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.     

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.     

Differential real-space renormalization of the Gaussian model in three dimensions

We show how the differential real-space renormalization method can be extended beyond two dimensions by applying it to the Gaussian model on the fcc and diamond lattice.     

Radiative mass generation in perturbation theory and the renormalization group

The mechanism of dynamical mass generation is investigated in perturbation theory for the spontaneously broken supersymmetry model of O'Raifeartaigh. The generated mass obeys an homogeneous renormalization group equation. The compatibility of the perturbation solution with the exact solutions spectrum of the renormalization group equation is shown.     

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.     

Minimal renormalization without ɛ-expansion: Amplitude functions in three dimensions

The minimal subtraction scheme and the Borel resummation method are used to calculate the amplitudes of renormalized correlation functions aboveT _c for then-vector model in three dimensions. Accurate representations are given for the effective amplitudes of the renormalized expressions of the correlation length, of the susceptibility and of the specific heat forn=1, 2, 3. The resummed higher-order contributions turn out to yield only small corrections to the low-order approximations. These results complement a recent calculation of renormalization-group functions and provide the basis for accurate analyses of the critical behavior in three dimensions including the amplitude functions.     

Fermion families and chirality through extra dimensions

We give a simple model to explain the origin of fermion families and chirality through the use of a domain wall-anti-domain wall pair placed in a five dimensional space-time. Received: 14 March 2002 / Revised version: 3 July 2002 / Published online: 27 September 2002 RID="a" ID="a" e-mail: erdem@likya.iyte.edu.tr     

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).     

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.     

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.     

High density quark matter and the renormalization group in QCD with two and three flavors

We consider the most general four fermion operators in QCD for two and three massless flavors and study their renormalization in the vicinity of the Fermi surface. We show that, asymptotically, the largest coupling corresponds to scalar diquark condensation. Asymptotically the direct and iterated (molecular) instanton interactions become equal. We provide simple arguments for the form of the operators that diagonalize the evolution equations. Some solutions of the flow equations exhibit instabilities arising out of purely repulsive interactions.     

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.     

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.     

Rapidity renormalization group

We introduce a systematic approach for the resummation of perturbative series which involves large logarithms not only due to large invariant mass ratios but large rapidities as well. A series of this form can appear in a variety of gauge theory observables. The formalism is utilized to calculate the jet broadening event shape in a systematic fashion to next-to-leading logarithmic order. An operator definition of the factorized cross section as well as a closed form of the next-to-leading-log cross section are presented. The result agrees with the data to within errors.     

Finite-size scaling and the renormalization group

Renormalization group calculations ind = 4 andd = 4 – are performed for a system of finite size. A form of mean-field theory is used which yields a rounded transition for a finite system, and this allows a sensible expansion in fluctuations. A combination of Ewald and Poisson sum techniques is used to produce explicit numerical results for the specific heat ind = 4 which, with the setting of two nonuniversal metrical factors and the fourth-order coupling constant may be compared with simulations. The numerical visibility of logarithmic corrections is investigated. The universal scaling function for the specific heat to relativeO() is also evaluated numerically.     

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.