Λ (4) QCD =.5 GeV and m K =1.7 GeV [6]. We show the numerical results of each amplitude for the B φk and B ± φk ± decays in Table I. The factorizable penguin amplitude Fe gives dominant contribution to the B φk decays. The factorizable annihilation penguin amplitude Fa p generates large strong phase. In the B ± φk ± modes, there are the contribution from f B Fa T and M a T. These tree amplitudes contribute a few percent to the whole amplitude, since the CKM matrix elements related to the tree amplitudes are very small. We predict that the branching ratios for the B φk decays are Br(B φk )= f B f K 19 MeV 16 MeV ( ), () Br(B ± φk ± )= f B f K 19 MeV 16 MeV ( ). (3) 4

5 The current experimental values are summarized in Table II. The values which are predicted in QCD are consistent with the current experimental data. However, our branching ratios have the theoretical error from the O(α s) corrections, the higher twist corrections, and the error of input parameters. Large uncertainties come from the meson decay constants, the shape parameter ω B,andm K. These parameters are fixed from the other modes(b Kπ, Dπ, etc.). We try to vary ω B from.36 GeV to.44 GeV, then we obtain Br(B ± φk ± )=( ) 1 6. Next, we set ω B =.4 and try to vary m K from 1.4 GeV to 1.8 GeV, then we obtain Br(B ± φk ± )=( ) 1 6. Now, we consider the ratio of the branching ratio for the B φk decay to the one for the B + φk + decay. The theoretical uncertainties from various parameters are small, since the parameters in a numerator cancel out those in a denominator. The difference between the two branching ratios comes from the B meson life times, the tree and electroweak penguin contributions in the annihilation amplitudes. The tree and electroweak penguin amplitudes in the annihilation diagrams are much smaller than the whole amplitudes. The tree amplitudes are suppressed by two factors. The one is that those are annihilation processes which are suppressed by helicity and the other is that the tree amplitudes are multiplied by the small CKM matrix element. We predict that this ratio is Br(B φk ) Br(B + φk + =.95, (4) ) where the theoretical uncertainties from m K and ω B are less than 1%. The experimental value of this ratio from the BELLE [] is Br(B φk )/Br(B + φk + )= ±.1. In the naive factorization assumption, the branching ratio is very sensitive to the effective number of colors Nc eff. If we set Nc eff =3, then the branching ratio is about where the scale of the Wilson coefficient is taken to M B /andf BK is.38 from the BSW model. Our predicted values are larger than those obtains from the naive factorization assumption. This is due to the enhancement of the Wilson coefficient for penguin. In QCD approach, the scale of the Wilson coefficients, which is equal to the hard scale t, can reach lower values than M B / in case of the naive factorization assumption, and the Wilson coefficients are then enhanced [6]. Therefore, our branching ratios are enhanced over those obtained from the naive factorization assumption. In the QCD-improved factorization, the branching ratios for the B φk decays are predicted as Br(B φk )=( ) 1 6 and Br(B φk )=( ) 1 6 with the annihilation effect [14]. Our predicted values are larger than these values. IV. SUMMARY In this paper, we calculate the B φk and B ± φk ± decays in QCD approach. Our predicted branching ratios agree with the current experimental data and are larger than the values obtained by the naive factorization assumption and the QCD-improved factorization. Because the Wilson coefficients for penguin operators are enhanced dynamically in QCD. Note added: After this work has been completed, we become aware of a similar calculation by Chen, Keum and Li [5]. Our results are in agreement. Acknowledgment The topic of this research was suggested by rofessor A.I. Sanda. The author thanks his QCD group members: Y.Y. Keum, E. Kou, T. Kurimoto, H-n. Li, T. Morozumi, R. Sinha, K. Ukai for useful discussions. The author thanks JSS for partial support. The work was supported in part by Grant-in Aid for Special roject Research (hysics of C violation); Grant-in Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan. 5

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