# An improved light-cone harmonic oscillator model for the pionic leading-twist distribution amplitude

Tao Zhong, Zhi-Hao Zhu, Hai-Bing Fu, Xing-Gang Wu, Tao Huang

aa r X i v : . [ h e p - ph ] F e b An improved light-cone harmonic oscillator model for the pionic leading-twistdistribution amplitude

Tao Zhong, ∗ Zhi-Hao Zhu, Hai-Bing Fu † ,

1, 3, ‡ Xing-Gang Wu, § and Tao Huang ¶ Department of Physics, Guizhou Minzu University, Guiyang 550025, P.R. China College of Physics and Materials Science, Henan Normal University, Xinxiang 453007, P.R. China Department of Physics, Chongqing University, Chongqing 401331, P.R. China Institute of High Energy Physics and Theoretical Physics Center for Science Facilities,Chinese Academy of Sciences, Beijing 100049, P.R. China (Dated: February 9, 2021)In this paper, we study the pion leading-twist distribution amplitude φ π ( x, µ ) by improving thetraditional light-cone harmonic oscillator model within the reconstruction of the function ϕ π ( x ).In order to constraining the model parameters, we calculate its moments h ξ n i π | µ in the frameworkof QCD background ﬁeld theory sum rule (BFTSR) up to 10 th order. Considering the fact that thesum rule of the 0 th moment h ξ i π | µ cannot be normalized, we suggest a more reasonable sum ruleformula for h ξ n i π | µ . Then, we obtain the values of h ξ n i π | µ with n = (2 , , , ,

10) at the initialscale µ = 1 GeV. The ﬁrst two moments are: h ξ i π | µ = 0 . ± . h ξ i π | µ = 0 . ± . a π ( µ ) = 0 . ± . a π ( µ ) = 0 . ± . h ξ n i π | µ , we obtained the appropriate model parameters byusing the least square method. The resultant behavior for twist-2 pion DA is more closely to theAdS/QCD and lattice result, but is narrower than that by Dyson-Schwinger equation. Furthermore,we calculate the pion-photon transition form factors (TFF) and B → π TFF within light-cone sumrule approach, which are conform with experimental and theoretical results.

PACS numbers: 12.38.-t, 12.38.Bx, 14.40.Aq

I. INTRODUCTION

Light meson light-cone distribution amplitudes (DAs)are universal nonperturbative objects, which describe themomentum fraction distributions of partons in a mesonfor a particular Fock state. Those DAs enter exclusiveprocesses based on the factorization theorems in the per-turbative QCD theory (pQCD), and therefore they arekey parameters in the QCD predictions for correspond-ing processes. In the standard treatment of exclusiveprocesses in QCD proposed by Brodsky and Lepage [1],cross sections are arranged according to diﬀerent twiststructures of meson DAs. In which the leading-twist DAcontribution usually dominates due to the contributionsfrom the higher twists are high power suppressed at shortdistance. Thereafter, the study of pionic leading-twistDA, which describes the momentum distribution of thevalence quarks in pion, has attracted much attention inthe literature.So far, a large number of studies on the pionicleading-twist DA rely on its Gegenbauer expansion se-ries [2, 3], the nonperturbative expansion coeﬃcients,denoted a πn ( µ ), are called Gegenbauer moments whichencode the long-distance dynamics at low energy scale( ∼ † Corresponding author ∗ Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] nonzero due to the isospin symmetry. In many applica-tions of pion leading-twist DA involving a high normal-ization scale, the higher Gegenbauer moment contribu-tions are suppressed due to the anomalous dimension of a πn ( µ ) grows with n , and only the lowest Gegenbauermoments are retained. Therefore, people usually adoptthe truncated form involving only the ﬁrst few termsin the Gegenbauer expansion series to be an approxi-mate form of φ π ( x, µ ). Those Gegenbauer moments canbe calculated directly via some non-perturbative meth-ods such as QCD sum rules [4–9] or lattice gauge the-ory [10–13] and so on. Using QCD sum rules to calcu-late a πn ( µ ) is realized by calculating h ξ n i π | µ . Recently,we realized that these calculations need to be improved.By QCD sum rules method, the analytic formula is for h ξ n i π | µ × h ξ i π | µ , but which is usual seen as the sumrules of h ξ n i π | µ due to normalization of φ π ( x, µ ). Infact, due to the incompleteness of our sum rules calcula-tion, the deviation of h ξ i π | µ from normalization mustbe considered. This motivates us to recalculate the mo-ments of pionic leading-twist DA with QCD sum rules.On the other hand, the truncated form mentionedabove does not seem to be enough to describe the behav-ior of DA under the low energy scale. A natural idea isto consider the contributions of higher-order Gegenbauerpolynomials, which requires the calculation of higher-order Gegenbauer moments. But there is a very seri-ous diﬃculty in doing so, that is, it is diﬃcult to getreliable higher a πn ( µ ). Through the mathematical rela-tionship between a πn ( µ ) and h ξ n i π | µ , we can ﬁnd thatwith the increase of order n , the reliability of a πn ( µ ) de-creases sharply, which makes our calculation of higher a πn ( µ ) meaningless. So people try to study the be-havior of φ π ( x, µ ) through other ways. In Ref. [14],in the framework of Dyson-Schwinger equations the au-thors obtain the pionic leading-twist DA (DS model) thatare concave and signiﬁcantly broader than the asymp-totic DA. Making use of the approximate bound statesolution of a hadron in terms of the quark model asthe starting point, Brodsky-Huang-Lepage (BHL) sug-gest the light-cone harmonic oscillator model (LCHOmodel) which is obtained by connecting the equal-timewavefunction (WF) in the rest frame and the WF inthe inﬁnite momentum frame [15]. Meanwhile, the holo-graphic Schr¨odinger equation for meson maps onto theﬁfth dimension of anti-de Sitter with QCD potential(AdS/QCD) [16].In this paper, we will study the pionic leading-twistDA φ π ( x, µ ) based on the improved LCHO model. Thedetermination of model parameters depends on the mo-ments h ξ n i π | µ rather than the Gegenbauer moments a πn ( µ ), and we will adopt a new method, that is, theleast square method ﬁtting, to determine the model pa-rameters directly. Especially, to get more accurate valuesof the moments h ξ n i π | µ , we will recalculate those mo-ments with the QCD sum rules in the framework of thebackground ﬁeld theory (BFTSR) and adopt a more rea-sonable and accurate sum rules formula for h ξ n i π | µ .The remaining parts of this paper are organized as fol-lows. In Sec. II A, we recalculate the pionic leading-twistDA moments by BFTSR. In Sec. II B, we give a briefoverview of the LCHO model, put forward the improvednew model, and introduce the method of least squaresﬁtting moment to get model parameters. Numerical re-sults are given in Sec. III. Section IV is reserved for asummary. II. THEORETICAL FRAMEWORKA. The BFTSR for the moments of φ π ( x, µ ) To derive the sum rules for the pionic leading-twist DAmoments h ξ n i π | µ , we adopt the following correlationfunction,Π ( n, π ( z, q ) = i Z d xe iq · x h | T { J n ( x ) J † (0) }| i = ( z · q ) n +2 I ( n, π ( q ) , (1)where z = 0, n = (0 , , , · · · ) since the odd momentsvanish due to the isospin symmetry, and the currents J n ( x ) = ¯ d ( x ) z/γ ( iz · ↔ D ) n u ( x ) , (2) J † (0) = ¯ u (0) z/γ d (0) . (3)In physical region, the correlation function (1) can becalculated by inserting a complete set of intermediatehadronic states. Combining the deﬁnition h | ¯ d (0) z/γ ( iz · ↔ D ) n u (0) | π ( q ) i = i ( z · q ) n +1 f π h ξ n i π | µ , (4)and the quark-hadron duality, the hadron expression of(1) can be obtained asIm I ( n, π ;had ( q ) = πδ ( q − m π ) f π h ξ n i π | µ + π π ( n + 1)( n + 3) θ ( q − s π ) , (5)where m π is the pion mass, f π is the decay constant, s π stands for the continuum threshold. In Eq. (4), themoments h ξ n i π | µ are deﬁned with the pionic leading-twist DA φ π ( x, µ ) as following: h ξ n i π | µ = Z dx (2 x − n φ π ( x, µ ) . (6)In the deep Euclidean region, we apply the operatorproduct expansion (OPE) for the correlation functionEq. (1). The corresponding calculation is performed inthe framework of BFTSR. For the basic assumption ofBFTSR, the corresponding Feynman rules, and the OPEcalculation technology, one can ﬁnd in Refs. [17, 18] fordetailed discussion.The hadron expression of correlation function (1) inthe physical region and its OPE in deep Euclidean re-gion can be matched with the dispersion relation. Af-ter applying the Borel transformation for both sides, thesum rules for the moments of the pionic leading-twist DA φ π ( x, µ ) can be obtained as: h ξ n i π | µ h ξ i π | µ f π M e m π /M = 34 π n + 1)( n + 3) (cid:16) − e − s π /M (cid:17) + ( m d + m u ) h ¯ qq i ( M ) + h α s G i ( M ) nθ ( n − π ( n + 1) − ( m d + m u ) h g s ¯ qσT Gq i ( M ) n + 118 + h g s ¯ qq i ( M ) n + 1)81 − h g s f G i ( M ) nθ ( n − π + h g s ¯ qq i ( M ) κ π × n − n + 25) (cid:16) − ln M µ (cid:17) + 3(17 n + 35) + θ ( n − h n (cid:16) − ln M µ (cid:17) + 49 n + 100 n + 56 n − n + 1) h ψ (cid:16) n + 12 (cid:17) − ψ (cid:16) n (cid:17) + ln 4 iio . (7)Where M is the Borel parameter, and for those vacuum condensates, we have taken: h ¯ qq i = h ¯ dd i = h ¯ uu i , h g s ¯ qσT Gq i = h g s ¯ dσT Gd i = h g s ¯ uσT Gu i , h g s ¯ qq i = h g s ¯ dd i = h g s ¯ uu i , h g s ¯ qq i = h g s ¯ dd i = h g s ¯ uu i , and with h ¯ ss i / h ¯ qq i = κ , g s X h g s ¯ ψψ i = (2 + κ ) h g s ¯ qq i , ( ψ = u, d, s ) . In the OPE calculation for the correlation function (1), we have corrected the mistake of a vacuum matrix element, h | G Aµν G Bρσ ; λτ | i , used in the previous work [18]. That is h | G Aµν G Bρσ ; λτ | i = δ AB n(cid:16) − X h g s ¯ ψψ i − h g s f G i (cid:17)(cid:2) g λτ ( g µσ g νρ − g µρ g νσ ) + g ρτ ( g µσ g νλ − g µλ g νσ )+ g στ ( g µλ g νρ − g µρ g νλ ) (cid:3) + (cid:16) − X h g s ¯ ψψ i + 1384 h g s f G i (cid:17)(cid:2) g µτ ( g ρν g σλ − g ρλ g νσ ) + g ντ × ( g ρλ g σµ − g ρµ g σλ ) (cid:3)o (8)It needs to be noted that, by taking n = 0 in Eq. (6) andconsidering the normalization of the pionic leading-twistDA φ π ( x, µ ), one can obtain the 0 th moment h ξ i π | µ = 1 . (9)Therefore, in many QCD sum rules calculation peopleusually substitute Eq. (9) as input directly into sum rules(7), and take Eq. (7) as the sum rules of the moments h ξ n i π | µ . This will bring extra deviation to the predictedvalues of h ξ n i π | µ , the reason is that the 0 th moment h ξ i π | µ in the l.h.s. of Eq. (7) is not strict that inEq. (9). By taking n = 0 in Eq. (7), one can obtainthe sum rule of h ξ i π | µ , h ξ i π | µ f π M e m π /M = 14 π (cid:16) α s π (cid:17)(cid:16) − e − s π /M (cid:17) + ( m d + m u ) h ¯ qq i ( M ) + h α s G i ( M ) π −

118 ( m d + m u ) × h g s ¯ qσT Gq i ( M ) + 481 h g s ¯ qq i ( M ) + h g s ¯ qq i ( M ) κ π × h − (cid:16) − ln M µ (cid:17) + 105 i . (10)Obviously, h ξ i π | µ in the l.h.s. of sum rule (7) cannot be normalized in the whole Borel parameter regions.The reason is that our calculation is not complete. Thehigh-order corrections and high-dimensional correctionshave not been calculated, and which are also impossibleto calculate completely. In fact, the authors of Ref. [8]discovered this more than 30 years ago. They obtain h ξ i π | µ ≃ .

83, and take f π h ξ i π | µ as normalization factor to calculate the values of h ξ i π | µ and h ξ i π | µ .In this paper, we argue that we need to further considerthe impact of the sum rule of h ξ i π | µ , Eq. (10), in thefull Borel parameter regions when using sum rule (7) tocalculate h ξ n i π | µ . Therefor, in order to obtain more ac-curate moments h ξ n i π | µ , we suggest the following form: h ξ n i π | µ = ( h ξ n i π | µ h ξ i π | µ ) | From Eq . (7) q h ξ i π | µ | From Eq . (10) . (11)Meanwhile, another advantage of Eq. (11) is that it canalso eliminate some systematic errors caused by the con-tinuum state, the absence of high dimensional conden-sates, the selection and determination of various inputparameters. B. The improved LCHO model for φ π ( x, µ ) Based on the BHL-description [15], the LCHO modelof the pion leading-twist WF has raised in Refs. [19, 20],and its form is:Ψ π ( x, k ⊥ ) = χ π ( x, k ⊥ )Ψ R π ( x, k ⊥ ) , (12)where k ⊥ is the pionic transverse momentum, χ π ( x, k ⊥ )stands for the spin-space wave function that comes fromthe Wigner-Melosh rotation, reads [21] χ π ( x, k ⊥ ) = m q q k ⊥ + m q , (13)andΨ R π ( x, k ⊥ ) = A π ϕ π ( x ) exp " − k ⊥ + m q β π x (1 − x ) , (14)indicates spatial WF. In Eqs. (13) and (14), m q is theconstitute quark mass, and we take m q = 200 MeV in thispaper. In Eq. (14), A π is the normalization constant,the k ⊥ -dependence part of the spatial WF Ψ R π ( x, k ⊥ )comes from the approximate bound-state solution in thequark model for pion [22] and determine the WF’s trans-verse distribution via the harmonious parameter β π , the x -dependence part ϕ π ( x ) dominates the WF’s longitu-dinal distribution. Using the relationship between thepionic leading-twist DA and WF, φ π ( x, µ ) = 2 √ f π Z | k ⊥ | ≤ µ d k ⊥ π Ψ π ( x, k ⊥ ) , (15)the leading-twist DA for pion, φ π ( x, µ ), can be ob-tained. That is, after integrating over the transversemomentum k ⊥ in Eq. (15), we have φ π ( x, µ ) = √ A π m q β π π / f π p x (1 − x ) ϕ π ( x ) × ( Erf "s m q + µ β π x (1 − x ) − Erf "s m q β π x (1 − x ) , (16)where Erf( x ) = 2 R x e − t dx/ √ π is the error function.The error function part in Eq. (16) comes from the k ⊥ -dependence part of the WF Ψ π ( x, k ⊥ ) and gives a goodendpoint behavior for φ π ( x, µ ), and ϕ π ( x ) dominatesthe broadness of φ π ( x, µ ). Obviously, the speciﬁc formof φ π ( x, µ ) is determined by the parameters A π , β π and the function ϕ π ( x ). There are two important con-straints [15] which can be used to constrain the parame-ters A π and β π , that is,(1) the WF normalization condition provided from theprocess π → µν , Z dx Z d k ⊥ π Ψ( x, k ⊥ ) = f π √ π → γγ decay ampli-tude, Z dx Ψ( x, k ⊥ = ) = √ f π . (18)Then the pionic leading-twist DA φ π ( x, µ ) only de-pends on the mathematical form of ϕ π ( x ). By solvingthe renormalization group equation of the pionic leading-twist DA, φ π ( x, µ ) can be written as the expansion form of the Gegenbauer series [2, 3]. Based on this, in ourprevious paper, ϕ π ( x ) is taken to be the linear super-position of the ﬁrst several Gegenbauer polynomials. Forexample, in Refs. [23–26], we take ϕ I2; π ( x ) = 1 + B × C / (2 x − , (19)and ϕ II2; π ( x ) = 1 + B × C / (2 x − B × C / (2 x −

1) (20)is adopted in Ref. [18]. For the former, when the value ofparameter B changes from 0 . .

6, the pionic leading-twist DA model, i.e. Eq. (16) can mimic the DA behaviorfrom asymptotic-like to CZ-like. And for the latter, wefurther consider the correction of 4 th order Gegenbauerpolynomial. The mathematical form of ϕ π ( x ) can usu-ally be determined in two ways. The ﬁrst one is to ex-tract ϕ π ( x ) from the experimental data of the exclusiveprocesses involving pion [23–26], such as semi-leptonicdecays B → πℓν ℓ and D → πℓν ℓ , the pion-photon tran-sition form factor F πγ ( Q ), and the exclusive process B → π π , etc.; the second one is to determine from themoments h ξ n i π | µ or the Gegenbauer moments a πn ( µ )of φ π ( x, µ ). In Ref. [18], we have adopted the secondmethod to determine the mathematical form of ϕ π ( x )and further the behavior of φ π ( x, µ ).In this paper, we will still make use of the secondmethod mentioned above to determine the behavior of φ π ( x, µ ), but we will improve it. The accuracy of the be-havior of φ π ( x, µ ) obtained by this method is restrictedby two aspects: the rationality of the constructed math-ematical form of ϕ π ( x ) and the accuracy of moments.In order to get better mathematical form of ϕ π ( x ), anatural idea is to add higher order Gegenbauer poly-nomial correction in ϕ II2; π ( x ), as we have done for the D, η c , B c , η b twist-2, 3 DAs in Refs. [27–30]. However,such improvement obviously destroys the beauty andconciseness of the model. Otherwise, we ﬁnd that theparameters B , B are close to the Gegenbauer moments a π ( µ ) , a π ( µ ) respectively. From the relationship be-tween h ξ n i π | µ and a πn ( µ ), it can be seen that the reli-ability of a πn ( µ ) calculated by QCD sum rules decreasessharply with the increase of order- n . In view of this,in this paper we will improve the mathematical form of ϕ π ( x ) by other way, as well as propose a new determi-nation method of model parameters.We notice that although it is diﬃcult to improve pionicleading-twist DA by introducing higher Gegenbauer poly-nomial correction, our goal is still to make it more reason-able and accurate by adjusting the behavior of φ π ( x, µ ).We ﬁnd that the factor p x (1 − x ) in Eq. (16) can regu-late DA’s behavior to some extent. Inspired by this, weintroduce a factor [ x (1 − x )] α π into WF’s longitudinaldistribution function ϕ π ( x ), i.e., ϕ III2; π ( x ) = [ x (1 − x )] α π . (21)In order to further apply our LCHO model to other me-son DA, and combine the form of ϕ I2; π ( x ), we propose amore complex form, ϕ IV2; π ( x ) = [ x (1 − x )] α π h B π C / (2 x − i , (22)the parameters α π and B π will be determined by ﬁt-ting the moments h ξ n i π | µ directly through the methodof least squares, and the values of moments h ξ n i π | µ come from Eq. (11) calculated under BFTSR in Sec. II A.In order to distinguish our LCHO model with ϕ III2; π ( x ) and ϕ IV2; π ( x ), and facilitate the discussion later, we will recordthe former as LCHO model (III) and the latter as LCHOmodel (IV).Considering a set of N independent measurements y i with the known variance σ i and the mean µ ( x i ; θ ) atknown points x i . The objective of the least squaresmethod is to obtain the best value of ﬁtting parameters θ by minimizing the likelihood function [31] χ ( θ ) = N X i =1 ( y i − µ ( x i , θ )) σ i . (23)As for the present case, the function µ ( x i ; θ ) indicatesthe pionic leading-twist DA moments h ξ n i π | µ deﬁnedby combining Eqs. (6, 16, 21) and θ = ( α π , B π ); Thetheoretical values of h ξ n i π | µ calculated by QCD sumrules in next section are assumed to be the value of y i and its variance σ i . The probability density function of χ can be obtained, f ( y ; n d ) = 1Γ (cid:16) n d (cid:17) n d / y nd − e − y , (24) n d is the number of degree-of-freedom. Then one canfurther calculate the following probability, P χ = Z ∞ χ f ( y ; n d ) dy. (25)The magnitude of the probability P χ ( P χ ∈ [0 , III. NUMERICAL ANALYSISA. Basic input parameters

To do the numerical calculation, we adopt the lat-est data from Particle Data Group (PDG) [31]: m π =139 . ± . f π = 130 . ± . u, d -quark are adoptedas m u = 2 . +0 . − . MeV and m d = 4 . +0 . − . MeV atscale µ = 2 GeV. Based on these latest values, we canupdate the vacuum condensates. • For the double-quark condensate, we adopt Gell-Mann-Oakes-Renner relation: m u h ¯ uu i + m d h ¯ dd i ≃ − f π m π − (1 . ± . × − GeV . (26)Combining with the u, d quark masses, we have: h ¯ qq i = (cid:0) − . +0 . − . (cid:1) × − GeV = (cid:0) − . +9 . − . (cid:1) MeV , (27)at scale µ = 2 GeV. • By combining Eqs. (26), (27) and the rela-tion h g s ¯ qσT Gq i = m h ¯ qq i with m = 0 . ± .

02 GeV [32], the quark-gluon mixed condensatewould be m u h g s ¯ uσT Gu i + m d h g s ¯ dσT Gd i = − (1 . ± . × − GeV , (28) h g s ¯ qσT Gq i = (cid:0) − . +0 . − . (cid:1) × − GeV . (29) • By adopting the data in Ref. [32], ρα s h ¯ qq i = (5 . ± . × − GeV , (30)with ρ ≃ −

4, and combining the value of thedouble-quark condensate in Eq. (27), the four-quark condensates can be obtained as: h g s ¯ qq i = (2 . +0 . − . ) × − GeV , (31)and h g s ¯ qq i = (7 . +2 . − . ) × − GeV , (32) • From Ref. [33], we have h α s G i = 0 . ± .

011 GeV , (33)and h g s f G i ≃ .

045 GeV . (34) • For the ratio κ = h ¯ ss i / h ¯ qq i , Ref. [34] gives: κ = 0 . ± . , (35) B. The renormalization group equation for theinput parameters and the moments h ξ n i π | µ In numerical calculation for the moments’ BFTSR(11), we take the scale µ = M as usual. From α s ( M z ) =0 . ± . M Z = 91 . ± . m c ( ¯ m c ) = 1 . ± .

02 GeV and ¯ m b ( ¯ m b ) =4 . +0 . − . GeV [31], under the 3-loop approximate solu-tion we predict Λ ( n f )QCD ≃ , ,

207 MeV for the num-ber of quark ﬂavors n f = 3 , ,

5, respectively.The renormalization group equations (RGE) of thequark mass and vacuum condensates are given as [35–37]: m q | µ = m q | µ (cid:20) α s ( µ ) α s ( µ ) (cid:21) − /β , h ¯ qq i| µ = h ¯ qq i| µ (cid:20) α s ( µ ) α s ( µ ) (cid:21) /β , h g s ¯ qσT Gq i| µ = h g s ¯ qσT Gq i| µ (cid:20) α s ( µ ) α s ( µ ) (cid:21) − / (3 β ) , h α s G i| µ = h α s G i| µ , h g s f G i| µ = h g s f G i| µ , (36)with β = (33 − n f ) /

3. Obviously, the double-gluon con-densate and the triple-gluon condensate are energy scaleindependent. From Eq. (8), one can ﬁnd that h g s ¯ qq i and h g s f G i have the same RGE. In other words, h g s ¯ qq i isalso energy scale independent, e.g., h g s ¯ qq i | µ = h g s ¯ qq i | µ . (37)Combining with the RGE of the double-quark condensateand Eq. (37), one can ﬁnd that the h g s ¯ qq i and h ¯ qq i havethe same energy scale evolution equation, e.g., h g s ¯ qq i | µ = h g s ¯ qq i | µ (cid:20) α s ( µ ) α s ( µ ) (cid:21) /β . (38)It should be noted that, according to the basic assump-tion of BFTSR, g s in all the above vacuum condensates isthe “coupling constant” between the background ﬁelds,which is diﬀerent from the one in pQCD, and should beabsorbed into vacuum condensates as part of these non-perturbative parameters.The RGE of the Gegenbauer moments of the pionleading-twist distribution amplitude is: a πn ( µ ) = a πn ( µ ) E n ( µ, µ ) , (39)with E n ( µ, µ ) = (cid:20) α s ( µ ) α s ( µ ) (cid:21) γ (0) n / (2 β ) . The LO anomalous dimension γ (0) n = 8 C F (cid:20) ψ ( n + 2) + γ E − − n + 1)( n + 2) (cid:21) , with C F = 4 /

3. Based on Eq. (39), the RGE of themoments h ξ n i π | µ can be obtained.With the BFTSR of the moments of the pionic leading-twist distribution amplitude φ π ( x, µ ) shown in Eqs. (7),(10) and (11), the values of h ξ n i π | µ can be calculated. TABLE I: The determined Borel windows and the corre-sponding pionic leading-twist DA moments h ξ n i π | µ with n = (2 , , , , n M h ξ n i π | µ . , . . , . . , . . , . . , . . , . . , . . , . . , . . , . By requiring that there is reasonable Borel window tonormalize h ξ i π | µ with Eq. (10), one can get the con-tinuum threshold parameter as about s π ≃ .

05 GeV . Inaddition to the traditional method to determine the con-tribution of the continuum state, the continuum methodcan limiting or overcoming model-dependence and draw-ing clean lines in connecting the data with QCD it-self [38]. To obtain the allowable Borel window for thesum rules of h ξ n i π | µ , we require that the continuumstate’s contribution and the dimension-six condensate’scontribution to be as small as possible, and the values for h ξ n i π | µ are stable in the Borel window. Based on thecriteria, the Dimension-six contribution for h ξ n i π | µ areprescribe a limit to less than 5% for all the n th-order.And the continue contribution for h ξ n i π | µ are restrictto 30%, 35%, 40%, 40%, 40% for n = (2 , , , ,

10) re-spectively.To have a deeper insight into the continuum state anddimension-six contribute to the pionic leading-twist DAmoments h ξ n i π | µ versus the Borel parameter M withinBFTSR approach, we present the curves in Fig. 1. Theshaded band indicate the Borel Window for h ξ n i π | µ for n = (2 , , , ,

10) respectively. The ﬁgure indicates that, • The dimension-six contributions are constraint inthe region <

5% guaranteed good convergence forthe BFTSR results. And the continue contribu-tions no more than 40% have agreement with thetraditional sum rule strictly. • Borel parameters associate with the region of BorelWindow become larger with the increase of index n .To study the inﬂuence of the Borel parameters to thepionic DA moments in the Borel window, we listed theresults h ξ n i π | µ changed with Borel windows in Table I.In which the h ξ n i π | µ changed less than 10% with theBorel windows, i.e. 4 . , . , . , . , .

1% for n = (2 , , , ,

10) respectively. Thus, the Borel win-dows for h ξ n i π | µ are stable to the BFTSR. Furthermore,the ﬁve curves of pionic leading-twist DA moments, i.e. h ξ n i π | µ for n = (2 , , , ,

10) versus the Borel parame-ter M are shown in Fig. 2. The ﬁgure indicate that: • The curves for h ξ n i π | µ changed sharply in thesmall Borel area especially for the M . FIG. 1: The continuum state and dimension-six contribute to the pionic leading-twist DA moments h ξ n i π | µ versus the Borelparameter M within BFTSR approach. The shaded band indicate the Borel Window for h ξ n i π | µ for n = (2 , , , , FIG. 2: The pionic leading-twist DA moments h ξ n i π | µ with n = (2 , , , ,

10) versus the Borel parameter M , whereall input parameters are set to be their central values. Inwhich, the shaded band indicate the Borel Windows for n = (2 , , , , • The values of h ξ n i π | µ became small with the in-creasement for order n . • The stable Borel parameter M for h ξ n i π | µ be-come larger with the increase of n . After taking all uncertainty sources into considera-tion, and adopting the RGE of moments mentioned inthe above subsection, the ﬁrst ﬁve nonvanishing valuesof h ξ n i π | µ , i.e. n = (2 , , , ,

10) within uncertaintiescoming from every input parameters are shown in Ta-ble II. In which, the factorization scale are taken boththe initial scale µ and typical scale µ = 2 GeV. As adeeper comparison, we also listed the Light-front Holo-graphic with B = 0 and B ≫ h x − i| µ = R dxx − φ π ( x, µ ) in Table II. In addition, in order toshow the advantages of new sum rules formula (11), thevalues of h ξ n i π | µ obtained by the formula combiningEq. (7) and Eq. (9) commonly used in literature is alsolisted in this table. From the table, we can get the con-clusions, • Up to 10th-order accuracy, we provide a completeseries results for h ξ n i π | µ within uncertainties. • For the n = (2 ,

4) cases, our results have a goodagreement with the DS model and Lattice results. • The inverse moment at µ = 2 GeV of our predictionis closely to the Playkurtic and NLC Sum Rules TABLE II: Our predictions for the ﬁrst ﬁve nonvanishing moments and inverse moment of the pion DA, compared to othertheoretical predictions. Meanwhile, the values obtained by the formula combining Eq. (7) and Eq. (9) is also shown. µ [GeV] h ξ i π | µ h ξ i π | µ h ξ i π | µ h ξ i π | µ h ξ i π | µ h x − i| µ BFTSR (This Work) 1 0.271(13) 0.138(10) 0.087(6) 0.064(7) 0.050(6) 3.95BFTSR (This Work) 2 0.254(10) 0.125(7) 0.077(6) 0.054(5) 0.041(4) 3.33Asymptotic ∞ B = 0) [39] 1,2 0.180, 0.185 0.067, 0.071 - - - 2.81,2.85LF Holographic ( B ≫

1) [39] 1,2 0.200, 0.200 0.085, 0.085 - - - 2.93,2.95LF Holographic [40] ∼ . +0 . − . . +0 . − . - - - 3 . +0 . − . LF Quark Model [42] ∼ . +0 . − . . +0 . − . - - - 3.16(9)Sum Rules [5] 2 0.343 0.181 - - - 4.25Dyson-Schwinger [RL, DB] [14] 2 0.280, 0.251 0.151, 0.128 - - - 5.5,4.6Lattice [48] 2 0.28(1)(2) - - - - -Lattice [49] 2 0.2361(41)(39) - - - - -Lattice [50] 2 0.27(4) - - - - -Lattice [51] 2 0.2077(43) - - - - -Lattice [53] 2 0.234(6)(6) - - - - -Lattice [54] 2 0.244(30) - - - - -Eq. (7)+Eq. (9) 1 0.303(19) 0.179(21) 0.128(16) 0.098(14) 0.082(20) - results. • Comparing the values in the ﬁrst and last row, onecan ﬁnd that the diﬀerences between correspond-ing moments are about 12%, 30%, 47%, 53% and64% for n = 2 , , , ,

10, respectively. These ra-tios can be regarded as the accuracy improved byadopting new sum rules formula (11). At the sametime, one can ﬁnd that those diﬀerences increasedwith the increase of the order- n . The reason isthat the Borel window moves to the right with theincrease of order- n (see Table I), and the devia-tion of the sum rule of 0 th moment, Eq. (10), fromnormalization, increases with the increase of Borelparameter. The errors in the ﬁrst row are signiﬁ-cantly less than that in the last row. The reasonis that the sum rules (11) can eliminate some sys-tematic errors caused by the selection and deter-mination of various input parameters. To calcu-late h ξ n i π | µ by combining Eq. (7) and Eq. (9),we have required that the continuum state contri-butions are less than 45%, 50%, 50%, 55%, 55%;and the dimension-six contributions are not morethan 10%, 15%, 15%, 15%, 15%, for the order- n = 2 , , , ,

10, respectively. Comparing the crite-rions adopted for sum rules (11) mentioned above,which are obviously much larger. This means thatthe sum rules (11) does eliminate some systematicerrors caused by the continuum state and the ab-sence of high dimensional condensates.Moreover, considering the low reliability of high orderGegenbauer moments, we only give the values of the sec-ond and forth Gegenbauer moments in this paper, they are a π ( µ ) = 0 . ± . ,a π ( µ ) = 0 . ± . . (40) C. The model parameters of the pionicleading-twist DA and applications

Combining the the normalization condition (17) andthe BFTSR (18) derived from π → γγ decay amplitude,making use of the least square method mentioned in Sec.II to ﬁt the values of moments h ξ n i π | µ shown in Table II,the parameters of our LCHO model(III) can be obtained: A π = 14 . − ,α π = − . ,β π = 0 . , (41)with χ /n d = 0 . / P χ = 0 . A π = 5 . − ,α π = − . ,B π = − . ,β π = 0 . , (42)with χ /n d = 0 . / P χ = 0 . B =1 [39], the DAs by the light-front constituent quark model(LFCQM) [52] and LQCD [53, 54] are also shown in FIG. 3: The pionic leading-twist DA curves in this work. For the panel-(a) and panel-(b), we present DS model [14], AdS/QCDmodel [39], the DAs by LFCQM [52] and LQCD [53, 54] as a comparison. For the panel-(c), our LCHO model (IV) at severaltypical energy scales e.g., µ = 1 , , , ,

100 GeV are given respectively.

Fig. 3. From the panel-(a) in Fig. 3, one can ﬁnd thatour LCHO model (III) is near ﬂat in region x ∈ [0 . , . h ξ n i π | µ inversely, which are 0 . . . .

063 and 0 .

048 for n = 2 , , , ,

10, respec-tively. Those values are very closely to that in Table II.By substituting our LCHO model (III) with parametersin Eqs. (41) into Eq. (6), h ξ n i π | µ can be calculated as0 . . . .

062 and 0 .

046 for n = 2 , , , , • Our LCHO model at µ is signiﬁcantly broaderthan the asymptotic form. • With the increase of scale µ , our pionic leading-twist DA model curve becomes narrower and closerto the asymptotic form. Especially, when the scale µ is lower than 2 GeV, our pionic leading-twist DAbehavior is more sensitive to µ , while when µ > µ . • Otherwise, we can get φ π (0 . , µ ) = 1.186, 1.414,1.475, 1.490, 1.498 for µ = 1 , , , ,

100 GeV re-spectively.As signiﬁcant applications, we recalculate the pion-photon TFF F πγ ( Q ) and the B → π TFF f B → π + ( q )with our pionic leading-twist DA model. For the pion-photon TFF F πγ ( Q ), It can be expressed as the sum (cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)(cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4) (cid:4)(cid:1)(cid:2)(cid:3)(cid:4) FIG. 4: The pion-photon TFF Q F πγ ( Q ) with ourmodel. For comparison, the experimental data reported byCELLO [56], CLEO [57, 58], BaBar [59] and Belle [60] collab-orations are shown. of the valence quark part contribution F (V) πγ ( Q ) and thenon-valence quark part contribution F (NV) πγ ( Q ), F πγ ( Q ) = F (V) πγ ( Q ) + F (NV) πγ ( Q ) , (43)where the corresponding analytical formula of F (V) πγ ( Q )and F (NV) πγ ( Q ) can be found in Refs. [19, 55]. Figure 4show the curve of Q F πγ ( Q ) versus Q by our pionicleading-twist DA model and the experimental data re-ported by CELLO [56], CLEO [57, 58], BaBar [59] andBelle [60] collaborations, and one can ﬁnd that our pre-diction is consistent with the BELLE data in large Q region.Furthermore, as another important application for thepion twist-2 LCDA, the TFF for the B → π decay pro-cesses should be analysis. We start with the following0correlation functionΠ µ ( p, q ) = i Z d xe iq · x h π + ( p ) | T { j µV ( x ) , j † B (0) }| i (44)with j µV ( x ) = ¯ u ( x ) γ µ (1 + γ ) b ( x ). For the currentof B -meson j † B (0), we choice the right-handed current j † B (0) = m b b (0) i (1 + γ ) d (0) which can highlight thetwist-2, 4 DAs contributions, and the twist-3 DAs contri-butions vanished. By following the standard proceduresof light-cone sum rule approach [61, 62], we can get the B → π TFF f B → π + ( q ), reads f B → π + ( q ) = e m B /M m B f B (cid:2) F ( q , M , s B )+ α s C F π F ( q , M , s B ) (cid:21) , (45)where C F = 4 / m B and f B are the B -meson mass anddecay constant respectively, s B is the continuum thresh-old. The LO contribution of the LCSR (45) is expressedas F ( q , M , s B )= m b f π Z u due − m b − q uuM (cid:26) φ π ( u ) u + 1 m b − q × (cid:20) − m b u m b − q ) d φ π ( u ) du + uψ π ( u )+ Z u dvψ π ( v ) − I π ( u ) (cid:21)(cid:27) , (46)and the NLO term of f B → π + ( q ) is F ( q , M , s B )= f π π Z s B m b dse − s/M Z du Im s T ( q , s, u ) φ π ( u ) . (47)Where m b is the b -quark mass, ¯ u = 1 − u , u = ( m b − q ) / ( s B − q ), φ π ( u ) and ψ π ( u ) are the pionic twist-4 DAs, and I π ( u ) is the combination function of fourpionic twist-4 DAs Ψ π ( u ), Φ π ( u ), e Ψ π ( u ) and e Φ π ( u ).For the expressions of those pionic twist-4 DAs, I π ( u ),and the imaginary part of the amplitude T , one canﬁnd in Ref. [62]. By taking µ = 3 GeV, M = 18 ± , s B = 35 . ± .

25 GeV , m B = 5 .

279 GeV, f B = 214 +7 − MeV [62], we can obtain f B → π + (0) = 0 . +0 . − . (48)This value is consistent with but larger than the one inRef. [62] by the conventional current correlation. Thediﬀerence between those two values is mainly due to thediﬀerence in the selected correlation function. Compar-ing Eqs. (45) - (47) above with Eqs. (4.4), (4.5), (4.7) inRef. [62], one can ﬁnd that the contributions from pionic twist-3 DAs disappeared, while the contributions of pio-nic twist-2,4 DAs doubled. Then the diﬀerence betweenthe twist-2 DA’s contribution and twist-3 DAs’ contribu-tions in the LCSR with the conventional current correla-tion can be used as the system error caused by adoptedthe chiral current correlation function. IV. SUMMARY

In this paper, we improve the traditional LCHO modelof pionic leading-twist DA φ π ( x, µ ) by introducing anew WF’s longitudinal distribution function, i.e., ϕ IV2; π in Eq. (22). At the same time, we improve the methodof determining the model parameters. The least squaremethod is used to ﬁt the moments h ξ n i π | µ directly todetermine those parameters. This makes it necessary andmeaningful to calculate higher-order moments. The moremoments, the stronger the constraint on DA behavior.We adopt the QCD sum rules based on the BFT tocalculate the moments h ξ n i π | µ , and the values of ﬁrstﬁve moments are h ξ i π | µ = 0 . ± . h ξ i π | µ =0 . ± . h ξ i π | µ = 0 . ± . h ξ i π | µ =0 . ± . h ξ i π | µ = 0 . ± . h ξ n i π | µ , making use of the least squaremethod, we obtain the behavior of φ π ( x, µ ), that is,Eqs. (16), (22) and (42). The probability judged thegoodness-of-ﬁt is very close to 1, and our model is good.Compared with our previous work, in addition to theimprovement of the LCHO model, there are three im-provements: i) The moments h ξ n i π | µ , rather than theGegenbauer moments a πn ( µ ), is used as constraint con-ditions to determine the model parameters; ii) The leastsquares method is used to ﬁt the moments h ξ n i π | µ to getthe appropriate model parameters; iii) We take Eq. (11)rather than Eq. (7) as the sum rules of h ξ n i π | µ , whichcan avoid the error caused by non-normalized 0 th moment h ξ i π | µ on the left side of Eq. (7), and make the accu-racy of the resulted values of h ξ n i π | µ to be increased bymore than 10%. Those improvements can be widely usedto QCD sum rules studies of other meson DA to obtainmore accurate DA’s behavior.As an application, we take our model to calculate thepion-photon TFF F πγ ( Q ) which are shown in Fig. 4.Our results have an agreement with the Belle predic-tions at large Q -region. Meanwhile, the B → π TFF f B → π + ( q ) is calculated up to NLO accuracy, which hasan agreement with other theoretical predictions. V. ACKNOWLEDGMENTS

This work was supported in part by the NationalNatural Science Foundation of China under GrantNo.11765007, No.11875122, No.11625520, No.11947406and No.12047564, the Project of Guizhou ProvincialDepartment of Science and Technology under Grant1No.KY[2019]1171, the Project of Guizhou Provincial De-partment of Education under Grant No.KY[2021]030, theChongqing Graduate Research and Innovation Founda-tion under Grant No.ydstd1912, the Fundamental Re- search Funds for the Central Universities under GrantNo.2020CQJQY-Z003, the Project of Guizhou MinzuUniversity under Grant No. GZMU[2019]YB19. [1] S. J. Brodsky and G. P. Lepage, in:

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