From short to long scales in the QCD vacuum

We study approximate decimations in SU(N) LGT that connect the short to long distance regimes, and provide both upper and lower bounds on the exact partition function. This leads to a representation of the exact partition function in terms of successive decimations. The implications for a derivation of confinement from first principles are discussed.Comment: 3 pages, talk presented at Lattice2003(topology

Hidden Symmetries of Large N QCD

The local SUSY symmetry of the loop dynamics of QCD is found. The remarkable thing is, there is no einbein-gravitino on this theory, which makes it a 1D topological supergravity, or locally SUSY quantum mechanics. Using this symmetry, we derive the large $N_c$ loop equation in momentum superloop space. Introducing as before the position operator \X{\mu} we argue that the superloop equation is equivalent to invariance of correlation functions of products of these operators with respect to certain quadrilinear transformation. The applications to meson and glueball sectors as well as the chiral symmetry breaking are discussed. The 1D field theory with Quark propagating around the loop in superspace is constructed.Comment: 40 pages, 1 Postscript figure, LaTe

Critical couplings and string tensions via lattice matching of RG decimations

We calculate critical couplings and string tensions in SU(2) and SU(3) pure lattice gauge theory by a simple and inexpensive technique of two-lattice matching of RG block transformations. The transformations are potential moving decimations generating plaquette actions with large number of group characters and exhibit rapid approach to a unique renormalized trajectory. Fixing the critical coupling $\beta_c(N_\tau)$ at one value of temporal lattice length $N_\tau$ by MC simulation, the critical couplings for any other value of $N_\tau$ are then obtained by lattice matching of the block decimations. We obtain $\beta_c(N_\tau)$ values over the range $N_\tau = 3 - 32$ and find agreement with MC simulation results to within a few percent in all cases. A similar procedure allows the calculation of string tensions with similarly good agreement with MC data.Comment: 12 pages, Latex, 1 figur

Some Remarks About Induced QCD

Migdal and Kazakov have suggested that lattice QCD with an adjoint representation scalar in the infinite coupling limit could induce QCD. I find an exact saddlepoint of this theory for infinite $N$ in the case of a quadratic scalar potential. I discuss some aspects of this solution and also show how the continuum D=1 matrix model with an arbitrary potential can be reproduced through this approach.Comment: 9 pages, PUPT-133

Quark Confinement and the Renormalization Group

Recent approaches to quark confinement are reviewed, with an emphasis on their connection to renormalization group methods. Basic concepts related to confinement are introduced: the string tension, Wilson loops and Polyakov lines, string breaking, string tension scaling laws, center symmetry breaking, and the deconfinement transition at non-zero temperature. Current topics discussed include confinement on $R^3\times S^1$, the real-space renormalization group, the functional renormalization group, and the Schwinger-Dyson equation approach to confinement.Comment: 22 pages; report from the INT Workshop "New applications of the renormalization group in nuclear, particle, and condensed matter physics", held February 22-26 201

Mixed Model of Induced QCD

The problems with the $Z_N$ symmetry breaking in the induced QCD are analyzed. We compute the Wilson loops in the strong coupling phase, but we do not find the $Z_N$ symmetry breaking, for arbitrary potential. We suggest to bypass this problem by adding to the model a heavy fermion field in a fundamental representation of $SU(N)$. Remarkably, the model still can be solved exactly by the Rieman-Hilbert method, for arbitrary number $N_f$ of flavors. At $N_f \ll N \rightarrow \infty$ there is a new regime, with two vacuum densities. The $Z_N$ symmetry breaking density satisfies the linear integral equation, with the kernel, depending upon the old density. The symmetry breaking requires certain eigenvalue condition, which takes some extra parameter adjustment of the scalar potential.Comment: 14 pages, Latex, no figures, ( after final debugging