Constraint of xi-moments calculated with QCD sum rules on the pion distribution amplitude models

So far, the behavior of the pionic leading-twist distribution amplitude (DA) $\phi_{2;\pi}(x,\mu)$ $-$ which is universal physical quantity and enters the high-energy processes involving pion based on the factorization theorem $-$ has not been completely consistent. The form of $\phi_{2;\pi}(x,\mu)$ is usually described by phenomenological models and constrained by the experimental data of the exclusive processes containing pion or the moments calculated with the QCD sum rules and lattice QCD theory. Obviously, an appropriate model is very important for us to determine the exact behavior of $\phi_{2;\pi}(x,\mu)$. In this paper, by adopting the least squares method to fit the $\xi$-moments calculated with QCD sum rules based on the background field theory, we perform an analysis for several commonly used models of the pionic leading-twist DA in the literature, such as the truncation form of the Gegenbauer polynomial series, the light-cone harmonic oscillator model, the form from the Dyson-Schwinger equations, the model from the light-front holographic AdS/QCD and a simple power-law parametrization form.