Lipschitz-Volume rigidity in Alexandrov geometry

We prove a Lipschitz-Volume rigidity theorem in Alexandrov geometry, that is, if a 1-Lipschitz map $f\colon X=\amalg X_\ell\to Y$ between Alexandrov spaces preserves volume, then it is a path isometry and an isometry when restricted to the interior of $X$. We furthermore characterize the metric structure on $Y$ with respect to $X$ when $f$ is also onto. This implies the converse of Petrunin's Gluing Theorem: if a gluing of two Alexandrov spaces via a bijection between their boundaries produces an Alexandrov space, then the bijection must be an isometry.Comment: This is the published version on AI

Ehrhart h∗h^*-vectors of hypersimplices

We consider the Ehrhart $h^*$-vector for the hypersimplex. It is well-known that the sum of the $h_i^*$ is the normalized volume which equals an Eulerian numbers. The main result is a proof of a conjecture by R. Stanley which gives an interpretation of the $h^*_i$ coefficients in terms of descents and excedances. Our proof is geometric using a careful book-keeping of a shelling of a unimodular triangulation. We generalize this result to other closely related polytopes

Globalization with probabilistic convexity

We introduce a notion of probabilistic convexity and generalize some classical globalization theorems in Alexandrov geometry. A weighted Alexandrov's lemma is developed as a basic tool.Comment: in Journal of Topology and Analysis, 201

A canonical expansion of the product of two Stanley symmetric functions

We study the problem of expanding the product of two Stanley symmetric functions $F_w\cdot F_u$ into Stanley symmetric functions in some natural way. Our approach is to consider a Stanley symmetric function as a stabilized Schubert polynomial $F_{w}=\lim_{n\to \infty}\mathfrak{S}_{1^{n}\times w}$, and study the behavior of the expansion of \s_{1^n\times w}\cdot\s_{1^n\times u} into Schubert polynomials, as $n$ increases. We prove that this expansion stabilizes and thus we get a natural expansion for the product of two Stanley symmetric functions. In the case when one permutation is Grassmannian, we have a better understanding of this stability. We then study some other related stable properties, which provides a second proof of the main result

Resummation with Wilson lines off the light cone

I review the resummation formalism for organizing large logarithms in perturbative expansion of collinear subprocesses through the variation of Wilson lines off the light cone. A master equation is derived, which involves the evolution kernel resulting from this variation. It is then demonstrated that all the known single- and double-logarithm summations for a parton distribution function or a transverse-momentum-dependentparton distribution can be reproduced from the master equation by applying appropriate soft-gluon approximations to the evolution kernel. Moreover, jet substructures, information which is crucial for particle identification at the Large Hadron Collider and usually acquired from event generators, can also be calculated in this formalism.Comment: 18 pages, 13 figures; invited review article to be published by Physics of Elementary Particles and Atomic Nucle

Non-dipolar Wilson links for quasi-parton distribution functions

We propose a modified definition for a quasi-parton distribution function (QPDF) with an equal-time correlator in the large momentum limit, whose two pieces of space-like Wilson links are oriented in orthogonal directions. It is explicitly shown at one-loop level that the linear divergence in the original QPDF with dipolar Wilson links, which complicates its matching to the standard light-cone parton distribution function (LPDF), is removed. The LPDF can then be extracted reliably from Euclidean lattice data for the QPDF with the non-dipolar Wilson links.Comment: 7 pages, 2 figure

Perturbative QCD analysis of b-hadron lifetimes

We develop perturbative QCD factorization theorems for inclusive b-hadron decays, in which radiative corrections characterized by the hadronic scale, the b-hadron mass, and the W boson mass are absorbed into a heavy hadron distribution function, a hard b quark decay amplitude, and a "harder" function, respectively. Double logarithmic corrections associated with a light energetic final-state quark, which appear at kinematic end points, are absorbed into a jet function. Various large logarithms contained in the above functions are summed to all orders, leading to the evolution factors among the three characteristic scales. The heavy hadron distribution function is identical to the one constructed in the framework of heavy quark effective theory. It is shown that hadron kinematics must be employed in factorization theorems, and that perturbative contributions, depending on hadron kinematics, distinguish the lifetimes of the b-hadrons $B_d, B_s$ and $\Lambda_b$. Assuming the same heavy-quark-effective-theory parametesr $\lambda_1$ for these hadrons, we predict the lifetimes $\tau(B_d)=1.56$ ps, $\tau(B_s)=1.46$ ps and $\tau(\Lambda_b)=1.22$ ps. We also predict the $B_u$ meson lifetime $\tau(B_u)=1.62$ ps by varying the B meson distribution function slightly. All the above results are consistent with experimental data.Comment: 18 pages in revtex, 1 figure in postscript fil

A modified BFKL equation with unitarity

We propose a modified Balitskii-Fadin-Kuraev-Lipatov equation from the viewpoint of the resummation technique, which satisfies the unitarity bound. The idea is to relax the strong rapidity ordering and to restrict phase space for real gluon emissions in the evaluation of the BFKL kernel. It is found that the gluon distribution function rises as a power of the Bjorken variable $x$, and then saturates at $x\to 0$. We estimate that the saturation begins to occur for $x< 10^{-4}$.Comment: Conclusion is revised. One figure is adde

Perturbative QCD analysis of exclusive BB meson decays

We review the perturbative QCD formalism for exclusive heavy meson decays, concentrating on the three-scale factorization theorem for nonleptonic processes. The formalism is then extended to the radiative decay $B\to K^*\gamma$, which occurs through penguin diagrams. It is observed that the contributions from the operators other than the penguin one $b\to s\gamma$ are not negligible. From the best fit to the experimental data of the branching ratio ${\cal B}(B\to K^*\gamma)$, we extract the $B$ meson wave function, which possesses a sharp peak in the region with a small momentum fraction.Comment: numerical outcomes are revise

PQCD factorizaiton of two-body B decays

I review the known approaches to two-body nonleptonic $B$ meson decays, including factorization assumption, modified factorization assumption, QCD factorization, and perturbative QCD factorization. Important phenomenological aspects of these approaches are emphasized.Comment: talks presented at the International Conference on Flavor Physics, Zhangjiajie, China, May, 2001, at the International Workshop on Exclusive $B$ Decays, Regensburg, Germany, July, 2001, at the International Conference on Light-Cone Physics, Trento, Italy, Sep. 2001, and at the 5th KEK International Topic Conference, KEK, Japan, Nov. 200