Shape evolution of electrodeposited bumps into deep cavities

Metal posts and finer pitch solder bumps are the indispensable microconnectors for chip size packaging and are formed by electrodeposition into deep cavities. It is difficult to stir inside these deep cavities. Natural convection due to density difference is effective in stirring inside cavity with 200 mum cathode width of aspect ratio of one. The bump shape increases toward lower side in a vertical cathode arrangement with placement angle of Theta = 90 degrees. This increase in bump height results from a collision of flow along the lower side of the resist sidewall which enlarges local current and thickens the lower edge of bumps. The effect of natural convection is also evident in the neighboring two cavities of 200 mum cathode width. The natural convection is not effective for cavities with less than 100 mum cathode width. The bump shapes become flat. Only diffusion occurs within these smaller than 100 mum cavities. (C) 2001 The Electrochemical Society. All rights reserved.</p

Flavor-Dependence and Higher Orders of Gauge-Independent Solutions in Strong Coupling Gauge Theory

The fermion flavor $N_f$ dependence of non-perturbative solutions in the strong coupling phase of the gauge theory is reexamined based on the interrelation between the inversion method and the Schwinger-Dyson equation approach. Especially we point out that the apparent discrepancy on the value of the critical coupling in QED will be resolved by taking into account the higher order corrections which inevitably lead to the flavor-dependence. In the quenched QED, we conclude that the gauge-independent critical point $\alpha_c=2\pi/3$ obtained by the inversion method to the lowest order will be reduced to the result $\alpha_c=\pi/3$ of the Schwinger-Dyson equation in the infinite order limit, but its convergence is quite slow. This is shown by adding the chiral-invariant four-fermion interaction.Comment: CHIBA-EP-72, 13 pages (including 1 Table), LaTex fil

Reformulating SU(N) Yang-Mills theory based on change of variables

We propose a new version of SU(N) Yang-Mills theory reformulated in terms of new field variables which are obtained by a nonlinear change of variables from the original Yang-Mills gauge field. The reformulated Yang-Mills theory enables us to study the low-energy dynamics by explicitly extracting the topological degrees of freedom such as magnetic monopoles and vortices to clarify the mechanism for quark confinement. The dual superconductivity in Yang-Mills theory is understood in a gauge-invariant manner, as demonstrated recently by a non-Abelian Stokes theorem for the Wilson loop operator, although the basic idea of this reformulation is based on the Cho-Faddeev-Niemi decomposition of the gauge potential.Comment: 51 pages, 1 figure; version to be published in Prog. Theor. Phys. Vol. 120, No.1 (2008

Magnetic condensation, Abelian dominance and instability of Savvidy vacuum

We show that a certain type of color magnetic condensation originating from magnetic monopole configurations is sufficient to provide the mass for off-diagonal gluons in the SU(2) Yang-Mills theory under the Cho--Faddeev--Niemi decomposition. We point out that the generated gluon mass can cure the instability of the Savvidy vacuum. In fact, such a novel type of magnetic condensation is shown to occur by calculating the effective potential. This enables us to explain the infrared Abelian dominance and monopole dominance by way of a non-Abelian Stokes theorem, which suggests the dual superconductivity picture of quark confinement. Finally, we discuss the implication to the Faddeev-Skyrme model with knot soliton as a low-energy effective theory of Yang-Mills theory.Comment: 14 pages, 2 figures; a version accepted in Phys. Lett. B, Main changes in sections 2.5 and 2.6. in order to explain the crucial idea bette

The `BRST-invariant' Condensate of Dimension Two in QCD

The status of the `BRST-invariant' condensate of mass dimension two in QCD is explained. The condensate is only invariant under an `on-shell' BRST symmetry which includes a partial gauge-fixing. The on-shell BRST symmetry represents the residual gauge symmetry under gauge transformations which preserve the partial gauge fixing. The gauge-invariant operators which correspond to the BRST-invariant condensate are identified in the Lorentz and maximal Abelian gauges and are shown to be invariant under the residual gauge transformations.Comment: 6 page

Running Flavor Number and Asymptotic Freedom in the Normal Phase of QED

In the normal phase (where no dynamical fermion mass generation occurs) of the D-dimensional quantum electrodynamics with $N_f$ flavors of fermions, we derive an integral equation which should be satisfied by (the inverse of) the wave function renormalization of the fermion in the Landau gauge. For this we use the inverse Landau-Khalatnikov transformation connecting the nonlocal gauge with the Landau gauge. This leads to a similar equation for the running flavor number in the framework of the $1/N_f$ resumed Schwinger-Dyson equation. Solving the equation analytically and numerically, we study the infrared behavior and the critical exponent of the 3-dimensional QED (QED$_3$). This confirms that the flavor number in QED$_3$ runs according to the $\beta$ function which is consistent with the asymptotic freedom as that in 4-dimensional QCD.Comment: 11 pages, 1 figure, latex, to appear in Phys. Lett.

Non-Abelian Stokes Theorem and Quark Confinement in SU(3) Yang-Mills Gauge Theory

We derive a new version of SU(3) non-Abelian Stokes theorem by making use of the coherent state representation on the coset space $SU(3)/(U(1)\times U(1))=F_2$, the flag space. Then we outline a derivation of the area law of the Wilson loop in SU(3) Yang-Mills theory in the maximal Abelian gauge (The detailed exposition will be given in a forthcoming article). This derivation is performed by combining the non-Abelian Stokes theorem with the reformulation of the Yang-Mills theory as a perturbative deformation of a topological field theory recently proposed by one of the authors. Within this framework, we show that the fundamental quark is confined even if $G=SU(3)$ is broken by partial gauge fixing into $H=U(2)$ just as $G$ is broken to $H=U(1) \times U(1)$. An origin of the area law is related to the geometric phase of the Wilczek-Zee holonomy for U(2). Abelian dominance is an immediate byproduct of these results and magnetic monopole plays the dominant role in this derivation.Comment: 14 pages, Latex, no figures, version accepted for publication in Mod. Phys. Lett. A (some comments are added in the final parts

Vacuum condensate of mass dimension 2 as the origin of mass gap and quark confinement (A review)

This is a brief summary of recent works on the possibility of vacuum condensate of mass dimension 2 in Yang-Mills theory as the gluon sector of QCD. In particular, I discuss the physical implications due to this condensate, especially, for the mass gap and quark confinement. This talk is mainly based on a recent paper [1] and papers in preparation [2].Comment: 13 pages, no figures, Talk presented at the workshop ``Structure and Reaction of Hadrons based on non-perturbative QCD" held at the Research Center of Nuclear Physics (RCNP), Osaka University, Japan, 23-24 July 200