acta physica slovaca

Abstract: An Exact Renormalisation Group (ERG) approach to non-Abelian gauge theories is discussed. We focus on the derivation of universal beta-functions and the choice of the initial effective action, the latter being a key input in the approach. To that end we establish the map between the renormalisation group (RG) scaling of the full theory and the anomalous scaling in an ERG approach. Then this map is used to sketch the derivation of the two loop $\beta$-function within a simple straightforward calculation. The implications for the choice of the initial effective action are discussed.

Abstract: A novel, non-power, expansion of QCD quantities replacing the standard perturbative expansion in powers of the renormalized couplant $a$ has recently been introduced and examined by two of us. Being obtained by analytic continuation in the Borel plane, the new expansion functions $W_{n}(a)$ share the basic analyticity properties with the expanded quantity. In this note we investigate the renormalization scale dependence of finite order sums of this new expansion for the phenomenologically interesting case of the $\tau$-lepton decay rate.

Abstract: We report on recent works aiming at describing the influence of non-magnetic impurities on the phase transitions in frustrated magnets. A two-loop calculation and a renormalization-group approach in the framework of the effective average action show that the phase transition, which is expected to be weakly of first order in the pure case, is turned into a continuous one in presence of impurities.

Abstract: The critical properties of statistical systems in thermal equilibrium are well understood, thanks to renormalization group analysis. For systems driven into \emph{non-equilibrium steady states}, many surprising new features appear. For example, when the Ising lattice gas is driven with biased diffusion or coupled to \emph{two} thermal baths, long range correlations exist at all temperatures. The second order phase transition still survives, but the associated universal properties are drastically different. After a brief overview of the phenomenology of driven lattice gases, applications of RG to the study of several specific systems will be presented.

Abstract: Many non-equilibrium systems display dynamic phase transitions from active to absorbing states, where fluctuations cease entirely. Based on a field theory representation of the master equation, the critical behavior can be analyzed by means of the renormalization group. The resulting universality classes for single-species systems are reviewed here. Generically, the critical exponents are those of directed percolation (Reggeon field theory), with critical dimension $d_c = 4$. Yet local particle number parity conservation in even-offspring branching and annihilating random walks implies an inactive phase (emerging below $d_c' \approx 4/3$) that is characterized by the power laws of the pair annihilation reaction, and leads to different critical exponents at the transition. For local processes without memory, the pair contact process with diffusion represents the only other non-trivial universality class. The consistent treatment of restricted site occupations and quenched random reaction rates are important open issues.

Abstract: In this brief article I show how the notion of coarse graining and the Renormalization Group enter naturally in the dynamics of genetic systems, in particular in the presence of recombination. I show how the latter induces a dynamics wherein coarse grained and fine grained degrees of freedom are naturally linked as a function of time leading to a hierarchical dynamics that has a Feynman-diagrammatic representation. I show how this coarse grained formulation can be exploited to obtain new results.

Abstract: The the simplest version of the Bak-Sneppen model of self-organi\-zed biological evolution with random interaction structure is considered. It`s dynamics is described in the framework of master equation. The master equations can be solved exactly both for infinite system and for finite one. The equation for pair correlation function are solved exactly for infinite system. The dynamical regime of self-organized criticality in this model appears to be similar to one of completely integrable systems. Analysis of main characteristics of dynamics take it possible to revive the most essential feature of dynamics.

Abstract: The effect of random velocity field on the kinetics of single-species and two-species annihilation reactions is analysed near two dimensions in the framework of the field-theoretic renormalisation group. Fluctuations of particle density are modeled within the approach of Doi. The random incompressible velocity field is generated by stochastically forced Navier-Stokes equation in which thermal fluctuations-relevant below two dimensions-are taken into account.

Abstract: The field theoretic renormalization group is applied to Kraichnan's model of a passive scalar quantity advected by the Gaussian velocity field with the pair correlation function $\propto\delta(t-t')/k^{d+\varepsilon}$. Inertial-range anomalous scaling for the structure functions and various pair correlators is established as a consequence of the existence in the corresponding operator product expansions of ``dangerous'' composite opera\-tors (powers of the local dissipation rate), whose {\it negative} critical dimensions determine anomalous exponents. The latter are calculated to order $\varepsilon^{3}$ of the $\varepsilon$ expansion (three-loop approximation).

Abstract: Fully developed turbulence with anisotropy is investigated by means of the renormalization group approach and double expansion regularization for dimensions $d\ge 2$. Modification of the standard minimal subtraction scheme has been used to analyze the restoration of the stability of the Kolmogorov scaling regime under a transition from $d=2$ to 3. The results are in qualitative agreement with results obtained in the framework of a typical analytical regularization scheme.

Abstract: Renormalization group analysis is applied to the two-dimensional Navier--Stokes vorticity equation driven by a Gaussian random stirring. The energy-range spectrum $C_K \varepsilon^{2/3}k^{-5/3}$ obtained in the one-loop approximation coincides with earlier double epsilon expansion results, with $C_K=3.634$. This result is in good agreement with the value $C_K=3.35$ obtained by direct numerical simulation of the two-dimensional turbulent energy cascade using the pseudospectral method.

Abstract: Using the field theoretic renormalization group and the operator product expansion the structure of the fluctuations of passively advected magnetic field in a given anisotropic stochastic environment is analyzed. Inertial-range anomalous scaling behaviour is studied, and explicit asymptotic expressions for structure functions are determined. The corresponding anomalous exponents are calculated in the first order in a small parameter of the model as functions of the anisotropy parameters. The negativeness of some exponents indicates a complex multifractal structure of the fluctuations of the passively advected magnetic field in such environment.

Abstract: The field theoretic renormalization group is applied to the stochastic Navier--Stokes equation that describes fully developed fluid turbulence. The complete two-loop calculation of the renormalization constant, the beta function and the fixed point is performed. The ultraviolet correction exponent, the Kolmogorov constant and the inertial-range skewness factor are derived to second order of the $\eps$ expansion.

Abstract: I review my explanation of the irreversibility of the renormalization-group flow in even dimensions greater than two and address new investigations and tests.

Abstract: Hamiltonian formulation of local quantum field theory in the Fock space requires renormalization and even if a method acceptable in quantum mechanics were found, one could ask how special relativity could be obtained from an effective theory with a small range of momenta while Lorentz boosts change momenta by arbitrarily large amounts. This talk explains the principles of the renormalization group procedure for Hamiltonians of effective particles in quantum field theory, and describes how this procedure leads to the Poincare algebra in the Fock space.

Abstract: The whole set of correctly commuting renormalized Poincar\'e generators is presented up to the terms of second order in the coupling constant in the case of $g\phi^3$ theory. It is explained how Poincar\'e group elements are obtained by the exponentiation of generators in perturbation theory. A dynamical Lorentz transformation of one-particle eigenstate of the effective Hamiltonian is shown as an example.

Abstract: Success of renormalization group (RG) approach frequently depends on a proper functional form (probe function) used for renormalization. The behavior of coarse grained variables and their distributions under RG transformation have been numerically investigated for a stationary but non-equilibrium process of low-temperature creep. The simulation allows the investigation of a respective model at two levels (microscopic and coarse grained ones) simultaneously. A remarkable result of the model is that one-particle distributions for coarse (block) variables become Gaussians for block size 4 and larger. The investigation of dispersion dependence on block size has revealed a long-range correlation described by a critical index close to 5/3 (instead of 2 for independent variables). It is concluded that a probe function for many-body (many-particle) distribution at RG approach should be of Gaussian form.

Abstract: A general form of the system of differential equations simulating the self-organized criticality is presented. Three physically important cases of this system are studied in detail. It is shown that the critical states of the systems under consideration are really self-organized.

Abstract: Basic elements of the exact renormalization group method and recent results within this approach are reviewed. Topics covered are the derivation of equations for the effective action and relations between them, derivative expansion, solutions of fixed point equations and the calculation of the critical exponents, construction of the $c$-function and a description of the chiral phase transition.

Abstract: The standard demand for the quantum partition function to be invariant under the renormalization group transformation results in a general class of exact renormalization group equations, different in the form of the kernel. Physical quantities should not be sensitive to the particular choice of the kernel. Such scheme independence is elegantly illustrated in the scalar case by showing that, even with a general kernel, the one-loop beta function may be expressed only in terms of the effective action vertices, and in this way the universal result is recovered.

Abstract: A manifestly gauge invariant exact renormalization group for pure $SU(N)$ Yang-Mills theory is proposed, along with the necessary gauge invariant regularisation which implements the effective cutoff. The latter is naturally incorporated by embedding the theory into a spontaneously broken $SU(N|N)$ super-gauge theory, which guarantees finiteness to all orders in perturbation theory. The effective action, from which one extracts the physics, can be computed whilst manifestly preserving gauge invariance at each and every step. As an example, we give an elegant computation of the one-loop $SU(N)$ Yang-Mills beta function, for the first time at finite $N$ without any gauge fixing or ghosts. It is also completely independent of the details put in by hand, \eg the choice of covariantisation and the cutoff profile, and, therefore, guides us to a procedure for streamlined calculations.

Abstract: Within the exact renormalisation group approach, it is shown that stability properties of the flow are controlled by the choice for the regulator. Equally, the convergence of the flow is enhanced for specific optimised choices for the regularisation. As an illustration, we exemplify our reasoning for $3d$ scalar theories at criticality. Implications for other theories are discussed.

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