Triple crossing numbers of graphs

We introduce the triple crossing number, a variation of crossing number, of a graph, which is the minimal number of crossing points in all drawings with only triple crossings of the graph. It is defined to be zero for a planar graph, and to be infinite unless a graph admits a drawing with only triple crossings. In this paper, we determine the triple crossing numbers for all complete multipartite graphs including all complete graphs.Comment: 34 pages, 53 figures: We reorganized the article and revised some argument

Transverse double-spin asymmetries for small QTQ_T Drell-Yan pair production in pppp and ppˉp\bar{p} collisions

We discuss transverse double-spin asymmetries for dimuon production at small transverse-momentum $Q_T$ in $pp$ and $p\bar{p}$ collisions. All order resummation of large logarithms relevant in small $Q_T$ region is performed at next-to-leading logarithmic (NLL) accuracy, and asymmetries at RHIC, J-PARC and GSI are calculated.Comment: 4 pages, 2 figures, Talk given at DIS06, April 20-24, 2006, Tsukuba, Japa

Unravelling soft components in the shape function for inclusive B decays

In the heavy-quark limit, the decay spectrum for radiative and semileptonic inclusive B-meson decays is determined by a universal structure function, i.e. the shape function. We study the constraints on the bilocal heavy-quark operator for the shape function from the QCD equations of motion and heavy-quark symmetry, and we obtain a new basis of nonlocal operators that reveals relevant soft components in the shape function. Those nonlocal operators represent the ``kinetic energy distribution'' of the b-quark inside the B-meson and the four-parton correlations with additional quarks and gluons. A corresponding local operator basis relevant for relating the shape function to HQET parameters is also given, and novel effects of the quark-gluon correlations are discussed.Comment: 19 pages PTPTeX. Typos corrected and improvements made to the text. References added. Published versio