Renormalization group operators for maps and universal scaling of universal scaling exponents

Renormalization group (RG) methods provide a unifying framework for understanding critical behaviour, such as transition to chaos in area-preserving maps and other dynamical systems, which have associated with them universal scaling exponents. Recently, de la Llave et al. (2007) [10] have formulated the Principle of Approximate Combination of Scaling Exponents (PACSE for short), which relates exponents for different criticalities via their combinatorial properties. The main objective of this paper is to show that certain integrable fixed points of RG operators for area-preserving maps do indeed follow the PACSE.     

Renormalization group equations for the Kobayashi-Maskawa matrix

The renormalization group equations for the parametrization-convention independent quadratic parameters |V _ij |² of the KM matrix are derived. Numerical analysis of these equations shows that the heavy quark family (t, b) tends to mix with the lighter families (c, s) and (u, d) with increasing energy, although the variation is very much slow. The CP-nonconservation effects are shown to get larger with energy.     

Renormalization group equations for effective field theories

We derive the renormalization group equations for a generic non-renormalizable theory. We show that the equations allow one to derive the structure of the leading divergences at any loop order in terms of one-loop diagrams only. In chiral perturbation theory, e.g., this means that one can obtain the series of leading chiral logs by calculating only one-loop diagrams. We discuss also the renormalization group equations for the subleading divergences, and the crucial role of counterterms that vanish at the equations of motion. Finally, we show that the renormalization group equations obtained here apply equally well also to renormalizable theories.Received: 5 September 2003, Published online: 20 November 2003     

Renormalization group equations for a two-dimensional nonlinear sigma-model with torsion

The renormalization group equations in the one-loop approximation are formulated for a two-dimensional nonlinear -model with torsion on a semisimple group manifold. Solutions of these equations are investigated for several particular cases.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 37–40, September, 1988.     

Renormalization group equations and the gaussian spin model

Exact renormalization group equations are constructed for the one and two dimensional gaussian spin model by direct elimination of spins on a sublattice. They lead to the correct values of the critical exponents α, γ, ν and η.     

Renormalization group functions for the radiative symmetry breaking scheme

We obtain the renormalization group (RG) functions for the massless scalar field theory where symmetry breaking occurs radiatively. After obtaining the effective potential for the radiative symmetry breaking scheme by finite transformations for the classical field and coupling constant, we obtain the corresponding RG functions from that of the minimal subtraction (MS) scheme.     

Universal behavior of crossover scaling functions for continuous phase transitions

We consider two different systems exhibiting a continuous phase transition into an absorbing state. Both models belong to the same universality class; i.e., they are characterized by the same scaling functions and the same critical exponents. Varying the range of interactions, we examine the crossover from the mean-field-like to the non-mean-field scaling behavior. A phenomenological scaling form is applied in order to describe the full crossover region, which spans several decades. Our results strongly support the hypothesis that the crossover function is universal.     

Renormalization group and relations between scattering amplitudes in a theory with different mass scales

In the Yukawa model with two different mass scales, the renormalization group equation is used to obtain relations between scattering amplitudes at low energies. By considering fermion-fermion scattering as an example, a basic one-loop renormalization group relation is derived which gives the possibility to reduce the problem to the scattering of light particles in the “external field” substituting a heavy virtual state. Applications of the results to problems of searches for new physics beyond the Standard Model are discussed.     

Renormalization group treatment of the scaling properties of finite systems with subleading long-range interaction

The finite size behavior of the susceptibility, Binder cumulant and some even moments of the magnetization of a fully finite O(n) cubic system of size L are analyzed and the corresponding scaling functions are derived within a field-theoretic ɛ-expansion scheme under periodic boundary conditions. We suppose a van der Waals type long-range interaction falling apart with the distance r as r ^{- (d + σ)}, where 2 < σ < 4, which does not change the short-range critical exponents of the system. Despite that the system belongs to the short-range universality class it is shown that above the bulk critical temperature T _c the finite-size corrections decay in a power-in-L, and not in an exponential-in-L law, which is normally believed to be a characteristic feature for such systems. Received 8 August 2001     

Renormalization group equations for softly broken supersymmetry: The most general case

The renormalization group equations are derived for the most general gauge theory with softly broken supersymmetry. When specialized to the case that global supersymmetry breakdown is induced by spontaneously broken supergravity, the results exhibit new features which are important when applied to models which go beyond the simplest standard model with minimal particle content. As an application the renormalization group equations are given for a standard model with four Higgs doublets as well as for the minimal standard model including generation mixing, both in the framework of broken supergravity.     

Renormalization equations for models with renormalizable and nonrenormalizable interactions

We study models including renormalizable and nonrenormalizable polynomial interactions. We derive the partial differential equations, which are relevant for the variation of parameters of the model. A supersymmetric model is considered as example.     

A low temperature expansion for classicalN-vector models. II. Renormalization group equations

This paper continues the analysis of the low temperature expansions for classicalN-vector models started in [1]. A main part of it is a derivation of renormalization group equations and a construction of their solutions. To do this we have to introduce “a fluctuation integral” connected with a next renormalization transformation, and to make its preliminary analysis. The results of the paper are summarized in theorems stating that the renormalization transformation preserves the space of densitites, or actions described inductively in [1]. This work has been partially supported by the NSF Grant DMS-9102639.     

Renormalization group calculation for cells with inequivalent spins

We have considered a real-space renormalization group transformation for a bidimensional Ising model, carrying out approximate calculations for cells where site spins do not play the same role. The dependence on the ratio between the number of intercell and intracell nearest-neighbour interactions has also been discussed.Fellow of the Consejo Nacional de Investigaciones Científicas y Técnicas de Argentina.Fellow of the Comisión de Investigaciones Científicas de la Provincia de Buenos Aires, Argentina.     

Renormalization group for matrix models with branching interactions

We develop a method to obtain the large-N renormalization group flows for matrix models of two-dimensional gravity plus branched polymers. This method gives precise results for the critical points and exponents for one-matrix models. We show that it can be generalized to two-matrix models and we recover the Ising critical points.     

Renormalization group functions for two-dimensional phase transitions: To the problem of singular contributions

According to the available publications, the field theoretical renormalization group approach in the two-dimensional case gives the critical exponents that differ from the known exact values. This property is associated with the existence of nonanalytic contributions in the renormalization group functions. The situation is analyzed in this work using a new algorithm for summing divergent series that makes it possible to determine the dependence of the results for the critical exponents on the expansion coefficients for the renormalization group functions. It has been shown that the exact values of all the exponents can be obtained with a reasonable form of the coefficient functions. These functions have small nonmonotonic sections or inflections, which are poorly reproduced in natural interpolations. It is not necessary to assume the existence of singular contributions in the renormalization group functions.     

Electromagnetic Coulomb gas with vector charges and “elastic” potentials: Renormalization group equations

We present a detailed derivation of the renormalization group equations for two-dimensional electromagnetic Coulomb gases whose charges lie on a triangular lattice (magnetic charges) and its dual (electric charges). The interactions between the charges involve both angular couplings and a new electromagnetic potential. This motivates the denomination of “elastic” Coulomb gas. Such elastic Coulomb gases arise naturally in the study of the continuous melting transition of two-dimensional solids coupled to a substrate, either commensurate or with quenched disorder.     

Renormalization group equations and the continuum limit of some renormalizable quantum field theories

The renormalization group equations for asymptotically and non-asymptotically free theories are discussed by exploiting the general properties of their integral curves. Some constraints are derived on the existence of a consistent continuum limit for pure Yang-Mills theories in an infinite quantization volume. An analogy between the far infrared behaviour of non-abelian gauge theories and the deep ultraviolet one of asymptotically free theories is discussed.     

Renormalization group theory for fluid and plasma turbulence

For the last several decades, renormalization group (RG, or RNG) methods have been applied to a wide variety of problems of turbulence in hydrodynamics and plasma physics. A comprehensive review of this work will be presented, covering RG methods in hydrodynamic turbulence and in turbulent systems with coupled fluctuating fields like magnetohydrodynamic (MHD) turbulence. This review will attempt to specifically consider several questions about RG: (1) Does RG provide an improvement over previous analytical theories like the direct interaction approximation, or is RG a useful simplification of those theories? (2) How are nonlocal, or ‘sweeping’ effects treated in RG formalisms, or are they ignored entirely? (3) Can RG theories treat both local and nonlocal interactions in turbulence?