Study of the pure annihilation B c → A 2 A 3 decays
aa r X i v : . [ h e p - ph ] N ov Study of the pure annihilation B c → A A decays Zhen-Jun Xiao , and Xin Liu
1. Department of Physics and Institute of Theoretical Physics,Nanjing Normal University, Nanjing,Jiangsu 210046, People’s Republic of China2. Department of Physics and Institute of Theoretical Physics,Xuzhou Normal University, Xuzhou,Jiangsu 221116, People’s Republic of China3. High Energy Section, ICTP, Strada Costiera 11, 34014 Trieste, Italy (Dated: June 24, 2018)
Abstract
In this work, we calculate the CP -averaged branching ratios and the polarization fractions ofthe charmless hadronic B c → A A decays within the framework of perturbative QCD(pQCD)approach, where A is either a light P or P axial-vector meson. These thirty two decaymodes can occur through the annihilation topology only. Based on the perturbative calculationsand phenomenological analysis, we find the following results: (a) the branching ratios of theconsidered thirty two B c → A A decays are in the range of 10 − to 10 − ; (b) B c → a b , K K +1 and some other decays have sizable branching ratios and can be measured at the LHCexperiments; (c) the branching ratios of B c → A ( P ) A ( P ) decays are generally much largerthan those of B c → A ( P ) A ( P ) decays with a factor around (10 ∼ B c → K K +1 decays are sensitive to the value of θ K , which will be tested by the runningLHC and forthcoming SuperB experiments; (e) the large longitudinal polarization contributionsgovern most considered decays and play the dominant role. PACS numbers: 13.25.Hw, 12.38.Bx, 14.40.Nd a [email protected] b [email protected] . INTRODUCTION From the point of structure, the B c meson is a ground state of ¯ bc system: which islikely an intermediate state of the ¯ cc and ¯ bb -quarkonia, but should be very different fromboth of them since B c meson carries flavor B = − C = 1. When compared with theheavy-light B q meson with q = ( u, d, s ), on the other hand, the decays of the B c mesonmust be rather different from those B u /B d /B s mesons since here both b and c can decaywhile the other serves as a spectator, or annihilating into pairs of leptons or light mesons(such as K + π , etc). Physicists therefore believe that the B c physics must be very richif the statistics reaches high level [1–3]. In recent years, many theoretical studies onthe production and decays of B c meson have been done [2, 3], based on for examplethe Operator Production Expansion [4], NRQCD[5], QCD Sum Rules[6], SU(3) flavorsymmetry[7], ISGW II model[8], QCD factorization approach [9], and the perturbativeQCD (pQCD) factorization approach[10–13].On the experimental side, it is impossible to find a pair of B + c B − c in the B-factoryexperiments (BaBar and Belle) since its mass is well above 6 GeV. Although the firstobservation of approximately 20 B c events in the B c → J/ Ψ lν decay mode was reported in1998 by the CDF collaboration [1], it was not until 2008 that two confirming observationsin excess of 5 σ significance were made by CDF and D0 collaboration [14] at Tevatron viatwo decay channels: the hadronic B c → J/ Ψ π + decay and the semileptonic B c → J/ Ψ l + ν l decay.At the LHC experiment, specifically the LHCb, one could expect around 5 × B c events per year[2, 3]. And therefore, besides the charmed decays with large branchingratios, many rare B c decays with a decay rate at the level of 10 − to 10 − can alsobe measured with a good precision at the LHC experiments[7]. This means that, many B c → h h decays ( h i are the light scalar(S), pseudo-scalar(P), vector(V), axial-vector(A)and tensor(T) mesons, made of light u, d, s quarks ) can be observed experimentally. Inthe SM, such decays can only occur via the annihilation type diagrams. The studies onthese pure annihilation B c decays may open a new window to understand the annihilationmechanism in B physics, an important but very difficult problem to be resolved.In 2004, by employing the low energy effective Hamiltonian [15] and the pQCD ap-proach [16–18], we studied the pure annihilation decays B s → ππ and presented the pQCDprediction for its branching ratio[19]: Br ( B s → π + π − ) = (4 . ± . × − , which wasconfirmed by a later theoretical calculation[20] and by a very recent CDF measurementwith a significance of 3 . σ [21]: Br ( B s → π + π − ) = (5 . ± . ± . × − . This goodagreement encourage us to extend our work to the case of B c decays. Although the charmquark c is massive (relative to the known light quarks u , d , and s ), the B c meson hasbeen treated as a heavy-light structure in this work because of the ratio m c /m B c ∼ . b quark ina B c meson. With this assumption, we also employ the k T factorization theorem to the b decay in B c meson, in a similar way as for the decays of B u and B d mesons.During past two years, based on the pQCD factorization approach, we have made asystematic study on the two-body charmless hadronic decays of B c → P P, P V, V V [10], B c → SP, SV [11] and B c → AP, AV [12, 13]. For all the considered pure annihilation B c decay channels, we calculated their CP-averaged branching ratios and longitudinal polar-ization fractions, and found some interesting results to be tested by the LHC experiments.In this paper, we extend our previous investigation further to the charmless hadronic2 c → AA decays. The axial-vector mesons involved are the following: a (1260) , b (1235) , K (1270) , K (1400) , f (1285) , f (1420) , h (1170) , h (1380) . (1)All the thirty two decay modes are the pure annihilation decay processes in the SM.The internal structure of the axial-vector mesons has been one of the hot topics inrecent years [22–24]. Although many efforts on both theoretical and experimental aspectshave been made [25–30] , we currently still know little about the nature of the axial-vectormesons. Our study will be helpful to understand the structure of these mesons.As one of the popular factorization tools based on the QCD dynamics, the pQCDapproach can be used to analytically calculate the annihilation type diagrams. Besidesthe good agreement between the pQCD prediction and the newest CDF measurement for Br ( B s → π + π − ), the pQCD prediction of Br ( B → D − s K + ) ≈ (4 . ± . × − forthe pure annihilation B decay as presented in Ref. [31] also be consistent well with thedata [30]. We therefore believed that the pQCD factorization approach is a powerful andconsistent framework to perform the calculation for the annihilation type B u,d,s decays,and extend our work to the cases of B c decays.The paper is organized as follows. In Sec. II, we give a brief review about the axial-vector meson spectroscopy, and the theoretical framework of the pQCD factorizationapproach. We perform the perturbative calculations for considered decay channels inSec. III. The analytic expressions of the decay amplitudes for all thirty two B c → AA decays are also collected in this section. The numerical results and phenomenologicalanalysis are given in Sec. IV. The main conclusions and a short summary are presentedin the last section. II. THEORETICAL FRAMEWORKA. Axial-vector mesons and mixings
In the quark model, there exist two distinct types of light parity-even p -wave axial-vector mesons, namely, P ( J P C = 1 ++ ) and P ( J P C = 1 + − ) states: P nonet : a (1260) , f (1285) , f (1420) and K A ; P nonet : b (1235) , h (1170) , h (1380) and K B . (2)In the SU(3) flavor limit, the above mesons can not mix with each other. Because the s quark is heavier than u, d quarks, the physical mass eigenstates K (1270) and K (1400)are not purely P or P states, but believed to be mixtures of K A and K B . Analogousto η and η ′ system, the flavor-singlet and flavor-octet axial-vector meson can also mixwith each other.The physical states K (1270) and K (1400) can be written as the mixtures of the K A and K B states: K (1270) = sin θ K K A + cos θ K K B ,K (1400) = cos θ K K A − sin θ K K B , (3) For the sake of simplicity, we will adopt the forms a , b , K ′ , K ′′ , f ′ , f ′′ , h ′ and h ′′ to denote theaxial-vector mesons a (1260), b (1235), K (1270), K (1400), f (1285), f (1420), h (1170) and h (1380)correspondingly in the following sections, unless otherwise stated. We will also use K , f ′ and h ′ todenote K (1270) and K (1400), f (1285) and f (1420), and h (1170) and h (1380) for convenienceunless otherwise stated explicitly. θ K is the mixing angle to be determined by the experiments. But we currently havelittle knowledge about θ K due to the absence of the relevant data, although it has beenstudied for a long time [22–24]. In this paper, for simplicity, we will adopt two referencevalues as those used in Ref. [24]: θ K = ± ◦ .Analogous to the η - η ′ mixing in the pseudoscalar sector, the h (1170) and h (1380)( P states) system can be mixed in terms of the pure singlet h and octet h , h (1170) = sin θ h + cos θ h ,h (1380) = cos θ h − sin θ h . (4)Likewise, f (1285) and f (1420) (the P states) will mix in the same way: f (1285) = sin θ f + cos θ f ,f (1420) = cos θ f − sin θ f . (5)where the flavor contents of h , and f , can be written as h = f = 1 √ (cid:0) ¯ uu + ¯ dd + ¯ ss (cid:1) ,h = f = 1 √ (cid:0) ¯ uu + ¯ dd − ss (cid:1) . (6)The values of the mixing angles θ , can be chosen as [24]: θ = 10 ◦ or 45 ◦ ; θ = 38 ◦ or 50 ◦ . (7) B. Formalism
In the pQCD factorization approach, the four annihilation Feynman diagrams for B c → A A decays are shown in Fig.1, where (a) and (b) are factorizable diagrams, while (c)and (d) are the non-factorizable ones. The initial ¯ b and c quarks annihilate into u and ¯ d/ ¯ s ,and then form a pair of light mesons by hadronizing with another pair of q ¯ q ( q = ( u, d, s ))produced perturbatively through the one-gluon exchange mechanism. Besides the short-distance contributions based on one-gluon-exchange, the q ¯ q pair can also be producedthrough strong interaction in non-perturbative regime (final state interaction(FSI), forexample). FIG. 1. The annihilation Feynman diagrams for B c → A A decays.(a) and (b) are factorizablediagrams; while (c) and (d) are the non-factorizable ones. B c → A A decays, the key point is to calculate the correspondingmatrix elements: M ∝ < A A |H eff | B c > (8)where the weak effective Hamiltonian H eff is given by [15] H eff = G F √ V ∗ cb V uD ( C ( µ ) O ( µ ) + C ( µ ) O ( µ ))] , (9)with the current-current operators O , , O = ¯ u β γ µ (1 − γ ) D α ¯ c β γ µ (1 − γ ) b α ,O = ¯ u β γ µ (1 − γ ) D β ¯ c α γ µ (1 − γ ) b α , (10)where V cb , V uD ( D = d, s ) are the CKM matrix elements, C i ( µ ) are Wilson coefficients atthe renormalization scale µ .Although the dominance of the one-gluon exchange diagram seems favored by the dataof B s → π + π − and B → D − s K + decays, according to the good agreement between ourcalculations based on the pQCD approach [19, 31] and the data [21, 30], we currently stilldo not know whether the short-distance or the non-perturbative contribution dominatefor B c annihilation decays. We here first assume that the short-distance contributionis dominant, and then calculate the matrix element in Eq. (8) by employing the pQCDapproach, provide the pQCD predictions for the branching ratios and longitudinal polar-ization fractions, and finally wait for the test by the LHC experiments.We work in the frame with the B c meson at rest, i.e., with the B c meson momentum P = m Bc √ (1 , , T ) in the light-cone coordinates. We assume that the A ( A ) meson movesin the plus (minus) z direction carrying the momentum P ( P ) and the polarization vector ǫ ( ǫ ). Then the two final state meson momenta can be written as P = m B c √ − r , r , T ) , P = m B c √ r , − r , T ) , (11)where r = m A /m B , and r = m A /m B . The longitudinal polarization vectors, ǫ L and ǫ L , can be defined as ǫ L = m B c √ m A (1 − r , − r , T ) , ǫ L = m B c √ m A ( − r , − r , T ) . (12)The transverse ones are parameterized as ǫ T = (0 , , T ), and ǫ T = (0 , , T ). Putting the(light-) quark momenta in B c , A and A mesons as k , k , and k , respectively, we canchoose k = ( x P +1 , , k T ) , k = ( x P +2 , , k T ) , k = (0 , x P − , k T ) . (13)Then the decay amplitude can be written conceptually as the following form, M ( B c → A A ) = < A A |H eff | B c > ∼ Z dx dx dx b db b db b db × Tr (cid:2) C ( t )Φ B c ( x , b )Φ A ( x , b )Φ A ( x , b ) H ( x i , b i , t ) S t ( x i ) e − S ( t ) (cid:3) (14)5here b i is the conjugate space coordinate of k iT , and t is the largest energy scale infunction H ( x i , b i , t ). The large logarithms ln( m W /t ) are included in the Wilson coefficients C ( t ). The large double logarithms (ln x i ) are summed by the threshold resummation [32],and they lead to S t ( x i ) which smears the end-point singularities on x i . The last term, e − S ( t ) , is the Sudakov form factor which suppresses the soft dynamics effectively [33]. Thusit makes the perturbative calculation of the hard part H applicable at intermediate scale,i.e., m B c scale. We will calculate analytically the function H ( x i , b i , t ) for the considereddecays at leading order(LO) in α s expansion and give the convoluted amplitudes in nextsection. III. THE DECAY AMPLITUDES IN THE PQCD APPROACH
For an axial-vector meson, there are three kinds of polarizations, namely, longitudinal( L ), normal ( N ), and transverse ( T ). The B c → A ( ǫ , P ) A ( ǫ , P ) decays are charac-terized by the polarization states of these axial-vector mesons. A. Decay Amplitudes with different polarization
The decay amplitudes M H are classified accordingly, with H = L, N, T , M H = ( m B c M L , m B c M N ǫ ∗ ( T ) ǫ ∗ ( T ) , i M T ǫ αβγρ ǫ ∗ α ( T ) ǫ ∗ β ( T ) P γ P ρ ) . (15)where ǫ ( T ) stands for the transverse polarization vector and we have adopted the nota-tion ǫ = 1. Based on the Feynman diagrams shown in Fig. 1, we can combine allcontributions to these considered decays and obtain the general expression of total decayamplitude as follows, M H ( B c → A A ) = V ∗ cb V uD (cid:8) f B c F A A fa ; H a + M A A na ; H C (cid:9) , (16)where a = C / C , while F A A fa ; H and M A A na ; H denote the Feynman amplitudes withthree polarizations for factorizable and nonfactorizable annihilation contributions, respec-tively.The explicit expressions of the function F A A fa ; H and M A A na ; H in the pQCD approach canbe written as the following form: F Lfa = 8 πC F m B c Z dx dx Z ∞ b db b db × (cid:8)(cid:2) x φ ( x ) φ ( x ) + 2 r r (cid:0) ( x + 1) φ s ( x ) + ( x − φ t ( x ) (cid:1) φ s ( x ) (cid:3) × E fa ( t a ) h fa (1 − x , x , b , b ) + E fa ( t b ) h fa ( x , − x , b , b ) × (cid:2) ( x − φ ( x ) φ ( x ) + 2 r r φ s ( x ) (cid:0) ( x − φ s ( x ) − x φ t ( x ) (cid:1)(cid:3)(cid:9) , (17) One should note that a here just stands for the combined Wilson coefficient, not the abbreviation foraxial-vector meson a (1260). Lna = 16 √ πC F m B c Z dx dx Z ∞ b db b db × { [( r c − x + 1) φ ( x ) φ ( x ) + r r ( φ s ( x )((3 r c + x − x + 1) × φ s ( x ) − ( r c − x − x + 1) φ t ( x )) + φ t ( x )(( r c − x − x + 1) φ s ( x )+( r c − x + x − φ t ( x )) (cid:1)(cid:3) E na ( t c ) h cna ( x , x , b , b ) − [( r b + r c + x − φ ( x ) φ ( x ) + r r ( φ s ( x )((4 r b + r c + x − x − × φ s ( x ) − ( r c + x + x − φ T ( x )) + φ t ( x )(( r c + x + x − φ s ( x ) − ( r c + x − x − φ t ( x )) (cid:1)(cid:3) E na ( t d ) h dna ( x , x , b , b ) (cid:9) , (18) F Nfa = 8 πC F m B c Z dx dx Z ∞ b db b db r r × { [( x + 1)( φ a ( x ) φ a ( x ) + φ v ( x ) φ v ( x )) + ( x − φ v ( x ) φ a ( x ) + φ a ( x ) φ v ( x ))] × E fa ( t a ) h fa (1 − x , x , b , b )+ [( x − φ a ( x ) φ a ( x ) + φ v ( x ) φ v ( x )) − x ( φ a ( x ) φ v ( x ) + φ v ( x ) φ a ( x ))] × E fa ( t b ) h fa ( x , − x , b , b ) } , (19) M Nna = 32 √ πC F m B c Z dx dx Z ∞ b db b db r r × { r c [ φ a ( x ) φ a ( x ) + φ v ( x ) φ v ( x )] E na ( t c ) h cna ( x , x , b , b ) − r b [ φ a ( x ) φ a ( x ) + φ v ( x ) φ v ( x )] E na ( t d ) h dna ( x , x , b , b ) (cid:9) , (20) F Tfa = 16 πC F m B c Z dx dx Z ∞ b db b db r r × { [( x + 1)( φ a ( x ) φ v ( x ) + φ v ( x ) φ a ( x )) + ( x − φ a ( x ) φ a ( x ) + φ v ( x ) φ v ( x ))] × E fa ( t a ) h fa (1 − x , x , b , b )+ [( x − φ a ( x ) φ v ( x ) + φ v ( x ) φ a ( x )) − x ( φ a ( x ) φ a ( x ) + φ v ( x ) φ v ( x ))] × E fa ( t b ) h fa ( x , − x , b , b ) } , (21) M Tna = 64 √ πC F m B c Z dx dx Z ∞ b db b db r r × { r c [ φ a ( x ) φ v ( x ) + φ v ( x ) φ a ( x )] E na ( t c ) h cna ( x , x , b , b ) − r b [ φ a ( x ) φ v ( x ) + φ v ( x ) φ a ( x )] E na ( t d ) h dna ( x , x , b , b ) (cid:9) . (22)where r = m A /m B c , r c ( b ) = m c ( b ) /m B c . The explicit expressions for the distributionamplitudes φ A , φ tA , φ sA , φ TA , φ vA and φ aA are given in the Appendix A. The definitions andexpressions of the hard functions ( h fa , h na ) , ( E fa , E na ) and hard scales ( t a , t b , t c , t d ) canbe found in Appendix B of Ref. [10] and references therein. B. Decay Amplitudes for the considered decay modes
Now we can write down the total decay amplitudes for all thirty two B c → A A decays. The decay amplitudes of the sixteen ∆ S = 0 decay modes are the following: √ M H ( B c → a +1 a ) = V ∗ cb V ud n f B c (cid:16) F a +1 a u fa ; H − F a d a +1 fa ; H (cid:17) a + (cid:16) M a +1 a u na ; H − M a d a +1 na ; H (cid:17) C o , (23)7 M H ( B c → b +1 b ) = V ∗ cb V ud n f B c (cid:16) F b +1 b u fa ; H − F b d b +1 fa ; H (cid:17) a + (cid:16) M b +1 b u na ; H − M b d b +1 na ; H (cid:17) C o , (24) √ M H ( B c → a +1 b ) = V ∗ cb V ud n f B c (cid:16) F a +1 b u fa ; H − F b d a +1 fa ; H (cid:17) a + (cid:16) M a +1 b u na ; H − M b d a +1 na ; H (cid:17) C o , (25) √ M H ( B c → b +1 a ) = V ∗ cb V ud n f B c (cid:16) F b +1 a u fa ; H − F a d b +1 fa ; H (cid:17) a + (cid:16) M b +1 a u na ; H − M a d b +1 na ; H (cid:17) C o , (26) M H ( B c → a +1 f ′ ) = V ∗ cb V ud (cid:26) cos θ √ h f B c (cid:16) F a +1 f u fa ; H + F f d a +1 fa ; H (cid:17) a + (cid:16) M a +1 f u na ; H + M f d a +1 na ; H (cid:17) C i + sin θ √ h f B c (cid:16) F a +1 f u fa ; H + F f d a +1 fa ; H (cid:17) a + (cid:16) M a +1 f u na ; H + M f d a +1 na ; H (cid:17) C i(cid:27) , (27) M H ( B c → a +1 f ′′ ) = V ∗ cb V ud (cid:26) − sin θ √ h f B c (cid:16) F a +1 f u fa ; H + F f d a +1 fa ; H (cid:17) a + (cid:16) M a +1 f u na ; H + M f d a +1 na ; H (cid:17) C i + cos θ √ h f B c (cid:16) F a +1 f u fa ; H + F f d a +1 fa ; H (cid:17) a + (cid:16) M a +1 f u na ; H + M f d a +1 na ; H (cid:17) C i(cid:27) , (28) M H ( B c → b +1 f ′ ) = M H ( B c → a +1 f ′ )( a → b ) , M H ( B c → b +1 f ′′ ) = M H ( B c → a +1 f ′′ )( a → b ) , (29) M H ( B c → a +1 h ′ ) = M H ( B c → a +1 f ′ )( f → h, θ → θ ) , M H ( B c → a +1 h ′′ ) = M H ( B c → a +1 f ′′ )( f → h, θ → θ ) , (30) M H ( B c → b +1 h ′ ) = M H ( B c → a +1 h ′ )( a → b ) , M H ( B c → b +1 h ′′ ) = M H ( B c → a +1 h ′′ )( a → b ) , (31) M H ( B c → K ′ K ′ + ) = V ∗ cb V ud n − sin θ K (cid:16) f B c F K A K A fa ; H a + M K A K A na ; H C (cid:17) − cos θ K sin θ K (cid:16) f B c F K A K B fa ; H a + M K A K B na ; H C (cid:17) + cos θ K sin θ K (cid:16) f B c F K B K A fa ; H a + M K B K A na ; H C (cid:17) + cos θ K (cid:16) f B c F K B K B fa ; H a + M K B K B na ; H C (cid:17)o , (32) M H ( B c → K ′ K ′′ + ) = V ∗ cb V ud n − cos θ K sin θ K (cid:16) f B c F K A K A fa ; H a + M K A K A na ; H C (cid:17) + sin θ K (cid:16) f B c F K A K B fa ; H a + M K A K B na ; H C (cid:17) + cos θ K (cid:16) f B c F K B K A fa ; H a + M K B K A na ; H C (cid:17) − cos θ K sin θ K (cid:16) f B c F K B K B fa ; H a + M K B K B na ; H C (cid:17)o , (33)8 H ( B c → K ′′ K ′ + ) = V ∗ cb V ud n cos θ K sin θ K (cid:16) f B c F K A K A fa ; H a + M K A K A na ; H C (cid:17) + cos θ K (cid:16) f B c F K A K B fa ; H a + M K A K B na ; H C (cid:17) + sin θ K (cid:16) f B c F K B K A fa ; H a + M K B K A na ; H C (cid:17) + cos θ K sin θ K (cid:16) f B c F K B K B fa ; H a + M K B K B na ; H C (cid:17)o , (34) M H ( B c → K ′′ K ′′ + ) = V ∗ cb V ud n cos θ K (cid:16) f B c F K A K A fa ; H a + M K A K A na ; H C (cid:17) − cos θ K sin θ K (cid:16) f B c F K A K B fa ; H a + M K A K B na ; H C (cid:17) + cos θ K sin θ K (cid:16) f B c F K B K A fa ; H a + M K B K A na ; H C (cid:17) − sin θ K (cid:16) f B c F K B K B fa ; H a + M K B K B na ; H C (cid:17)o ; (35)The decay amplitudes of the sixteen ∆ S = 1 decay modes are of the form: M H ( B c → K ′ a +1 ) = √ M H ( B c → K ′ + a )= V ∗ cb V us n sin θ K h f B c F K A a +1 fa ; H a + M K A a +1 na ; H C i + cos θ K h f B c F K B a +1 fa ; H a + M K B a +1 na ; H C io , (36) M H ( B c → K ′′ a +1 ) = √ M H ( B c → K ′′ + a )= V ∗ cb V us n cos θ K [ f B c F K A a +1 fa ; H a + M K A a +1 na ; H C ] − sin θ K [ f B c F K B a +1 fa ; H a + M K B a +1 na ; H C ] o , (37) M H ( B c → K ′ b +1 ) = √ M H ( B c → K ′ + b ) = M H ( B c → K ′ a +1 )( a → b ) , (38) M H ( B c → K ′′ b +1 ) = √ M H ( B c → K ′′ + b ) = M H ( B c → K ′′ a +1 )( a → b ) , (39) M H ( B c → K ′ + f ′ ) = V ∗ cb V us × (cid:26) cos θ sin θ K √ h f B c (cid:16) F K A f u fa ; H + F f s K A fa ; H (cid:17) a + (cid:16) M K A f u na ; H + M f s K A na ; H (cid:17) C i + sin θ sin θ K √ h f B c (cid:16) F K A f u fa ; H − F f s K A fa ; H (cid:17) a + (cid:16) M K A f u na ; H − M f s K A na ; H (cid:17) C i + cos θ cos θ K √ h f B c (cid:16) F K B f u fa ; H + F f s K B fa ; H (cid:17) a + (cid:16) M K B f u na ; H + M f s K B na ; H (cid:17) C i + cos θ K sin θ √ h f B c (cid:16) F K B f u fa ; H − F f s K B fa ; H (cid:17) a + (cid:16) M K B f u na ; H − M f s K B na ; H (cid:17) C i(cid:27) , (40)9 H ( B c → K ′ + f ′′ ) = V ∗ cb V us × (cid:26) − sin θ sin θ K √ h f B c (cid:16) F K A f u fa ; H + F f s K A fa ; H (cid:17) a + (cid:16) M K A f u na ; H + M f s K A na ; H (cid:17) C i + cos θ sin θ K √ h f B c (cid:16) F K A f u fa ; H − F f s K A fa ; H (cid:17) a + (cid:16) M K A f u na ; H − M f s K A na ; H (cid:17) C i − cos θ K sin θ √ h f B c (cid:16) F K B f u fa ; H + F f s K B fa ; H (cid:17) a + (cid:16) M K B f u na ; H + M f s K B na ; H (cid:17) C i + cos θ K cos θ √ h f B c (cid:16) F K B f u fa ; H − F f s K B fa ; H (cid:17) a + (cid:16) M K B f u na ; H − M f s K B na ; H (cid:17) C i(cid:27) , (41) M H ( B c → K ′′ + f ′ ) = V ∗ cb V us × (cid:26) cos θ cos θ K √ h f B c (cid:16) F K A f u fa ; H + F f s K A fa ; H (cid:17) a + (cid:16) M K A f u na ; H + M f s K A na ; H (cid:17) C i + cos θ K sin θ √ h f B c (cid:16) F K A f u fa ; H − F f s K A fa ; H (cid:17) a + (cid:16) M K A f u na ; H − M f s K A na ; H (cid:17) C i − cos θ sin θ K √ h f B c (cid:16) F K B f u fa ; H + F f s K B fa ; H (cid:17) a + (cid:16) M K B f u na ; H + M f s K B na ; H (cid:17) C i − sin θ K sin θ √ h f B c (cid:16) F K B f u fa ; H − F f s K B fa ; H (cid:17) a + (cid:16) M K B f u na ; H − M f s K B na ; H (cid:17) C i(cid:27) , (42) M H ( B c → K ′′ + f ′′ ) = V ∗ cb V us × (cid:26) − cos θ K sin θ √ f B c ( F K A f u fa ; H + F f s K A fa ; H ) a + ( M K A f u na ; H + M f s K A na ; H ) C ]+ cos θ cos θ K √ f B c ( F K A f u fa ; H − F f s K A fa ; H ) a + ( M K A f u na ; H − M f s K A na ; H ) C ]+ sin θ sin θ K √ f B c ( F K B f u fa ; H + F f s K B fa ; H ) a + ( M K B f u na ; H + M f s K B na ; H ) C ] − cos θ sin θ K √ f B c ( F K B f u fa ; H − F f s K B fa ; H ) a + ( M K B f u na ; H − M f s K B na ; H ) C ] (cid:27) , (43) M H ( B c → K ′ + h ′ ) = M H ( B c → K ′ + f ′ )( f → h, θ → θ ) , (44) M H ( B c → K ′ + h ′′ ) = M H ( B c → K ′ + f ′′ )( f → h, θ → θ ) , (45) M H ( B c → K ′′ + h ′ ) = M H ( B c → K ′′ + f ′ )( f → h, θ → θ ) , (46) M H ( B c → K ′′ + h ′′ ) = M H ( B c → K ′′ + f ′′ )( f → h, θ → θ ) . (47) IV. NUMERICAL RESULTS AND DISCUSSIONS
In this section, we will calculate numerically the BRs and polarization fractions forthose considered thirty two B c → A A decay modes. First of all, the central values ofthe input parameters to be used are the following.10asses (GeV): m W = 80 . m B c = 6 . m b = 4 . m c = 1 . m a = 1 . m K A = 1 . m f = 1 . m f = 1 . m b = 1 . m K B = 1 . m h = 1 . m h = 1 .
37; (48)Decay constants (GeV): f a = 0 . f K A = 0 . f f = 0 . f f = 0 . f b = 0 . f K B = 0 . f h = 0 . f h = 0 . f B c = 0 . B c meson lifetime:Λ ( f =4)MS = 0 .
250 GeV , τ B c = 0 .
46 ps . (50)For the CKM matrix elements we use A = 0 .
814 and λ = 0 . ρ = 0 .
135 and ¯ η = 0 . B c → A A decays, the decay rate can be written explicitly as,Γ = G F | P c | πm B c X σ = L,T M ( σ ) † M ( σ ) (51)where | P c | ≡ | P | = | P | is the momentum of either of the outgoing axial-vector mesons.The polarization fractions f L ( || , ⊥ ) can be defined as [35], f L ( || , ⊥ ) = |A L ( || , ⊥ ) | |A L | + |A || | + |A ⊥ | , (52)where the amplitudes A i ( i = L, || , ⊥ ) are defined as, A L = − ξm B c M L , A k = ξ √ m B c M N , A ⊥ = ξm A m A p r − M T , (53)for the longitudinal, parallel, and perpendicular polarizations, respectively, with the nor-malization factor ξ = q G F P c / (16 πm B c Γ) and the ratio r = P · P / ( m A m A ). Theseamplitudes satisfy the relation, |A L | + |A k | + |A ⊥ | = 1 . (54)following the summation in Eq. (51).By using the analytic expressions for the complete decay amplitudes and the inputparameters as given explicitly in Eqs. (23)-(50), we calculate and then present the pQCDpredictions for the CP -averaged BRs and longitudinal polarization fractions (LPFs) ofthe considered decays with errors in Tables I-V. The dominant errors arise from theuncertainties of charm quark mass m c = 1 . ± .
15 GeV and the combined Gegenbauermoments a i of the axial-vector meson distribution amplitudes, respectively.11 ABLE I. The pQCD predictions of BRs and LPFs for B c → ( a , b )( a , b ) decays. The sourceof the dominant errors is explained in the text.∆ S = 0 ∆ S = 0Decay modes BRs (10 − ) LPFs (%) Decay modes BRs (10 − ) LPFs (%)B c → a +1 a . c → b +1 b . c → a +1 b . +0 . − . ( m c ) +1 . − . ( a i ) 92 . +1 . − . B c → b +1 a . +0 . − . ( m c ) +1 . − . ( a i ) 91 . +2 . − . A. The pQCD predictions for ∆ S = 0 decays In Table I and II, we show the pQCD predictions for the branching ratios and thelongitudinal polarization fractions of the sixteen ∆ S = 0 decays.For both the B c → a +1 a and b +1 b decays, since the quark structure of a and b are the same one , ( u ¯ u − d ¯ d ) / √
2, the contributions from u ¯ u and d ¯ d components to thecorresponding decay amplitude as shown in Eqs.(23,24) will interfere destructively, andtherefore will cancel each other exactly at leading order and result in the zero BRs forthese two channels, as illustrated in the Table I. For the possible high order contributions,they will also cancel each other due to the isospin symmetry between u and d quarks. Asfor the non-perturbative part, we currently do not know how to calculate it reliably. Butwe generally believe that it is small in magnitude for B meson decays. Consequently, wethink that a nonzero measurement for the branching ratios of these two decays may be asignal of the effects of new physics beyond the SM.For B c → a +1 b and B c → b +1 a decays, however, the pQCD predictions for their BRsare rather large, as given in Table I Br ( B c → a +1 b ) = Br ( B c → b +1 a ) ≈ . × − . (55)Besides B c → a +1 b and b +1 a decays, other six ∆ S = 0 decays, such as the B c → b h and B c → K K +1 decays, also have a large branching ratios at the 10 − level, as listedin Table II. According to the studies in Ref. [7], these B c decay modes with a branchingratio at 10 − level could be measured at the LHC experiments [7].Besides the large branching ratio at 10 − level, the B c → K K +1 decay modes alsohave a strong dependence on the value of the mixing angle θ K , as shown by the numbersin Table II. If these channels are measured at LHC experiments with enough precision,one can determine the θ K by compare the pQCD predictions with the data. In order toreduce the effects of the choice of input parameters, we define the ratio of the branchingratios between relevant decay modes: Br ( B c → K (1270) K (1400) + ) pQCD Br ( B c → K (1270) K (1270) + ) pQCD ≈ (cid:26) . , for θ K = 45 ◦ , . , for θ K = − ◦ ; (56) Br ( B c → K (1270) K (1400) + ) pQCD Br ( B c → K (1400) K (1270) + ) pQCD ≈ (cid:26) . , for θ K = 45 ◦ , . , for θ K = − ◦ ; (57)12 ABLE II. Same as Table I but for B c → ( a +1 , b +1 )( f ′ , h ′ ) decays.∆ S = 0 θ = 38 ◦ θ = 50 ◦ Decay modes BRs (10 − ) LPFs (%) BRs (10 − ) LPFs (%)B c → a (1260) + f (1285) 6 . +1 . − . ( m c ) +0 . − . ( a i ) 83 . +2 . − . . +1 . − . ( m c ) +0 . − . ( a i ) 84 . +2 . − . B c → a (1260) + f (1420) × a . +0 . − . ( m c ) +0 . − . ( a i ) 56 . +43 . − . . +0 . − . ( m c ) +1 . − . ( a i ) 78 . +7 . − . ∆ S = 0 θ = 38 ◦ θ = 50 ◦ Decay modes BRs (10 − ) LPFs (%) BRs (10 − ) LPFs (%)B c → b (1235) + f (1285) 2 . +4 . − . ( m c ) +1 . − . ( a i ) 65 . +28 . − . . +4 . − . ( m c ) +1 . − . ( a i ) 68 . +21 . − . B c → b (1235) + f (1420) 1 . +0 . − . ( m c ) +0 . − . ( a i ) 100 . ± . . +0 . − . ( m c ) +1 . − . ( a i ) 100 . +0 . − . ∆ S = 0 θ = 10 ◦ θ = 45 ◦ Decay modes BRs (10 − ) LPFs (%) BRs (10 − ) LPFs (%)B c → a (1260) + h (1170) 1 . +0 . − . ( m c ) +0 . − . ( a i ) 86 . +2 . − . . +0 . − . ( m c ) +0 . − . ( a i ) 73 . +7 . − . B c → a (1260) + h (1380) ×
10 1 . +0 . − . ( m c ) +1 . − . ( a i ) 68 . +23 . − . . +0 . − . ( m c ) +2 . − . ( a i ) 100 . +0 . − . ∆ S = 0 θ = 10 ◦ θ = 45 ◦ Decay modes BRs (10 − ) LPFs (%) BRs (10 − ) LPFs (%)B c → b (1235) + h (1170) 8 . +3 . − . ( m c ) +3 . − . ( a i ) 96 . +1 . − . . +4 . − . ( m c ) +4 . − . ( a i ) 96 . +0 . − . B c → b (1235) + h (1380) 2 . +0 . − . ( m c ) +1 . − . ( a i ) 100 . +0 . − . . +0 . − . ( m c ) +0 . − . ( a i ) 100 . ± . S = 0 θ K = 45 ◦ θ K = − ◦ Decay modes BRs (10 − ) LPFs (%) BRs (10 − ) LPFs (%)B c → K (1270) K (1270) + . +0 . − . ( m c ) +1 . − . ( a i ) 99 . +0 . − . . +1 . − . ( m c ) +4 . − . ( a i ) 71 . +16 . − . B c → K (1400) K (1400) + . +1 . − . ( m c ) +4 . − . ( a i ) 72 . +15 . − . . +0 . − . ( m c ) +1 . − . ( a i ) 99 . +0 . − . B c → K (1270) K (1400) + . +1 . − . ( m c ) +3 . − . ( a i ) 96 . +3 . − . . +0 . − . ( m c ) +2 . − . ( a i ) 94 . +3 . − . B c → K (1400) K (1270) + . +0 . − . ( m c ) +2 . − . ( a i ) 94 . +3 . − . . +1 . − . ( m c ) +3 . − . ( a i ) 96 . +3 . − . a Here, the factor 10 is specifically used for the BRs. The following one has the same meaning. Br ( B c → K (1400) K (1400) + ) pQCD Br ( B c → K (1270) K (1270) + ) pQCD ≈ (cid:26) . , for θ K = 45 ◦ , . , for θ K = − ◦ ; (58)The LHC experiments can measure these ratios with a better precision than that for adirect measurement of branching ratios for individual decays. We suggest such measure-ments as a way to determine the mixing angle θ K at LHC. B. The pQCD predictions for ∆ S = 1 decays In Table III, IV and V, we show the pQCD predictions for the branching ratios andthe longitudinal polarization fractions of the sixteen ∆ S = 1 decays.First of all, when compared with those ∆ S = 0 decays, these ∆ S = 1 decays are CKMsuppressed due to the factor | V us /V ud | ∼ .
04, as can be seen easily from the expressionsfor the decay amplitudes as given in Eqs.(23) to (47). The pQCD predictions for thebranching ratios of these B c decays are at the level of 10 − to 10 − , much smaller than13 ABLE III. Same as Table I but for B c → K a , K b decays.∆ S = 1 θ K = 45 ◦ θ K = − ◦ Decay modes BRs (10 − ) LPFs (%) BRs (10 − ) LPFs (%)B c → K (1270) a (1260) + . +1 . − . ( m c ) +4 . − . ( a i ) 79 . +12 . − . . +1 . − . ( m c ) +3 . − . ( a i ) 99 . +0 . − . B c → K (1400) a (1260) + . +1 . − . ( m c ) +3 . − . ( a i ) 100 . +0 . − . . +1 . − . ( m c ) +4 . − . ( a i ) 81 . +12 . − . B c → K (1270) + a (1260) . +0 . − . ( m c ) +2 . − . ( a i ) 79 . +12 . − . . +0 . − . ( m c ) +1 . − . ( a i ) 99 . +0 . − . B c → K (1400) + a (1260) . +0 . − . ( m c ) +1 . − . ( a i ) 100 . +0 . − . . +0 . − . ( m c ) +2 . − . ( a i ) 81 . +12 . − . ∆ S = 1 θ K = 45 ◦ θ K = − ◦ Decay modes BRs (10 − ) LPFs (%) BRs (10 − ) LPFs (%)B c → K (1270) b (1235) + . +0 . − . ( m c ) +1 . − . ( a i ) 91 . +5 . − . . +0 . − . ( m c ) +0 . − . ( a i ) 100 . +0 . − . B c → K (1400) b (1235) + . +0 . − . ( m c ) +0 . − . ( a i ) 100 . ± . . +0 . − . ( m c ) +1 . − . ( a i ) 93 . +5 . − . B c → K (1270) + b (1235) . +0 . − . ( m c ) +0 . − . ( a i ) 91 . +4 . − . . +0 . − . ( m c ) +0 . − . ( a i ) 100 . +0 . − . B c → K (1400) + b (1235) . +0 . − . ( m c ) +0 . − . ( a i ) 100 . ± . . +0 . − . ( m c ) +0 . − . ( a i ) 93 . +5 . − . that for those ∆ S = 0 decays. Most of them, for example B c → K a and K f decayswith BRs around 10 − or less, are hardly to be detected even at the LHC experiments.For the B c → K b decays, the pQCD predictions for the BRs are in the order of 10 − ,much larger than the BRs of the B c → K a decays, since the P meson behaves verydifferent from the P state. From the numerical values in Table III, we can also definethe following ratio Br ( B c → K (1270) b +1 ) pQCD Br ( B c → K (1270) + b ) pQCD ≈ Br ( B c → K (1400) b +1 ) pQCD Br ( B c → K (1400) + b ) pQCD ≈ θ K = ± ◦ . Such decays have a weak dependence on the variation of θ K .In Table IV, we show the pQCD predictions for the BRs and LPFs for B c → K +1 f ′ decays with θ = 38 ◦ (1st entry) and θ = 50 ◦ (2nd entry), respectively. In Table V,similarly, we show the pQCD predictions for the BRs and LPFs for B c → K +1 h ′ decayswith θ = 10 ◦ (1st entry) and θ = 45 ◦ (2nd entry), respectively.One can see from the numerical results in these two tables that all B c → K +1 ( f ′ , h ′ )decays have a weak or moderate dependence on the mixing angles θ and θ . It is difficultto measure θ and θ through the considered B c decays.For B c → K +1 h (1380) decays, the pQCD predictions for their BRs show a relativelystrong dependence on the mixing angle θ K . The LHC measurement of these decays mayalso help to constrain the size and sign of θ K .Frankly speaking, the theoretical predictions in the pQCD factorization approach stillhave large theoretical errors induced by the large uncertainties of many input parametersand the meson distribution amplitudes. Any progress in reducing the error of inputparameters will help us to improve the precision of the pQCD predictions.It is worth of stressing that we here calculated only the short-distance contributions inthe considered decay modes and do not consider the possible long-distance contributions,such as the rescattering effects, although they may be large and affect the theoreticalpredictions. Strictly speaking, it is the task after the first measurements of the B c mesondecays and thus beyond the scope of this work.14 ABLE IV. Same as Table I but for B c → K +1 f ′ decays with θ = 38 ◦ (1st entry) and θ =50 ◦ (2nd entry), respectively.∆ S = 1 θ K = 45 ◦ θ K = − ◦ Decay modes BRs (10 − ) LPFs (%) BRs (10 − ) LPFs (%)B c → K (1270) + f (1285) 1 . +0 . − . ( m c ) +2 . − . ( a i )1 . +1 . − . ( m c ) +2 . − . ( a i ) 65 . +27 . − . . +22 . − . . +0 . − . ( m c ) +1 . − . ( a i )1 . +0 . − . ( m c ) +1 . − . ( a i ) 96 . +2 . − . . +2 . − . B c → K (1400) + f (1285) 1 . +0 . − . ( m c ) +1 . − . ( a i )1 . +0 . − . ( m c ) +1 . − . ( a i ) 96 . +2 . − . . +4 . − . . +0 . − . ( m c ) +1 . − . ( a i )1 . +1 . − . ( m c ) +2 . − . ( a i ) 65 . +27 . − . . +21 . − . B c → K (1270) + f (1420) 0 . +0 . − . ( m c ) +0 . − . ( a i )0 . +0 . − . ( m c ) +0 . − . ( a i ) 81 . +13 . − . . +16 . − . . +0 . − . ( m c ) +1 . − . ( a i )4 . +0 . − . ( m c ) +1 . − . ( a i ) 71 . +4 . − . . +4 . − . B c → K (1400) + f (1420) 4 . +0 . − . ( m c ) +1 . − . ( a i )4 . +0 . − . ( m c ) +1 . − . ( a i ) 71 . +4 . − . . +4 . − . . +0 . − . ( m c ) +0 . − . ( a i )0 . +0 . − . ( m c ) +0 . − . ( a i ) 81 . +13 . − . . +16 . − . TABLE V. Same as Table I but for B c → K +1 h ′ decays θ = 10 ◦ (1st entry) and θ = 45 ◦ (2ndentry), respectively.∆ S = 1 θ K = 45 ◦ θ K = − ◦ Decay modes BRs (10 − ) LPFs (%) BRs (10 − ) LPFs (%)B c → K (1270) + h (1170) 1 . +0 . − . ( m c ) +1 . − . ( a i )0 . +0 . − . ( m c ) +0 . − . ( a i ) 94 . +2 . − . . +6 . − . . +0 . − . ( m c ) +1 . − . ( a i )0 . +0 . − . ( m c ) +0 . − . ( a i ) 98 . +0 . − . . +7 . − . B c → K (1400) + h (1170) 1 . +0 . − . ( m c ) +1 . − . ( a i )0 . +0 . − . ( m c ) +0 . − . ( a i ) 98 . +0 . − . . +7 . − . . +0 . − . ( m c ) +1 . − . ( a i )0 . +0 . − . ( m c ) +0 . − . ( a i ) 94 . +2 . − . . +6 . − . B c → K (1270) + h (1380) 0 . +0 . − . ( m c ) +0 . − . ( a i )1 . +0 . − . ( m c ) +1 . − . ( a i ) 98 . +0 . − . . +0 . − . . +0 . − . ( m c ) +0 . − . ( a i )2 . +1 . − . ( m c ) +1 . − . ( a i ) 89 . +2 . − . . +1 . − . B c → K (1400) + h (1380) 1 . +0 . − . ( m c ) +0 . − . ( a i )2 . +1 . − . ( m c ) +1 . − . ( a i ) 89 . +2 . − . . +1 . − . . +0 . − . ( m c ) +0 . − . ( a i )1 . +0 . − . ( m c ) +1 . − . ( a i ) 98 . +0 . − . . +0 . − . V. SUMMARY
In this paper, we studied the thirty two charmless hadronic B c → A A decays by em-ploying the pQCD factorization approach. These considered decay channels can only occurvia the annihilation type diagrams in the SM. The pQCD predictions for the CP -averagedbranching ratios and longitudinal polarization fractions are analyzed phenomenologically.From our perturbative evaluations and phenomenological analysis, we found the fol-lowing results:1. The branching ratios of the considered thirty two B c → AA decays are in the rangeof 10 − to 10 − ; B c → a b , K K +1 and some other decays have sizable branchingratios ( ∼ − ) and can be measured at the LHC experiments;15. The branching ratios of B c → A ( P ) A ( P ) decays are generally much larger thanthose of B c → A ( P ) A ( P ) decays with a factor around (10 ∼ P and P states;3. For B c → AA decays, the branching ratios of ∆ S = 0 processes are generally muchlarger than those of ∆ S = 1 ones. Such differences are mainly induced by the CKMfactors involved: V ud ∼ V us ∼ .
22 for the latter ones.4. The branching ratios of B c → K K +1 decays are sensitive to the value of θ K , whichwill be tested by the running LHC and forthcoming SuperB experiments;5. The LPFs is larger than 80% for almost all decay modes. That means that thesepure annihilation decays of B c meson are dominated by the longitudinal polarizationfraction.These charmless hadronic B c meson decays will provide an important platform forstudying the mechanism of annihilation contributions, understanding the helicity struc-ture of these considered channels and the content of the axial-vector mesons. ACKNOWLEDGMENTS
Z.J. Xiao is very grateful to the high energy section of ICTP, Italy, where part of thiswork was done, for warm hospitality and financial support. This work is supported bythe National Natural Science Foundation of China under Grant No. 10975074, and No.10735080; by the Project on Graduate Students’ Education and Innovation of JiangsuProvince, under Grant No. CX09B − Appendix A: Wave functions and distribution amplitudes
For the wave function of the heavy B c meson, we adopt the form (see Ref. [10], andreferences therein) as follows,Φ B c ( x ) = i √ P/ + m B c ) γ φ B c ( x )] αβ . (A1)where the distribution amplitude φ B c is of the form [36] in the nonrelativistic limit, φ B c ( x ) = f B c √ δ ( x − m c /m B c ) . (A2)In fact, we know little about φ B c for heavy B c meson. Because of embracing b and c quarks simultaneously, B c meson can be approximated as a non-relativistic bound state.At the non-relativistic limit, the leading 2-particle distribution amplitude φ B c can beapproximated by delta function [36], fixing the light-cone momenta of the quarks accordingto their masses. According to Ref. [36], this form will become a smooth function afterconsidering the evolution effect from relativistic gluon exchange.16or the wave function of axial-vector meson, the longitudinal( L ) and transverse( T )polarizations are involved, and can be written as,Φ LA ( x ) = 1 √ γ (cid:8) m A ǫ/ ∗ LA φ A ( x ) + ǫ/ ∗ LA P/φ tA ( x ) + m A φ sA ( x ) (cid:9) αβ , (A3)Φ TA ( x ) = 1 √ γ (cid:8) m A ǫ/ ∗ TA φ vA ( x ) + ǫ/ ∗ TA P/φ TA ( x ) + m A iǫ µνρσ γ γ µ ǫ ∗ νT n ρ v σ φ aA ( x ) (cid:9) αβ , (A4)where ǫ L,TA denotes the longitudinal and transverse polarization vectors of axial-vectormeson, satisfying P · ǫ = 0 in each polarization, x denotes the momentum fraction carriedby quark in the meson, and n = (1 , , T ) and v = (0 , , T ) are dimensionless light-likeunit vectors. We here adopt the convention ǫ = 1 for the Levi-Civita tensor ǫ µναβ .The twist-2 distribution amplitudes φ A ( x ) and φ TA ( x ) in Eqs.(A3,A4) can be parame-terized as [24, 29]: φ A ( x ) = 3 f A √ x (1 − x ) (cid:20) a k A + 3 a k A (2 x −
1) + a k A
32 (5(2 x − − (cid:21) , (A5) φ TA ( x ) = 3 f A √ x (1 − x ) (cid:20) a ⊥ A + 3 a ⊥ A (2 x −
1) + a ⊥ A
32 (5(2 x − − (cid:21) , (A6)Here, the definition of these distribution amplitudes φ A ( x ) and φ TA ( x ) satisfy the followingnormalization relations: Z φ P ( x ) = f P √ , Z φ T P ( x ) = a ⊥ P f P √ Z φ P ( x ) = a || P f P √ , Z φ T P ( x ) = f P √ . (A7)where a || P = 1 and a ⊥ P = 1 have been used.As for the twist-3 distribution amplitudes in Eqs.(A3,A4), we use the followingform [29]: φ tA ( x ) = 3 f A √ (cid:26) a ⊥ A (2 x − + 12 a ⊥ A (2 x − x − − (cid:27) , (A8) φ sA ( x ) = 3 f A √ ddx (cid:8) x (1 − x )( a ⊥ A + a ⊥ A (2 x − (cid:9) . (A9) φ vA ( x ) = 3 f A √ (cid:26) a k A (1 + (2 x − ) + a k A (2 x − (cid:27) , (A10) φ aA ( x ) = 3 f A √ ddx n x (1 − x )( a k A + a k A (2 x − o . (A11)where f A is the decay constant of the relevant axial-vector meson. When the axial-vectormesons are K A and K B , x in the distribution amplitudes stands for the momentumfraction carrying by the s quark. 17he Gegenbauer moments have been studied extensively in the literatures (see Ref. [24]and references therein), here we adopt the following values: a || a = − . ± . a ⊥ a = − . ± . a || b = − . ± . a || f = − . ± . a ⊥ f = − . ± . a || h = − . ± . a || f = − . ± . a ⊥ f = − . ± . a || h = − . ± . a || K A = 0 . ± . a || K A = − . ± . a ⊥ K A = 0 . ± . a ⊥ K A = − . ± . a || K B = 0 . ± . a || K B = − . ± . a || K B = 0 . ± . a ⊥ K B = 0 . ± . . (A12) [1] F. Abe et al (CDF Collaboration), Phys. Rev. Lett. et al., (Quarkonium Working Group), CERN-2005-005, arXiv:0412158[hep-ph].[3] N. Brambilla et al , Eur. Phys. J. C , 1534 (2011), arXiv:1010.5827v3[hep-ph].[4] I.I. Bigi, Phys. Lett. B , 105 (1996); M. Beneke and G. Buchalla, Phys. Rev. D ,4991 (1996).[5] C.H. Chang and Y.Q. Chen, Phys. Lett. B , 3399 (1994); C.H. Chang, Y.Q. Chen andR.J. Oakes, Phys. Rev. D , 4344 (1996).[6] V.V. Kiselev, A.E. Kovalsky and A.K. Likhoded, Nucl. Phys. B , 353 (2000); V.V. Kise-lev, J. Phys. G , 1445 (2003).[7] S. Descotes-Genon, J. He, E. Kou and P. Robbe, Phys. Rev. D , 114031 (2009).[8] N. Sharma, Phys. Rev. D , 014027 (2010); N. Sharma and R.C. Verma, Phys. Rev. D , 094014 (2010); N. Sharma, R. Dhir and R.C. Verma, Phys. Rev. D , 014007 (2011).[9] J.F. Sun et al. , Phys. Rev. D , 074013 (2008); Phys. Rev. D , 114004 (2008); Eur.Phys. J. C , 107 (2009); Y.L. Yang, J.F. Sun and N. Wang, Phys. Rev. D , 074012(2010).[10] X. Liu, Z.J. Xiao and C.D. L¨u, Phys. Rev. D , 014022(2010).[11] X. Liu and Z.J. Xiao, Phys. Rev. D , 054029 (2010).[12] X. Liu and Z.J. Xiao, Phys. Rev. D , 074017(2010).[13] X. Liu and Z.J. Xiao, J. Phys. G 38, 035009 (2011).[14] T. Aaltonen et al., (CDF Collaboration), Phys. Rev. Lett. , 182002 (2008);V.M. Abazov et al., (D0 Collaboration), Phys. Rev. Lett. , 012001 (2008).[15] G. Buchalla, A.J. Buras and M.E. Lautenbacher, Rev. Mod. Phys. , 1125 (1996).[16] Y.Y. Keum, H.N. Li and A.I. Sanda, Phys. Lett. B , 6 (2001); Phys. Rev. D ,054008 (2001).[17] C.D. L¨u, K. Ukai and M.Z. Yang, Phys. Rev. D , 074009 (2001).[18] H.N. Li, Prog. Part. & Nucl. Phys. , 85 (2003), and reference therein.[19] Y. Li, C.D. L¨u, Z.J. Xiao, and X.Q. Yu, Phys. Rev. D , 034009 (2004).[20] A. Ali et al., Phys. Rev. D , 074018 (2007).[21] M.J. Morello et al., , (CDF Collaboration), CDF public note 10498 (2011).
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