Heavy Pseudoscalar Leading-Twist Distribution Amplitudes within QCD Theory in Background Fields

In this paper, we study the leading-twist distribution amplitude (DA) of the heavy pseudoscalars (HPs), such as $\eta_c$, $\eta_b$ and $B_c$, within the QCD theory in the background fields. New sum rules up to dimension-six condensates for both the HP decay constants and their leading-twist DA moments are presented. From the sum rules for the HP decay constants, we obtain $f_{\eta_c} = 453 \pm 4 \textrm{MeV}$, $f_{B_c} = 498 \pm 14 \textrm{MeV}$, and $f_{\eta_b} = 811 \pm 34 \textrm{MeV}$. Basing on the sum rules for the HPs' leading-twist DA moments, we construct a new model for the $\eta_c$, $\eta_b$ and $B_c$ leading-twist DAs. Our present HP DA model can also be adaptable for the light pseudo-scalar DAs, such as the pion and kaon DAs. Thus, it shall be applicable for a wide range of QCD exclusive processes. As an application, we apply the $\eta_c$ leading-twist DA to calculate the $B_c \to \eta_c$ transition form factor $f_+^{B_c \to \eta_c}(q^2)$. At the maximum recoil region, we obtain $f_+^{B_c \to \eta_c}(0) = 0.612^{+0.053}_{-0.052}$. After further extrapolating the TFF $f_+^{B_c \to \eta_c}(q^2)$ to its allowable $q^2$ region, we predict the branching ratio for the semi-leptonic decay $B_c \to \eta_c l \nu$. We obtain ${\cal B}(B_c \to \eta_c l \nu)=\left(7.70^{+1.65}_{-1.48}\right) \times 10^{-3}$ for massless leptons, which is consistent with the LCSRs estimation obtained in the literature.

The HP leading-twist DA at the scale µ can be expanded in Gegenbauer polynomials as [12]: where a HP n (µ) stands for the n th -order Gegenbauer moment, and the odd moments should be zero for the η c and η b mesons.When the scale µ tends to infinity, the DA φ HP (µ, x) shall evolve into its asymptotic form 6x(1 − x) [13].Since the typical energy scale of a specific process is always finite, it is interesting to know the φ HP behavior at any finite scale.
It is reasonable to assume that the η c and η b DAs have similar behaviors.As for the η c leading-twist DA, several models have been suggested in the literature [9,[14][15][16][17][18][19][20].For examples, Bondar and Chernyak [14] proposed a phenomenological model for the η c leading-twist DA (the BC model) as a try to resolve the disagreement between the experimental observations and the NRQCD prediction on the production cross section of e + e − → J/Ψ + η c ; Braguta, Likhoded and Luchinsky [15] proposed a model for the η c leading-twist DA (the BLL model) based on the moments calculated under the QCD Shifman-Vainshtein-Zakharov (SVZ) sum rules up to dimension-four condensates.As for the B c meson, one usually adopts a naive δ-like model for its leading-twist DA φ Bc [21].
In this paper, we study the HP leading-twist DAs within the SVZ sum rules [22] under the background field theory (BFT) [23].As the basic assumption of the SVZ sum rules, the quark condensate qq , the gluon condensate G 2 and etc., reflect the nonperturbative property in QCD.It is noted that the BFT provides a selfconsistent description for those vacuum condensates and provides a systematic way to achieve the goal of the SVZ sum rules [23].The HP DAs are more involved than the light pseudoscalar DAs, since we have to take the quark mass effect in the calculation.Recently, within the framework of BFT, we have for the first time calculated the quark propagator and vertex operator (z • ↔ D) n with full mass dependence up to dimension-six operators [24].Thus we are facing the chance of deriving a more precise sum rules for the HP DA moments and a precise HP DA behavior.For convenience, based on the BHL-prescription for constructing the meson wavefunctions [25], we suggest a general model for the HP leadingtwist wavefunctions and their DAs.
As an application of the suggested DA model, we apply the η c leading-twist DA to calculate the B c → η c transition form factor (TFF) f Bc→ηc + (q 2 ) within the LC-SRs.It is the key component for the semi-leptonic decay B c → η c lν.It is also the only TFF for the decay if the generated leptons are massless.By adopting the conventional correlator for the LCSRs, similar to the B → π TFFs [26], the TFF f Bc→ηc + (q 2 ) shall be formulated as the function involving the η c leading-twist DA, twist-3 DA, and other higher-twist DAs.The higher-twist DAs follow the power suppression rule in large scale region, however they may have sizable contributions to the TFF in the intermediate energy regions, similar to the pionic cases of the B → π TFFs and the pion TFFs [27].At present, the η c higher-twist DAs are still with great uncertainty, thus the possible LCSRs with η c various twist DAs shall inversely greatly dilute our understanding of the leading-twist DA behaviors.To cure the problem, we adopt the chiral correlator suggested in Ref. [10] to do our calculation, and we find that the most uncertain twist-3 DAs can be eliminated, then we can see more clearly on how the leading-twist DA affects f Bc→ηc + (q 2 ).The remaining parts of the paper are organized as follows.In Sec.II, the QCD SVZ sum rules for the HP decay constants and the HP leading-twist DA moments are given within the framework of BFT.A new model for the HP leading-twist DAs are also suggested here.Numerical results are presented in Sec.III.Sec.IV is reserved for a summary.

A. SVZ Sum Rules for the HP Decay Constants
To obtain the SVZ sum rules for the HP decay constants, we take the following correlation function Here the pseudo-scalar current where , respectively.The HP decay constant f HP is defined as where m HP stands for the HP mass and m 1(2) is the mass of Q 1(2) quark.Following the standard sum rules procedures, the correlation function (2) can be inserted by a completed set of intermediate hadronic states in the physical region.It can also be treated in the framework of the operator product expansion (OPE) in the deep Euclidean region simultaneously.Those two results can be related by the dispersion relation subtractions, (5) where t min = (m 1 + m 2 ) 2 .Then, the sum rules can be achieved by applying the Borel transform for both side of Eq.( 5).
More explicitly, on the one hand, we do the OPE for the correlator (2), Π QCD (q 2 ), within the framework of BFT.For the purpose, we first apply the following replacement for the quark fields in Eq.( 2), where Q 1 and Q 2 in the right-hand-side of Eq.( 6) stand for the quark background fields, η 1 and η 2 are the corresponding quantum fluctuations (quantum fields) on the background field.The quantum fields interacts with each other according to the Feynman rule of BFT [23], for example, the quantum quark-anti-quark pair can be contracted as a propagator; while the remaining background fields shall be kept to form the various vacuum matrix elements.Fig. 1 shows the Feynman diagrams for determining the HP decay constant up to dimension-six operators, in which the newly derived quark propagator with up to dimension-six operators has been adopted [24].In Fig. 1, the big dot stands for the vertex operators iγ 5 in the current (3), the cross symbol attached to the gluon line indicates the tensor of the local gluon background field, and "n" indicates the n th -order covariant derivative.Fig. 1.(a1) provides the perturbative contribution, Fig. 1.(b1,b2,c2) provide the contributions proportional to the dimension-four condensate α s G 2 , and the remaining thirteen diagrams Fig. 1.(d1-f1) provide the contributions proportional to the dimensionsix condensate g 3 s f G 3 .Here, α s G 2 and g 3 s f G 3 are abbreviations for the condensates 0 α s G A µν G Aµν 0 and 0 g 3 s f ABC G Aµν G Bρ ν G C ρµ 0 , respectively, where the color indices A, B, C = (1, 2, • • • , 8).Then, we can directly derive the explicit expression for Π QCD (q 2 ) from Fig. 1, which is rather lengthy and shall not be presented here for simplicity.
On the other hand, with the help of the definition (4), the hadronic spectrum representation of the correlator (2) can be written as where s HP is the continue threshold parameter, θ is the usual step function, and ρ cont stands for the hadron spectrum density from the continuous states.Due to the quark-hadron duality, ρ cont can be written as As a combination of Eqs.(5,7,8), we are ready to derive the SVZ sum rules for the HP decay constant.After further applying the Borel transformation to suppress both the unknown continuous states' contributions and the higher dimensional condensates' contributions, the final SVZ sum rules for the HP decay constant reads FIG.1: Feynman diagrams for the HP decay constant.The big dot stands for the vertex operators iγ5 in the current (3), the cross symbol attached to the gluon line indicates the tensor of the local gluon background field, and "n" indicates the n th -order covariant derivative.
where M is the Borel parameter, the operator LM = lim −q 2 ,n→∞; n stands for the usual Borel transformation operator.The perturbative part have been studied up to one-loop level by Ref. [28], and we have t is Spence function.Moreover, for the parts proportional to the dimension-four and dimension-six condensates, we have In deriving these sum rules, we have adopted the M Sscheme to deal with the infrared divergences.During the calculation, we have to deal with the following vacuum matrix elements in D-dimensional space (D = 4 − 2ǫ): The formulae for relating these matrix elements with the conventional condensates under the D-dimensional space have been given in the Appendix B of Ref. [24].For simplicity, we do not present them here, and the interesting readers may turn to this reference for detailed technology.

B. SVZ Sum Rules for the Moments of the HP
Leading-Twist DA The HP leading-twist DA φ HP is defined as where ξ = 2u − 1. Expanding the left-hand-side of Eq.( 13) near z = 0 and writing the exponent in righthand-side of Eq.( 13) as power series, we obtain the definition of the DA moments 0 Q1 (0) zγ 5 (iz where is the n th -order moment of φ HP .The 0 th -order moment gives the normalization condition for φ HP .Setting n = 0 in Eq.( 14), one can get To derive sum rules for the φ HP moments, we consider the following correlation function: where z 2 = 0, and the two currents . Similar to Sec.II.A, we can deduce the SVZ sum rules for the moments ξ n HP .Fig. 2 shows the corresponding Feynman diagrams for deriving the moments ξ n HP .In Fig. 2, the left big dot and the right big dot stand for the vertex operators zγ 5 (z • ↔ D) n and zγ 5 in the currents J n (x) and J † 0 (0), respectively; the cross symbol attached to the gluon line indicates the tensor of the local gluon background field, and "n" indicates n th -order covariant derivative.In different to Fig. 1, there are seven Feynman diagrams that have not been shown in Fig. 2, because they have no contribution for the moments ξ n HP due to their quark loops explicitly lead to Tr[• • •] = 0. Fig. 2.(a1) provides the perturbative contribution, Fig. 2.(b1-d1) provide the double-gluon condensate contribution and the remaining twenty-three diagrams provide the triple-gluon condensate contribution.Furthermore, comparing with Fig. 1, we have some extra diagrams for the present case, i.e.Following the standard SVZ procedures of the sum rules, the final sum rules for the moments of the HP leading-twist DA can be written as where Up to 6 th -order, the moments ξ n HP and the Gegenbauer moments a HP n at the same scale µ can be related via the following equations: Thus, inversely, we can derive the Gegenbauer moments a HP n from the above sum rules for ξ n HP .Usually, the Gegenbauer moments a HP n are known for an initial scale µ 0 around Λ QCD , which can be evolved from any scale µ via the equation where For the running coupling, we adopt [29] with C. A Model for the HP Leading-Twist DAs The meson DA can be derived from its light-cone wavefunction by integrating out its transverse components.Thus, it is helpful to construct a HP leading-twist wavefunction and then get its DA.For the purpose, one may assume that the HPs wavefunctions have similar form as those of the pseudoscalars kaon with SU f (3)-breaking effect [30] and the D meson or B meson [10,31].Based on the BHL-prescription [25], the HP wavefunction can be constructed as where k ⊥ is the transverse momentum, χ HP (x, k ⊥ ) is the spin-space wavefunction and Ψ R HP (x, k ⊥ ) stands for the spatial wavefunction.The spin-space wavefunction χ HP (x, k ⊥ ) takes the form [32] where m1,2 are the constituent quark masses for the HP.m1 = mb and m2 = mc for the case of B c meson, m1 = m2 = mc ( mb ) for the case of η c (η b ).We take mc = 1.8GeV and mb = 4.7GeV to do our numerical calculations.It is noted that different choices of mc or mb will lead to quite small differences to the HP DA.Because mb , mc >> Λ QCD , the spin-space wavefunction χ HP tends to 1 for the heavy scalars, thus, one may omit such factor as a simplified model.The spatial wavefunction Ψ R HP (x, k ⊥ ) takes the form where A HP is normalization constant.The parameter β HP is a harmonious parameter that dominantly determines the wavefunction transverse distributions.The function ϕ HP (x) dominantly dominates the wavefunction's longitudinal distribution, whose behavior is further dominated by its first several Gegenbauer polynomials.
By keeping up to 6 th -order Gegenbauer moments, it can be expansion as in which B HP 1,3,5 should be 0 for the case of η c or η b DA, due to the fact that the η c or η b DA should be unchanged over the transformation x ↔ (1 − x).
Using the relationship between the HP leading-twist DA and the HP wavefunction, (35) we can obtain the required leading-twist DA for the HP.That is, after integrating over the transverse momentum for the wavefunction (31), we obtain where m = m1 (1 − x)+ m2 x, µ 0 ∼ Λ QCD is the factorization scale, and the error function Erf(x) = 2 √ π x 0 e −t 2 dt.The wavefunction parameters A HP , B HP n and β HP can be determined by the following constraints: The decay constant f HP can be determined by the sum rules (9).
• The probability of finding the leading Fock state | Q 1 Q2 in the HP Fock state expansion, Equivalently, one can replace the constraint (38) by the average value of the squared transverse momentum k 2 ⊥ HP , which is measurable and is defined as The experimental measurements on k 2 ⊥ HP are not available at the present.We adopt the constraint (38) and take P ηc ≃ 0.8 [18,32] and P Bc ∼ P η b ≃ 1 [31] to do the calculation.The choice of P η b ∼ P Bc > P ηc is reasonable, since with the increase of the constituent quark masses, the valence Fock state occupies a bigger fraction in hadron and the probability of finding the valence Fock state will be close to unity in the non-relativistic limit.We have checked that all the wavefunction parameters change very slightly by varying P Bc from 1.0 to 0.9, which indicates that the B c meson already reaches the non-relativistic limit.
• The Gegenbauer moments can also be derived from the following definition They should be equal to the Gegenbauer moments determined from the values of ξ n HP , which can be determined from the sum rules (19).
Using these constraints, one can strictly determine the wavefunction parameters A HP , B HP n and β HP at an initial scale µ 0 .These parameters are scale dependent, one can obtain their values at any scale µ via the following evolution equation [13] x where δ h1 h2 = 1 when the Q 1 and Q2 have opposite helicities and δ h1 h2 = 0 for other cases.

B. The HP Decay Constants
To set the threshold parameter s HP and the allowable Borel window for the sum rules (9), we require that the continuum contribution to be less than 30%, and the values for f HP are stable in the Borel window.We obtain s ηc = 18GeV 2 , s Bc = 45GeV 2 and s η b = 90GeV 2 .Our predictions for the HP decay constants f HP under the allowable Borel windows are put in Table I, where all other input parameters are taken as their central values.
We put the curves for the decay constants f ηc , f Bc and f η b versus the Borel parameter M 2 in Fig. 3, where the shaded bands indicate the uncertainties from the input parameters m HP , m c,b , α s G 2 and g 3 s f G 3 .By taking all uncertainty errors into consideration and adding them in quadrature, our final predictions on f HP are put in Table II.As a comparison, some typical estimations on the HP decay constants derived under various approaches [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49] are also presented.Table II shows that our present estimations on HP decay constants agree with those derived under the Lattice QCD [49], especially for f Bc and f η b .

C. The HP Leading-Twist DAs
First, we calculate the HP leading-twist DA moments ξ n HP with the SVZ sum rules (19).As suggested by Braguta etal.[15], we set the continue threshold to be infinity.We adopt the ratio f 2 HP ξ n HP /(f 2 HP ξ 0 HP ) to derive the n th -moment ξ n HP instead of directly calculating ξ n HP .Due to the theoretical uncertainty sources for f HP and ξ n HP are mutually correlated with each other, such a treatment result in a much smaller theoretical uncertainty.Our results are presented in Table III, in which the HP leading-twist DA moments ξ n HP up to 6 th -order are presented.We take the Borel window Bc and ξ n η b , respectively.Fig. 4 shows the stability The HP leading-twist DA moments ξ n HP up to 6 th -order.The errors are squared average of those from all the input parameters, such as the Borel parameter, the condensates and the bound state parameters.The scale µ is set to be mc( mc) for ηc and mb ( mb ) for Bc and η b .

TABLE IV:
The HP leading-twist DA Gegenbauer moments a HP n up to 6 th -order, which are derived from ξ n HP via the relations (23,24,25,26,27,28).The scale µ is set to be mc( mc) for ηc and mb ( mb ) for Bc and η b .Second, we adopt the relationship between the moments ξ n HP and the Gegenbauer moments a HP n , i.e.Eqs.(23,24,25,26,27,28), to derive the Gegenbauer moments a HP n from Table III.The results for the Gegenbauer moments a HP n are shown in Table IV.Third, we determine all the input parameters A HP , B HP n and β HP for the HP leading-twist DA model (36).Using the central values for the Gegenbauer moments a HP n listed in Table IV, we obtain, at the scale µ = mb ( mb ), for the η c leading-twist DA; and  36), the BC model [14] and the BLL model [15], respectively.µ = mc( mc).
for the η b leading-twist DA.All those three HPs' leadingtwist DAs are presented in Fig. 5.The φ ηc (x, µ) is broader than φ η b (x, µ), and both of them are symmetric, while the φ Bc (x, µ) is non-symmetrical, which is consistent with the fact that its constitute c-and b-quarks are different.
Finally, we take the η c leading-twist DA as an explicit example to show the HP DA properties in detail.Fig. 6 presents a comparison of our η c leading-twist DA model (36) with those of the BC model [14] and the BLL model [15].Our DA model is broader in shape than that of the BLL model, but narrower than that of the BC model.Fig. 7 shows how φ ηc (x, µ) changes with the scale, in which four typical values, i.e. µ = 1.275GeV, 4.18GeV, 10GeV and 100GeV, are adopted.From Fig. 7, one may observe that with increment of the scale µ, the φ ηc (x, µ) becomes broader and broader, which shall finally tends to the asymptotic form for µ = ∞ limit.

D. An Application of the Leading-Twist DA φη c
As an application, in this subsection, we calculate the B c → η c TFF f Bc→ηc + (q 2 ) by using our present η c DA model (36).) versus the Borel parameter M 2 at several typical q 2 .All the input parameters are taken to be their central values.
As has been discussed in the Introduction, it is helpful to apply the LCSRs approach with chiral current correlator to calculate f Bc→ηc + (q 2 ) [10].Thus the most uncertain twist-3 DAs' contributions are eliminated, and we can see more clearly the properties of the leading-twist DA.Following the standard way as programmed in Ref. [10], we obtain +twist-4 and higher-twist terms, (46) where ū = 1 − u, and We take the η c leading-twist DA φ ηc (u) at the scale µ ≃ mb ( mb ) to do the calculation.We adopt the same criteria as those of Ref. [10] to determine the Borel window of the process and we take the continuum threshold to be s 0 = 42GeV 2 .The determined Borel window is M 2 = (20−35)GeV 2 , in which the TFF also has a good stability as shown by Fig. 8. ) FIG. 9: The TFF f Bc→ηc + (q 2 ) versus q 2 , in which the shaded hand indicates its uncertainties.framework of the QCD sum rules, the QCD LCSRs, and the pQCD factorization approaches.The QCD SVZ sum rules provides one of the most effective approaches for exclusive processes, which separates the short-and longdistance quark-gluon interaction, and parameterizes the latter as a series of non-perturebative vacuum condensates.The BFT provides a systematic method for achieving the goal of SVZ sum rules and also provides a physical picture for the vacuum condensates.As a sequential work of Ref. [24], in this paper, we have made a detailed study on the HP leading-twist DAs together with the HP decay constants under the framework of BFT up to dimension-six condensates.
Using the sum rules (9), we obtain f ηc = 453 ± 4MeV, f Bc = 498 ± 14MeV and f η b = 811 ± 34MeV.These values are in agreement with those derived by the Lattice QCD [49].Using the sum rules (19), we calculate the first several moments for the HP leadingtwist DA, which are presented in Table III.Using the relations (23,24,25,26,27,28), we further obtain −0.3 [7] the Gegenbauer moments up to 6 th -order.More explicitly, the non-zero Gegenbauer moments for φ ηc are: a 2 ( mc ( mc )) = −0.372± 0.027, a 4 ( mc ( mc )) = 0.124 ± 0.029 and a 6 ( mc ( mc )) = −0.025± 0.017; the nonzero Gegenbauer moments for φ η b are: a 2 ( mb ( mb )) = −0.387± 0.019, a 4 ( mb ( mb )) = 0.136 ± 0.022 and a 6 ( mb ( mb )) = −0.028± 0.013; the non-zero Gegenbauer moments for φ Bc are: a 1 ( mb ( mb )) = 0.466 ± 0.038, a 2 ( mb ( mb )) = −0.053± 0.016, a 3 ( mb ( mb )) = −0.106± 0.018, a 4 ( mb ( mb )) = −0.028± 0.010, a 5 ( mb ( mb )) = −0.017± 0.002, a 6 ( mb ( mb )) = −0.014± 0.001.Here, the errors are squared average of those from the uncertainties of the Borel parameter, the condensates, and the bound state parameters.The Gegenbauer moments at any other scale can be obtained via evolution.The meson DA is of non-perturbative nature, thus, it is helpful to have a general model for all the related HPs.Based on the BHL-prescription [25], we have suggested a model (36) for the HP leading-twist DAs.The model parameters of φ HP (x, µ) are determined with three reasonable constraints together with the newly obtained HP decay constants and Gegenbauer moments.The behaviors of the η c , B c and η b leading-twist DAs are presented in Fig. 5.It has been shown that the φ ηc and φ η b are symmetric and are close in shape; while, the φ Bc is nonsymmetrical and quite different from the naive δ-model, i.e. φ Bc (x) ∝ δ(x− mb /m Bc ), suggested in Ref. [21].Our present HP DA model can also be adaptable for the light pseudo-scalar DAs, such as pion and kaon DAs.Thus, it shall be applicable for a wide range of QCD exclusive processes.With more and more data available, we may get more definite conclusions on the behaviors of the pseudoscalar DAs, and then achieve a more accurate theoretical prediction on those processes.
As an application for the η c leading-twist DA φ ηc , we study the TFF f Bc→ηc + (q 2 ) within the LCSRs.It is noted that the branching ratio Br(B c → η c lν) strongly depends on the TFF f Bc→ηc + (q 2 ), thus a more accurate TFF shall result in a more accurate branching ratio.At the maximum recoil point, we obtain f Bc→ηc + (0) = 0.612 +0.053 −0.052 .Furthermore, by using the extrapolated TFF, we predict the branching ratio of the semi-leptonic decay B c → η c lν, i.e., Br(B c → η c lν) = 7.70 +1.65 −1.48 × 10 −3 , which is consistent with previous LCSRs prediction [54] and the quark model result [47,50].

FIG. 2 :
FIG. 2: Feynman diagrams for the moments of the HP leading-twist DA.The left big dot and the right big dot stand for the vertex operators zγ5(z • ↔ D) n and zγ5 in the currents Jn(x) and J † 0 (0), respectively.The cross symbol attached to the gluon line indicates the tensor of the local gluon background field, and "n" indicates n th -order covariant derivative.

FIG. 3 :
FIG.3:The HP decay constants versus the Borel parameter M 2 .The shaded band indicates the uncertainty.

FIG. 7 :
FIG. 7:The running of the ηc leading-twist DA.The dashed, the dash-dot, the solid and the dotted lines are for µ = 1.275GeV, 4.18GeV, 10GeV and 100GeV, respectively.

2 FIG. 8 :
FIG.8:The TFF f Bc→ηc + (q 2 ) versus the Borel parameter M 2 at several typical q 2 .All the input parameters are taken to be their central values.

TABLE I :
The HP decay constants for sη c = 18GeV 2 , sB c = 45GeV 2 and sη b = 90GeV 2 under the allowable Borel windows, where all the other input parameters are taken to be their central values.

TABLE V :
(49)fitted parameters a and b for the TFF extrapolation(49).The lowest, middle and the highest TFFs determined from the LCSRs (46) are adopted for such a determination.

TABLE VI :
The branching ratio of Bc → ηclν (in unit %).As a comparison, we also present those derived by the LCSRs, the quark model (QM), the pQCD, the QCD relativistic potential model (RPM) and the NRQCD approaches.Approach Br(Bc → ηclν)