Spatially resolved, long-slit echellograms at different position angles of the bright, low excita... more Spatially resolved, long-slit echellograms at different position angles of the bright, low excitation planetary nebula NGC 40 indicate that the higher the gas excitation, the faster the radial motion, thus confirming the overturn of the Wilson law already suggested by Sabbadin & Hamzaoglu (1982). New reduction procedures, giving the radial trends of the electron density and of the ionic and chemical abundances, were applied to NGC 40; they show that: - the radial matter distribution has a sharp ``bell'' profile with peaks up to 4000 cm-3; - the ionization structure is peculiar, indicating the presence of chemical composition gradients within the nebula: the innermost regions, hydrogen depleted, are essentially constituted of photospheric material ejected at high velocity by the WC8 nucleus. Moreover, detailed H+, O++ and N+ tomographic maps, giving the spatial ionic distributions at four position angles, are presented and discussed within the interacting winds evolutionary model.
We consider a fractal Wilson loop $<F_{P}>$ and present physical arguments why this should be a r... more We consider a fractal Wilson loop $<F_{P}>$ and present physical arguments why this should be a relevant observable in nature. We show for non-compact SU(2) lattice gauge theory in the next to leading order of strong coupling expansion that $<F_{P}>$ obeys an area law behavior and is gauge invariant.
There is still confusion about the correct form of the area law for the baryonic Wilson loop (BWL... more There is still confusion about the correct form of the area law for the baryonic Wilson loop (BWL) of QCD. Strong-coupling (i.e., finite lattice spacing in lattice gauge theory) approximations suggest the form exp[-KAY], where K is the qq¯ string tension and AY is the global minimum area, generically a three-bladed area with the blades joined along a Steiner line (Y configuration). However, the correct answer is exp[-(K/2)(A12+A13+A23)], where, e.g., A12 is the minimal area between quark lines 1 and 2 (Δ configuration). This second answer was given long ago, based on certain approximations, and is also strongly favored in lattice computations. In the present work, we derive the Δ law from the usual vortex-monopole picture of confinement, and show that, in any case, because of the 1/2 in the Δ law, this law leads to a larger value for the BWL (smaller exponent) than does the Y law. We show that the three-bladed, strong-coupling surfaces, which are infinitesimally thick in the limit of zero lattice spacing, survive as surfaces to be used in the non-Abelian Stokes' theorem for the BWL, which we derive, and lead via this Stokes' theorem to the correct Δ law. Finally, we extend these considerations, including perturbative contributions, to gauge groups SU(N), with N>3.
The $Q \bar{Q}$ semirelativistic interaction in QCD can be simply expressed in terms of the Wilso... more The $Q \bar{Q}$ semirelativistic interaction in QCD can be simply expressed in terms of the Wilson loop and its functional derivatives. In this approach we present the $Q \bar{Q}$ potential up to order $1/m^2$ using the expressions for the Wilson loop given by the Wilson Minimal Area Law (MAL), the Stochastic Vacuum Model (SVM) and Dual QCD (DQCD). We confirm the original results given in the different frameworks and obtain new contributions. In particular we calculate up to order $1/m^2$ the complete velocity dependent potential in the SVM. This allows us to show that the MAL model is entirely contained in the SVM. We compare and discuss also the SVM and the DQCD potentials. It turns out that in these two very different models the spin-orbit potentials show up the same leading non-perturbative contributions and 1/r corrections in the long-range limit.
Spatially resolved, long-slit echellograms at different position angles of the bright, low excita... more Spatially resolved, long-slit echellograms at different position angles of the bright, low excitation planetary nebula NGC 40 indicate that the higher the gas excitation, the faster the radial motion, thus confirming the overturn of the Wilson law already suggested by Sabbadin & Hamzaoglu (1982). New reduction procedures, giving the radial trends of the electron density and of the ionic and chemical abundances, were applied to NGC 40; they show that: - the radial matter distribution has a sharp ``bell'' profile with peaks up to 4000 cm-3; - the ionization structure is peculiar, indicating the presence of chemical composition gradients within the nebula: the innermost regions, hydrogen depleted, are essentially constituted of photospheric material ejected at high velocity by the WC8 nucleus. Moreover, detailed H+, O++ and N+ tomographic maps, giving the spatial ionic distributions at four position angles, are presented and discussed within the interacting winds evolutionary model.
We consider a fractal Wilson loop $<F_{P}>$ and present physical arguments why this should be a r... more We consider a fractal Wilson loop $<F_{P}>$ and present physical arguments why this should be a relevant observable in nature. We show for non-compact SU(2) lattice gauge theory in the next to leading order of strong coupling expansion that $<F_{P}>$ obeys an area law behavior and is gauge invariant.
There is still confusion about the correct form of the area law for the baryonic Wilson loop (BWL... more There is still confusion about the correct form of the area law for the baryonic Wilson loop (BWL) of QCD. Strong-coupling (i.e., finite lattice spacing in lattice gauge theory) approximations suggest the form exp[-KAY], where K is the qq¯ string tension and AY is the global minimum area, generically a three-bladed area with the blades joined along a Steiner line (Y configuration). However, the correct answer is exp[-(K/2)(A12+A13+A23)], where, e.g., A12 is the minimal area between quark lines 1 and 2 (Δ configuration). This second answer was given long ago, based on certain approximations, and is also strongly favored in lattice computations. In the present work, we derive the Δ law from the usual vortex-monopole picture of confinement, and show that, in any case, because of the 1/2 in the Δ law, this law leads to a larger value for the BWL (smaller exponent) than does the Y law. We show that the three-bladed, strong-coupling surfaces, which are infinitesimally thick in the limit of zero lattice spacing, survive as surfaces to be used in the non-Abelian Stokes' theorem for the BWL, which we derive, and lead via this Stokes' theorem to the correct Δ law. Finally, we extend these considerations, including perturbative contributions, to gauge groups SU(N), with N>3.
The $Q \bar{Q}$ semirelativistic interaction in QCD can be simply expressed in terms of the Wilso... more The $Q \bar{Q}$ semirelativistic interaction in QCD can be simply expressed in terms of the Wilson loop and its functional derivatives. In this approach we present the $Q \bar{Q}$ potential up to order $1/m^2$ using the expressions for the Wilson loop given by the Wilson Minimal Area Law (MAL), the Stochastic Vacuum Model (SVM) and Dual QCD (DQCD). We confirm the original results given in the different frameworks and obtain new contributions. In particular we calculate up to order $1/m^2$ the complete velocity dependent potential in the SVM. This allows us to show that the MAL model is entirely contained in the SVM. We compare and discuss also the SVM and the DQCD potentials. It turns out that in these two very different models the spin-orbit potentials show up the same leading non-perturbative contributions and 1/r corrections in the long-range limit.
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