Loop Integrals in Three Outstanding Gauges: Feynman, Light-Cone, and Coulomb

We apply the negative dimensional integration method (NDIM) to three outstanding gauges: Feynman, light-cone, and Coulomb gauges. Our aim is to show that NDIM is a very suitable technique to deal with loop integrals, regardless of which gauge choice that originated them. In the Feynman gauge we perform scalar two-loop four-point massless integrals; in the light-cone gauge we calculate scalar two-loop integrals contributing to two-point functions without any kind of prescriptions, since NDIM can abandon such devices - this calculation is the first test of our prescriptionless method beyond one-loop order; and finally, for the Coulomb gauge we consider a four-propagator massless loop integral, in the split-dimensional regularization context. © 2001 Academic Press.

First results for the Coulomb gauge integrals using NDIM

The Coulomb gauge has at least two advantages over other gauge choices in that bound states between quarks and studies of confinement are easier to understand in this gauge. However, perturbative calculations, namely Feynman loop integrations, are not well defined (there are the so-called energy integrals) even within the context of dimensional regularization. Leibbrandt and Williams proposed a possible cure to such a problem by splitting the space-time dimension into D = ω + ρ, i.e., introducing a specific parameter ρ to regulate the energy integrals. The aim of our work is to apply the negative dimensional integration method (NDIM) to the Coulomb gauge integrals using the recipe of split-dimension parameters and present complete results - finite and divergent parts - to the one- and two-loop level for arbitrary exponents of the propagators and dimension.

Evaluating residues and integrals through negative dimensional integration method (NDIM)

The standard way of evaluating residues and some real integrals through the residue theorem (Cauchy's theorem) is well-known and widely applied in many branches of Physics. Herein we present an alternative technique based on the negative dimensional integration method (NDIM) originally developed to handle Feynman integrals. The advantage of this new technique is that we need only to apply Gaussian integration and solve systems of linear algebraic equations, with no need to determine the poles themselves or their residues, as well as obtaining a whole class of results for differing orders of poles simultaneously.

Limit cycles of cubic polynomial differential systems with rational first integrals of degree 2

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

The 2ómega-dimensional light-cone integrals with momentum shift

A class of light-cone integrals typical to one-loop calculations in the two-component formalism is considered. For the particular cases considered, convergence is verified though the results cannot be expressed as a finite sum of elementary functions. © 1988 American Institute of Physics.

Efeitos da eletroestimulação do músculo vasto medial oblíquo em portadores de síndrome da dor patelofemoral: uma análise eletromiográfica

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Estimates for the Poisson kernel and Hardy spaces on compact manifolds

We study Hardy spaces on the boundary of a smooth open subset or R-n and prove that they can be defined either through the intrinsic maximal function or through Poisson integrals, yielding identical spaces. This extends to any smooth open subset of R-n results already known for the unit ball. As an application, a characterization of the weak boundary values of functions that belong to holomorphic Hardy spaces is given, which implies an F. and M. Riesz type theorem. (C) 2004 Elsevier B.V. All rights reserved.

Global dynamics of the Rikitake system

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

On the global dynamics of the Rabinovich system

In this paper by using the Poincare compactification in R(3) make a global analysis of the Rabinovich system(x) over dot = hy - v(1)x + yz, (y) over dot = hx - v(2)y - xz, (z) over dot = -v(3)z + xy,with (x, y, z) is an element of R(3) and ( h, v(1), v(2), v(3)) is an element of R(4). We give the complete description of its dynamics on the sphere at infinity. For ten sets of the parameter values the system has either first integrals or invariants. For these ten sets we provide the global phase portrait of the Rabinovich system in the Poincare ball (i.e. in the compactification of R(3) with the sphere S(2) of the infinity). We prove that for convenient values of the parameters the system has two families of singularly degenerate heteroclinic cycles. Then changing slightly the parameters we numerically found a four wings butterfly shaped strange attractor.

Analytical attitude prediction of spin stabilized spacecrafts perturbed by magnetic residual torque

An analytical approach for spin-stabilized spacecraft attitude prediction is presented for the influence of the residual magnetic torques. Assuming an inclined dipole model for the Earth's magnetic field, an analytical averaging method is applied to obtain the mean residual torque every orbital period. The orbit mean anomaly is utilized to compute the average components of residual torque in the spacecraft body frame reference system. The theory is developed for time variations in the orbital elements, and non-circular orbits, giving rise to many curvature integrals. It is observed that the residual magnetic torque does not have component along the spin axis. The inclusion of this torque on the rotational motion differential equations of a spin stabilized spacecraft yields conditions to derive an analytical solution. The solution shows that residual torque does not affect the spin velocity magnitude, contributing only for the precession and the drift of the spin axis of the spacecraft. (c) 2005 COSPAR. Published by Elsevier Ltd. All rights reserved.

The light-cone gauge without prescriptions

Feynman integrals in the physical light-cone gauge are more difficult to solve than their covariant counterparts. The difficulty is associated with the presence of unphysical singularities due to the inherent residual gauge freedom in the intermediate boson propagators constrained within this gauge choice. In order to circumvent these non-physical singularities, the headlong approach has always been to call for mathematical devices - prescriptions - some successful and others not. A more elegant approach is to consider the propagator from its physical point of view, that is, an object obeying basic principles such as causality. Once this fact is realized and carefully taken into account, the crutch of prescriptions can be avoided altogether. An alternative, third approach, which for practical computations could dispense with prescriptions as well as avoiding the necessity of careful stepwise consideration of causality, would be of great advantage. and this third option is realizable within the context of negative dimensions, or as it has been coined, the negative dimensional integration method (NDIM).

Massless and massive one-loop three-point functions in negative dimensional approach

In this article we present the complete massless and massive one-loop triangle diagram results using the negative dimensional integration method (NDIM). We consider the following cases: massless internal fields; one massive, two massive with the same mass m and three equal masses for the virtual particles. Our results are given in terms of hypergeometric and hypergeometric-type functions of the external momenta (and masses for the massive cases) where the propagators in the Feynman integrals are raised to arbitrary exponents and the dimension of the space-time is D. Our approach reproduces the known results; it produces other solutions as yet unknown in the literature as well. These new solutions occur naturally in the context of NDIM revealing a promising technique to solve Feynman integrals in quantum field theories.

A possible way to relate the covariantization and the negative dimensional integration method in the light-cone gauge

Here we present a possible way to relate the method of covariantizing the gauge-dependent pole and the negative dimensional integration method for computing Feynman integrals pertinent to the light-cone gauge fields. Both techniques are applicable to the algebraic light-cone gauge and dispense with prescriptions to treat the characteristic poles.

Negative-dimensional integration revisited

Feynman diagrams are the best tool we have to study perturbative quantum field theory. For this very reason the development of any new technique that allows us to compute Feynman integrals is welcome. By the middle of the 1980s, Halliday and Ricotta suggested the possibility of using negative-dimensional integrals to tackle the problem. The aim of this work is to revisit the technique as such and check on its possibilities. For this purpose, we take a box diagram integral contributing to the photon-photon scattering amplitude in quantum electrodynamics using the negative-dimensional integration method. Our approach enables us to quickly reproduce the known results as well as six other solutions as yet unknown in the literature. These six new solutions arise quite naturally in the context of negative-dimensional integration method, revealing a promising technique to handle Feynman integrals.