Heavy Pseudoscalar Twist-3 Distribution Amplitudes within QCD Theory in Background Fields

In this paper, we study the properties of the twist-3 distribution amplitude (DA) of the heavy pseudo-scalars such as $\eta_c$, $B_c$ and $\eta_b$. New sum rules for the twist-3 DA moments \left_{\rm HP} and \left_{\rm HP} up to sixth orders and up to dimension-six condensates are deduced under the framework of the background field theory. Based on the sum rules for the twist-3 DA moments, we construct a new model for the two twist-3 DAs of the heavy pseudo-scalar with the help of the Brodsky-Huang-Lepage prescription. Furthermore, we apply them to the $B_c\to\eta_c$ transition form factor ($f^{B_c\to\eta_c}_+(q^2)$) within the light-cone sum rules approach, and the results are comparable with other approaches. It has been found that the twist-3 DAs $\phi^P_{3;\eta_c}$ and $\phi^\sigma_{3;\eta_c}$ are important for a reliable prediction of $f^{B_c\to\eta_c}_+(q^2)$. For example, at the maximum recoil region, we have $f^{B_c\to\eta_c}_+(0) = 0.674 \pm 0.066$, in which those two twist-3 terms provide $\sim33\%$ and $\sim22\%$ contributions. Also we calculate the branching ratio of the semi-leptonic decay $B_c \to\eta_c l\nu$ as $Br(B_c \to\eta_c l\nu) = \left( 9.31^{+2.27}_{-2.01} \right) \times 10^{-3}$.Comment: 12 pages, 16 figure

The Algorithmic Complexity of Bondage and Reinforcement Problems in bipartite graphs

Let $G=(V,E)$ be a graph. A subset $D\subseteq V$ is a dominating set if every vertex not in $D$ is adjacent to a vertex in $D$. The domination number of $G$, denoted by $\gamma(G)$, is the smallest cardinality of a dominating set of $G$. The bondage number of a nonempty graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with domination number larger than $\gamma(G)$. The reinforcement number of $G$ is the smallest number of edges whose addition to $G$ results in a graph with smaller domination number than $\gamma(G)$. In 2012, Hu and Xu proved that the decision problems for the bondage, the total bondage, the reinforcement and the total reinforcement numbers are all NP-hard in general graphs. In this paper, we improve these results to bipartite graphs.Comment: 13 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:1109.1657; and text overlap with arXiv:1204.4010 by other author