Study of the Υ(1S) {\to} DP decays
Yueling Yang, Mingfei Duan, Junliang Lu, Jinshu Huang, Junfeng Sun
aa r X i v : . [ h e p - ph ] F e b Study of the
Υ(1 S ) → DP decays Yueling Yang, Mingfei Duan, Junliang Lu, Jinshu Huang, and Junfeng Sun Institute of Particle and Nuclear Physics,Henan Normal University, Xinxiang 453007, China School of Physics and Electronic Engineering,Nanyang Normal University, Nanyang 473061, China
Abstract
Inspired by the potential prospects of high-luminosity dedicated colliders and the high enthu-siasms in searching for new physics in the flavor sector at the intensity frontier, the Υ(1 S ) → D − π + , D π and D − s K + weak decays are studied with the perturbative QCD approach. It isfound within the standard model that the branching ratios for the concerned processes are tiny,about O (10 − ), and far beyond the detective ability of current experiments unless there existssome significant enhancements from a novel interaction. b quark weak decays are ideal places to explore NP effects, because atthe intensity frontier, there will be more than 10 b ¯ b pairs with 300 ab − dataset at LHCb[1, 2] and about 5 × b ¯ b pairs with 50 ab − dataset at Belle-II [2, 3] in the near future.The b rare decays usually have tiny branching ratios within SM, and are often used to searchfor NP, because an obvious deviation from SM predictions might be a smoking gun of NP.Of course, the precondition is a comprehensive investigation into specific processes withinSM. According to the future experimental prospects, in this paper, we will study the Υ(1 S ) → DP decays (here P = π and K ) within SM in order to offer a ready reference for futureanalysis.The Υ(1 S ) meson is one of the b ¯ b bound states (bottomonium). The mass of Υ(1 S ) me-son, m Υ(1 S ) = 9460 . b ¯ b pairs into three gluons, with branching ratio B r (Υ → ggg ) = 81 . Υ = 54 . .
25) keV [4]. To date, the sum of measured branching ratio of 100 exclusivehadronic modes is only about 1.2% [4]. Besides the strong and electromagnetic transitions,the Υ(1 S ) meson can decay through the weak interactions, for example, the flavor non-conservation processes of Υ(1 S ) → DP decays. It is estimated that the branching ratios ofΥ(1 S ) weak decays should usually be very small, about 2 /τ B Γ Υ ∼ O (10 − ), where τ B andΓ Υ are the lifetime of B meson and width of Υ meson. Within SM, the Υ(1 S ) → DP decaysare induced by the W ± exchange, which is illustrated in Fig. 1. ¯ c ¯ bb u Υ W ± ¯ d π + d D − G FIG. 1: The Feynman diagram for the Υ(1 S ) → D − π + decay within SM. A sample of 10 Υ(1 S ) events has been collected at resonances by the Belle detector [8].A much more number of Υ(1 S ) with great precision is expected at the running Belle-II andupgraded LHCb experiments. Besides the direct production via e + e − → Υ(1 S ), the Υ(1 S )2eson can also be produced via the Υ( nS ) → ππ Υ(1 S ) and η Υ(1 S ) transitions (where n ≥
2) and initial state radiation processes e + e − → ππ Υ(1 S ). The huge amount of data makethe study of Υ(1 S ) weak decay interesting and worthwhile, although very challenging.With the help of the excellent performance of Belle-II and LHCb detectors, and the assis-tance of sophisticated analysis technology and methods, events of the Υ(1 S ) → DP decaysshould in principle be easily selected. On the one hand, the final states carry definite energiesand momenta in the rest frame of the Υ(1 S ) meson; on the other hand, the identification ofa single charmed meson is free from inefficiently double tagging, and provides a conclusiveevidence of the Υ(1 S ) weak decay. And what’s more, the phenomenon of an abnormallylarge production rate of a single charmed mesons would be a hint of NP.As far as we know, the Υ(1 S ) → DP decays have not been studied seriously yet. From theexperimental point of view, inadequate data samples and tiny branching ratios might be themain considerations. From the theoretical point of view, one of the principal problems is howto properly calculate the hadron transition matrix elements due to our limited informationsabout the hadronization mechanisms, the long-distance contributions, and so on.From Fig. 1, it is clearly seen that there are simultaneously many scales involved in thetheoretical calculation of the Υ(1 S ) → DP decays, such as the mass of W gauge boson m W , the b quark mass m b and the QCD characteristic scale Λ QCD . In general, differentdynamics correspond to different scales. Here, we will adopt the commonly acknowledgedtreatment by using the effective theory. The effective Hamiltonian in charge of the Υ(1 S ) → DP decays is written as [9], H eff = G F √ V ub V ∗ cb X i =1 f i C i ( µ ) O i ( µ ) + H . c ., (1)where G F ≃ . × − GeV − [4] is the Fermi coupling constant. The Cabibbo-Kobayashi-Maskawa (CKM) factor | V ub V ∗ cb | = 1 . × − [4]. The factor f i = +1 for tree operators O , and − O − , respectively. The Wilson coefficients C i are calcu-lable with the renormalization group improved perturbation theory at the scale of m W , andthen evolved to the scale of µ . The operators describing the local interactions among fourquarks are defined to be, O = (cid:2) ¯ u α γ µ (1 − γ ) b α (cid:3) (cid:2) ¯ b β γ µ (1 − γ ) c β (cid:3) , (2) O = (cid:2) ¯ u α γ µ (1 − γ ) b β (cid:3) (cid:2) ¯ b β γ µ (1 − γ ) c α (cid:3) , (3)3 = (cid:2) ¯ u α γ µ (1 − γ ) c α (cid:3) X q (cid:2) ¯ q β γ µ (1 − γ ) q β (cid:3) , (4) O = (cid:2) ¯ u α γ µ (1 − γ ) c β (cid:3) X q (cid:2) ¯ q β γ µ (1 − γ ) q α (cid:3) , (5) O = (cid:2) ¯ u α γ µ (1 − γ ) c α (cid:3) X q (cid:2) ¯ q β γ µ (1 + γ ) q β (cid:3) , (6) O = (cid:2) ¯ u α γ µ (1 − γ ) c β (cid:3) X q (cid:2) ¯ q β γ µ (1 + γ ) q α (cid:3) , (7) O = (cid:2) ¯ u α γ µ (1 − γ ) c α (cid:3) X q Q q (cid:2) ¯ q β γ µ (1 + γ ) q β (cid:3) , (8) O = (cid:2) ¯ u α γ µ (1 − γ ) c β (cid:3) X q Q q (cid:2) ¯ q β γ µ (1 + γ ) q α (cid:3) , (9) O = (cid:2) ¯ u α γ µ (1 − γ ) c α (cid:3) X q Q q (cid:2) ¯ q β γ µ (1 − γ ) q β (cid:3) , (10) O = (cid:2) ¯ u α γ µ (1 − γ ) c β (cid:3) X q Q q (cid:2) ¯ q β γ µ (1 − γ ) q α (cid:3) , (11)where α and β are color indices and the sum over repeated indices is understood. Q q is theelectric charge of quark q in the unit of | e | , and q ∈ { u , d , c , s , b } .With the interaction Hamiltonian of Eq.(1), the decay amplitudes for the Υ(1 S ) → DP decays can be written as, A (Υ → DP ) = h DP |H eff | Υ i = G F √ V ub V ∗ cb X i =1 f i C i ( µ ) h DP | O i ( µ ) | Υ i . (12)It is seen that the decay amplitudes of Eq.(12) are clearly factorized into four parts: thecouplings of weak interactions G F , the CKM factors V ub V ∗ cb , the Wilson coefficients C i sum-marizing the physical contributions above the scale of µ , and the hadronic matrix elements(HMEs) h O i i = h DP | O i ( µ ) | Υ i containing the physical contributions below the scale of µ .The product of the first three parts, G F , V ub V ∗ cb and C i , can be regarded as the effective cou-pling of operators O i , and has been well known. The HMEs h O i i describing the transitionsfrom quarks to participating hadrons are the core and difficulty of theoretical calculations.In addition, the QCD radiative corrections to HMEs should be included in order to obtaina physical amplitude by cancelling the scale µ dependence of the Wilson coefficients.Recently, many QCD-inspired phenomenological models, such as the QCD factoriza-tion (QCDF) approach [10–15] based on the collinear approximation, and the perturbativeQCD (pQCD) approach [16–22] based on k T factorization, have been successfully applied to4xclusive nonleptonic B meson decay processes. Using these models, HMEs have a simplestructure. They are generally expressed as the convolution of scattering sub-amplitudes aris-ing from hard gluon exchanges among quarks and the wave functions (WFs) reflecting thenonperturbative contributions. The scattering sub-amplitudes are in principle calculable or-der by order with the perturbative theory. WFs are universal and process-independent, andcould be obtained by nonperturbative methods or from data. So the theoretical calculationof HMEs becomes reasonably practical. The W ± exchange topology of Fig. 1 correspondsto annihilation topologies (see Fig. 2) within the effective theory of Eq.(1). Another twounknown parameters or more will be introduced to deal with the endpoint divergences of theannihilation amplitudes with the QCDF approach [23–31]. While the transverse momentumeffects and Sudakov factors are considered to settle the endpoint contributions of quark scat-tering amplitudes and hadronic WFs with the pQCD approach [16–22]. In this paper, we willinvestigate the Υ(1 S ) → DP decays with the pQCD approach. The master pQCD formulafor decay amplitudes could be factorized into three parts : the hard contributions above thescale of µ incorporated into the Wilson coefficients C i , the perturbatively calculable quarkscattering amplitudes H near the scale of µ , and the long-distribution contributions belowthe scale of µ incorporated into hadronic WFs Φ. A i = Z dx dx dx db db db C i ( t i ) H i ( x , x , x , b , b , b )Φ Υ ( x , b ) e − S Υ Φ D ( x , b ) e − S D Φ P ( x , b ) e − S P , (13)where x i is the longitudinal momentum fraction of the valence quark, b i is the conjugatevariable of the transverse momentum, and e − S i is the Sudakov factor.With the convention of Refs. [32–36], the relevant mesonic WFs and distribution ampli-tudes (DAs) are defined as follows. h | ¯ b α (0) b β ( z ) | Υ( p , ǫ k ) i = f Υ Z d k e − i k · z (cid:8) ǫ k Υ (cid:2) m Υ φ v Υ − 6 p φ t Υ (cid:3)(cid:9) βα , (14) h D ( p ) | ¯ q α ( z ) c β (0) | i = − i f D Z d k e + i k · z (cid:8) γ (cid:0) p + m D (cid:1) φ D (cid:9) βα , (15) h P ( p ) | ¯ q α ( z ) u β (0) | i = − i f P Z d k e + i k · z (cid:8) γ (cid:2) p φ aP + µ P φ pP − µ P (cid:0) n − n + − (cid:1) φ tP (cid:3)(cid:9) βα , (16)where f Υ , f D and f P are decay constants. µ P = 1 . ± . n + = (1 , ,
0) and n − = (0 , ,
0) are the light cone vectors, and satisfy the relations of n ± = 05nd n + · n − = 1. In the rest frame of the Υ(1 S ) meson, the kinematic variables are definedas follows. p Υ = p = m Υ √ (cid:0) , , (cid:1) , (17) p D = p = m Υ √ (cid:0) , r D , (cid:1) , (18) p P = p = m Υ √ (cid:0) , − r D , (cid:1) , (19) k = m Υ √ (cid:0) x , x , ~k T (cid:1) , (20) k = m Υ √ (cid:0) x , , ~k T (cid:1) , (21) k = m Υ √ (cid:0) , x (1 − r D ) , ~k T (cid:1) , (22) ǫ k Υ = 1 √ (cid:0) , − , (cid:1) , (23)where k i , x i and ~k iT are respectively the momentum, longitudinal momentum fraction andtransverse momentum, as shown in Fig. 2(a). The mass ratio r D = m D m Υ . ǫ k Υ is the longitu-dinal polarization vector. The explicit DA expressions [32–36] are as follows. φ v Υ ( x ) = A x ¯ x exp (cid:8) − m b β x ¯ x (cid:9) , (24) φ t Υ ( x ) = B ξ exp (cid:8) − m b β x ¯ x (cid:9) , (25) φ D ( x ) = C x ¯ x exp (cid:8) − β (cid:0) m q x + m c ¯ x (cid:1)(cid:9) , (26) φ D ( x, b ) = 6 x ¯ x (cid:8) − C D ξ (cid:9) exp (cid:8) − ω D b (cid:9) , (27) φ aP ( x ) = 6 x ¯ x (cid:8) a P C / ( ξ ) + a P C / ( ξ ) (cid:9) , (28) φ pP ( x ) = 1 + 3 ρ P + − ρ P − a P + 18 ρ P + a P + 32 ( ρ P + + ρ P − ) (1 − a P + 6 a P ) ln( x )+ 32 ( ρ P + − ρ P − ) (1 + 3 a P + 6 a P ) ln(¯ x ) − ( 32 ρ P − − ρ P + a P + 27 ρ P − a P ) C / ( ξ )+ (30 η P − ρ P − a P + 15 ρ P + a P ) C / ( ξ ) , (29) φ tP ( x ) = 32 ( ρ P − − ρ P + a P + 6 ρ P − a P )6 C / ( ξ ) (cid:8) ρ P + − ρ P − a P + 24 ρ P + a P + 32 ( ρ P + + ρ P − ) (1 − a P + 6 a P ) ln( x )+ 32 ( ρ P + − ρ P − ) (1 + 3 a P + 6 a P ) ln(¯ x ) (cid:9) − ρ P + a P − ρ P − a P ) C / ( ξ ) , (30)where ¯ x = 1 − x and ξ = x − ¯ x = 2 x − β = m b α s ( m b ) and β = m D α s ( m D ) are theshape parameters of DAs for the Υ(1 S ) and D mesons. a Pi and C mn ( ξ ) are the Gegenbauermoment and Gegenbauer polynomials. The other shape parameters of pseudoscalar DAs are ρ P + = m P µ P , ρ K − ≃ m s µ K , ρ π − = 0, and η P = f P f P µ P . The parameters A , B and C in Eq.(24),Eq.(25) and Eq.(26) can be determined by the normalization conditions. Z dx φ v,t Υ ( x ) = 1 , (31) Z dx φ D q ( x ) = 1 . (32)It should be pointed out there are many models for DAs of the D mesons, for example,Eq.(30) of Ref.[36]. In this paper, we take the typical models of Eq.(26) and Eq.(27) forexamples to illustrate the model dependence of the results. For the scenario I of Eq.(26),the mass of light quark is m u,d = 310 MeV and m s = 510 MeV [37]. For the scenario II ofEq.(27), the shape parameters C D = 0 . ω D = 0 . D u,d meson and C D =0 . ω D = 0 . D s meson [36]. b ( k )¯ b Υ ¯ cd ( k ) D − u ¯ d ( k ) π + G b ¯ b Υ ¯ cd D − u ¯ d π + G b ¯ b Υ ¯ cd D − u ¯ d π + G b ¯ b Υ ¯ cd D − u ¯ d π + G (a) (b) (c) (d)FIG. 2: The Feynman diagram for the Υ(1 S ) → D − π + decay with the pQCD approach, where (a,b)are factorizable diagrams, and (c,d) are nonfactorizable diagrams. The dots denote appropriateinteractions, and the dashed circles denote quark scattering amplitudes. The lowest order Feynman diagrams for the Υ(1 S ) → Dπ decay with the pQCD approachare shown in Fig. 2. After a series of calculation with the pQCD formula of Eq.(13), the7xpressions of the decay amplitude and branching ratio are written as follows. A (Υ → DP ) = F G F √ V ub V ∗ cb (cid:8)(cid:0) a − a − a + 12 a + 12 a (cid:1) (cid:0) A LLa + A LLb (cid:1) + (cid:0) C − C + 12 C (cid:1) (cid:0) A LLc + A LLd (cid:1) − (cid:0) C − C (cid:1) (cid:0) A LRc + A LRd (cid:1)(cid:9) , (33) B r = p cm π m Γ Υ |A (Υ → DP ) | , (34)where the factor F = 1 √ P = π , and F = +1 for P = π + and K + . p cm is the center-of-mass momentum of final states in the rest frame of the Υ(1 S ) meson. The building blocks ofamplitudes A ji are listed in Appendix A. With the input parameters in Table I, the numericalresults of the branching ratios for the Υ(1 S ) → DP decays are summarized in Table II. TABLE I: The values of the input parameters, where their central values are regarded as the defaultinputs unless otherwise specified. The numbers in parentheses are errors.mass and decay constants of the particles [4] m π = 134 .
98 MeV, m D = 1864 . f π = 130 . .
2) MeV, m π ± = 139 .
57 MeV, m D ± = 1869 . f K = 155 . m K ± = 493 .
68 MeV, m D ± s = 1969 . .
4) MeV, f π = 0 . × − GeV [34], m b = 4 . f D = 212 . f K = 0 . × − GeV [34], m c = 1 . f D s = 249 . f Υ(1 S ) = 676 . .
7) MeV [38],Gegenbauer moments at the scale of µ = 1 GeV [34] a π = 0, a π = 0 . a K = 0 . a K = 0 . S ) → DP decays in the unit of 10 − . The uncertaintiescome from the shape parameters of all participating hadronic DAs, including m b , m c , µ P and a P for scenario I and m b , C D ± . ω D ± .
04 GeV, µ P and a P for scenario II, respectively.mode D − π + D π D − s K + scenario I 1 . +0 . − . . +0 . − . . +0 . − . scenario II 0 . +0 . − . . +0 . − . . +0 . − . (1) For each specific process, the branching ratios of scenario I is larger than those ofscenario II. The branching ratios are sensitive to the DA models for the D mesons.82) Because of the relations among decay constants, i.e. , f D s > f D and f K > f π , there isa clear hierarchical pattern among branching ratios. B r (Υ(1 S ) → D s K ) > B r (Υ(1 S ) → D d π ) > B r (Υ(1 S ) → D u π ) . (35)In addition, there is a relation, B r (Υ(1 S ) → D d π ) ≈ B r (Υ(1 S ) → D u π ), because of the quarkcompositions of electrically neutral pion, i.e. , | π i = | u ¯ u i − | d ¯ d i√ | V ub V ∗ cb | and decay constants,respectively.(4) There are many possible reasons for the small branching ratios. (a) Almost all of thedecay width of the Υ(1 S ) meson come from the strong and electromagnetic interactions.The Υ(1 S ) weak decays are strongly suppressed by the Fermi coupling constant G F ∼ − ,compared with the couplings of α s ∼ − and α em ∼ − . (b) The annihilation amplitudesare generally power suppressed based on the power counting rule in the heavy quark limit[11]. (c) These processes are highly suppressed by the CKM factor of | V ub V ∗ cb | ∼ − . (d)According to the conservation law of angular momentum, only the P -wave contributionallows in the decay amplitudes. (e) These decays are suppressed by color due to the W exchange between quarks of different final states.(5) The branching ratios for the Υ(1 S ) → DP decays within SM are at the order of10 − , which are too small to be measurable in the near future. Of course, it is possible thatsome extraordinary effects from NP may significantly enhance these branching ratios, andproduce an observable phenomena.In summary, considering the developmental opportunities and important challenges atthe high-luminosity dedicated heavy-flavor factories in the future, the exclusive two-bodynonleptonic Υ(1 S ) decays through the weak interactions into final states including only onecharmed meson, Υ(1 S ) → DP , are studied for the first time with the pQCD approach. Ourresults show that (1) the Υ(1 S ) → D s K decay has relatively large occurrence probabilityamong the concerned processes; (2) the branching ratios for the Υ(1 S ) → DP decay are tiny, O (10 − ), and impossible to measure at Belle-II and LHCb during the next decades. Oneexperimental signal of the Υ(1 S ) → D s K and/or Dπ decays will be an obvious deviationfrom the SM prediction and an omen of NP.9 cknowledgments The work is supported by the National Natural Science Foundation of China (Grant Nos.11705047, 11981240403, U1632109 and 11547014).
Appendix A: Building blocks of decay amplitudes
For the sake of convenience in writing, some shorthands are used. There should alwaysbe a Sudakov factor corresponding to each WF with the pQCD approach. So the shorthandsare φ v,t Υ = φ v,t Υ ( x ) e − S Υ , φ D = φ D ( x ) e − S D , φ aP = φ aP ( x ) e − S P and φ p,tP = µ P m Υ φ p,tP ( x ) e − S P . a i = (cid:26) C i + 1 N c C i +1 , for odd i ; C i + 1 N c C i − , for even i. (A1)According to the pQCD formula of Eq.(13), the amplitude building block A ji should be afunction of the Wilson coefficients C i . That is to say, the expression C k A ji in Eq.(33) shouldactually be A ji [ C k ]. As to the amplitude building block A ji , the subscript i corresponds tothe indices of Fig.2, and the superscript j refers to the two possible Dirac structures Γ ⊗ Γ of the operator (¯ q Γ q )(¯ q Γ q ), namely j = LL for γ µ (1 − γ ) ⊗ γ µ (1 − γ ) and j = LR for γ µ (1 − γ ) ⊗ γ µ (1 + γ ). The expressions of A ji are written as follows. C = m f Υ f D f P π C F N c , (A2) A LLa = C Z dx dx Z ∞ db db α s ( t a ) H ab ( α g , β a , b , b ) C i ( t a ) S t ( x ) φ D (cid:8) φ aP (1 − r D ) x + 2 r D φ pP (cid:2) x − (1 − r D ) (cid:3)(cid:9) , (A3) A LLb = C Z dx dx Z ∞ db db α s ( t b ) H ab ( α g , β b , b , b ) C i ( t b ) S t ( x ) φ D (cid:8) φ aP (1 − r D ) (cid:2) x (1 − r D ) − r D (cid:3) + r D (1 − r D ) ( x − ¯ x ) (cid:2) φ pP + φ tP (cid:3) + 2 r D φ tP (cid:9) , (A4) A LLc = C N c Z dx dx dx Z ∞ db db α s ( t c ) H cd ( α g , β c , b , b ) C i ( t c ) φ D (cid:8) φ v Υ (cid:2) φ aP (1 − r D ) (cid:0) x (1 + r D ) − x r D − x (1 − r D ) (cid:1) r D φ pP (cid:0) x − x (1 − r D ) (cid:1) + r D φ tP (cid:0) x − x − x (1 − r D ) (cid:1)(cid:3) − φ t Υ (cid:2) φ aπ (1 − r D ) + 4 r D φ tP (cid:3)(cid:9) b = b , (A5) A LRc = C N c Z dx dx dx Z ∞ db db α s ( t c ) H cd ( α g , β c , b , b ) C i ( t c ) φ D (cid:8) φ v Υ (cid:2) φ aP (1 − r D ) ( x − x ) + r D φ pP (cid:0) x − x (1 − r D ) (cid:1) − r D φ tP (cid:0) x − x − x (1 − r D ) (cid:1)(cid:3) + 12 φ t Υ φ aP (1 − r D ) (cid:9) b = b , (A6) A LLd = − A LRc ( x → ¯ x , t c → t d , β c → β d ) , (A7) A LRd = − A LLc ( x → ¯ x , t c → t d , β c → β d ) , (A8) H ab ( α, β, b i , b j ) = − π b i b j (cid:8) J ( b j √ α ) + i Y ( b j √ α ) (cid:9)(cid:8) θ ( b i − b j ) (cid:2) J ( b i p β ) + i Y ( b i p β ) (cid:3) J ( b j p β ) + ( b i ↔ b j ) (cid:9) , (A9) H cd ( α, β, b , b ) = b b (cid:8) iπ θ ( β ) (cid:2) J ( b p β ) + i Y ( b p β ) (cid:3) + θ ( − β ) K ( b p − β ) (cid:9) iπ (cid:8) θ ( b − b ) (cid:2) J ( b √ α ) + i Y ( b √ α ) (cid:3) J ( b √ α ) + ( b ↔ b ) (cid:9) , (A10) S Υ = s ( x , p +1 , /b ) + 2 Z t /b dµµ γ q , (A11) S D = s ( x , p +2 , /b ) + s (¯ x , p +2 , /b ) + 2 Z t /b dµµ γ q , (A12) S P = s ( x , p − , /b ) + s (¯ x , p − , /b ) + 2 Z t /b dµµ γ q , (A13) α g = m (1 − r D ) x x , (A14) β a = m (1 − r D ) x , (A15) β b = m (1 − r D ) x , (A16) β c = α g − m (1 − r D ) x x − m x x , (A17) β d = α g − m (1 − r D ) ¯ x x − m ¯ x x , (A18) t a,b = max( p β a,b , /b , /b ) , (A19) t c,d = max( √ α g , q | β c,d | , /b , /b ) , (A20)11here I , J , K and Y are Bessel functions. The expression of s ( x, Q, b ) can be found inRef.[18]. γ q = − α s π is the quark anomalous dimension. [1] I. Bediaga et al. (LHCb Collaboration), arXiv:1808.08865.[2] J. Albrecht, F. Bernlochner, M. Kenzie et al. , arXiv:1709.10308.[3] K. Kou et al. , Prog. Theor. Exp. Phys. 2019, 123C01 (2019).[4] P. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020).[5] S. Okubo, Phys. Lett. 5, 165 (1963).[6] G. Zweig, CERN-TH-401, 402, 412 (1964).[7] J. Iizuka, Prog. Theor. Phys. Suppl. 37-38, 21 (1966).[8] Ed. A. Bevan et al. , Eur. Phys. J. C 74, 3026 (2014).[9] G. Buchalla, A. Buras, M. Lautenbacher, Rev. Mod. Phys. 68, 1125, (1996).[10] M. Beneke, G. Buchalla, M. Neubert, C. Sachrajda, Phys. Rev. Lett. 83, 1914 (1999).[11] M. Beneke, G. Buchalla, M. Neubert, C. Sachrajda, Nucl. Phys. B 591, 313 (2000).[12] M. Beneke, G. Buchalla, M. Neubert, C. Sachrajda, Nucl. Phys. B 606, 245 (2001).[13] D. Du, D. Yang, G. Zhu, Phys. Lett. B 488, 46 (2000).[14] D. Du, D. Yang, G. Zhu, Phys. Lett. B 509, 263 (2001).[15] D. Du, D. Yang, G. Zhu, Phys. Rev. D 64, 014036 (2001).[16] H. Li, H. Yu, Phys. Rev. Lett. 74, 4388 (1995).[17] H. Li, Phys. Lett. B 348, 597 (1995).[18] H. Li, Phys. Rev. D 52, 3958 (1995).[19] Y. Keum, H. Li, Phys. Rev. D 63, 074006 (2001).[20] Y. Keum, H. Li, A. Sanda, Phys. Rev. D 63, 054008 (2001).[21] C. L¨u, K. Ukai, M. Yang, Phys. Rev. D 63, 074009 (2001).[22] H. Li, K. Ukai, Phys. Lett. B 555, 197 (2003).[23] D. Du, H. Gong, J. Sun, D. Yang, G. Zhu, Phys. Rev. D 65, 074001 (2002).[24] D. Du, H. Gong, J. Sun, D. Yang, G. Zhu, Phys. Rev. D 65, 094025 (2002).Erratum, Phys. Rev. D 66, 079904 (2002).[25] J. Sun, G. Zhu, D. Du, Phys. Rev. D 68, 054003 (2003).[26] M. Beneke, M. Neubert, Nucl. Phys. B 675, 333 (2003).
27] M. Beneke, J. Rohrer, D. Yang, Nucl. Phys. B 774, 64 (2007).[28] Q. Chang, J. Sun, Y. Yang, X. Li, Phys. Rev. D 90, 054019 (2014).[29] Q. Chang, X. Hu, J. Sun, Y. Yang, Phys. Rev. D 91, 074026 (2015).[30] Q. Chang, J. Sun, Y. Yang, X. Li, Phys. Lett. B 740, 56 (2015).[31] J. Sun, Q. Chang, X. Hu, Y. Yang, Phys. Lett. B 743, 444 (2015).[32] J. Sun, Y. Yang, Q. Li et al. , Phys. Lett. B 752, 322 (2016).[33] J. Sun, Y. Yang, Q. Li et al. , Int. J. Mod. Phys. A 31, 1650061 (2016).[34] P. Ball, V. Braun, A. Lenz, JHEP 0605, 004 (2006).[35] T. Kurimoto, H. Li, A. Sanda, Phys. Rev. D 65, 014007 (2001).[36] R. Li, C. L¨u, H. Zou, Phys. Rev. D 78, 014018 (2008).[37] A. Kamal,
Particle Physics , (Springer, 2014) p.298.[38] J. Sun, Q. Li, Y. Yang et al. , Phys. Rev. D 92, 074028 (2015)., Phys. Rev. D 92, 074028 (2015).