Collective Fields for QCD

A gauge-symmetric approach to effective Lagrangians is described with special emphasis on derivations of effective low-energy Lagrangians from QCD. The examples we discuss are based on exact rewritings of cut-off QCD in terms of new collective degrees of freedom. These cut-off Lagrangians are thus ``effective'' in the sense that they explicitly contain some of the physical long-distance degrees of freedom from the outset.(Talk presented by P.H. Damgaard at the workshop on ``Quantum Field Theoretical Methods in High Energy Physics'', Kyffhauser, Germany, Sept. 1993. To appear in those proceedings).Comment: LaTeX, 12 pages, CERN--TH-7035/9

Patterns of Spontaneous Chiral Symmetry Breaking in Vectorlike Gauge Theories

It has been conjectured that spontaneous chiral symmetry breaking in strongly coupled vectorlike gauge theories falls into only three different classes, depending on the gauge group and the representations carried by the fermions. We test this proposal by studying SU(2), SU(3) and SU(4) lattice gauge theories with staggered fermions in different irreducible representations. Staggered fermions away from the continuum limit should, for all complex representations, still belong to the continuum class of spontaneous symmetry breaking. But for all real and pseudo-real representations we show that staggered fermions should belong to incorrect symmetry breaking classes away from the continuum, thus generalizing previous results. As an unambiguous signal for whether chiral symmetry breaks, and which breaking pattern it follows, we look at the smallest Dirac eigenvalue distributions. We find that the patterns of symmetry breaking are precisely those conjectured.Comment: LaTeX, 17 pages. Typos in eq (17) correcte

Interference Phenomenon for the Faddeevian Regularization of 2D Chiral Fermionic Determinants

The classification of the regularization ambiguity of 2D fermionic determinant in three different classes according to the number of second-class constraints, including the new faddeevian regularization, is examined and extended. We found a new and important result that the faddeevian class, with three second-class constraints, possess a free continuous one parameter family of elements. The criterion of unitarity restricts the parameter to the same range found earlier by Jackiw and Rajaraman for the two-constraints class. We studied the restriction imposed by the interference of right-left modes of the chiral Schwinger model ($\chi QED_{2}$) using Stone's soldering formalism. The interference effects between right and left movers, producing the massive vectorial photon, are shown to constrain the regularization parameter to belong to the four-constraints class which is the only non-ambiguous class with a unique regularization parameter.Comment: 15 pages, Revtex. Final version to be published in Phys. Rev.

The large N limit of four dimensional Yang-Mills field coupled to adjoint fermions on a single site lattice

We consider the large N limit of four dimensional SU(N) Yang-Mills field coupled to adjoint fermions on a single site lattice. We use perturbative techniques to show that the Z^4_N center-symmetries are broken with naive fermions but they are not broken with overlap fermions. We use numerical techniques to support this result. Furthermore, we present evidence for a non-zero chiral condensate for one and two Majorana flavors at one value of the lattice gauge coupling.Comment: 21 pages, 13 figures; a reference added; version to be published in JHEP, small clarifications and references adde

Chaotic Behaviour of Renormalisation Flow in a Complex Magnetic Field

It is demonstrated that decimation of the one dimensional Ising model, with periodic boundary conditions, results in a non-linear renormalisation transformation for the couplings which can lead to chaotic behaviour when the couplings are complex. The recursion relation for the couplings under decimation is equivalent to the logistic map, or more generally the Mandelbrot map. In particular an imaginary external magnetic field can give chaotic trajectories in the space of couplings. The magnitude of the field must be greater than a minimum value which tends to zero as the critical point T=0 is approached, leading to a gap equation and associated critical exponent which are identical to those of the Lee-Yang edge singularity in one dimension.Comment: 8 pages, 2 figures, PlainTeX, corrected some typo

Smooth Bosonization as a Quantum Canonical Transformation

We consider a 1+1 dimensional field theory which contains both a complex fermion field and a real scalar field. We then construct a unitary operator that, by a similarity transformation, gives a continuum of equivalent theories which smoothly interpolate between the massive Thirring model and the sine-Gordon model. This provides an implementation of smooth bosonization proposed by Damgaard et al. as well as an example of a quantum canonical transformation for a quantum field theory.Comment: 20 pages, revte

Spectrum of the U(1) staggered Dirac operator in four dimensions

We compare the low-lying spectrum of the staggered Dirac operator in the confining phase of compact U(1) gauge theory on the lattice to predictions of chiral random matrix theory. The small eigenvalues contribute to the chiral condensate similar as for the SU(2) and SU(3) gauge groups. Agreement with the chiral unitary ensemble is observed below the Thouless energy, which is extracted from the data and found to scale with the lattice size according to theoretical predictions.Comment: 5 pages, 3 figure

Character Expansions for the Orthogonal and Symplectic Groups

Formulas for the expansion of arbitrary invariant group functions in terms of the characters for the Sp(2N), SO(2N+1), and SO(2N) groups are derived using a combinatorial method. The method is similar to one used by Balantekin to expand group functions over the characters of the U(N) group. All three expansions have been checked for all N by using them to calculate the known expansions of the generating function of the homogeneous symmetric functions. An expansion of the exponential of the traces of group elements, appearing in the finite-volume gauge field partition functions, is worked out for the orthogonal and symplectic groups.Comment: 20 pages, in REVTE

Unstable Modes in Three-Dimensional SU(2) Gauge Theory

We investigate SU(2) gauge theory in a constant chromomagnetic field in three dimensions both in the continuum and on the lattice. Using a variational method to stabilize the unstable modes, we evaluate the vacuum energy density in the one-loop approximation. We compare our theoretical results with the outcomes of the numerical simulations.Comment: 24 pages, REVTEX 3.0, 3 Postscript figures included. (the whole postscript file (text+figures) is available on request from [email protected]