Exposing the QCD Splitting Function with CMS Open Data

The splitting function is a universal property of quantum chromodynamics (QCD) which describes how energy is shared between partons. Despite its ubiquitous appearance in many QCD calculations, the splitting function cannot be measured directly since it always appears multiplied by a collinear singularity factor. Recently, however, a new jet substructure observable was introduced which asymptotes to the splitting function for sufficiently high jet energies. This provides a way to expose the splitting function through jet substructure measurements at the Large Hadron Collider. In this letter, we use public data released by the CMS experiment to study the 2-prong substructure of jets and test the 1 -> 2 splitting function of QCD. To our knowledge, this is the first ever physics analysis based on the CMS Open Data.Comment: 7 pages, 5 figures; v2: references updated and figure formatting improved; v3: approximate version to appear in PR

Classical dimer model with anisotropic interactions on the square lattice

We discuss phase transitions and the phase diagram of a classical dimer model with anisotropic interactions defined on a square lattice. For the attractive region, the perturbation of the orientational order parameter introduced by the anisotropy causes the Berezinskii-Kosterlitz-Thouless transitions from a dimer-liquid to columnar phases. According to the discussion by Nomura and Okamoto for a quantum-spin chain system [J. Phys. A 27, 5773 (1994)], we proffer criteria to determine transition points and also universal level-splitting conditions. Subsequently, we perform numerical diagonalization calculations of the nonsymmetric real transfer matrices up to linear dimension specified by L=20 and determine the global phase diagram. For the repulsive region, we find the boundary between the dimer-liquid and the strong repulsion phases. Based on the dispersion relation of the one-string motion, which exhibits a two-fold ``zero-energy flat band'' in the strong repulsion limit, we give an intuitive account for the property of the strong repulsion phase.Comment: 11 pages, 8 figure

Dressed gluon exponentiation

Perturbative and non-perturbative aspects of differential cross-sections close to a kinematic threshold are studied applying ``dressed gluon exponentiation'' (DGE). The factorization property of soft and collinear gluon radiation is demonstrated using the light-cone axial gauge: it is shown that the singular part of the squared matrix element for the emission of an off-shell gluon off a nearly on-shell quark is universal. We derive a generalized splitting function that describes the emission probability and show how Sudakov logs emerge from the phase-space boundary where the gluon transverse momentum vanishes. Both soft and collinear logs associated with a single dressed gluon are computed through a single integral over the running-coupling to any logarithmic accuracy. The result then serves as the kernel for exponentiation. The divergence of the perturbative series in the exponent indicates specific non-perturbative corrections. We identify two classes of observables according to whether the radiation is from an initial-state quark, as in the Drell-Yan process, or a final-state quark, forming a jet with a constrained invariant mass, as in fragmentation functions, event-shape variables and deep inelastic structure functions.Comment: 28 page

Galois-stability for Tame Abstract Elementary Classes

We introduce tame abstract elementary classes as a generalization of all cases of abstract elementary classes that are known to permit development of stability-like theory. In this paper we explore stability results in this context. We assume that \K is a tame abstract elementary class satisfying the amalgamation property with no maximal model. The main results include: (1) Galois-stability above the Hanf number implies that \kappa(K) is less than the Hanf number. Where \kappa(K) is the parallel of \kapppa(T) for f.o. T. (2) We use (1) to construct Morley sequences (for non-splitting) improving previous results of Shelah (from Sh394) and Grossberg & Lessmann. (3) We obtain a partial stability-spectrum theorem for classes categorical above the Hanf number.Comment: 23 page

On the universal cover and the fundamental group of an RCD∗(K,N)RCD^*(K,N)-space

The main goal of the paper is to prove the existence of the universal cover for $RCD^*(K,N)$-spaces. This generalizes earlier work of C. Sormani and the second named author on the existence of universal covers for Ricci limit spaces. As a result, we also obtain several structure results on the (revised) fundamental group of such spaces. These are the first topological results for $RCD^{*}(K,N)$-spaces without extra structural-topological assumptions (such as semi-local simple connectedness).Comment: Final version to appear in Journal f\"ur die Reine und Angewandte Mathemati