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.     

The renormalization group in the problem of turbulent convection of a passive scalar impurity with nonlinear diffusion

The problem of turbulent mixing of a passive scalar impurity is studied within the renormalization-group approach to the stochastic theory of developed turbulence for the case where the diffusion coefficient is an arbitrary function of the impurity concentration. Such a problem incorporates an infinite number of coupling constants (“charges”). A one-loop calculation shows that in the infinite-dimensional space of the charges there is a two-dimensional surface of fixed points of the renormalization-group equations. When the surface has an IR-stability region, the problem has scaling with universal critical dimensionalities, corresponding to the phenomenological laws of Kolmogorov and Richardson, but with nonuniversal (i.e., depending on the Prandtl number and the explicit form of the nonlinearity in the diffusion equation) scaling functions, amplitude factors in the power laws, and value of the “effective Prandtl turbulence number.” Zh. éksp. Teor. Fiz. 112, 1649–1663 (November 1997)     

Non-trivial fixed points of the renormalization group in six dimensions

We start a systematic analysis of supersymmetric field theories in six dimensions. We find necessary conditions for the existence of non-trivial interacting fixed points. String theory provides us with examples of such theories. We conjecture that there are many other examples.     

Linear and nonlinear dynamic scaling relations in the renormalization group theory

Dynamic scaling relations in the presence of an external field are derived by extending Hubbard's static arguments to non-equilibrium systems using the RG theory. This confirms microscopically the scaling relations of the linear and non-linear critical slowing down.     

Real-space renormalization group study of a two-dimensional nonlinear massless field

The magnetic susceptibility of CuF₂·2H₂O has been measured as a function of magnetic field from 1.5 to 10 K. The spin-flop transition was observed and its value extrapolated to zero temperature is H_SF(0) = 30.5 kOe. This critical field is in very good agreement with data obtained from zero field measurements.     

A proof of the irreversibility of renormalization group flows in four dimensions

We present a proof of the irreversibility of renormalization group flows, i.e. the c-theorem for unitary, renormalizable theories in four (or generally even) dimensions. Using Ward identities for scale transformations and spectral representation arguments, we show that the c-function based on the trace of the energy-momentum tensor (originally suggested by Cardy) decreases monotonically along renormalization group trajectories. At fixed points this c-function is stationary and coincides with the coefficient of the Euler density in the trace anomaly, while away from fixed points its decrease is due to the decoupling of positive-norm massive modes.     

Disorder-induced quantum-Griffiths-like features from a non-conventional renormalization group analysis near four dimensions

We investigate the role played by symmetry conserving quenched disorder on quantum criticality of a variety of d-dimensional systems with a continuous symmetry order parameter. We employ a non-standard procedure which combines a preliminary reduction to an effective classical random problem and a successive conventional renormalization group treatment. Solving the effective flow equations to first order in ε=4−d and then restoring the original coupling parameters, for d<4 we find a quantum critical point scenario exhibiting unusual features, which remind us of some predictions of the quantum Griffiths phase model.     

Polymer diffusion in quenched disorder: A renormalization group approach

We study the diffusion of polymers through quenched short-range correlated random media by renormalization group (RG) methods, which allow us to derive universal predictions in the limit of long chains and weak disorder. We take local quenched random potentials with second momentv and the excluded-volume interactionu of the chain segments into account. We show that our model contains the relevant features of polymer diffusion in random media in the RG sense if we focus on the local entropic effects rather than on the topological constraints of a quenched random medium. The dynamic generating functional and the general structure of its perturbation expansion inu andv are derived. The distribution functions for the center-of-mass motion and the internal modes of one chain and for the correlation of the center of mass motions of two chains are calculated to one-loop order. The results allow for sufficient cross-checks to have trust in the one-loop renormalizability of the model. The general structure as well as the one-loop results of the integrated RG flow of the parameters are discussed. Universal results can be found for the effective static interactionwu–v0 and for small effective disorder coupling on the intermediate length scalel. As a first physical prediction from our analysis, we determine the general nonlinear scaling form of the chain diffusion constant and evaluate it explicitly as for .     

Anomalous diffusion for a correlated process with long jumps

We discuss diffusion properties of a dynamical system, which is characterised by long-tail distributions and finite correlations. The particle velocity has the stable Lévy distribution; it is assumed as a jumping process (the kangaroo process) with a variable jumping rate. Both the exponential and the algebraic form of the covariance–defined for the truncated distribution–are considered. It is demonstrated by numerical calculations that the stationary solution of the master equation for the case of power-law correlations decays with time, but a simple modification of the process makes the tails stable. The main result of the paper is a finding that–in contrast to the velocity fluctuations–the position variance may be finite. It rises with time faster than linearly: the diffusion is anomalously enhanced. On the other hand, a process which follows from a superposition of the Ornstein–Uhlenbeck–Lévy processes always leads to position distributions with a divergent variance which means accelerated diffusion.     

Analysis of a continuous field theory in two dimensions with use of the density matrix renormalization group

A formulation of the Ginzburg-Landau-Wilson version of the partition function of a system with a continuously varying order parameter as a transfer matrix calculation allows for the application of methods based on the density matrix renormalization group (DMRG) to the calculation of the free energy of the O(1) model. The essence of both the mapping and the DMRG calculation is laid out, along with results that validate this strategy.     

Causality and the renormalization group

We suppose that for the invariant coupling constant (ICC) the spectral representation of the Källen-Lehmann type is valid. By combining this conjecture with the general solution of the functional renormalization group (RG) equation it is possible to analyze the type of singularity in the coupling constant at g=0. For logarithmic models it is of the form exp (-1/g).     

On renormalization and complex space-time dimensions

The method of using the dimension of space-time as a complex parameter introduced recently to regularize Feynman amplitudes is extended to an arbitrary Feynman graph. The method has promise of being particularly well-suited to gauge theories. It is shown how the renormalized amplitude, together with the Lagrangian counter-terms, may be extracted directly, following the method of analytic renormalization.     

Application of renormalization group techniques to transport in the presence of nonlinear sources and sinks

The paper considers the problem of spreading of a localized distribution due to diffusion in the presence of nonlinear sources and sinks modeling annihilation and creation of material in chemical reactions. The evolution of the parameters characterizing the problem, namely, the amplitude and radius of the distribution, has been investigated under the assumption that the distribution is self-similar. These parameters have been calculated using the renormalization-group method with renormalization of the diffusion coefficient and the total amount of material. As a result, it is possible to classify various regimes of the asymptotic behavior of the system at long times according to the sign of nonlinear interaction constant and the spatial dimensionality. In particular, the conditions under which the regime of asymptotic freedom and the peaking regime are realized have been found. The renormalization group method not only allows us to calculate the exponents of functions with power-law behavior, but also to track the transition to the asymptotic regime and calculate numerical coefficients. Zh. éksp. Teor. Fiz. 115, 1497–1510 (April 1999)     

Analyticity and renormalization group

We prove the inequalities ψ(y, α) ?α, |α_s(d/dα_s)(β(α_s)/α_s| ? 1 (for the Paterman-Stueckelberg-Gell-Mann-Low functions in QED and QCD) and γ₀(α_s ? 1 (for the anomalous dimension of the gauge-invariant operator O(x)). The consequences of the inequalities are discussed: for modern energies, comparison of theoretical and experimental moments of deep inelastic structure functions has a meaning only for N ? 7 (singlet case) and N ? 50 (non-singlet case); perturbation theory in QCD has a meaning only for α_rms ? 0.45.     

Φ3 theory in six dimensions and the renormalization group

Φ³ field theory in six space-time dimensions is used as a testing ground for the renormalization group. The consistency of the new method devised by 't Hooft is verified at the two-loop level by exhibiting certain cancellations among single pole coefficients, and also relations among double and single pole coefficients. This calculation is contrasted in efficiency with one using the Gell-Mann-Low equation in dimensionally regularized form for which a solution is obtained at the one-loop level, without neglecting mass dependence.The theory shares with asymptotically free ones the virtue of an asymptotically vanishing effective coupling constant. Indeed, the next to leading term in the beta function is also of opposite sign to the coupling constant.