aa r X i v : . [ h e p - ph ] F e b Three Parton Corrections in B → P P decays
Tsung-Wen Yeh ∗ Department of Science Application And Dissemination,National Taichung University, Taichung 403, Taiwan
Abstract
The 1 /m b corrections from the three parton q ¯ qg Fock state of the final state light meson in B → P P decays are evaluated by means of a collinear expansion method. The impacts of thesecorrections on the CP averaged branching ratios of the B → πK decays are analyzed. PACS numbers: 13.25.HwKeywords: power corrections, QCD factorization, B decays ∗ Electronic address: [email protected] . INTRODUCTION The QCD factorization [1, 2, 3] has been widely used to investigate the charmless hadronic B decays. For an operator O i of the weak effective Hamiltonian, the matrix element for¯ B → M M decays under the QCD factorization is found to be expressible as h M M | O i | ¯ B i = X j F B → M j ( m ) Z duT Iij ( u )Φ M ( u ) + ( M ↔ M )+ Z dξdudvT IIi ( ξ, u, v )Φ B ( ξ )Φ M ( u )Φ M ( v ) , (1)where T I ( II ) denote the parton amplitudes and Φ B , Φ M , Φ M represent the light-cone dis-tribution amplitudes (LCDAs) for the initial state ¯ B meson and the final state M and M mesons, respectively. The parton amplitudes T I ( II ) contain short distance interactions in-volved in the decay processes. The LCDAs Φ B , Φ M , and Φ M are introduced to account forthe long distance interactions. The F B → M j ( m ) with j = + , B → M transitionform factors. The meson state vector | M i i , i = 1 ,
2, for the meson M i is composed of Fockstates with different number of partons | M i i = | q ¯ q i M i + | q ¯ qg i M i + · · · . (2)So far, most applications of the factorization formula Eq. (1) are limited to leading Fockstate | q ¯ q i M i of the light mesons. However, the three parton Fock state | q ¯ qg i M i of the M i meson can also contribute.The corrections related to the higher Fock state are usually classified as subleading twistcontributions, since their contributions are suppressed by factors of O (1 /m nb ) with n ≥ m b denotes the b quark mass. Within QCD fac-torization, one can employ the Feynman-diagram approach or the effective-theory approachfor studies of subleading twist contributions. There exist established Feynman-diagram ap-proaches for processes other than hadronic decays, such as the calculation scheme for theinclusive hard scattering processes [5, 6, 7] or the method for the exclusive hard scatter-ing processes [29, 30, 31, 32, 33]. However, a systematic Feynman-diagram approach forcharmless hadronic B decays is still inaccessible. On the other hand, the effective-theoryapproaches for charmless hadronic B decays have been extensively investigated in recentyears [34, 35, 36, 37, 38, 39]. 2n this paper, a calculation scheme based on the Feynman-diagram approach for charm-less hadronic B decays will be developed. We will concentrate on the construction of thiscalculation scheme and apply the constructed method to calculate the tree level three par-ton corrections. The O ( α s ) three parton corrections is also desirable to understand theirfactorization properties. Since the related analysis is tedious, we plan to present the rele-vant calculations in our another preparing work [44]. The organization of this paper is asfollowing. In Section II, the calculation scheme will be constructed. The factorization of thetree level three parton corrections into the partonic and hadronic parts will be outlined. InSection III, the analysis on how the tree level three parton corrections can be factorized intoits partonic and hadronic parts will be described in detail. In Section IV, we will apply theresults of the Section III to make predictions for the branching ratios of B → πK decays.The last section devotes for discussion and conclusion. II. COLLINEAR EXPANSION AT TREE LEVEL
In this section, we will generalize the collinear expansion method [5, 6, 7] to calculate thethree parton corrections from the Fock state | q ¯ qg i of the meson M in the decay ¯ B → M M .There exist other types of power corrections, such as the power corrections from soft gluonsor renormalons. We identify these as non-partonic power corrections. For these powercorrections, our proposed scheme may not be useful. However, to include these non-partonicpower corrections requires further assumptions beyond the factorization. For example, thesoft gluon power corrections are better determined by nonperturbative theories, such as theQCD sum rules or lattice QCD. In this work, we only investigate how the partonic (or thedynamic) power corrections can be included into the QCD factorization in a consistent way.The collinear expansion method arises from a motivation of generalizing the leading twistfactorization theorem for the hard scattering processes to include the corrections from highFock states of the target hadrons. The original idea of the collinear expansion method wasproposed by Polizer at 1980 [4]. The systematical method was developed by Ellis, Furman-ski and Petronzio (EFP) in their pioneer works [5, 6]. Using the collinear expansion, theEFP group showed that, for the DIS processes, the twist-4 power suppressed correctionscan be factorized into short distance and long distance parts, which are in a similar factor-ized form as the leading twist contributions. However, in the EFP’s approach, the parton3nterpretation for the twist-4 corrections are lost. To recover the parton model picture,Qiu then introduced a Feynman-diagram approach [7] to re-formula the EFP’s method. Inthis Feynman-diagram language, a parton model interpretation for the twist-4 correctionsbecomes trivial.To begin with, we first express the matrix element of an operator O i of the weak effectiveHamiltonian H eff of the standard model for the hadronic decays ¯ B → M M in terms ofparton model amplitudes h M M ) | O i | ¯ B i = X j =+ , F B → M j ( m M ) Z d l (2 π ) Tr[ T Iij ( l )Φ M ( l )] + ( M ↔ M )+ X j =+ , F B → M j ( m M ) Z d l (2 π ) Z d l (2 π ) Tr[ T Iij,µ ( l , l )Φ µM ( l , l )] + ( M ↔ M )+ Z d l B (2 π ) d l M (2 π ) d l M (2 π ) Tr[ T IIi ( l N , l M , l M )Φ B ( l B )Φ M ( l M )Φ M ( l M )] , (3)where F B → M j denote the form factors for ¯ B → M l ¯ ν transition. The form factor F j aredefined as h M ( p ) | ¯ qγ µ b | ¯ B ( P b ) i = F + ( q )( p + P b ) µ + F + ( q ) − F ( q ) q q µ (4)where q = P b − p and F + ( q ) = F ( q ) under the limit q →
0. The parton amplitudes T Iij ( l ), T Iij,µ ( l , l ) and T IIi ( l B , l M , l M ) are defined to describe the hard scattering centerinvolving four parton, five parton, and six parton interactions corresponding to those dia-grams depicted in Fig. 1(a)-(c), respectively. Note that there are also other types of partonamplitudes involving five or six parton interactions not being presented in Fig. 1, which canbe attributed to either the physical form factors, or to be of higher twist than three. Wehave neglected these contributions in Eq. (3). The Tr symbol denotes the trace operationapplied on the color and spin indices. For convenience, we employ the light-cone coordinatesystem such that P µB = ( p µ + q µ ) with two light-like vectors q µ = ( q + , q − , q i ⊥ ) = ( Q, ,
0) and p µ = ( p + , p − , p i ⊥ ) = (0 , Q,
0) with Q = m B / √
2, which are defined as the momenta carriedby the final state M and M mesons, respectively. The M meson is defined to receivethe spectator quark of the bottom meson. The M meson is defined as the emitted mesonproduced from the hard scattering center. The hadron amplitudes Φ M and Φ µM ( l , l ) are4efined as Φ M ( l ) = Z d ye il · y h M | ¯ q ( y ) q (0) | i , (5)Φ µM ( l , l ) = Z d y Z d ze il · y e i ( l − l ) · z h M | ¯ q ( y )( − gA µ ( z )) q (0) | i . (6)In our language, Eq. (3) contain leading, sub-leading and higher twist contributions. TheEq. (3) becomes meaningful only if the leading twist contributions can be separated from thesub-leading twist contributions. For this purpose, we employ the Qiu’s Feynman-diagramcollinear expansion approach [7] to expand each Feynman diagram in a twist by twist man-ner. As the loop corrections are considered, the twist expansion then interplays with theexpansion in α s . To be specific, we choose the following expansion strategy.1. All possible Feynman diagrams ordered in α s are first drawn.2. According to the collinear expansion (developed below), each Feynman diagram oforder O ( α ns ) with n ≥ α s order are added up together.4. The factorization properties of the final expression with a specific twist and a specific α s order are analyzed.The last one is important for us to derive a meaningful perturbation theory beyond theleading twist.For latter uses, we define the soft, collinear and hard loop parton momenta. We let thesoft momentum scale as ( l + , l − , l ⊥ ) ∼ ( λ, λ, λ ), the collinear momentum scale as ( l + , l − , l ⊥ ) ∼ ( Q, λ /Q, λ ), and the hard momentum scale as ( l + , l − , l ⊥ ) ∼ ( Q, Q, Q ). The scale variablesare defined as Q ∼ m b and λ ∼ Λ QCD . For a collinear loop parton, it is convenient toparametrize its momentum l µ into its components proportional to the meson momentum q µ ,the light-cone vector n ν , and the transversal directions l µ = n · lq µ + l + l ⊥ n · l n µ + l µ ⊥ , (7)where the vector n µ satisfies n · q = 1, n · l ⊥ = 0, and n = 0. For convenience, we furtherdefine the collinear component, ˆ l µ , the on-shell component, l µL , and the off-shell component,5 µS of the momentum l µ as ˆ l µ = n · lq µ ,l µL = ˆ l µ + l ⊥ n · l n µ + l µ ⊥ ,l µS = l n · l n µ . (8)In the above expansion of the parton momentum into different parts, we have assumed m M i = 0, i = 1 ,
2, and q = 0. The loop partons except of the bottom quark are assumedmassless for simplicity. The contributions from non-vanishing light quark masses are taken ascorrections. Because the light quark mass contributions are relatively negligible as comparedto the bottom quark mass, we also neglect the light quark mass effects in the followingcalculations.According to the parametrization in Eq. (7), a parton propagator can be separated intoits long distance part and short distance part (the special propagator). If we write the loopparton propagator as [7] F ( y, z ) = Z d l (2 π ) e il · ( y − z ) [ F L ( l ) + F S ( l )]= F L ( y, z ) + F S ( y, z ) , (9)where F L ( l ) = i/l L l , F S ( l ) = i/n n · l . (10)The F L ( l ) propagator corresponds to the long distance part of the propagator, since F L ( y, z ) ∝ θ ( y − z ). The F S ( l ) propagator represents the short distance part because F S ( y, z ) ∝ δ ( y − z ). We now describe one important property of the long distance propaga-tor F L ( l ). In a parton amplitude, the F L ( l ) may contact with a /qn µ component of a vertex γ µ . Whenever this happens, the /qn µ vertex will extract one short distance propagator F S ( l )and one interaction vertex iγ ν from the relevant hadron amplitude [7] i/l L l /q = i/l L l ( iγ ν ) i/n n · l /q ( l − ˆ l ) ν . (11)The momentum factor ( l − ˆ l ) ν is then absorbed by the hadron amplitude due to the Wardidentity [7]. We now explain how the identity Eq. (11) can be obtained by a simple manipu-lation. We first insert an identity 1 = ( /l ) /l into the left hand side of Eq. (11) and expresse6ach /l into /l L + /l S to obtain i/l L l /l/ll /q = i/l L l ( /l L + /l S )( /l L + /l S ) l /q . (12)Since ( /l L ) = 0 = ( /l S ) , the above equation then becomes i/l L l /l/ll /q = i/l L l ( /l L /l S + /l S /l L ) l /q , (13)where the first term /l L /l S in the right hand side leads to a vanishing result as it contacts withthe term i/l L /l term. The only contribution can only come from the second term /l S /l L in theright hand side of Eq. (13). In addition, the /l L can be expanded in the terms proportionalto /q , /n , /l ⊥ . This gives /l S /l L /q = l /n n · l ( n · l/q + l ⊥ /n n · l + /l ⊥ ) /q . Due to /q = /n = 0, it further reduces to l /n n · l ( /l ⊥ ) /q . By substituting the above back into Eq. (12), Eq. (11) is then obtained by noting that /l L ( iγ α )( i/n ) /q ( l − ˆ l ) α = /l L /n/l ⊥ /q . Using Eq. (11), one can systematically include the effects from the non-collinearity and theoff-shellness of the collinear partons. This property of the long distance part of the partonpropagator plays an important role in our following analysis, and its meaning will becomemore clear after we have investigated real cases lately.According to the parton model, the hadron amplitudes are defined as the probability forfinding the on-shell partons inside the hadron. The parton amplitudes are then required tocontain only the on-shell components of the external parton momenta. However, according toEq. (8), either the on-shell momentum l L or the collinear momentum ˆ l can be assigned for anon-shell parton. Therefore, there arise two factorization schemes, the collinear factorization[8, 9, 10, 11, 12, 13](QCD factorization) and the k T factorization [14, 15, 16, 17] (PQCDfactorization [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]). In the k T factorization scheme, an on-shell parton carries a momentum l L . On the other hand, in the collinear factorization scheme,an on-shell parton carries a momentum ˆ l . In this work, we follow the QCD factorizationto use the collinear factorization scheme as our basics. Our proposed collinear expansionmethod is composed of following steps: 7. Use scale analysis for the parton amplitudes according to the scales of parton momentato find out the leading regions of the parton momentum configuration.2. The parton amplitudes are expanded into a Taylor series with respect to the leadingregions of parton momenta.3. The expanded parton amplitudes are substituted back into the contraction with thehadron amplitudes to extract relevant contributions up to specific twist order.4. The factorization of parton momentum integrals is accomplished by means of integraltransformations (See, for example, Eq. (24)).5. The color structure of the parton amplitude is extracted to be attributed to the hadronamplitudes to complete the color factorization.6. The factorization of spin indices is completed by means of Fierz transformation.7. The property of the long distance parton propagator is used to extract higher twistcontributions.We are now ready to discuss the collinear expansion. First, we order the parton ampli-tudes in α s T Iij ( l ) = T I (0) ij + T I (1) ij ( l ) + O ( α s ) , (14) T Iij,µ ( l , l ) = T I (0) ij,µ ( l , l ) + T I (1) ij,µ ( l , l ) + O ( α s ) , (15) T IIi ( l B , l M , l M ) = T II (1) i ( l B , l M , l M ) + O ( α s ) , (16)where the superscription (0) and (1) are used to denote the relevant parton amplitude ofzeroth and first order in α s , respectively. The parton amplitude T I (0) ij is just the tree vertex¯Γ i δ ij in the diagram as depicted in Fig. 2. There are vertex and penguin diagrams for T I (1) ij amplitudes as depicted in Fig. 4. The parton amplitude T I (0) ij,µ describes the tree diagramsas depicted in Fig. 3, in which two quark partons and one gluon parton from the meson M are interacting with a local four fermion operator O i = (¯ q Γ i b )(¯ q ¯Γ i q ) with Γ i (¯Γ i ) the Diracgamma matrix for the operator. The parton amplitude T IIi starts from O ( α s ) diagrams asdepicted in Fig. 5 and is denoted as T II (1) . 8e first expand the parton amplitudes T I (0) ij,µ with respect to the collinear components oftheir relevant parton momenta as T I (0) ij,µ ( l , l ) = T I (0) ij,µ (ˆ l , ˆ l ) + X k =1 ∂T I (0) ij,µ ∂l νk (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l k =ˆ l k ( l k − ˆ l k ) ν + · · · , (17)where T I (0) ij,µ (ˆ l , ˆ l ) = (( iγ µ ) i/ ˆ l ˆ l ¯Γ i + ¯Γ i − i/ ¯ˆ l ¯ˆ l ( − iγ µ )) δ ij , (18) ∂T I (0) ij,µ ∂l νk (ˆ l , ˆ l ) = (( iγ µ ) i/ ˆ l ˆ l ( iγ ν ) i/ ˆ l ˆ l ¯Γ i δ k − ¯Γ i − i/ ¯ˆ l ¯ˆ l ( − iγ ν ) − i/ ¯ˆ l ¯ˆ l ( − iγ µ ) δ k ) δ ij . (19)The expansion series are then substituted back into the convolution integrals with the hadronamplitudes for further analysis. The reason why one can expand the parton amplitudes withrespect to the relevant collinear momenta will be explained in detail below. A. Expansion with T I (0) ij The expression for the contributions associated with T I (0) ij is written as X j =+ , F B → M j ( m M ) Z d l (2 π ) Tr[ T I (0) ij Φ M ( l )] , (20)where the loop momentum l is carried by the loop parton in Fig. 1(a). Since the partonamplitude T I (0) ij is independent of l , we propose to use the following integral identity totransform the expression into a form consistent with the parton model picture Z dxδ ( x − n · l )= Z dx Z ∞ dλ π e iλ ( x − n · l ) = 1 . (21)The transformed result appears as Z dx Tr[ T I (0) ij Φ M ( x )] (22)where Φ M ( x ) = Z ∞ dλ π e iλx h M | ¯ q ( λn ) q (0) | i . (23)9he following integral transformation has been used in the above to simplify the expression Z d l (2 π ) Z d ye il · ( y − λn ) G ( y, Z d yδ (4) ( y − λn ) G ( y, G ( λn, , (24)where G ( y,
0) denotes any function of the coordinates. Two comments for the above integraltransformations Eqs. (21) and (24) are necessary. First, the momentum fraction x for theparton of the meson M is introduced. Second, the quark field ¯ q ( λn ) is ordered in light-conedirection n . This implies that the hadron amplitude Φ M ( x ) is defined on the light-cone n = 0 where n µ is a null light-cone vector. By using the above integral transformations, theparton amplitude and the hadron amplitude are only related by the momentum fraction x .Because the parton amplitude T I (0) ij is equal to ¯Γ i δ ij , the integral over x is then associatedwith Φ M ( x ).The factorization of spin indices depends on the structure of ¯Γ i . For ( V − A )( V ± A )operators, ¯Γ i = ( V ± A ) and can be expanded into /q (1 ± γ ), /n (1 ± γ ) and γ ⊥ γ . If M isa pseudo-scalar meson, only the axial vector part can contribute. However, only /nγ leadsto leading twist contributions. The /qγ will result in twist-4 contributions and γ ⊥ γ willnot contribute. For other types of meson, similar considerations can be made. We nowexplain how the /qγ part can contribute. The long distance part of the parton propagatorcan interact with vertex /qγ to have i/l L l /q = i/l L l ( iγ α ) i/n n · l /q ( l − ˆ l ) α . (25)It is also applicable for the other parton propagator of the anti-quark line. The short distancepart of the parton propagator i/n/ (2 n · l ) and the vertex iγ α are absorbed into the partonamplitude. This results in Z dx Tr[ T ij,αβ ( x, x, x ) w αα ′ w ββ ′ Φ α ′ β ′ M ,∂ ( x, x, x )] (26)where w αα ′ = g αα ′ − q α n α ′ , T ij,αβ ( x, x, x ) ≡ ( iγ α ) i/n n · l T I (0) ij − i/n n · ¯ l ( − iγ β ) , Φ α ′ β ′ M ,∂ ( x, x, x ) = Z ∞ dλ π e iλx h M | ¯ q ( λn ) i∂ α ′ ( λn ) i∂ β ′ ( λn ) q (0) | i . (27)10here are corresponding contributions from the two gluon insertion diagrams depicted inFig. 6, whose expression is written as Z d l (2 π ) d l (2 π ) d l (2 π ) Tr[ T I (0) ij,αβ ( l , l , l ) w αα ′ w ββ ′ Φ αβM ,A ( l , l , l )] (28)where we have employed the light-cone gauge n · A = 0 for the gluon fields, and the partonamplitude and hadron amplitude are expressed as T I (0) ij,αβ ( l , l , l ) = ( iγ α ) i/n n · l T I (0) − i/n n · ¯ l ( − iγ β )Φ αβM ,A ( l , l , l ) = Z d z Z d y Z d we il · y e i ( l − l ) · z e i ( l − l ) · w ×h M | ¯ q ( y )( − gA α ( z ))( − gA β ( w )) q (0) | i (29)Since T I (0) ij,αβ ( l , l , l ) can be replaced by T I (0) ij,αβ ( x , x , x ) straightforwardly, the momentumintegrations over l , l , l can be transformed into the integrations over x , x , x . We thenobtain Z dx dx dx Tr[ T I (0) ij,αβ ( x , x , x )Φ αβM ,A ( x , x , x )] . (30)The combination of Eq. (26) and Eq. (30) gives Z dx dx dx Tr[ T I (0) ij,αβ ( x , x , x ) w αα ′ w ββ ′ Φ α ′ β ′ M ,D ( x , x , x )] (31)where Φ αβM ,D ( x , x , x ) = Z dλ π Z dη π Z dω π e iλx e iη ( x − x ) e iω ( x − x ) ×h M | ¯ q ( λn )( iD α ( ηn ))( iD β ( ωn )) q (0) | i (32)with iD α = i∂ α − gA α being the covariant derivative. Since /n is of O ( Q − ), T I (0) ij,αβ is of O ( Q − )as the scale of T I (0) ij being of O (1). The relevant contributions are of higher than twist-4.The above example is to show that, using the collinear expansion, one can calculate the treelevel higher twist corrections from the dynamical partons in a systematic way. Because weonly intend to calculate the twist-3 corrections, we will not further explore the contributionsof twist order higher than three. For − S − P )( S + P ) operators, ¯Γ i = γ . Up to twist-3,the expression appears as Z dx Tr[ T I (0) ij Φ M ( x )] = − Z dx Tr[ T I (0) ij /qγ ]Tr[ /nγ Φ M ( x )]+ 14 Z dx Tr[ T I (0) ij γ ]Tr[ γ Φ M ( x )] . (33)11y identifying Tr[ /nγ Φ M ( x )] and Tr[ γ Φ M ( x )] as the twist-2 and twist-3 two parton LCDAsof the M meson Tr[ /nγ Φ M ( x )] = − if M φ tw M ( x ) , (34)Tr[ γ Φ M ( x )] = − if M µ χ φ tw M ,P ( x ) , (35)where µ M χ = m M / ( ¯ m q + ¯ m ¯ q ) with ¯ m q and ¯ m ¯ q the current quark masses and m M the mesonmass, we recover the naive factorization result up to twist-3 order. B. Expansion with T I (0) ij,µ We show the light-cone gauge n · A = 0 and the covariant gauge ∂ · A = 0 for theexpansion with T I (0) ij,µ . Because the analysis is tedious, we outline the procedure, here, andleave the details for next section. The first step is to take a power counting for the partonamplitude T I (0) ij,µ . There are three interesting regions. The first region is composed of eithertwo soft loop parton momenta, or one soft loop parton momentum and one collinear loopparton momentum. The T I (0) ij,µ in the first region is counted as λ − . The second region iscomposed of two collinear loop parton momenta. The T I (0) ij,µ in the second region is countedas Qλ − . The third region is composed of either one collinear loop parton momentum andone hard loop parton momentum, or two hard loop parton momenta. In the third region,the T I (0) ij,µ is counted as Q − . We conclude that the region composed of two collinear loopparton momenta is dominant.Let’s, first, consider the light-cone gauge n · A = 0. The expansion series of T I (0) ij,µ ( l , l )with respect to ˆ l i , i = 1 ,
2, are written as T I (0) ij,µ ( l , l ) = T I (0) ij,µ (ˆ l , ˆ l ) + X k =1 , T I (0) ijk,µν (ˆ l , ˆ l k , ˆ l )( l k − ˆ l k ) ν + · · · , (36)where T I (0) ijk,µν (ˆ l , ˆ l k , ˆ l ) are defined by assuming the low energy theories T I (0) ijk,µν (ˆ l , ˆ l k , ˆ l ) = ∂T I (0) ij,µ ∂l νk (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l =ˆ l ,l =ˆ l . (37)The expansion series are then substituted back into the convolution integrals. Since thegauge condition n · A = 0 with a light-cone vector n µ satisfying n · q = 1, n = 0, and n · l ⊥ = 0, the first term T I (0) ij,µ (ˆ l , ˆ l ) leads to the result Z d l (2 π ) Z d l (2 π ) Tr[ T I (0) ij,µ (ˆ l , ˆ l ) w µα Φ αM ( l , l )] (38)12here we have introduced w µα = g µα − q µ n α . Similarly, we employ the following integralidentities to simplify the momentum integrations Z dx Z dx Z ∞ dλ π Z ∞ dη π e iλ ( x − n · l ) e iη ( x − n · l ) = 1 (39)The expression appears as Z dx Z dx Tr[ T I (0) ij,µ ( x , x ) w µα Φ αM ( x , x )] (40)in which T I (0) ij,µ ( x , x ) = T I (0) ij,µ (ˆ l , ˆ l ) (cid:12)(cid:12)(cid:12) ˆ l = x , ˆ l = x , (41)Φ αM ( x , x ) = Z ∞ dλ π Z ∞ dη π e iλx e iη ( x − x ) ×h M | ¯ q ( λn )( − gA µ ( ηn )) q (0) | i . (42)In the above equations, we have used the following transformation for any function G ( y, z, y µ and z µ Z d l (2 π ) Z d l (2 π ) Z d y Z d ze il · ( y − z − λn ) e il · ( z − ηn ) G ( y, z, G ( λn, ηn, . (43)For tree diagrams, only color singlet operators can contribute at twist-3. The remainingtask is to finish the factorization of spin indices. For ( V − A )( V ± A ) operators , the relatedcontributions are of twist-4, which are beyond our accuracy. For − S − P )( S + P ) operators,the related contributions are of twist-3. To obtain the collinear limit ˆ l i → x i , i = 1 ,
2, for T I (0) ij,µ ( x , x ) , we have used the following substitution for the parton propagators i ˆ /l ˆ l → i/n n · l . (44)This is because the vertex iγ µ in T I (0) ij,µ ( x , x ) is transversal and only the off-shell part ofthe parton propagators can contribute. See further explanations in the next section. Theterms associated with T I (0) ijk,µν are of twist-4 and higher. They are neglected accordingly. Thefactorization of spin indices results in18 Z dx Z ¯ x dx Tr[ T I (0) ij,µ σ αβ γ ] w µµ ′ Tr[ σ αβ γ Φ µ ′ M ( x , x )] . (45)The explicit expression for Eq. (45) is left to the next section.13e now consider the expansion with covariant gauge ∂ · A = 0. Since the factorizationsof the momentum integrals and color indices are independent of gauge condition, we cango through to consider the factorization of spin indices. The first term in the expansionof T I (0) ij,µ ( x , x ), under covariant gauge, is related to the gauge invariant phase factor ofthe related two parton amplitudes. The gluon fields A µ in Φ µM can be expanded as A µ = n · Aq µ + q · An µ + d µα A α . The contraction Tr[ T I (0) ij,µ ( x , x ) q µ n · Φ M ( x , x )] leads to Z dx Z dx Tr[¯Γ i n · k n · Φ M ( x , x )] , (46)where k = l − l being the gluon momentum and n · Φ M ( x , x ) = Z ∞ dλ π Z ∞ dη π e iλx e iη ( x − x ) h M | ¯ q ( λn )( − gn · A ( ηn )) q (0) | i . (47)The terms with q · An µ vanish since the covariant gauge condition ∂ · A = 0. The terms withthe contraction Tr[ T I (0) ij,µ ( x , x ) d µα Φ αM ( x , x )] are of higher twist than twist-3 and will beneglected. With the above considerations, the contraction Tr[ T I (0) ij,µ ( x , x )Φ µM ( x , x )] leadsto contributions of twist-2 or higher than twist-3.We next consider the contraction Tr[ T I (0) ijk,µν ( x , x )( l k − ˆ l k ) ν Φ µM ( x , x )], which can berewritten as Tr[ T I (0) ijk,µν ( x , x k , x ) w νν ′ Φ ν ′ µM ( x , x k , x )] (48)with Φ ν ′ µM ( x , x k , x ) ≡ Z dλ π Z dη π e iλx e iη ( x − x ) h M | ¯ q ( λn ) igG ν ′ µ ( ηn ) q (0) | i . (49)Note that only transversal part d ν ⊥ ,β ( l k − ˆ l k ) β of the ( l k − ˆ l k ) ν can contribute at twist-3. For( V − A )( V ± A ) operators, the contributions are of twist-4. For − S − P )( S + P ) operators,the result appears as18 Z dx Z ¯ x dx Z dx k Tr[ T I (0) ijk,µν ( x , x k , x ) d ν ⊥ ,ν ′ σ αβ γ ][ σ αβ γ Φ ν ′ µM ( x , x k , x )] × ( δ ( x k − x ) + δ ( x k − x )) . (50)The reader may have noticed that the terms in the expansion series of T I (0) ij,µ ( l , l ) in Eq. (36)are of different twist order under the covariant or the light-cone gauge. For example, thetwist-3 contributions are from the the first term in the expansion series of T I (0) ij,µ ( l , l ) under14he light-cone gauge. On the other hand, under the covariant gauge, the twist-3 contribu-tions are from the the second term in the expansion series of T I (0) ij,µ ( l , l ). Since the partonamplitude and hadron amplitude under the collinear expansion are required to be gauge in-variant, respectively, this feature of the collinear expansion method can be used as a guidingprinciple for calculations. III. TWIST-3 CORRECTIONS
In this section, we make a more detail descriptions for the twist-3 contributions from thethree parton Fock state q ¯ qg of the M meson. The amplitude for the three parton q ¯ qg of M interacting with the operator O at the tree level for ¯ B → M M decays is written as h M | ¯ q (0)(1 − γ ) b (0) | ¯ B i× Z d y Z d z Z d l (2 π ) d k (2 π ) e il · z e ik · ( y − z ) h M | ¯ q ( z )[( − ig/A ( y )) i ( /l + /k )( l + k ) + iǫ (1 + γ )+ − i ( /q − /l + /k )( q − l + k ) + iǫ (+ ig/A ( y ))(1 + γ )] q (0) | i . (51)The l and k denote the momenta carried by the q quark and g gluon fields in Fig. 3(a) and(b). We first employ the light-cone gauge n · A ( y ) = 0. The gluonic fields A α ( y ) represents A α,a ( y ) T a with the color matrix T a in the fundamental representation P T a T b = δ ab /
2. Torelate to the previous introduced collinear expansion, we recast the convolution integrationpart of Eq. (51) into the form Z d l (2 π ) d k (2 π ) Tr[ T I (0) µ ( k, l ) w µµ ′ Φ µ ′ ( k, l )] (52)where the parton amplitude T I (0) µ ( k, l ) is defined as T I (0) µ ( k, l ) = [( iγ µ ) i ( /l + /k )( l + k ) + iǫ + − i ( /q − /l + /k )( q − l + k ) + iǫ ( − iγ µ )](1 + γ ) (53)and the meson amplitude Φ µ ′ ( k, l )Φ µ ′ ( k, l ) = Z d y Z d ze il · z e ik · ( y − z ) h M | ¯ q ( z )( − gA µ ′ ( y )) q (0) | i . (54)The tensor w µµ ′ = g µµ ′ − q µ n µ ′ has been introduced. Note that, for convenience, we have madea change of variables for the loop parton momenta l = l µ and k µ = ( l − l ) µ . We assume15hat the emitted M meson is highly energetic. As shown in last section, the dominante con-figuration is composed of collinear l and l . This allows us to expand the parton amplitude T I (0) ij,α ( k, l ) with respect to ˆ l = xq and ˆ k = ( x ′ − x ) qT I (0) ij,µ ( k, l ) = T I (0) µ (ˆ k, ˆ l ) + ∂T I (0) ij,µ ( k, l ) ∂l ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l =ˆ l,k =ˆ k ( l − ˆ l ) ν + ∂T I (0) ij,µ ( k, l ) ∂k ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l =ˆ l,k =ˆ k ( k − ˆ k ) ν + · · · . (55)Substituting the first term back into the convolution integrations gives Z dx Z dx ′ Tr[ T I (0) ij,µ ( x, x ′ ) w µµ ′ Φ µ ′ ( x, x ′ )] (56)where Φ µ ′ ( x ′ , x ) = Z d l (2 π ) δ ( x − l · n ) Z d k (2 π ) δ ( x ′ − x − k · n ) Z d y Z d z × e il · z e ik · y h M | ¯ q ( z )( − gA µ ′ ( y )) q (0) | i . (57)In the above collinear limit step T I (0) ij,µ ( l, k ) → T I (0) ij,µ ( x, x ′ ), there arises an infrared divergenceas x ′ →
0, which is from the denominators of virtual quark propagators ix ′ /q ( x ′ q ) + iǫ . (58)We regularize this divergence by the following method. Since the full quark propagator withmomentum l ′ = l + k can be decomposed into its long distance part and short distance partas i/l ( l + k ) + iǫ = i/l ′ L ( l ′ ) + iǫ + i/n n · l ′ . (59)The long distance part gives vanishing result upto twist-3. The short distance part isabsorbed by the parton amplitude. The divergence is then regularized by replacing thequark propagators with its corresponding special propagators [7, 40] ix ′ /q ( x ′ q ) + iǫ → i/n x + iǫ x ′ − xx ′ − x + iǫ . (60)The introduction of a special propagator for an on-shell fermion propagator is due to thefact that the fermion propagators in Fig. 3(a) and 3(b) become on-shell and divergent afterthe collinear expansion. The divergent part of these propagators leads to long distancecontributions that should be included into the twist-2 distribution amplitude for the M meson. However, there are also finite contact part of these propagators, which leads to16ontributions of one twist higher. The more detailed explanation about the meaning of thespecial propagator refers to [7, 40].Under light-cone gauge n · A ( y ) = 0, it is convenient to transform the gluon fields A µ ( y )into its field strength G νµ ( y ) by using following replacement A µ ′ ( y ) → in ν G νµ ′ ( y )( x ′ − x ) , (61)and Φ µ ′ M ( x, x ′ ) → in ν x ′ − x Φ νµ ′ M ( x, x ′ ) . (62)The factor in ν / ( x ′ − x ) is then absorbed by T I (0) µ ( x, x ′ ) into the form T I (0) µν ( x, x ′ ) ≡ T I (0) µ ( x, x ′ ) in ν x ′ − x . (63)The factorization of the spin indices gives18 Z dx Z dx ′ Tr[ T I (0) µν ( x, x ′ ) σ αβ γ ] w µµ ′ Tr[ σ αβ γ Φ νµ ′ ( x, x ′ )] + · · · , (64)in which other spin decompositions give higher twist contributions.The numerators in the contraction Tr[ T I (0) µν σ αβ γ ] can give terms proportional to n ν n µ ( q α n β − n α q β ), n ν d ⊥ ,µµ ′′ ( q α n β − n α q β ), and n ν d ⊥ ,µµ ′′ ǫ ⊥ ,αβ . The transversal tensors d ⊥ ,αβ and ǫ ⊥ ,αβ are defined as d ⊥ ,αβ = q α n β + q β n α − g αβ and ǫ ⊥ ,αβ = ǫ αβηλ q η n λ . The trace of d ⊥ ,αβ is defined to be negative d α ⊥ ,α = −
2. Since ν and µ indices in Φ νµM are antisymmetricunder µ ↔ ν , the terms proportional to n ν n µ ( q α n β − n α q β ) then vanish. For those termsproportional to n ν d ⊥ ,µµ ′′ ( q α n β − n α q β ), as they are contracted with Tr[ σ αβ γ Φ νµ ′ ( x, x ′ )], the q α factor in n ν d ⊥ ,µµ ′′ ( q α n β − n α q β ) results in twist-4 contributions by using the property ofthe long distance propagator of the quark fields. The terms proportional to n ν d ⊥ ,µµ ′′ ǫ ⊥ ,αβ lead to twist-3 contributions. The final result appears as Z dx Z dx ′ G βµ ( x, x ′ ) n µ n β ( x ′ − x ) x , (65)where the function G βµ ( x, x ′ ) is defined as G βµ ( x, x ′ ) = Z d l (2 π ) δ ( x − l · n ) Z d k (2 π ) δ ( x ′ − x − k · n ) Z d y Z d z × e il · z e ik · y h M | ¯ q ( z ) σ µα γ w αα ′ gG βα ′ ( y ) q (0) | i . (66)17ote that we have used the G -parity symmetry x ↔ ¯ x ′ to simplify the above result. Thisassumption is valid for π mesons, but may not be appropriate for the K or η mesons.Therefore, it is noted that, in the above result, there exist symmetry breaking effects for K and η mesons. However, we will ignore such a corrections from the symmetry breaking inthe following calculations. By referring to the definition [41] h M | ¯ q ( z ) σ µν γ gG αβ ( y ) q (0) | i = − i f M m M m q + m ¯ q ( q α q µ d ⊥ ,νβ − q α q ν d ⊥ ,µβ − q β q µ d ⊥ ,να + q β q ν d ⊥ ,αµ ) T ( z, y ) + · · · , (67)where T ( z, y ) = Z dx Z ¯ x dx ′ e − ixq · z e − i ( x − x ′ ) q · y T ( x, x ′ ) , (68)we can arrive at the result Z dx Z dx ′ G βµ ( x, x ′ ) n µ n β ( x ′ − x ) x = − if M m M m q + m ¯ q Z dx Z dx ′ T ( x, x ′ )( x ′ − x ) x . (69)By using the normalization for h M | ¯ q (0)(1 − γ ) b (0) | ¯ B i , it is easy to derive the tree levelthree parton contributions for operator O as h O i − gluon = 2 A G M m M m b ( m q + m ¯ q ) h O i f (70)with A G M = 2 Z dx Z ¯ x dx ′ T M ( x ′ , x )( x ′ − x ) x . (71)We now explain the expansion with the covariant gauge ∂ · A = 0. We first decompose A µ ( y ) into its longitudinal and transversal components as A µ ( y ) = n · A ( y ) q µ + d µ ⊥ ,µ ′ A µ ′ ( y ).The transversal part d µ ⊥ ,µ ′ A µ ′ ( y ) results in contributions of higher than twist-3 . The lon-gitudinal part n · A ( y ) q α gives twist-3 contributions. Similar to the light-cone gauge, weneed to transform the gluon fields into its field strength. Here, it needs one transversalmomentum k ⊥ factor from expansion of the parton amplitude T I (0) µ ( l, k ) in Eq. (55). Thecontraction of T I (0) µ ( x, x ′ ) with q µ n · Φ M ( x, x ′ ) leads to two parton gauge phase factorTr[ T I (0) µ ( x, x ′ ) q µ n · Φ M ( x, x ′ )] = Tr[ T I (0) ( x ′ ) − T I (0) ( x ) x ′ − x n · Φ M ( x, x ′ )] (72)18t is convenient to write ( k − ˆ k ) ρ = d ρ ⊥ ρ ′ ( k − ˆ k ) ρ ′ + q · kn ρ . Only transversal part k ρ ⊥ = d ρ ⊥ ρ ′ ( k − ˆ k ) ρ ′ contributes at twist-3. This is because the term ∂T I (0) µ /∂k ν can have termsproportional to g µν and σ µν . The terms related to q · kn ν leads to twist-4 contributions.For the transversal part k ⊥ , only σ µν terms can contribute. Let the k ρ ⊥ factor absorbedinto Φ µ ( l, k ) and use the replacement k ν ⊥ A µ ( y ) → − iG νµ ( y ), we can derive the result h O i t =31 − gluon = − Z dx Z dx ′ Tr[ ∂T I (0) µ ( x, x ′ ) ∂k ν G νµ ( x, x ′ )] ×h M | ¯ q (0)(1 − γ ) b (0) | ¯ B i , (73)where ∂T I (0) µ ( x ′ , x ) ∂k ν = − iσ µν ( x ′ − x ) xq (1 + γ ) (74)and G νµ ( x, x ′ ) = Z d l (2 π ) δ ( x − l · n ) Z d k (2 π ) δ ( x ′ − x − k · n ) Z d y Z d z × e il · z e ik · y h M | ¯ q ( z ) igG νµ ( y ) q (0) | i . (75)The contraction of σ µν with G νµ ( x, x ′ ) givesTr[ iσ µν G νµ ( x, x ′ )] = − if M m M q ( m q + m ¯ q ) T ( x, x ′ ) . (76)Note that the q factor in the denominator of ∂T I (0) µ ( x ′ , x ) /∂k ν is cancelled by the q factorin the numerator of Tr[ iσ νµ G νµ ( x, x ′ )]. It is easy to see that Eq. (73) is equal to the resultderived from the light-cone gauge. This explicitly shows the gauge invariance of the threeparton contributions.There are related diagrams, such as those in Fig. 3(c) and 3(d). Because the spectatorquark of the ¯ B meson can carry only a soft momentum, this makes the relevant contributionsassociated with Fig. 3(c) and 3(d) dominated by soft gluons as the form factors F B → M + , .In addition, the relevant contributions are of O ( m − b ) with respect to the leading twistamplitude. It can be understood as following. The sum of the lower parts of the diagarmsin Fig. 3(c) and 3(d) is proportional to h M | ¯ q (cid:20) p ν + γ ν /k p · k Γ i − Γ i P bν − /kγ ν P b · k (cid:21) b | ¯ B i , (77)19here k is the momentum of the gluon from M and the equation of motions for b and q quarks have been used. After taking the collinear limit, k → x ′ q , we write the expression as Aq ν + B µν q µ , (78)where A ≡ x ′ Q h M | ¯ q Γ i | ¯ B i , (79) B µν ≡ x ′ Q h M | ¯ q ( γ ν γ µ Γ i + Γ i γ µ γ ν ) b | ¯ B i . (80)For light-cone gauge, only A term contributes. As for the covariant gauge, only B µν termcontributes. The only contributions come from ( V − A )( V ± A ) operators. This implies thatthe upper parts of the diagarms in Fig. 3(c) and 3(d) are proportional to the twist-4 LCDAof M . The combination of the upper and the lower parts gives a O ( m − b ) contributions withrespect to the leading twist amplitude.There are possibilities that the additional gluon of the M meson can interact with thespectator quark of the ¯ B meson. Since the spectator quark carries a soft momentum, themomentum conservation at the interaction vertex prevents the momentum of the gluon frombeing collinear to the M meson’s momentum. Therefore, there require additional radiativegluons interacting between the other parton lines and the spectator quark line to make themomentum of the gluon to be collinear to the M meson’s momentum. This results incontributions of order O ( α s ). We identify the relevant contributions as O ( α s ) three partoncorrections. As mentioned previously, we plan to discuss these contributions in other places[44].The total twist-3 contribution from operator O is then equal to h O i t =3 = 2(1 + A G M ) m M m b ( m q + m ¯ q ) h O i f . (81)For operator O , there are similar results. 20 V. APPLICATIONS
For penguin dominant B → πK decays, the relevant decay amplitudes under QCDfactorization are parametrized as the following [3] A ( B − → π − ¯ K ) = λ p (cid:20) ( a p − a p ) + r Kχ ( a p − a p ) (cid:21) A πK +( λ u b + ( λ u + λ c )( b + b EW )) B πK , −√ A ( B − → π K − ) = [ λ u a + λ p ( a p + a p ) + λ p r Kχ ( a p + a p )] A πK +[ λ u a + λ p
32 ( − a + a )] A Kπ +( λ u b + ( λ u λ c )( b + b EW )) B πK , − A ( ¯ B → π + K − ) = [ λ u a + λ p ( a p + a p ) + λ p r Kχ ( a p + a p )] A πK +(( λ u + λ c )( b − b EW )) B πK √ A ( ¯ B → π ¯ K ) = A ( B − → π − ¯ K ) + √ A ( B − → π K − ) − A ( ¯ B → π + K − ) (82)where λ p = V pb V ∗ ps , a i ≡ a i ( πK ), and λ p a pi = λ u a ui + λ c a ci . The CP conjugation of decayamplitudes are obtained by replacing λ p → λ ∗ p for the above amplitudes. The factorizedmatrix elements are defined as A πK = i G F √ m B − m π ) F B → π ( m K ) f K ,A Kπ = i G F √ m B − m K ) F B → K ( m π ) f π . (83)The form factors are defined h P ( p ) | ¯ qγ µ b | ¯ B i = F B → P + ( q )( P µB + p µ ) + [ F B → P ( q ) − F B → P + ( q )] m B − m P q q µ . (84)The form factors coincide as q = 0, F B → P + (0) = F B → P (0). The expressions for the param-eters a i are referred to [2, 3]. For numerical calculations, we will use the following inputparametersΛ (5)¯ MS = 0 . , m b ( m b ) = 4 . m c ( m b ) = 1 . , m s (2GeV) = 0 . , | V cb | = 0 . , | V ub /V cb | = 0 . , γ = 70 ◦ , τ ( B − ) = 1 . ,τ ( B d ) = 1 . , f π = 131MeV , f K = 160MeV , f B = 200MeV ,F B → π = 0 . , F B → K = 0 . . (85)For λ u and λ c , we take the following convention for their parametrization λ u λ c = tan θ c R b e − iγ (86)21here tan θ c = λ − λ ,R b = 1 − λ / λ | V ub V cb | ,λ = | V us | . (87)The value of λ is taken as 0 . a i , i = 1 , · · · ,
10, and b j , j = 1 , · · · ,
3, calculated at the scale m b = 4 . a = 0 .
995 + 0 . i , a = 0 . − . i , a = − .
003 + 0 . i ,a u = − . − . i a c = − .
030 + 0 . i , a = 0 . − . i ,r Kχ a u = − . − . i , r Kχ a c = − . − . i , a /α = 0 .
007 + 0 . i ,r Kχ a u /α = 0 . − . i , r Kχ a c /α = 0 . − . i , a /α = − . − . i ,a u /α = − .
175 + 0 . i , a c /α = − .
175 + 0 . i , r A b = 0 . ,r A b = − . , r A b = − . , r A b EW /α = − . , (88)in which r A = B πK A πK = f B f π m B F B → π (0) , (89)and r Kχ = 2 m K m b ( m q + m s ) . (90)The tree level three parton contributions modify the parameters r Kχ as r Kχ (1 + A G M ) withparameter A G M defined in Eq. (71). The value of A G M depends on the model of the threeparton distribution amplitude T M ( x, x ′ ). Here we employ the model derived from the light-cone sum rule [41] T ( x, x ′ ) = 360 ηxx ′ ( x − x ′ ) (1 + ω x − x ′ ) − , (91)where the parameters are assumed to be η = 0 .
015 and ω = − . M = π, K, or η . Thisgive us A G M = 0 . B → πK decay is given by this expressions Br ( ¯ B → πK ) = τ B πm B | A ( ¯ B → πK ) | . (92)22e can use the above formula to predict CP averaged branching ratios for B → πK decays.The predictions with three parton corrections in units of 10 − are given as Br ( B − → π − ¯ K ) = 19 . ,Br ( B − → π K − ) = 10 . ,Br ( ¯ B → π + K − ) = 16 . ,Br ( ¯ B → π ¯ K ) = 7 . . (93)For comparison, we also list the predictions with only two parton contributions in units of10 − in the following, Br ( B − → π − ¯ K ) = 11 . ,Br ( B − → π K − ) = 6 . ,Br ( ¯ B → π + K − ) = 9 . ,Br ( ¯ B → π ¯ K ) = 4 . . (94)For reference, we enlist the experimental data in units of 10 − summarized by the HFAGgroup [42] Br ( B − → π − ¯ K ) = 23 . ± . ,Br ( B − → π K − ) = 12 . ± . Br ( ¯ B → π + K − ) = 19 . ± . Br ( ¯ B → π ¯ K ) = 10 . ± . . (95)By comparing the predictions with or without the three parton corrections, one may noticethat the the predicted branching ratios are significantly enhanced by about 1 . ∼ . B → πK decays made here are consistent with thefindings of previous studies using QCD factorization approach [3, 43]. The two partonpredictions made in [43] are much lower than the experimental data under the QCD factor-ization approach. Because the calculations of the three parton corrections were inaccessiblein their studies, this led them to conclude that the QCD factorization is impossible to ex-plain the penguin dominant ¯ B → πK decays. The two parton predictions made in [3] for¯ B → πK decays are still lower than the data. Only extending the predictions by using23xtreme limits of input parameters can make the predictions to be consistent with the mea-surements. This seems not a reasonable solution from the theoretical point of view. On theother hand, as shown in the above, our approach has shown that the predictions with threeparton contributions are more close to the data than the two parton predictions. Althoughthe predictions with three parton contributions are still lower than the experimental data,the O ( α s ) corrections can improve the predictions. V. DISCUSSIONS AND CONCLUSIONS
The significance of the three parton contributions for penguin dominant ¯ B → πK decayscan also be seen from a phenomenological point of view. By appropriate arrangement, theparameters a i can be calculated for B → ππ decays. For the pure penguin B − → π − ¯ K ,its dominant contributions arise from the | a c ( πK ) + r Kχ a c ( πK ) | term. The uncertainty dueto the form factors can be eliminated by considering the ratio between the decay rates of B − → π − ¯ K and B − → π − π as the following (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a c ( πK ) + r Kχ a c ( πK ) a ( ππ ) + a ( ππ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | V ub || V cb | f π f K (cid:20) Γ( B − → π − ¯ K )2Γ( B − → π − π ) (cid:21) / = 0 . ± . , (96)where the error comes from the branching ratios. In the above, we have used the branchingratio Br ( B − → π − π ) = 5 . ± .
4. According to QCD factorization calculations, | a ( ππ ) + a ( ππ ) | = 1 .
17 and | a c ( πK ) + r Kχ a c ( πK ) | = 0 .
08 with only two parton contributions. Thisgives the prediction of the ratio to be 0 . r Kχ then becomes r Kχ (1 + A G K )and makes | a c ( πK ) + r Kχ (1 + A G K ) a c ( πK ) | = 0 . . B → πK decays under the QCD factorization approach.In order to make sure that the three parton contributions are indeed significant and alsocompatible with the QCD factorization, an important task is to finish O ( α s ) calculationsfor the three parton contributions. The related work in this direction has been proceededand will be reported in our another preparing paper [44].24here are similar three parton contributions having been calculated under the light-cone sum rule [45, 46]. In [45], the twist-3 three parton contributions associated with softgluons are calculated in the framework of light-cone sum rule. The contributions are shownnegligible in ¯ B → ππ decays. In [46], the contributions from the diagrams similar to Fig. 3(c)and 3(d) were calculated for B → πω decays. They are found to be twist-4 and vanishing.In addition, significant effects were found due to the three parton Fock state of the π in the B → πω decays. Since they are dominated by soft gluons, it is better determined by QCDsum rule. As shown in [46], in the Euclidean region of ( p + q ) , the relevant contributionsare from the twist-3 and twist-4 three parton LCDAs of the π . As mentioned before, weidentify these power corrections as non-partonic ones. From the theoretical point of view,we suggest that the partonic and non-partonic power corrections should be distinquishedunder the QCD factorization, although they may be equally important in phenomenology.For comparison, we employ the replacing rules for the a i coefficients [46] to account forthe three parton effects from the M meson. The rule is a i → a i + [1 + ( − δ i + δ i ] C i − f / ,a i − → a i − + ( − δ i + δ i C i f , (97)where i = 1 , · · · ,
5, and C i are the Wilson coefficients calculated at the scale µ h = 1 .
45 GeV,and f = 0 .
12, which is assumed to be universal. With these three parton corrections, thepredicted CP averaged branching ratios for ¯ B → πK in units of 10 − are Br ( B − → π − ¯ K ) = 9 . ,Br ( B − → π K − ) = 5 . ,Br ( ¯ B → π + K − ) = 8 . ,Br ( ¯ B → π ¯ K ) = 3 . , (98)which becomes smaller than those predictions in Eq. (94). Acknowledgments
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