General Solution of the non-abelian Gauss law and non-abelian analogs of the Hodge decomposition

General solution of the non-abelian Gauss law in terms of covariant curls and gradients is presented. Also two non-abelian analogs of the Hodge decomposition in three dimensions are addressed. i) Decomposition of an isotriplet vector field $V_{i}^{a}(x)$ as sum of covariant curl and gradient with respect to an arbitrary background Yang-Mills potential is obtained. ii) A decomposition of the form $V_{i}^{a}=B_{i}^{a}(C)+D_{i}(C) \phi^{a}$ which involves non-abelian magnetic field of a new Yang-Mills potential C is also presented. These results are relevant for duality transformation for non-abelian gauge fields.Comment: 6 pages, no figures, revte

A numerical study of Goldstone-mode effects and scaling functions of the three-dimensional O(2) model

We investigate numerically the three-dimensional O(2) model on 8^3-160^3 lattices as a function of the magnetic field H. In the low-temperature phase we verify the H-dependence of the magnetization M induced by the Goldstone modes and determine M in the thermodynamic limit on the coexistence line both by extrapolation and by chiral perturbation theory. We compute two critical amplitudes from the scaling behaviours on the coexistence line and on the critical line. In both cases we find negative corrections to scaling. With additional high temperature data we calculate the scaling function and show that it has a smaller slope than that of the O(4) model. For future tests of QCD lattice data we study as well finite-size-scaling functions.Comment: Lattice 2000 (Spin Models), minor typographic errors fixe

Infrared Behaviour of Systems With Goldstone Bosons

We develop various complementary concepts and techniques for handling quantum fluctuations of Goldstone bosons.We emphasise that one of the consequences of the masslessness of Goldstone bosons is that the longitudinal fluctuations also have a diverging susceptibility characterised by an anomalous dimension $(d-2)$ in space-time dimensions $2<d<4$.In $d=4$ these fluctuations diverge logarithmically in the infrared region.We show the generality of this phenomenon by providing three arguments based on i). Renormalization group flows, ii). Ward identities, and iii). Schwinger-Dyson equations.We obtain an explicit form for the generating functional of one-particle irreducible vertices of the O(N) (non)--linear $\sigma$--models in the leading 1/N approximation.We show that this incorporates all infrared behaviour correctly both in linear and non-linear $\sigma$-- models. Our techniques provide an alternative to chiral perturbation theory.Some consequences are discussed briefly.Comment: 28 pages,2 Figs, a new section on some universal features of multipion processes has been adde

Equation of state and Goldstone-mode effects of the three-dimensional O(2) model

We investigate numerically the three-dimensional O(2) model on 8^3-160^3 lattices as a function of the magnetic field H. In the low-temperature phase we verify the H-dependence of the magnetization M induced by Goldstone modes and determine M in the thermodynamic limit both by extrapolation and by chiral perturbation theory. This enables us to calculate the corresponding critical amplitude. At T_c the critical scaling behaviour of the magnetization as a function of H is used to determine another critical amplitude. In both cases we find negative corrections-to-scaling. Our low-temperature results are well described by the perturbative form of the model's magnetic equation of state, with coefficients determined nonperturbatively from our data. The O(2) scaling function for the magnetization is found to have a smaller slope than the one for the O(4) model.Comment: 15 pages, Latex2e, Fig.6b replaced, several comments and two references added, final version for Phys. Lett.

Topologically Massive Non-Abelian Gauge Theories: Constraints and Deformations

We study the relationship between three non-Abelian topologically massive gauge theories, viz. the naive non-Abelian generalization of the Abelian model, Freedman-Townsend model and the dynamical 2-form theory, in the canonical framework. Hamiltonian formulation of the naive non-Abelian theory is presented first. The other two non-Abelian models are obtained by deforming the constraints of this model. We study the role of the auxiliary vector field in the dynamical 2-form theory in the canonical framework and show that the dynamical 2-form theory cannot be considered as the embedded version of naive non-Abelian model. The reducibility aspect and gauge algebra of the latter models are also discussed.Comment: ReVTeX, 17 pp; one reference added, version published in Phys. Rev.

Scaling and Goldstone effects in a QCD with two flavours of adjoint quarks

We study QCD with two Dirac fermions in the adjoint representation at finite temperature by Monte Carlo simulations.In such a theory the deconfinement and chiral phase transitions occur at different temperatures. We locate the second order chiral transition point at beta_c=5.624(2) and show that the scaling behaviour of the chiral condensate in the vicinity of beta_c is in full agreeement with that of the 3d O(2) universality class, and to a smaller extent comparable to the 3d O(6) class. From the previously determined first order deconfinement transition point beta_d=5.236(3) and the two-loop beta function we find the ratio T_c/T_d = 7.8(2). In the region between the two phase transitions we explicitly confirm the quark mass dependence of the chiral condensate which is expected due to the existence of Goldstone modes like in 3d O(N) spin models. At the deconfinement transition the condensate shows a gap, and below beta_d, it is nearly mass-independent for fixed beta.Comment: 28 pages, 12 figures, Latex2

Recent Developments in Lattice QCD

I review the current status of lattice QCD results. I concentrate on new analytical developments and on numerical results relevant to phenomenology.Comment: 35 pages, 4 figures (Figures are excerpted from others' work and are not included) Uses harvmac.te

Critical behaviour and scaling functions of the three-dimensional O(6) model

We numerically investigate the three-dimensional O(6) model on 12^3 to 120^3 lattices within the critical region at zero magnetic field, as well as at finite magnetic field on the critical isotherm and for several fixed couplings in the broken and the symmetric phase. We obtain from the Binder cumulant at vanishing magnetic field the critical coupling J_c=1.42865(3). The universal value of the Binder cumulant at this point is g_r(J_c)=-1.94456(10). At the critical coupling, the critical exponents \gamma=1.604(6), \beta=0.425(2) and \nu=0.818(5) are determined from a finite-size-scaling analysis. Furthermore, we verify predicted effects induced by massless Goldstone modes in the broken phase. The results are well described by the perturbative form of the model's equation of state. Our O(6)-result is compared to the corresponding Ising, O(2) and O(4) scaling functions. Finally, we study the finite-size-scaling behaviour of the magnetisation on the pseudocritical line.Comment: 13 pages, 20 figures, REVTEX, fixed an error in the determination of R_\chi and changed the corresponding line in figure 13

New Variables for Classical and Quantum Gravity in all Dimensions II. Lagrangian Analysis

We rederive the results of our companion paper, for matching spacetime and internal signature, by applying in detail the Dirac algorithm to the Palatini action. While the constraint set of the Palatini action contains second class constraints, by an appeal to the method of gauge unfixing, we map the second class system to an equivalent first class system which turns out to be identical to the first class constraint system obtained via the extension of the ADM phase space performed in our companion paper. Central to our analysis is again the appropriate treatment of the simplicity constraint. Remarkably, the simplicity constraint invariant extension of the Hamiltonian constraint, that is a necessary step in the gauge unfixing procedure, involves a correction term which is precisely the one found in the companion paper and which makes sure that the Hamiltonian constraint derived from the Palatini Lagrangian coincides with the ADM Hamiltonian constraint when Gauss and simplicity constraints are satisfied. We therefore have rederived our new connection formulation of General Relativity from an independent starting point, thus confirming the consistency of this framework.Comment: 42 pages. v2: Journal version. Some nonessential sign errors in section 2 corrected. Minor clarification