Abstract
We suggest to probe the pion light-cone distribution amplitude, applying a dispersion relation for the pion electromagnetic form factor. Instead of the standard dispersion relation, we use the equation between the spacelike form factor $F_\pi(Q^2)$ and the integrated modulus of the timelike form factor. For $F_\pi(Q^2)$, the QCD light-cone sum rule with a dominant twist-2 term is used. Adopting for the pion twist-2 distribution amplitude a certain combination of the first few Gegenbauer polynomials, it is possible to fit their coefficients $a_{2,4,6,...}$ (Gegenbauer moments) from this equation, employing the measured pion timelike form factor. For the exploratory fit we use the data of the BaBar collaboration. The results definitely exclude the asymptotic twist-2 distribution amplitude. Also the model with a single $a_2\neq 0$ is disfavoured by the fit. Considering the models with $a_{n>2}\neq 0$, we find that the fitted values of the second and fourth Gegenbauer moments cover the intervals $a_2 (1 \mbox{GeV}) = (0.22 - 0.33) $, $a_4 (1 \mbox{GeV}) = (0.12 - 0.25) $. The higher moments starting from $a_{8}$ are consistent with zero, albeit with large uncertainties. The spacelike pion form factor obtained in two different ways, from the dispersion relation and from the light-cone sum rule, agrees, within uncertainties, with the measurement by the Jefferson Lab $F_\pi$ collaboration.
Highlights
The light-cone distribution amplitudes (DAs) are the key elements in several QCD methods used to describe the hard exclusive scattering and heavy hadron decays
The purpose of this paper is to demonstrate that the timelike form factor FπðsÞ can provide an additional information about the pion DAs
To avoid the uncertainty induced by the restoration of the imaginary part from the measured modulus of the timelike form factor, we suggest to use a modified dispersion relation in which the spacelike form factor is equal to the integral over jFπðsÞj2
Summary
The light-cone distribution amplitudes (DAs) are the key elements in several QCD methods used to describe the hard exclusive scattering and heavy hadron decays. For the processes involving a pion, the accuracy of this description depends first of all on our knowledge of the leading twist-2 pion DA. The expansion of this DA in orthogonal Gegenbauer polynomials reduces the necessary input to their coefficients anðn 1⁄4 2; 4; 6; ...Þ known as the Gegenbauer moments. In many practical applications only the low moments are retained in the pion twist-2 DA. Such a parametrization is justified by the fact that the anomalous dimension of the multiplicatively renormalizable moment an grows with the number n, so that the contributions of higher moments to the processes with a large momentum scale are suppressed. For an accurate description of the pion DA, the knowledge of the first few Gegenbauer moments at a certain normalization scale is of utmost importance
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