A window on infrared QCD with small expansion parameters

Lattice simulations of the QCD correlation functions in the Landau gauge have established two remarkable facts. First, the coupling constant in the gauge sector — defined, e.g., in the Taylor scheme— remains finite and moderate at all scales, suggesting that some kind of perturbative description should be valid down to infrared momenta. Second, the gluon propagator reaches a finite nonzero value at vanishing momentum, corresponding to a gluon screening mass. We review recent studies which aim at describing the long-distance properties of Landau gauge QCD by means of the perturbative Curci-Ferrari model. The latter is the simplest deformation of the Faddeev-Popov Lagrangian in the Landau gauge that includes a gluon screening mass at tree-level. There are, by now, strong evidences that this approach successfully describes many aspects of the infrared QCD dynamics. In particular, several correlation functions were computed at one- and two-loop orders and compared with ab-initio lattice simulations. The typical error is of the order of ten percent for a one-loop calculation and drops to few percents at two loops. We review such calculations in the quenched approximation as well as in the presence of dynamical quarks. In the latter case, the spontaneous breaking of the chiral symmetry requires to go beyond a coupling expansion but can still be described in a controlled approximation scheme in terms of small parameters. We also review applications of the approach to nonzero temperature and chemical potential.

Redefining the Graph Edit Distance

Graph edit distance has been used since 1983 to compare objects in machine learning when these objects are represented by attributed graphs instead of vectors. In these cases, the graph edit distance is usually applied to deduce a distance between attributed graphs. This distance is defined as the minimum amount of edit operations (deletion, insertion and substitution of nodes and edges) needed to transform a graph into another. Since now, it has been stated that the distance properties have to be applied [(1) non-negativity (2) symmetry (3) identity and (4) triangle inequality] to the involved edit operations in the process of computing the graph edit distance to make the graph edit distance a metric. In this paper, we show that there is no need to impose the triangle inequality in each edit operation. This is an important finding since in pattern recognition applications, the classification ratio usually maximizes in the edit operation combinations (deletion, insertion and substitution of nodes and edges) that the triangle inequality is not fulfilled.

O(2) spin model and the Kosterlitz–Thouless’s phase transition

At low temperature, the large distance properties of the O(2) spin lattice model can be described by the O(2) non-linear σ-model. The latter model is free and massless in two dimensions. The origin of this peculiarity can be found in the local structure of the field manifold: for N = 2, the O(N) sphere reduces to a circle, which cannot be distinguished locally from a straight line. Because the physical fields are sin θ or cos θ, or equivalently e± iθ, instead of θ, a field renormalization is necessary, and temperature-dependent anomalous dimensions are generated. However, the free θ action cannot describe the long-distance properties of the lattice model for all temperatures, since a high temperature analysis of the corresponding spin model shows that the correlation length is finite at high temperature, and thus a phase transition is required. In fact, it is necessary to take into account the property that θ is a cyclic variable. This condition is irrelevant at low temperature, but when the temperature increases, classical configurations with singularities at isolated points, around which θ varies by a multiple of 2π become important. The action of these configurations (vortices) can be identified with the energy of a neutral Coulomb gas, which exhibits a transition between a low temperature of bound neutral molecules and a high temperature phase of a plasma of free charges. The Coulomb gas can be mapped onto the sine-Gordon (sG) model, mapping in which the low- and high-temperature regions of the models are exchanged. This correspondence helps to understand some properties of the famous Kosterlitz-Thouless (KT) phase transition, which separates an infinite correlation length phase without order, the low-temperature phase of the O(2) spin model, from a finite correlation length phase, the high-temperature phase of the O(2) spin model.

Ultraviolet divergences: Effective field theory (EFT)

Only local relativistic quantum field theories (QFT) are considered: the action that appears in the field integral is the integral of a classical Lagrangian density, function of fields and their derivatives (taken at the same point). Physical quantities can be calculated as power series in the various interactions. As a consequence of locality, infinities appear in perturbative calculations, due to short-distance singularities, or after Fourier transformation, to integrals diverging at large momenta: one speaks of ultraviolet (UV) divergences. These divergences are peculiar to local QFT: in contrast to classical mechanics or non-relativistic quantum mechanics (QM) with a finite number of particles, a straightforward construction of a QFT of point-like objects with contact interactions is impossible. A local QFT, in a straightforward formulation, is an incomplete theory. It is an effective theory, which eventually (perhaps at the Planck's scale?), to be embedded in some non-local theory, which renders the full theory finite, but where the non-local effects affect only short-distance properties (an operation sometimes called UV completion). The impossibility to define a QFT without an explicit reference to an external short scale is an indication of a non-decoupling between short- and long-distance physics. The forms of divergences are investigated to all orders in perturbation theory using power counting arguments.

Wiener Index of Edge Thorny Graphs of Catacondensed Benzenoids

The Wiener index is a topological index of a molecular graph, defined as the sum of distances between all pairs of its vertices. Benzenoid graphs include molecular graphs of polycyclic aromatic hydrocarbons. An edge thorny graph G is constructed from a catacondensed benzenoid graph H by attaching new graphs to edges of a perfect matching of H. A formula for the Wiener index of G is derived. The index of the resulting graph does not contain distance characteristics of elements of H and depends on the Wiener index of H and distance properties of the attached graphs.

Theoretical properties of nearest-neighbor distance distributions and novel metrics for high dimensional bioinformatics data

The performance of nearest-neighbor feature selection and prediction methods depends on the metric for computing neighborhoods and the distribution properties of the underlying data. The effects of the distribution and metric, as well as the presence of correlation and interactions, are reflected in the expected moments of the distribution of pairwise distances. We derive general analytical expressions for the mean and variance of pairwise distances for Lq metrics for normal and uniform random data with p attributes and m instances. We use extreme value theory to derive results for metrics that are normalized by the range of each attribute (max – min). In addition to these expressions for continuous data, we derive similar analytical formulas for a new metric for genetic variants (categorical data) in genome-wide association studies (GWAS). The genetic distance distributions account for minor allele frequency and transition/transversion ratio. We introduce a new metric for resting-state functional MRI data (rs-fMRI) and derive its distance properties. This metric is applicable to correlation-based predictors derived from time series data. Derivations assume independent data, but empirically we also consider the effect of correlation. These analytical results and new metrics can be used to inform the optimization of nearest neighbor methods for a broad range of studies including gene expression, GWAS, and fMRI data. The summary of distribution moments and detailed derivations provide a resource for understanding the distance properties for various metrics and data types.