arXiv subjects
Publications and source records attributed to C. Wetterich.
The scaling form of the critical equation of state is computed for $O(N)$-symmetric models. We employ a method based on an exact flow equation for a coarse grained free energy. A suitable truncation is solved numerically.
Non-perturbative exact flow equations describe the scale dependence of the effective average action. We present a numerical solution for an approximate form of the flow equation for the potential in a three-dimensional N-component scalar field theory. The critical behaviour, with associated critical exponents, can be inferred with good accuracy.
The scale dependence of an effective average action for mesons and quarks is described by a nonperturbative flow equation. The running couplings lead to spontaneous chiral symmetry breaking. We argue that for strong Yukawa coupling between quarks and mesons the low momentum physics is essentially determined by infrared fixed points. This allows us to establish relations between various parameters related to the meson potential. The results for $f_π$ and $\VEV{\olpsiψ}$ are not very sensitive to the poorly known details of the quark--meson effective action at scales where the mesonic bound states form. For realistic constituent quark masses we find $f_π$ around $100\MeV$.
Two-scalar theories at high temperature exhibit a rich spectrum of possible critical behaviour, with a second or first order phase transition. In the vicinity of the critical temperature one can observe critical exponents, tricritical points and crossover behaviour. None of these phenomena are visible to high temperature perturbation theory.
We investigate the phase transition in the three-dimensional abelian Higgs model for N complex scalar fields, using the gauge-invariant average action Γ_{k}. The dependence of Γ_{k} on the effective infra-red cut-off k is described by a non-perturbative flow equation. The transition turns out to be first- or second-order, depending on the ratio between scalar and gauge coupling. We look at the fixed points of the theory for various N and compute the critical exponents of the model. Comparison with results from the ε-expansion shows a rather poor convergence for ε=1 even for large N. This is in contrast to the surprisingly good results of the ε-expansion for pure scalar theories. Our results suggest the existence of a parameter range with a second-order transition for all N, including the case of the superconductor phase transition for N=1.
We employ nonperturbative flow equations for the description of the effective action in Yang-Mills theories. We find that the perturbative vacuum with vanishing gauge field strength does not correspond to the minimum of the Euclidean effective action. The true ground state is characterized by a nonvanishing gluon condensate.
We investigate the O(N) symmetric linear $σ$-model in two dimensions by means of an exact nonperturbative evolution equation. The perturbative infrared divergences are absent in this formulation. We use a simple approximative solution of the flow equation which corresponds to a derivative expansion for the effective action. For N=2 this gives a good picture of the Kosterlitz-Thouless phase transition.
A quantitative discussion of nonperturbative effects for the high temperature electroweak phase transition is presented. We propose a method for the computation of the temperature dependent effective scalar potential that takes into account the running of the effective gauge coupling. Compared to perturbation theory we find a moderate decrease of the critical temperature and an important change in the strength of the first order transition. We conclude that perturbation theory gives a misleading picture of the dynamics of the transition.
We explore the Coleman-Weinberg phase transition in regions outside the validity of perturbation theory. For this purpose we study a Euclidean field theory with two scalars and discrete symmetry in four dimensions. The phase diagram is established by a numerical solution of a suitable truncation of exact non-perturbative flow equations. We find regions in parameter space where the phase transition (in dependence on the mass term) is of the second or the first order, separated by a triple point. Our quantitative results for the first order phase transition compare well to the standard perturbative Coleman-Weinberg calculation of the effective potential.
We perform a non-perturbative study of the Coleman-Weinberg phase transition in scalar QED. Our method permits a consistent treatment of the effective potential near the origin, a region not accessible to perturbation theory. As a result, we establish reliably the first order character of the phase transition for arbitrary values of the scalar coupling.
Within the Euclidean effective action approach we propose criteria for the ground state of QCD. Despite a nonvanishing field strength the ground state should be invariant with respect to modified Poincaré transformations consisting of a combination of translations and rotations with suitable gauge transformations. We have found candidate states for QCD with four or more colours. The formation of gluon condensates shows similarities with the Higgs phenomenon.
A nonperturbative method is proposed for the approximative solution of the exact evolution equation which describes the scale dependence of the effective average action. It consists of a combination of exact evolution equations for independent couplings with renormalization group improved one loop expressions of secondary couplings. Our method is illustrated by an example: We compute the beta-function of the quartic coupling lambda of an O(N) symmetric scalar field theory to order lambda^3 as well as the anomalous dimension to order lambda^2 using only one loop expressions and find agreement with the two loop perturbation theory. We also treat the case of very strong coupling and confirm the existence of a "triviality bound".
Evolution equations describe the effect of consecutively integrating out all quantum fluctuations with momenta larger than some infrared cutoff scale k. We develop a formalism for the introduction of collective degrees of freedom at some intermediate scale and derive the corresponding evolution equations. This allows to account for the appearance of bound states at some characteristic length scale as, for example, the mesons in QCD. The vacuum properties, including condensates of composite operators, can be directly infered from the effective action for k->0. We compute in a simple QCD-motivated model with four-quark interactions the chiral condensate \barψψand the effective action for pions including f_π. A full treatment of QCD along these lines seems feasible but still requires substantial work. (5 figures available by ordinary mail upon request)
Resubmitted due to LATEXing problems
We compute the critical behaviour of three-dimensional scalar theories using a new exact non-perturbative evolution equation. Our values for the critical exponents agree well with previous precision estimates.
We study, with various methods (standard large N evaluation of the functional integral for the effective potential, solution of the Schwinger-Dyson equations), the high temperature phase transition for the $N$-component $ϕ^4$ theory in the large $N$ limit. Our results fully confirm a previous investigation of the problem, for arbitrary $N$, with the method of the average potential which employs renormalization group ideas. The phase transition is of the second order with an effectively three-dimensional critical behaviour. }