We revisit the problem of gauge invariance in Coleman-Weinberg model which a $U(1)$ symmetry is driven spontaneously broken by radiative corrections. It was noticed previous work that masses this are not invariant at one-loop order. In our analysis, we use dressed propagators scalars include resummation self-energy correction to tree-level propagator. calculate vector meson using these propagat...

We provide an in-depth study of pseudo-spectral numerical methods associated with modeling the self-assembly of molten mixed polymer brushes in the framework of self-consistent field theory (SCFT). SCFT of molten polymer brushes has proved numerically challenging in the past because of sharp features that arise in the self-consistent pressure field at the grafting surface due to the chain end t...

Leading nonperturbative corrections to the quark and gluon propagators are derived following the assumption that the nonperturbative QCD vacuum can be described in terms of non-vanishing vacuum expectation values for the composite operators [T&I and [GEY]. Th e nonperturbative quark propagator can be described in terms of a running quark mass and a running normalization function. These quantiti...

We obtain the Hausdorff dimension, h = 2− 2s, for particles with fractional spins in the interval, 0 ≤ s ≤ 0.5, such that the manifold is characterized by a topological invariant given by, W = h + 2s − 2p. This object is related to fractal properties of the path swept out by fractional spin particles, the spin of these particles, and the genus ( number of anyons ) of the manifold. We prove that...