Revisiting the production of J/ψ+ η c via the e + e − annihilation within the QCD light-cone sum rules
Long Zeng, Hai-Bing Fu, Dan-Dan Hu, Ling-Li Chen, Wei Cheng, Xing-Gang Wu
aa r X i v : . [ h e p - ph ] F e b Revisiting the production of
J/ψ + η c via the e + e − annihilation within the QCDlight-cone sum rules Long Zeng , Hai-Bing Fu , , , ∗ Dan-Dan Hu , and Ling-Li Chen Department of Physics, Guizhou Minzu University, Guiyang 550025, P.R. China
Wei Cheng , † Institute of Theoretical Physics, Chinese Academy of Sciences, P.O.Box 2735, Beijing 100190, P.R. China and CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, P.R. China
Xing-Gang Wu ‡ Department of Physics, Chongqing University, Chongqing 401331, P.R. China (Dated: February 24, 2021)We make a detailed study on the typical production channel of double charmoniums, e + e − → J/ψ + η c , at the center-of-mass collision energy √ s = 10 .
58 GeV. The key component of the processis the form factor F VP ( q ), which has been calculated within the QCD light-cone sum rules (LCSR).To improve the accuracy of the derived LCSR, we keep the J/ψ light-cone distribution amplitudeup to twist-4 accuracy. Total cross sections for e + e − → J/ψ + η c at three typical factorizationscales are σ | µ s = 22 . +3 . − . fb, σ | µ k = 21 . +3 . − . fb and σ | µ = 21 . +3 . − . fb, respectively. Thefactorization scale dependence is small, and those predictions are consistent with the BABAR andBelle measurements within errors. I. INTRODUCTION
Double charmonium production at the B -factories hasattracted large attention of experimentalists and theo-rists for a long time. At the beginning of this century, to-tal cross section of e + e − → J/ψ + η c at the center-of-masscollision energy √ s = 10 .
58 GeV was firstly reported bythe Belle Collaboration, σ ( e + e − → J/ψ + η c ) × B ≥ =33 . +7 . − . ± . B ≥ being the branching ratio of η c into four or more charged tracks [1], which was updateto σ ( e + e − → J/ψ + η c ) × B ≥ = 25 . ± . ± . σ ( e + e − → J/ψ + η c ) × B ≥ = 17 . ± . +1 . − . fb [3].Those measurements have severe discrepancy with theleading-order (LO) predictions based on the nonrelativis-tic QCD (NRQCD) factorization theory, which are withinthe range of 2 . ∼ . σ = 18 . − σ = 17 . +8 . − . fb [8] by furtherincluding relativistic corrections. A recent scale-invariantNRQCD prediction has been given in Ref.[9] by applyingthe principle of maximum conformality (PMC) [10–13],which gives σ = 20 . +3 . − . fb, where the uncertainties aresquared averages of the errors due to uncertainties fromthe charm-quark mass and the quarkonium wavefunctionat the origin. Thus, it could be treated as another suc-cessful application of NRQCD. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected]
The total cross-section of e + e − → J/ψ + η c has alsobeen studied by using the light-cone formalism [14–17].Within the light-cone formalism, the amplitude of theprocess can be factorized as the perturbatively calcula-ble short-distance part and the non-perturbative light-cone distribution amplitudes (LCDAs), which results in σ = 14 . +11 . − . fb [18]. The electromagnetic form fac-tor F VP ( q ) dominates the light-cone formalism, whichcan be calculated by using the QCD light-cone sum rules(LCSR). In Ref.[19], after applying the operator produc-tion expansion (OPE) near the light cone and takingthe η c leading-twist LCDA into account, the authors ob-tained a large factorization scale dependent total cross-section. By choosing the factorization scale as µ s =5 . σ | µ s = 25 . ± .
55 fb;and by setting the factorization as µ k = 3 . σ | µ k = 13 . ± .
32 fb. Aphysical observable should be independent to the choiceof factorization scale, and in the present paper, we shalladopt the LCSR approach to reanalyze the process andits factorization scale dependence.The LCSR prediction should be independent to anychoice of the correlator, an example for the QCD sumrules prediction of the B -meson constant f B under vari-ous choices of the correlator has been given in Ref.[20]. Itis helpful to show whether other choices of correlator canalso explain the data. As a new attempt, in the presentpaper, we shall adopt different correlator from Ref.[19]to do the LCSR calculation, in which the J/ψ
LCDAsother than the η c LCDAs shall be introduced. To improvethe accuracy, we shall keep the
J/ψ
LCDAs up to twist-4 accuracy, i.e., the resultant form factor F VP ( q ) willcontain φ λ J/ψ ( x ), φ λ J/ψ ( x ), φ λ J/ψ ( x ), ψ ⊥ J/ψ ( x ) with λ = ( k , ⊥ ), which correspond to longitudinal and trans-verse distributions, respectively.The remaining parts of the paper are organized as fol-lows. In Sec. II, we present the calculation technologyfor dealing with the form factor F VP ( q ) up to twist-4accuracy within the LCSR approach. Our choices of the J/ψ
LCDAs shall also be given here. In Sec. III, thephenomenological results and discussions are presented.Section IV is reserved for a summary.
II. THEORETICAL FRAMEWORKA. Cross section for e + + e − → J/ψ + η c In this subsection, we give a brief review on how to cal-culate the cross-section of the process e + ( p ) + e − ( p ) → J/ψ ( p ) + η c ( p ), which can be written as [21] σ = 14 E E v rel Z d ~p d ~p (2 π ) E (2 π ) E (2 π ) × δ ( p + p − p − p ) |M| , (1)where p i = ( E i , ~p i ) stands for the four-momentum of theinitial or final particle, and the relative velocity betweenpositron and electron, v rel = | ~p /E − ~p /E | . |M| isthe squared absolute value of the matrix element, wherethe color states and spin projections of the initial andfinal particles have been summed up and those of theinitial particles have been averaged. The matrix element M can be written as M = i Z d x × h V P | T (cid:8) Q c J cµ ( x ) A µ ( x ) , ¯ e (0) γ ν e (0) A ν (0) (cid:9) | e + e − i . (2)Hereafter, to simplify the notation, we set V = J/ψ and P = η c . The c -quark electromagnetic current J cµ ( x ) =¯ c ( x ) γ µ c ( x ). Then, we obtain |M| = 2 Q c | F VP ( q ) | p | p | s (cid:2) θ (cid:3) , (3)where θ is the scattering angle, Q c = 2 / c -quark, s = − q = ( p + p ) or ( p + p ) , | p | is themagnitude of the three-momentum of one of the final-state mesons in the center-of-mass frame.The form factor F VP ( q ) is defined through the follow-ing matrix element [15] h J/ψ ( p , λ ) , η c ( p ) | J Vµ | i = ε µναβ ˜ ǫ ∗ ( λ ) ν p α p β F VP ( q ) , (4)where ǫ ν is the polarization vector of J/ψ . Neglectingthe spin-flitting effects, we have m η c = m J/ψ , and thecross section becomes σ = πα Q c − m J/ψ s ! / | F VP ( q ) | . (5) B. The form factor F VP ( q ) within the QCD LCSR To derive the form factor F VP ( q ) within the QCDLCSR approach, we start with the following two-pointcorrelation function (correlator)Π µν ( p, q ) = i Z d xe iq · x h V ( p, λ ) | T { J Vµ ( x ) , J Aν (0) }| i , (6)where q and p are four-momentum of the virtual photonand J/ψ . The current J Aν ( x ) = ¯ c ( x ) γ ν γ c ( x ) is the c -quark axial-vector current.On the one hand, we deal with the hadronic represen-tation of the correlator. It can be calculated by insertinga complete set of the intermediate hadronic states intothe correlator, e.g.Π µν ( p, q ) = h V ( p, λ ) | J Vµ (0) | P ( p − q ) ih P ( p − q ) | J Pν (0) | i m P − ( p − q ) + 1 π Z ∞ s ds ImΠ µν s − ( p − q ) , (7)where ǫ ν is the polarization vector of J/ψ and s is thecontinuum threshold parameter, whose value could be setnear the squared mass of the lowest vector charmoniumstate. The dispersion integration in Eq.(7) contains thecontributions from the higher resonances and the contin-uum states. The matrix element h V ( p, λ ) | J Vµ (0) | P ( p − q ) i and h P ( p − q ) | J Aν (0) | i are defined as h V ( p, λ ) | J Vµ (0) | P ( p − q ) i = ε µναβ ˜ ǫ ∗ ( λ ) ν q α p β F VP ( q ) , (8) h | J Aν (0) | P ( p − q ) i = if P ( p − q ) ν , (9)where f P is the η c decay constant. Inserting Eqs.(8, 9)into Eq.(7), we obtainΠ Had µν ( p, q ) = ε µναβ ˜ ǫ ∗ ( λ ) α p β m P f P F VP ( q ) m P − ( p − q ) + 1 π Z ∞ s ds F µν ( q ) s − ( p − q ) . (10)On the other hand, the correlator in the large space-likeregion, i.e. ( p + q ) − m c ≪ q ∼ O (1 GeV) ≪ m c for the momentum transfer, corresponds to the T -product of quark currents near small light-cone distance x →
0, which can be treated by operator product expan-sion (OPE) with the coefficients being pQCD calculable.For such purpose, we contract the two c -quark fields andwrite down a free c -quark propagator with gluon field S c ( x,
0) = h | c iα ( x )¯ c jβ (0) | i as follows [22, 23] h | c iα ( x )¯ c jβ (0) | i = − i Z d k (2 π ) e − ik · x (cid:26) δ ij /k + m c m c − k + g s Z dv G µν ( vx ) (cid:18) λ (cid:19) ij (cid:20) /k + m c m c − k ) σ µν + 1 m c − k vx µ γ ν (cid:21)(cid:27) αβ . (11)Substituting Eq.(11) into the correlator, one needs todeal with the matrix elements of the nonlocal operatorsbetween vector meson and vacuum state, that is, h V ( p, λ ) | ¯ q ( x ) σ µν q (0) | i = if ⊥ V Z du ˜ ǫ iup · x × (cid:26) (˜ ǫ ∗ ( λ ) µ p ν − ˜ ǫ ∗ ( λ ) ν p µ ) (cid:20) φ ⊥ V ( u ) + m V x φ ⊥ V ( u ) (cid:21) + ( p µ x ν − p ν x µ ) ˜ ǫ ∗ ( λ ) · x ( p · x ) m V (cid:20) φ k V ( u ) − φ ⊥ V ( u ) − ψ ⊥ V ( u ) (cid:21) + 12 (˜ ǫ ∗ ( λ ) µ x ν − ˜ ǫ ∗ ( λ ) ν x µ ) m V p · x (cid:20) ψ ⊥ V ( u ) − φ ⊥ V ( u ) (cid:21)(cid:27) , (12) h V ( p, λ ) | ¯ q ( x ) γ µ q (0) | i = m V f k V Z due iup · x × (cid:26) ˜ ǫ ∗ ( λ ) µ φ ⊥ V ( u ) + ˜ ǫ ∗ ( λ ) · xp · x p µ (cid:20) φ k V ( u ) + φ ⊥ V ( u ) (cid:21) + ˜ ǫ ∗ ( λ ) · x ( p · x ) p µ m V x φ k V ( u ) − x µ ˜ ǫ ∗ ( λ ) · x ( p · x ) m V × (cid:20) ψ k V ( u ) + φ k V ( u ) − φ ⊥ V ( u ) (cid:21)(cid:27) , (13)and h V ( p, λ ) | ¯ q ( x ) iγ µ gG αβ ( vx ) q (0) | i = p µ (˜ ǫ ∗ ( λ ) ⊥ α p β − ˜ ǫ ∗ ( λ ) ⊥ β p α ) × f k V m V Φ k V ( v, p · x ) + ( p α g ⊥ µβ − p β g ⊥ µα ) ˜ ǫ ∗ ( λ ) · xp · x × f k V m V Φ k V ( v, p · x ) + p µ ( p α x β − p β x α ) ˜ ǫ ∗ ( λ ) · xp · x × f k V m V Ψ k V ( v, p · x ) + . . . . (14)The J/ψ
LCDAs φ k , ⊥ V ( u ), φ k , ⊥ V ( u ) and φ k , ⊥ V ( u )/ ψ ⊥ V ( u )stand for the two-particles twist-2, twist-3 and twist-4ones, respectively; and the J/ψ
LCDAs Φ k V ( v, p · x ) andΦ k V ( v, p · x ) / Ψ k V ( v, p · x ) stand for the three-particlestwist-3 and twist-4 ones, respectively.Inserting the above LCDAs into the correlator (6), andcompleting the integration over x and k , we can derivethe OPE representation of the correlator. By equatingboth phenomenological and theoretical sides of the cor-relator and employ the usual Borel transform B M Π( q ) = lim − q ,n →∞− q /n = M ( − q ) n +1 n ! (cid:18) ddq (cid:19) n Π( q ) , (15)the LCSR for the form factors F VP ( q ) can be obtained,which reads F VP ( q ) = m V m P f P (cid:26) Z due ( m P − s ( u )) /M (cid:26) m c m V f ⊥ V (cid:20) um V Θ( c ( u, s )) φ ⊥ V ( u ) − m c u M × ˜˜Θ( c ( u, s )) φ ⊥ V ( u ) − u M ˜Θ( c ( u, s )) I L ( u ) − uM ˜Θ( c ( u, s )) H ( u ) (cid:21) + f k V × (cid:20) Θ( c ( u, s )) φ ⊥ V ( u ) + 1 u Θ( c ( u, s )) A ( u ) − m V (cid:18) m c u M ˜˜Θ( c ( u, s )) + 1 u M × ˜Θ( c ( u, s )) (cid:19) B ( u ) (cid:27) + f k V Z D α i Z dv e ( m P − s ( X )) /M (cid:20) m V (2 v + 1) 1 XM × ˜Θ( c ( X, s )) + (4 v + 1)( m V − m P + q ) 14 X M ˜Θ( c ( X, s )) (cid:21) Φ k V ( α ) (cid:27) , (16)where α i = ( α , α , α ), s ( X ) = [ m c − ¯ X ( q − Xm V )] /X with X = α + vα and ¯ X = (1 − X ). The integrationover x can be done by transforming the x µ in the nomina-tor to i∂/∂ ( up µ ), or equivalently to − i∂/∂q µ , and maketransformation1 p · x φ ( u ) → − i Z u dvφ ( v ) ≡ − i Φ( u ) . (17)The simplified distribution functions I L ( u ), H ( u ), A ( u ) and B ( u ) are defined as: I L ( u ) = Z u dv Z v dw (cid:20) φ k V ( w ) − φ ⊥ V ( w ) − ψ ⊥ V ( w ) (cid:21) ,H ( u ) = Z u dv (cid:20) ψ ⊥ V ( v ) − φ ⊥ V ( v ) (cid:21) ,A ( u ) = Z u dv h φ k V ( u ) + φ ⊥ V ( u ) i , B ( u ) = Z u dvφ k V ( u ) . (18)The Θ( c ( u, s )) with c ( u, s ) = us − m b + ¯ uq − u ¯ um V isthe conventional step function, ˜Θ[ c ( u, s )] and ˜˜Θ[ c ( u, s )]take the following form Z duu M e − s ( u ) /M ˜Θ( c ( u, s )) f ( u )= Z u duu M e − s ( u ) /M f ( u ) + δ ( c ( u , s )) , (19) Z du u M e − s ( u ) /M ˜˜Θ( c ( u, s )) f ( u )= Z u du u M e − s ( u ) /M f ( u ) + ∆( c ( u , s )) , (20)where δ ( c ( u, s )) = e − s /M f ( u ) C , ∆( c ( u, s )) = e − s /M (cid:20) u M f ( u ) C − u C ddu (cid:18) f ( u ) u C (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) u = u (cid:21) , C = m b + u m V − q and u is the solution of c ( u , s ) =0 with 0 ≤ u ≤ C. The
J/ψ
LCDAs
The important components for the form factor F VP ( q )are the gauge-independent and process-independentLCDAs, which can be derived from the wavefunctionby integrating over the transverse components. For the J/ψ
LCDAs, we start from the following Brodsky-Huang-Lepage (BHL) [25]
J/ψ longitudinal/transverse twist-2wavefunction, ψ λ J ; ψ ( x, k ⊥ ) = χ J/ψ ( k ⊥ ) ψ λ,R J ; ψ ( x, k ⊥ ) , (21)where k ⊥ stands for the transverse momentum, χ J/ψ ( k ⊥ )is the spin-space wavefunction which can be taken as theform χ J/ψ ( k ⊥ ) = ˆ m c / p k ⊥ + ˆ m c . The ˆ m c = 1 . ψ λ,R J ; ψ ( x, k ⊥ ) can be written as: ψ λ,R J ; ψ ( x, k ⊥ ) = A λJ/ψ exp (cid:20) − β λ J/ψ k ⊥ + ˆ m c x ¯ x (cid:21) , (22)where ¯ x = 1 − x , A λJ/ψ is normalization constant, and β λJ/ψ is the harmonic parameter that dominantly deter-mines the wavefunction transverse distributions. The LCDA can be obtained by integrating over the transversemomentum of the wavefunction, i.e. φ λ J/ψ ( x, µ ) = 2 √ f λJ/ψ Z | k ⊥ | ≤ µ d k ⊥ π ψ λ J ; ψ ( x, k ⊥ ) . (23)where µ = ˆ m c = 1 . φ λ J/ψ ( x, µ ) = √ A λJ/ψ ˆ m c β λJ/ψ π / f λJ/ψ √ x ¯ x × (cid:26) Erf (cid:20)s ˆ m c + µ µ x ¯ x (cid:21) − Erf (cid:20)s ˆ m c µ x ¯ x (cid:21)(cid:27) , (24)where λ = ⊥ , k , and the error function Erf( x ) =2 R x e − t dt/ √ π . For the non-leading twist-3 wavefunc-tion, we take the heavy quarkonium the light-front 1 S -Coulomb form [15] ψ Coulomb3;
J/ψ ∼ (cid:20) k ⊥ + (1 − x ¯ x ) ˆ m c x ¯ x + q B (cid:21) − (25)with q B is the Bohr momentum. After integrating withthe transverse momentum k ⊥ , the fully expression canbe written as φ λ J/ψ ( x, v ) = c i ( v ) φ λ, Asy . J/ψ ( x ) (cid:20) x ¯ x − x ¯ x (1 − v ) (cid:21) − v (26)where the mean heavy quark velocity v = q B / ˆ m c ≪ v ≃ .
30 [26] to do the numerical anal-ysis. The twist-3 LCDAs are normalized to 1, i.e. R φ λ J/ψ ( x, v ) = 1. Finally, the twist-3 LCDAs takesthe following form: φ k J/ψ ( x ) = 10 . ξ (cid:20) x ¯ x − . x ¯ x (cid:21) . ,φ ⊥ J/ψ ( x ) = 1 . ξ ) (cid:20) x ¯ x − . x ¯ x (cid:21) . , (27)where ξ = 2 x −
1. The twist-3 LCDAs φ λ J/ψ ( x ) can alsobe derived from the twist-2 LCDAs φ λ J/ψ ( x ) by usingthe Wandzura-Wilczek approximation [27, 28]. Howeverwe observe that the contribution of LCDAs from the end-point region x ∼ , S -Coulomb form for the twist-3wavefunction which is usually taken in the literature todeal with the double charmonium production.Because the terms involving the twist-4 LCDAs arequite small in comparison to the twist-2 and twist-3terms, so the uncertainties from the twist-4 LCDAs them-selves could be negligible; thus we shall employ the twist-4 LCDAs φ λ J/ψ ( x ) and ψ ⊥ J/ψ ( x ) without charm-quarkmass effect that have been suggested by P. Ball and V.M.Braun [29] to do the numerical calculation. III. NUMERICAL ANALYSISA. Input parameters and the
J/ψ
LCDAs
To do the numerical calculation, we neglect the spin-flipping effect for the charmoniums and set the mass of η c or J/ψ to be the same, m η c = m J/ψ = 3 .
097 GeV [21].As for the
J/ψ decay constant f k J/ψ , we extract it fromits leptonic decay width Γ(
J/ψ → e + e − ) by using thefollowing relation [30] f k J/ψ = 3 m J/ψ πα c J/ψ Γ( J/ψ → e + e − ) , (28)where α = 1 /
137 and c J/ψ = 4 /
9. Taking the PDGaveraged value, Γ(
J/ψ → e + e − ) = 5 . f k J/ψ = 416 . f ⊥ J/ψ is taken as 0 . η c decay constant f η c = 0 . A λ and β λ arefixed by two criteria: • The normalization condition of the twist-2 LCDA,i.e. Z φ λ J/ψ ( x, µ ) dx = 1 . (29) • The Gegenbauer moment a λn and the twist-2 LCDAcan be related via the following relation, a λn ; J/ψ ( µ ) = Z dxφ λ J/ψ ( x, µ ) C / n (2 x − Z x ¯ x [ C / n (2 x − . (30)One can derive the Gegenbauer moments a λn ; J/ψ ( µ )of φ λ J/ψ by using their relationship to the mo-ments, h ξ λn ; J/ψ i = R dx (2 x − n φ λ J/ψ ( x, µ ). Moreexplicitly, we have h ξ λ J/ψ i = 15 (cid:18) a λ J/ψ (cid:19) . (31)The first moments of φ λ J/ψ has been calculatedby Ref.[33], e.g., h ξ k J/ψ i = 0 . ± . h ξ ⊥ J/ψ i = 0 . ± . µ = 1 . a λn ; J/ψ ( µ )can be obtained via the QCD evolution. At the NLOaccuracy, we have a λn ; J/ψ ( µ ) = a λn ; J/ψ ( µ ) E NLO n ; J/ψ + α s ( µ )4 π n − X k =0 a λk ; J/ψ ( µ ) L γ (0) k / (2 β ) d (1) nk . (32) TABLE I: Two parameters of the
J/ψ longitudinal and trans-verse wavefunctions at the scale µ = 1 . A λJ/ψ β λJ/ψ φ k J/ψ
458 0.682 φ ⊥ J/ψ
526 0.667
Here µ is the initial scale, µ is the required scale, and E NLO n ; J/ψ = L γ (0) n / (2 β ) × (cid:26) γ (1) n β − γ (0) n β πβ (cid:2) α s ( µ ) − α s ( µ ) (cid:3)(cid:27) , (33)where L = α s ( µ ) /α s ( µ ), β = 11 − n f / β =102 − n f / n f being the active flavor numbers. γ (0) n stands for the anomalous dimensions to NLO accu-racy, γ (0) n is the diagonal two-loop anomalous dimension,and the mixing coefficients d (1) nk with k ≤ n − J/ψ longitudi-nal and transverse wavefunctions at the scale µ = 1 . a k ( µ ) = − .
321 and a ⊥ ( µ ) = − . J/ψ lon-gitudinal and transverse twist-2 LCDAs at the scale µ = 1 . φ asy . =6 x ¯ x . Fig. 1 indicates that all the LCDA models prefera single-peaked behavior, the BC and PM LCDAs areclose in shape. Our present LCDA has a slightly sharperpeak around x ∼ . x ∼ ,
1. We find that the shape of φ k J/ψ ( x, µ ) LCDAs within uncertainties is almost thesame as that of the BFTSR in the whole regions. B. e + e − → J/ψ + η c cross section To derive the numerical results of F VP ( q ), we needto fix the magnitudes of the effective threshold param-eter s and the Borel parameter M . As for s , we set s = 3 . GeV [37] which is close to the squared massof ψ (2 S ). As for the Borel parameter M , we set it inthe range M ∈ [39 ,
41] GeV . In this Borel window,not only the contributions of the higher resonance statesand continuum states are greatly suppressed, but also the M -dependence is effectively suppressed [19].As for the factorization scale µ of e + + e − → J/ψ + η c ,to discuss the factorization scale dependence, in additionto the previously choice of µ = µ , we also take another FIG. 1: The
J/ψ twist-2 LCDAs φ λ J/ψ ( x, µ ) at the scale µ = 1 . λ = ( k , ⊥ ) stand for the longitudinal (Leftdiagram) and the transverse (Right diagram) parts, respectively. As a comparison, the asymptotic form, the BFTSR [36], theQCD SR [33], the BC model [15], and the potential model [17] are also presented. two frequently choices to do our calculation, i.e. µ = µ k ≈ √ k ≈ .
46 GeV, which is determined by fixing thecoupling constant h α s ( k ) i ≈ .
263 and the mean valueof h Z km i ≈ .
80 [15]; and µ = µ s ≈ √ s/ ≈ FIG. 2: Total cross-section of e + + e − → J/ψ + η c at differ-ent factorization scale within the LCSR approach. The solid,dashed and dotted lines are the central values, which corre-spond to the J/ψ distribution amplitude at the scale µ = µ , µ k and µ s , respectively. The shaded bands are their errorsfrom all inputs parameters. Using those inputs together with the total cross sec-tion (5), we calculate the total cross-sections of e + e − → J/ψ + η c under three different factorization scales, andwe put their values versus the Borel parameter M inFig. 2. Fig. 2 confirms that the total cross-sectionchanges slightly within the allowable Borel widow, be-cause the higher-twist terms are 1 /M -power suppressed. TABLE II: Uncertainties of the total cross section of e + + e − → J/ψ + η c caused by the mentioned input parameterswithin the QCD LCSR approach. µ s µ k µ ∆ M = ± +2 . − .
17 +2 . − .
15 +2 . − . ∆ s = ± +1 . − .
79 +1 . − .
72 +1 . − . ∆ m c = ± +1 . .
80 +1 . − .
91 +1 . − . ∆ f η c = ± +0 . − .
29 +0 . − .
29 +0 . − . ∆ f k J/ψ = ± +0 . − .
11 +0 . − .
11 +0 . − . ∆ f ⊥ J/ψ = ± +0 . − .
47 +0 . − .
46 +0 . − . ∆ h ξ k J/ψ i = ± +0 . − .
01 +0 . − .
00 +0 . − . ∆ h ξ ⊥ J/ψ i = ± +0 . − .
27 +0 . − .
13 +0 . − . To have a clear look at the errors coming from all theinput parameters, we list the errors caused by each pa-rameter in Table II. When discussing the error from oneinput parameter, all the other input parameters are setto be their central values. By adding up all the errorsin mean square, our final LCSR predictions for the to-tal cross-section of e + + e − → J/ψ + η c at three typicalfactorization scales are σ | µ s = 22 . +3 . − . fb , (34) σ | µ k = 21 . +3 . − . fb , (35) σ | µ = 21 . +3 . − . fb . (36)Those cross-sections are close to each other, indicat-ing the factorization scale dependence is small. Thusby properly dealing with the QCD evolution effect, theLCSR predictions shall be slightly affected by differentchoice of factorization scale. IV. SUMMARY (cid:1)(cid:1) (cid:2)(cid:2)(cid:3)(cid:3)(cid:4)(cid:4)(cid:5)(cid:5)(cid:6)(cid:6)(cid:7)(cid:7) (cid:8)(cid:8) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)
10 15 20 25 30 35 40
FIG. 3: Total cross section of e + + e − → J/ψ + η c at dif-ferent factorization scales within the LCSR approach. Themarks represent the corresponding central values, and linesare the errors from the variation of all inputs parameters. Asa comparison, the Belle data [2], the BaBar data [3], the NLONRQCD prediction (NLO-I) [7], the NRQCD prediction withNLO radiative and relativistic corrections (NLO-II) [8], andthe PMC NLO NRQCD prediction [9] are also presented. In this paper, we have investigated the total cross-section for e + e − → J/ψ + η c within the QCD LCSRapproach. We put a comparison of total cross-sectionwith other theoretical and experimental predictions inFig. 3. Fig. 3 shows that our results are in consistentwith the BaBar and Belle measurements and also thePMC NRQCD prediction within errors. Thus the LCSRapproach also provides a helpful and reliable approach todeal with the high-energy processes involving charmoni-ums. Acknowledgments : We are grateful to Dr. TaoZhong and Xu-Chang Zheng for helpful discussions andvaluable suggestions. Hai-Bing Fu would like to thankthe Institute of Theoretical Physics in Chongqing Univer-sity for kind hospitality. This work was supported in partby the National Natural Science Foundation of China un-der Grant No.11765007, No.11947406 and No.11625520,the Project of Guizhou Provincial Department of Sci-ence and Technology under Grant No.KY[2019]1171, theProject of Guizhou Provincial Department of Educa-tion under Grant No.KY[2021]030 and No.KY[2021]003,the China Postdoctoral Science Foundation under GrantNo.2019TQ0329 and No.2020M670476, and the Funda-mental Research Funds for the Central Universities underGrant No.2020CQJQY-Z003. [1] K. Abe et al. [ Belle
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