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.
Renormalization scheme dependence in the case of a QCD non-power perturbative expansion
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.
Continuous phase transition induced by impurities in frustrated magnets
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.
Renormalization group analysis of driven diffusive systems
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.
Dynamic phase transitions in diffusion-limited reactions
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.
The renormalization group and the dynamics of genetic systems
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.
Self-organized criticality in simple model of evolution: exact description of scaling laws
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.
Anomalous transport processes in chemically active random environment
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.
Renormalization group, operator product expansion and anomalous
scaling in models of passive turbulent advection
Author(s):
L. Ts. Adzhemyan, N. V. Antonov, A. N. Vasil'ev
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).
Influence of anisotropy on the scaling regimes in fully developed turbulence
Author(s):
J. Busa, M. Hnatic, E. Jurcisinova, M. Jurcisin, M. Stehlik
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.
Field theory model for two-dimensional turbulence: vorticity-based approach
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.
Advection of vector admixture by turbulent flows with strong anisotropy
Author(s):
M. Hnatic, M. Jurcisin, A. Mazzino, S. Sprinc
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.
Renormalization group in the statistical theory of turbulence: two-loop approximation
Author(s):
L. Ts. Adzhemyan, N. V. Antonov, M. V. Kompaniets, A. N. Vasil'ev
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.
Sum rules for trace anomalies and irreversibility of the renormalization-group flow
I review my explanation of the irreversibility of the
renormalization-group flow in even dimensions greater than two and address
new investigations and tests.
Renormalization group and special relativity in the Fock space
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.
Example of a renormalization group solution for Poincare transformation in the Fock space
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.
Numerical renormgroup analysis of self-organized criticality in low temperature creep model
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.
Self-organization of the critical state in physical systems described by differential equations
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.
Exact renormalization group as a tool for non-perturbative studies
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.
A demonstration of one-loop scheme independence in a generalised scalar exact renormalization group
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.
Towards a manifestly gauge invariant and universal calculus for Yang-Mills theory
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.
Convergence and stability of the renormalisation group
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.
