Determination of the pion distribution amplitude

Abstract

Right now, we have not enough knowledge to determine the hadron distribution amplitudes (DAs) which are universal physical quantities in the high energy processes involving hadron for applying pQCD to exclusive processes. Even for the simplest pion, one can't discriminate from different DA models. Inversely, one expects that processes involving pion can in principle provide strong constraints on the pion DA. For example, the pion-photon transition form factor (TFF) can get accurate information of the pion wave function or DA, due to the single pion in this process. However, the data from Belle and BABAR have a big difference on TFF in high $Q^2$ regions, at present, they are helpless for determining the pion DA. At the present paper, we think it is still possible to determine the pion DA as long as we perform a combined analysis of the most existing data of the processes involving pion such as $\pi \to \mu \bar{\nu}$, $\pi^0 \to \gamma \gamma$, $B\to \pi l \nu$, $D \to \pi l \nu$, and etc. Based on the revised light-cone harmonic oscillator model, a convenient DA model has been suggested, whose parameter $B$ which dominates its longitudinal behavior for $\phi_{\pi}(x,\mu^2)$ can be determined in a definite range by those processes. A light-cone sum rule analysis of the semi-leptonic processes $B \to \pi l \nu$ and $D \to \pi l \nu$ leads to a narrow region $B = [0.01,0.14]$, which indicate a slight deviation from the asymptotic DA. Then, one can predict the behavior of the pion-photon TFF in high $Q^2$ regions which can be tested in the future experiments. Following this way it provides the possibility that the pion DA will be determined by the global fit finally.Comment: 9 pages, 6 figure

Revisiting Quantum Discord for Two-Qubit X States: The Error Bound to an Analytical Formula

“…Table I shows the quantum discord of 3 element POVM, that of 2 projective measurements and that of measurement obtained by Ali et al(δ 2 ) respectively. As we can see, δ 3,min for ρ AB 1 becomes 0.123010 which is 0.001613 less than the minimum value of the quantum discord obtained from 2 projective measurement in [15](The quantum discord obtained here is a little different from the result of [15]. It is because the quantum discord obtained in ref.…”

“…[14] tried to find an * Electronic address: mslab.nk@gmail.com † Electronic address: yyhkwon@hanyang.com analytic formula of the quantum discord for symmetric X state. Recently, by using von Neumann measurement Y. Huang showed that the result of [12] may be valid with some worst-case error [15]. In this report we show that three element POVM can provide a better quantum discord.…”

“…The results to quantum discord in terms of the natural logarithm(log e ) can be found in Appendix.) In addition, δ 3,min for ρ AB 2 becomes 0.107873 which is 0.000075 less than the minimum value of the quantum discord obtained from 2 projective measurement in [15]. Furthermore δ 3,min for ρ AB [14]; however, it is known that the formula may not be optimal.…”

“…Quantum discord for a Bell-diagonal state was analytically obtained [10]. Much research has had a focus on finding the quantum discord of the X state [12][13][14][15][16][17][18].…”

Revisiting the pion leading-twist distribution amplitude within the QCD background field theory

“…n (2x − 1) , (1) where the C 3/2 n (2x − 1) are Gegenbauer polynomials and the a π n (µ) are Gegenbauer moments. Theoretically, the Gegenbauer moments have been calculated by QCD * Electronic address: zhongtao@ihep.ac.cn † Electronic address: wuxg@cqu.edu.cn ‡ Electronic address: zgwang@aliyun.com § Electronic address: huangtao@ihep.ac.cn sum rules [9][10][11][12][13] and the lattice gauge theory [14][15][16][17].…”