QCD effective charges from lattice data

We use recent lattice data on the gluon and ghost propagators, as well as the Kugo-Ojima function, in order to extract the non-perturbative behavior of two particular definitions of the QCD effective charge, one based on the pinch technique construction, and one obtained from the standard ghost-gluon vertex. The construction relies crucially on the definition of two dimensionful quantities, which are invariant under the renormalization group, and are built out of very particular combinations of the aforementioned Green's functions. The main non-perturbative feature of both effective charges, encoded in the infrared finiteness of the gluon propagator and ghost dressing function used in their definition, is the freezing at a common finite (non-vanishing) value, in agreement with a plethora of theoretical and phenomenological expectations. We discuss the sizable discrepancy between the freezing values obtained from the present lattice analysis and the corresponding estimates derived from several phenomenological studies, and attribute its origin to the difference in the gauges employed. A particular toy calculation suggests that the modifications induced to the non-perturbative gluon propagator by the gauge choice may indeed account for the observed deviation of the freezing values.Comment: 23 pages, 7 figure

Infrared finite effective charge of QCD

We show that the gauge invariant treatment of the Schwinger-Dyson equations of QCD leads to an infrared finite gluon propagator, signaling the dynamical generation of an effective gluon mass, and a non-enhanced ghost propagator, in qualitative agreement with recent lattice data. The truncation scheme employed is based on the synergy between the pinch technique and the background field method. One of its most powerful features is that the transversality of the gluon self-energy is manifestly preserved, exactly as dictated by the BRST symmetry of the theory. We then explain, for the first time in the literature, how to construct non-perturbatively a renormalization group invariant quantity out of the conventional gluon propagator. This newly constructed quantity serves as the natural starting point for defining a non-perturbative effective charge for QCD, which constitutes, in all respects, the generalization in a non-Abelian context of the universal QED effective charge. This strong effective charge displays asymptotic freedom in the ultraviolet, while in the low-energy regime it freezes at a finite value, giving rise to an infrared fixed point for QCD. Some possible pitfalls related to the extraction of such an effective charge from infrared finite gluon propagators, such as those found on the lattice, are briefly discussed.Comment: Invited talk given at LIGHT CONE 2008 Relativistic Nuclear and Particle Physics, July 7-11 2008 Mulhouse, Franc

Nonperturbative gluon and ghost propagators for d=3 Yang-Mills

We study a manifestly gauge invariant set of Schwinger-Dyson equations to determine the nonperturbative dynamics of the gluon and ghost propagators in $d=3$ Yang-Mills. The use of the well-known Schwinger mechanism, in the Landau gauge, leads to the dynamical generation of a mass for the gauge boson (gluon in $d=3$), which, in turn, gives rise to an infrared finite gluon propagator and ghost dressing function. The propagators obtained from the numerical solution of these nonperturbative equations are in very good agreement with the results of $SU(2)$ lattice simulations.Comment: 25 pages, 8 figure

The gluon mass generation mechanism: a concise primer

We present a pedagogical overview of the nonperturbative mechanism that endows gluons with a dynamical mass. This analysis is performed based on pure Yang-Mills theories in the Landau gauge, within the theoretical framework that emerges from the combination of the pinch technique with the background field method. In particular, we concentrate on the Schwinger-Dyson equation satisfied by the gluon propagator and examine the necessary conditions for obtaining finite solutions within the infrared region. The role of seagull diagrams receives particular attention, as do the identities that enforce the cancellation of all potential quadratic divergences. We stress the necessity of introducing nonperturbative massless poles in the fully dressed vertices of the theory in order to trigger the Schwinger mechanism, and explain in detail the instrumental role of these poles in maintaining the Becchi-Rouet-Stora-Tyutin symmetry at every step of the mass-generating procedure. The dynamical equation governing the evolution of the gluon mass is derived, and its solutions are determined numerically following implementation of a set of simplifying assumptions. The obtained mass function is positive definite, and exhibits a power law running that is consistent with general arguments based on the operator product expansion in the ultraviolet region. A possible connection between confinement and the presence of an inflection point in the gluon propagator is briefly discussed.Comment: 37 pages, 11 figures. Based on the talk given at the Workshop Dyson-Schwinger equations in modern mathematics and physics, ECT* (Trento) 22-26 September 2014. Review article contribution to the special issue of Frontiers of Physics (Eds. M. Pitschmann and C. D. Roberts

Schwinger mechanism in linear covariant gauges

In this work we explore the applicability of a special gluon mass generating mechanism in the context of the linear covariant gauges. In particular, the implementation of the Schwinger mechanism in pure Yang-Mills theories hinges crucially on the inclusion of massless bound-state excitations in the fundamental nonperturbative vertices of the theory. The dynamical formation of such excitations is controlled by a homogeneous linear Bethe-Salpeter equation, whose nontrivial solutions have been studied only in the Landau gauge. Here, the form of this integral equation is derived for general values of the gauge-fixing parameter, under a number of simplifying assumptions that reduce the degree of technical complexity. The kernel of this equation consists of fully-dressed gluon propagators, for which recent lattice data are used as input, and of three-gluon vertices dressed by a single form factor, which is modelled by means of certain physically motivated Ans\"atze. The gauge-dependent terms contributing to this kernel impose considerable restrictions on the infrared behavior of the vertex form factor; specifically, only infrared finite Ans\"atze are compatible with the existence of nontrivial solutions. When such Ans\"atze are employed, the numerical study of the integral equation reveals a continuity in the type of solutions as one varies the gauge-fixing parameter, indicating a smooth departure from the Landau gauge. Instead, the logarithmically divergent form factor displaying the characteristic "zero crossing", while perfectly consistent in the Landau gauge, has to undergo a dramatic qualitative transformation away from it, in order to yield acceptable solutions. The possible implications of these results are briefly discussed.Comment: 27 pages, 9 figures; v2: typos corrected, version matching the published on

Indirect determination of the Kugo-Ojima function from lattice data

We study the structure and non-perturbative properties of a special Green's function, u(q), whose infrared behavior has traditionally served as the standard criterion for the realization of the Kugo-Ojima confinement mechanism. It turns out that, in the Landau gauge, u(q) can be determined from a dynamical equation, whose main ingredients are the gluon propagator and the ghost dressing function, integrated over all physical momenta. Using as input for these two (infrared finite) quantities recent lattice data, we obtain an indirect determination of u(q). The results of this mixed procedure are in excellent agreement with those found previously on the lattice, through a direct simulation of this function. Most importantly, in the deep infrared the function deviates considerably from the value associated with the realization of the aforementioned confinement scenario. In addition, the dependence of u(q), and especially of its value at the origin, on the renormalization point is clearly established. Some of the possible implications of these results are briefly discussed.Comment: 25 pages, 10 figures; v2: typos corrected, expanded version that matches the published articl

Coloring, location and domination of corona graphs

A vertex coloring of a graph $G$ is an assignment of colors to the vertices of $G$ such that every two adjacent vertices of $G$ have different colors. A coloring related property of a graphs is also an assignment of colors or labels to the vertices of a graph, in which the process of labeling is done according to an extra condition. A set $S$ of vertices of a graph $G$ is a dominating set in $G$ if every vertex outside of $S$ is adjacent to at least one vertex belonging to $S$. A domination parameter of $G$ is related to those structures of a graph satisfying some domination property together with other conditions on the vertices of $G$. In this article we study several mathematical properties related to coloring, domination and location of corona graphs. We investigate the distance-$k$ colorings of corona graphs. Particularly, we obtain tight bounds for the distance-2 chromatic number and distance-3 chromatic number of corona graphs, throughout some relationships between the distance-$k$ chromatic number of corona graphs and the distance-$k$ chromatic number of its factors. Moreover, we give the exact value of the distance-$k$ chromatic number of the corona of a path and an arbitrary graph. On the other hand, we obtain bounds for the Roman dominating number and the locating-domination number of corona graphs. We give closed formulaes for the $k$-domination number, the distance-$k$ domination number, the independence domination number, the domatic number and the idomatic number of corona graphs.Comment: 18 page

Charging Interacting Rotating Black Holes in Heterotic String Theory

We present a formulation of the stationary bosonic string sector of the whole toroidally compactified effective field theory of the heterotic string as a double Ernst system which, in the framework of General Relativity describes, in particular, a pair of interacting spinning black holes; however, in the framework of low--energy string theory the double Ernst system can be particularly interpreted as the rotating field configuration of two interacting sources of black hole type coupled to dilaton and Kalb--Ramond fields. We clarify the rotating character of the $B_{t\phi}$--component of the antisymmetric tensor field of Kalb--Ramond and discuss on its possible torsion nature. We also recall the fact that the double Ernst system possesses a discrete symmetry which is used to relate physically different string vacua. Therefore we apply the normalized Harrison transformation (a charging symmetry which acts on the target space of the low--energy heterotic string theory preserving the asymptotics of the transformed fields and endowing them with multiple electromagnetic charges) on a generic solution of the double Ernst system and compute the generated field configurations for the 4D effective field theory of the heterotic string. This transformation generates the $U(1)^n$ vector field content of the whole low--energy heterotic string spectrum and gives rise to a pair of interacting rotating black holes endowed with dilaton, Kalb--Ramond and multiple electromagnetic fields where the charge vectors are orthogonal to each other.Comment: 15 pages in latex, revised versio