Renormalization and topological susceptibility on the lattice: SU(2) Yang-Mills theory

The renormalization functions involved in the determination of the topological susceptibility in the SU(2) lattice gauge theory are extracted by direct measurements, without relying on perturbation theory. The determination exploits the phenomenon of critical slowing down to allow the separation of perturbative and non-perturbative effects. The results are in good agreement with perturbative computations.Comment: 12 pages + 4 figures (PostScript); report no. IFUP-TH 10/9

Testing the heating method with perturbation theory

The renormalization constants present in the lattice evaluation of the topological susceptibility can be non-perturbatively calculated by using the so-called heating method. We test this method for the $O(3)$ non-linear $\sigma$-model in two dimensions. We work in a regime where perturbative calculations are exact and useful to check the values obtained from the heating method. The result of the test is positive and it clarifies some features concerning the method. Our procedure also allows a rather accurate determination of the first perturbative coefficients.Comment: 15 pages, LaTeX file, needs RevTeX style. Tarred, gzipped, uuencode

A critical comparison of different definitions of topological charge on the lattice

A detailed comparison is made between the field-theoretic and geometric definitions of topological charge density on the lattice. Their renormalizations with respect to continuum are analysed. The definition of the topological susceptibility, as used in chiral Ward identities, is reviewed. After performing the subtractions required by it, the different lattice methods yield results in agreement with each other. The methods based on cooling and on counting fermionic zero modes are also discussed.Comment: 12 pages (LaTeX file) + 7 (postscript) figures. Revised version. Submitted to Phys. Rev.

High energy parton-parton amplitudes from lattice QCD and the stochastic vacuum model

Making use of the gluon gauge-invariant two-point correlation function, recently determined by numerical simulation on the lattice in the quenched approximation and the stochastic vacuum model, we calculate the elementary (parton-parton) amplitudes in both impact-parameter and momentum transfer spaces. The results are compared with those obtained from the Kr\"{a}mer and Dosch ansatz for the correlators. Our main conclusion is that the divergences in the correlations functions suggested by the lattice calculations do not affect substantially the elementary amplitudes. Phenomenological and semiempirical information presently available on elementary amplitudes is also referred to and is critically discussed in connection with some theoretical issues.Comment: Text with 11 pages in LaTeX (twocolumn form), 10 figures in PostScript (psfig.tex used). Replaced with changes, Fig.1 modified, two references added, some points clarified, various typos corrected. Version to appear in Phys. Rev.

Color confinement and dual superconductivity of the vacuum. III

It is demonstrated that monopole condensation in the confined phase of SU(2) and SU(3) gauge theories is independent of the specific Abelian projection used to define the monopoles. Hence the dual excitations which condense in the vacuum to produce confinement must have magnetic U(1) charge in all the Abelian projections. Some physical implications of this result are discussed.Comment: 6 pages, 5 postscript figure

Color confinement and dual superconductivity in full QCD

We report on evidence that confinement is related to dual superconductivity of the vacuum in full QCD, as in quenched QCD. The vacuum is a dual superconductor in the confining phase, whilst the U(1) magnetic symmetry is realized a la Wigner in the deconfined phase.Comment: 4 pages, 4 eps figure

Improved lattice operators: the case of the topological charge density

We analyze the properties of a class of improved lattice topological charge density operators, constructed by a smearing-like procedure. By optimizing the choice of the parameters introduced in their definition, we find operators having (i) a better statistical behavior as estimators of the topological charge density on the lattice, i.e. less noisy; (ii) a multiplicative renormalization much closer to one; (iii) a large suppression of the perturbative tail (and other unphysical mixings) in the corresponding lattice topological susceptibility.Comment: 11 pages (REVTEX) + 4 (uuencoded) figure

Recognizing and Drawing IC-planar Graphs

IC-planar graphs are those graphs that admit a drawing where no two crossed edges share an end-vertex and each edge is crossed at most once. They are a proper subfamily of the 1-planar graphs. Given an embedded IC-planar graph $G$ with $n$ vertices, we present an $O(n)$-time algorithm that computes a straight-line drawing of $G$ in quadratic area, and an $O(n^3)$-time algorithm that computes a straight-line drawing of $G$ with right-angle crossings in exponential area. Both these area requirements are worst-case optimal. We also show that it is NP-complete to test IC-planarity both in the general case and in the case in which a rotation system is fixed for the input graph. Furthermore, we describe a polynomial-time algorithm to test whether a set of matching edges can be added to a triangulated planar graph such that the resulting graph is IC-planar

The non-Abelian dual Meissner effect as color-alignment in SU(2) lattice gauge theory

A new gauge (m-gauge) condition is proposed by means of a generalization of the Maximal Abelian gauge (MAG). The new gauge admits a space time dependent embedding of the residual U(1) into the SU(2) gauge group. This embedding is characterized by a color vector $\vec{m}(x)$. It turns out that this vector only depends of gauge invariant parts of the link configurations. Our numerical results show color ferromagnetic correlations of the $\vec{m}(x)$ field in space-time. The correlation length scales towards the continuum limit. For comparison with the MAG, we introduce a class of gauges which smoothly interpolates between the MAG and the m-gauge. For a wide range of the gauge parameter, the vacuum decomposes into regions of aligned vectors $\vec{m}$. The ''neutral particle problem'' of MAG is addressed in the context of the new gauge class.Comment: 15 pages, 6 figures, LaTeX using eps

3D Visibility Representations of 1-planar Graphs

We prove that every 1-planar graph G has a z-parallel visibility representation, i.e., a 3D visibility representation in which the vertices are isothetic disjoint rectangles parallel to the xy-plane, and the edges are unobstructed z-parallel visibilities between pairs of rectangles. In addition, the constructed representation is such that there is a plane that intersects all the rectangles, and this intersection defines a bar 1-visibility representation of G.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017