A fixed point for quantum gravity

Using the 1/N expansion a fixed point of the renormalization group is found for quantized gravitational theories which is non-trivial in all dimensions, d, including four. Using the fixed point it is shown how Einstein's theory can be renormalized for 3<d<4. In four dimensions the pure Einstein theory does not exist, but the R + C_μναβ² theory does. It is shown how gravitational theories whose quantum lagrangians are scale invariant may be renormalized such that the scale invariance is broken only by the choice of the critical renormalization group trajectory. A comparison is made with the renormalization of four-fermion and Yukawa theories in 4?? dimensions which suggests that quantum gravity might exist in four dimensions even if those theories do not.     

Monte Carlo renormalization group study of φ4 field theory

A previously proposed general method for evaluating block renormalized coupling constants within the framework of the Monte Carlo renormalization group (MCRG) is applied to φ⁴ field theory. The flow diagrams, fixed points, and critical exponents are determined in two, three and four dimensions. Results in four dimensions are consistent with the idea that φ⁴ field theory is trivial (non-interacting) in the continuum limit. The possibility of using MCRG techniques to ascertain whether a general non-asymptotically free theory is trivial or not is also discussed.     

On the triviality of λϕd4 theories and the approach to the critical point in d(−) > 4 dimensions

It is shown that one- and two-component λ|?|⁴ theories and non-linear σ-models in five or more dimensions approach free or generalized free fields in the continuum (scaling) limit, and that in four dimensions the same result holds, provided there is infinite field strength renormalization, as expected. Some critical exponents for the lattice theories in five or more dimensions are shown to be mean field. The main tools are Symanzik's polymer representation of scalar field theories and correlation inequalities.     

On the renormalization of the Kardar-Parisi-Zhang equation

The Kardar-Parisi-Zhang (KPZ) equation of nonlinear stochastic growth in d dimensions is studied using the mapping onto a system of directed polymers in a quenched random medium. The polymer problem is renormalized exactly in a minimally subtracted perturbation expansion about d = 2. For the KPZ roughening transition in dimensions d > 2, this renormalization group yields the dynamic exponent z = 2 and the roughness exponent χ = 0, which are exact to all orders in ε ≡ (2 − d)/2. The expansion becomes singular in d = 4. If this singularity persists in the strong-coupling phase, it indicates that d = 4 is the upper critical dimension of the KPZ equation. Further implications of this perturbation theory for the strong-coupling phase are discussed. In particular, it is shown that the correlation functions and the coupling constant defined in minimal subtraction develop an essential singularity at the strong-coupling fixed point.     

Two-parameter expansion in the renormalization-group analysis of turbulence

The renormalization of the solution of the Navier-Stokes equation for randomly stirred fluid with long-range correlations of the driving force is analysed near two dimensions. It is shown that a local term must be added to the correlation function of the random force for the correct renormalization of the model at two dimensions. The interplay of the short-range and long-range terms in the large-scale behaviour of the model is analysed near two dimensions by the field-theoretic renormalization group. A regular expansion in 2ε=d-2 and δ=2-λ is constructed, whered is the space dimension and λ the exponent of the powerlike correlation function of the driving force. It is shown that in spite of the additional divergences, the asymptotic behaviour of the model near two dimensions is the same as in higher dimensions, contrary to recent conjectures based on an incorrect renormalization procedure.     

Two-loop renormalization of the Finkel’stein theory: The specific heat

We explore the two-loop renormalization of the specific heat for an interacting disordered electron system in the case of broken time reversal symmetry. Within the nonlinear sigma model approach we derive the two-loop result for the anomalous dimension which controls scaling of the specific heat with temperature. As an example, we elaborate the metal-insulator transition in d=2+?$d=2+?$ dimensions for the case of broken time reversal and spin rotational symmetries and in the presence of Coulomb interaction. In this situation scaling of the specific heat is determined by the anomalous dimension of the Finkel’stein operator which is the eigenoperator of the renormalization group complementary to the eigenoperator corresponding to the second moment of the local density of states. We find that the absolute values of the anomalous dimensions of these operators differ beyond one-loop approximation contrary to the noninteracting case.     

Monte Carlo renormalization group study of φ field theory

A previously proposed general method for evaluating block renormalized coupling constants within the framework of the Monte Carlo renormalization group (MCRG) is applied to φ⁴ field theory. The flow diagrams, fixed points, and critical exponents are determined in two, three and four dimensions. Results in four dimensions are consistent with the idea that φ⁴ field theory is trivial (non-interacting) in the continuum limit. The possibility of using MCRG techniques to ascertain whether a general non-asymptotically free theory is trivial or not is also discussed.     

Triviality of Hierarchical Models with Small Negative φ4 Coupling in Four Dimensions

The Kadanoff-Wilson renormalization group (RG) for a class of hierarchical spin models including small negative φ⁴ terms in four dimensions are studied by using Gawędzki and Kupiainen's analysis. We prove triviality for the class, namely prove existence of critical trajectory that leads to the Gaussian fixed point.     

Two-parameter expansion in the renormalization-group analysis of turbulence

The renormalization of the solution of the Navier-Stokes equation for randomly stirred fluid with long-range correlations of the driving force is analysed near two dimensions. It is shown that a local term must be added to the correlation function of the random force for the correct renormalization of the model at two dimensions. The interplay of the short-range and long-range terms in the large-scale behaviour of the model is analysed near two dimensions by the field-theoretic renormalization group. A regular expansion in 2ε=d-2 and δ=2-λ is constructed, whered is the space dimension and λ the exponent of the powerlike correlation function of the driving force. It is shown that in spite of the additional divergences, the asymptotic behaviour of the model near two dimensions is the same as in higher dimensions, contrary to recent conjectures based on an incorrect renormalization procedure.     

Dynamically driven renormalization group

We present a detailed discussion of a novel dynamical renormalization group scheme: the dynamically driven renormalization group (DDRG). This is a general renormalization method developed for dynamical systems with non-equilibrium critical steady state. The method is based on a real-space renormalization scheme driven by a dynamical steady-state condition which acts as a feedback on the transformation equations. This approach has been applied to open nonlinear systems such as self-organized critical phenomena, and it allows the analytical evaluation of scalling dimensions and critical exponents. Equilibrium models at the critical point can also be considered. The explicit application to some models and the corresponding results are discussed.     

On renormalizable two-dimensional σ-models with dynamical charges

(a) The general conditions are stated under which a non-linear σ-model in two dimensions has dynamical charges of the type found by Lüscher and Pohlmeyer for the O (N)-model. (b) he renormalization of a large class of σ-models including CPⁿ-models and chiral fields on Grassmann manifolds is discussed.     

Ground state pressure and energy density of an interacting homogeneous Bose gas in two dimensions

We consider an interacting homogeneous Bose gas at zero temperature in two spatial dimensions. The properties of the system can be calculated as an expansion in powers of g, where g is the coupling constant. We calculate the ground state pressure and the ground state energy density to second order in the quantum loop expansion. The renormalization group is used to sum up leading and subleading logarithms from all orders in perturbation theory. In the dilute limit, the renormalization group improved pressure and energy density are expansions in powers of the T ^2B and T ^2Bln(T ^2B), respectively, where T ^2B is the two-body T-matrix. Received 19 April 2002 Published online 13 August 2002     

NLO evolution for large scale distances,positivity constraints and the low-energy model of the nucleon

We present a computationally reliable and accurate method for solving the Gribov-Lipatov-Altarelli-Parisi equations at next to leading order, both in the non-singlet and in the singlet case. It requires solving numerically the renormalization group equations for the anomalous dimensions of composite operators in the complex plane, and finally performing an inverse Mellin transformation. In this way the group property of renormalization is exactly preserved, i.e. performing two successive scale transformations coincides exactly with a direct one making parton distributions independent of the integration path used to connect two different scales. This is relevant when large scale differences are involved and makes upward or downward evolution fully equivalent. Thus, it becomes possible to evolve the known parton distributions and leading twist contributions to the structure functions from Q² = m_b² to the lowest possible scale imposed by positivity and unitarity.     

Progress in lattice gauge theory

Two topics of lattice gauge theory are reviewed. They include string tension and β-function calculations by strong coupling Hamiltonian methods for SU(3) gauge fields in 3 + 1 dimensions, and a 1/N-expansion for discrete gauge and spin systems in all dimensions. The SU(3) calculations give solid evidence for the coexistence of quark confinement and asymptotic freedom in the renormalized continuum limit of the lattice theory. The crossover between weak and strong coupling behavior in the theory is seen to be a weak coupling but non-perturbative effect. Quantitative relationships between perturbative and non-perturbative renormalization schemes are obtained for the O(N) nonlinear sigma models in 1 + 1 dimensions as well as the range theory in 3 + 1 dimensions. Analysis of the strong coupling expansion of the β-function for gauge fields suggests that it has cuts in the complex 1/g²-plane. A toy model of such a cut structure which naturally explains the abruptness of the theory's crossover from weak to strong coupling is presented. The relation of these cuts to other approaches to gauge field dynamics is discussed briefly.The dynamics underlying first order phase transitions in a wide class of lattice gauge theories is exposed by considering a class of models-P(N) gauge theories - which are soluble in the N → ∞ limit and have non-trivial phase diagrams. The first order character of the phase transitions in Potts spin systems for N #62; 4 in 1 + 1 dimensions is explained in simple terms which generalizes to P(N) gauge systems in higher dimensions. The phase diagram of Ising lattice gauge theory coupled to matter fields is obtained in a $\text{1}\text{N}$ expansion. A one-plaquette model (1 time-0 space dimensions) with a first-order phase transitions in the N → ∞ limit is discussed.     

Renormalization group functions of the φ<Superscript>4</Superscript> theory from high-temperature expansions

It has been previously shown that calculation of the renormalization group (RG) functions of scalar ϕ⁴ theory reduces to analysis of thermodynamic properties of the Ising model. Using high-temperature expansions for the latter, RG functions of the four-dimensional theory can be calculated for arbitrary coupling constant g with an accuracy of 10⁻⁴ for the Gell-Mann-Low function β(g) and with an accuracy of 10⁻³–10⁻² for anomalous dimensions. The expansions of the renormalization group functions up to the 13th order in g ^−1/2 have been obtained.     

Improved renormalization group equations with dimensional regularization and the ε expansions

Due to the absence of dimensional cut-off parameters in the dimensional regularization scheme, vanishing of the renormalized mass of the scalar boson implies vanishing of its renormalized mass; thus the masses of both bosons and fermions in renormalizable field theories can be made finite by multiplicative mass renormalizations. The improved renormalization group equations in D dimensions are derived in such a way that both the large (or the small) momentum limits and the Wilson ? expansions can be uniformly treated for the fermion as well as the boson cases. We discuss the improved equations for φ₆³ theory, φ₄⁴ theory, quantumelectrodynamics, massive vector-gluon model, and non-Abelian guage theories incorporating fermions. For the latter three classes of theories, the gauge dependent problem of the coefficient functions in the improved renormalization group equations is discussed.     

The renormalization group of the model ofA 4-coupling in the abstract approach of quantum field theory

For the model ofA ⁴-interaction the postulates of the renormalization group are stated within the abstract approach of quantum field theory. In the massive case these postulates follow if an on-shell formulation of the model is assumed to exist. For the massless model the postulates of the renormalization group imply that the propagator has a pole at momentum zero. Consequently there is no dynamic mass generation and the propagator is normalizable on the mass shell. It is shown that theS-matrix elements scale with canonical dimensions. A general method of rescaling parameter values is developed which takes into account the possibility of propagator zeros and stationary points of the effective coupling.     

Dimensional regularization of massless theories in spherical space-time

The use of space-time curvature as an infra-red cut-off has been suggested for massless theories. In this paper we investigate the renormalization of massless theories in a spherical space-time (Euclidean version of de Sitter space) using dimensional regularization. Naive expectations are confirmed, namely that the coupling constant and wave-function renormalizations are independent of the curvature. Furthermore the curvature does not induce divergent mass terms or vacuum field values as would be possible on purely dimensional grounds. Although we have investigated only scalar field theories, φ⁴ theory in four dimensions and φ³ theory in six, these results are encouraging for an application of the method to gauge theories.Formally massless theories are conformally invariant so the formulation of the theory in a spherical space ought to be equivalent to its formulation in flat space. In fact the renormalization procedure breaks conformal invariance and removes this equivalence. We show that to achieve the flat space limit it is necessary to invoke the aid of the renormalization group. Thus the zero curvature limit can be achieved for infra-red stable theories (φ₄⁴) but not for infra-red unstable theories (φ₆³ as might be expected.     

Anomalous dimensions and the renormalization group equations

An interpretation is given of scale and anomalous dimensions in the framework of the renormalization group, and the renormalization group equations are derived which are regarded to represent the conservation of these scale dimensions. By the use of continuous dimensional regularization all coefficient functions appearing in these equations and in the Callan-Symanzik equations are explicitly expressed in terms of the residues of the single poles at n = 4 as well as the finite part of renormalization counter-terms.