Direct CP violation for \bar{B}_s^0 \to {π^+}{π^-}K^{*0} in perturbative QCD
DDirect CP violation for ¯ B s → π + π − K ∗ in perturbative QCD Sheng-Tao Li ∗ , Gang L¨u † Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOE),Central China Normal University, Wuhan, Hubei 430079, China College of Science, Henan University of Technology, Zhengzhou 450001, China
In perturbative QCD approach, based on the first order of isospin symmetry breaking, we studythe direct CP violation in the decay of ¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ . An interesting mechanism isapplied to enlarge the CP violating asymmetry involving the charge symmetry breaking between ρ and ω . We find that the CP violation is large by the ρ − ω mixing mechanism when the invariantmasses of the π + π − pairs is in the vicinity of the ω resonance. For the decay process of ¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ , the maximum CP violation can reach − . ρ − ω mixing into account, we calculate the branching ratio for ¯ B s → ρ ( ω ) K ∗ . We also discuss thepossibility of observing the predicted CP violation asymmetry at the LHC. PACS numbers:
I. INTRODUCTION
Charge-Parity ( CP ) violation is an open problem, even though it has been known in the Neutral kaon systems formore than five decades [1]. The study of CP violation in the heavy quark systems is important to our understandingof both particle physics and the evolution of the early universe. Within the standard model (SM), CP violationis related to the non-zero weak complex phase angle from the Cabibbo-Kobayashi-Maskawa (CKM) matrix, whichdescribes the mixing of the three generations of quarks [2, 3]. Theoretical studies predicted large CP violation inthe B meson system [4–6]. In recent years, the LHCb collaboration has measured sizable direct CP asymmetries inthe phase space of the three-body decay channels of B ± → π ± π + π − and B ± → K ± π + π − [7–9]. These processesare also valuable for studying the mechanism of multi-body heavy meson decays. Hence, more attention has beenfocused on the non-leptonic B meson three-body decays channels in searching for CP violation, both theoreticallyand experimentally.Direct CP violation in b hadron decays occurs through the interference of at least two amplitudes with differentweak phase φ and strong phase δ . The weak phase difference φ is determined by the CKM matrix elements, while thestrong phase can be produced by the hadronic matrix elements and interference between the intermediate states. Thehadronic matrix elements are not still well determined by the theoretical approach. The mechanism of two-body B decay is still not quite clear, although many physicists are devoted to this field. Many factorization approaches havebeen developed to calculate the two-body hadronic decays, such as the naive factorization approach [10–13], the QCDfactorization (QCDF) [14–18], perturbative QCD (pQCD) [19–21], and soft-collinear effective theory (SCET) [22–24]. ∗ [email protected] † [email protected] a r X i v : . [ h e p - ph ] F e b Most factorization approaches are based on heavy quark expansion and light-cone expansion in which only the leadingpower or part of the next to leading power contributions are calculated to compare with the experiments. However,the different methods may present different strong phases so as to affect the value of the CP violation. Meanwhile, inorder to have a large signal of CP violation, we need appeal to some phenomenological mechanism to obtain a largestrong phase δ . In Refs. [25–30], the authors studied direct CP violation in hadronic B (include B s and Λ b ) decaysthrough the interference of tree and penguin diagrams, where ρ - ω mixing was used for this purpose in the past fewyears and focused on the naive factorization and QCD factorization approaches. This mechanism was also appliedto generalize the pQCD approach to the three-body non-leptonic decays in B , ± → π , ± π + π − and B c → D +( s ) π + π − where even larger CP violation may be possible [31, 32]. In this paper, we will investigate direct CP violation of thedecay process ¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ involving the same mechanism in the pQCD approach.Three-body decays of heavy B mesons are more complicated than the two-body decays as they receive both resonantand non-resonant contributions. Unlike the two-body case, to date we still do not have effective theories for hadronicthree-body decays, though attempts along the framework of pQCD and QCDF have been used in the past [33–36]. Asa working starting point, we intend to study ρ - ω mixing effect in three-body decays of the B meson. The ρ - ω mixingmechanism is caused by the isospin symmetry breaking from the mixing between the u and d flavors [37, 38]. In Ref.[39], the authors studied the ρ − ω mixing and the pion form factor in the time-like region, where ρ − ω mixing comesfrom three part contributions: two from the direct coupling of the quasi-two-body decay of ¯ B s → ρK ∗ → π + π − K ∗ and ¯ B s → ωK ∗ → π + π − K ∗ and the other from the interference of ¯ B s → ωK ∗ → ρK ∗ → π + π − K ∗ mixing.Generally speaking, the amplitudes of their contributions: ¯ B s → ρK ∗ → π + π − K ∗ > ¯ B s → ωK ∗ → ρK ∗ → π + π − K ∗ > ¯ B s → ωK ∗ → π + π − K ∗ . ω → π + π − and ω → ρ → π + π − were used to obtain the (effective) mixingmatrix element (cid:101) Π ρω ( s ) [40–42]. The magnitude has been determined by the pion form factor through the data fromthe cross section of e + e − → π + π − in the ρ and ω resonance region [39, 42–45]. Recently, isospin symmetry breakingwas discussed by incorporating the vector meson dominance (VMD) model in the weak decay process of the meson[27, 32, 46–48]. However, one can find that ρ − ω mixing produces the large CP violation from the effect of isospinsymmetry breaking in the three and four bodies decay process. Hence, in this paper, we shall follow the method ofRefs. [27, 32, 46–48] to investigate the decay process of ¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ by isospin symmetry breaking.The remainder of this paper is organized as follows. In Sec. II we will present the form of the effective Hamiltonianand briefly introduce the pQCD framework and wave functions. In Sec. III we give the calculating formalism anddetails of the CP violation from ρ − ω mixing in the decay process ¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ . In Sec. IV wecalculate the branching ratio for decay process of ¯ B s → ρ ( ω ) K ∗ . In Sec. V we show the input parameters. Wepresent the numerical results in Sec. VI. Summary and discussion are included in Sec. VII. The related functiondefined in the text are given in Appendix. II. THE FRAMEWORK
Based on the operator product expansion, the effective weak Hamiltonian for the decay processes ¯ B s → ρ ( ω ) K ∗ can be expressed as [49] H eff = G F √ (cid:40) V ub V ∗ ud (cid:104) C ( µ ) Q u ( µ ) + C ( µ ) Q u ( µ ) (cid:105) − V tb V ∗ td (cid:104) (cid:88) i =3 C i ( µ ) Q i ( µ ) (cid:105)(cid:41) + H.c. , (1)where G F represents the Fermi constant, C i ( µ ) (i=1,...,10) are the Wilson coefficients, and V ub , V ud , V tb , and V td arethe CKM matrix element. The operators O i have the following forms: O u = ¯ d α γ µ (1 − γ ) u β ¯ u β γ µ (1 − γ ) b α ,O u = ¯ dγ µ (1 − γ ) u ¯ uγ µ (1 − γ ) b,O = ¯ dγ µ (1 − γ ) b (cid:88) q (cid:48) ¯ q (cid:48) γ µ (1 − γ ) q (cid:48) ,O = ¯ d α γ µ (1 − γ ) b β (cid:88) q (cid:48) ¯ q (cid:48) β γ µ (1 − γ ) q (cid:48) α ,O = ¯ dγ µ (1 − γ ) b (cid:88) q (cid:48) ¯ q (cid:48) γ µ (1 + γ ) q (cid:48) ,O = ¯ d α γ µ (1 − γ ) b β (cid:88) q (cid:48) ¯ q (cid:48) β γ µ (1 + γ ) q (cid:48) α ,O = 32 ¯ dγ µ (1 − γ ) b (cid:88) q (cid:48) e q (cid:48) ¯ q (cid:48) γ µ (1 + γ ) q (cid:48) ,O = 32 ¯ d α γ µ (1 − γ ) b β (cid:88) q (cid:48) e q (cid:48) ¯ q (cid:48) β γ µ (1 + γ ) q (cid:48) α ,O = 32 ¯ dγ µ (1 − γ ) b (cid:88) q (cid:48) e q (cid:48) ¯ q (cid:48) γ µ (1 − γ ) q (cid:48) ,O = 32 ¯ d α γ µ (1 − γ ) b β (cid:88) q (cid:48) e q (cid:48) ¯ q (cid:48) β γ µ (1 − γ ) q (cid:48) α , (2)where α and β are SU(3) color indices, e q (cid:48) is the electric charge of quark q (cid:48) in the unit of | e | , and the sum extend over q (cid:48) = u, d, s, c or b quarks. In Eq. (2) O u and O u are tree operators, O – O are QCD penguin operators and O – O are the operators associated with electroweak penguin diagrams.The Wilson coefficient C i ( µ ) in Eq. (1) describes the coupling strength for a given operator and summarizes thephysical contributions from scales higher than µ [50]. They are calculable perturbatively with the renormalizationgroup improved perturbation theory. Usually, the scale µ is chosen to be of order O ( m b ) for B meson decays. Sincewe work in the leading order of perturbative QCD ( O ( α s )), it is consistent to use the leading order Wilson coefficients.So, we use numerical values of C i ( m b ) as follow [19, 21]: C = − . , C = 1 . ,C = 0 . , C = − . ,C = 0 . , C = − . ,C = 0 . , C = 0 . ,C = − . , C = 0 . . (3)The combinations a – a of the Wilson coefficients are defined as usual [51–54]: a = C + C / , a = C + C / ,a i = C i + C i ± / , ( i = 3 − , (4)where the upper (lower) sign applies, when i is odd (even).For the two-body decay processes of ¯ B s → M M , we denote the emitted or annihilated meson as M while therecoiling meson is M . The meson M ( ρ or ω ) and the final-state meson M ( K ∗ ) move along the direction of n = (1 , , T ) and v = (0 , , T ) in the light-cone coordinates, respectively. The decay amplitude can be expressedas the convolution of the wave functions φ B s , φ M and φ M and the hard scattering kernel T H in the pQCD. ThepQCD factorization theorem has been developed for the two-body non-leptonic heavy meson decays, based on theformalism of Botts, Lepage, Brodsky and Sterman [55–58]. The basic idea of the pQCD approach is that it takes intoaccount the transverse momentum of the valence quarks in the hadrons which results in the Sudakov factor in thedecay amplitude. Then, the decay channels of ¯ B s → ρ ( ω ) K ∗ are conceptually written as the following: A ( ¯ B s → ρ ( ω ) K ∗ ) = (cid:90) d k d k d k Tr (cid:2) C ( t ) φ B s ( k ) φ M ( k ) φ M ( k ) T H ( k , k , k , t ) (cid:3) , (5)where k i ( i = 1 , ,
3) are momentum of light quark in each meson. Tr denotes the trace over Dirac structure andcolor indices. C ( t ) is the short distance Wilson coefficients at the hard scale t . The meson wave functions φ B s and φ M ( m = 2 , T H ( k , k , k , t ) describes the four quarkoperator and the spectator quark connected by a hard gluon, which can be perturbatively calculated including allpossible Feynman diagrams of the factorizable and non-factorizable contributions without end-point singularity.The ρ ( ω ) and K ∗ mesons are treated as a light-light system. At the B s meson rest frame, they are moving veryfast. We define the ratios r K ∗ = M K ∗ M Bs , r ρ = M ρ M Bs and r ω = M ω M Bs . In the limit M K ∗ , M ρ , M ω →
0, one can dropthe terms of proportional to r K ∗ , r ρ , r ω safely. The symbols P B , P and P refer to the ¯ B s meson momentum, the ρ ( ω ) meson momentum, and the final-state K ∗ meson momentum, respectively. The momenta of the participatingmesons in the rest frame of the B s meson can be written as: P B = M B s √ , , T ) , P = M B s √ , , T ) , P = M B s √ , , T ) . (6)One can denote the light (anti-)quark momenta k , k and k for the initial meson ¯ B s , and the final mesons ρ ( ω )and K ∗ , respectively. We can choose: k = ( x M B s √ , , k ⊥ ) , k = ( x M B s √ , , k ⊥ ) , k = (0 , x M B s √ , k ⊥ ) , (7)where x , x and x are the momentum fraction. k ⊥ , k ⊥ and k ⊥ refer to the transverse momentum of the quark,respectively. To extract the helicity amplitudes, we parameterize the following longitudinal polarization vectors ofthe ρ ( ω ) and K ∗ as following: (cid:15) ( L ) = P M ρ ( ω ) − M ρ ( ω ) P · v v, (cid:15) ( L ) = P M K ∗ − M K ∗ P · n n, (8)which satisfy the orthogonality relationship of (cid:15) ( L ) · P = (cid:15) ( L ) · P = 0, and the normalization of (cid:15) ( L ) = (cid:15) ( L ) = − (cid:15) ( T ) = (0 , , T ) , (cid:15) ( T ) = (0 , , T ) . (9)Within the pQCD framework, both the initial and the final state meson wave functions and distribution amplitudesare important as non-perturbative input parameters. For the B s meson, the wave function of the meson can beexpressed as Φ B s = i √ (cid:54) P B s + M B s ) γ φ B s ( k ) , (10)where the distribution amplitude φ B s is shown in Refs. [59–61]: φ B s ( x, b ) = N B s x (1 − x ) exp (cid:20) − M B s x ω b −
12 ( ω b b ) (cid:21) . (11)The shape parameter ω b is a free parameter and N B s is a normalization factor. Based on the studies of the light-conesum rule, lattice QCD or be fitted to the measurements with good precision [62], we take ω b = 0 .
50 GeV for the B s meson. The normalization factor N B s depends on the values of the shape parameter ω b and decay constant f B s ,which is defined through the normalization relation (cid:82) dxφ B s ( x,
0) = f B s / (2 √ ρ , ω or K ∗ ), φ V , φ TV , φ tV , φ sV , φ vV , and φ aV , can be written in thefollowing form [63, 64]: φ ρ ( x ) = 3 f ρ √ x (1 − x ) (cid:104) . C / ( t ) (cid:105) , (12) φ ω ( x ) = 3 f ω √ x (1 − x ) (cid:104) . C / ( t ) (cid:105) , (13) φ K ∗ ( x ) = 3 f K ∗ √ x (1 − x ) (cid:104) . C / ( t ) + 0 . C / ( t ) (cid:105) , (14) φ Tρ ( x ) = 3 f Tρ √ x (1 − x ) (cid:104) . C / ( t ) (cid:105) , (15) φ Tω ( x ) = 3 f Tω √ x (1 − x ) (cid:104) . C / ( t ) (cid:105) , (16) φ TK ∗ ( x ) = 3 f TK ∗ √ x (1 − x ) (cid:104) . C / ( t ) + 0 . C / ( t ) (cid:105) , (17) φ tV ( x ) = 3 f TV √ t , (18) φ sV ( x ) = 3 f TV √ − t ) , (19) φ vV ( x ) = 3 f V √ t ) , (20) φ aV ( x ) = 3 f V √ − t ) , (21)where t = 2 x −
1. Here f ( T ) V is the decay constant of the vector meson with longitudinal(transverse) polarization. TheGegenbauer polynomials C νn ( t ) can be defined as [65, 66]: C / ( t ) = 3 t (22) C / ( t ) = 32 (5 t − . (23) III. CP VIOLATION IN ¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ DECAY PROCESSA. Formalism
The decay width Γ for the processes of ¯ B s → ρ ( ω ) K ∗ is given byΓ = P c πM B s (cid:88) σ = L,T A ( σ ) † A ( σ ) , (24)where P c is the absolute value of the three-momentum of the final state mesons. The decay amplitude A ( σ ) which isdecided by QCD dynamics will be calculated later in pQCD factorization approach. The superscript σ denotes thehelicity states of the two vector mesons with the longitudinal (transverse) components L(T). The amplitude A ( σ ) forthe decays B s ( P B s ) → V ρ ( ω ) ( P , (cid:15) ∗ µ ) + V K ∗ ( P , (cid:15) ∗ µ ) can be decomposed as follows [66–69]: A ( σ ) = M B s A L + M B s A N (cid:15) ∗ ( σ = T ) · (cid:15) ∗ ( σ = T ) + iA T (cid:15) αβγρ (cid:15) ∗ α ( σ ) (cid:15) ∗ β ( σ ) P γ P ρ , (25)where (cid:15) ∗ is the polarization vector of the vector meson. The amplitude A i ( i refer to the three kinds of polarizations,longitudinal (L), normal (N) and transverse (T)) can be written as M B s A L = a (cid:15) ∗ ( L ) · (cid:15) ∗ ( L ) + bM M (cid:15) ∗ ( L ) · P (cid:15) ∗ ( L ) · P ,M B s A N = a ,A T = cM M , (26)where a , b and c are the Lorentz-invariant amplitudes. M and M are the masses of the vector mesons ρ ( ω ) and K ∗ , respectively.The longitudinal H and transverse H ± of helicity amplitudes can be expressed H = M B s A L ,H ± = M B s A N ∓ M M (cid:112) κ − A T , (27)where H and H ± are the penguin-level and tree-level helicity amplitudes of the decay process ¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ from the three kinds of polarizations, respectively. The helicity summation satisfy the relation (cid:88) σ = L,R A ( σ ) † A ( σ ) = | H | + | H + | + | H − | . (28)In the vector meson dominance model [70, 71], the photon propagator is dressed by coupling to vector mesons. Basedon the same mechanism, ρ − ω mixing was proposed and later gradually applied to B meson physics [29, 39, 46, 72].According to the effective Hamiltonian, the amplitude A ( ¯ A ) for the three-body decay process ¯ B s → π + π − K ∗ ( B s → π + π − ¯ K ∗ ) can be written as [46]: A = (cid:10) π + π − K ∗ | H T | ¯ B s (cid:11) + (cid:10) π + π − K ∗ | H P | ¯ B s (cid:11) , (29)¯ A = (cid:10) π + π − ¯ K ∗ | H T | B s (cid:11) + (cid:10) π + π − ¯ K ∗ | H P | B s (cid:11) , (30)where H T and H P are the Hamiltonian for the tree and penguin operators, respectively.The relative magnitude and phases between the tree and penguin operator contribution are defined as follows: A = (cid:10) π + π − K ∗ | H T | ¯ B s (cid:11) [1 + re i ( δ + φ ) ] , (31)¯ A = (cid:10) π + π − ¯ K ∗ | H T | B s (cid:11) [1 + re i ( δ − φ ) ] , (32)where δ and φ are strong and weak phases differences, respectively. The weak phase difference φ can be expressedas a combination of the CKM matrix elements, and it is φ = arg[( V tb V ∗ td ) / ( V ub V ∗ ud )] for the b → d transition. Theparameter r is the absolute value of the ratio of tree and penguin amplitudes: r ≡ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:10) π + π − K ∗ | H P | ¯ B s (cid:11)(cid:10) π + π − K ∗ | H T | ¯ B s (cid:11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (33)The parameter of CP violating asymmetry, A CP , can be written as A CP = | A | − | ¯ A | | A | + | ¯ A | = − T r sin δ + T r + sin δ + + T − r − sin δ − ) sin φ (cid:80) i =0+ − T i (1 + r i + 2 r i cos δ i cos φ ) , (34)where T i ( i = 0 , + , − ) represent the tree-level helicity amplitudes of the decay process ¯ B s → π + π − K ∗ from H , H + and H − of the Eq. (27), respectively. r j ( j = 0 , + , − ) refer to the absolute value of the ratio of tree and penguinamplitude for the three kinds of polarizations, respectively. δ k ( k = 0 , + , − ) are the relative strong phases between thetree and penguin operator contributions from three kinds of helicity amplitudes, respectively. We can see explicitlyfrom Eq. (34) that both weak and strong phase differences are needed to produce CP violation. In order to obtaina large signal for direct CP violation, we intend to apply the ρ − ω mixing mechanism, which leads to large strongphase differences in hadron decays. ¯ B s K ∗ ρ I π + π − +¯ B s K ∗ π + π − Π ρω ω I ρ I FIG. 1: The diagram for the ¯ B s → π + π − K ∗ decay with the ρ − ω mixing mechanism for the first order of isospin violationin the isospin representation With the ρ − ω mixing mechanism, the process of the ¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ decay is shown in Fig.1. In theisospin representation, the decay amplitude M ¯ B s → ρ ( ω ) → π + π − in Fig.1 can be written as [31, 39, 47, 73] M ¯ B s → π + π − = M ρ I → ππ s ρ M ¯ B s → ρ I + M ρ I → ππ s ρ Π ρω s ω M ¯ B s → ω I . (35)Introducing the (cid:15) = Π ρω s ρ − s ω [31, 39, 47, 73], We have identified the physical amplitudes as M ρ → ππ = M ρ I → ππ , (36) M ω → ππ = (cid:15)M ρ I → ππ , (37) M ¯ B s → ρ = M ¯ B s → ρ I − (cid:15)M ¯ B s → ω I , (38) M ¯ B s → ω = M ¯ B s → ω I − (cid:15)M ¯ B s → ρ I . (39)From the physical representation, we can obtain the decay amplitude M ¯ B s → π + π − = M ρ → ππ s ρ M ¯ B s → ρ + M ρ → ππ Π ρω s ρ s ρ − s ω s ρ s ω M ¯ B s → ω I (40)where O ( (cid:15) ) corrections is neglected, and M ρ → ππ = g ρ , M ¯ B s → ρ = t iρ or p iρ and M ¯ B s → ω = t iω or p iω are used. So, wecan get (cid:101) Π ρω = Π ρω s ρ s ρ − s ω . (cid:101) Π ρω is the effective ρ − ω mixing amplitude which also effectively includes the direct coupling ω → π + π − . At the first order of isospin violation, we have the following tree and penguin amplitudes when theinvariant mass of π + π − pair is near the ω resonance mass [26, 46]: (cid:10) π + π − K ∗ | H T | ¯ B s (cid:11) = g ρ s ρ s ω (cid:101) Π ρω t iω + g ρ s ρ t iρ , (41) (cid:10) π + π − K ∗ | H P | ¯ B s (cid:11) = g ρ s ρ s ω (cid:101) Π ρω p iω + g ρ s ρ p iρ , (42)where t iρ ( p iρ ) and t iω ( p iω ) are the tree (penguin)-level helicity amplitudes for ¯ B s → ρ K ∗ and ¯ B s → ωK ∗ , respectively.The amplitudes t iρ , t iω , p iρ and p iω can be found in Sec. III B. g ρ is the coupling constant for the decay process ρ → π + π − . s V , m V and Γ V ( V = ρ or ω ) is the inverse propagator, mass and decay width of the vector meson V ,respectively. s V can be expressed as s V = s − m V + i m V Γ V , (43)with √ s being the invariant masses of the π + π − pairs. The ρ − ω mixing parameter (cid:101) Π ρω ( s ) = Re (cid:101) Π ρω ( m ω )+Im (cid:101) Π ρω ( m ω )are [74] Re (cid:101) Π ρω ( m ω ) = − ±
440 MeV , Im (cid:101) Π ρω ( m ω ) = − ± . (44)From Eqs. (29), (31), (41) and (42) one has re iδ i e iφ = (cid:101) Π ρω p iω + s ω p iρ (cid:101) Π ρω t iω + s ω t iρ , (45)Defining [25, 75] p iω t iρ ≡ r (cid:48) e i ( δ iq + φ ) , t iω t iρ ≡ αe iδ iα , p iρ p iω ≡ βe iδ iβ , (46)where δ iα , δ iβ and δ iq are strong phases of the decay process ¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ from the three kinds ofpolarizations, respectively. One finds the following expression from Eqs. (45) and (46): re iδ i = r (cid:48) e iδ iq (cid:101) Π ρω + βe iδ iβ s ω (cid:101) Π ρω αe iδ iα + s ω . (47) αe iδ iα , βe iδ iβ , and r (cid:48) e iδ iq will be calculated in the perturbative QCD approach. In order to obtain the CP violatingasymmetry in Eq. (34), A CP , sin φ and cos φ are needed. φ is determined by the CKM matrix elements. In the0Wolfenstein parametrization [76], the weak phase φ comes from [ V tb V ∗ td /V ub V ∗ ud ]. One hassin φ = η (cid:113) [ ρ (1 − ρ ) − η ] + η , (48)cos φ = ρ (1 − ρ ) − η (cid:113) [ ρ (1 − ρ ) − η ] + η , (49)where the same result has been used for b → d transition from Ref. [28, 77]. B. Calculation details
We can decompose the decay amplitudes for the decay processes ¯ B s → ρ ( ω ) K ∗ in terms of tree and penguincontributions depending on the CKM matrix elements of V ub V ∗ ud and V tb V ∗ td . From Eqs. (34), (45) and (46), in leadingorder to obtain the formulas of the CP violation, we need calculate the amplitudes t ρ , p ρ , t ω and p ω in perturbativeQCD approach. The relevant function can be found in the Appendix from the perturbative QCD approach. ¯ B s K ∗ ρ ( ω ) ( a ) ¯ B s K ∗ ρ ( ω ) ( b ) ¯ B s K ∗ ρ ( ω ) ( c ) ¯ B s K ∗ ρ ( ω ) ( d ) ¯ B s K ∗ ρ ( ω ) ( e ) ¯ B s K ∗ ρ ( ω ) ( f ) ¯ B s K ∗ ρ ( ω ) ( g ) ¯ B s K ∗ ρ ( ω ) ( h ) FIG. 2: Leading order Feynman diagrams for ¯ B s → ρ ( ω ) K ∗ In the pQCD, there are eight types of the leading order Feynman diagrams contributing to ¯ B s → ρ ( ω ) K ∗ decays,which are shown in Fig.2. The first row is for the emission-type diagrams, where the first two diagrams in Fig.2 (a)(b)are called factorizable emission diagrams and the last two diagrams in Fig.2 (c)(d) are called non-factorizable emissiondiagrams [67, 78]. The second row is for the annihilation-type diagrams, where the first two diagrams in Fig.2 (e)(f)are called factorizable annihilation diagrams and the last two diagrams in Fig.2 (g)(h) are called non-factorizableannihilation diagrams [61, 79]. The relevant decay amplitudes can be easily obtained by these hard gluon exchangediagrams and the Lorenz structures of the mesons wave functions. Through calculating these diagrams, the formulasof ¯ B s → ρK ∗ or ¯ B s → ωK ∗ are similar to those of B → φK ∗ and B s → K ∗− K ∗ + [78, 80]. We just need to replacesome corresponding Wilson coefficients, wave functions and corresponding parameters.With the Hamiltonian equation (1), depending on CKM matrix elements of V ub V ∗ ud and V tb V ∗ td , the tree dominantdecay amplitudes A ( i ) for ¯ B s → ρK ∗ in pQCD can be written as √ A ( i ) ( ¯ B s → ρ K ∗ ) = V ub V ∗ ud T iρ − V tb V ∗ td P iρ , (50)1where the superscript i denote different helicity amplitudes L, N and T . The longitudinal t ρ ( ω ) , transverse t ± ρ ( ω ) ofhelicity amplitudes satisfy relationship from Eq. (27). The amplitudes of the tree and penguin diagrams can bewritten as T iρ = t iρ /V ub V ∗ ud and P iρ = p iρ /V tb V ∗ td , respectively. The formula for the tree level amplitude is T iρ = G F √ (cid:110) f ρ F LL,iB s → K ∗ [ a ] + M LL,iB s → K ∗ [ C ] (cid:111) , (51)where f ρ refers to the decay constant of ρ meson. The penguin level amplitude are expressed in the following P iρ = − G F √ (cid:110) f ρ F LL,iB s → K ∗ (cid:20) − a + 32 a + 32 a + 12 a (cid:21) − M LR,iB s → K ∗ (cid:20) − C + 12 C (cid:21) + M LL,iB s → K ∗ (cid:20) − C + 12 C + 32 C (cid:21) − M SP,iB s → K ∗ (cid:20) C (cid:21) + f B s F LL,iann (cid:20) − a + 12 a (cid:21) − f B s F SP,iann (cid:20) − a + 12 a (cid:21) + M LL,iann (cid:20) − C + 12 C (cid:21) − M LR,iann (cid:20) − C + 12 C (cid:21) (cid:27) . (52)The tree dominant decay amplitude for ¯ B s → ωK ∗ can be written as √ A i ( ¯ B s → ωK ∗ ) = V ub V ∗ ud T iω − V tb V ∗ td P iω , (53)where T iω = t iω /V ub V ∗ us and P iω = p iω /V tb V ∗ ts which refer to the tree and penguin amplitude, respectively. We can givethe tree level contribution in the following T iω = G F √ (cid:110) f ω F LL,iB s → K ∗ [ a ] + M LL,iB s → K ∗ [ C ] (cid:111) , (54)where f ω refers to the decay constant of ω meson. The penguin level contribution are given as following P iω = G F √ (cid:26) f ω F LL,iB s → K ∗ (cid:20) a + a + 2 a + 12 a + 12 a − a (cid:21) − M LR,iB s → K ∗ (cid:20) C − C (cid:21) + M LL,iB s → K ∗ (cid:20) C + 2 C − C + 12 C (cid:21) − M SP,iB s → K ∗ (cid:20) C + 12 C (cid:21) + f B s F LL,iann (cid:20) a − a (cid:21) − f B s F SP,iann (cid:20) a − a (cid:21) + M LL,iann (cid:20) C − C (cid:21) − M LR,iann (cid:20) C − C (cid:21) (cid:27) (55)Based on the definition of Eq. (46), we can get αe iδ iα = t iω t iρ , (56) βe iδ iβ = p iρ p iω , (57) r (cid:48) e iδ iq = P iω T iρ × (cid:12)(cid:12)(cid:12)(cid:12) V tb V ∗ td V ub V ∗ ud (cid:12)(cid:12)(cid:12)(cid:12) , (58)where (cid:12)(cid:12)(cid:12)(cid:12) V tb V ∗ td V ub V ∗ ud (cid:12)(cid:12)(cid:12)(cid:12) = (cid:113) [ ρ (1 − ρ ) − η ] + η (1 − λ /
2) ( ρ + η ) . (59)2From above equations, the new strong phases δ iα , δ iβ and δ iq are obtained from tree and penguin diagram contributionsby the ρ − ω interference. Substituting Eqs. (56), (57) and (58) into (47), we can obtain total strong phase δ i in theframework of pQCD. Then in combination with Eqs. (48) and (49) the CP violating asymmetry can be obtained. IV. BRANCHING RATIO OF ¯ B s → ρ ( ω ) K ∗ Based on the relationship of Eqs. (24) and (28), we can calculate the decay rates for the processes of ¯ B s → ρ ( ω ) K ∗ by using the following expression: Γ = P c πM B s ( | H | + | H + | + | H − | ) , (60)where P c = (cid:112) [ M B s − ( M ρ/ω + M K ∗ ) ][ M B s − ( M ρ/ω − M K ∗ ) ]2 M B s (61)is the c.m. momentum of the product particle and H i ( i = 0 , + , − ) are helicity amplitudes.In this case we take into account the ρ − ω mixing contribution to the branching ratio, since we are working to thefirst order of isospin violation. The derivation is straightforward and we can explicitly express the branching ratio forthe processes ¯ B s → ρ ( ω ) K ∗ [28, 81]: BR ( ¯ B s → ρ ( ω ) K ∗ ) = τ B s P c πM B s ( | H ρω | + | H ρω + | + | H ρω − | ) , (62)where τ B s is the lifetime of the B s meson and H ρωi ( i =0 , + , − ) = ( | V ub V ∗ ud | T iρ − | V tb V ∗ td | P iρ ) + ( | V ub V ∗ ud | T iω − | V tb V ∗ td | P iω ) (cid:101) Π ρω ( s ρ − M ω ) + i M ω Γ ω (63)take into account the helicity amplitudes of the ρ meson and ω meson contribution involved in the tree and penguindiagrams. V. INPUT PARAMETERS
The CKM matrix, which elements are determined from experiments, can be expressed in terms of the Wolfensteinparameters A , ρ , λ and η [76, 82]: − λ λ Aλ ( ρ − i η ) − λ − λ Aλ Aλ (1 − ρ − i η ) − Aλ , (64)3 TABLE I: Input parameters
Parameters Input data ReferencesFermi constant (in GeV − ) G F = 1 . × − . [83] M B s = 5366 . , τ B s = 1 . × − s,M ρ (770) = 775 . , Γ ρ (770) = 149 . , Masses and decay widths (in MeV) M ω (782) = 782 . , Γ ω (782) = 8 . , [83] M π = 139 . , M K ∗ = 895 . .f ρ = 215 . ± . , f Tρ = 165 ± , Decay constants (in MeV) f ω = 196 . ± . , f Tω = 145 ± , [64, 84, 85] f K ∗ = 217 ± , f TK ∗ = 185 ± . where O ( λ ) corrections are neglected. The latest values for the parameters in the CKM matrix are [83]: λ = 0 . ± . , A = 0 . +0 . − . , ¯ ρ = 0 . +0 . − . , ¯ η = 0 . ± . , (65)where ¯ ρ = ρ (1 − λ , ¯ η = η (1 − λ . (66)From Eqs. (65) and (66) we have 0 . < ρ < . , . < η < . . (67) VI. THE NUMERICAL RESULTS OF CP VIOLATION AND BRANCHING RATIOA. CP violation via ρ − ω mixing in ¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ We have investigated the CP violating asymmetry, A CP , for the ¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ of the three-bodydecay process in the perturbative QCD. The numerical results of the CP violating asymmetry are shown for the¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ decay process in Fig. 3. It is found that the CP violation can be enhanced via ρ − ω mixing for the decay channel ¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ when the invariant mass of π + π − pair is in the vicinityof the m ω resonance within perturbative QCD scheme.The CP violating asymmetry depends on the weak phase difference φ from CKM matrix elements and the strongphase difference δ in the Eq. (34). The CKM matrix elements, which relate to ¯ ρ , A , ¯ η and λ , are given in Eq. (65).The uncertainties due to the CKM matrix elements are mostly from ρ and η since λ is well determined. Hence wetake the central value of λ = 0 .
226 in Eq. (67). In the numerical calculations for the ¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ ρ , η and λ = 0 .
226 vary among the limiting values. The numerical results are shown from Fig. 3with the different parameter values of CKM matrix elements. The solid line, dot line and dash line corresponds tothe maximum, middle, and minimum CKM matrix element for the decay channel of ¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ ,respectively. We find the numberical results of the CP violation is not sensitive to the CKM matrix elements forthe different values of ρ and η . In Fig. 3, we show the plot of CP violation as a function of √ s in the perturbativeQCD. From the figure, one can see the CP violation parameter is dependent on √ s and changes rapidly by the ρ − ω mixing mechanism when the invariant mass of π + π − pair is in the vicinity of the m ω resonance. From the numericalresults, it is found that the CP violating asymmetry is large and ranges from -50 .
19% to 43 .
02% via the ρ − ω mixingmechanism for the process. The maximum CP violating parameter can reach -48 . +1 . − . % for the decay channel of¯ B s → π + π − K ∗ in the case of ( ρ , η ). This error corresponds to the CKM parameters. - - FIG. 3: The CP violation, A cp , as a function of √ s for different CKM matrix elements. The solid line, dot line and dashline corresponds to the maximum, middle, and minimum CKM matrix element for the decay channel of ¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ , respectively. From Eq. (34), one can find that the CP violating parameter is related to r and sin δ . In Fig. 4 and Fig. 5, weshow the plots of sin δ (sin δ + and sin δ − ) and r ( r + and r − ) as a function of √ s , respectively. We can see that the ρ − ω mixing mechanism produces a large sin δ (sin δ + and sin δ − ) in the vicinity of the ω resonance. As can be seen - - FIG. 4: Plot of sin δ as a function of √ s corresponding to central parameter values of CKM matrix elements for ¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ . The solid line, dot line and dash line corresponds to sin δ , sin δ + and sin δ − , respectively. FIG. 5: Plot of r as a function of √ s corresponding to central parameter values of CKM matrix elements for ¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ . The solid line, dot line and dash line corresponds to r , r + and r − , respectively. × - × - × - × - FIG. 6: The branching ratio, BR ( ¯ B s → ρK ∗ ), as a function of √ s for different CKM matrix elements. The solid line, dot lineand dash line corresponds to the maximum, middle, and minimum CKM matrix element for the decay channel of ¯ B s → ρ K ∗ ,respectively. from Fig. 4, the plots vary sharply in the cases of sin δ , sin δ + and sin δ − in the range of the resonance. Meanwhile,sin δ + and sin δ − change weakly compared with the sin δ . It can be seen from Fig. 5 that r , r + and r − change morerapidly when the π + π − pairs in the vicinity of the ω resonance.We have shown that the ρ − ω mixing does enhance the direct CP violating asymmetry and provide a mechanismfor large CP violation in the perturbative QCD factorization scheme. In other words, it is important to see whetherit is possible to observe this large CP violating asymmetry in experiments. This depends on the branching ratio forthe decay channel of ¯ B s → ρ K ∗ . We will study this problem in the next section. B. Branching ratio via ρ − ω mixing in ¯ B s → ρ ( ω ) K ∗ In the pQCD, we calculate the value of the branching ratio via ρ − ω mixing mechanism for the decay channel¯ B s → ρ ( ω ) K ∗ . The numerical result is shown for the decay process in Fig. 6. Based on a reasonable parameterrange, we obtain the maximum branching ratio of ¯ B s → ρ ( ω ) K ∗ as (3 . +0 . − . ) × − , which is consistent with theresult in [80, 86]. The error comes from CKM parameters. On the other hand, although we calculate the branching6ratio due to ρ − ω mixing in the pQCD factorization scheme, we find that the contribution of ρ − ω mixing to thebranching ratio of ¯ B s → ρ ( ω ) K ∗ is small and can be neglected. However, the ρ − ω mixing mechanism producesnew strong phase differences. This is why the Fig. 6 presents a tiny effect for the branching ratio of ¯ B s → ρ ( ω ) K ∗ when the invariant mass of π + π − pair is around 1.1 GeV.The Large Hadron Collider (LHC) is a proton-proton collider that has started at the European Organization forNuclear Research (CERN). With the designed center-of-mass energy 14 TeV and luminosity L = 10 cm − s − , theLHC provides a high energy frontier at TeV-level scale and an opportunity to further improve the consistency test forthe CKM matrix. LHCb is a dedicated heavy flavor physics experiments and one of the main projects of LHC. Its maingoal is to search for indirect evidence of new physics in CP violation and rare decays in the interactions of beauty andcharm hadrons systems, by looking for the effects of new particles in decay processes that are precisely predicted in theSM. Such studies can help us to comprehend the matter-antimatter asymmetry of the universe. Recently, the LHCbcollaboration found clear evidence for direct CP violation in some three-body decay channels of B meson. Large CP violation is obtained for the decay channels of B ± → π ± π + π − in the localized phase spaces region m π + π − low < . and m π + π − high >
15 GeV [7, 87]. A zoom of the π + π − invariant mass from the B + → π + π + π − decay processis shown the region m π + π − low < zone in the Figure 8 (zone I) of the Ref. [87]. In addition, the branchingratio of ¯ B s → π + π − φ is probed in the π + π − invariant mass range 400 < m ( π + π − ) < [88]. In thenext years, we expect the LHCb Collaboration to collect date for detecting our prediction of CP violation from the¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ decay process when the invariant mass of π + π − is in the vicinity of the ω resonance.At the LHC, the b -hadrons come from pp collisions. The possible asymmetry between the numbers of the b -hadrons H b and those of their anti-particles ¯ H b has been studied by using the intrinsic heavy quark model and the Lund stringfragmentation model [89, 90]. It has been shown that this asymmetry can only reach values of a few percents. In thefollowing discussion, we will ignore this small asymmetry and give the numbers of H b ¯ H b pairs needed for observingour prediction of the CP violating asymmetries. These numbers depend on both the magnitudes of the CP violatingasymmetries and the branching ratios of heavy hadron decays which are model dependent. For one-standard-deviation(1 σ ) signature and three-standard deviation (3 σ ) signature, the numbers of H b ¯ H b pairs we need [91–93] N H b ¯ H b (1 σ ) ∼ BR ( ¯ B s → ρ K ∗ ) A CP (1 − A CP ) (68)and N H b ¯ H b (3 σ ) ∼ BR ( ¯ B s → ρ K ∗ ) A CP (1 − A CP ) , (69)where A CP is the CP violation in the process of ¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ . Now, we can estimate the possibilityto observe CP violation. The branching ratio for ¯ B s → ρ ( ω ) K ∗ is of order 10 − , then the number N H b ¯ H b (1 σ ) ∼ for 1 σ signature and 10 for 3 σ signature. Theoretically, in order to achieve the current experiments on b -hadrons,which can only provide about 10 B s ¯ B s pairs. Therefore, it is very possible to observe the large CP violation for¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ when the invariant masses of π + π − pairs are in the vicinity of the ω resonance inexperiments at the LHC.7 VII. SUMMARY AND CONCLUSION
In this paper, we have studied the direct CP violation for the decay process of ¯ B s → ρ ( ω ) K ∗ → π + π − K ∗ inperturbative QCD. It has been found that, by using ρ − ω mixing, the CP violation can be enhanced at the areaof ω resonance. There is the resonance effect via ρ − ω mixing which can produce large strong phase in this decayprocess. As a result, one can find that the maximum CP violation can reach -50 .
19% when the invariant mass of the π + π − pair is in the vicinity of the ω resonance. Furthermore, taking ρ − ω mixing into account, we have calculatedthe branching ratio of the decays of ¯ B s → ρ ( ω ) K ∗ . We have also given the numbers of B s ¯ B s pairs required forobserving our prediction of the CP violating asymmetries at the LHC experiments.In our calculation there are some uncertainties. The major uncertainties come from the input parameters. Inparticular, these include the CKM matrix element, the particle mass, the perturbative QCD approach and the hadronicparameters (decay constants, the wave functions, the shape parameters and etc). We expect that our predictions willprovide useful guidance for future experiments. Acknowledgements
This work was supported by National Natural Science Foundation of China (Project Numbers 11605041), and theResearch Foundation of the young core teacher from Henan province.
Appendix: Related functions defined in the text
In this appendix we present explicit expressions of the factorizable and non-factorizable amplitudes in PerturbativeQCD [19–21, 59]. The factorizable amplitudes F LL,iB s → K ∗ ( a i ), F LL,iann ( a i ) and F SP,iann ( a i ) (i=L,N,T) are written as f M F LL,LB s → K ∗ ( a i ) = 8 πC F M B s f M (cid:90) dx dx (cid:90) ∞ b db b db φ B s ( x , b ) (cid:110) a i ( t a ) E e ( t a ) × (cid:2) r (1 − x )( φ s ( x ) + φ t ( x )) + (1 + x ) φ ( x ) (cid:3) h e ( x , x , b , b )+2 r φ s ( x ) a i ( t (cid:48) a ) E e ( t (cid:48) a ) h e ( x , x , b , b ) (cid:111) , (70) f M F LL,NB s → K ∗ ( a i ) = 8 πC F M B s f M r (cid:90) dx dx (cid:90) ∞ b db b db φ B s ( x , b ) (cid:110) h e ( x , x , b , b ) × E e ( t a ) a i ( t a ) (cid:2) r φ v ( x ) + r x ( φ v ( x ) − φ a ( x )) + φ T ( x ) (cid:3) + r [ φ v ( x ) + φ a ( x )] E e ( t (cid:48) a ) a i ( t (cid:48) a ) h e ( x , x , b , b ) (cid:111) , (71) f M F LL,TB s → K ∗ ( a i ) = 16 πC F M B s f M r (cid:90) dx dx (cid:90) ∞ b db b db φ B s ( x , b ) (cid:110) h e ( x , x , b , b ) × (cid:2) r φ v ( x ) − r x ( φ v ( x ) − φ a ( x )) + φ T ( x ) (cid:3) E e ( t a ) a i ( t a )+ r [ φ v ( x ) + φ a ( x )] E e ( t (cid:48) a ) a i ( t (cid:48) a ) h e ( x , x , b , b ) (cid:111) , (72)8 f B s F LL,Lann ( a i ) = 8 πC F M B s f B s (cid:90) dx dx (cid:90) ∞ b db b db (cid:110) a i ( t c ) E a ( t c )[( x − φ ( x ) φ ( x ) − r r φ s ( x ) φ s ( x ) + 2 r r x φ s ( x )( φ s ( x ) − φ t ( x ))] h a ( x , − x , b , b )+[ x φ ( x ) φ ( x ) + 2 r r ( φ s ( x ) − φ t ( x )) φ s ( x ) + 2 r r x ( φ s ( x )+ φ t ( x )) φ s ( x )] a i ( t (cid:48) c ) E a ( t (cid:48) c ) h a (1 − x , x , b , b ) (cid:111) , (73) f B s F LL,Nann ( a i ) = − πC F M B s f B s r r (cid:90) dx dx (cid:90) ∞ b db b db (cid:110) E a ( t c ) a i ( t c ) h a ( x , − x , b , b )) × [ x ( φ v ( x ) φ a ( x ) + φ a ( x ) φ v ( x )) + (2 − x ) ( φ v ( x ) φ v ( x ) + φ a ( x ) φ a ( x ))] − h a (1 − x , x , b , b )[(1 + x )( φ a ( x ) φ a ( x ) + φ v ( x ) φ v ( x )) − (1 − x )( φ a ( x ) φ v ( x )) + φ v ( x ) φ a ( x )] E a ( t (cid:48) c ) a i ( t (cid:48) c ) (cid:111) , (74) f B s F LL,Tann ( a i ) = − πC F M B s f B s r r (cid:90) dx dx (cid:90) ∞ b db b db (cid:110)(cid:104) x ( φ a ( x ) φ a ( x ) + φ v ( x ) φ v ( x ))+(2 − x )( φ a ( x ) φ v ( x ) + φ v ( x ) φ a ( x )) (cid:105) E a ( t c ) a i ( t c ) h a ( x , − x , b , b )+ h a (1 − x , x , b , b )[(1 − x )( φ a ( x ) φ a ( x ) + φ v ( x ) φ v ( x )) − (1 + x )( φ a ( x ) φ v ( x ) + φ v ( x ) φ a ( x ))] E a ( t (cid:48) c ) a i ( t (cid:48) c ) (cid:111) , (75) f B s F SP,Lann ( a i ) = 16 πC F M B s f B s (cid:90) dx dx (cid:90) ∞ b db b db (cid:110) [( x − r φ ( x )( φ s ( x ) + φ t ( x )) − r φ s ( x ) φ ( x )] a i ( t c ) E a ( t c ) h a ( x , − x , b , b ) − [2 r φ ( x ) φ s ( x )+ r x ( φ s ( x ) − φ t ( x )) φ ( x )] a i ( t (cid:48) c ) E a ( t (cid:48) c ) h a (1 − x , x , b , b ) (cid:111) , (76) f B s F SP,Tann ( a i ) = 2 f B s F SP,Nann ( a i )= − πC F M B s f B s (cid:90) dx dx (cid:90) ∞ b db b db (cid:110) r ( φ a ( x ) + φ v ( x )) φ T ( x ) × E a ( t c ) a i ( t c ) h a ( x , − x , b , b ) + r φ T ( x ) × ( φ v ( x ) − φ a ( x )) E a ( t (cid:48) c ) a i ( t (cid:48) c ) h a (1 − x , x , b , b ) (cid:111) , (77)with the color factor C F = 3 / a i represents the corresponding Wilson coefficients for specific decay channels and f M , f B s refer to the decay constants of M ( ρ or ω ) and ¯ B s mesons. In the above functions, r ( r ) = M V ( M V ) /M B s and φ ( φ ) = φ ρ/ω ( φ K ∗ ), with M B s and M V ( m V ) being the masses of the initial and final states.The non-factorizable amplitudes M LL,iB s → K ∗ ( a i ), M LR,iB s → K ∗ ( a i ), M SP,iB s → K ∗ ( a i ), M LL,iann ( a i ) and M LR,iann ( a i )(i=L,N,T) are9written as M LL,LB s → K ∗ ( a i ) = 32 πC F M B s / √ (cid:90) dx dx dx (cid:90) ∞ b db b db φ B s ( x , b ) φ ( x ) × (cid:110)(cid:104) (1 − x ) φ ( x ) − r x ( φ s ( x ) − φ t ( x )) (cid:105) a i ( t b ) E (cid:48) e ( t b ) × h n ( x , − x , x , b , b ) + h n ( x , x , x , b , b ) × (cid:104) r x ( φ s ( x ) + φ t ( x )) − ( x + x ) φ ( x ) (cid:105) a i ( t (cid:48) b ) E (cid:48) e ( t (cid:48) b ) (cid:111) , (78) M LL,NB s → K ∗ ( a i ) = 32 πC F M B s r / √ (cid:90) dx dx dx (cid:90) ∞ b db b db φ B s ( x , b ) × (cid:110) (cid:2) x ( φ v ( x ) + φ a ( x )) φ T ( x ) − r ( x + x )( φ a ( x ) φ a ( x ) + φ v ( x ) φ v ( x )) (cid:3) × h n ( x , x , x , b , b ) E (cid:48) e ( t (cid:48) b ) a i ( t (cid:48) b ) + (1 − x )( φ v ( x ) + φ a ( x )) φ T ( x ) × E (cid:48) e ( t b ) a i ( t b ) h n ( x , − x , x , b , b ) (cid:111) , (79) M LL,TB s → K ∗ ( a i ) = 64 πC F M B s r / √ (cid:90) dx dx dx (cid:90) ∞ b db b db φ B s ( x , b ) (cid:110) E (cid:48) e ( t (cid:48) b ) a i ( t (cid:48) b ) × (cid:2) x ( φ v ( x ) + φ a ( x )) φ T ( x ) − r ( x + x )( φ v ( x ) φ a ( x )+ φ a ( x ) φ v ( x )) (cid:3) h n ( x , x , x , b , b ) + (1 − x )[ φ v ( x ) + φ a ( x )] φ T ( x ) × E (cid:48) e ( t b ) a i ( t b ) h n ( x , − x , x , b , b ) (cid:111) , (80) M LR,LB s → K ∗ ( a i ) = 32 πC F M B s r / √ (cid:90) dx dx dx (cid:90) ∞ b db b db φ B s ( x , b ) × (cid:110) h n ( x , − x , x , b , b ) (cid:104) (1 − x ) φ ( x ) (cid:0) φ s ( x ) + φ t ( x ) (cid:1) + r x (cid:0) φ s ( x ) − φ t ( x ) (cid:1) (cid:0) φ s ( x ) + φ t ( x ) (cid:1) +(1 − x ) r (cid:0) φ s ( x ) + φ t ( x ) (cid:1) (cid:0) φ s ( x ) − φ t ( x ) (cid:1) (cid:105) a i ( t b ) E (cid:48) e ( t b ) − h n ( x , x , x , b , b ) (cid:104) x φ ( x )( φ s ( x ) − φ t ( x ))+ r x ( φ s ( x ) − φ t ( x ))( φ s ( x ) − φ t ( x ))+ r x ( φ s ( x ) + φ t ( x ))( φ s ( x ) + φ t ( x )) (cid:105) a i ( t (cid:48) b ) E (cid:48) e ( t (cid:48) b ) (cid:111) , (81) M LR,TB s → K ∗ ( a i ) = 2 M LR,NB s → K ∗ ( a i )= 64 πC F M B s / √ (cid:90) dx dx dx (cid:90) ∞ b db b db φ B s ( x , b ) × r x φ T ( x )( φ v ( x ) − φ a ( x )) × (cid:110) E (cid:48) e ( t b ) a i ( t b ) h n ( x , − x , x , b , b ) + E (cid:48) e ( t (cid:48) b ) a i ( t (cid:48) b ) h n ( x , x , x , b , b ) (cid:111) , (82)0 M SP,LB s → K ∗ ( a i ) = 32 πC F M B s / √ (cid:90) dx dx dx (cid:90) ∞ b db b db φ B s ( x , b ) φ ( x ) × (cid:110)(cid:104) ( x − x − φ ( x ) + r x ( φ s ( x ) + φ t ( x )) (cid:105) × a i ( t b ) E (cid:48) e ( t b ) h n ( x , − x , x , b , b ) + a i ( t (cid:48) b ) E (cid:48) e ( t (cid:48) b ) × (cid:104) x φ ( x ) + r x ( φ t ( x ) − φ s ( x )) (cid:105) h n ( x , x , x , b , b ) (cid:111) , (83) M SP,NB s → K ∗ ( a i ) = 32 πC F M B s / √ (cid:90) dx dx dx (cid:90) ∞ b db b db φ B s ( x , b ) r × (cid:110) x ( φ v ( x ) − φ a ( x )) φ T ( x ) E (cid:48) e ( t (cid:48) b ) a i ( t (cid:48) b ) h n ( x , x , x , b , b )+ h n ( x , − x , x , b , b )[(1 − x )( φ v ( x ) − φ a ( x )) φ T ( x ) − r (1 − x + x )( φ v ( x ) φ v ( x ) − φ a ( x ) φ a ( x ))] E (cid:48) e ( t b ) a i ( t b ) (cid:111) , (84) M SP,TB s → K ∗ ( a i ) = 64 πC F M B s / √ (cid:90) dx dx dx (cid:90) ∞ b db b db φ B s ( x , b ) r × (cid:110) x ( φ v ( x ) − φ a ( x )) φ T ( x ) E (cid:48) e ( t (cid:48) b ) a i ( t (cid:48) b ) h n ( x , x , x , b , b )+ h n ( x , − x , x , b , b )[(1 − x )( φ v ( x ) − φ a ( x )) φ T ( x ) − r (1 − x + x )( φ v ( x ) φ a ( x ) − φ a ( x ) φ v ( x ))] E (cid:48) e ( t b ) a i ( t b ) (cid:111) , (85) M LL,Lann ( a i ) = 32 πC F M B s / √ (cid:90) dx dx dx (cid:90) ∞ b db b db φ B s ( x , b ) (cid:110) h na ( x , x , x , b , b ) × (cid:104) r r x ( φ s ( x ) − φ t ( x ))( φ s ( x ) + φ t ( x )) − ( x φ ( x ) φ ( x ) + 4 r r φ s ( x ) φ s ( x ))+ r r (1 − x )( φ s ( x ) + φ t ( x ))( φ s ( x ) − φ t ( x )) (cid:105) a i ( t d ) E (cid:48) a ( t d ) + h (cid:48) na ( x , x , x , b , b ) × (cid:104) (1 − x ) φ ( x ) φ ( x ) + (1 − x ) r r ( φ s ( x ) + φ t ( x ))( φ s ( x ) − φ t ( x ))+ x r r ( φ s ( x ) − φ t ( x ))( φ s ( x ) + φ t ( x )) (cid:105) a i ( t (cid:48) d ) E (cid:48) a ( t (cid:48) d ) (cid:111) , (86) M LL,Nann ( a i ) = − πC F M B s r r / √ (cid:90) dx dx dx (cid:90) ∞ b db b db φ B s ( x , b )[ φ v ( x ) φ v ( x )+ φ a ( x ) φ a ( x )] E (cid:48) a ( t d ) a i ( t d ) h na ( x , x , x , b , b ) , (87) M LL,Tann ( a i ) = − πC F M B s r r / √ (cid:90) dx dx dx (cid:90) ∞ b db b db φ B s ( x , b )[ φ v ( x ) φ a ( x )+ φ a ( x ) φ v ( x )] E (cid:48) a ( t d ) a i ( t d ) h na ( x , x , x , b , b ) , (88)1 M LR,Lann ( a i ) = 32 πC F M B s / √ (cid:90) dx dx dx (cid:90) ∞ b db b db φ B s ( x , b ) (cid:110) h na ( x , x , x , b , b ) × (cid:104) r ( x − φ s ( x ) + φ t ( x )) φ ( x ) + r (1 + x ) φ ( x )( φ s ( x ) − φ t ( x )) (cid:105) a i ( t d ) E (cid:48) a ( t d )+ h (cid:48) na ( x , x , x , b , b ) (cid:104) − r x (cid:0) φ s ( x ) + φ t ( x ) (cid:1) φ ( x )+ r (1 − x ) φ ( x )( φ s ( x ) − φ t ( x )) (cid:105) a i ( t (cid:48) d ) E (cid:48) a ( t (cid:48) d ) (cid:111) , (89) M LR,Tann ( a i ) = 2 M LR,Nann ( a i )= − πC F M B s / √ (cid:90) dx dx dx (cid:90) ∞ b db b db φ B s ( x , b ) (cid:110) h (cid:48) na ( x , x , x , b , b ) × (cid:2) r x ( φ v ( x ) + φ a ( x )) φ T ( x ) − r (1 − x ) φ T ( x )( φ v ( x ) − φ a ( x )) (cid:3) E (cid:48) a ( t (cid:48) d ) a i ( t (cid:48) d )+ (cid:2) r (2 − x )( φ v ( x ) + φ a ( x )) φ T ( x ) − r (1 + x ) φ T ( x )( φ v ( x ) − φ a ( x )) (cid:3) × E (cid:48) a ( t d ) a i ( t d ) h na ( x , x , x , b , b ) (cid:111) . (90)The hard scale t are chosen as the maximum of the virtuality of the internal momentum transition in the hardamplitudes, including 1 /b i : t a = max {√ x M B s , /b , /b } , (91) t (cid:48) a = max {√ x M B s , /b , /b } , (92) t b = max {√ x x M B s , (cid:112) | − x − x | x M B s , /b , /b } , (93) t (cid:48) b = max {√ x x M B s , (cid:112) | x − x | x M B s , /b , /b } , (94) t c = max {√ − x M B s , /b , /b } , (95) t (cid:48) c = max {√ x M B s , /b , /b } , (96) t d = max { (cid:112) x (1 − x ) M B s , (cid:112) − (1 − x − x ) x M B s , /b , /b } , (97) t (cid:48) d = max { (cid:112) x (1 − x ) M B s , (cid:112) | x − x | (1 − x ) M B s , /b , /b } . (98)The function h, coming from the Fourier transform of hard part H, are written as h e ( x , x , b , b ) = (cid:2) θ ( b − b ) I ( √ x M B s b ) K ( √ x M B s b )+ θ ( b − b ) I ( √ x M B s b ) K ( √ x M B s b ) (cid:3) K ( √ x x M B s b ) S t ( x ) , (99) h n ( x , x , x , b , b ) = [ θ ( b − b ) K ( √ x x M B s b ) I ( √ x x M B s b )+ θ ( b − b ) K ( √ x x M B s b ) I ( √ x x M B s b )] × (cid:40) iπ H (1)0 ( (cid:112) ( x − x ) x M B s b ) , x − x < K ( (cid:112) ( x − x ) x M B s b ) , x − x > , (100)2 h a ( x , x , b , b ) = ( iπ S t ( x ) (cid:104) θ ( b − b ) H (1)0 ( √ x M B s b ) J ( √ x M B s b )+ θ ( b − b ) H (1)0 ( √ x M B s b ) J ( √ x M B s b ) (cid:105) H (1)0 ( √ x x M B s b ) , (101) h na ( x , x , x , b , b ) = iπ (cid:104) θ ( b − b ) H (1)0 ( (cid:112) x (1 − x ) M B s b ) J ( (cid:112) x (1 − x ) M B s b )+ θ ( b − b ) H (1)0 ( (cid:112) x (1 − x ) M B s b ) J ( (cid:112) x (1 − x ) M B s b ) (cid:105) × K ( (cid:112) − (1 − x − x ) x M B s b ) , (102) h (cid:48) na ( x , x , x , b , b ) = iπ (cid:104) θ ( b − b ) H (1)0 ( (cid:112) x (1 − x ) M B s b ) J ( (cid:112) x (1 − x ) M B s b )+ θ ( b − b ) H (1)0 ( (cid:112) x (1 − x ) M B s b ) J ( (cid:112) x (1 − x ) M B s b ) (cid:105) × (cid:40) iπ H (1)0 ( (cid:112) ( x − x )(1 − x ) M B s b ) , x − x < K ( (cid:112) ( x − x )(1 − x ) M B s b ) , x − x > , (103)where J and Y are the Bessel function with H (1)0 ( z ) = J ( z ) + i Y ( z ).The threshold re-sums factor S t follows the parameterized S t ( x ) = 2 c Γ(3 / c ) √ π Γ(1 + c ) [ x (1 − x )] c , (104)where the parameter c = 0 . E ( (cid:48) ) e and E ( (cid:48) ) a entering in the expressions for the matrix elements are given by E e ( t ) = α s ( t ) exp[ − S B ( t ) − S M ( t )] , E (cid:48) e ( t ) = α s ( t ) exp[ − S B ( t ) − S M ( t ) − S M ( t )] | b = b , (105) E a ( t ) = α s ( t ) exp[ − S M ( t ) − S M ( t )] , E (cid:48) a ( t ) = α s ( t ) exp[ − S B ( t ) − S M ( t ) − S M ( t )] | b = b , (106)in which the Sudakov exponents are defined as S B ( t ) = s (cid:18) x M B s √ , b (cid:19) + 53 (cid:90) t /b d ¯ µ ¯ µ γ q ( α s (¯ µ )) , (107) S M ( t ) = s (cid:18) x M B s √ , b (cid:19) + s (cid:18) (1 − x ) M B s √ , b (cid:19) + 2 (cid:90) t /b d ¯ µ ¯ µ γ q ( α s (¯ µ )) , (108) S M ( t ) = s (cid:18) x M B s √ , b (cid:19) + s (cid:18) (1 − x ) M B s √ , b (cid:19) + 2 (cid:90) t /b d ¯ µ ¯ µ γ q ( α s (¯ µ )) , (109)3where γ q = − α s /π is the anomalous dimension of the quark. The explicit form for the function s ( Q, b ) is: s ( Q, b ) = A (1) β ˆ q ln (cid:18) ˆ q ˆ b (cid:19) − A (1) β (cid:16) ˆ q − ˆ b (cid:17) + A (2) β (cid:18) ˆ q ˆ b − (cid:19) − (cid:20) A (2) β − A (1) β ln (cid:18) e γ E − (cid:19)(cid:21) ln (cid:18) ˆ q ˆ b (cid:19) + A (1) β β ˆ q (cid:34) ln(2ˆ q ) + 1ˆ q − ln(2ˆ b ) + 1ˆ b (cid:35) + A (1) β β (cid:104) ln (2ˆ q ) − ln (2ˆ b ) (cid:105) , (110)where the variables are defined by ˆ q ≡ ln[ Q/ ( √ , ˆ b ≡ ln[1 / ( b Λ)] , (111)and the coefficients A ( i ) and β i are β = 33 − n f , β = 153 − n f ,A (1) = 43 , A (2) = 679 − π − n f + 83 β ln( 12 e γ E ) , (112)with n f is the number of the quark flavors and γ E is the Euler constant.4 [1] J. H. Christenson, J. W. Cronin, V. L. Fitch and R. Turlay, Phys. Rev. Lett. , 138-140 (1964).[2] N. Cabibbo, Phys. Rev. Lett. , 531-533 (1963).[3] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. , 652-657 (1973).[4] A. B. Carter and A. I. Sanda, Phys. Rev. Lett. , 952 (1980).[5] A. B. Carter and A. I. Sanda, Phys. Rev. D , 1567 (1981).[6] I. I. Y. Bigi and A. I. Sanda, Nucl. Phys. B , 85-108 (1981).[7] R. Aaij et al. [LHCb], Phys. Rev. Lett. , no.1, 011801 (2014).[8] R. Aaij et al. [LHCb], Phys. Rev. D , no.11, 112004 (2014).[9] I. Bediaga and C. G¨obel, Prog. Part. Nucl. Phys. , 103808 (2020).[10] D. Fakirov and B. Stech, Nucl. Phys. B , 315-326 (1978).[11] N. Cabibbo and L. Maiani, Phys. Lett. B , 418 (1978).[12] M. Wirbel, B. Stech and M. Bauer, Z. Phys. C , 637 (1985).[13] M. Bauer, B. Stech and M. Wirbel, Z. Phys. C , 103 (1987).[14] M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Phys. Rev. Lett. , 1914-1917 (1999).[15] M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Nucl. Phys. B , 313-418 (2000).[16] C. T. Sachrajda, Acta Phys. Polon. B , 1821-1834 (2001).[17] M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Nucl. Phys. B , 245-321 (2001).[18] M. Beneke and M. Neubert, Nucl. Phys. B , 333-415 (2003).[19] C. D. Lu, K. Ukai and M. Z. Yang, Phys. Rev. D , 074009 (2001).[20] Y. Y. Keum, H. n. Li and A. I. Sanda, Phys. Lett. B , 6-14 (2001).[21] Y. Y. Keum, H. N. Li and A. I. Sanda, Phys. Rev. D , 054008 (2001).[22] C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart, Phys. Rev. D , 114020 (2001).[23] C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. Lett. , 201806 (2001).[24] C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. D , 054022 (2002).[25] R. Enomoto and M. Tanabashi, Phys. Lett. B , 413-421 (1996).[26] X. H. Guo and A. W. Thomas, Phys. Rev. D , 096013 (1998).[27] X. H. Guo, O. M. A. Leitner and A. W. Thomas, Phys. Rev. D , 056012 (2001).[28] O. M. A. Leitner, X. H. Guo and A. W. Thomas, Eur. Phys. J. C , 215-226 (2003).[29] G. Lu, B. H. Yuan and K. W. Wei, Phys. Rev. D , 014002 (2011).[30] G. Lu, Z. H. Zhang, X. Y. Liu and L. Y. Zhang, Int. J. Mod. Phys. A , 2899-2912 (2011).[31] G. L¨u, W. L. Zou, Z. H. Zhang and M. H. Weng, Phys. Rev. D , no.7, 074005 (2013).[32] G. L¨u, S. T. Li and Y. T. Wang, Phys. Rev. D , no.3, 034040 (2016).[33] S. Kr¨ankl, T. Mannel and J. Virto, Nucl. Phys. B , 247-264 (2015).[34] W. F. Wang, H. C. Hu, H. n. Li and C. D. L¨u, Phys. Rev. D , no.7, 074031 (2014).[35] C. H. Chen and H. n. Li, Phys. Lett. B , 258-265 (2003).[36] J. J. Qi, Z. Y. Wang, X. H. Guo, Z. H. Zhang and C. Wang, Phys. Rev. D , no.7, 076010 (2019).[37] H. Fritzsch and A. S. Muller, Nucl. Phys. B Proc. Suppl. , 273-276 (2001).[38] H. Fritzsch, arXiv:hep-ph/0106273.[39] H. B. O’Connell, B. C. Pearce, A. W. Thomas and A. G. Williams, Prog. Part. Nucl. Phys. , 201-252 (1997).[40] A. Bernicha, G. Lopez Castro and J. Pestieau, Phys. Rev. D , 4454-4461 (1994). [41] K. Maltman, H. B. O’Connell and A. G. Williams, Phys. Lett. B , 19-24 (1996).[42] H. B. O’Connell, A. W. Thomas and A. G. Williams, Nucl. Phys. A , 559-569 (1997).[43] H. B. O’Connell, Austral. J. Phys. , 255-262 (1997).[44] C. E. Wolfe and K. Maltman, Phys. Rev. D , 114024 (2009).[45] C. E. Wolfe and K. Maltman, Phys. Rev. D , 077301 (2011).[46] S. Gardner, H. B. O’Connell and A. W. Thomas, Phys. Rev. Lett. , 1834-1837 (1998).[47] X. H. Guo and A. W. Thomas, Phys. Rev. D , 116009 (2000).[48] G. L¨u, J. Q. Lei, X. H. Guo, Z. H. Zhang and K. W. Wei, Adv. High Energy Phys. , 785648 (2014).[49] G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. , 1125-1144 (1996).[50] A. J. Buras, arXiv:hep-ph/9806471.[51] A. Ali, G. Kramer and C. D. Lu, Phys. Rev. D , 094009 (1998).[52] A. Ali, G. Kramer and C. D. Lu, Phys. Rev. D , 014005 (1999).[53] Y. Y. Keum and H. n. Li, Phys. Rev. D , 074006 (2001).[54] C. D. Lu and M. Z. Yang, Eur. Phys. J. C , 275-287 (2002).[55] C. H. V. Chang and H. n. Li, Phys. Rev. D , 5577-5580 (1997).[56] T. W. Yeh and H. n. Li, Phys. Rev. D , 1615-1631 (1997).[57] G. P. Lepage and S. J. Brodsky, Phys. Rev. D , 2157 (1980).[58] J. Botts and G. F. Sterman, Nucl. Phys. B , 62-100 (1989).[59] A. Ali, G. Kramer, Y. Li, C. D. Lu, Y. L. Shen, W. Wang and Y. M. Wang, Phys. Rev. D , 074018 (2007).[60] Y. Li, C. D. Lu, Z. J. Xiao and X. Q. Yu, Phys. Rev. D , 034009 (2004).[61] J. J. Wang, D. T. Lin, W. Sun, Z. J. Ji, S. Cheng and Z. J. Xiao, Phys. Rev. D , no.7, 074046 (2014).[62] H. n. Li, Prog. Part. Nucl. Phys. , 85-171 (2003).[63] P. Ball and V. M. Braun, Nucl. Phys. B , 201-238 (1999).[64] P. Ball and R. Zwicky, Phys. Rev. D , 014029 (2005).[65] Y. Y. Fan, W. F. Wang, S. Cheng and Z. J. Xiao, Phys. Rev. D , no.9, 094003 (2013).[66] H. W. Huang, C. D. Lu, T. Morii, Y. L. Shen, G. Song and Jin-Zhu, Phys. Rev. D , 014011 (2006).[67] J. Zhu, Y. L. Shen and C. D. Lu, J. Phys. G , 101-110 (2006).[68] C. D. Lu, Y. l. Shen and J. Zhu, Eur. Phys. J. C , 311-317 (2005).[69] H. n. Li and S. Mishima, Phys. Rev. D , 054025 (2005).[70] Y. Nambu, Phys. Rev. , 1366-1367 (1957).[71] J. J. Sakurai, Conf. Proc. C , 91-104 (1969).[72] S. Gardner, H. B. O’Connell and A. W. Thomas, AIP Conf. Proc. , no.1, 383-386 (1997).[73] X. H. Guo, G. Lu and Z. H. Zhang, Eur. Phys. J. C , 223-244 (2008).[74] G. L¨u, Y. T. Wang and Q. Q. Zhi, Phys. Rev. D , no.1, 013004 (2018).[75] R. Enomoto and M. Tanabashi, arXiv:hep-ph/9706340.[76] L. Wolfenstein, Phys. Rev. Lett. , 562-564 (1964).[77] Z. J. Ajaltouni, O. M. A. Leitner, P. Perret, C. Rimbault and A. W. Thomas, Eur. Phys. J. C , 215-233 (2003).[78] C. H. Chen, Y. Y. Keum and H. n. Li, Phys. Rev. D , 054013 (2002).[79] Y. L. Shen, W. Wang, J. Zhu and C. D. Lu, Eur. Phys. J. C , 877-887 (2007).[80] Z. T. Zou, A. Ali, C. D. Lu, X. Liu and Y. Li, Phys. Rev. D , 054033 (2015).[81] G. L¨u, Z. H. Zhang, X. H. Guo, J. C. Lu and S. M. Yan, Eur. Phys. J. C , no.8, 2519 (2013).[82] L. Wolfenstein, Phys. Rev. Lett. , 1945 (1983).[83] P. A. Zyla et al. [Particle Data Group], PTEP , no.8, 083C01 (2020). [84] A. Bharucha, D. M. Straub and R. Zwicky, JHEP , 098 (2016).[85] X. Liu, Z. J. Xiao and Z. T. Zou, Phys. Rev. D , no.11, 113005 (2016).[86] D. C. Yan, X. Liu and Z. J. Xiao, Nucl. Phys. B , 17-39 (2018).[87] A. C. dos Reis [LHCb], J. Phys. Conf. Ser. , no.4, 042001 (2016).[88] R. Aaij et al. [LHCb], Phys. Rev. D , no.1, 012006 (2017).[89] E. Norrbin, arXiv:hep-ph/9909437.[90] G. Altarelli and M. L. Mangano, “1999 CERN Workshop on standard model physics (and more) at the LHC, CERN,Geneva, Switzerland, 25-26 May: Proceedings,” CERN Yellow Reports: Conference Proceedings (2000).[91] D. s. Du, Phys. Rev. D34