Study of C parity violating and strangeness changing J/ψ {\to} PP weak decays
Yueling Yang, Junliang Lu, Mingfei Duan, Jinshu Huang, Junfeng Sun
aa r X i v : . [ h e p - ph ] F e b Study of the C parity violating andstrangeness changing J/ψ → P P weak decays
Yueling Yang, Junliang Lu, Mingfei Duan, 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
The
J/ψ weak decays are rare but possible within the standard model of elementary particles.Inspired by the potential prospects and sensitivity at the future intensity frontier, the
J/ψ → P P weak decays are studied with the perturbative QCD approach, where P denote the ground SU (3)pseudoscalar meson. It is found that the C parity violating J/ψ → η ( ′ ) η ( ′ ) decays have relativelylarge branching ratios, about the order of 10 − , which might be within the measurement capabilityand sensitivity of the future STCF experiment. . INTRODUCTION Now, nearly fifty years after the discovery of the
J/ψ particle in 1974 [1, 2], charmoniumphysics continues to be an interesting and exciting subject of research, because of manyrecent observations of exotic resonances beyond our comprehension, such as XYZs [3].The
J/ψ particle, a system consisting of the charmed quark and antiquark pair c ¯ c , isthe lowest orthocharmonium state with the well established quantum number of J P C =1 −− [3]. With the same J P C of the photon, the
J/ψ particle can be directly produced by e + e − annihilation. Up to date, there are more than 10 J/ψ events available with BESIIIdetector [4]. Considering the large
J/ψ production cross section σ ∼ nb [5], it isexpected that more than 10 J/ψ will be accumulated at the planning Super Tau CharmFacility (STCF) with 3 ab − on-resonance dataset in the future. The huge amount of dataprovides a good opportunity for studying the properties of J/ψ particle, understanding thestrong interactions and hadronic dynamics, exploring novel phenomena, and searching fornew physics (NP) beyond the standard model (SM).The mass of the
J/ψ particle, m ψ = 3096 . J/ψ hadronic decays through the annihilation of c ¯ c quark into gluons are ofhigher order in the quark-gluon coupling α s and are therefore severely suppressed by thephenomenological Okubo-Zweig-Iizuka (OZI) rule [6–8]. The OZI suppression results in (1)the electromagnetic decay ratio has the same order of magnitude as its strong decay ratio, B r ( J/ψ → γ ∗ → ℓ + ℓ − + hadrons) ≈
25% and B r ( J/ψ → ggg ) ≈
64% [3]; and (2) the small de-cay width, Γ ψ = 92 . ± . J/ψ hadronic decays, the simplest hadronicfinal states are two pseudoscalar mesons. However, only three decay modes,
J/ψ → π + π − , K + K − and K L K S , have been measured until now. The other decay modes, such as J/ψ → π π , π η ( ′ ) and η ( ′ ) η ( ′ ) , are forbidden explicitly by charge-conjugation ( C ) invariance, andthe J/ψ → πK , η ( ′ ) K decays are forbidden by the conservation of the strangeness quantumnumber. In fact, both the C parity violating modes and the flavor changing modes can beinduced by the weak interactions rather than the strong and electromagnetic interactionswithin SM. It is estimated that the branching ratios of J/ψ weak decays should usuallybe very small, about 2 /τ D Γ ψ ∼ O (10 − ), where τ D and Γ ψ are the lifetime of D mesonand full width of the J/ψ particle. Inspired by the potentials of BESIII and future STCPexperiments, in this paper, we will study the
J/ψ decays into two pseudoscalar mesons via2he weak interactions. In addition, as far as we know, there is no theoretical investigationon the C parity violating and strangeness changing J/ψ → P P weak decays yet. Our studywill provide a ready reference for future experimental analysis.
II. THE EFFECTIVE HAMILTONIAN
The effective Hamiltonian in charge of the
J/ψ → P P weak decay is written as [9], H eff = G F √ X q ,q V cq V ∗ cq (cid:8) C ( µ ) O ( µ ) + C ( µ ) O ( µ ) (cid:9) + h . c ., (1)where G F ≃ . × − GeV − [3] is the Fermi coupling constant; V cq , is the Cabibbo-Kobayashi-Maskawa (CKM) element, and q , ∈ { d , s } . The latest values of CKM elementsfrom data are | V cd | = 0 . ± .
004 and | V cs | = 0 . ± .
011 [3]. The factorization scale µ separates the physical contributions into the short- and long-distance parts. The Wilsoncoefficients C , summarize the short-distance physical contributions above the scales of µ . They are computable with the perturbative field theory at the scale of the W ± bosonmass m W , and then evolved to a characteristic scale of µ for c quark decay based on therenormalization group equations. ~C ( µ ) = U ( µ, m b ) M ( m b ) U ( m b , m W ) ~C ( m W ) , (2)where the explicit expression of the evolution matrix U f ( µ f , µ i ) and the threshold matchingmatrix M ( m b ) can be found in Ref. [9]. The operators describing the local interactionsamong four quarks are defined as, O = (cid:2) ¯ c α γ µ (1 − γ ) q ,α (cid:3) (cid:2) ¯ q ,β γ µ (1 − γ ) c β (cid:3) , (3) O = (cid:2) ¯ c α γ µ (1 − γ ) q ,β (cid:3) (cid:2) ¯ q ,β γ µ (1 − γ ) c α (cid:3) , (4)where α and β are color indices.It should be pointed out that the contributions of penguin operators being proportionalto the CKM factors V uq V ∗ uq + V cq V ∗ cq = − V tq V ∗ tq , are not considered here, because theyare suppressed by a factor of λ against the tree contributions being proportional to V cq V ∗ cq ,where the Wolfenstein parameter λ ≈ . A ( J/ψ → P P ) = G F √ X q ,q V cq V ∗ cq X i =1 C i ( µ ) h P P | O i ( µ ) | J/ψ i , (5)3here the hadron transition matrix elements (HMEs) h P P | O i ( µ ) | J/ψ i = h O i i relate thequark operators with the concerned hadrons. With inadequate understanding of hadronzi-ation mechanism, the remaining and most critical theoretical work is to properly computeHMEs. And it is not difficult to imagine that the main uncertainties will come from HMEscontaining nonperturbative contributions. III. HADRON TRANSITION MATRIX ELEMENTS
In the past few years, many phenomenological models, such as the QCD factorization(QCDF) approach [10–15] and the perturbative QCD (pQCD) approach [16–22], have beenfully developed and widely employed in evaluation of HMEs. According to these phenomeno-logical models, HMEs are usually written as the convolution of scattering amplitudes andthe hadronic wave functions (WFs). The scattering amplitudes and WFs reflect the contri-butions at the quark and hadron levels, respectively. The scattering amplitudes describingthe interactions between hard gluons and quarks are perturbatively calculable. WFs rep-resenting the momentum distribution of compositions in hadron are regarded as processindependent and universal, and could be obtained by nonperturbative methods or fromdata. A potential disadvantage of the QCDF approach [10–15] in the practical calculationis that the annihilation contributions cannot be computed self-consistently, and other phe-nomenological parameters are introduced to deal with the soft endpoint divergences usingthe collinear approximation. With the pQCD approach [16–22], in order to regularize theendpoint contributions of the QCD radiative corrections to HMEs, the transverse momen-tum are suggested to be retained within the scattering amplitudes on the one hand, and onthe other hand a Sudakov factor is introduced expressly for WFs of all involved hadrons. Fi-nally, the pQCD decay amplitudes are expressed as the convolution integral of three parts :the ultra-hard contributions embodied by Wilson coefficients C i , hard scattering amplitudes H and soft part contained in hadronic WFs Φ. A i = Y j Z dx j db j C i ( t i ) H i ( t i , x j , b j ) Φ j ( x j , b j ) e − S j , (6)where x j is the longitudinal momentum fraction of the valence quark, b j is the conjugatevariable of the transverse momentum, and e − S j is the Sudakov factor. In this paper, we willadopt the pQCD approach to investigate the J/ψ → P P weak decays within SM.4
V. KINEMATIC VARIABLES
For the
J/ψ → P P weak decays, the valence quarks of the final states are entirelydifferent from those of the initial state. There is only annihilation configurations. So the
J/ψ → P P weak decays provide us with some typical processes to closely scrutinize thepure annihilation contributions. As one example, the Feynman diagram for
J/ψ → π − K + decay are shown in Fig. 1. c ( k )¯ cψ ¯ s ( k ) u K + d ¯ u ( k ) π − G c ¯ cψ ¯ su K + d ¯ u π − G c ¯ cψ ¯ su K + d ¯ u π − G c ¯ cψ ¯ su K + d ¯ u π − G (a) (b) (c) (d)FIG. 1: Feynman diagrams for the J/ψ → π − K + 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 scattering amplitudes. It is convenient to use the light-cone vectors to define the kinematic variables. In the restframe of the
J/ψ particle, one has p ψ = p = m ψ √ , , , (7) p K = p = m ψ √ , , , (8) p π = p = m ψ √ , , , (9) k = x p + (0 , , ~k ⊥ ) , (10) k = x p +2 + (0 , , ~k ⊥ ) , (11) k = x p − + (0 , , ~k ⊥ ) , (12) ǫ k ψ = 1 √ , − , , (13)where k i , x i and ~k i ⊥ are respectively the momentum, longitudinal momentum fraction andtransverse momentum; the quark momentum k i is illustrated in Fig. 1 (a); ǫ k ψ is the lon-gitudinal polarization vector of the J/ψ particle, and satisfies with both the normalizationcondition ǫ k ψ · ǫ k ψ = − ǫ k ψ · p ψ = 0; all hadrons are on mass shell, i.e. , p = m ψ , p = 0 and p = 0. 5 . HADRONIC WAVE FUNCTIONS With the convention of Refs. [23–26], the WFs and distribution amplitudes (DAs) aredefined as follows. h | ¯ c α (0) c β ( z ) | ψ ( p , ǫ k ) i = 14 f ψ Z dk e + i k · z (cid:8) ǫ k (cid:2) m ψ φ vψ − 6 p φ tψ (cid:3)(cid:9) βα , (14) h K ( p ) | ¯ u α (0) s β ( z ) | i = − i f K Z dk e − i k · z (cid:8) γ (cid:2) p φ aK + µ K φ pK − µ K (cid:0) n + n − − (cid:1) φ tK (cid:3)(cid:9) βα , (15) h π ( p ) | ¯ d α (0) u β ( z ) | i = − i f π Z dk e − i k · z (cid:8) γ (cid:2) p φ aπ + µ π φ pπ − µ π (cid:0) n − n + − (cid:1) φ tπ (cid:3)(cid:9) βα , (16)where f ψ , f K and f π are decay constants; µ K,π = 1 . ± . n + = (1 , ,
0) and n − = (0 , ,
0) are the null vectors; φ aP and φ p,tP are twist-2 and twist-3. Theexplicit expressions of φ v,tψ and φ a,p,tP can be found in Ref. [23] and Refs. [25, 26], respectively.We collect these DAs as follows. φ vψ ( x ) = A x ¯ x exp (cid:0) − m c ω x ¯ x (cid:1) , (17) φ tψ ( x ) = B (¯ x − x ) exp (cid:0) − m c ω x ¯ x (cid:1) , (18) φ aP ( x ) = 6 x ¯ x (cid:8) a P C / ( ξ ) + a P C / ( ξ ) (cid:9) , (19) 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 ) , (20) φ tP ( x ) = 32 ( ρ P − − ρ P + a P + 6 ρ P − a P ) − 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 )6 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 ) , (21)where ¯ x = 1 − x and ξ = x − ¯ x . ω = m c α s ( m c ) is the shape parameter. The parameters A in Eq.(17) and B in Eq.(18) are determined by the normalization conditions, Z φ v,tψ ( x ) dx = 1 . (22)The meaning and definition of other parameters can refer to Refs. [25, 26]. VI. DECAY AMPLITUDES
When the final states include the isoscalar η or/and η ′ , we assume that the componentsof glueball, charmonium or bottomonium are negligible. The physical η and η ′ states arethe mixtures of the SU (3) octet and singlet states. In our calculation, we will adopt thequark-flavor basis description proposed in Ref. [27], i.e. , ηη ′ = cos φ − sin φ sin φ cos φ η q η s , (23)where the mixing angle is φ = (39 . ± . ◦ , and η q = ( u ¯ u + d ¯ d ) / √ η s = s ¯ s . In addition,we assume that DAs of η q and η s are the same as those of pion, but with different decayconstants and mass [27–29], f q = (1 . ± . f π , (24) f s = (1 . ± . f π , (25) m η q = m η cos φ + m η ′ sin φ − √ f s f q ( m η ′ − m η ) cos φ sin φ, (26) m η s = m η sin φ + m η ′ cos φ − f q √ f s ( m η ′ − m η ) cos φ sin φ. (27)For the C parity violating J/ψ decays, the amplitudes are √ A ( J/ψ → π π ) = G F √ V cd V ∗ cd (cid:8) a A ab ( π, π ) + C A cd ( π, π ) (cid:9) , (28) A ( J/ψ → π η q ) = − G F √ V cd V ∗ cd (cid:8) a (cid:2) A ab ( π, η q ) + A ab ( η q , π ) (cid:3) C (cid:2) A cd ( π, π ) + A cd ( η q , π ) (cid:3)(cid:9) , (29) A ( J/ψ → π η ) = A ( J/ψ → π η q ) cos φ, (30) A ( J/ψ → π η ′ ) = A ( J/ψ → π η q ) sin φ, (31) A ( J/ψ → η s η s ) = √ G F V cs V ∗ cs (cid:8) a A ab ( η s , η s ) + C A cd ( η s , η s ) (cid:9) , (32) A ( J/ψ → η q η q ) = G F √ V cd V ∗ cd (cid:8) a A ab ( η q , η q ) + C A cd ( η q , η q ) (cid:9) , (33) √ A ( J/ψ → ηη ) = A ( J/ψ → η q η q ) cos φ + A ( J/ψ → η s η s ) sin φ, (34) A ( J/ψ → ηη ′ ) = (cid:8) A ( J/ψ → η q η q ) − A ( J/ψ → η s η s ) (cid:9) sin φ cos φ, (35) √ A ( J/ψ → η ′ η ′ ) = A ( J/ψ → η q η q ) sin φ + A ( J/ψ → η s η s ) cos φ. (36)For the strangeness changing J/ψ decays, the amplitudes are A ( J/ψ → π − K + ) = G F √ V cs V ∗ cd (cid:8) a A ab ( π, K ) + C A cd ( π, K ) (cid:9) , (37) A ( J/ψ → π K ) = − G F V cs V ∗ cd (cid:8) a A ab ( π, K ) + C A cd ( π, K ) (cid:9) , (38) A ( J/ψ → K η s ) = G F √ V cs V ∗ cd (cid:8) a A ab ( K, η s ) + C A cd ( K, η s ) (cid:9) , (39) A ( J/ψ → K η q ) = G F V cs V ∗ cd (cid:8) a A ab ( η q , K ) + C A cd ( η q , K ) (cid:9) , (40) A ( J/ψ → K η ) = A ( J/ψ → K η q ) cos φ − A ( J/ψ → K η s ) sin φ, (41) A ( J/ψ → K η ′ ) = A ( J/ψ → K η q ) sin φ + A ( J/ψ → K η s ) cos φ, (42)where coefficient a = C + C /N c and the color number N c = 3; The amplitude buildingblocks A ij are listed in Appendix A. From the above amplitudes, it is seen that if the J/ψ → π π and πη ( ′ ) decays were experimentally observed, then the CKM element V cd could beconstrained or extracted. VII. NUMERICAL RESULTS AND DISCUSSION
The branching ratio is defined as follows. B r = p cm π m ψ Γ ψ |A ( J/ψ → P P ) | , (43)8 ABLE I: The values of the input parameters, where their central values will be regarded as thedefault inputs unless otherwise specified.mass, width and decay constants of the particles [3] m π = 134 .
98 MeV, m K = 497 .
61 MeV, f π = 130 . ± . m π ± = 139 .
57 MeV, m K ± = 493 .
68 MeV, f K = 155 . ± . m η = 547 .
86 MeV, m η ′ = 957 .
78 MeV, f ψ = 395 . ± . m c = 1 . ± .
07 GeV, m J/ψ = 3096 . ψ = 92 . ± . µ = 1 GeV [25] a π = 0, a π = 0 . ± . a K = 0 . ± . a K = 0 . ± . J/ψ → P P weak decays, where the uncertainties come frommesonic DAs, including the parameters of m c , µ P and a P .the C parity violating decay modesmode B r mode B rJ/ψ → π π (1 . +0 . − . ) × − J/ψ → ηη (1 . +0 . − . ) × − J/ψ → π η (2 . +0 . − . ) × − J/ψ → ηη ′ (3 . +0 . − . ) × − J/ψ → π η ′ (1 . +0 . − . ) × − J/ψ → η ′ η ′ (2 . +0 . − . ) × − the strangeness changing decay modesmode B r mode B rJ/ψ → π − K + (0 . +0 . − . ) × − J/ψ → K η (0 . +0 . − . ) × − J/ψ → π K (0 . +0 . − . ) × − J/ψ → K η ′ (1 . +0 . − . ) × − where p cm is the center-of-mass momentum of final states in the rest frame of the J/ψ particle. Using the inputs in Table I, the numerical results of branching ratios are obtainedand listed in Table II. Our comments on the results are listed as follows.(1) The
J/ψ → η ( ′ ) η ( ′ ) decays are Cabibbo-favored. The J/ψ → Kπ and Kη ( ′ ) de-cays are singly Cabibbo-suppressed. The J/ψ → π π and πη ( ′ ) decays are doublyCabibbo-suppressed. So there is a hierarchical structure, i.e. , B r ( J/ψ → η ( ′ ) η ( ′ ) ) ∼ O (10 − ), B r ( J/ψ → Kπ, Kη ( ′ ) ) ∼ O (10 − − − ) and B r ( J/ψ → π π , πη ( ′ ) ) ∼ O (10 − ).(2) Compared with the external W -emission induced J/ψ → D ( s ) M decays [23, 30], the9nternal W -exchange induced J/ψ → P P decays are color-suppressed because the two lightvalence quarks of the effective operators belong to different final states. So the branchingratios for
J/ψ → P P decays are less than those for
J/ψ → D ( s ) M decays with the sameCKM factors by one or two orders of magnitude.(3) The nonperturbative mesonic DAs are the essential parameters of the amplitudes withthe pQCD approach. One of the main theoretical uncertainties arising from participatingDAs is given in Table II. Besides, there are many other influence factors. For example, thedecay constant f ψ and width Γ ψ will bring 2.5% and 3% uncertainties to branching ratios.(4) Branching ratios for the J/ψ → η ( ′ ) η ( ′ ) decays can reach up to the order of 10 − , whichare far beyond the measurement precision and capability of current BESIII experiment, butmight be accessible at the future high-luminosity STCF experiment. It will be very difficultand challenging but interesting to search for the J/ψ → P P weak decays experimentally.An observation of the phenomenon of an abnormally large occurrence probability would bea hint of NP.
VIII. SUMMARY
Within the SM, the C parity violating J/ψ → π π and π η ( ′ ) decays and the strangenesschanging J/ψ → Kπ and Kη ( ′ ) decays are legitimate and possible only through the weakinteractions, but they are very rare. In this paper, based on the lastest progress and furtureprospects of the J/ψ physics at high-luminosity collider, we studied the
J/ψ → P P weakdecays with the pQCD approach for the first time. It is found that branching ratios for the
J/ψ → η ( ′ ) η ( ′ ) decays can reach up to the order of 10 − , which might be measurable by thefuture STCF experiment. Acknowledgments
The work is supported by the National Natural Science Foundation of China (Grant Nos.11705047, U1632109 and 11547014). 10 ppendix A: Amplitude building blocks
From the definition of Eq.(15) and Eq.(16), it is clearly seen that the twist-3 DAs arealways accompanied by a chiral mass µ P . With the pQCD approach, a Sudakov factor isintroduced for each of hadronic WFs.We take the J/ψ → K + π − decay as an example. For the sake of simplicity, we use thefollowing shorthand forms. φ v,tψ = φ v,tψ ( x ) e − S ψ , (A1) φ aK = φ aK ( x ) e − S K , (A2) φ p,tK = µ K m ψ φ p,tK ( x ) e − S K , (A3) φ aπ = φ aπ ( x ) e − S π , (A4) φ p,tπ = µ π m ψ φ p,tπ ( x ) e − S π , (A5)where the definitions of Sudakov factors are S ψ = s ( x , p +1 , b ) + 2 Z t /b dµµ γ q , (A6) S K = s ( x , p +2 , b ) + s (¯ x , p +2 , /b ) + 2 Z t /b dµµ γ q , (A7) S π = s ( x , p − , b ) + s (¯ x , p − , /b ) + 2 Z t /b dµµ γ q . (A8)The expression of s ( x, Q, b ) can be found in Ref. [18]. γ q = − α s /π is the quark anomalousdimension. In addition, the decay amplitudes are always the functions of Wilson coefficient C i . It should be understood that the shorthand C i A jk ( π, K ) = π C F N c m ψ f ψ f K f π (cid:8) A j ( C i ) + A k ( C i ) (cid:9) , (A9)where the color factor C F = 4 / N c = 3. The subscripts j and k ofbuilding block A j ( k ) correspond to the indices of Fig. 1. The expressions of A i are writtenas follows. A a = Z dx dx Z ∞ db db H ab ( α g , β a , b , b ) α s ( t a ) C i ( t a ) S t (¯ x ) (cid:8) φ aK φ aπ ¯ x − φ pπ (cid:2) φ pK x + φ tK (1 + ¯ x ) (cid:3)(cid:9) , (A10)11 b = Z dx dx Z ∞ db db H ab ( α g , β b , b , b ) α s ( t b ) C i ( t b ) S t ( x ) (cid:8) φ aK φ aπ x − φ pK (cid:2) φ pπ ¯ x − φ tπ (1 + x ) (cid:3)(cid:9) , (A11) A c = 1 N c Z dx dx dx Z ∞ db db H cd ( α g , β c , b , b ) α s ( t c ) C i ( t c ) (cid:8) φ vψ (cid:2) φ aK φ aπ ( x − x ) + ( φ pK φ pπ − φ tK φ tπ ) (¯ x − x )+( φ pK φ tπ − φ tK φ pπ ) (2 x − ¯ x − x ) (cid:3) − φ tψ (cid:2) φ aK φ aπ + 2 φ pK φ tπ (cid:3)(cid:9) b = b , (A12) A d = 1 N c Z dx dx dx Z ∞ db db H cd ( α g , β d , b , b ) α s ( t d ) C i ( t d ) (cid:8) φ vψ (cid:2) φ aK φ aπ ( x − x ) + ( φ pK φ pπ − φ tK φ tπ ) ( x − ¯ x )+( φ pK φ tπ − φ tK φ pπ ) (2 ¯ x − ¯ x − x ) (cid:3) − φ tψ (cid:2) φ aK φ aπ − φ tK φ pπ (cid:3)(cid:9) b = b , (A13) 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) , (A14) H cd ( α, β, b , b )= b b (cid:8) i π θ ( β ) (cid:2) J ( b p β ) + i Y ( b p β ) (cid:3) + θ ( − β ) K ( b p − β ) (cid:9) i π (cid:8) θ ( b − b ) (cid:2) J ( b √ α ) + i Y ( b √ α ) (cid:3) J ( b √ α ) + ( b ↔ b ) (cid:9) , (A15)where I , J , K and Y are Bessel functions. The parametrization of the Sudakov factor S t ( x ) can be found in Ref. [22]. The virtualities of gluons and quarks are α g = m ψ ¯ x x , (A16) β a = m ψ ¯ x , (A17) β b = m ψ x , (A18) β c = α g − m ψ x (¯ x + x ) , (A19) β d = α g − m ψ ¯ x (¯ x + x ) , (A20)12 a,b = max( p β a,b , /b , /b ) , (A21) t c,d = max( √ α g , q | β c,d | , /b , /b ) . (A22) [1] J. Aubert et al. , Phys. Rev. Lett. 33, 1404 (1974).[2] J. Augustin et al. , Phys. Rev. Lett. 33, 1406 (1974).[3] P. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020).[4] http://english.ihep.cas.cn/bes/doc/2250.html.[5] M. Ablikim et al. (BESIII Collaboration), Nucl. Instr. Meth. Phys. Res. A 614, 345 (2010).[6] S. Okubo, Phys. Lett. 5, 165 (1963).[7] G. Zweig, CERN-TH-401, 402, 412 (1964).[8] J. Iizuka, Prog. Theor. Phys. Suppl. 37-38, 21 (1966).[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] Y. Yang, J. Sun, J. Gao, Q. Chang, J. Huang, G. Lu, Int. J. Mod. Phys. A 31, 1650161 (2016).[24] T. Kurimoto, H. Li, A. Sanda, Phys. Rev. D 65, 014007 (2001).[25] P. Ball, V. Braun, A. Lenz, JHEP 0605, 004 (2006).
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