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.     

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.     

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.     

On Kadanoff's approximate renormalization group transformation

The fixed point structure resulting from the approximate renormalization group equations obtained by shifting bonds on the square Ising lattice is considered as a function of a free parameter h appearing in the definition of these equations. Next to the fixed point S considered by Kadanoff which is located in a symmetry plane two other “critical” fixed points A and B are found for h0.726. At the value h = 0.741, A crosses the fixed point S and vanishes together with the fixed point B at h = 0.726. Furthermore correction terms to the eigenvalues of the linearized renormalization group equations as obtained by Kadanoff are considered which arise if one chooses h to be optimal at all points of the coupling parameter space.     

The non-perturbative renormalization group in the ordered phase

We study some analytical properties of the solutions of the non-perturbative renormalization group flow equations for a scalar field theory with Z₂ symmetry in the ordered phase, i.e. at temperatures below the critical temperature. The study is made in the framework of the local potential approximation. We show that the required physical discontinuity of the magnetic susceptibility χ(M) at M=±M₀ (M₀ spontaneous magnetization) is reproduced only if the cut-off function which separates high and low energy modes satisfies to some restrictive explicit mathematical conditions; we stress that these conditions are not satisfied by a sharp cut-off in dimensions of space d<4.By generalizing a method proposed earlier by Bonanno and Lacagnina [Nucl. Phys. B 693 (2004) 36] to any kind of cut-off we propose to solve numerically the renormalization group flow equations for the threshold functions rather than for the local potential. It yields an algorithm sufficiently robust and precise to extract universal as well as non-universal quantities from numerical experiments at any temperature, in particular at sub-critical temperatures in the ordered phase. Numerical results obtained for the φ⁴ potential with three different cut-off functions are reported and compared. The data confirm our theoretical predictions concerning the analytical behavior of χ(M) at M=±M₀.Fixed point solutions of the adimensioned renormalization group flow equations are also obtained in the same vein, that is by solving the fixed points equations and the associated eigenvalue problem for the threshold functions rather than for the potential. We report high precision data for the odd and even spectra of critical exponents for different cut-offs obtained in this way.     

Exact renormalization group and Φ-derivable approximations

We show that the so-called Φ-derivable approximations can be combined with the exact renormalization group to provide efficient non-perturbative approximation schemes. On the one hand, the Φ-derivable approximations allow for a simple truncation of the infinite hierarchy of the renormalization group flow equations. On the other hand, the flow equations turn the non-linear equations that derive from the Φ-derivable approximations into an initial value problem, offering new practical ways to solve these equations.     

Supersymmetric renormalization group flows

The functional renormalization group equation for the quantum effective action is a powerful tool to investigate non-perturbative phenomena in quantum field theories. We discuss the application of manifest supersymmetric flow equations to the N = 1 Wess-Zumino model in two and three dimensions and the linear O(N) sigma model in three dimensions in the large-N limit. The former is a toy model for dynamical supersymmetry breaking, the latter for an exactly solvable field theory.     

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.     

Renormalized Normal Product and equations of motion of Φ4-coupling

The notion of a Renormalized Normal Product (RNP) in Euclidean space of 1 ≤ r ≤ 4 dimensions is introduced for a Φ⁴-model in a nonperturbative approach. The essential ingredients used for this purpose are the composite operators defined in perturbation theory and the renormalized G-convolution product constructed in the axiomatic field theory framework in Euclidean momentum space. Convergent equations of motion for the connected Green's functions are established where the interaction term is represented by the RNP. The corresponding renormalization constants are defined as boundary values of the RNP by imposing “physical” renormalization conditions. In the special case of 2-dimensions it is proved that these equations conserve analyticity and algebraic properties (in complex Minkowski space of 2-momenta) coming from the first principles of general local field theory, together with properties of asymptotic behaviour at infinity (in Euclidean space of 2-momenta).     

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.     

Predictions from high scale mixing unification hypothesis

Starting with ‘high scale mixing unification’ hypothesis, we investigate the renormalization group evolution of mixing parameters and masses for both Dirac and Majorana-type neutrinos. Following this hypothesis, the PMNS mixing parameters are taken to be identical to the CKM ones at a unifying high scale. Then, they are evolved to a low scale using MSSM renormalization group equations. For both types of neutrinos, the renormalization group evolution naturally results in a non-zero and small value of leptonic mixing angle ??₁₃. One of the important predictions of this analysis is that, in both cases, the mixing angle ??₂₃ turns out to be non-maximal for most of the parameter range. We also elaborate on the important differences between Dirac and Majorana neutrinos within our framework and how to experimentally distinguish between the two scenarios. Furthermore, for both cases, we also derive constraints on the allowed parameter range for the SUSY breaking and unification scales, for which this hypothesis works. The results can be tested by the present and future experiments.     

Shadows of the planck scale: scale dependence of compactification geometry

By studying the effects of the shape moduli associated with toroidal compactifications, we demonstrate that Planck-sized extra dimensions can cast significant "shadows" over low-energy physics. These shadows distort our perceptions of the compactification geometry associated with large extra dimensions and place a fundamental limit on our ability to probe the geometry of compactification by measuring Kaluza-Klein states. We also find that compactification geometry is effectively renormalized as a function of energy scale, with "renormalization group equations" describing the "flow" of geometric parameters such as compactification radii and shape angles as functions of energy.     

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.     

The average action for scalar fields near phase transitions

We compute the average action for scalar fields in two, three and four dimensions, including the effects of wave function renormalization. A study of the one loop evolution equations for the scale dependence of the average action gives a unified picture of the qualitatively different behaviour in various dimensions for discrete as well as abelian and nonabelian continuous symmetry. The different phases and the phase transitions can be infered from the evolution equation.     

ß-functions and the exact renormalization group

We relate ß-functions to the flow of relevant couplings in the exact renormalization group. The specific case of a cutoff γφ⁴ theory in four dimensions is discussed in detail. The underlying idea of convergence of the flow of effective lagrangians is developed to identify the ß-functions. A perturbative calculation of the ß-functions using the exact flow equations is sketched. The analysis may be extended to any system with a cutoff.     

Weak localization and coulomb interaction in disordered systems

Interacting electrons, diffusing in a two-dimensional (2d) disordered system, are studied. The renormalization group equations, including both localization effects and Coulomb correlations, are derived. We encounter a qualitatively new situation: the constants describing electron interaction diverge as a result of the renormalization when a certain scale is achieved, whereas the resistence proves to be finite. Calculation of the spin density correlation function reveals that the system exhibits a tendency for spin density rearrangement.     

Computing the Scaling Exponents in Fluid Turbulence from First Principles: Demonstration of Multiscaling

We develop a consistent closure procedure for the calculation of the scaling exponents ζ _n of the nth-order correlation functions in fully developed hydro-dynamic turbulence, starting from first principles. The closure procedure is constructed to respect the fundamental rescaling symmetry of the Euler equation. The starting point of the procedure is an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents ζ _n . This hierarchy was discussed in detail in a recent publication by V. S. L'vov and I. Procaccia. The scaling exponents in this set of equations cannot be found from power counting. In this paper we present in detail the lowest non-trivial closure of this infinite set of equations, and prove that this closure leads to the determination of the scaling exponents from solvability conditions. The equations under consideration after this closure are nonlinear integro-differential equations, reflecting the nonlinearity of the original Navier–Stokes equations. Nevertheless they have a very special structure such that the determination of the scaling exponents requires a procedure that is very similar to the solution of linear homogeneous equations, in which amplitudes are determined by fitting to the boundary conditions in the space of scales. The renormalization scale that is necessary for any anomalous scaling appears at this point. The Hölder inequalities on the scaling exponents select the renormalization scale as the outer scale of turbulence L. We demonstrate that the solvability condition of our equations leads to non-Kolmogorov values of the scaling exponents ζ _n . Finally, we show that this solutions is a first approximation in a systematic series of improving approximations for the calculation of the anomalous exponents in turbulence.     

One-loop string corrections to the effective field theory

We calculate the parity-conserving one-loop string corrections to the bosonic part of the 10-dimensional effective field theory for the heterotic string. There is no renormalization of the Chamseddine-Chapline-Manton supergravity action, but terms of fourth order in the curvature tensors are generated. Most of these are of the same forms as those found at the tree level of the string topological expansion, such as (TrR²)², (TrR²)(TrF²) and (TrF²)². However, we also find a non-zero Tr(F⁴) coupling which was not present at the tree level. We comment on the phenomenological implications of these results, which include absences of renormalization of the Kähler metric and of the gauge kinetic function of the effective field theory in four dimensions after compactification, and a modification of the classical equations to be obeyed by the manifold of compactification.     

Dynamic Monte Carlo renormalization group

A new and simple method of applying the idea of real space renormalization group theory to the analysis of Monte Carlo configurations is proposed and applied to the Glauber kinetic Ising model in two and three dimensions, and to the Kawasaki model in two dimensions. Our method, if correct, utilizes how the system approaches its equilibrium; in contrast to most other Monte Carlo investigations there is no need to wait until equilibrium is established. The renormalization analysis takes only a small fraction of the computer time needed to produce the Monte Carlo configurations, and the results are obtained as the system relaxes atT =T _c , the critical temperature. The values obtained for the dynamical critical exponent,z, are 2.12 (d=2) and 2.11 (d=3) for the Glauber model, the 3.90 for the two-dimensional Kawasaki model. These results are in good agreement with those obtained by other methods but with smaller error bars in three dimensions.