Gegenbauer Coefficients Research Articles

Dispersive derivation of the pion distribution amplitude

We derive the dependence of the leading-twist pion light-cone distribution amplitude (LCDA) on a parton momentum fraction $x$ by directly solving the dispersion relations for the moments with inputs from the operator product expansion (OPE) of the corresponding correlation function. It is noticed that these dispersion relations must be organized into those for the Gegenbauer coefficients first in order to avoid the ill-posed problem appearing in the conversion from the moments to the Gegenbauer coefficients. Given the values of various condensates in the OPE, we find solutions for the pion LCDA, which are convergent and stable in the Gegenbauer expansion. Moreover, the solution from summing contributions up to 18 Gegenbauer polynomials is smooth, and can be well approximated by a function proportional to ${x}^{p}(1\ensuremath{-}x{)}^{p}$ with $p\ensuremath{\approx}0.45$ at the scale $\ensuremath{\mu}=2\text{ }\text{ }\mathrm{GeV}$. Turning off the condensates, we get the asymptotic form for the pion LCDA as expected. We then solve for the pion LCDA at a different scale $\ensuremath{\mu}=1.5\text{ }\text{ }\mathrm{GeV}$ with the condensate inputs at this $\ensuremath{\mu}$, and demonstrate that the result is consistent with the one obtained by evolving the Gegenbauer coefficients from $\ensuremath{\mu}=2\text{ }\text{ }\mathrm{GeV}$ to 1.5 GeV. That is, our formalism is compatible with the QCD evolution. The strength of the above framework that goes beyond analyses limited to only the first few moments of a LCDA in conventional QCD sum rules is highlighted. The precision of our results can be improved systematically by including higher-order and higher-power terms in the OPE.

Improved calculation of the γ*γ → π process at low Q2 using LCSR’s and renormalization-group summation

We study two versions of lightcone sum rules to calculate the γ*γ → π0 transition form factor (TFF) within QCD. While the standard version is based on fixed-order perturbation theory by means of a power-series expansion in the strong coupling, the new method incorporates radiative corrections by renormalization-group summation and generates an expansion within a generalized fractional analytic perturbation theory involving only analytic couplings. Using this scheme, we determine the relative nonperturbative parameters and the first two Gegenbauer coefficients of the pion distribution amplitude (DA) to obtain TFF predictions in good agreement with the preliminary BESIII data, while the best-fit pion DA satisfies the most recent lattice constraints on the second moment of the pion DA at the three-loop level.

Light-cone distribution amplitudes of light mesons with QED effects

We discuss the generalization of the leading-twist light-cone distribution amplitude for light mesons including QED effects. This generalization was introduced to describe virtual collinear photon exchanges at the strong-interaction scale ΛQCD in the factorization of QED effects in non-leptonic B-meson decays. In this paper we study the renormalization group evolution of this non-perturbative function. For charged mesons, in particular, this exhibits qualitative differences with respect to the well-known scale evolution in QCD only, especially regarding the endpoint-behaviour. We analytically solve the evolution equation to first order in the electromagnetic coupling αem, which resums large logarithms in QCD on top of a fixed-order expansion in αem. We further provide numerical estimates for QED corrections to Gegenbauer coefficients as well as inverse moments relevant to (QED-generalized) factorization theorems for hard exclusive processes.

Extending the application of the light-cone sum rules method to low momenta using QCD renormalization-group summation: Theory and phenomenology

We show that using renormalization-group summation to generate the QCD radiative corrections to the $\pi-\gamma$ transition form factor, calculated with lightcone sum rules (LCSR), renders the strong coupling free of Landau singularities while preserving the QCD form-factor asymptotics. This enables a reliable applicability of the LCSR method to momenta well below 1 GeV$^2$. This way, one can use the new preliminary BESIII data with unprecedented accuracy below 1.5 GeV$^2$ to fine tune the prefactor of the twist-six contribution. Using a combined fit to all available data below 3.1 GeV$^2$, we are able to determine all nonperturbative scale parameters and a few Gegenbauer coefficients entering the calculation of the form factor. Employing these ingredients, we determine a pion distribution amplitude with conformal coefficients $(b_2,b_4)$ that agree at the $1\sigma$ level with the data for $Q^2 \leqslant 3.1$ GeV$^2$ and fulfill at the same time the lattice constraints on $b_2$ at N$^3$LO together with the constraints from QCD sum rules with nonlocal condensates.The form-factor prediction calculated herewith reproduces the data below 1 GeV$^2$ significantly better than analogous predictions based on a fixed-order power-series expansion in the strong coupling constant.

High-order Gegenbauer integral spectral element method integrated with an adaptive Chebyshev optimization strategy for solving linear singularly perturbed differential equations

We propose a high-order adaptive spectral element method based on Gegenbauer polynomial basis for solving singularly perturbed differential equations with Dirichlet boundary conditions and a boundary layer at one end. The method reduces the singularly perturbed boundary value problem into a single system of linear algebraic equations when the solution is smooth on the solution domain, or two systems of linear algebraic equations when the solution has a thin boundary layer. The core of the new adaptive strategy lies in constructing a mathematical model of the last few Gegenbauer coefficients in the Gegenbauer truncated series approximating the second-order derivative of the solution, which can be readily converted into a minimization problem in a least-squares sense and solved using fast optimization techniques. An error and convergence analysis of the method for sufficiently smooth solutions is presented. A numerical test example with various parameter settings is presented to verify the accuracy, effectiveness, and applicability of the proposed method. Numerical comparisons with other rival methods in the literature are included to demonstrate further the power of the proposed method.

On some implications of the BaBar data on the γ⁎η′ transition form factor

At leading-twist accuracy the form factors for the transitions from a virtual photon to the η or η′ can be expanded into a power series of the variable ω, being related to the difference of the two photon virtualities. The series possess the remarkable feature that only the Gegenbauer coefficients of the meson distribution amplitudes of order l≤m contribute to the term ∼ωm. Thus, for ω→0 only the asymptotic meson distribution amplitude contributes, allowing for a test of the mixing of the η and η′ decay constants. Employing the Gegenbauer coefficients determined in an analysis of the form factors in the real photon limit, we present predictions for the γ⁎η and γ⁎η′ form factors and compare them to the BaBar data.

Semigroup asymptotics, the Funk-Hecke identity and the Gegenbauer coefficients associated with the spherical Laplacian

A trace formulation of the Maclaurin spectral coefficients of the Schwartzian kernel of functions of the spherical Laplacian is given. A class of polynomials $\mathscr {P}^\nu _l(X)$ $(l \ge 0$, $\nu \gt -1/2)$ linking to the classical Gegenbauer polynomials through a differential-spectral identity is introduced, and its connection to the above spectral coefficients and their asymptotics analyzed. The paper discusses some applications of these ideas combined with the Funk-Hecke identity and semigroup techniques to geometric and variational-energy inequalities on the sphere and presents some examples.

Theoretical Description and Measurement of the Pion–Photon Transition Form Factor

Detailed predictions for the scaled pion-photon transition form factor are given, derived with the method of light-cone sum rules and using pion distribution amplitudes with two and three Gegenbauer coefficients obtained from QCD sum rules with nonlocal condensates. These predictions agree well with all experimental data that are compatible with QCD scaling (and collinear factorization), but disagree with the high-$Q^2$ data of the BaBar Collaboration that grow with the momentum. A good agreement of our predictions with results obtained from AdS/QCD models and Dyson-Schwinger computations is found.

Strictly and non-strictly positive definite functions on spheres

Isotropic positive definite functions on spheres play important roles in spatial statistics, where they occur as the correlation functions of homogeneous random fields and star-shaped random particles. In approximation theory, strictly positive definite functions serve as radial basis functions for interpolating scattered data on spherical domains. We review characterizations of positive definite functions on spheres in terms of Gegenbauer expansions and apply them to dimension walks, where monotonicity properties of the Gegenbauer coefficients guarantee positive definiteness in higher dimensions. Subject to a natural support condition, isotropic positive definite functions on the Euclidean space $\mathbb{R}^3$, such as Askey's and Wendland's functions, allow for the direct substitution of the Euclidean distance by the great circle distance on a one-, two- or three-dimensional sphere, as opposed to the traditional approach, where the distances are transformed into each other. Completely monotone functions are positive definite on spheres of any dimension and provide rich parametric classes of such functions, including members of the powered exponential, Mat\'{e}rn, generalized Cauchy and Dagum families. The sine power family permits a continuous parameterization of the roughness of the sample paths of a Gaussian process. A collection of research problems provides challenges for future work in mathematical analysis, probability theory and spatial statistics.

The η (η′) gamma transition form factor and the gluon–gluon distribution amplitude

The ηγ and η′γ transition form factors are analyzed to leading-twist accuracy and next-to-leading order of perturbative QCD. Using an η–η′ mixing scheme and all currently available experimental data the lowest Gegenbauer coefficients of the distribution amplitudes for the valence octet and singlet and the gluon–gluon Fock components are extracted. Predictions for the g*g*η′ vertex function are presented. We also comment on the new BELLE results for the πγ transition form factor.

Can we understand an auxetic pion-photon transition form factor within QCD?

A state-of-the-art analysis of the pion-photon transition form factor is presented based on an improved theoretical calculation that includes the effect of a finite virtuality of the quasireal photon in the method of light-cone sum rules. We carry out a detailed statistical analysis of the existing experimental data using this method and by employing pion distribution amplitudes with up to three Gegenbauer coefficients ${a}_{2}$, ${a}_{4}$, ${a}_{6}$. Allowing for an error range in the coefficient ${a}_{6}\ensuremath{\approx}0$, the theoretical predictions for ${\ensuremath{\gamma}}^{*}\ensuremath{\gamma}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}$ obtained with nonlocal QCD sum rules are found to be in good agreement with all data that support a scaling behavior of the transition form factor at higher ${Q}^{2}$, like those of the Belle Collaboration. The data on ${\ensuremath{\gamma}}^{*}\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\eta}/{\ensuremath{\eta}}^{\ensuremath{'}}$ from CLEO and BABAR are also reproduced, while there is a strong conflict with the auxetic trend of the BABAR data above $10\text{ }\text{ }{\mathrm{GeV}}^{2}$. The broader implications of these findings are discussed.

The expansion in Gegenbauer polynomials: A simple method for the fast computation of the Gegenbauer coefficients

We present a simple and fast algorithm for the computation of the Gegenbauer transform, which is known to be very useful in the development of spectral methods for the numerical solution of ordinary and partial differential equations of physical interest. We prove that the coefficients of the expansion of a function f(x) in Gegenbauer (also known as ultraspherical) polynomials coincide with the Fourier coefficients of a suitable integral transform of the function f(x). This allows to compute N Gegenbauer coefficients in O(Nlog2N) operations by means of a single Fast Fourier Transform of the integral transform of f(x). We also show that the inverse Gegenbauer transform is expressible as the Abel-type transform of a suitable Fourier series. This fact produces a novel algorithm for the fast evaluation of Gegenbauer expansions.

Pion-photon transition: The new QCD frontier

We perform a detailed analysis of all existing data (CELLO, CLEO, BaBar) on the pion-photon transition form factor by means of light-cone sum rules in which we include the NLO QCD radiative corrections and the twist-four contributions. The NNLO radiative correction together with the twist-six contribution are also taken into account in terms of theoretical uncertainties. Keeping only the first two Gegenbauer coefficients $a_2$ and $a_4$, we show that the $1\sigma$ error ellipse of all data up to 9 GeV$^2$ greatly overlaps with the set of pion distribution amplitudes obtained from nonlocal QCD sum rules---within the range of uncertainties due to twist-four. This remains valid also for the projection of the $1\sigma$ error ellipsoid on the $(a_2,a_4)$ plane when including $a_6$. We argue that it is not possible to accommodate the high-$Q^2$ tail of the BaBar data with the same accuracy, despite opposite claims by other authors, and conclude that the BaBar data still pose a challenge to QCD.

Pion distribution amplitude extracted from the experimental data with the local duality sum rule

The photon-to-pion transition form factor is investigated using the form of the renormalon-based twist-four pion distribution amplitude (DA) in the framework of the light-cone local-duality QCD sum rule, which, with suitable parameters, is insensitive to the higher-order Gegenbauer coefficients. With a careful determination for the insertion parameters so that the contribution from the higher-order Gegenbauer expansions is suppressed, the best-fit central values of the first two nontrivial Gegenbauer coefficients of the pion distribution amplitude are extracted out from the CLEO data to be ${a}_{2}(1\text{ }\text{ }{\mathrm{GeV}}^{2})=0.145\ifmmode\pm\else\textpm\fi{}0.055$ and ${a}_{4}(1\text{ }\text{ }{\mathrm{GeV}}^{2})=\ensuremath{-}(0.125\ifmmode\pm\else\textpm\fi{}0.085)$, respectively. The rescaled photon-to-pion transition form factor with our best-fit parameters is consistent very well with both the CELLO data and the prediction of the interpolation formula in all the experimental accessible region of the momentum transfer. The shape of the pion distribution amplitude based on the two-parameter model favors the camel-like type, where the near-end-point values are suppressed more than the asymptotic DA, and satisfies the midpoint constraint from light-cone sum rules approximately.

HIGHER TWIST EFFECTS IN PHOTON–PHOTON COLLISIONS

In this article, we investigate the contribution of the high twist Feynman diagrams to the large-pT single pseudoscalar and vector mesons inclusive production cross section in two-photon collisions and we present the general formulae for the high and leading twist differential cross sections. The pion wave function where two non-trivial Gegenbauer coefficients a2 and a4 have been extracted from the CLEO data, Braun–Filyanov pion wave function, the asymptotic and the Chernyak–Zhitnitsky wave functions are all used in the calculations. For ρ-meson we used the Ball–Braun wave function. The results of the calculations reveal that the high twist cross sections, the ratio R, the dependence transverse momentum pT and the rapidity y of meson in the Φ CLEO (x, Q2) wave function case is very close to the Φ asy (x) asymptotic wave function case. It is shown that the high twist contribution to the cross section depends on the choice of the meson wave functions.

Self-consistent Gaussian model of nonperturbative QCD vacuum

We show that the minimal Gaussian model of nonlocal vacuum quark and quark-gluon condensates in QCD violates the transverse character of the correlator of two vector currents. We suggest the improved Gaussian model of the nonperturbative QCD vacuum, which respects QCD equations of motion and minimizes the revealed gauge-invariance breakdown. We obtain the refined values of pion distribution amplitude (DA) conformal moments 〈ξ2N〉π (N = 1, ..., 5) using the improved QCD vacuum model, including the inverse moment 〈x −1〉π, being inaccessible if one uses the standard QCD SR. We construct the allowed region for Gegenbauer coefficients a 2 and a 4 of the pion DA for two values of the QCD vacuum nonlocality parameter, λ 2 q = 0.4 and 0.5 GeV2.

Twist-3 distribution amplitudes of scalar mesons from QCD sum rules

We study the twist-3 distribution amplitudes for scalar mesons made up of two valence quarks based on QCD sum rules. By choosing the proper correlation functions, we derive the moments of the scalar mesons up to the first two orders. Making use of these moments, we then calculate the first two Gegenbauer coefficients for twist-3 distribution amplitudes of scalar mesons. It is found that the second Gegenbauer coefficients of scalar density twist-3 distribution amplitudes for ${K}_{0}^{*}$ and ${f}_{0}$ mesons are quite close to that for ${a}_{0}$, which indicates that the SU(3) symmetry breaking effect is tiny here. However, this effect could not be neglected for the forth Gegenbauer coefficients of scalar twist-3 distribution amplitudes between ${a}_{0}$ and ${f}_{0}$. Besides, we also observe that the first two Gegenbauer coefficients corresponding to the tensor current twist-3 distribution amplitudes for all the ${a}_{0}$, ${K}_{0}^{*}$ and ${f}_{0}$ are very small. The renormalization group evolution of condensates, quark masses, decay constants and moments are considered in our calculations. As a by-product, it is found that the masses for isospin $I=1$, $\frac{1}{2}$ scalar mesons are around $1.27\ensuremath{\sim}1.41\text{ }\text{ }\mathrm{GeV}$ and $1.44\ensuremath{\sim}1.56\text{ }\text{ }\mathrm{GeV}$ respectively, while the mass for isospin state composed of $\overline{s}s$ is $1.62\ensuremath{\sim}1.73\text{ }\text{ }\mathrm{GeV}$.

Measurement of the photon light-cone wave function by diffractive dissociation

The measurement of the pion light-cone wave function is revisited and results for the Gegenbauer coefficients are presented. Mesurements of the photon electromagnetic and hadronic wave functions are described and results are presented.

HIGHER TWIST EFFECTS IN PROTON-PROTON COLLISIONS

In this article, we investigate the contribution of the high twist Feynman diagrams to the large-pT pion production cross section in proton-proton collisions and we present the general formulae for the high and leading twist differential cross sections. The pion wave function where two non-trivial Gegenbauer coefficients a2 and a4 have been extracted from the CLEO data, two other pion model wave functions, P2, P3, the asymptotic and the Chernyak-Zhitnitsky wave functions are used in the calculations. The results of all the calculations reveal that the high twist cross sections, the ratios R, r, the dependence transverse momentum pT and the rapidity y of pion in the Φ CLEO (x,Q2) wave function case is very close to the Φ asy (x) asymptotic wave function case. It is shown that the high twist contribution to the cross section depends on the choice of the meson wave functions.

Tagging the pion quark structure in QCD

We combine the constraints on the pion quark structure available from perturbative QCD, nonperturbative QCD (nonlocal QCD sum rules and light-cone sum rules) with the analysis of current data on ${F}_{\ensuremath{\pi}\ensuremath{\gamma}{\ensuremath{\gamma}}^{*}}({Q}^{2})$, including recent high-precision lattice calculations of the second moment of the pion's distribution amplitude. We supplement these constraints with those extracted from the renormalon approach by means of the twist-four contributions to the pion distribution amplitude in order to further increase stability with respect to related theoretical uncertainties. We show which regions in the space of the first two nontrivial Gegenbauer coefficients ${a}_{2}$ and ${a}_{4}$ of all these constraints overlap, tagging this way the pion structure to the highest degree possible at present.