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To date, the behavior of the pionic leading-twist distribution amplitude (DA) which is a universal physical quantity and is introduced into high-energy processes involving pions based on the factorization theorem is not completely consistent. The form of is usually described by phenomenological models and constrained by the experimental data on exclusive processes containing pions or the moments computed using QCD sum rules and the lattice QCD theory. Evidently, an appropriate model is extremely important to determine the exact behavior of . In this paper, by adopting the least squares method to fit the ξ-moments calculated using QCD sum rules based on the background field theory, we perform an analysis on several commonly used models of the pionic leading-twist DA in the literature; these include the truncation form of the Gegenbauer polynomial series, the light-cone harmonic oscillator model, the form extracted from the Dyson-Schwinger equations, the model from the light-front holographic AdS/QCD, and a simple power-law parametrization form.

In this study, α+β Ti-Al-V-Mo-Nb alloys with the addition of multiple elements that are suitable for laser additive manufacturing (LAM) were designed according to a Ti-6Al-4V cluster formula. This formula can be expressed as 12[Al-Ti12](AlTi2)+5[Al-Ti14]((Mo, V, Nb)2Ti), in which Mo and Nb were added into the alloys partially instead of V to give alloys with nominal compositions of Ti-6.01Al-3.13V-1.43Nb, Ti-5.97Al-2.33V-2.93Mo, and Ti-5.97Al-2.33V-2.20Mo-0.71Nb (wt.%). The microstructures and mechanical properties of the as-deposited and heat-treated samples prepared via LAM were examined. The sizes of the β columnar grains and α laths in the Nb-containing samples are found to be larger than those of the Ti-6Al-4V alloy, whereas Mo- or Mo/Nb-added alloys contain finer grains. It indicates that Nb gives rise to coarsened β columnar grains and α laths, while Mo significantly refines them. Furthermore, the single addition of Nb improves the elongation, whereas the single addition of Mo enhances the strength of the alloys. The simultaneous addition of Mo/Nb significantly improves the comprehensive mechanical properties of the alloys, leading to the best properties with an ultimate tensile strength of 1,070 MPa, a yield strength of 1,004 MPa, an elongation of 9%, and micro-hardness of 355 HV. The fracture modes of all the alloys are ductile-brittle mixed fracture.

In this paper, we study the pion leading-twist distribution amplitude $\phi_{2;\pi}(x,\mu)$ by improving the traditional light-cone harmonic oscillator model within the reconstruction of the function $\varphi_{2;\pi}(x)$. In order to constraining the model parameters, we calculate its moments $\langle\xi^n\rangle_{2;\pi}|_\mu$ in the framework of QCD background field theory sum rule (BFTSR) up to $10^{\rm th}$ order. Considering the fact that the sum rule of the $0^{\rm th}$ moment $\langle\xi^0\rangle_{2;\pi}|_\mu$ cannot be normalized, we suggest a more reasonable sum rule formula for $\langle\xi^n\rangle_{2;\pi}|_\mu$. Then, we obtain the values of $\langle\xi^n\rangle_{2;\pi}|_{\mu_0}$ with $n=(2,4,6,8,10)$ at the initial scale $\mu_0 = 1~{\rm GeV}$. The first two moments are: $\langle \xi^2\rangle_{2;\pi}|_{\mu_0} = 0.271 \pm 0.013$, $\langle\xi^4\rangle_{2;\pi}|_{\mu_0} = 0.138 \pm 0.010$; and the corresponding Gegenbauer moments are $a^{2;\pi}_2(\mu_0) = 0.206 \pm 0.038$, $a^{2;\pi}_4(\mu_0) = 0.047 \pm 0.011$, respectively. After fitting the moments $\langle\xi^n\rangle_{2;\pi}|_{\mu}$, we obtained the appropriate model parameters by using the least square method. The resultant behavior for twist-2 pion DA is more closely to the AdS/QCD and lattice result, but is narrower than that by Dyson-Schwinger equation. Furthermore, we calculate the pion-photon transition form factors (TFF) and $B\to\pi$ TFF within light-cone sum rule approach, which are conform with experimental and theoretical results., Comment: 15 pages, 6 figures, matches the version to appear in PRD